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diff --git a/42216-0.zip b/42216-0.zip Binary files differdeleted file mode 100644 index 70c4e08..0000000 --- a/42216-0.zip +++ /dev/null diff --git a/42216-8.txt b/42216-8.txt deleted file mode 100644 index 0a9148b..0000000 --- a/42216-8.txt +++ /dev/null @@ -1,2003 +0,0 @@ -The Project Gutenberg EBook of On the History of Gunter's Scale and the -Slide Rule during the Seventeenth Century, by Florian Cajori - -This eBook is for the use of anyone anywhere at no cost and with -almost no restrictions whatsoever. You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org - - -Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century - -Author: Florian Cajori - -Release Date: February 26, 2013 [EBook #42216] - -Language: English - -Character set encoding: ISO-8859-1 - -*** START OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - - - - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - IN - MATHEMATICS - - Vol. 1, No. 9, pp. 187-209 February 17, 1920 - - - - - ON THE HISTORY OF GUNTER'S SCALE AND - THE SLIDE RULE DURING THE - SEVENTEENTH CENTURY - - - BY - FLORIAN CAJORI - - - UNIVERSITY OF CALIFORNIA PRESS - BERKELEY - - - - - TABLE OF CONTENTS - - - PAGE - I. Introduction 187 - II. Innovations in Gunter's Scale 188 - Changes introduced by Edmund Wingate 188 - Changes introduced by Milbourn 189 - Changes introduced by Thomas Brown and John Brown 190 - Changes introduced by William Leybourn 192 - III. Richard Delamain's "Grammelogia" 192 - Different editions or impressions 194 - Description of Delamain's instrument of 1630 195 - Delamain's later designs, and directions for using his - instruments 197 - IV. Controversy between Oughtred and Delamain on the invention of - the circular slide rule 199 - V. Independence and priority of invention 203 - VI. Oughtred's "Gauging Line," 1633 206 - VII. Other seventeenth century slide rules 207 - - - - - I. INTRODUCTION - - - In my history of the slide rule[1], and my article on its invention[2] it -is shewn that William Oughtred and not Edmund Wingate is the inventor, that -Oughtred's circular rule was described in print in 1632, his rectilinear -rule in 1633. Richard Delamain is referred to as having tried to -appropriate the invention to himself[3] and as having written a scurrilous -pamphlet against Oughtred. All our information about Delamain was taken -from De Morgan,[4] who, however, gives no evidence of having read any of -Delamain's writings on the slide rule. Through Dr. Arthur Hutchinson of -Pembroke College, Cambridge, I learned that Delamain's writings on the -slide rule were available. In this article will be given: First, some -details of the changes introduced during the seventeenth century in the -design of Gunter's scale by Edmund Wingate, Milbourn, Thomas Brown, John -Brown and William Leybourn; second, an account of Delamain's book of 1630 -on the slide rule which antedates Oughtred's first publication (though -Oughtred's date of invention is earlier than the date of Delamain's alleged -invention) and of Delamain's later designs of slide rules; third, an -account of the controversy between Delamain and Oughtred; fourth, an -account of a later book on the slide rule written by William Oughtred, and -of other seventeenth century books on the slide rule. - - - - - II. INNOVATIONS IN GUNTER'S SCALE - - - Changes introduced by Wingate - - We begin with Anthony Wood's account of Wingate's introduction of -Gunter's scale into France.[5] - - In 1624 he transported into France the rule of proportion, having a - little before been invented by Edm. Gunter of Gresham Coll. and - communicated it to most of the chiefest mathematicians then residing in - Paris: who apprehending the great benefit that might accrue thereby, - importun'd him to express the use thereof in the French tongue. Which - being performed accordingly, he was advised by monsieur Alleawne the - King's chief engineer to dedicate his book to monsieur the King's only - brother, since duke of Orleans. Nevertheless the said work coming forth - as an abortive (the publishing thereof being somewhat hastened, by reason - an advocate of Dijon in Burgundy began to print some uses thereof, which - Wingate had in a friendly way communicated to him) especially in regard - Gunter himself had learnedly explained its use in a far larger volume.[6] - -Gunter's scale, which Wingate calls the "rule of proportion," contained, as -described in the French edition of 1624, four lines: (1) A single line of -numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot -in length, divided into 12 inches and tenths of inches, also a line, one -foot in length, divided into tenths and hundredths. - -The English editions of this book which appeared in 1623 and 1628 are -devoid of interest. The editions of 1645 and 1658 contain an important -innovation.[7] In the preface the reasons why this instrument has not been -used more are stated to be: (1) the difficulty of drawing the lines with -exactness, (2) the trouble of working thereupon by reason (sometimes) of -too large an extent of the compasses, (3) the fact that the instrument is -not readily portable. The drawing of Wingate's arrangement of the scale in -the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains -five parallel lines, about 66 cm. long, each having the divisions of one -line marked on one side and of another line on the other side. Thus each -line carries two graduations: (1) A single logarithmic line of numbers; (2) -a logarithmic line of numbers thrice repeated; (3) the first scale -repeated, but beginning with the graduations which are near the middle of -the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3; -(4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of -tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000 -equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of -inches; (10) a scale consisting of three kinds, viz., a gauge line, a line -of chords, and a foot measure, divided into 1000 equal parts. - -Important are the first and second scales, by which cube root extraction -was possible "by inspection only, without the aid of pen or compass;" -similarly the third and fourth scales, for square roots. This innovation is -due to Wingate. The 1645 edition announces that the instrument was made in -brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in -Hosier Lane. - - - Changes introduced by Milbourn - -William Leybourn, in his The Line of Proportion or Numbers, Commonly called -Gunter's Line, Made Easie, London, 1673, says in his preface "To the -Reader:" - - The Line of Proportion or Numbers, commonly called (by Artificers) - Gunter's Line, hath been discoursed of by several persons, and variously - applied to divers uses; for when Mr. Gunter had brought it from the - Tables to a Line, and written some Uses thereof, Mr. Wingate added divers - Lines of several lengths, thereby to extract the Square or Cube Roots, - without doubling or trebling the distance of the Compasses: After him Mr. - Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral - Line, thereby enlarging the divisions of the Line. - -On pages 127 and 128 Leybourn adds: - - Again, One T. Browne, a Maker of Mathematical Instruments, made it in a - Serpentine or Spiral Line, composed of divers Concentrick Circles, - thereby to enlarg the divisions, which was the contrivance of one Mr. - Milburn a Yorkshire Gentleman, who writ thereof, and communicated his - Uses to the aforesaid Brown, who (since his death) attributed it to - himself: But whoever was the contriver of it, it is not without - inconvenience; for it can in no wise be made portable; and besides - (instead of compasses) an opening Joynt with thirds [threads] must be - placed to move upon the Centre of the Instrument, without which no - proportion can be wrought. - -This Mr. Milburn is probably the person named in the diary of the -antiquarian, Elias Ashmole, on August 13 [1646?]; "I bought of Mr. Milbourn -all his Books and Mathematical Instruments."[8] Charles Hutton[9] says that -Milburne of Yorkshire designed the spiral form about 1650. This date is -doubtless wrong, for Thomas Browne who, according to Leybourn, got the -spiral form of line from Milbourn, is repeatedly mentioned by William -Oughtred in his Epistle[10] printed some time in 1632 or 1633. Oughtred -does not mention Milbourn, and says (page 4) that the spiral form "was -first hit upon by one Thomas Browne a Joyner, . . . the serpentine -revolution being but two true semicircles described on severall -centers."[11] - - - Changes introduced by Thomas Brown and John Brown - -Thomas Brown did not publish any description of his instrument, but his -son, John Brown, published in 1661 a small book,[12] in which he says -(preface) that he had done "as Mr. Oughtred with Gunter's Rule, to a -sliding and circular form; and as my father Thomas Brown into a Serpentine -form; or as Mr. Windgate in his Rule of Proportion." He says also that -"this brief touch of the Serpentine-line I made bold to assert, to see if I -could draw out a performance of that promise, that hath been so long -unperformed by the promisers thereof." Accordingly in Chapter XX he gives a -description of the serpentine line, "contrived in five (or rather 15) -turn." Whether this description, printed in 1661, exactly fits the -instrument as it was developed in 1632, we have no means of knowing. John -Brown says: - - 1. First next the center is two circles divided one into 60, the other - into 100 parts, for the reducing of minutes to 100 parts, and the - contrary. - - 2. You have in seven turnes two inpricks, and five in divisions, the - first Radius of the sines (or Tangents being neer the matter, alike to - the first three degrees,) ending at 5 degrees and 44 minutes. - - 3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in - three margents in divisions, and the line of versed sines in pricks, - under the line of Tangents, according to Mr. Gunter's cross-staff: the - sines and Tangents beginning at 5 degrees, and 44 minutes where the other - ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the - line of numbers beginning at 10, and proceeding to 100, being one entire - Radius, and graduated into as many divisions as the largeness of the - instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to - 100 into 20 parts in one unit of increase, but the Tangents are divided - into single minutes from the beginning to the end, both in the first, - second and third Radiusses, and the sines into minutes; also from 30 - minutes to 40 degrees, and from 40 to 60, into every two minutes, and - from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the - rest as many as can be well discovered. - - The versed sines are set after the manner of Mr. Gunter's Cross-staff, - and divided into every 10th minutes beginning at 0, and proceeding to 156 - going backwards under the line of Tangents. - - 4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is - the secants to 51 degrees, being nothing else but the sines reitterated - beyond 90. - - 5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85 - degrees, whereby all trouble of backward working is avoided. - - 6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and - also a line of natural sines, for finding of differences in sines, for - finding hour and Azimuth. - - 7. Seventhly, next the verge or outermost edge is a line of equal parts - to get the Logarithm of any number, or the Logarithm sine and Tangent of - any ark or angle to four figures besides the carracteristick. - - 8. Eightly and lastly, in the space place between the ending of the - middle five turnes, and one half of the circle are three prickt lines - fitted for reduction. The uppermost being for shillings, pence and - farthings. The next for pounds, and ounces, and quarters of small - Averdupoies weight. The last for pounds, shillings and pence, and to be - used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay - the hair or edge of one of the legs of the index on 16. 3 1/2 in the line - of 1. s. d. and the hair shall cut on the equal parts 81 16; and the - contrary, if you have a decimal fraction, and would reduce it to a proper - fraction, the like may you do for shillings, and pence, and pounds, and - ounces. - - The uses of the lines follow. - - As to the use of these lines, I shall in this place say but little, and - that for two reasons. First, because this instrument is so contrived, - that the use is sooner learned then any other, I speak as to the manner, - and way of using it, because by means of first second and third - radiusses, in sines and Tangents, the work is always right on, one way or - other, according to the Canon whatsoever it be, in any book that treats - of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in - Oughtred from page 64 to 107. - - Secondly, and more especially, because the more accurate, and large - handling thereof is more then promised, if not already performed by more - abler pens, and a large manuscript thereof by my Sires meanes, provided - many years ago, though to this day not extant in print; so for his sake I - claiming my interest therein, make bold to present you with these few - lines, in order to the use of them: And first note, - - 1. Which soever of the two legs is set to the first term in the question, - that I call the first leg always, and the other being set to the second - term, I call the second leg . . . - -The exact nature of the contrivance with the "two legs" is not described, -but it was probably a flat pair of compasses, attached to the metallic -surface on which the serpentine line was drawn. In that case the instrument -was a slide rule, rather than a form of Gunter's line. In his publication -of 1661, as also in later publications,[13] John Brown devoted more space -to Gunter's scales, requiring the use of a separate pair of compasses, than -to slide rules. - - - Changes introduced by William Leybourn - -The same remark applies to William Leybourn who, after speaking of Seth -Partridge's slide rule, returns to forms of Gunter's scale, saying:[14] - - There is yet another way of disposing of this Line of Proportion, by - having one Line of the full length of the Ruler, and another Line of the - same Radius broken in two parts between 3 and 4; so that in working your - Compasses never go off of the Line: This is one of the best contrivances, - but here Compasses must be used. These are all the Contrivances that I - have hitherto seen of these Lines: That which I here speak of, and will - shew how to use, is only two Lines of one and the same Radius, being set - upon a plain Ruler of any length (the larger the better) having the - beginning of one Line, at the end of the other, the divisions of each - Line being set so close together, that if you find any number upon one of - the Lines, you may easily see what number stands against it on the other - Line. This is all the Variation. . . . - - Example 1. If a Board be 1 Foot 64 parts broad, how much in length of - that Board will make a Foot Square? Look upon one of your Lines (it - matters not which) for 1 Foot 64 parts, and right against it on the other - Line you shall find 61; and so many parts of a Foot will make a Foot - square of that Board. - -This contrivance solves the equation 1.64x=1, yielding centesimal parts of -a foot. - -James Atkinson[15] speaks of "Gunter's scale" as "usually of Boxwood . . . -commonly 2 ft. long, 1 1/2 inch broad" and "of two kinds: long Gunter or -single Gunter, and the sliding Gunter. It appears that during the -seventeenth century (and long after) the Gunter's scale was a rival of the -slide rule. - - - - - III. RICHARD DELAMAIN'S GRAMMELOGIA - - -We begin with a brief statement of the relations between Oughtred and -Delamain. At one time Delamain, a teacher of mathematics in London, was -assisted by Oughtred in his mathematical studies. In 1630 Delamain -published the Grammelogia, a pamphlet describing a circular slide rule and -its use. In 1631 he published another tract, on the Horizontall -Quadrant.[16] In 1632 appeared Oughtred's Circles of Proportion[17] -translated into English from Oughtred's Latin manuscript by another pupil, -William Forster, in the preface of which Forster makes the charge (without -naming Delamain) that "another . . . went about to pre-ocupate" the new -invention. This led to verbal disputes and to the publication by Delamain -of several additions to the Grammelogia, describing further designs of -circular slide rules and also stating his side of the bitter controversy, -but without giving the name of his antagonist. Oughtred's Epistle was -published as a reply. Each combatant accuses the other of stealing the -invention of the circular slide rule and the horizontal quadrant. - -The two title-pages of the edition of the Grammelogia in the British Museum - in London which we have called "Grammelogia IV." - - - Different editions or impressions - -There are at least five different editions, or impressions, of the -Grammelogia which we designate, for convenience, as follows: - - Grammelogia I, 1630. One copy in the Cambridge University Library.[18] - - Grammelogia II, I have not seen a copy of this. - - Grammelogia III, One copy in the Cambridge University Library.[19] - - Grammelogia IV, One copy in the British Museum, another in the Bodleian - Library, Oxford.[20] - - Grammelogia V, One copy in the British Museum. - -In Grammelogia I the first three leaves and the last leaf are without -pagination. The first leaf contains the title-page; the second leaf, the -dedication to the King and the preface "To the Reader;" the third leaf, the -description of the Mathematical Ring. Then follow 22 numbered pages. -Counting the unnumbered pages, there are altogether 30 pages in the -pamphlet. Only the first three leaves of this pamphlet are omitted in -Grammelogia IV and V. - -In Grammelogia III the Appendix begins with a page numbered 52 and bears -the heading "Conclusion;" it ends with page 68, which contains the same two -poems on the mathematical ring that are given on the last page of -Grammelogia I but differs slightly in the spelling of some of the words. -The 51 pages which must originally have preceded page 52, we have not seen. -The edition containing these we have designated Grammelogia II. The reason -for the omission of these 51 pages can only be conjectured. In Oughtred's -Epistle (p. 24), it is stated that Delamain had given a copy of the -Grammelogia to Thomas Brown, and that two days later Delamain asked for the -return of the copy, "because he had found some things to be altered -therein" and "rent out all the middle part." Delamain labored "to recall -all the bookes he had given forth, (which were many) before the sight of -Brownes Lines." These spiral lines Oughtred claimed that Delamain had -stolen from Brown. The title-page and page 52 are the only parts of the -Appendix, as given in Grammelogia III, that are missing in the Grammelogia -IV and V. - -Grammelogia IV answers fully to the description of Delamain's pamphlet -contained in Oughtred's Epistle. It was brought out in 1632 or 1633, for -what appears to be the latest part of it contains a reference (page 99) to -the Grammelogia I (1630) as "being now more then two yeares past." -Moreover, it refers to Oughtred's Circles of Proportion, 1632, and -Oughtred's reply in the Epistle was bound in the Circles of Proportion -having the Addition of 1633. For convenience of reference we number the two -title-pages of Grammelogia IV, "page (1)" and "page (2)," as is done by -Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page -numbers which we assign will be placed in parentheses, to distinguish them -from the page numbers which are printed in Grammelogia IV. The pages -(44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the -same as the pages 53-68. Thus only thirty-eight pages have page numbers -printed on them. The pages (67) and (83) are identical in wording, except -for some printer's errors; they contain verses in praise of the Ring, and -have near the bottom the word "Finis." Also, pages (22) and (23) are -together identical in wording with page (113), which is set up in finer -type, containing an advertisement of a part of Grammelogia IV explaining -the mode of graduating the circular rules. There are altogether six parts -of Grammelogia IV which begin or end by an address to the reader, thus: "To -the Reader," "Courteous Reader," or "To the courteous and benevolent Reader -. . .," namely the pages (8), (22), (68), (89), (90), (108). In his Epistle -(page 2), Oughtred characterizes the make up of the book in the following -terms: - - In reading it . . . I met with such a patchery and confusion of - disjoynted stuffe, that I was striken with a new wonder, that any man - should be so simple, as to shame himselfe to the world with such a - hotch-potch. - -Grammelogia V differs from Grammelogia IV in having only the second -title-page. The first title-page may have been torn off from the copy I -have seen. A second difference is that the page with the printed numeral 22 -in Grammelogia IV has after the word "Finis" the following notice: - - This instrument is made in Silver, or Brasse for the Pocket, or at any - other bignesse, over against Saint Clements Church without Temple Barre, - by Elias Allen. - -This notice occurs also on page 22 of Grammelogia I and III, but is omitted -from page 22 of Grammelogia V. - - - Description of Delamain's instrument of 1630 - -In his address to King Charles I, in his Grammelogia I, Delamain emphasizes -the ease of operating with his slide rule by stating that it is "fit for -use . . . as well on Horse backe as on Foot." Speaking "To the Reader," he -states that he has "for many yeares taught the Mathematicks in this Towne," -and made efforts to improve Gunter's scale "by some Motion, so that the -whole body of Logarithmes might move proportionally the one to the other, -as occasion required. This conceit in February last [1629] I struke upon, -and so composed my Grammelogia or Mathematicall Ring; by which only with an -ocular inspection, there is had at one instant all proportionalls through -the said body of Numbers." He dates his preface "first of January, 1630." -The fifth and sixth pages contain his "Description of the Grammelogia," the -term Grammelogia being applied to the instrument, as well as to the book. -His description is as follows: - - The parts of the Instrument are two Circles, the one moveable, and the - other fixed; The moveable is that unto which is fastened a small pin to - move it by; the other Circle may be conceived to be fixed; The - circumference of the moveable Circle is divided into unequall parts, - charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures - doe represent themselves, or such numbers unto which a Cipher or Ciphers - are added, and are varied as the occasion falls out in the speech of - Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or - 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c. - -After elaborating this last point and explaining the decimal subdivisions -on the scales of the movable circle, he says that "the numbers and -divisions on the fixed Circle, are the very same that the moveable are, . -." There is no drawing of the slide rule in this publication. The -twenty-two numbered pages give explanations of the various uses to which -the instrument can be put: "How to performe the Golden Rule" (pp. 1-3), -"Further uses of the Golden Rule" (pp. 4-6), "Notions or Principles -touching the disposing or ordering of the Numbers in the Golden Rule in -their true places upon the Grammelogia" (pp. 7-11), "How to divide one -number by another" (pp. 12, 13), "to multiply one Number by another" (pp. -14, 15), "To find Numbers in continuall proportion" (pp. 16, 17), "How to -extract the Square Root," "How to extract the Cubicke Root" (pp. 18-21), -"How to performe the Golden Rule" (the rule of proportion) is explained -thus: - - Seeke the first number in the moveable, and bring it to the second number - in the fixed, so right against the third number in the moveable, is the - answer in the fixed. - - If the Interest of 100. li. be 8. li. in the yeare, what is the Interest - of 65. li. for the same time. - - Bring 100. in the moveable to 8. in the fixed, so right against 65. in - the moveable is 5.2. in the fixed, and so much is the Interest of 65. li. - for the yeare at 8. li. for 100. li. per annum. - - The Instrument not removed, you may at one instant right against any - summe of money in the moveable, see the Interest thereof in the fixed: - the reason of this is from the Definition of Logarithmes. - -These are the earliest known printed instructions on the use of a slide -rule. It will be noticed that the description of the instrument at the -opening makes no references to logarithmic lines for the trigonometric -functions; only the line of numbers is given. Yet the title-page promised -the "resolution of Plaine and Sphericall Triangles." Page 22 throws light -upon this matter: - - If there be composed three Circles of equal thicknesse, A.B.C. so that - the inner edge of D [should be B] and the outward edge of A bee - answerably graduated with Logarithmall signes [sines], and the outward - edge of B and the inner edge of A with Logarithmes; and then on the - backside be graduated the Logarithmall Tangents, and againe the - Logarithmall signes oppositly to the former graduations, it shall be - fitted for the resolution of Plaine and Sphericall Triangles. - -After twelve lines of further remarks on this point he adds: - - Hence from the forme, I have called it a Ring, and Grammelogia by - annoligie of a Lineary speech; which Ring, if it were projected in the - convex unto two yards Diameter, or thereabouts, and the line Decupled, it - would worke Trigonometrie unto seconds, and give proportionall numbers - unto six places only by an ocular inspection, which would compendiate - Astronomicall calculations, and be sufficient for the Prosthaphaeresis of - the Motions: But of this as God shall give life and ability to health and - time. - -The unnumbered page following page 22 contains the patent and copyright on -the instrument and book: - - Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs - an Instrument called Grammelogia, or The Mathematicall Ring, together - with a Booke so intituled, expressing the use thereof, being his owne - Invention; we of our Gracious and Princely favour have granted unto the - said Richard Delamain and his Assignes, Privilege, Licence, and - Authority, for the sole Making, Printing and Selling of the said - Instrument and Booke: straightly forbidding any other to Make, Imprint, - or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument - or Booke within any our Dominions, during the space of ten yeares next - ensuing the date hereof, upon paine of Our high displeasure. Given under - our hand and Signet at our Palace of Westminster, the fourth day of - January, in the sixth yeare of our Raigne. - - - Delamain's later designs, and directions for using his instruments - -In the Appendix of Grammelogia III, on page 52 is given a description of an -instrument promised near the end of Grammelogia I: - - That which I have formerly delivered hath been onely upon one of the - Circles of my Ring, simply concerning Arithmeticall Proportions, I will - by way of Conclusion touch upon some uses of the Circles, of Logarithmall - Sines, and Tangents, which are placed on the edge of both the moveable - and fixed Circles of the Ring in respect of Geometricall Proportions, but - first of the description of these Circles. - - First, upon the side that the Circle of Numbers is one, are graduated on - the edge of the moveable, and also on the edge of the fixed the - Logarithmall Sines, for if you bring 1. in the moveable amongst the - Numbers to 1. in the fixed, you may on the other edge of the moveable and - fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto - 6.6. and each degree subdivided, and then over the former divisions and - figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5. - 4. 3. 2. 1. each of those divided by small points. - - Secondly, (if the Ring is great) neere the outward edge of this side of - the fixed against the Numbers, are the usuall divisions of a Circle, and - the points of the Compasse: serving for observation in Astronomy, or - Geometry, and the sights belonging to those divisions, may be placed on - the moveable Circle. - - Thirdly, opposite to those Sines on the other side are the Logarithmall - Tangents, noted alike both in the moveable and fixed thus - 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or - divisions serve also for their Complements to 90. so 40 gr. stands for - 50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both - in the moveable and fixed is also divided into parts. As for the degrees - which are under 6. viz. 5.4.3.2.1. they are noted with small figures over - this divided Circle from 45.40.35.30.25. &c. and each of those degrees - divided into parts by small points both in the moveable and fixed. - - Fourthly, on the other edge of the moveable on the same side is another - graduation of Tangents, like that formerly described. And opposite unto - it, in the fixed is a Graduation of Logarithmall sines in every thing - answerable to the first descrition of Sines on the other side. - - Fifthly, on the edge of the Ring is graduated a parte of the quator, - numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees - of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree - both of the quator, and Meridian are subdivided into parts; these two - graduated Circles serve to resolve such Questions which concerne - Latitude, Longitude, Rumb, and Distance, in Nauticall operations. - - Sixthly, to the concave of the Ring may be added a Circle to be elevated - or depressed for any Latitude, representing the quator, and so divided - into houres and parts with an Axis, to shew both the houre, and Azimuth, - and within this Circle may be hanged a Box, and Needle with a Socket for - a staffe to slide into it, and this accommodated with scrue pines to - fasten it to the Ring and staffe, or to take it off at pleasure. - -The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and -V make no reference to the dispute with Oughtred and may, therefore, be -assumed to have been published before the appearance of Oughtred's Circles -of Proportion. On page 53, "To the Reader," he says: - - . . . you may make use of the Projection of the Circles of the Ring upon - a Plaine, having the feet of a paire of compasses (but so that they be - flat) to move on the Center of that Plaine, and those feet to open and - shut as a paire of Compasses . . . now if the feet bee opened to any two - termes or numbers in that Projection, then may you move the first foot to - the third number, and the other foot shall give the Answer; . . . it hath - pleased some to make use of this way. But in this there is a double - labour in respect to that of the Ring, the one in fitting those feet unto - the numbers assigned, and the other by moving them about, in which a man - can hardly accommodate the Instrument with one hand, and expresse the - Proportionals in writing with the other. By the Ring you need not but - bring one number to another, and right against any other number is the - Answer without any such motion. . . . upon that [the Ring] I write, - shewing some uses of those Circles amongst themselves, and conjoyned with - others . . . in Astronomy, Horolographie, in plaine Triangles applyed to - Dimensions, Navigation, Fortification, etc. . . . But before I come to - Construction, I have thought it convenient by way introduction, to - examine the truth of the graduation of those Circles . . . - -These are the words of a practical man, interested in the mechanical -development of his instrument. He considers not only questions of -convenience but also of accuracy. The instrument has, or may have now, also -lines of sines and tangents. To test the accuracy of the circles of -Numbers, "bring any number in the moveable to halfe of that number in the -fixed: so any number or part in the fixed shall give his double in the -moveable, and so may you trie of the thirds, fourths &c. of numbers, vel -contra," (p. 54). On page 55 are given two small drawings, labelled, "A -Type of the Ringe and Scheme of this Logarithmicall projection, the use -followeth. These Instruments are made in Silver or Brasse by John Allen -neare the Sauoy in the Strand." - - - IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE - CIRCULAR SLIDE RULE - -Delamain's publication of 1630 on the 'Mathematicall Ring' does not appear -at that time to have caused a rupture between him and Oughtred. When in -1631 Delamain brought out his Horizontall Quadrant, the invention of which -Delamain was afterwards charged to have stolen from Oughtred, Delamain was -still in close touch with Oughtred and was sending Oughtred in the Arundell -House, London, the sheets as they were printed. Oughtred's reference to -this in his Epistle (p. 20) written after the friendship was broken, is as -follows: - - While he was printing his tractate of the Horizontall quadrant, although - he could not but know that it was injurious to me in respect of my free - gift to Master Allen, and of William Forster, whose translation of my - rules was then about to come forth: yet such was my good nature, and his - shamelessnesse, that every day, as any sheet was printed, hee sent, or - brought the same to mee at my chamber in Arundell house to peruse which I - lovingly and ingenuously did, and gave him my judgment of it. - -Even after Forster's publication of Oughtred's Circles of Proportion, 1632, -Oughtred had a book, A canon of Sines Tangents and Secants, which he had -borrowed from Delamain and was then returning (Epistle, page (5)). The -attacks which Forster, in the preface to the Circles of Proportion, made -upon Delamain (though not naming Delamain) started the quarrel. Except for -Forster and other pupils of Oughtred who urged him on to castigate -Delamain, the controversy might never have arisen. Forster expressed -himself in part as follows: - - . . . being in the time of the long vacation 1630, in the Country, at the - house of the Reverend, and my most worthy friend, and Teacher, Mr. - William Oughtred (to whose instruction I owe both my initiation, and - whole progresse in these Sciences.) I vpon occasion of speech told him of - a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made - (such as it vsually called Mr. Gunter's Ruler) 6 feet long, to be vsed - with a payre of beame-compasses. "He answered that was a poore invention, - and the performance very troublesome: But, said he, seeing you are taken - with such mechanicall wayes of Instruments, I will shew you what deuises - I have had by mee these many yeares." And first, hee brought to mee two - Rulers of that sort, to be vsed by applying one to the other, without any - compasses: and after that hee shewed mee those lines cast into a circle - or Ring, with another moueable circle vpon it. I seeing the great - expeditenesse of both those wayes; but especially, of the latter, wherein - it farre excelleth any other Instrument which hath bin knowne; told him, - I wondered that hee could so many yeares conceale such vseful inuentions, - not onely from the world, but from my selfe, to whom in other parts and - mysteries of Art, he had bin so liberall. He answered, "That the true way - of Art is not by Instruments, but by Demonstration: and that it is a - preposterous course of vulgar Teachers, to begin with Instruments, and - not with the Sciences, and so in-stead of Artists, to make their - Schollers only doers of tricks, and as it were Iuglers: to the despite of - Art, losse of precious time, and betraying of willing and industrious - wits, vnto ignorance and idlenesse. That the vse of Instruments is indeed - excellent, if a man be an Artist: but contemptible, being set and opposed - to Art. And lastly, that he meant to commend to me, the skill of - Instruments, but first he would haue me well instructed in the Sciences. - He also shewed me many notes, and Rules for the vse of those circles, and - of his Horizontall Instrument, (which he had proiected about 30 yeares - before) the most part written in Latine. All which I obtained of him - leaue to translate into English, and make publique, for the vse, and - benefit of such as were studious, and louers of these excellent Sciences. - - Which thing while I with mature, and diligent care (as my occasions would - give me leaue) went about to doe: another to whom the Author in a louing - confidence discouered this intent, using more hast then good speed, went - about to preocupate; of which vntimely birth, and preuenting (if not - circumuenting) forwardnesse, I say no more: but aduise the studious - Reader, onely so farre to trust, as he shal be sure doth agree to truth & - Art. - -While in this dedication reference is made to a slide rule or "ring" with a -"moveable circle," the instrument actually described in the Circles of -Proportion consists of fixed circles "with an index to be opened after the -manner of a paire of Compasses." Delamain, as we have seen, had decided -preference for the moveable circle. To Oughtred, on the other hand, one -design was about as good as the other; he was more of a theorist and -repeatedly expressed his contempt for mathematical instruments. In his -Epistle (page (25)), he says he had not "the one halfe of my intentions -upon it" (the rule in his book), nor one with a "moveable circle and a -thread, but with an opening Index at the centre (if so be that bee cause -enough to make it to bee not the same, but another Instrument) for my part -I disclaime it: it may go seeke another Master: which for ought I know, -will prove to be Elias Allen himselfe: for at his request only I altered a -little my rules from the use of the moveable circle and the thread, to the -two armes of an Index." - -All parts of Delamain's Grammelogia IV, except pages 1-22 and 53-68 -considered above, were published after the Circles of Proportion, for they -contain references to the ill treatment that Delamain felt or made believe -that he felt, that he had received in the book published by Oughtred and -Forster. Oughtred's reference to teachers whose scholars are "doers of -tricks," "Iuglers," and Forster's allusion to "another to whom the Author -in a loving confidence" explained the instrument and who "went about to -preocupate" it, are repeatedly mentioned. Delamain says, (page (89)) that -at first he did not intend to express himself in print, "but sought peace -and my right by a private and friendly way." Oughtred's account of -Delamain's course is that of an "ill-natured man" with a "virulent tongue," -"sardonical laughter" and "malapert sawsiness." Contrasting Forster and -Delamain, he says that, of the former he "had the very first moulding" and -made him feel that "the way of Art" is "by demonstration." But Delamain was -"already corrupted with doing upon Instruments, and quite lost from ever -being made an Artist." (Epistle page (27)). Repeatedly does Oughtred assert -Delamain's ignorance of mathematics. The two men were evidently of wholly -different intellectual predilections. That Delamain loved instruments is -quite evident, and we proceed to describe his efforts to improve the -circular slide rule. - -The Grammelogia IV is dedicated to King Charles I. Delamain says: - - . . . Everything hath his beginning, and curious Arts seldome come to the - height at the first; It was my promise then to enlarge the invention by a - way of decuplating the Circles, which I now present unto your sacred - Majestie as the quintessence and excellencie there of . . . - -His enlarged circular rules are illustrated in the Bodleian Library copy of -Grammelogia IV by four diagrams, two of them being the two drawings on the -two title-pages at the beginning of the Grammelogia IV, 4 inches in -external diameter, and exhibiting eleven concentric circular lines carrying -graduations of different sorts. In the second of these designs all circles -are fixed. The other two drawings are each 10 3/4 inches in external -diameter and exhibit 18 concentric circular lines; the folded sheet of the -first of these drawings is inserted between pages (23) and (24), the second -folded sheet between pages (83) and (84). All circles of this second -instrument are fixed. Counting in the two small drawings in Grammelogia -III, there are in all six drawings of slide rules in the Bodleian -Grammelogia IV. On pages (24) to (43) Delamain explains the graduation of -slide rules. He takes first a rule which has one circle of equal parts, -divided into 1000 equal divisions. From a table of logarithms he gets log 2 -= 0.301; from the number 301 in the circle of equal parts he draws a line -to the center of the circle and marks the intersection with the circles of -numbers by the figure 2. Thus he proceeds with log 3, log 4, and so on; -also with log sin x and log tan x. For log sin x he uses two circles, the -first (see page (27)) for angles from 34' 24" to 5^o 44' 22", the second -circle from 5^o 44' 22" to 90^o. The drawings do not show the seconds. He -suggests many different designs of rules. On page (29) he says: - - For the single projection of the Circles of my Ring, and the dividing and - graduating of them: which may bee so inserted upon the edges of Circles - of mettle turned in the forme of a Ring, so that one Circle may moove - betweene two fixed, by helpe of two stayes, then may there be graduated - on the face of the Ring, upon the outer edge of the mooveable and inner - edge of the fixed, the Circle of Numbers, then upon the inner edge of - that mooveable Circle, and the outward edge of that inner fixed Circle - may be inserted the Circle of Sines, and so according to the description - of those that are usually made. - -In addition to these lines he proceeds to mention the circle giving the -ordinary division into degrees and minutes, and two circles of tangents on -the other side of the rule. - -Next Delamain explains an arrangement of all the graduation on one side of -the rule by means of "a small channell in the innermost fixed Circle, in -which may be placed a small single Index, which may have sufficient length -to reach from the innermost edge of the Mooveable Circle, unto the outmost -edge of the fixed Circle, which may be mooved to and fro at pleasure, in -the channell, which Index may serve to shew the opposition of Numbers" (p. -(31)). From this it is clear that the invention of the "runner" goes back -to the very first writers on the slide rule. - -After describing a modification of the above arrangement, he adds, "many -other formes might be deliverd, about this single projection" (p. (32)). - -Proceeding to the "enlarging" of the circles in the Ring, to, say, the -"Quadruple to that which is single, that is, foure times greater," the -"equall parts" are distributed over four circles instead of only one -circle, but the general method of graduation is the same as before (p. -(33)); there being now four circles carrying the logarithms of numbers, and -so on. Next he points out "severall wayes how the Circles of the -Mathematicall Ring (being inlarged) may be accommodated for practicall -use:" (1) The Circles are all fixed in a plain and movable flat compasses -(or better, a movable semicircle) are used for fixing any two positions; -(2) There is a "double projection" of each logarithmic line "inlarged on a -Plaine," one fixed, the other movable, as shown in his first figure on the -title-page, a single index only being used; (3) use of "my great Cylinder -which I have long proposed (in which all the Circles are of equall -greatnesse,) and it may be made of any magnitude or capacity, but for a -study (hee that will be at the charge) it may be of a yard diameter and of -such an indifferent length that it may containe 100 or more Circles fixed -parallel one to the other on the Cylinder, having a space betweene each of -them, so that there may bee as many mooveable Circles, as there are fixed -ones, and these of the mooveable linked, or fastened together, so that they -may all moove together by the fixed ones in these spaces, whose edges both -of the fixed, and mooveable being graduated by helpe of a single Index will -shew the proportionalls by opposition in this double Projection, or by a -double Index in a single Projection" (p. (36)). - -Next follows the detailed description of his Ring "on a Plaine, according -to the diagramme that was given the King (for a view of that projection) -and afterwards the Ring it selve." The diagram is the large one which we -mentioned as inserted between pages (23) and (24). The instrument has two -circles, one moveable, upon each of which are described 13 distinct -circular graduations. The lines on the fixed circle are: "The Circle of -degrees and calendar," E. "Circle of equall parts, and part of the Equator, -and Meridian," TT. "The Circle of Tangents," S. "The Circle of Sines," D. -"The Circle of Decimals," N. "The Circle of Numbers." The lines on the -movable circle are: N. "The Circle of Numbers," E. "The Circle of equated -figures, and bodies," S. "The Circle of Sines," TT. "The Circle of -Tangents," Y. "The Circle of time, yeares, and monethes." - -On pages (84)-(88) Delamain explains an enlargement of his Ring for -computations involving the sines of angles near to 90^o. On page (86) he -says: - - I have continued the Sines of the Projection unto two severall - revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr. - (being the last revolution of the decuplation of the former, or the - hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s. - and ends at 90.gr. (being the last of a ternary of decuplated - revolutions, or the thousand part of that Projection) and may bee thus - used. - -He explains the manner of using these extra graduations. Thus he claims to -have attained degrees of accuracy which enabled him to do what "some one" -had declared "could not bee done." It is hardly necessary to point out that -Delamain's Grammelogia IV suggests designs of slide rules which inventors -two hundred or more years later were endeavouring to produce. Which of -Delamain's designs of rules were actually made and used, he does not state -explicitly. He refers to a rule 18 inches in diameter as if it had been -actually constructed (pages (86), (88)). Oughtred showed no appreciation of -such study in designing and ridiculed Delamain's efforts, in his Epistle. - -Additional elucidations of his designs of rules, along with explanations of -the relations of his work to that of Gunter and Napier, and sallies -directed against Oughtred and Forster, are contained on pages (8)-(21) of -his Grammelogia IV. - - - - - V. INDEPENDENCE AND PRIORITY OF INVENTION - - -The question of independence and priority of invention is discussed by -Delamain more specifically on pages (89)-(113); Oughtred devotes his entire -Epistle to it. It is difficult to determine definitely which publication is -the later, Delamain's Grammelogia IV or Oughtred's Epistle. Each seems to -quote from the other. Probably the explanation is that the two publications -contain arguments which were previously passed from one antagonist to the -other by word of mouth or by private letter. Oughtred refers in his Epistle -(p. (12)) to a letter from Delamain. We believe that the Epistle came after -Delamain's Grammelogia IV. Delamain claims for himself the invention of the -circular slide rule. He says in his Grammelogia IV. (p. (99)), "when I had -a sight of it, which was in February, 1629 (as I specified in my Epistle) I -could not conceale it longer, envying my selfe, that others did not tast of -that which I found to carry with it so delightfull and pleasant a goate -[taste] . . ." Delamain asserts (without proof) that Oughtred "never saw it -as he now challengeth it to be his invention, untill it was so fitted to -his hand, and that he made all his practise on it after the publishing of -my Booke upon my Ring, and not before; so it was easie for him or some -other to write some uses of it in Latin after Christmas, 1630 and not the -Sommer before, as is falsely alledged by some one . . ." (p. (91)). -Delamain's accusation of theft on the part of Oughtred cannot be seriously -considered. Oughtred's reputation as a mathematician and his standing in -his community go against such a supposition. Moreover, William Forster is a -witness for Oughtred. The fact that Oughtred had the mastery of the -rectilinear slide rule as well, while Delamain in 1630 speaks only of the -circular rule, weighs in Oughtred's favour. - -Oughtred says he invented the slide rule "above twelve yeares agoe," that -is, about 1621, and "I with mine owne hand made me two such Circles, which -I have used ever since, as my occasions required," (Epistle p. (22)). On -the same page, he describes his mode of discovery thus: - - I found that it required many times too great a paire of Compasses [in - using Gunter's line], which would bee hard to open, apt to slip, and - troublesome for use. I therefore first devised to have another Ruler with - the former: and so by setting and applying one to the other, I did not - onely take away the use of Compasses, but also make the worke much more - easy and expedite: when I should not at all need the motion of my hand, - but onely the glancing at my sight: and with one position of the Rulers, - and view of mine eye, see not one onely, but the manifold proportions - incident unto the question intended. But yet this facility also wanted - not some difficulty especially in the line of tangents, when one arch was - in the former mediety of the quadrant, and the other in the latter: for - in this case it was needful that either one Ruler must bee as long againe - as the other; or else that I must use an inversion of the Ruler, and - regression. By this consideration I first of all saw that if those lines - upon both Rulers were inflected into two circles, that of the tangents - being in both doubled, and that those two Circles should move one upon - another; they with a small thread in the center to direct the sight, - would bee sufficient with incredible and wonderfull facility to worke all - questions of Trigonometry . . . - -Oughtred said that he had no desire to publish his invention, but in the -vacation of 1630 finally promised William Forster to let him bring out a -translation. Oughtred claims that Delamain got the invention from him at -Alhallontide [November 1], 1630, when they met in London. The accounts of -that meeting we proceed to give in double column. - - Delamain's Statement - Grammelogia IV, page (98) - - ". . . about Alhalontide 1630. (as our Authors reporteth) was the time he - was circumvented, and then his intent in a loving manner (as before) he - opened unto me, which particularly I will dismantle in the very naked - truth: for, wee being walking together some few weekes before Christmas, - upon Fishstreet hill, we discoursed upon sundry things Mathematicall, - both Theoreticall and Practicall, and of the excellent inventions and - helpes that in these dayes were produced, amongst which I was not a - little taken with that of the Logarythmes, commending greatly the - ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, - in the lines of proportion, extracted from these Logarythmes for ordinary - Practicall uses; He replyed unto me (in these very words) What will yov - say to an Invention that I have, which in a lesse extent of the Compasses - shall worke truer then that of Mr. Gunters Ruler, I asked him then of - what forme it was, he answered with some pause (which no doubt argued his - suspition of mee that I might conceive it) that it was Arching-wise, but - now hee sayes that hee told mee then, it was Circular (but were I put to - my oath to avoid the guilt of Conscience I would conclude in the former.) - At which immediately I answered, I had the like my selfe, and so we - discoursed not a word more touching that subject . . . Then after my - coming home I sent him a sight of my Projection drawne in Pastboard: Now - admit I had not the Invention of my Ring before I discoursed . . . it was - not so facil for mee . . . to raise and compose so complete, and absolute - an Instrument from so small a principle, or glimpse of light . . ." - - Oughtred's Statement - Epistle, page (23) - - "Shortly after my gift to Elias Allen, I chanced to meet with Richard - Delamain in the street (it was at Alhallontide) and as we walked together - I told him what an Instrument I had given to Master Allen, both of the - Logarithmes projected into circles, which being lesse then one foot - diameter would performe as much as one of Master Gunters Rulers of sixe - feet long: and also of the Prostaphaereses of the Plannets and second - motions. Such an invention have I said he: for now his intentions (that - is his ambition) beganne to worke: . . . But he saith, Then after my - comming home I sent him a sight of my projection drawne in past-board. - See how notoriously he jugleth without an Instrument. Then after: how - long after? a sight of my projection: of how much? More then seven weekes - after on December 23, he sent to mee the line of numbers onely set upon a - circle: . . . and so much onely he presented to his Majesty: but as for - Sine or tangent of his, there was not the least shew of any. Neither - could he give to Master Allen any direction for the composure of the - circles of his Ring, or for the division of them: as upon his oath Master - Allen will testify how hee misled him, and made him labour in vain above - three weeks together, until Master Allen himselfe found out his ignorance - and mistaking, which is more cleare then is possible with any impudence - to be outfaced." - -Oughtred makes a further statement (Epistle, p. (24)) as follows: - - Delamain hearing that Brown with his Serpentine had another line by which - he could worke to minutes in the 90 degree of sines . . . gave the [his] - booke to Browne: who in thankfulnesse could not but gratify Delamain with - his Lines also: and teach him the use of them, but especially of the - great Line: with this caution on both sides, that one should not meddle - with the others invention. Two dayes after Delamain . . . because he had - found some things to be altered therin, . . . asked for the booke . . . - but as soone as he had got it in his hands he rent out all the middle - part with the two Schemes & put them up in his pocket & went his way . . - . and . . . laboureth to recall all the bookes he had given forth . . . - And shortly after this he got a new Printer (who was ignorant of his - former Schemes) to print him new: giving him an especiall charge of the - outermost line newly graven in the Plate, which indeed is Brownes very - line: and then altering his book . . . - -This and other statements made by Oughtred seem damaging to Delamain's -reputation. But it is quite possible that Oughtred's guesses as to -Delamain's motives are wrong. Moreover, some of Oughtred's statements are -not first hand knowledge with him, but mere hearsay. One may accept his -first hand facts and still clear Delamain of wrong doing. There is always -danger that rival claimants of an invention or discovery will proceed on -the assumption that no one else could possibly have come independently upon -the same devices that they themselves did; the history of science proves -the opposite. Seldom is an invention of any note made by only one man. We -do not feel competent to judge Delamain's case. We know too little about -him as a man. We incline to the opinion that the hypothesis of independent -invention is the most plausible. At any rate, Delamain figures in the -history of the slide rule as the publisher of the earliest book thereon and -as an enthusiastic and skillful designer of slide rules. - -The effect of this controversy upon interested friends was probably small. -Doubtless few people read both sides. Oughtred says:[21] "this scandall . . -. hath with them, to whom I am not knowne, wrought me much prejudice and -disadvantage . ." Aubrey,[22] a friend of Oughtred, refers to Delamain "who -was so sawcy to write against him" and remembers having seen "many yeares -since, twenty or more good verses made" against Delamain. Another friend of -Oughtred, William Robinson, who had seen some of Delamain's publications, -but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before -the appearance of the latter's Epistle: - - I cannot but wonder at the indiscretion of Rich. Delamain, who being - conscious to himself that he is but the pickpurse of another man's wit, - would thus inconsiderately provoke and awake a sleeping lion . . . he - hath so weakly (though in my judgment, vaingloriously enough) commended - his own labour . . .[23] - -Delamain presented King Charles I with one of his sun-dials, also with a -manuscript and, later, with a printed copy of his book of 1630. A drawing -of his improved slide rule was sent to the King and the Grammelogia IV is -dedicated to him. The King must have been favorably impressed, for Delamain -was appointed tutor to the King in mathematics. His widow petitioned the -House of Lords in 1645 for relief; he had ten children.[24] - -Anthony Wood states that Charles I, on the day of his execution, commanded -his friend Thomas Herbert "to give his son the duke of York his large -ring-sundial of silver, a jewel his maj. much valued." Anthony Wood adds, -"it was invented and made by Rich. Delamaine a very able mathematician, who -projected it, and in a little printed book did shew its excellent use in -resolving many questions in arithmetic and other rare operations to be -wrought by it in the mathematics."[25] - - - - - VI. OUGHTRED'S GAUGING LINE, 1633 - - -It has not been generally known, hitherto, that Oughtred designed a -rectilinear slide rule for gauging and published a description thereof in -1633.[26] In his Circles of Proportion, chapter IX, Oughtred had offered a -closer approximation than that of Gunter for the capacity of casks. The -Gauger of London expostulated with Oughtred for presuming to question -anything that Gunter had written. The ensuing discussion led to an -invitation extended by the Company of Vintners to the instrument maker -Elias Allen to request Oughtred to design a gauging rod.[27] This he did, -and Allen received an order for "threescore" instruments. On page 19 -Oughtred describes his 'Gauging Rod:' - - It consisteth of two rulers of brasse about 32 ynches of length, which - also are halfe an ynch broad, and a quarter of an ynch thick . . . At one - end of both those rulers are two little sockets of brasse fastened on - strongly: by which the rulers are held together, and made to move one - upon another, and to bee drawne out unto any length, as occasion shall - require: and when you have them at the just length, there is upon one of - the sockets a long Scrue-pin to scrue them fast. - -There are graduations on three sides of the rulers, one graduation being -the logarithmic line of numbers. He says (p. 39), "the maner of computing -the Gauge-divisions I have concealed." W. Robinson, who was a friend of -Oughtred, wrote him as follows:[28] - - I have light upon your little book of artificial gauging, wherewith I am - much taken, but I want the rod, neither could I get a sight of one of - them at the time, because Mr. Allen had none left . . . I forgot to ask - Mr. Allen the price of one of them, which if not much I would have one of - them." Oughtred annotated this passage thus: "Or in wood, if any be made - in wood by Thompson or any other." - -Another of Oughtred's admirers, Sir Charles Cavendish, wrote, on February -11, 1635 thus:[29] - - I thank you for your little book, but especially for the way of - calculating the divisions of your gauging rod. I wish, both for their own - sakes and yours, that the citizens were as capable of the acuteness of - this invention, as they are commonly greedy of gain, and then I doubt not - but they would give you a better recompense than I doubt now they will. - -On April 20, 1638, we find Oughtred giving Elias Allen directions[30] -"about the making of the two rulers." As in 1633,[31] so now, Oughtred -takes one ruler longer than the other. This 1633 instrument was used also -as "a crosse-staffe to take the height of the Sunne, or any Starre above -the Horizon, and also their distances." The longer ruler was called staffe, -the shorter transversarie. While in 1633 he took the lengths of the two in -the ratio "almost 3 to 2," in 1638, he took "the transversary three -quarters of the staff's length, . . . that the divisions may be larger." - - - - - VII. OTHER SEVENTEENTH CENTURY SLIDE RULES - - -In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard, -Henry Coggeshall, W. Hunt and Sir Isaac Newton.[32] Of Partridge's Double -Scale of Proportion, London, I have examined a copy dated 1661, which is -the earliest date for this book that I have seen. As far as we know, 1661 -is the earliest date of publications on the slide rule, since Oughtred and -Delamain. But it would not be surprising if the intervening 28 years were -found not so barren as they seem at present. The 1661 and 1662 impressions -of Partridge are identical, except for the date on the title-page. William -Leybourn, who printed Partridge's book, speaks in high appreciation of it -in his own book.[33] - -In 1661 was published also John Brown's first book, Description and Use of -a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use -of "Mr. Whites rule" for the measuring of board and timber, round and -square. He calls this a "sliding rule." The existence, in 1661, of a -"Whites rule" indicates activities in designing of which we know as yet -very little. In his book of 1761, previously quoted, Brown gives a drawing -of "White's sliding rule" (p. 193); also a special contrivance of his own, -as indicated by him in these words: - - A further improvement of the Triangular Quadrant, as I have made it - several times, with a sliding Cover on the in-side, when made hollow, to - carry Ink, Pens, and Compasses; then on the sliding Cover, and Edges, is - put the Line of Numbers, according to Mr. White's first Contrivance for - manner of operation; but much augmented, and made easie, by John Brown. - -He gives no drawing of his "triangular quadrant," hence his account of it -is unsatisfactory. He explains the use of "gage-points." His placing -logarithmic lines on the edges of instrument boxes was outdone in oddity -later by Everard who placed them on tobacco-boxes.[34] In Brown's -publication of 1704 the White slide rule is given again, "being as neat and -ready a way as ever was used." He tells also of a "glasier's sliding rule." -William Leybourn explains in 1673 how Wingate's double and triple lines for -squaring and cubing, or square and cube root, can be used on slide -rules.[35] - -Beginning early in the history of the slide rule, when Oughtred designed -his "gauging rod," we notice the designing of rules intended for very -special purposes. Another such contrivance, which enjoyed long popularity, -was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677, -a booklet of 35 pages. Coggeshall says in his preface: - - For what can be more ready and easie, then having set twelve to the - length, to see the Content exactly against the Girt or Side of the - Square. Whereas on Mr. Partridge's Scale the Content is the Sixth Number, - which is far more troublesome then [even] with Compasses. - -One line on Coggeshall's rule begins with 4 and extends to 40, these -numbers being the "Girt" (a quarter of the circumference), which in -ordinary practice of measuring round timber lies between 4 inches and 40 -inches. This "Girt line" slides "against the line of Numbers in two -Lengths, to which it is exactly equal." A second edition, 1682, shows some -changes in the rule, as well as an enlargement and change of title of the -book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent, -London, 1682. In this, the description of the rule is given thus: - - There are four Lines on each flat of this Rule; two next the outward - edges, which are Lines of Measure; and two next the inward edges, which - are Lines of Proportion. On one flat, next the inward edges, is the - Square-line [Girt-line in round timber measurement] with the Line of - Numbers his fellow. Next the outward, a Line of Inches divided into - Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 - to 24 on the other. On the other flat, next the inward edges, is the - double Scale of Numbers [for solving proportions]. Next the outward on - one Rule a Line of Inches divided each into ten parts; and this for - gauging, etc. On the other a foot divided into 100 parts. - -Later further changes were introduced in Coggeshall's rule.[36] - -It is worthy of note that Coggeshall's slide rule book, The Art of -Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p. -473; hence Leupold's description[37] of the rectilinear slide rule in his -Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not -the earliest reference to the rectilinear rule found in German -publications. The above date is earlier even than Biler's reference to a -circular slide rule in his Descriptio instrumenti mathematici universalis -of 1696. - -Two noted slide rules for gauging were described by Tho. Everard, -Philomath, in his Stereometry made easie, London, 1684. He designates his -lines by the capital letters A, B, C, D, E. On the first instrument, A on -the rule, and B and C on the slide, have each two radiuses of numbers, D -has only one, while E has three. The second rule is described in an -Appendix; it is one foot long, with two slides enabling the rule to be -extended to 3 feet. - -Everard's instruments were made in London by Isaac Carver who, soon after, -himself wrote a sixteen-page Description and Use of a New Sliding Rule, -projected from the Tables in the Gauger's Magazine, London, 1687, which was -"printed for William Hunt" and bound in one volume with a book by Hunt, -called The Gauger's Magazine, London, 1687. This appears to be the same -William Hunt who later brought out descriptions of his own of slide rules. -The instrument described by Carver "consists of three pieces, two whereof -are moveable to be drawn out till the whole be 36 inches long." It has -several non-logarithmic graduations, together with logarithmic lines marked -A, B, C, D, of which A, B, C are "double lines," and D a "single line" used -for squares and square roots. It is designed for the determination of the -vacuity of a "spheroidal cask lying," a "spheroidal cask standing," and a -"parabolical cask lying." - -Another seventeenth century writer on the slide rule is John Atkinson, whom -we have mentioned earlier. He says:[38] "The Lines of Numbers, Sines and -Tangents, are set double, that is, one on each side, as the middle piece -slides: which middle piece is so contrived, to slip to and fro easily, to -slide out, and to be put in any side uppermost, in order to bring those -Lines together (or against one another) most proper for solving the -Question, wrought by Sliding-Gunter." - -The data presented in this article show that, while the earliest slide -rules were of the circular type, the later slide rules of the seventeenth -century were of the rectilinear type.[39] - - January 12, 1915. - - - - - Footnotes - - -[1]F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments, - New York, 1909, pp. 7-14, also Addenda i-vi. - -[2]F. Cajori, "On the Invention of the Slide Rule," in Colorado College - Publication, Engineering Series Vol. 1, 1910. An abstract of this is - given in Nature (London), Vol. 82, 1909, p. 267. - -[3]F. Cajori, History etc., p. 14. - -[4]Art. "Slide Rule" in the Penny Cyclopaedia and in the English - Cyclopaedia [Arts and Sciences]. - -[5]Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817, - p. 423. - -[6]The full title of the book which Wingate published on this subject in - Paris is as follows: - - L'Vsage | de la | Reigle de | Proportion | en l'Arithmetique & | - Geometrie. | Par Edmond Vvingate, | Gentil-homme Anglois. | - - Ean s philomaths, es polymaths. - - In tenui, sed no tenuis vsusve, laborne. | - - A Paris, | Chez Melchior Mondiere, | demeurant en l'Isle du Palais, | - la | ru de Harlay aux deux Viperes. | M. DC. XXIV. | Auec Priuilege du - Roy. | - - Back of the title page is the announcement: - - Notez que la Reigle de Proportion en toutes faons se vend Paris chez - Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces, - demeurant en l'Isle du Palais sur le Quay qui regarde la Megisserie - l'Espic d'or. - -[7]The title-page of the edition of 1658 is as follows: - - The Use of the Rule of Proportion in Arithmetick & Geometrie. First - published at Paris in the French tongue, and dedicated to Monsieur, the - then king's onely Brother (now Duke of Orleance). By Edm. Wingate, an - English Gent. And now translated into English by the Author. Whereinto - is now also inserted the Construction of the same Rule, & a farther use - thereof . . . 2nd edition inlarged and amended. London, 1658. - -[8]Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; - Drawn up by himself by way of Diary. With Appendix of original Letters. - Publish'd by Charles Burman, Esq., London, 1717, p. 23. - -[9]Mathematical Tables, 1811, p. 36, and art. "Gunter's Line" in his Phil. - and Math. Dictionary, London, 1815. - -[10]To the English Gentrie, and all others studious of the Mathematicks, - which shall bee readers hereof. The just Apologie of Wil: Ovghtred, - against the slaunderous insimulations of Richard Delamain, in a Pamphlet - called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum - projectio circularis. We shall refer to this document as Epistle. It was - published without date in 32 unnumbered pages of fine print, and was - bound in with Oughtred's Circles of Proportion, in the editions of 1633 - and 1639. In the 1633 edition it is inserted at the end of the volume - just after the Addition vnto the Vse of the Instrument etc., and in that - of 1639 immediately after the preface. It was omitted from the Oxford - edition of 1660. The Epistle was also published separately. There is a - separate copy in the British Museum, London. Aubrey, in his Brief Lives, - edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, "He - writt a stitch't pamphlet about 163(?4) against . . . Delamaine." - -[11]Thomas Browne is mentioned by Stone in his Mathematical Instruments, - London 1723, p. 16. See also Cajori, History of the Slide Rule, New - York, 1909, p. 15. - -[12]The Description and Use of a Joynt-Rule: . . . also the use of Mr. - White's Rule for measuring of Board and Timber, round and square; With - the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and - Versed Sines. By J. Brown, Philom., London, 1661. - -[13]A Collection of Centers and Useful Proportions on the Line of Numbers, - by John Brown, 1662(?), 16 pages; Description and Use of the Triangular - Quadrant, by John Brown, London, 1671; Wingate's Rule of Proportion in - Arithmetick and Geometry: or Gunter's Line. Newly rectified by Mr. Brown - and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The - Description and Use of the Carpenter's-Rule: Together with the Use of - the Line of Numbers commonly call'd Gunter's-Line, by John Brown, - London, 1704. - -[14]William Leybourn, op. cit., pp. 129, 130, 132, 133. - -[15]James Atkinson's edition of Andrew Wakely's The Mariners Compass - Rectified, London, 1694 [Wakely's preface dated 1664, Atkinson's - preface, 1693]. Atkinson adds An Appendix containing Use of Instruments - most useful in Navigation. Our quotation is from this Appendix, p. 199. - -[16]R. Delamain, The Making, Description, and Use of a small portable - Instrument . . . called a Horizontall Quadrant, etc., London, 1631. - -[17]Oughtred's description of his circular slide rule of 1632 and his - rectilinear slide rule of 1633, as well as a drawing of the circular - slide rule, are reproduced in Cajori's History of the Slide Rule, - Addenda, pp. ii-vi. - -[18]The full title of the Grammelogia I is as follows: - - Grammelogia | or, | The Mathematicall Ring. | Shewing (any reasonable - Capacity that hath | not Arithmeticke) how to resolve and worke | all - ordinary operations of Arithmeticke. | And those which are most - difficult with greatest | facilitie: The extraction of Roots, the - valuation of | Leases, &c. The measuring of Plaines | and Solids. | With - the resolution of Plaine and Sphericall | Triangles. | And that onely by - an Ocular Inspection, | and a Circular Motion. | Naturae secreta tempus - aperit. | London printed by John Haviland, 1630. - -[19]Grammelogia III is the same as Grammelogia I, except for the addition - of an appendix, entitled: - - De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit - nullo (praescripto tempore) vitae | vsuram nobis ingeniique Deus. | - London, | - - . . . The next line or two of this title-page which probably contained - the date of publication, were cut off by the binder in trimming the - edges of this and several other pamphlets for binding into one volume. - -[20]Grammelogia IV has two title pages. The first is Mirifica Logarithmoru' - Projectio Circularis. There follows a diagram of a circular slide rule, - with the inscription within the innermost ring: Nil Finis, Motvs, - Circvlvs vllvs Habet. The second title page is as follows: - - Grammelogia | Or, the Mathematicall Ring. | Extracted from the - Logarythmes, and projected Circular: Now published in the | inlargement - thereof unto any magnitude fit for use: shewing any reason- | able - capacity that hath not Arithmeticke how to resolve and worke, | all - ordinary operations of Arithmeticke: | And those that are most difficult - with greatest facilitie, the extracti- | on of Rootes, the valuation of - Leases, &c. the measuring of Plaines and Solids, | with the resolution - of Plaine and Sphericall Triangles applied to the | Practicall parts of - Geometrie, Horologographie, Geographie | Fortification, Navigation, - Astronomie, &c. | And that onely by an ocular inspection, and a Circular - motion, Invented and first published, by R. Delamain, Teacher, and - Student of the Mathematicks. | Naturae secreta tempus aperit. | - - There is no date. There follows the diagram of a second circular slide - rule, with the inscription within the innermost ring: Typus proiectionis - Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630 - promisi. There are numerous drawings in the Grammelogia, all of which, - excepting the drawings of slide rules on the engraved title-pages of - Grammelogia IV and V, were printed upon separate pieces of paper and - then inserted by hand into the vacant spaces on the printed pages - reserved for them. Some drawings are missing, so that the Bodleian - Grammelogia IV differs in this respect slightly from the copy in the - British Museum and from the British Museum copy of Grammelogia V. - -[21]Epistle, p. (8). - -[22]Aubrey, op. cit., Vol. II., p. 111. - -[23]Rigaud, Correspondence of Scientific Men during the 17th Century, Vol. - I, Oxford, 1841, p. 11. - -[24]Dictionary of National Biography, Art. "Delamain, Richard." See also - Rev. Charles J. Robinson, Taylors' School, from A.D. 1562 to 1874, Vol. - I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b; - Sixth Report of the Royal Commission on Historical Manuscripts, Part I, - Report and Appendix, London, 1877. In this Appendix, p. 82, we read the - following: - - Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. - Petitioner's husband was servant to the King, and one of His Majesty's - engineers for the fortification of the kingdom, and his tutor in - mathematical arts; but upon the breaking out of the war he deserted the - Court, and was called by the State to several employments, in fortifying - the towns of Northampton, Newport, and Abingdon; and was also abroad - with the armies as Quartermaster-General of the Foot, and therein died. - Petitioner is left a disconsolate widow with ten children, the four - least of whom are now afflicted with sickness, and petitioner has - nothing left to support them. There are several considerable sums of - money due to the petitioner, as well from the King as the State. Prays - that she may have some relief amongst other widows. See L. J., VII. 6. - 657. - -[25]Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London, - 1820, p. 34. - -[26]The New Artificial Gauging Line or Rod: together with rules concerning - the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London, - 1633. The copy we have seen is in the Bodleian Library, Oxford. The book - is small sized and has 40 pages. - -[27]Oughtred, op. cit., p. 11. - -[28]S. J. Rigaud, Correspondence of Scientific Men of the 17th Century, - Oxford, Vol. I, 1841, p. 17. - -[29]Rigaud, loc. cit., p. 22. - -[30]Rigaud, loc. cit., pp. 30, 31. - -[31]Oughtred, An Addition vnto the Vse of the Instrument called the Circles - of Proportion, London, 1633, p. 63. - -[32]F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22, - Addenda, pp. vi-ix. - -[33]W. Leybourn, op. cit., 1673, Preface, and pp. 128-29. - -[34]Cajori op. cit., Addenda, p. ix. - -[35]William Leybourn, op. cit., 1673, p. 35. - -[36]See Cajori, op. cit., pp. 20, 28, Addenda, p. ix. - -[37]See F. Cajori, "A Note on the History of the Slide Rule," Bibliotheca - mathematica, 3 F., Vol. 10, pp. 161-163. - -[38]John Atkinson, op. cit., 1694, p. 204. - -[39]Probably the oldest slide rule now in existence is owned by St. John's - College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in - diameter. It was exhibited along with other instruments in May, 1919. - According to the Catalogue of a Loan Exhibition of Early Scientific - Instruments in Oxford, opened May 16, 1919, the instrument is inscribed - with the name of the maker ("Elias Allen fecit") and with the name of - the donor, Georgius Barkham. It is dated 1635, which is only three years - after the first publication of Oughtred's description of his circular - slide rule. It is stated in the Catalogue: "Unfortunately all the - movable parts but the base-plate and a couple of thumb-screws are - missing. The face of the instrument is engraved with Oughtred's - Horizontal Instrument. The back is engraved with eleven Circles of - Proportion as described in Arthur Haughton's book, a copy of which was - presented to St. John's College by George Barkham, to explain the use of - the instrument." As Arthur Haughton's Oxford edition of Oughtred's - Circles of Proportion did not appear until 1660, it would seem that the - instrument was probably not presented to the College before 1660. As far - as is known, the next oldest slide rule is of the year 1654, kept in the - South Kensington Museum, London, and is described in Nature of March 5, - 1914. It is a rectilinear rule, "of boxwood, well made, and bound - together with brass at the two ends. It is of the square type, a little - more than 2 ft. in length, and bears the logarithmic lines first - described by Edmund Gunter. Of these, the num, sin and tan lines are - arranged in pairs, identical and contiguous, one line in each pair being - on the fixed part, and the other on the slide." The instrument is - inscribed, "Made by Robert Bissaker for T. W., 1654." Nowhere else have - we seen reference to Robert Bissaker. His slide rule seems to antedate - the "Whites rule" mentioned above. [This foot-note was added on October - 15, 1919.] - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - - -Note.--The University of California Publications are offered in exchange -for the publication of learned societies and institutions, universities and -libraries. Complete lists of all the publications of the University will be -sent upon request. For sample copies, lists of publications or other -information, address the Manager of the University of California Press, -Berkeley, California, U. 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You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org - - -Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century - -Author: Florian Cajori - -Release Date: February 26, 2013 [EBook #42216] - -Language: English - -Character set encoding: UTF-8 - -*** START OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - - - - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - - - - - -</pre> <div class="box"> <p class="center"><span class="small">UNIVERSITY OF CALIFORNIA PUBLICATIONS</span> @@ -1439,377 +1402,7 @@ Volume XXII.</p> -<pre> - - - - - -End of the Project Gutenberg EBook of On the History of Gunter's Scale and -the Slide Rule during the Seventeent, by Florian Cajori - -*** END OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - -***** This file should be named 42216-h.htm or 42216-h.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/4/2/2/1/42216/ - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - -Updated editions will replace the previous one--the old editions -will be renamed. - -Creating the works from public domain print editions means that no -one owns a United States copyright in these works, so the Foundation -(and you!) can copy and distribute it in the United States without -permission and without paying copyright royalties. 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Thus, we do not necessarily -keep eBooks in compliance with any particular paper edition. - -Most people start at our Web site which has the main PG search facility: - - www.gutenberg.org - -This Web site includes information about Project Gutenberg-tm, -including how to make donations to the Project Gutenberg Literary -Archive Foundation, how to help produce our new eBooks, and how to -subscribe to our email newsletter to hear about new eBooks. - - -</pre> +<div>*** END OF THE PROJECT GUTENBERG EBOOK 42216 ***</div> </body> </html> diff --git a/42216.txt b/42216.txt deleted file mode 100644 index fbf097c..0000000 --- a/42216.txt +++ /dev/null @@ -1,2003 +0,0 @@ -The Project Gutenberg EBook of On the History of Gunter's Scale and the -Slide Rule during the Seventeenth Century, by Florian Cajori - -This eBook is for the use of anyone anywhere at no cost and with -almost no restrictions whatsoever. You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org - - -Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century - -Author: Florian Cajori - -Release Date: February 26, 2013 [EBook #42216] - -Language: English - -Character set encoding: ASCII - -*** START OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - - - - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - IN - MATHEMATICS - - Vol. 1, No. 9, pp. 187-209 February 17, 1920 - - - - - ON THE HISTORY OF GUNTER'S SCALE AND - THE SLIDE RULE DURING THE - SEVENTEENTH CENTURY - - - BY - FLORIAN CAJORI - - - UNIVERSITY OF CALIFORNIA PRESS - BERKELEY - - - - - TABLE OF CONTENTS - - - PAGE - I. Introduction 187 - II. Innovations in Gunter's Scale 188 - Changes introduced by Edmund Wingate 188 - Changes introduced by Milbourn 189 - Changes introduced by Thomas Brown and John Brown 190 - Changes introduced by William Leybourn 192 - III. Richard Delamain's "Grammelogia" 192 - Different editions or impressions 194 - Description of Delamain's instrument of 1630 195 - Delamain's later designs, and directions for using his - instruments 197 - IV. Controversy between Oughtred and Delamain on the invention of - the circular slide rule 199 - V. Independence and priority of invention 203 - VI. Oughtred's "Gauging Line," 1633 206 - VII. Other seventeenth century slide rules 207 - - - - - I. INTRODUCTION - - - In my history of the slide rule[1], and my article on its invention[2] it -is shewn that William Oughtred and not Edmund Wingate is the inventor, that -Oughtred's circular rule was described in print in 1632, his rectilinear -rule in 1633. Richard Delamain is referred to as having tried to -appropriate the invention to himself[3] and as having written a scurrilous -pamphlet against Oughtred. All our information about Delamain was taken -from De Morgan,[4] who, however, gives no evidence of having read any of -Delamain's writings on the slide rule. Through Dr. Arthur Hutchinson of -Pembroke College, Cambridge, I learned that Delamain's writings on the -slide rule were available. In this article will be given: First, some -details of the changes introduced during the seventeenth century in the -design of Gunter's scale by Edmund Wingate, Milbourn, Thomas Brown, John -Brown and William Leybourn; second, an account of Delamain's book of 1630 -on the slide rule which antedates Oughtred's first publication (though -Oughtred's date of invention is earlier than the date of Delamain's alleged -invention) and of Delamain's later designs of slide rules; third, an -account of the controversy between Delamain and Oughtred; fourth, an -account of a later book on the slide rule written by William Oughtred, and -of other seventeenth century books on the slide rule. - - - - - II. INNOVATIONS IN GUNTER'S SCALE - - - Changes introduced by Wingate - - We begin with Anthony Wood's account of Wingate's introduction of -Gunter's scale into France.[5] - - In 1624 he transported into France the rule of proportion, having a - little before been invented by Edm. Gunter of Gresham Coll. and - communicated it to most of the chiefest mathematicians then residing in - Paris: who apprehending the great benefit that might accrue thereby, - importun'd him to express the use thereof in the French tongue. Which - being performed accordingly, he was advised by monsieur Alleawne the - King's chief engineer to dedicate his book to monsieur the King's only - brother, since duke of Orleans. Nevertheless the said work coming forth - as an abortive (the publishing thereof being somewhat hastened, by reason - an advocate of Dijon in Burgundy began to print some uses thereof, which - Wingate had in a friendly way communicated to him) especially in regard - Gunter himself had learnedly explained its use in a far larger volume.[6] - -Gunter's scale, which Wingate calls the "rule of proportion," contained, as -described in the French edition of 1624, four lines: (1) A single line of -numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot -in length, divided into 12 inches and tenths of inches, also a line, one -foot in length, divided into tenths and hundredths. - -The English editions of this book which appeared in 1623 and 1628 are -devoid of interest. The editions of 1645 and 1658 contain an important -innovation.[7] In the preface the reasons why this instrument has not been -used more are stated to be: (1) the difficulty of drawing the lines with -exactness, (2) the trouble of working thereupon by reason (sometimes) of -too large an extent of the compasses, (3) the fact that the instrument is -not readily portable. The drawing of Wingate's arrangement of the scale in -the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains -five parallel lines, about 66 cm. long, each having the divisions of one -line marked on one side and of another line on the other side. Thus each -line carries two graduations: (1) A single logarithmic line of numbers; (2) -a logarithmic line of numbers thrice repeated; (3) the first scale -repeated, but beginning with the graduations which are near the middle of -the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3; -(4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of -tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000 -equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of -inches; (10) a scale consisting of three kinds, viz., a gauge line, a line -of chords, and a foot measure, divided into 1000 equal parts. - -Important are the first and second scales, by which cube root extraction -was possible "by inspection only, without the aid of pen or compass;" -similarly the third and fourth scales, for square roots. This innovation is -due to Wingate. The 1645 edition announces that the instrument was made in -brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in -Hosier Lane. - - - Changes introduced by Milbourn - -William Leybourn, in his The Line of Proportion or Numbers, Commonly called -Gunter's Line, Made Easie, London, 1673, says in his preface "To the -Reader:" - - The Line of Proportion or Numbers, commonly called (by Artificers) - Gunter's Line, hath been discoursed of by several persons, and variously - applied to divers uses; for when Mr. Gunter had brought it from the - Tables to a Line, and written some Uses thereof, Mr. Wingate added divers - Lines of several lengths, thereby to extract the Square or Cube Roots, - without doubling or trebling the distance of the Compasses: After him Mr. - Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral - Line, thereby enlarging the divisions of the Line. - -On pages 127 and 128 Leybourn adds: - - Again, One T. Browne, a Maker of Mathematical Instruments, made it in a - Serpentine or Spiral Line, composed of divers Concentrick Circles, - thereby to enlarg the divisions, which was the contrivance of one Mr. - Milburn a Yorkshire Gentleman, who writ thereof, and communicated his - Uses to the aforesaid Brown, who (since his death) attributed it to - himself: But whoever was the contriver of it, it is not without - inconvenience; for it can in no wise be made portable; and besides - (instead of compasses) an opening Joynt with thirds [threads] must be - placed to move upon the Centre of the Instrument, without which no - proportion can be wrought. - -This Mr. Milburn is probably the person named in the diary of the -antiquarian, Elias Ashmole, on August 13 [1646?]; "I bought of Mr. Milbourn -all his Books and Mathematical Instruments."[8] Charles Hutton[9] says that -Milburne of Yorkshire designed the spiral form about 1650. This date is -doubtless wrong, for Thomas Browne who, according to Leybourn, got the -spiral form of line from Milbourn, is repeatedly mentioned by William -Oughtred in his Epistle[10] printed some time in 1632 or 1633. Oughtred -does not mention Milbourn, and says (page 4) that the spiral form "was -first hit upon by one Thomas Browne a Joyner, . . . the serpentine -revolution being but two true semicircles described on severall -centers."[11] - - - Changes introduced by Thomas Brown and John Brown - -Thomas Brown did not publish any description of his instrument, but his -son, John Brown, published in 1661 a small book,[12] in which he says -(preface) that he had done "as Mr. Oughtred with Gunter's Rule, to a -sliding and circular form; and as my father Thomas Brown into a Serpentine -form; or as Mr. Windgate in his Rule of Proportion." He says also that -"this brief touch of the Serpentine-line I made bold to assert, to see if I -could draw out a performance of that promise, that hath been so long -unperformed by the promisers thereof." Accordingly in Chapter XX he gives a -description of the serpentine line, "contrived in five (or rather 15) -turn." Whether this description, printed in 1661, exactly fits the -instrument as it was developed in 1632, we have no means of knowing. John -Brown says: - - 1. First next the center is two circles divided one into 60, the other - into 100 parts, for the reducing of minutes to 100 parts, and the - contrary. - - 2. You have in seven turnes two inpricks, and five in divisions, the - first Radius of the sines (or Tangents being neer the matter, alike to - the first three degrees,) ending at 5 degrees and 44 minutes. - - 3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in - three margents in divisions, and the line of versed sines in pricks, - under the line of Tangents, according to Mr. Gunter's cross-staff: the - sines and Tangents beginning at 5 degrees, and 44 minutes where the other - ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the - line of numbers beginning at 10, and proceeding to 100, being one entire - Radius, and graduated into as many divisions as the largeness of the - instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to - 100 into 20 parts in one unit of increase, but the Tangents are divided - into single minutes from the beginning to the end, both in the first, - second and third Radiusses, and the sines into minutes; also from 30 - minutes to 40 degrees, and from 40 to 60, into every two minutes, and - from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the - rest as many as can be well discovered. - - The versed sines are set after the manner of Mr. Gunter's Cross-staff, - and divided into every 10th minutes beginning at 0, and proceeding to 156 - going backwards under the line of Tangents. - - 4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is - the secants to 51 degrees, being nothing else but the sines reitterated - beyond 90. - - 5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85 - degrees, whereby all trouble of backward working is avoided. - - 6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and - also a line of natural sines, for finding of differences in sines, for - finding hour and Azimuth. - - 7. Seventhly, next the verge or outermost edge is a line of equal parts - to get the Logarithm of any number, or the Logarithm sine and Tangent of - any ark or angle to four figures besides the carracteristick. - - 8. Eightly and lastly, in the space place between the ending of the - middle five turnes, and one half of the circle are three prickt lines - fitted for reduction. The uppermost being for shillings, pence and - farthings. The next for pounds, and ounces, and quarters of small - Averdupoies weight. The last for pounds, shillings and pence, and to be - used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay - the hair or edge of one of the legs of the index on 16. 3 1/2 in the line - of 1. s. d. and the hair shall cut on the equal parts 81 16; and the - contrary, if you have a decimal fraction, and would reduce it to a proper - fraction, the like may you do for shillings, and pence, and pounds, and - ounces. - - The uses of the lines follow. - - As to the use of these lines, I shall in this place say but little, and - that for two reasons. First, because this instrument is so contrived, - that the use is sooner learned then any other, I speak as to the manner, - and way of using it, because by means of first second and third - radiusses, in sines and Tangents, the work is always right on, one way or - other, according to the Canon whatsoever it be, in any book that treats - of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in - Oughtred from page 64 to 107. - - Secondly, and more especially, because the more accurate, and large - handling thereof is more then promised, if not already performed by more - abler pens, and a large manuscript thereof by my Sires meanes, provided - many years ago, though to this day not extant in print; so for his sake I - claiming my interest therein, make bold to present you with these few - lines, in order to the use of them: And first note, - - 1. Which soever of the two legs is set to the first term in the question, - that I call the first leg always, and the other being set to the second - term, I call the second leg . . . - -The exact nature of the contrivance with the "two legs" is not described, -but it was probably a flat pair of compasses, attached to the metallic -surface on which the serpentine line was drawn. In that case the instrument -was a slide rule, rather than a form of Gunter's line. In his publication -of 1661, as also in later publications,[13] John Brown devoted more space -to Gunter's scales, requiring the use of a separate pair of compasses, than -to slide rules. - - - Changes introduced by William Leybourn - -The same remark applies to William Leybourn who, after speaking of Seth -Partridge's slide rule, returns to forms of Gunter's scale, saying:[14] - - There is yet another way of disposing of this Line of Proportion, by - having one Line of the full length of the Ruler, and another Line of the - same Radius broken in two parts between 3 and 4; so that in working your - Compasses never go off of the Line: This is one of the best contrivances, - but here Compasses must be used. These are all the Contrivances that I - have hitherto seen of these Lines: That which I here speak of, and will - shew how to use, is only two Lines of one and the same Radius, being set - upon a plain Ruler of any length (the larger the better) having the - beginning of one Line, at the end of the other, the divisions of each - Line being set so close together, that if you find any number upon one of - the Lines, you may easily see what number stands against it on the other - Line. This is all the Variation. . . . - - Example 1. If a Board be 1 Foot 64 parts broad, how much in length of - that Board will make a Foot Square? Look upon one of your Lines (it - matters not which) for 1 Foot 64 parts, and right against it on the other - Line you shall find 61; and so many parts of a Foot will make a Foot - square of that Board. - -This contrivance solves the equation 1.64x=1, yielding centesimal parts of -a foot. - -James Atkinson[15] speaks of "Gunter's scale" as "usually of Boxwood . . . -commonly 2 ft. long, 1 1/2 inch broad" and "of two kinds: long Gunter or -single Gunter, and the sliding Gunter. It appears that during the -seventeenth century (and long after) the Gunter's scale was a rival of the -slide rule. - - - - - III. RICHARD DELAMAIN'S GRAMMELOGIA - - -We begin with a brief statement of the relations between Oughtred and -Delamain. At one time Delamain, a teacher of mathematics in London, was -assisted by Oughtred in his mathematical studies. In 1630 Delamain -published the Grammelogia, a pamphlet describing a circular slide rule and -its use. In 1631 he published another tract, on the Horizontall -Quadrant.[16] In 1632 appeared Oughtred's Circles of Proportion[17] -translated into English from Oughtred's Latin manuscript by another pupil, -William Forster, in the preface of which Forster makes the charge (without -naming Delamain) that "another . . . went about to pre-ocupate" the new -invention. This led to verbal disputes and to the publication by Delamain -of several additions to the Grammelogia, describing further designs of -circular slide rules and also stating his side of the bitter controversy, -but without giving the name of his antagonist. Oughtred's Epistle was -published as a reply. Each combatant accuses the other of stealing the -invention of the circular slide rule and the horizontal quadrant. - -The two title-pages of the edition of the Grammelogia in the British Museum - in London which we have called "Grammelogia IV." - - - Different editions or impressions - -There are at least five different editions, or impressions, of the -Grammelogia which we designate, for convenience, as follows: - - Grammelogia I, 1630. One copy in the Cambridge University Library.[18] - - Grammelogia II, I have not seen a copy of this. - - Grammelogia III, One copy in the Cambridge University Library.[19] - - Grammelogia IV, One copy in the British Museum, another in the Bodleian - Library, Oxford.[20] - - Grammelogia V, One copy in the British Museum. - -In Grammelogia I the first three leaves and the last leaf are without -pagination. The first leaf contains the title-page; the second leaf, the -dedication to the King and the preface "To the Reader;" the third leaf, the -description of the Mathematical Ring. Then follow 22 numbered pages. -Counting the unnumbered pages, there are altogether 30 pages in the -pamphlet. Only the first three leaves of this pamphlet are omitted in -Grammelogia IV and V. - -In Grammelogia III the Appendix begins with a page numbered 52 and bears -the heading "Conclusion;" it ends with page 68, which contains the same two -poems on the mathematical ring that are given on the last page of -Grammelogia I but differs slightly in the spelling of some of the words. -The 51 pages which must originally have preceded page 52, we have not seen. -The edition containing these we have designated Grammelogia II. The reason -for the omission of these 51 pages can only be conjectured. In Oughtred's -Epistle (p. 24), it is stated that Delamain had given a copy of the -Grammelogia to Thomas Brown, and that two days later Delamain asked for the -return of the copy, "because he had found some things to be altered -therein" and "rent out all the middle part." Delamain labored "to recall -all the bookes he had given forth, (which were many) before the sight of -Brownes Lines." These spiral lines Oughtred claimed that Delamain had -stolen from Brown. The title-page and page 52 are the only parts of the -Appendix, as given in Grammelogia III, that are missing in the Grammelogia -IV and V. - -Grammelogia IV answers fully to the description of Delamain's pamphlet -contained in Oughtred's Epistle. It was brought out in 1632 or 1633, for -what appears to be the latest part of it contains a reference (page 99) to -the Grammelogia I (1630) as "being now more then two yeares past." -Moreover, it refers to Oughtred's Circles of Proportion, 1632, and -Oughtred's reply in the Epistle was bound in the Circles of Proportion -having the Addition of 1633. For convenience of reference we number the two -title-pages of Grammelogia IV, "page (1)" and "page (2)," as is done by -Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page -numbers which we assign will be placed in parentheses, to distinguish them -from the page numbers which are printed in Grammelogia IV. The pages -(44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the -same as the pages 53-68. Thus only thirty-eight pages have page numbers -printed on them. The pages (67) and (83) are identical in wording, except -for some printer's errors; they contain verses in praise of the Ring, and -have near the bottom the word "Finis." Also, pages (22) and (23) are -together identical in wording with page (113), which is set up in finer -type, containing an advertisement of a part of Grammelogia IV explaining -the mode of graduating the circular rules. There are altogether six parts -of Grammelogia IV which begin or end by an address to the reader, thus: "To -the Reader," "Courteous Reader," or "To the courteous and benevolent Reader -. . .," namely the pages (8), (22), (68), (89), (90), (108). In his Epistle -(page 2), Oughtred characterizes the make up of the book in the following -terms: - - In reading it . . . I met with such a patchery and confusion of - disjoynted stuffe, that I was striken with a new wonder, that any man - should be so simple, as to shame himselfe to the world with such a - hotch-potch. - -Grammelogia V differs from Grammelogia IV in having only the second -title-page. The first title-page may have been torn off from the copy I -have seen. A second difference is that the page with the printed numeral 22 -in Grammelogia IV has after the word "Finis" the following notice: - - This instrument is made in Silver, or Brasse for the Pocket, or at any - other bignesse, over against Saint Clements Church without Temple Barre, - by Elias Allen. - -This notice occurs also on page 22 of Grammelogia I and III, but is omitted -from page 22 of Grammelogia V. - - - Description of Delamain's instrument of 1630 - -In his address to King Charles I, in his Grammelogia I, Delamain emphasizes -the ease of operating with his slide rule by stating that it is "fit for -use . . . as well on Horse backe as on Foot." Speaking "To the Reader," he -states that he has "for many yeares taught the Mathematicks in this Towne," -and made efforts to improve Gunter's scale "by some Motion, so that the -whole body of Logarithmes might move proportionally the one to the other, -as occasion required. This conceit in February last [1629] I struke upon, -and so composed my Grammelogia or Mathematicall Ring; by which only with an -ocular inspection, there is had at one instant all proportionalls through -the said body of Numbers." He dates his preface "first of January, 1630." -The fifth and sixth pages contain his "Description of the Grammelogia," the -term Grammelogia being applied to the instrument, as well as to the book. -His description is as follows: - - The parts of the Instrument are two Circles, the one moveable, and the - other fixed; The moveable is that unto which is fastened a small pin to - move it by; the other Circle may be conceived to be fixed; The - circumference of the moveable Circle is divided into unequall parts, - charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures - doe represent themselves, or such numbers unto which a Cipher or Ciphers - are added, and are varied as the occasion falls out in the speech of - Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or - 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c. - -After elaborating this last point and explaining the decimal subdivisions -on the scales of the movable circle, he says that "the numbers and -divisions on the fixed Circle, are the very same that the moveable are, . -." There is no drawing of the slide rule in this publication. The -twenty-two numbered pages give explanations of the various uses to which -the instrument can be put: "How to performe the Golden Rule" (pp. 1-3), -"Further uses of the Golden Rule" (pp. 4-6), "Notions or Principles -touching the disposing or ordering of the Numbers in the Golden Rule in -their true places upon the Grammelogia" (pp. 7-11), "How to divide one -number by another" (pp. 12, 13), "to multiply one Number by another" (pp. -14, 15), "To find Numbers in continuall proportion" (pp. 16, 17), "How to -extract the Square Root," "How to extract the Cubicke Root" (pp. 18-21), -"How to performe the Golden Rule" (the rule of proportion) is explained -thus: - - Seeke the first number in the moveable, and bring it to the second number - in the fixed, so right against the third number in the moveable, is the - answer in the fixed. - - If the Interest of 100. li. be 8. li. in the yeare, what is the Interest - of 65. li. for the same time. - - Bring 100. in the moveable to 8. in the fixed, so right against 65. in - the moveable is 5.2. in the fixed, and so much is the Interest of 65. li. - for the yeare at 8. li. for 100. li. per annum. - - The Instrument not removed, you may at one instant right against any - summe of money in the moveable, see the Interest thereof in the fixed: - the reason of this is from the Definition of Logarithmes. - -These are the earliest known printed instructions on the use of a slide -rule. It will be noticed that the description of the instrument at the -opening makes no references to logarithmic lines for the trigonometric -functions; only the line of numbers is given. Yet the title-page promised -the "resolution of Plaine and Sphericall Triangles." Page 22 throws light -upon this matter: - - If there be composed three Circles of equal thicknesse, A.B.C. so that - the inner edge of D [should be B] and the outward edge of A bee - answerably graduated with Logarithmall signes [sines], and the outward - edge of B and the inner edge of A with Logarithmes; and then on the - backside be graduated the Logarithmall Tangents, and againe the - Logarithmall signes oppositly to the former graduations, it shall be - fitted for the resolution of Plaine and Sphericall Triangles. - -After twelve lines of further remarks on this point he adds: - - Hence from the forme, I have called it a Ring, and Grammelogia by - annoligie of a Lineary speech; which Ring, if it were projected in the - convex unto two yards Diameter, or thereabouts, and the line Decupled, it - would worke Trigonometrie unto seconds, and give proportionall numbers - unto six places only by an ocular inspection, which would compendiate - Astronomicall calculations, and be sufficient for the Prosthaphaeresis of - the Motions: But of this as God shall give life and ability to health and - time. - -The unnumbered page following page 22 contains the patent and copyright on -the instrument and book: - - Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs - an Instrument called Grammelogia, or The Mathematicall Ring, together - with a Booke so intituled, expressing the use thereof, being his owne - Invention; we of our Gracious and Princely favour have granted unto the - said Richard Delamain and his Assignes, Privilege, Licence, and - Authority, for the sole Making, Printing and Selling of the said - Instrument and Booke: straightly forbidding any other to Make, Imprint, - or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument - or Booke within any our Dominions, during the space of ten yeares next - ensuing the date hereof, upon paine of Our high displeasure. Given under - our hand and Signet at our Palace of Westminster, the fourth day of - January, in the sixth yeare of our Raigne. - - - Delamain's later designs, and directions for using his instruments - -In the Appendix of Grammelogia III, on page 52 is given a description of an -instrument promised near the end of Grammelogia I: - - That which I have formerly delivered hath been onely upon one of the - Circles of my Ring, simply concerning Arithmeticall Proportions, I will - by way of Conclusion touch upon some uses of the Circles, of Logarithmall - Sines, and Tangents, which are placed on the edge of both the moveable - and fixed Circles of the Ring in respect of Geometricall Proportions, but - first of the description of these Circles. - - First, upon the side that the Circle of Numbers is one, are graduated on - the edge of the moveable, and also on the edge of the fixed the - Logarithmall Sines, for if you bring 1. in the moveable amongst the - Numbers to 1. in the fixed, you may on the other edge of the moveable and - fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto - 6.6. and each degree subdivided, and then over the former divisions and - figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5. - 4. 3. 2. 1. each of those divided by small points. - - Secondly, (if the Ring is great) neere the outward edge of this side of - the fixed against the Numbers, are the usuall divisions of a Circle, and - the points of the Compasse: serving for observation in Astronomy, or - Geometry, and the sights belonging to those divisions, may be placed on - the moveable Circle. - - Thirdly, opposite to those Sines on the other side are the Logarithmall - Tangents, noted alike both in the moveable and fixed thus - 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or - divisions serve also for their Complements to 90. so 40 gr. stands for - 50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both - in the moveable and fixed is also divided into parts. As for the degrees - which are under 6. viz. 5.4.3.2.1. they are noted with small figures over - this divided Circle from 45.40.35.30.25. &c. and each of those degrees - divided into parts by small points both in the moveable and fixed. - - Fourthly, on the other edge of the moveable on the same side is another - graduation of Tangents, like that formerly described. And opposite unto - it, in the fixed is a Graduation of Logarithmall sines in every thing - answerable to the first descrition of Sines on the other side. - - Fifthly, on the edge of the Ring is graduated a parte of the AEquator, - numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees - of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree - both of the AEquator, and Meridian are subdivided into parts; these two - graduated Circles serve to resolve such Questions which concerne - Latitude, Longitude, Rumb, and Distance, in Nauticall operations. - - Sixthly, to the concave of the Ring may be added a Circle to be elevated - or depressed for any Latitude, representing the AEquator, and so divided - into houres and parts with an Axis, to shew both the houre, and Azimuth, - and within this Circle may be hanged a Box, and Needle with a Socket for - a staffe to slide into it, and this accommodated with scrue pines to - fasten it to the Ring and staffe, or to take it off at pleasure. - -The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and -V make no reference to the dispute with Oughtred and may, therefore, be -assumed to have been published before the appearance of Oughtred's Circles -of Proportion. On page 53, "To the Reader," he says: - - . . . you may make use of the Projection of the Circles of the Ring upon - a Plaine, having the feet of a paire of compasses (but so that they be - flat) to move on the Center of that Plaine, and those feet to open and - shut as a paire of Compasses . . . now if the feet bee opened to any two - termes or numbers in that Projection, then may you move the first foot to - the third number, and the other foot shall give the Answer; . . . it hath - pleased some to make use of this way. But in this there is a double - labour in respect to that of the Ring, the one in fitting those feet unto - the numbers assigned, and the other by moving them about, in which a man - can hardly accommodate the Instrument with one hand, and expresse the - Proportionals in writing with the other. By the Ring you need not but - bring one number to another, and right against any other number is the - Answer without any such motion. . . . upon that [the Ring] I write, - shewing some uses of those Circles amongst themselves, and conjoyned with - others . . . in Astronomy, Horolographie, in plaine Triangles applyed to - Dimensions, Navigation, Fortification, etc. . . . But before I come to - Construction, I have thought it convenient by way introduction, to - examine the truth of the graduation of those Circles . . . - -These are the words of a practical man, interested in the mechanical -development of his instrument. He considers not only questions of -convenience but also of accuracy. The instrument has, or may have now, also -lines of sines and tangents. To test the accuracy of the circles of -Numbers, "bring any number in the moveable to halfe of that number in the -fixed: so any number or part in the fixed shall give his double in the -moveable, and so may you trie of the thirds, fourths &c. of numbers, vel -contra," (p. 54). On page 55 are given two small drawings, labelled, "A -Type of the Ringe and Scheme of this Logarithmicall projection, the use -followeth. These Instruments are made in Silver or Brasse by John Allen -neare the Sauoy in the Strand." - - - IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE - CIRCULAR SLIDE RULE - -Delamain's publication of 1630 on the 'Mathematicall Ring' does not appear -at that time to have caused a rupture between him and Oughtred. When in -1631 Delamain brought out his Horizontall Quadrant, the invention of which -Delamain was afterwards charged to have stolen from Oughtred, Delamain was -still in close touch with Oughtred and was sending Oughtred in the Arundell -House, London, the sheets as they were printed. Oughtred's reference to -this in his Epistle (p. 20) written after the friendship was broken, is as -follows: - - While he was printing his tractate of the Horizontall quadrant, although - he could not but know that it was injurious to me in respect of my free - gift to Master Allen, and of William Forster, whose translation of my - rules was then about to come forth: yet such was my good nature, and his - shamelessnesse, that every day, as any sheet was printed, hee sent, or - brought the same to mee at my chamber in Arundell house to peruse which I - lovingly and ingenuously did, and gave him my judgment of it. - -Even after Forster's publication of Oughtred's Circles of Proportion, 1632, -Oughtred had a book, A canon of Sines Tangents and Secants, which he had -borrowed from Delamain and was then returning (Epistle, page (5)). The -attacks which Forster, in the preface to the Circles of Proportion, made -upon Delamain (though not naming Delamain) started the quarrel. Except for -Forster and other pupils of Oughtred who urged him on to castigate -Delamain, the controversy might never have arisen. Forster expressed -himself in part as follows: - - . . . being in the time of the long vacation 1630, in the Country, at the - house of the Reverend, and my most worthy friend, and Teacher, Mr. - William Oughtred (to whose instruction I owe both my initiation, and - whole progresse in these Sciences.) I vpon occasion of speech told him of - a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made - (such as it vsually called Mr. Gunter's Ruler) 6 feet long, to be vsed - with a payre of beame-compasses. "He answered that was a poore invention, - and the performance very troublesome: But, said he, seeing you are taken - with such mechanicall wayes of Instruments, I will shew you what deuises - I have had by mee these many yeares." And first, hee brought to mee two - Rulers of that sort, to be vsed by applying one to the other, without any - compasses: and after that hee shewed mee those lines cast into a circle - or Ring, with another moueable circle vpon it. I seeing the great - expeditenesse of both those wayes; but especially, of the latter, wherein - it farre excelleth any other Instrument which hath bin knowne; told him, - I wondered that hee could so many yeares conceale such vseful inuentions, - not onely from the world, but from my selfe, to whom in other parts and - mysteries of Art, he had bin so liberall. He answered, "That the true way - of Art is not by Instruments, but by Demonstration: and that it is a - preposterous course of vulgar Teachers, to begin with Instruments, and - not with the Sciences, and so in-stead of Artists, to make their - Schollers only doers of tricks, and as it were Iuglers: to the despite of - Art, losse of precious time, and betraying of willing and industrious - wits, vnto ignorance and idlenesse. That the vse of Instruments is indeed - excellent, if a man be an Artist: but contemptible, being set and opposed - to Art. And lastly, that he meant to commend to me, the skill of - Instruments, but first he would haue me well instructed in the Sciences. - He also shewed me many notes, and Rules for the vse of those circles, and - of his Horizontall Instrument, (which he had proiected about 30 yeares - before) the most part written in Latine. All which I obtained of him - leaue to translate into English, and make publique, for the vse, and - benefit of such as were studious, and louers of these excellent Sciences. - - Which thing while I with mature, and diligent care (as my occasions would - give me leaue) went about to doe: another to whom the Author in a louing - confidence discouered this intent, using more hast then good speed, went - about to preocupate; of which vntimely birth, and preuenting (if not - circumuenting) forwardnesse, I say no more: but aduise the studious - Reader, onely so farre to trust, as he shal be sure doth agree to truth & - Art. - -While in this dedication reference is made to a slide rule or "ring" with a -"moveable circle," the instrument actually described in the Circles of -Proportion consists of fixed circles "with an index to be opened after the -manner of a paire of Compasses." Delamain, as we have seen, had decided -preference for the moveable circle. To Oughtred, on the other hand, one -design was about as good as the other; he was more of a theorist and -repeatedly expressed his contempt for mathematical instruments. In his -Epistle (page (25)), he says he had not "the one halfe of my intentions -upon it" (the rule in his book), nor one with a "moveable circle and a -thread, but with an opening Index at the centre (if so be that bee cause -enough to make it to bee not the same, but another Instrument) for my part -I disclaime it: it may go seeke another Master: which for ought I know, -will prove to be Elias Allen himselfe: for at his request only I altered a -little my rules from the use of the moveable circle and the thread, to the -two armes of an Index." - -All parts of Delamain's Grammelogia IV, except pages 1-22 and 53-68 -considered above, were published after the Circles of Proportion, for they -contain references to the ill treatment that Delamain felt or made believe -that he felt, that he had received in the book published by Oughtred and -Forster. Oughtred's reference to teachers whose scholars are "doers of -tricks," "Iuglers," and Forster's allusion to "another to whom the Author -in a loving confidence" explained the instrument and who "went about to -preocupate" it, are repeatedly mentioned. Delamain says, (page (89)) that -at first he did not intend to express himself in print, "but sought peace -and my right by a private and friendly way." Oughtred's account of -Delamain's course is that of an "ill-natured man" with a "virulent tongue," -"sardonical laughter" and "malapert sawsiness." Contrasting Forster and -Delamain, he says that, of the former he "had the very first moulding" and -made him feel that "the way of Art" is "by demonstration." But Delamain was -"already corrupted with doing upon Instruments, and quite lost from ever -being made an Artist." (Epistle page (27)). Repeatedly does Oughtred assert -Delamain's ignorance of mathematics. The two men were evidently of wholly -different intellectual predilections. That Delamain loved instruments is -quite evident, and we proceed to describe his efforts to improve the -circular slide rule. - -The Grammelogia IV is dedicated to King Charles I. Delamain says: - - . . . Everything hath his beginning, and curious Arts seldome come to the - height at the first; It was my promise then to enlarge the invention by a - way of decuplating the Circles, which I now present unto your sacred - Majestie as the quintessence and excellencie there of . . . - -His enlarged circular rules are illustrated in the Bodleian Library copy of -Grammelogia IV by four diagrams, two of them being the two drawings on the -two title-pages at the beginning of the Grammelogia IV, 4 inches in -external diameter, and exhibiting eleven concentric circular lines carrying -graduations of different sorts. In the second of these designs all circles -are fixed. The other two drawings are each 10 3/4 inches in external -diameter and exhibit 18 concentric circular lines; the folded sheet of the -first of these drawings is inserted between pages (23) and (24), the second -folded sheet between pages (83) and (84). All circles of this second -instrument are fixed. Counting in the two small drawings in Grammelogia -III, there are in all six drawings of slide rules in the Bodleian -Grammelogia IV. On pages (24) to (43) Delamain explains the graduation of -slide rules. He takes first a rule which has one circle of equal parts, -divided into 1000 equal divisions. From a table of logarithms he gets log 2 -= 0.301; from the number 301 in the circle of equal parts he draws a line -to the center of the circle and marks the intersection with the circles of -numbers by the figure 2. Thus he proceeds with log 3, log 4, and so on; -also with log sin x and log tan x. For log sin x he uses two circles, the -first (see page (27)) for angles from 34' 24" to 5^o 44' 22", the second -circle from 5^o 44' 22" to 90^o. The drawings do not show the seconds. He -suggests many different designs of rules. On page (29) he says: - - For the single projection of the Circles of my Ring, and the dividing and - graduating of them: which may bee so inserted upon the edges of Circles - of mettle turned in the forme of a Ring, so that one Circle may moove - betweene two fixed, by helpe of two stayes, then may there be graduated - on the face of the Ring, upon the outer edge of the mooveable and inner - edge of the fixed, the Circle of Numbers, then upon the inner edge of - that mooveable Circle, and the outward edge of that inner fixed Circle - may be inserted the Circle of Sines, and so according to the description - of those that are usually made. - -In addition to these lines he proceeds to mention the circle giving the -ordinary division into degrees and minutes, and two circles of tangents on -the other side of the rule. - -Next Delamain explains an arrangement of all the graduation on one side of -the rule by means of "a small channell in the innermost fixed Circle, in -which may be placed a small single Index, which may have sufficient length -to reach from the innermost edge of the Mooveable Circle, unto the outmost -edge of the fixed Circle, which may be mooved to and fro at pleasure, in -the channell, which Index may serve to shew the opposition of Numbers" (p. -(31)). From this it is clear that the invention of the "runner" goes back -to the very first writers on the slide rule. - -After describing a modification of the above arrangement, he adds, "many -other formes might be deliverd, about this single projection" (p. (32)). - -Proceeding to the "enlarging" of the circles in the Ring, to, say, the -"Quadruple to that which is single, that is, foure times greater," the -"equall parts" are distributed over four circles instead of only one -circle, but the general method of graduation is the same as before (p. -(33)); there being now four circles carrying the logarithms of numbers, and -so on. Next he points out "severall wayes how the Circles of the -Mathematicall Ring (being inlarged) may be accommodated for practicall -use:" (1) The Circles are all fixed in a plain and movable flat compasses -(or better, a movable semicircle) are used for fixing any two positions; -(2) There is a "double projection" of each logarithmic line "inlarged on a -Plaine," one fixed, the other movable, as shown in his first figure on the -title-page, a single index only being used; (3) use of "my great Cylinder -which I have long proposed (in which all the Circles are of equall -greatnesse,) and it may be made of any magnitude or capacity, but for a -study (hee that will be at the charge) it may be of a yard diameter and of -such an indifferent length that it may containe 100 or more Circles fixed -parallel one to the other on the Cylinder, having a space betweene each of -them, so that there may bee as many mooveable Circles, as there are fixed -ones, and these of the mooveable linked, or fastened together, so that they -may all moove together by the fixed ones in these spaces, whose edges both -of the fixed, and mooveable being graduated by helpe of a single Index will -shew the proportionalls by opposition in this double Projection, or by a -double Index in a single Projection" (p. (36)). - -Next follows the detailed description of his Ring "on a Plaine, according -to the diagramme that was given the King (for a view of that projection) -and afterwards the Ring it selve." The diagram is the large one which we -mentioned as inserted between pages (23) and (24). The instrument has two -circles, one moveable, upon each of which are described 13 distinct -circular graduations. The lines on the fixed circle are: "The Circle of -degrees and calendar," E. "Circle of equall parts, and part of the Equator, -and Meridian," TT. "The Circle of Tangents," S. "The Circle of Sines," D. -"The Circle of Decimals," N. "The Circle of Numbers." The lines on the -movable circle are: N. "The Circle of Numbers," E. "The Circle of equated -figures, and bodies," S. "The Circle of Sines," TT. "The Circle of -Tangents," Y. "The Circle of time, yeares, and monethes." - -On pages (84)-(88) Delamain explains an enlargement of his Ring for -computations involving the sines of angles near to 90^o. On page (86) he -says: - - I have continued the Sines of the Projection unto two severall - revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr. - (being the last revolution of the decuplation of the former, or the - hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s. - and ends at 90.gr. (being the last of a ternary of decuplated - revolutions, or the thousand part of that Projection) and may bee thus - used. - -He explains the manner of using these extra graduations. Thus he claims to -have attained degrees of accuracy which enabled him to do what "some one" -had declared "could not bee done." It is hardly necessary to point out that -Delamain's Grammelogia IV suggests designs of slide rules which inventors -two hundred or more years later were endeavouring to produce. Which of -Delamain's designs of rules were actually made and used, he does not state -explicitly. He refers to a rule 18 inches in diameter as if it had been -actually constructed (pages (86), (88)). Oughtred showed no appreciation of -such study in designing and ridiculed Delamain's efforts, in his Epistle. - -Additional elucidations of his designs of rules, along with explanations of -the relations of his work to that of Gunter and Napier, and sallies -directed against Oughtred and Forster, are contained on pages (8)-(21) of -his Grammelogia IV. - - - - - V. INDEPENDENCE AND PRIORITY OF INVENTION - - -The question of independence and priority of invention is discussed by -Delamain more specifically on pages (89)-(113); Oughtred devotes his entire -Epistle to it. It is difficult to determine definitely which publication is -the later, Delamain's Grammelogia IV or Oughtred's Epistle. Each seems to -quote from the other. Probably the explanation is that the two publications -contain arguments which were previously passed from one antagonist to the -other by word of mouth or by private letter. Oughtred refers in his Epistle -(p. (12)) to a letter from Delamain. We believe that the Epistle came after -Delamain's Grammelogia IV. Delamain claims for himself the invention of the -circular slide rule. He says in his Grammelogia IV. (p. (99)), "when I had -a sight of it, which was in February, 1629 (as I specified in my Epistle) I -could not conceale it longer, envying my selfe, that others did not tast of -that which I found to carry with it so delightfull and pleasant a goate -[taste] . . ." Delamain asserts (without proof) that Oughtred "never saw it -as he now challengeth it to be his invention, untill it was so fitted to -his hand, and that he made all his practise on it after the publishing of -my Booke upon my Ring, and not before; so it was easie for him or some -other to write some uses of it in Latin after Christmas, 1630 and not the -Sommer before, as is falsely alledged by some one . . ." (p. (91)). -Delamain's accusation of theft on the part of Oughtred cannot be seriously -considered. Oughtred's reputation as a mathematician and his standing in -his community go against such a supposition. Moreover, William Forster is a -witness for Oughtred. The fact that Oughtred had the mastery of the -rectilinear slide rule as well, while Delamain in 1630 speaks only of the -circular rule, weighs in Oughtred's favour. - -Oughtred says he invented the slide rule "above twelve yeares agoe," that -is, about 1621, and "I with mine owne hand made me two such Circles, which -I have used ever since, as my occasions required," (Epistle p. (22)). On -the same page, he describes his mode of discovery thus: - - I found that it required many times too great a paire of Compasses [in - using Gunter's line], which would bee hard to open, apt to slip, and - troublesome for use. I therefore first devised to have another Ruler with - the former: and so by setting and applying one to the other, I did not - onely take away the use of Compasses, but also make the worke much more - easy and expedite: when I should not at all need the motion of my hand, - but onely the glancing at my sight: and with one position of the Rulers, - and view of mine eye, see not one onely, but the manifold proportions - incident unto the question intended. But yet this facility also wanted - not some difficulty especially in the line of tangents, when one arch was - in the former mediety of the quadrant, and the other in the latter: for - in this case it was needful that either one Ruler must bee as long againe - as the other; or else that I must use an inversion of the Ruler, and - regression. By this consideration I first of all saw that if those lines - upon both Rulers were inflected into two circles, that of the tangents - being in both doubled, and that those two Circles should move one upon - another; they with a small thread in the center to direct the sight, - would bee sufficient with incredible and wonderfull facility to worke all - questions of Trigonometry . . . - -Oughtred said that he had no desire to publish his invention, but in the -vacation of 1630 finally promised William Forster to let him bring out a -translation. Oughtred claims that Delamain got the invention from him at -Alhallontide [November 1], 1630, when they met in London. The accounts of -that meeting we proceed to give in double column. - - Delamain's Statement - Grammelogia IV, page (98) - - ". . . about Alhalontide 1630. (as our Authors reporteth) was the time he - was circumvented, and then his intent in a loving manner (as before) he - opened unto me, which particularly I will dismantle in the very naked - truth: for, wee being walking together some few weekes before Christmas, - upon Fishstreet hill, we discoursed upon sundry things Mathematicall, - both Theoreticall and Practicall, and of the excellent inventions and - helpes that in these dayes were produced, amongst which I was not a - little taken with that of the Logarythmes, commending greatly the - ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, - in the lines of proportion, extracted from these Logarythmes for ordinary - Practicall uses; He replyed unto me (in these very words) What will yov - say to an Invention that I have, which in a lesse extent of the Compasses - shall worke truer then that of Mr. Gunters Ruler, I asked him then of - what forme it was, he answered with some pause (which no doubt argued his - suspition of mee that I might conceive it) that it was Arching-wise, but - now hee sayes that hee told mee then, it was Circular (but were I put to - my oath to avoid the guilt of Conscience I would conclude in the former.) - At which immediately I answered, I had the like my selfe, and so we - discoursed not a word more touching that subject . . . Then after my - coming home I sent him a sight of my Projection drawne in Pastboard: Now - admit I had not the Invention of my Ring before I discoursed . . . it was - not so facil for mee . . . to raise and compose so complete, and absolute - an Instrument from so small a principle, or glimpse of light . . ." - - Oughtred's Statement - Epistle, page (23) - - "Shortly after my gift to Elias Allen, I chanced to meet with Richard - Delamain in the street (it was at Alhallontide) and as we walked together - I told him what an Instrument I had given to Master Allen, both of the - Logarithmes projected into circles, which being lesse then one foot - diameter would performe as much as one of Master Gunters Rulers of sixe - feet long: and also of the Prostaphaereses of the Plannets and second - motions. Such an invention have I said he: for now his intentions (that - is his ambition) beganne to worke: . . . But he saith, Then after my - comming home I sent him a sight of my projection drawne in past-board. - See how notoriously he jugleth without an Instrument. Then after: how - long after? a sight of my projection: of how much? More then seven weekes - after on December 23, he sent to mee the line of numbers onely set upon a - circle: . . . and so much onely he presented to his Majesty: but as for - Sine or tangent of his, there was not the least shew of any. Neither - could he give to Master Allen any direction for the composure of the - circles of his Ring, or for the division of them: as upon his oath Master - Allen will testify how hee misled him, and made him labour in vain above - three weeks together, until Master Allen himselfe found out his ignorance - and mistaking, which is more cleare then is possible with any impudence - to be outfaced." - -Oughtred makes a further statement (Epistle, p. (24)) as follows: - - Delamain hearing that Brown with his Serpentine had another line by which - he could worke to minutes in the 90 degree of sines . . . gave the [his] - booke to Browne: who in thankfulnesse could not but gratify Delamain with - his Lines also: and teach him the use of them, but especially of the - great Line: with this caution on both sides, that one should not meddle - with the others invention. Two dayes after Delamain . . . because he had - found some things to be altered therin, . . . asked for the booke . . . - but as soone as he had got it in his hands he rent out all the middle - part with the two Schemes & put them up in his pocket & went his way . . - . and . . . laboureth to recall all the bookes he had given forth . . . - And shortly after this he got a new Printer (who was ignorant of his - former Schemes) to print him new: giving him an especiall charge of the - outermost line newly graven in the Plate, which indeed is Brownes very - line: and then altering his book . . . - -This and other statements made by Oughtred seem damaging to Delamain's -reputation. But it is quite possible that Oughtred's guesses as to -Delamain's motives are wrong. Moreover, some of Oughtred's statements are -not first hand knowledge with him, but mere hearsay. One may accept his -first hand facts and still clear Delamain of wrong doing. There is always -danger that rival claimants of an invention or discovery will proceed on -the assumption that no one else could possibly have come independently upon -the same devices that they themselves did; the history of science proves -the opposite. Seldom is an invention of any note made by only one man. We -do not feel competent to judge Delamain's case. We know too little about -him as a man. We incline to the opinion that the hypothesis of independent -invention is the most plausible. At any rate, Delamain figures in the -history of the slide rule as the publisher of the earliest book thereon and -as an enthusiastic and skillful designer of slide rules. - -The effect of this controversy upon interested friends was probably small. -Doubtless few people read both sides. Oughtred says:[21] "this scandall . . -. hath with them, to whom I am not knowne, wrought me much prejudice and -disadvantage . ." Aubrey,[22] a friend of Oughtred, refers to Delamain "who -was so sawcy to write against him" and remembers having seen "many yeares -since, twenty or more good verses made" against Delamain. Another friend of -Oughtred, William Robinson, who had seen some of Delamain's publications, -but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before -the appearance of the latter's Epistle: - - I cannot but wonder at the indiscretion of Rich. Delamain, who being - conscious to himself that he is but the pickpurse of another man's wit, - would thus inconsiderately provoke and awake a sleeping lion . . . he - hath so weakly (though in my judgment, vaingloriously enough) commended - his own labour . . .[23] - -Delamain presented King Charles I with one of his sun-dials, also with a -manuscript and, later, with a printed copy of his book of 1630. A drawing -of his improved slide rule was sent to the King and the Grammelogia IV is -dedicated to him. The King must have been favorably impressed, for Delamain -was appointed tutor to the King in mathematics. His widow petitioned the -House of Lords in 1645 for relief; he had ten children.[24] - -Anthony Wood states that Charles I, on the day of his execution, commanded -his friend Thomas Herbert "to give his son the duke of York his large -ring-sundial of silver, a jewel his maj. much valued." Anthony Wood adds, -"it was invented and made by Rich. Delamaine a very able mathematician, who -projected it, and in a little printed book did shew its excellent use in -resolving many questions in arithmetic and other rare operations to be -wrought by it in the mathematics."[25] - - - - - VI. OUGHTRED'S GAUGING LINE, 1633 - - -It has not been generally known, hitherto, that Oughtred designed a -rectilinear slide rule for gauging and published a description thereof in -1633.[26] In his Circles of Proportion, chapter IX, Oughtred had offered a -closer approximation than that of Gunter for the capacity of casks. The -Gauger of London expostulated with Oughtred for presuming to question -anything that Gunter had written. The ensuing discussion led to an -invitation extended by the Company of Vintners to the instrument maker -Elias Allen to request Oughtred to design a gauging rod.[27] This he did, -and Allen received an order for "threescore" instruments. On page 19 -Oughtred describes his 'Gauging Rod:' - - It consisteth of two rulers of brasse about 32 ynches of length, which - also are halfe an ynch broad, and a quarter of an ynch thick . . . At one - end of both those rulers are two little sockets of brasse fastened on - strongly: by which the rulers are held together, and made to move one - upon another, and to bee drawne out unto any length, as occasion shall - require: and when you have them at the just length, there is upon one of - the sockets a long Scrue-pin to scrue them fast. - -There are graduations on three sides of the rulers, one graduation being -the logarithmic line of numbers. He says (p. 39), "the maner of computing -the Gauge-divisions I have concealed." W. Robinson, who was a friend of -Oughtred, wrote him as follows:[28] - - I have light upon your little book of artificial gauging, wherewith I am - much taken, but I want the rod, neither could I get a sight of one of - them at the time, because Mr. Allen had none left . . . I forgot to ask - Mr. Allen the price of one of them, which if not much I would have one of - them." Oughtred annotated this passage thus: "Or in wood, if any be made - in wood by Thompson or any other." - -Another of Oughtred's admirers, Sir Charles Cavendish, wrote, on February -11, 1635 thus:[29] - - I thank you for your little book, but especially for the way of - calculating the divisions of your gauging rod. I wish, both for their own - sakes and yours, that the citizens were as capable of the acuteness of - this invention, as they are commonly greedy of gain, and then I doubt not - but they would give you a better recompense than I doubt now they will. - -On April 20, 1638, we find Oughtred giving Elias Allen directions[30] -"about the making of the two rulers." As in 1633,[31] so now, Oughtred -takes one ruler longer than the other. This 1633 instrument was used also -as "a crosse-staffe to take the height of the Sunne, or any Starre above -the Horizon, and also their distances." The longer ruler was called staffe, -the shorter transversarie. While in 1633 he took the lengths of the two in -the ratio "almost 3 to 2," in 1638, he took "the transversary three -quarters of the staff's length, . . . that the divisions may be larger." - - - - - VII. OTHER SEVENTEENTH CENTURY SLIDE RULES - - -In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard, -Henry Coggeshall, W. Hunt and Sir Isaac Newton.[32] Of Partridge's Double -Scale of Proportion, London, I have examined a copy dated 1661, which is -the earliest date for this book that I have seen. As far as we know, 1661 -is the earliest date of publications on the slide rule, since Oughtred and -Delamain. But it would not be surprising if the intervening 28 years were -found not so barren as they seem at present. The 1661 and 1662 impressions -of Partridge are identical, except for the date on the title-page. William -Leybourn, who printed Partridge's book, speaks in high appreciation of it -in his own book.[33] - -In 1661 was published also John Brown's first book, Description and Use of -a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use -of "Mr. Whites rule" for the measuring of board and timber, round and -square. He calls this a "sliding rule." The existence, in 1661, of a -"Whites rule" indicates activities in designing of which we know as yet -very little. In his book of 1761, previously quoted, Brown gives a drawing -of "White's sliding rule" (p. 193); also a special contrivance of his own, -as indicated by him in these words: - - A further improvement of the Triangular Quadrant, as I have made it - several times, with a sliding Cover on the in-side, when made hollow, to - carry Ink, Pens, and Compasses; then on the sliding Cover, and Edges, is - put the Line of Numbers, according to Mr. White's first Contrivance for - manner of operation; but much augmented, and made easie, by John Brown. - -He gives no drawing of his "triangular quadrant," hence his account of it -is unsatisfactory. He explains the use of "gage-points." His placing -logarithmic lines on the edges of instrument boxes was outdone in oddity -later by Everard who placed them on tobacco-boxes.[34] In Brown's -publication of 1704 the White slide rule is given again, "being as neat and -ready a way as ever was used." He tells also of a "glasier's sliding rule." -William Leybourn explains in 1673 how Wingate's double and triple lines for -squaring and cubing, or square and cube root, can be used on slide -rules.[35] - -Beginning early in the history of the slide rule, when Oughtred designed -his "gauging rod," we notice the designing of rules intended for very -special purposes. Another such contrivance, which enjoyed long popularity, -was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677, -a booklet of 35 pages. Coggeshall says in his preface: - - For what can be more ready and easie, then having set twelve to the - length, to see the Content exactly against the Girt or Side of the - Square. Whereas on Mr. Partridge's Scale the Content is the Sixth Number, - which is far more troublesome then [even] with Compasses. - -One line on Coggeshall's rule begins with 4 and extends to 40, these -numbers being the "Girt" (a quarter of the circumference), which in -ordinary practice of measuring round timber lies between 4 inches and 40 -inches. This "Girt line" slides "against the line of Numbers in two -Lengths, to which it is exactly equal." A second edition, 1682, shows some -changes in the rule, as well as an enlargement and change of title of the -book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent, -London, 1682. In this, the description of the rule is given thus: - - There are four Lines on each flat of this Rule; two next the outward - edges, which are Lines of Measure; and two next the inward edges, which - are Lines of Proportion. On one flat, next the inward edges, is the - Square-line [Girt-line in round timber measurement] with the Line of - Numbers his fellow. Next the outward, a Line of Inches divided into - Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 - to 24 on the other. On the other flat, next the inward edges, is the - double Scale of Numbers [for solving proportions]. Next the outward on - one Rule a Line of Inches divided each into ten parts; and this for - gauging, etc. On the other a foot divided into 100 parts. - -Later further changes were introduced in Coggeshall's rule.[36] - -It is worthy of note that Coggeshall's slide rule book, The Art of -Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p. -473; hence Leupold's description[37] of the rectilinear slide rule in his -Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not -the earliest reference to the rectilinear rule found in German -publications. The above date is earlier even than Biler's reference to a -circular slide rule in his Descriptio instrumenti mathematici universalis -of 1696. - -Two noted slide rules for gauging were described by Tho. Everard, -Philomath, in his Stereometry made easie, London, 1684. He designates his -lines by the capital letters A, B, C, D, E. On the first instrument, A on -the rule, and B and C on the slide, have each two radiuses of numbers, D -has only one, while E has three. The second rule is described in an -Appendix; it is one foot long, with two slides enabling the rule to be -extended to 3 feet. - -Everard's instruments were made in London by Isaac Carver who, soon after, -himself wrote a sixteen-page Description and Use of a New Sliding Rule, -projected from the Tables in the Gauger's Magazine, London, 1687, which was -"printed for William Hunt" and bound in one volume with a book by Hunt, -called The Gauger's Magazine, London, 1687. This appears to be the same -William Hunt who later brought out descriptions of his own of slide rules. -The instrument described by Carver "consists of three pieces, two whereof -are moveable to be drawn out till the whole be 36 inches long." It has -several non-logarithmic graduations, together with logarithmic lines marked -A, B, C, D, of which A, B, C are "double lines," and D a "single line" used -for squares and square roots. It is designed for the determination of the -vacuity of a "spheroidal cask lying," a "spheroidal cask standing," and a -"parabolical cask lying." - -Another seventeenth century writer on the slide rule is John Atkinson, whom -we have mentioned earlier. He says:[38] "The Lines of Numbers, Sines and -Tangents, are set double, that is, one on each side, as the middle piece -slides: which middle piece is so contrived, to slip to and fro easily, to -slide out, and to be put in any side uppermost, in order to bring those -Lines together (or against one another) most proper for solving the -Question, wrought by Sliding-Gunter." - -The data presented in this article show that, while the earliest slide -rules were of the circular type, the later slide rules of the seventeenth -century were of the rectilinear type.[39] - - January 12, 1915. - - - - - Footnotes - - -[1]F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments, - New York, 1909, pp. 7-14, also Addenda i-vi. - -[2]F. Cajori, "On the Invention of the Slide Rule," in Colorado College - Publication, Engineering Series Vol. 1, 1910. An abstract of this is - given in Nature (London), Vol. 82, 1909, p. 267. - -[3]F. Cajori, History etc., p. 14. - -[4]Art. "Slide Rule" in the Penny Cyclopaedia and in the English - Cyclopaedia [Arts and Sciences]. - -[5]Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817, - p. 423. - -[6]The full title of the book which Wingate published on this subject in - Paris is as follows: - - L'Vsage | de la | Reigle de | Proportion | en l'Arithmetique & | - Geometrie. | Par Edmond Vvingate, | Gentil-homme Anglois. | - - Ean es philomathes, ese polymathes. - - In tenui, sed no tenuis vsusve, laborne. | - - A Paris, | Chez Melchior Mondiere, | demeurant en l'Isle du Palais, | a - la | rue de Harlay aux deux Viperes. | M. DC. XXIV. | Auec Priuilege du - Roy. | - - Back of the title page is the announcement: - - Notez que la Reigle de Proportion en toutes facons se vend a Paris chez - Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces, - demeurant en l'Isle du Palais sur le Quay qui regarde la Megisserie a - l'Espic d'or. - -[7]The title-page of the edition of 1658 is as follows: - - The Use of the Rule of Proportion in Arithmetick & Geometrie. First - published at Paris in the French tongue, and dedicated to Monsieur, the - then king's onely Brother (now Duke of Orleance). By Edm. Wingate, an - English Gent. And now translated into English by the Author. Whereinto - is now also inserted the Construction of the same Rule, & a farther use - thereof . . . 2nd edition inlarged and amended. London, 1658. - -[8]Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; - Drawn up by himself by way of Diary. With Appendix of original Letters. - Publish'd by Charles Burman, Esq., London, 1717, p. 23. - -[9]Mathematical Tables, 1811, p. 36, and art. "Gunter's Line" in his Phil. - and Math. Dictionary, London, 1815. - -[10]To the English Gentrie, and all others studious of the Mathematicks, - which shall bee readers hereof. The just Apologie of Wil: Ovghtred, - against the slaunderous insimulations of Richard Delamain, in a Pamphlet - called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum - projectio circularis. We shall refer to this document as Epistle. It was - published without date in 32 unnumbered pages of fine print, and was - bound in with Oughtred's Circles of Proportion, in the editions of 1633 - and 1639. In the 1633 edition it is inserted at the end of the volume - just after the Addition vnto the Vse of the Instrument etc., and in that - of 1639 immediately after the preface. It was omitted from the Oxford - edition of 1660. The Epistle was also published separately. There is a - separate copy in the British Museum, London. Aubrey, in his Brief Lives, - edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, "He - writt a stitch't pamphlet about 163(?4) against . . . Delamaine." - -[11]Thomas Browne is mentioned by Stone in his Mathematical Instruments, - London 1723, p. 16. See also Cajori, History of the Slide Rule, New - York, 1909, p. 15. - -[12]The Description and Use of a Joynt-Rule: . . . also the use of Mr. - White's Rule for measuring of Board and Timber, round and square; With - the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and - Versed Sines. By J. Brown, Philom., London, 1661. - -[13]A Collection of Centers and Useful Proportions on the Line of Numbers, - by John Brown, 1662(?), 16 pages; Description and Use of the Triangular - Quadrant, by John Brown, London, 1671; Wingate's Rule of Proportion in - Arithmetick and Geometry: or Gunter's Line. Newly rectified by Mr. Brown - and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The - Description and Use of the Carpenter's-Rule: Together with the Use of - the Line of Numbers commonly call'd Gunter's-Line, by John Brown, - London, 1704. - -[14]William Leybourn, op. cit., pp. 129, 130, 132, 133. - -[15]James Atkinson's edition of Andrew Wakely's The Mariners Compass - Rectified, London, 1694 [Wakely's preface dated 1664, Atkinson's - preface, 1693]. Atkinson adds An Appendix containing Use of Instruments - most useful in Navigation. Our quotation is from this Appendix, p. 199. - -[16]R. Delamain, The Making, Description, and Use of a small portable - Instrument . . . called a Horizontall Quadrant, etc., London, 1631. - -[17]Oughtred's description of his circular slide rule of 1632 and his - rectilinear slide rule of 1633, as well as a drawing of the circular - slide rule, are reproduced in Cajori's History of the Slide Rule, - Addenda, pp. ii-vi. - -[18]The full title of the Grammelogia I is as follows: - - Grammelogia | or, | The Mathematicall Ring. | Shewing (any reasonable - Capacity that hath | not Arithmeticke) how to resolve and worke | all - ordinary operations of Arithmeticke. | And those which are most - difficult with greatest | facilitie: The extraction of Roots, the - valuation of | Leases, &c. The measuring of Plaines | and Solids. | With - the resolution of Plaine and Sphericall | Triangles. | And that onely by - an Ocular Inspection, | and a Circular Motion. | Naturae secreta tempus - aperit. | London printed by John Haviland, 1630. - -[19]Grammelogia III is the same as Grammelogia I, except for the addition - of an appendix, entitled: - - De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit - nullo (praescripto tempore) vitae | vsuram nobis ingeniique Deus. | - London, | - - . . . The next line or two of this title-page which probably contained - the date of publication, were cut off by the binder in trimming the - edges of this and several other pamphlets for binding into one volume. - -[20]Grammelogia IV has two title pages. The first is Mirifica Logarithmoru' - Projectio Circularis. There follows a diagram of a circular slide rule, - with the inscription within the innermost ring: Nil Finis, Motvs, - Circvlvs vllvs Habet. The second title page is as follows: - - Grammelogia | Or, the Mathematicall Ring. | Extracted from the - Logarythmes, and projected Circular: Now published in the | inlargement - thereof unto any magnitude fit for use: shewing any reason- | able - capacity that hath not Arithmeticke how to resolve and worke, | all - ordinary operations of Arithmeticke: | And those that are most difficult - with greatest facilitie, the extracti- | on of Rootes, the valuation of - Leases, &c. the measuring of Plaines and Solids, | with the resolution - of Plaine and Sphericall Triangles applied to the | Practicall parts of - Geometrie, Horologographie, Geographie | Fortification, Navigation, - Astronomie, &c. | And that onely by an ocular inspection, and a Circular - motion, Invented and first published, by R. Delamain, Teacher, and - Student of the Mathematicks. | Naturae secreta tempus aperit. | - - There is no date. There follows the diagram of a second circular slide - rule, with the inscription within the innermost ring: Typus proiectionis - Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630 - promisi. There are numerous drawings in the Grammelogia, all of which, - excepting the drawings of slide rules on the engraved title-pages of - Grammelogia IV and V, were printed upon separate pieces of paper and - then inserted by hand into the vacant spaces on the printed pages - reserved for them. Some drawings are missing, so that the Bodleian - Grammelogia IV differs in this respect slightly from the copy in the - British Museum and from the British Museum copy of Grammelogia V. - -[21]Epistle, p. (8). - -[22]Aubrey, op. cit., Vol. II., p. 111. - -[23]Rigaud, Correspondence of Scientific Men during the 17th Century, Vol. - I, Oxford, 1841, p. 11. - -[24]Dictionary of National Biography, Art. "Delamain, Richard." See also - Rev. Charles J. Robinson, Taylors' School, from A.D. 1562 to 1874, Vol. - I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b; - Sixth Report of the Royal Commission on Historical Manuscripts, Part I, - Report and Appendix, London, 1877. In this Appendix, p. 82, we read the - following: - - Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. - Petitioner's husband was servant to the King, and one of His Majesty's - engineers for the fortification of the kingdom, and his tutor in - mathematical arts; but upon the breaking out of the war he deserted the - Court, and was called by the State to several employments, in fortifying - the towns of Northampton, Newport, and Abingdon; and was also abroad - with the armies as Quartermaster-General of the Foot, and therein died. - Petitioner is left a disconsolate widow with ten children, the four - least of whom are now afflicted with sickness, and petitioner has - nothing left to support them. There are several considerable sums of - money due to the petitioner, as well from the King as the State. Prays - that she may have some relief amongst other widows. See L. J., VII. 6. - 657. - -[25]Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London, - 1820, p. 34. - -[26]The New Artificial Gauging Line or Rod: together with rules concerning - the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London, - 1633. The copy we have seen is in the Bodleian Library, Oxford. The book - is small sized and has 40 pages. - -[27]Oughtred, op. cit., p. 11. - -[28]S. J. Rigaud, Correspondence of Scientific Men of the 17th Century, - Oxford, Vol. I, 1841, p. 17. - -[29]Rigaud, loc. cit., p. 22. - -[30]Rigaud, loc. cit., pp. 30, 31. - -[31]Oughtred, An Addition vnto the Vse of the Instrument called the Circles - of Proportion, London, 1633, p. 63. - -[32]F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22, - Addenda, pp. vi-ix. - -[33]W. Leybourn, op. cit., 1673, Preface, and pp. 128-29. - -[34]Cajori op. cit., Addenda, p. ix. - -[35]William Leybourn, op. cit., 1673, p. 35. - -[36]See Cajori, op. cit., pp. 20, 28, Addenda, p. ix. - -[37]See F. Cajori, "A Note on the History of the Slide Rule," Bibliotheca - mathematica, 3 F., Vol. 10, pp. 161-163. - -[38]John Atkinson, op. cit., 1694, p. 204. - -[39]Probably the oldest slide rule now in existence is owned by St. John's - College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in - diameter. It was exhibited along with other instruments in May, 1919. - According to the Catalogue of a Loan Exhibition of Early Scientific - Instruments in Oxford, opened May 16, 1919, the instrument is inscribed - with the name of the maker ("Elias Allen fecit") and with the name of - the donor, Georgius Barkham. It is dated 1635, which is only three years - after the first publication of Oughtred's description of his circular - slide rule. It is stated in the Catalogue: "Unfortunately all the - movable parts but the base-plate and a couple of thumb-screws are - missing. The face of the instrument is engraved with Oughtred's - Horizontal Instrument. The back is engraved with eleven Circles of - Proportion as described in Arthur Haughton's book, a copy of which was - presented to St. John's College by George Barkham, to explain the use of - the instrument." As Arthur Haughton's Oxford edition of Oughtred's - Circles of Proportion did not appear until 1660, it would seem that the - instrument was probably not presented to the College before 1660. As far - as is known, the next oldest slide rule is of the year 1654, kept in the - South Kensington Museum, London, and is described in Nature of March 5, - 1914. It is a rectilinear rule, "of boxwood, well made, and bound - together with brass at the two ends. It is of the square type, a little - more than 2 ft. in length, and bears the logarithmic lines first - described by Edmund Gunter. Of these, the num, sin and tan lines are - arranged in pairs, identical and contiguous, one line in each pair being - on the fixed part, and the other on the slide." The instrument is - inscribed, "Made by Robert Bissaker for T. W., 1654." Nowhere else have - we seen reference to Robert Bissaker. His slide rule seems to antedate - the "Whites rule" mentioned above. [This foot-note was added on October - 15, 1919.] - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - - -Note.--The University of California Publications are offered in exchange -for the publication of learned societies and institutions, universities and -libraries. Complete lists of all the publications of the University will be -sent upon request. For sample copies, lists of publications or other -information, address the Manager of the University of California Press, -Berkeley, California, U. 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Current, Volume XXII. - - - - - Transcriber's Notes - - ---Inserted subchapter headings from the Table of Contents into the - text. - ---Corrected two page numbers in the Table of Contents - ---Corrected a Greek quotation in footnote 6 by reference to the - original sources. - - - - - - - -End of the Project Gutenberg EBook of On the History of Gunter's Scale and -the Slide Rule during the Seventeenth Century, by Florian Cajori - -*** END OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - -***** This file should be named 42216.txt or 42216.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/4/2/2/1/42216/ - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - -Updated editions will replace the previous one--the old editions -will be renamed. - -Creating the works from public domain print editions means that no -one owns a United States copyright in these works, so the Foundation -(and you!) can copy and distribute it in the United States without -permission and without paying copyright royalties. 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Thus, we do not necessarily -keep eBooks in compliance with any particular paper edition. - -Most people start at our Web site which has the main PG search facility: - - www.gutenberg.org - -This Web site includes information about Project Gutenberg-tm, -including how to make donations to the Project Gutenberg Literary -Archive Foundation, how to help produce our new eBooks, and how to -subscribe to our email newsletter to hear about new eBooks. diff --git a/42216.zip b/42216.zip Binary files differdeleted file mode 100644 index 974e2e4..0000000 --- a/42216.zip +++ /dev/null diff --git a/old/42216-0.txt b/old/42216-0.txt deleted file mode 100644 index 91d9c5c..0000000 --- a/old/42216-0.txt +++ /dev/null @@ -1,2002 +0,0 @@ -The Project Gutenberg EBook of On the History of Gunter's Scale and the -Slide Rule during the Seventeenth Centur, by Florian Cajori - -This eBook is for the use of anyone anywhere at no cost and with -almost no restrictions whatsoever. You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org - - -Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century - -Author: Florian Cajori - -Release Date: February 26, 2013 [EBook #42216] - -Language: English - -Character set encoding: UTF-8 - -*** START OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - - - - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - IN - MATHEMATICS - - Vol. 1, No. 9, pp. 187-209 February 17, 1920 - - - - - ON THE HISTORY OF GUNTER’S SCALE AND - THE SLIDE RULE DURING THE - SEVENTEENTH CENTURY - - - BY - FLORIAN CAJORI - - - UNIVERSITY OF CALIFORNIA PRESS - BERKELEY - - - - - TABLE OF CONTENTS - - - PAGE - I. Introduction 187 - II. Innovations in Gunter’s Scale 188 - Changes introduced by Edmund Wingate 188 - Changes introduced by Milbourn 189 - Changes introduced by Thomas Brown and John Brown 190 - Changes introduced by William Leybourn 192 - III. Richard Delamain’s “Grammelogia” 192 - Different editions or impressions 194 - Description of Delamain’s instrument of 1630 195 - Delamain’s later designs, and directions for using his - instruments 197 - IV. Controversy between Oughtred and Delamain on the invention of - the circular slide rule 199 - V. Independence and priority of invention 203 - VI. Oughtred’s “Gauging Line,” 1633 206 - VII. Other seventeenth century slide rules 207 - - - - - I. INTRODUCTION - - - In my history of the slide rule[1], and my article on its invention[2] it -is shewn that William Oughtred and not Edmund Wingate is the inventor, that -Oughtred’s circular rule was described in print in 1632, his rectilinear -rule in 1633. Richard Delamain is referred to as having tried to -appropriate the invention to himself[3] and as having written a scurrilous -pamphlet against Oughtred. All our information about Delamain was taken -from De Morgan,[4] who, however, gives no evidence of having read any of -Delamain’s writings on the slide rule. Through Dr. Arthur Hutchinson of -Pembroke College, Cambridge, I learned that Delamain’s writings on the -slide rule were available. In this article will be given: First, some -details of the changes introduced during the seventeenth century in the -design of Gunter’s scale by Edmund Wingate, Milbourn, Thomas Brown, John -Brown and William Leybourn; second, an account of Delamain’s book of 1630 -on the slide rule which antedates Oughtred’s first publication (though -Oughtred’s date of invention is earlier than the date of Delamain’s alleged -invention) and of Delamain’s later designs of slide rules; third, an -account of the controversy between Delamain and Oughtred; fourth, an -account of a later book on the slide rule written by William Oughtred, and -of other seventeenth century books on the slide rule. - - - - - II. INNOVATIONS IN GUNTER’S SCALE - - - Changes introduced by Wingate - - We begin with Anthony Wood’s account of Wingate’s introduction of -Gunter’s scale into France.[5] - - In 1624 he transported into France the rule of proportion, having a - little before been invented by Edm. Gunter of Gresham Coll. and - communicated it to most of the chiefest mathematicians then residing in - Paris: who apprehending the great benefit that might accrue thereby, - importun’d him to express the use thereof in the French tongue. Which - being performed accordingly, he was advised by monsieur Alleawne the - King’s chief engineer to dedicate his book to monsieur the King’s only - brother, since duke of Orleans. Nevertheless the said work coming forth - as an abortive (the publishing thereof being somewhat hastened, by reason - an advocate of Dijon in Burgundy began to print some uses thereof, which - Wingate had in a friendly way communicated to him) especially in regard - Gunter himself had learnedly explained its use in a far larger volume.[6] - -Gunter’s scale, which Wingate calls the “rule of proportion,” contained, as -described in the French edition of 1624, four lines: (1) A single line of -numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot -in length, divided into 12 inches and tenths of inches, also a line, one -foot in length, divided into tenths and hundredths. - -The English editions of this book which appeared in 1623 and 1628 are -devoid of interest. The editions of 1645 and 1658 contain an important -innovation.[7] In the preface the reasons why this instrument has not been -used more are stated to be: (1) the difficulty of drawing the lines with -exactness, (2) the trouble of working thereupon by reason (sometimes) of -too large an extent of the compasses, (3) the fact that the instrument is -not readily portable. The drawing of Wingate’s arrangement of the scale in -the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains -five parallel lines, about 66 cm. long, each having the divisions of one -line marked on one side and of another line on the other side. Thus each -line carries two graduations: (1) A single logarithmic line of numbers; (2) -a logarithmic line of numbers thrice repeated; (3) the first scale -repeated, but beginning with the graduations which are near the middle of -the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3; -(4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of -tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000 -equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of -inches; (10) a scale consisting of three kinds, viz., a gauge line, a line -of chords, and a foot measure, divided into 1000 equal parts. - -Important are the first and second scales, by which cube root extraction -was possible “by inspection only, without the aid of pen or compass;” -similarly the third and fourth scales, for square roots. This innovation is -due to Wingate. The 1645 edition announces that the instrument was made in -brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in -Hosier Lane. - - - Changes introduced by Milbourn - -William Leybourn, in his The Line of Proportion or Numbers, Commonly called -Gunter’s Line, Made Easie, London, 1673, says in his preface “To the -Reader:” - - The Line of Proportion or Numbers, commonly called (by Artificers) - Gunter’s Line, hath been discoursed of by several persons, and variously - applied to divers uses; for when Mr. Gunter had brought it from the - Tables to a Line, and written some Uses thereof, Mr. Wingate added divers - Lines of several lengths, thereby to extract the Square or Cube Roots, - without doubling or trebling the distance of the Compasses: After him Mr. - Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral - Line, thereby enlarging the divisions of the Line. - -On pages 127 and 128 Leybourn adds: - - Again, One T. Browne, a Maker of Mathematical Instruments, made it in a - Serpentine or Spiral Line, composed of divers Concentrick Circles, - thereby to enlarg the divisions, which was the contrivance of one Mr. - Milburn a Yorkshire Gentleman, who writ thereof, and communicated his - Uses to the aforesaid Brown, who (since his death) attributed it to - himself: But whoever was the contriver of it, it is not without - inconvenience; for it can in no wise be made portable; and besides - (instead of compasses) an opening Joynt with thirds [threads] must be - placed to move upon the Centre of the Instrument, without which no - proportion can be wrought. - -This Mr. Milburn is probably the person named in the diary of the -antiquarian, Elias Ashmole, on August 13 [1646?]; “I bought of Mr. Milbourn -all his Books and Mathematical Instruments.”[8] Charles Hutton[9] says that -Milburne of Yorkshire designed the spiral form about 1650. This date is -doubtless wrong, for Thomas Browne who, according to Leybourn, got the -spiral form of line from Milbourn, is repeatedly mentioned by William -Oughtred in his Epistle[10] printed some time in 1632 or 1633. Oughtred -does not mention Milbourn, and says (page 4) that the spiral form “was -first hit upon by one Thomas Browne a Joyner, . . . the serpentine -revolution being but two true semicircles described on severall -centers.”[11] - - - Changes introduced by Thomas Brown and John Brown - -Thomas Brown did not publish any description of his instrument, but his -son, John Brown, published in 1661 a small book,[12] in which he says -(preface) that he had done “as Mr. Oughtred with Gunter’s Rule, to a -sliding and circular form; and as my father Thomas Brown into a Serpentine -form; or as Mr. Windgate in his Rule of Proportion.” He says also that -“this brief touch of the Serpentine-line I made bold to assert, to see if I -could draw out a performance of that promise, that hath been so long -unperformed by the promisers thereof.” Accordingly in Chapter XX he gives a -description of the serpentine line, “contrived in five (or rather 15) -turn.” Whether this description, printed in 1661, exactly fits the -instrument as it was developed in 1632, we have no means of knowing. John -Brown says: - - 1. First next the center is two circles divided one into 60, the other - into 100 parts, for the reducing of minutes to 100 parts, and the - contrary. - - 2. You have in seven turnes two inpricks, and five in divisions, the - first Radius of the sines (or Tangents being neer the matter, alike to - the first three degrees,) ending at 5 degrees and 44 minutes. - - 3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in - three margents in divisions, and the line of versed sines in pricks, - under the line of Tangents, according to Mr. Gunter’s cross-staff: the - sines and Tangents beginning at 5 degrees, and 44 minutes where the other - ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the - line of numbers beginning at 10, and proceeding to 100, being one entire - Radius, and graduated into as many divisions as the largeness of the - instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to - 100 into 20 parts in one unit of increase, but the Tangents are divided - into single minutes from the beginning to the end, both in the first, - second and third Radiusses, and the sines into minutes; also from 30 - minutes to 40 degrees, and from 40 to 60, into every two minutes, and - from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the - rest as many as can be well discovered. - - The versed sines are set after the manner of Mr. Gunter’s Cross-staff, - and divided into every 10th minutes beginning at 0, and proceeding to 156 - going backwards under the line of Tangents. - - 4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is - the secants to 51 degrees, being nothing else but the sines reitterated - beyond 90. - - 5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85 - degrees, whereby all trouble of backward working is avoided. - - 6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and - also a line of natural sines, for finding of differences in sines, for - finding hour and Azimuth. - - 7. Seventhly, next the verge or outermost edge is a line of equal parts - to get the Logarithm of any number, or the Logarithm sine and Tangent of - any ark or angle to four figures besides the carracteristick. - - 8. Eightly and lastly, in the space place between the ending of the - middle five turnes, and one half of the circle are three prickt lines - fitted for reduction. The uppermost being for shillings, pence and - farthings. The next for pounds, and ounces, and quarters of small - Averdupoies weight. The last for pounds, shillings and pence, and to be - used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay - the hair or edge of one of the legs of the index on 16. 3½ in the line of - 1. s. d. and the hair shall cut on the equal parts 81 16; and the - contrary, if you have a decimal fraction, and would reduce it to a proper - fraction, the like may you do for shillings, and pence, and pounds, and - ounces. - - The uses of the lines follow. - - As to the use of these lines, I shall in this place say but little, and - that for two reasons. First, because this instrument is so contrived, - that the use is sooner learned then any other, I speak as to the manner, - and way of using it, because by means of first second and third - radiusses, in sines and Tangents, the work is always right on, one way or - other, according to the Canon whatsoever it be, in any book that treats - of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in - Oughtred from page 64 to 107. - - Secondly, and more especially, because the more accurate, and large - handling thereof is more then promised, if not already performed by more - abler pens, and a large manuscript thereof by my Sires meanes, provided - many years ago, though to this day not extant in print; so for his sake I - claiming my interest therein, make bold to present you with these few - lines, in order to the use of them: And first note, - - 1. Which soever of the two legs is set to the first term in the question, - that I call the first leg always, and the other being set to the second - term, I call the second leg . . . - -The exact nature of the contrivance with the “two legs” is not described, -but it was probably a flat pair of compasses, attached to the metallic -surface on which the serpentine line was drawn. In that case the instrument -was a slide rule, rather than a form of Gunter’s line. In his publication -of 1661, as also in later publications,[13] John Brown devoted more space -to Gunter’s scales, requiring the use of a separate pair of compasses, than -to slide rules. - - - Changes introduced by William Leybourn - -The same remark applies to William Leybourn who, after speaking of Seth -Partridge’s slide rule, returns to forms of Gunter’s scale, saying:[14] - - There is yet another way of disposing of this Line of Proportion, by - having one Line of the full length of the Ruler, and another Line of the - same Radius broken in two parts between 3 and 4; so that in working your - Compasses never go off of the Line: This is one of the best contrivances, - but here Compasses must be used. These are all the Contrivances that I - have hitherto seen of these Lines: That which I here speak of, and will - shew how to use, is only two Lines of one and the same Radius, being set - upon a plain Ruler of any length (the larger the better) having the - beginning of one Line, at the end of the other, the divisions of each - Line being set so close together, that if you find any number upon one of - the Lines, you may easily see what number stands against it on the other - Line. This is all the Variation. . . . - - Example 1. If a Board be 1 Foot 64 parts broad, how much in length of - that Board will make a Foot Square? Look upon one of your Lines (it - matters not which) for 1 Foot 64 parts, and right against it on the other - Line you shall find 61; and so many parts of a Foot will make a Foot - square of that Board. - -This contrivance solves the equation 1.64x=1, yielding centesimal parts of -a foot. - -James Atkinson[15] speaks of “Gunter’s scale” as “usually of Boxwood . . . -commonly 2 ft. long, 1½ inch broad” and “of two kinds: long Gunter or -single Gunter, and the sliding Gunter. It appears that during the -seventeenth century (and long after) the Gunter’s scale was a rival of the -slide rule. - - - - - III. RICHARD DELAMAIN’S GRAMMELOGIA - - -We begin with a brief statement of the relations between Oughtred and -Delamain. At one time Delamain, a teacher of mathematics in London, was -assisted by Oughtred in his mathematical studies. In 1630 Delamain -published the Grammelogia, a pamphlet describing a circular slide rule and -its use. In 1631 he published another tract, on the Horizontall -Quadrant.[16] In 1632 appeared Oughtred’s Circles of Proportion[17] -translated into English from Oughtred’s Latin manuscript by another pupil, -William Forster, in the preface of which Forster makes the charge (without -naming Delamain) that “another . . . went about to pre-ocupate” the new -invention. This led to verbal disputes and to the publication by Delamain -of several additions to the Grammelogia, describing further designs of -circular slide rules and also stating his side of the bitter controversy, -but without giving the name of his antagonist. Oughtred’s Epistle was -published as a reply. Each combatant accuses the other of stealing the -invention of the circular slide rule and the horizontal quadrant. - -The two title-pages of the edition of the Grammelogia in the British Museum - in London which we have called “Grammelogia IV.” - - - Different editions or impressions - -There are at least five different editions, or impressions, of the -Grammelogia which we designate, for convenience, as follows: - - Grammelogia I, 1630. One copy in the Cambridge University Library.[18] - - Grammelogia II, I have not seen a copy of this. - - Grammelogia III, One copy in the Cambridge University Library.[19] - - Grammelogia IV, One copy in the British Museum, another in the Bodleian - Library, Oxford.[20] - - Grammelogia V, One copy in the British Museum. - -In Grammelogia I the first three leaves and the last leaf are without -pagination. The first leaf contains the title-page; the second leaf, the -dedication to the King and the preface “To the Reader;” the third leaf, the -description of the Mathematical Ring. Then follow 22 numbered pages. -Counting the unnumbered pages, there are altogether 30 pages in the -pamphlet. Only the first three leaves of this pamphlet are omitted in -Grammelogia IV and V. - -In Grammelogia III the Appendix begins with a page numbered 52 and bears -the heading “Conclusion;” it ends with page 68, which contains the same two -poems on the mathematical ring that are given on the last page of -Grammelogia I but differs slightly in the spelling of some of the words. -The 51 pages which must originally have preceded page 52, we have not seen. -The edition containing these we have designated Grammelogia II. The reason -for the omission of these 51 pages can only be conjectured. In Oughtred’s -Epistle (p. 24), it is stated that Delamain had given a copy of the -Grammelogia to Thomas Brown, and that two days later Delamain asked for the -return of the copy, “because he had found some things to be altered -therein” and “rent out all the middle part.” Delamain labored “to recall -all the bookes he had given forth, (which were many) before the sight of -Brownes Lines.” These spiral lines Oughtred claimed that Delamain had -stolen from Brown. The title-page and page 52 are the only parts of the -Appendix, as given in Grammelogia III, that are missing in the Grammelogia -IV and V. - -Grammelogia IV answers fully to the description of Delamain’s pamphlet -contained in Oughtred’s Epistle. It was brought out in 1632 or 1633, for -what appears to be the latest part of it contains a reference (page 99) to -the Grammelogia I (1630) as “being now more then two yeares past.” -Moreover, it refers to Oughtred’s Circles of Proportion, 1632, and -Oughtred’s reply in the Epistle was bound in the Circles of Proportion -having the Addition of 1633. For convenience of reference we number the two -title-pages of Grammelogia IV, “page (1)” and “page (2),” as is done by -Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page -numbers which we assign will be placed in parentheses, to distinguish them -from the page numbers which are printed in Grammelogia IV. The pages -(44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the -same as the pages 53-68. Thus only thirty-eight pages have page numbers -printed on them. The pages (67) and (83) are identical in wording, except -for some printer’s errors; they contain verses in praise of the Ring, and -have near the bottom the word “Finis.” Also, pages (22) and (23) are -together identical in wording with page (113), which is set up in finer -type, containing an advertisement of a part of Grammelogia IV explaining -the mode of graduating the circular rules. There are altogether six parts -of Grammelogia IV which begin or end by an address to the reader, thus: “To -the Reader,” “Courteous Reader,” or “To the courteous and benevolent Reader -. . .,” namely the pages (8), (22), (68), (89), (90), (108). In his Epistle -(page 2), Oughtred characterizes the make up of the book in the following -terms: - - In reading it . . . I met with such a patchery and confusion of - disjoynted stuffe, that I was striken with a new wonder, that any man - should be so simple, as to shame himselfe to the world with such a - hotch-potch. - -Grammelogia V differs from Grammelogia IV in having only the second -title-page. The first title-page may have been torn off from the copy I -have seen. A second difference is that the page with the printed numeral 22 -in Grammelogia IV has after the word “Finis” the following notice: - - This instrument is made in Silver, or Brasse for the Pocket, or at any - other bignesse, over against Saint Clements Church without Temple Barre, - by Elias Allen. - -This notice occurs also on page 22 of Grammelogia I and III, but is omitted -from page 22 of Grammelogia V. - - - Description of Delamain’s instrument of 1630 - -In his address to King Charles I, in his Grammelogia I, Delamain emphasizes -the ease of operating with his slide rule by stating that it is “fit for -use . . . as well on Horse backe as on Foot.” Speaking “To the Reader,” he -states that he has “for many yeares taught the Mathematicks in this Towne,” -and made efforts to improve Gunter’s scale “by some Motion, so that the -whole body of Logarithmes might move proportionally the one to the other, -as occasion required. This conceit in February last [1629] I struke upon, -and so composed my Grammelogia or Mathematicall Ring; by which only with an -ocular inspection, there is had at one instant all proportionalls through -the said body of Numbers.” He dates his preface “first of January, 1630.” -The fifth and sixth pages contain his “Description of the Grammelogia,” the -term Grammelogia being applied to the instrument, as well as to the book. -His description is as follows: - - The parts of the Instrument are two Circles, the one moveable, and the - other fixed; The moveable is that unto which is fastened a small pin to - move it by; the other Circle may be conceived to be fixed; The - circumference of the moveable Circle is divided into unequall parts, - charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures - doe represent themselves, or such numbers unto which a Cipher or Ciphers - are added, and are varied as the occasion falls out in the speech of - Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or - 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c. - -After elaborating this last point and explaining the decimal subdivisions -on the scales of the movable circle, he says that “the numbers and -divisions on the fixed Circle, are the very same that the moveable are, . -.” There is no drawing of the slide rule in this publication. The -twenty-two numbered pages give explanations of the various uses to which -the instrument can be put: “How to performe the Golden Rule” (pp. 1-3), -“Further uses of the Golden Rule” (pp. 4-6), “Notions or Principles -touching the disposing or ordering of the Numbers in the Golden Rule in -their true places upon the Grammelogia” (pp. 7-11), “How to divide one -number by another” (pp. 12, 13), “to multiply one Number by another” (pp. -14, 15), “To find Numbers in continuall proportion” (pp. 16, 17), “How to -extract the Square Root,” “How to extract the Cubicke Root” (pp. 18-21), -“How to performe the Golden Rule” (the rule of proportion) is explained -thus: - - Seeke the first number in the moveable, and bring it to the second number - in the fixed, so right against the third number in the moveable, is the - answer in the fixed. - - If the Interest of 100. li. be 8. li. in the yeare, what is the Interest - of 65. li. for the same time. - - Bring 100. in the moveable to 8. in the fixed, so right against 65. in - the moveable is 5.2. in the fixed, and so much is the Interest of 65. li. - for the yeare at 8. li. for 100. li. per annum. - - The Instrument not removed, you may at one instant right against any - summe of money in the moveable, see the Interest thereof in the fixed: - the reason of this is from the Definition of Logarithmes. - -These are the earliest known printed instructions on the use of a slide -rule. It will be noticed that the description of the instrument at the -opening makes no references to logarithmic lines for the trigonometric -functions; only the line of numbers is given. Yet the title-page promised -the “resolution of Plaine and Sphericall Triangles.” Page 22 throws light -upon this matter: - - If there be composed three Circles of equal thicknesse, A.B.C. so that - the inner edge of D [should be B] and the outward edge of A bee - answerably graduated with Logarithmall signes [sines], and the outward - edge of B and the inner edge of A with Logarithmes; and then on the - backside be graduated the Logarithmall Tangents, and againe the - Logarithmall signes oppositly to the former graduations, it shall be - fitted for the resolution of Plaine and Sphericall Triangles. - -After twelve lines of further remarks on this point he adds: - - Hence from the forme, I have called it a Ring, and Grammelogia by - annoligie of a Lineary speech; which Ring, if it were projected in the - convex unto two yards Diameter, or thereabouts, and the line Decupled, it - would worke Trigonometrie unto seconds, and give proportionall numbers - unto six places only by an ocular inspection, which would compendiate - Astronomicall calculations, and be sufficient for the Prosthaphaeresis of - the Motions: But of this as God shall give life and ability to health and - time. - -The unnumbered page following page 22 contains the patent and copyright on -the instrument and book: - - Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs - an Instrument called Grammelogia, or The Mathematicall Ring, together - with a Booke so intituled, expressing the use thereof, being his owne - Invention; we of our Gracious and Princely favour have granted unto the - said Richard Delamain and his Assignes, Privilege, Licence, and - Authority, for the sole Making, Printing and Selling of the said - Instrument and Booke: straightly forbidding any other to Make, Imprint, - or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument - or Booke within any our Dominions, during the space of ten yeares next - ensuing the date hereof, upon paine of Our high displeasure. Given under - our hand and Signet at our Palace of Westminster, the fourth day of - January, in the sixth yeare of our Raigne. - - - Delamain’s later designs, and directions for using his instruments - -In the Appendix of Grammelogia III, on page 52 is given a description of an -instrument promised near the end of Grammelogia I: - - That which I have formerly delivered hath been onely upon one of the - Circles of my Ring, simply concerning Arithmeticall Proportions, I will - by way of Conclusion touch upon some uses of the Circles, of Logarithmall - Sines, and Tangents, which are placed on the edge of both the moveable - and fixed Circles of the Ring in respect of Geometricall Proportions, but - first of the description of these Circles. - - First, upon the side that the Circle of Numbers is one, are graduated on - the edge of the moveable, and also on the edge of the fixed the - Logarithmall Sines, for if you bring 1. in the moveable amongst the - Numbers to 1. in the fixed, you may on the other edge of the moveable and - fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto - 6.6. and each degree subdivided, and then over the former divisions and - figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5. - 4. 3. 2. 1. each of those divided by small points. - - Secondly, (if the Ring is great) neere the outward edge of this side of - the fixed against the Numbers, are the usuall divisions of a Circle, and - the points of the Compasse: serving for observation in Astronomy, or - Geometry, and the sights belonging to those divisions, may be placed on - the moveable Circle. - - Thirdly, opposite to those Sines on the other side are the Logarithmall - Tangents, noted alike both in the moveable and fixed thus - 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or - divisions serve also for their Complements to 90. so 40 gr. stands for - 50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both - in the moveable and fixed is also divided into parts. As for the degrees - which are under 6. viz. 5.4.3.2.1. they are noted with small figures over - this divided Circle from 45.40.35.30.25. &c. and each of those degrees - divided into parts by small points both in the moveable and fixed. - - Fourthly, on the other edge of the moveable on the same side is another - graduation of Tangents, like that formerly described. And opposite unto - it, in the fixed is a Graduation of Logarithmall sines in every thing - answerable to the first descrition of Sines on the other side. - - Fifthly, on the edge of the Ring is graduated a parte of the Æquator, - numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees - of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree - both of the Æquator, and Meridian are subdivided into parts; these two - graduated Circles serve to resolve such Questions which concerne - Latitude, Longitude, Rumb, and Distance, in Nauticall operations. - - Sixthly, to the concave of the Ring may be added a Circle to be elevated - or depressed for any Latitude, representing the Æquator, and so divided - into houres and parts with an Axis, to shew both the houre, and Azimuth, - and within this Circle may be hanged a Box, and Needle with a Socket for - a staffe to slide into it, and this accommodated with scrue pines to - fasten it to the Ring and staffe, or to take it off at pleasure. - -The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and -V make no reference to the dispute with Oughtred and may, therefore, be -assumed to have been published before the appearance of Oughtred’s Circles -of Proportion. On page 53, “To the Reader,” he says: - - . . . you may make use of the Projection of the Circles of the Ring upon - a Plaine, having the feet of a paire of compasses (but so that they be - flat) to move on the Center of that Plaine, and those feet to open and - shut as a paire of Compasses . . . now if the feet bee opened to any two - termes or numbers in that Projection, then may you move the first foot to - the third number, and the other foot shall give the Answer; . . . it hath - pleased some to make use of this way. But in this there is a double - labour in respect to that of the Ring, the one in fitting those feet unto - the numbers assigned, and the other by moving them about, in which a man - can hardly accommodate the Instrument with one hand, and expresse the - Proportionals in writing with the other. By the Ring you need not but - bring one number to another, and right against any other number is the - Answer without any such motion. . . . upon that [the Ring] I write, - shewing some uses of those Circles amongst themselves, and conjoyned with - others . . . in Astronomy, Horolographie, in plaine Triangles applyed to - Dimensions, Navigation, Fortification, etc. . . . But before I come to - Construction, I have thought it convenient by way introduction, to - examine the truth of the graduation of those Circles . . . - -These are the words of a practical man, interested in the mechanical -development of his instrument. He considers not only questions of -convenience but also of accuracy. The instrument has, or may have now, also -lines of sines and tangents. To test the accuracy of the circles of -Numbers, “bring any number in the moveable to halfe of that number in the -fixed: so any number or part in the fixed shall give his double in the -moveable, and so may you trie of the thirds, fourths &c. of numbers, vel -contra,” (p. 54). On page 55 are given two small drawings, labelled, “A -Type of the Ringe and Scheme of this Logarithmicall projection, the use -followeth. These Instruments are made in Silver or Brasse by John Allen -neare the Sauoy in the Strand.” - - - IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE - CIRCULAR SLIDE RULE - -Delamain’s publication of 1630 on the ‘Mathematicall Ring’ does not appear -at that time to have caused a rupture between him and Oughtred. When in -1631 Delamain brought out his Horizontall Quadrant, the invention of which -Delamain was afterwards charged to have stolen from Oughtred, Delamain was -still in close touch with Oughtred and was sending Oughtred in the Arundell -House, London, the sheets as they were printed. Oughtred’s reference to -this in his Epistle (p. 20) written after the friendship was broken, is as -follows: - - While he was printing his tractate of the Horizontall quadrant, although - he could not but know that it was injurious to me in respect of my free - gift to Master Allen, and of William Forster, whose translation of my - rules was then about to come forth: yet such was my good nature, and his - shamelessnesse, that every day, as any sheet was printed, hee sent, or - brought the same to mee at my chamber in Arundell house to peruse which I - lovingly and ingenuously did, and gave him my judgment of it. - -Even after Forster’s publication of Oughtred’s Circles of Proportion, 1632, -Oughtred had a book, A canon of Sines Tangents and Secants, which he had -borrowed from Delamain and was then returning (Epistle, page (5)). The -attacks which Forster, in the preface to the Circles of Proportion, made -upon Delamain (though not naming Delamain) started the quarrel. Except for -Forster and other pupils of Oughtred who urged him on to castigate -Delamain, the controversy might never have arisen. Forster expressed -himself in part as follows: - - . . . being in the time of the long vacation 1630, in the Country, at the - house of the Reverend, and my most worthy friend, and Teacher, Mr. - William Oughtred (to whose instruction I owe both my initiation, and - whole progresse in these Sciences.) I vpon occasion of speech told him of - a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made - (such as it vsually called Mr. Gunter’s Ruler) 6 feet long, to be vsed - with a payre of beame-compasses. “He answered that was a poore invention, - and the performance very troublesome: But, said he, seeing you are taken - with such mechanicall wayes of Instruments, I will shew you what deuises - I have had by mee these many yeares.” And first, hee brought to mee two - Rulers of that sort, to be vsed by applying one to the other, without any - compasses: and after that hee shewed mee those lines cast into a circle - or Ring, with another moueable circle vpon it. I seeing the great - expeditenesse of both those wayes; but especially, of the latter, wherein - it farre excelleth any other Instrument which hath bin knowne; told him, - I wondered that hee could so many yeares conceale such vseful inuentions, - not onely from the world, but from my selfe, to whom in other parts and - mysteries of Art, he had bin so liberall. He answered, “That the true way - of Art is not by Instruments, but by Demonstration: and that it is a - preposterous course of vulgar Teachers, to begin with Instruments, and - not with the Sciences, and so in-stead of Artists, to make their - Schollers only doers of tricks, and as it were Iuglers: to the despite of - Art, losse of precious time, and betraying of willing and industrious - wits, vnto ignorance and idlenesse. That the vse of Instruments is indeed - excellent, if a man be an Artist: but contemptible, being set and opposed - to Art. And lastly, that he meant to commend to me, the skill of - Instruments, but first he would haue me well instructed in the Sciences. - He also shewed me many notes, and Rules for the vse of those circles, and - of his Horizontall Instrument, (which he had proiected about 30 yeares - before) the most part written in Latine. All which I obtained of him - leaue to translate into English, and make publique, for the vse, and - benefit of such as were studious, and louers of these excellent Sciences. - - Which thing while I with mature, and diligent care (as my occasions would - give me leaue) went about to doe: another to whom the Author in a louing - confidence discouered this intent, using more hast then good speed, went - about to preocupate; of which vntimely birth, and preuenting (if not - circumuenting) forwardnesse, I say no more: but aduise the studious - Reader, onely so farre to trust, as he shal be sure doth agree to truth & - Art. - -While in this dedication reference is made to a slide rule or “ring” with a -“moveable circle,” the instrument actually described in the Circles of -Proportion consists of fixed circles “with an index to be opened after the -manner of a paire of Compasses.” Delamain, as we have seen, had decided -preference for the moveable circle. To Oughtred, on the other hand, one -design was about as good as the other; he was more of a theorist and -repeatedly expressed his contempt for mathematical instruments. In his -Epistle (page (25)), he says he had not “the one halfe of my intentions -upon it” (the rule in his book), nor one with a “moveable circle and a -thread, but with an opening Index at the centre (if so be that bee cause -enough to make it to bee not the same, but another Instrument) for my part -I disclaime it: it may go seeke another Master: which for ought I know, -will prove to be Elias Allen himselfe: for at his request only I altered a -little my rules from the use of the moveable circle and the thread, to the -two armes of an Index.” - -All parts of Delamain’s Grammelogia IV, except pages 1-22 and 53-68 -considered above, were published after the Circles of Proportion, for they -contain references to the ill treatment that Delamain felt or made believe -that he felt, that he had received in the book published by Oughtred and -Forster. Oughtred’s reference to teachers whose scholars are “doers of -tricks,” “Iuglers,” and Forster’s allusion to “another to whom the Author -in a loving confidence” explained the instrument and who “went about to -preocupate” it, are repeatedly mentioned. Delamain says, (page (89)) that -at first he did not intend to express himself in print, “but sought peace -and my right by a private and friendly way.” Oughtred’s account of -Delamain’s course is that of an “ill-natured man” with a “virulent tongue,” -“sardonical laughter” and “malapert sawsiness.” Contrasting Forster and -Delamain, he says that, of the former he “had the very first moulding” and -made him feel that “the way of Art” is “by demonstration.” But Delamain was -“already corrupted with doing upon Instruments, and quite lost from ever -being made an Artist.” (Epistle page (27)). Repeatedly does Oughtred assert -Delamain’s ignorance of mathematics. The two men were evidently of wholly -different intellectual predilections. That Delamain loved instruments is -quite evident, and we proceed to describe his efforts to improve the -circular slide rule. - -The Grammelogia IV is dedicated to King Charles I. Delamain says: - - . . . Everything hath his beginning, and curious Arts seldome come to the - height at the first; It was my promise then to enlarge the invention by a - way of decuplating the Circles, which I now present unto your sacred - Majestie as the quintessence and excellencie there of . . . - -His enlarged circular rules are illustrated in the Bodleian Library copy of -Grammelogia IV by four diagrams, two of them being the two drawings on the -two title-pages at the beginning of the Grammelogia IV, 4 inches in -external diameter, and exhibiting eleven concentric circular lines carrying -graduations of different sorts. In the second of these designs all circles -are fixed. The other two drawings are each 10¾ inches in external diameter -and exhibit 18 concentric circular lines; the folded sheet of the first of -these drawings is inserted between pages (23) and (24), the second folded -sheet between pages (83) and (84). All circles of this second instrument -are fixed. Counting in the two small drawings in Grammelogia III, there are -in all six drawings of slide rules in the Bodleian Grammelogia IV. On pages -(24) to (43) Delamain explains the graduation of slide rules. He takes -first a rule which has one circle of equal parts, divided into 1000 equal -divisions. From a table of logarithms he gets log 2 = 0.301; from the -number 301 in the circle of equal parts he draws a line to the center of -the circle and marks the intersection with the circles of numbers by the -figure 2. Thus he proceeds with log 3, log 4, and so on; also with log sin -x and log tan x. For log sin x he uses two circles, the first (see page -(27)) for angles from 34′ 24″ to 5° 44′ 22″, the second circle from 5° 44′ -22″ to 90°. The drawings do not show the seconds. He suggests many -different designs of rules. On page (29) he says: - - For the single projection of the Circles of my Ring, and the dividing and - graduating of them: which may bee so inserted upon the edges of Circles - of mettle turned in the forme of a Ring, so that one Circle may moove - betweene two fixed, by helpe of two stayes, then may there be graduated - on the face of the Ring, upon the outer edge of the mooveable and inner - edge of the fixed, the Circle of Numbers, then upon the inner edge of - that mooveable Circle, and the outward edge of that inner fixed Circle - may be inserted the Circle of Sines, and so according to the description - of those that are usually made. - -In addition to these lines he proceeds to mention the circle giving the -ordinary division into degrees and minutes, and two circles of tangents on -the other side of the rule. - -Next Delamain explains an arrangement of all the graduation on one side of -the rule by means of “a small channell in the innermost fixed Circle, in -which may be placed a small single Index, which may have sufficient length -to reach from the innermost edge of the Mooveable Circle, unto the outmost -edge of the fixed Circle, which may be mooved to and fro at pleasure, in -the channell, which Index may serve to shew the opposition of Numbers” (p. -(31)). From this it is clear that the invention of the “runner” goes back -to the very first writers on the slide rule. - -After describing a modification of the above arrangement, he adds, “many -other formes might be deliverd, about this single projection” (p. (32)). - -Proceeding to the “enlarging” of the circles in the Ring, to, say, the -“Quadruple to that which is single, that is, foure times greater,” the -“equall parts” are distributed over four circles instead of only one -circle, but the general method of graduation is the same as before (p. -(33)); there being now four circles carrying the logarithms of numbers, and -so on. Next he points out “severall wayes how the Circles of the -Mathematicall Ring (being inlarged) may be accommodated for practicall -use:” (1) The Circles are all fixed in a plain and movable flat compasses -(or better, a movable semicircle) are used for fixing any two positions; -(2) There is a “double projection” of each logarithmic line “inlarged on a -Plaine,” one fixed, the other movable, as shown in his first figure on the -title-page, a single index only being used; (3) use of “my great Cylinder -which I have long proposed (in which all the Circles are of equall -greatnesse,) and it may be made of any magnitude or capacity, but for a -study (hee that will be at the charge) it may be of a yard diameter and of -such an indifferent length that it may containe 100 or more Circles fixed -parallel one to the other on the Cylinder, having a space betweene each of -them, so that there may bee as many mooveable Circles, as there are fixed -ones, and these of the mooveable linked, or fastened together, so that they -may all moove together by the fixed ones in these spaces, whose edges both -of the fixed, and mooveable being graduated by helpe of a single Index will -shew the proportionalls by opposition in this double Projection, or by a -double Index in a single Projection” (p. (36)). - -Next follows the detailed description of his Ring “on a Plaine, according -to the diagramme that was given the King (for a view of that projection) -and afterwards the Ring it selve.” The diagram is the large one which we -mentioned as inserted between pages (23) and (24). The instrument has two -circles, one moveable, upon each of which are described 13 distinct -circular graduations. The lines on the fixed circle are: “The Circle of -degrees and calendar,” E. “Circle of equall parts, and part of the Equator, -and Meridian,” TT. “The Circle of Tangents,” S. “The Circle of Sines,” D. -“The Circle of Decimals,” N. “The Circle of Numbers.” The lines on the -movable circle are: N. “The Circle of Numbers,” E. “The Circle of equated -figures, and bodies,” S. “The Circle of Sines,” TT. “The Circle of -Tangents,” Y. “The Circle of time, yeares, and monethes.” - -On pages (84)-(88) Delamain explains an enlargement of his Ring for -computations involving the sines of angles near to 90°. On page (86) he -says: - - I have continued the Sines of the Projection unto two severall - revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr. - (being the last revolution of the decuplation of the former, or the - hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s. - and ends at 90.gr. (being the last of a ternary of decuplated - revolutions, or the thousand part of that Projection) and may bee thus - used. - -He explains the manner of using these extra graduations. Thus he claims to -have attained degrees of accuracy which enabled him to do what “some one” -had declared “could not bee done.” It is hardly necessary to point out that -Delamain’s Grammelogia IV suggests designs of slide rules which inventors -two hundred or more years later were endeavouring to produce. Which of -Delamain’s designs of rules were actually made and used, he does not state -explicitly. He refers to a rule 18 inches in diameter as if it had been -actually constructed (pages (86), (88)). Oughtred showed no appreciation of -such study in designing and ridiculed Delamain’s efforts, in his Epistle. - -Additional elucidations of his designs of rules, along with explanations of -the relations of his work to that of Gunter and Napier, and sallies -directed against Oughtred and Forster, are contained on pages (8)-(21) of -his Grammelogia IV. - - - - - V. INDEPENDENCE AND PRIORITY OF INVENTION - - -The question of independence and priority of invention is discussed by -Delamain more specifically on pages (89)-(113); Oughtred devotes his entire -Epistle to it. It is difficult to determine definitely which publication is -the later, Delamain’s Grammelogia IV or Oughtred’s Epistle. Each seems to -quote from the other. Probably the explanation is that the two publications -contain arguments which were previously passed from one antagonist to the -other by word of mouth or by private letter. Oughtred refers in his Epistle -(p. (12)) to a letter from Delamain. We believe that the Epistle came after -Delamain’s Grammelogia IV. Delamain claims for himself the invention of the -circular slide rule. He says in his Grammelogia IV. (p. (99)), “when I had -a sight of it, which was in February, 1629 (as I specified in my Epistle) I -could not conceale it longer, envying my selfe, that others did not tast of -that which I found to carry with it so delightfull and pleasant a goate -[taste] . . .” Delamain asserts (without proof) that Oughtred “never saw it -as he now challengeth it to be his invention, untill it was so fitted to -his hand, and that he made all his practise on it after the publishing of -my Booke upon my Ring, and not before; so it was easie for him or some -other to write some uses of it in Latin after Christmas, 1630 and not the -Sommer before, as is falsely alledged by some one . . .” (p. (91)). -Delamain’s accusation of theft on the part of Oughtred cannot be seriously -considered. Oughtred’s reputation as a mathematician and his standing in -his community go against such a supposition. Moreover, William Forster is a -witness for Oughtred. The fact that Oughtred had the mastery of the -rectilinear slide rule as well, while Delamain in 1630 speaks only of the -circular rule, weighs in Oughtred’s favour. - -Oughtred says he invented the slide rule “above twelve yeares agoe,” that -is, about 1621, and “I with mine owne hand made me two such Circles, which -I have used ever since, as my occasions required,” (Epistle p. (22)). On -the same page, he describes his mode of discovery thus: - - I found that it required many times too great a paire of Compasses [in - using Gunter’s line], which would bee hard to open, apt to slip, and - troublesome for use. I therefore first devised to have another Ruler with - the former: and so by setting and applying one to the other, I did not - onely take away the use of Compasses, but also make the worke much more - easy and expedite: when I should not at all need the motion of my hand, - but onely the glancing at my sight: and with one position of the Rulers, - and view of mine eye, see not one onely, but the manifold proportions - incident unto the question intended. But yet this facility also wanted - not some difficulty especially in the line of tangents, when one arch was - in the former mediety of the quadrant, and the other in the latter: for - in this case it was needful that either one Ruler must bee as long againe - as the other; or else that I must use an inversion of the Ruler, and - regression. By this consideration I first of all saw that if those lines - upon both Rulers were inflected into two circles, that of the tangents - being in both doubled, and that those two Circles should move one upon - another; they with a small thread in the center to direct the sight, - would bee sufficient with incredible and wonderfull facility to worke all - questions of Trigonometry . . . - -Oughtred said that he had no desire to publish his invention, but in the -vacation of 1630 finally promised William Forster to let him bring out a -translation. Oughtred claims that Delamain got the invention from him at -Alhallontide [November 1], 1630, when they met in London. The accounts of -that meeting we proceed to give in double column. - - Delamain’s Statement - Grammelogia IV, page (98) - - “. . . about Alhalontide 1630. (as our Authors reporteth) was the time he - was circumvented, and then his intent in a loving manner (as before) he - opened unto me, which particularly I will dismantle in the very naked - truth: for, wee being walking together some few weekes before Christmas, - upon Fishstreet hill, we discoursed upon sundry things Mathematicall, - both Theoreticall and Practicall, and of the excellent inventions and - helpes that in these dayes were produced, amongst which I was not a - little taken with that of the Logarythmes, commending greatly the - ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, - in the lines of proportion, extracted from these Logarythmes for ordinary - Practicall uses; He replyed unto me (in these very words) What will yov - say to an Invention that I have, which in a lesse extent of the Compasses - shall worke truer then that of Mr. Gunters Ruler, I asked him then of - what forme it was, he answered with some pause (which no doubt argued his - suspition of mee that I might conceive it) that it was Arching-wise, but - now hee sayes that hee told mee then, it was Circular (but were I put to - my oath to avoid the guilt of Conscience I would conclude in the former.) - At which immediately I answered, I had the like my selfe, and so we - discoursed not a word more touching that subject . . . Then after my - coming home I sent him a sight of my Projection drawne in Pastboard: Now - admit I had not the Invention of my Ring before I discoursed . . . it was - not so facil for mee . . . to raise and compose so complete, and absolute - an Instrument from so small a principle, or glimpse of light . . .” - - Oughtred’s Statement - Epistle, page (23) - - “Shortly after my gift to Elias Allen, I chanced to meet with Richard - Delamain in the street (it was at Alhallontide) and as we walked together - I told him what an Instrument I had given to Master Allen, both of the - Logarithmes projected into circles, which being lesse then one foot - diameter would performe as much as one of Master Gunters Rulers of sixe - feet long: and also of the Prostaphaereses of the Plannets and second - motions. Such an invention have I said he: for now his intentions (that - is his ambition) beganne to worke: . . . But he saith, Then after my - comming home I sent him a sight of my projection drawne in past-board. - See how notoriously he jugleth without an Instrument. Then after: how - long after? a sight of my projection: of how much? More then seven weekes - after on December 23, he sent to mee the line of numbers onely set upon a - circle: . . . and so much onely he presented to his Majesty: but as for - Sine or tangent of his, there was not the least shew of any. Neither - could he give to Master Allen any direction for the composure of the - circles of his Ring, or for the division of them: as upon his oath Master - Allen will testify how hee misled him, and made him labour in vain above - three weeks together, until Master Allen himselfe found out his ignorance - and mistaking, which is more cleare then is possible with any impudence - to be outfaced.” - -Oughtred makes a further statement (Epistle, p. (24)) as follows: - - Delamain hearing that Brown with his Serpentine had another line by which - he could worke to minutes in the 90 degree of sines . . . gave the [his] - booke to Browne: who in thankfulnesse could not but gratify Delamain with - his Lines also: and teach him the use of them, but especially of the - great Line: with this caution on both sides, that one should not meddle - with the others invention. Two dayes after Delamain . . . because he had - found some things to be altered therin, . . . asked for the booke . . . - but as soone as he had got it in his hands he rent out all the middle - part with the two Schemes & put them up in his pocket & went his way . . - . and . . . laboureth to recall all the bookes he had given forth . . . - And shortly after this he got a new Printer (who was ignorant of his - former Schemes) to print him new: giving him an especiall charge of the - outermost line newly graven in the Plate, which indeed is Brownes very - line: and then altering his book . . . - -This and other statements made by Oughtred seem damaging to Delamain’s -reputation. But it is quite possible that Oughtred’s guesses as to -Delamain’s motives are wrong. Moreover, some of Oughtred’s statements are -not first hand knowledge with him, but mere hearsay. One may accept his -first hand facts and still clear Delamain of wrong doing. There is always -danger that rival claimants of an invention or discovery will proceed on -the assumption that no one else could possibly have come independently upon -the same devices that they themselves did; the history of science proves -the opposite. Seldom is an invention of any note made by only one man. We -do not feel competent to judge Delamain’s case. We know too little about -him as a man. We incline to the opinion that the hypothesis of independent -invention is the most plausible. At any rate, Delamain figures in the -history of the slide rule as the publisher of the earliest book thereon and -as an enthusiastic and skillful designer of slide rules. - -The effect of this controversy upon interested friends was probably small. -Doubtless few people read both sides. Oughtred says:[21] “this scandall . . -. hath with them, to whom I am not knowne, wrought me much prejudice and -disadvantage . .” Aubrey,[22] a friend of Oughtred, refers to Delamain “who -was so sawcy to write against him” and remembers having seen “many yeares -since, twenty or more good verses made” against Delamain. Another friend of -Oughtred, William Robinson, who had seen some of Delamain’s publications, -but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before -the appearance of the latter’s Epistle: - - I cannot but wonder at the indiscretion of Rich. Delamain, who being - conscious to himself that he is but the pickpurse of another man’s wit, - would thus inconsiderately provoke and awake a sleeping lion . . . he - hath so weakly (though in my judgment, vaingloriously enough) commended - his own labour . . .[23] - -Delamain presented King Charles I with one of his sun-dials, also with a -manuscript and, later, with a printed copy of his book of 1630. A drawing -of his improved slide rule was sent to the King and the Grammelogia IV is -dedicated to him. The King must have been favorably impressed, for Delamain -was appointed tutor to the King in mathematics. His widow petitioned the -House of Lords in 1645 for relief; he had ten children.[24] - -Anthony Wood states that Charles I, on the day of his execution, commanded -his friend Thomas Herbert “to give his son the duke of York his large -ring-sundial of silver, a jewel his maj. much valued.” Anthony Wood adds, -“it was invented and made by Rich. Delamaine a very able mathematician, who -projected it, and in a little printed book did shew its excellent use in -resolving many questions in arithmetic and other rare operations to be -wrought by it in the mathematics.”[25] - - - - - VI. OUGHTRED’S GAUGING LINE, 1633 - - -It has not been generally known, hitherto, that Oughtred designed a -rectilinear slide rule for gauging and published a description thereof in -1633.[26] In his Circles of Proportion, chapter IX, Oughtred had offered a -closer approximation than that of Gunter for the capacity of casks. The -Gauger of London expostulated with Oughtred for presuming to question -anything that Gunter had written. The ensuing discussion led to an -invitation extended by the Company of Vintners to the instrument maker -Elias Allen to request Oughtred to design a gauging rod.[27] This he did, -and Allen received an order for “threescore” instruments. On page 19 -Oughtred describes his ‘Gauging Rod:’ - - It consisteth of two rulers of brasse about 32 ynches of length, which - also are halfe an ynch broad, and a quarter of an ynch thick . . . At one - end of both those rulers are two little sockets of brasse fastened on - strongly: by which the rulers are held together, and made to move one - upon another, and to bee drawne out unto any length, as occasion shall - require: and when you have them at the just length, there is upon one of - the sockets a long Scrue-pin to scrue them fast. - -There are graduations on three sides of the rulers, one graduation being -the logarithmic line of numbers. He says (p. 39), “the maner of computing -the Gauge-divisions I have concealed.” W. Robinson, who was a friend of -Oughtred, wrote him as follows:[28] - - I have light upon your little book of artificial gauging, wherewith I am - much taken, but I want the rod, neither could I get a sight of one of - them at the time, because Mr. Allen had none left . . . I forgot to ask - Mr. Allen the price of one of them, which if not much I would have one of - them.” Oughtred annotated this passage thus: “Or in wood, if any be made - in wood by Thompson or any other.” - -Another of Oughtred’s admirers, Sir Charles Cavendish, wrote, on February -11, 1635 thus:[29] - - I thank you for your little book, but especially for the way of - calculating the divisions of your gauging rod. I wish, both for their own - sakes and yours, that the citizens were as capable of the acuteness of - this invention, as they are commonly greedy of gain, and then I doubt not - but they would give you a better recompense than I doubt now they will. - -On April 20, 1638, we find Oughtred giving Elias Allen directions[30] -“about the making of the two rulers.” As in 1633,[31] so now, Oughtred -takes one ruler longer than the other. This 1633 instrument was used also -as “a crosse-staffe to take the height of the Sunne, or any Starre above -the Horizon, and also their distances.” The longer ruler was called staffe, -the shorter transversarie. While in 1633 he took the lengths of the two in -the ratio “almost 3 to 2,” in 1638, he took “the transversary three -quarters of the staff’s length, . . . that the divisions may be larger.” - - - - - VII. OTHER SEVENTEENTH CENTURY SLIDE RULES - - -In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard, -Henry Coggeshall, W. Hunt and Sir Isaac Newton.[32] Of Partridge’s Double -Scale of Proportion, London, I have examined a copy dated 1661, which is -the earliest date for this book that I have seen. As far as we know, 1661 -is the earliest date of publications on the slide rule, since Oughtred and -Delamain. But it would not be surprising if the intervening 28 years were -found not so barren as they seem at present. The 1661 and 1662 impressions -of Partridge are identical, except for the date on the title-page. William -Leybourn, who printed Partridge’s book, speaks in high appreciation of it -in his own book.[33] - -In 1661 was published also John Brown’s first book, Description and Use of -a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use -of “Mr. Whites rule” for the measuring of board and timber, round and -square. He calls this a “sliding rule.” The existence, in 1661, of a -“Whites rule” indicates activities in designing of which we know as yet -very little. In his book of 1761, previously quoted, Brown gives a drawing -of “White’s sliding rule” (p. 193); also a special contrivance of his own, -as indicated by him in these words: - - A further improvement of the Triangular Quadrant, as I have made it - several times, with a sliding Cover on the in-side, when made hollow, to - carry Ink, Pens, and Compasses; then on the sliding Cover, and Edges, is - put the Line of Numbers, according to Mr. White’s first Contrivance for - manner of operation; but much augmented, and made easie, by John Brown. - -He gives no drawing of his “triangular quadrant,” hence his account of it -is unsatisfactory. He explains the use of “gage-points.” His placing -logarithmic lines on the edges of instrument boxes was outdone in oddity -later by Everard who placed them on tobacco-boxes.[34] In Brown’s -publication of 1704 the White slide rule is given again, “being as neat and -ready a way as ever was used.” He tells also of a “glasier’s sliding rule.” -William Leybourn explains in 1673 how Wingate’s double and triple lines for -squaring and cubing, or square and cube root, can be used on slide -rules.[35] - -Beginning early in the history of the slide rule, when Oughtred designed -his “gauging rod,” we notice the designing of rules intended for very -special purposes. Another such contrivance, which enjoyed long popularity, -was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677, -a booklet of 35 pages. Coggeshall says in his preface: - - For what can be more ready and easie, then having set twelve to the - length, to see the Content exactly against the Girt or Side of the - Square. Whereas on Mr. Partridge’s Scale the Content is the Sixth Number, - which is far more troublesome then [even] with Compasses. - -One line on Coggeshall’s rule begins with 4 and extends to 40, these -numbers being the “Girt” (a quarter of the circumference), which in -ordinary practice of measuring round timber lies between 4 inches and 40 -inches. This “Girt line” slides “against the line of Numbers in two -Lengths, to which it is exactly equal.” A second edition, 1682, shows some -changes in the rule, as well as an enlargement and change of title of the -book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent, -London, 1682. In this, the description of the rule is given thus: - - There are four Lines on each flat of this Rule; two next the outward - edges, which are Lines of Measure; and two next the inward edges, which - are Lines of Proportion. On one flat, next the inward edges, is the - Square-line [Girt-line in round timber measurement] with the Line of - Numbers his fellow. Next the outward, a Line of Inches divided into - Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 - to 24 on the other. On the other flat, next the inward edges, is the - double Scale of Numbers [for solving proportions]. Next the outward on - one Rule a Line of Inches divided each into ten parts; and this for - gauging, etc. On the other a foot divided into 100 parts. - -Later further changes were introduced in Coggeshall’s rule.[36] - -It is worthy of note that Coggeshall’s slide rule book, The Art of -Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p. -473; hence Leupold’s description[37] of the rectilinear slide rule in his -Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not -the earliest reference to the rectilinear rule found in German -publications. The above date is earlier even than Biler’s reference to a -circular slide rule in his Descriptio instrumenti mathematici universalis -of 1696. - -Two noted slide rules for gauging were described by Tho. Everard, -Philomath, in his Stereometry made easie, London, 1684. He designates his -lines by the capital letters A, B, C, D, E. On the first instrument, A on -the rule, and B and C on the slide, have each two radiuses of numbers, D -has only one, while E has three. The second rule is described in an -Appendix; it is one foot long, with two slides enabling the rule to be -extended to 3 feet. - -Everard’s instruments were made in London by Isaac Carver who, soon after, -himself wrote a sixteen-page Description and Use of a New Sliding Rule, -projected from the Tables in the Gauger’s Magazine, London, 1687, which was -“printed for William Hunt” and bound in one volume with a book by Hunt, -called The Gauger’s Magazine, London, 1687. This appears to be the same -William Hunt who later brought out descriptions of his own of slide rules. -The instrument described by Carver “consists of three pieces, two whereof -are moveable to be drawn out till the whole be 36 inches long.” It has -several non-logarithmic graduations, together with logarithmic lines marked -A, B, C, D, of which A, B, C are “double lines,” and D a “single line” used -for squares and square roots. It is designed for the determination of the -vacuity of a “spheroidal cask lying,” a “spheroidal cask standing,” and a -“parabolical cask lying.” - -Another seventeenth century writer on the slide rule is John Atkinson, whom -we have mentioned earlier. He says:[38] “The Lines of Numbers, Sines and -Tangents, are set double, that is, one on each side, as the middle piece -slides: which middle piece is so contrived, to slip to and fro easily, to -slide out, and to be put in any side uppermost, in order to bring those -Lines together (or against one another) most proper for solving the -Question, wrought by Sliding-Gunter.” - -The data presented in this article show that, while the earliest slide -rules were of the circular type, the later slide rules of the seventeenth -century were of the rectilinear type.[39] - - January 12, 1915. - - - - - Footnotes - - -[1]F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments, - New York, 1909, pp. 7-14, also Addenda i-vi. - -[2]F. Cajori, “On the Invention of the Slide Rule,” in Colorado College - Publication, Engineering Series Vol. 1, 1910. An abstract of this is - given in Nature (London), Vol. 82, 1909, p. 267. - -[3]F. Cajori, History etc., p. 14. - -[4]Art. “Slide Rule” in the Penny Cyclopaedia and in the English - Cyclopaedia [Arts and Sciences]. - -[5]Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817, - p. 423. - -[6]The full title of the book which Wingate published on this subject in - Paris is as follows: - - L’Vsage | de la | Reigle de | Proportion | en l’Arithmetique & | - Geometrie. | Par Edmond Vvingate, | Gentil-homme Anglois. | - - Εἂν ἧς φιλεµαθὴς, ἕση ἥση πολυµαθὴς. - - In tenui, sed nõ tenuis vsusve, laborne. | - - A Paris, | Chez Melchior Mondiere, | demeurant en l’Isle du Palais, | à - la | ruë de Harlay aux deux Viperes. | M. DC. XXIV. | Auec Priuilege du - Roy. | - - Back of the title page is the announcement: - - Notez que la Reigle de Proportion en toutes façons se vend à Paris chez - Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces, - demeurant en l’Isle du Palais sur le Quay qui regarde la Megisserie à - l’Espic d’or. - -[7]The title-page of the edition of 1658 is as follows: - - The Use of the Rule of Proportion in Arithmetick & Geometrie. First - published at Paris in the French tongue, and dedicated to Monsieur, the - then king’s onely Brother (now Duke of Orleance). By Edm. Wingate, an - English Gent. And now translated into English by the Author. Whereinto - is now also inserted the Construction of the same Rule, & a farther use - thereof . . . 2nd edition inlarged and amended. London, 1658. - -[8]Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; - Drawn up by himself by way of Diary. With Appendix of original Letters. - Publish’d by Charles Burman, Esq., London, 1717, p. 23. - -[9]Mathematical Tables, 1811, p. 36, and art. “Gunter’s Line” in his Phil. - and Math. Dictionary, London, 1815. - -[10]To the English Gentrie, and all others studious of the Mathematicks, - which shall bee readers hereof. The just Apologie of Wil: Ovghtred, - against the slaunderous insimulations of Richard Delamain, in a Pamphlet - called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum - projectio circularis. We shall refer to this document as Epistle. It was - published without date in 32 unnumbered pages of fine print, and was - bound in with Oughtred’s Circles of Proportion, in the editions of 1633 - and 1639. In the 1633 edition it is inserted at the end of the volume - just after the Addition vnto the Vse of the Instrument etc., and in that - of 1639 immediately after the preface. It was omitted from the Oxford - edition of 1660. The Epistle was also published separately. There is a - separate copy in the British Museum, London. Aubrey, in his Brief Lives, - edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, “He - writt a stitch’t pamphlet about 163(?4) against . . . Delamaine.” - -[11]Thomas Browne is mentioned by Stone in his Mathematical Instruments, - London 1723, p. 16. See also Cajori, History of the Slide Rule, New - York, 1909, p. 15. - -[12]The Description and Use of a Joynt-Rule: . . . also the use of Mr. - White’s Rule for measuring of Board and Timber, round and square; With - the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and - Versed Sines. By J. Brown, Philom., London, 1661. - -[13]A Collection of Centers and Useful Proportions on the Line of Numbers, - by John Brown, 1662(?), 16 pages; Description and Use of the Triangular - Quadrant, by John Brown, London, 1671; Wingate’s Rule of Proportion in - Arithmetick and Geometry: or Gunter’s Line. Newly rectified by Mr. Brown - and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The - Description and Use of the Carpenter’s-Rule: Together with the Use of - the Line of Numbers commonly call’d Gunter’s-Line, by John Brown, - London, 1704. - -[14]William Leybourn, op. cit., pp. 129, 130, 132, 133. - -[15]James Atkinson’s edition of Andrew Wakely’s The Mariners Compass - Rectified, London, 1694 [Wakely’s preface dated 1664, Atkinson’s - preface, 1693]. Atkinson adds An Appendix containing Use of Instruments - most useful in Navigation. Our quotation is from this Appendix, p. 199. - -[16]R. Delamain, The Making, Description, and Use of a small portable - Instrument . . . called a Horizontall Quadrant, etc., London, 1631. - -[17]Oughtred’s description of his circular slide rule of 1632 and his - rectilinear slide rule of 1633, as well as a drawing of the circular - slide rule, are reproduced in Cajori’s History of the Slide Rule, - Addenda, pp. ii-vi. - -[18]The full title of the Grammelogia I is as follows: - - Gram̄elogia | or, | The Mathematicall Ring. | Shewing (any - reasonable Capacity that hath | not Arithmeticke) how to resolve and - worke | all ordinary operations of Arithmeticke. | And those which are - most difficult with greatest | facilitie: The extraction of Roots, the - valuation of | Leases, &c. The measuring of Plaines | and Solids. | With - the resolution of Plaine and Sphericall | Triangles. | And that onely by - an Ocular Inspection, | and a Circular Motion. | Naturae secreta tempus - aperit. | London printed by John Haviland, 1630. - -[19]Grammelogia III is the same as Grammelogia I, except for the addition - of an appendix, entitled: - - De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit - nullo (praescripto tempore) vitae | vsuram nobis ingeniique Deus. | - London, | - - . . . The next line or two of this title-page which probably contained - the date of publication, were cut off by the binder in trimming the - edges of this and several other pamphlets for binding into one volume. - -[20]Grammelogia IV has two title pages. The first is Mirifica Logarithmoru’ - Projectio Circularis. There follows a diagram of a circular slide rule, - with the inscription within the innermost ring: Nil Finis, Motvs, - Circvlvs vllvs Habet. The second title page is as follows: - - Grammelogia | Or, the Mathematicall Ring. | Extracted from the - Logarythmes, and projected Circular: Now published in the | inlargement - thereof unto any magnitude fit for use: shewing any reason- | able - capacity that hath not Arithmeticke how to resolve and worke, | all - ordinary operations of Arithmeticke: | And those that are most difficult - with greatest facilitie, the extracti- | on of Rootes, the valuation of - Leases, &c. the measuring of Plaines and Solids, | with the resolution - of Plaine and Sphericall Triangles applied to the | Practicall parts of - Geometrie, Horologographie, Geographie | Fortification, Navigation, - Astronomie, &c. | And that onely by an ocular inspection, and a Circular - motion, Invented and first published, by R. Delamain, Teacher, and - Student of the Mathematicks. | Naturae secreta tempus aperit. | - - There is no date. There follows the diagram of a second circular slide - rule, with the inscription within the innermost ring: Typus proiectionis - Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630 - promisi. There are numerous drawings in the Grammelogia, all of which, - excepting the drawings of slide rules on the engraved title-pages of - Grammelogia IV and V, were printed upon separate pieces of paper and - then inserted by hand into the vacant spaces on the printed pages - reserved for them. Some drawings are missing, so that the Bodleian - Grammelogia IV differs in this respect slightly from the copy in the - British Museum and from the British Museum copy of Grammelogia V. - -[21]Epistle, p. (8). - -[22]Aubrey, op. cit., Vol. II., p. 111. - -[23]Rigaud, Correspondence of Scientific Men during the 17th Century, Vol. - I, Oxford, 1841, p. 11. - -[24]Dictionary of National Biography, Art. “Delamain, Richard.” See also - Rev. Charles J. Robinson, Taylors’ School, from A.D. 1562 to 1874, Vol. - I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b; - Sixth Report of the Royal Commission on Historical Manuscripts, Part I, - Report and Appendix, London, 1877. In this Appendix, p. 82, we read the - following: - - Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. - Petitioner’s husband was servant to the King, and one of His Majesty’s - engineers for the fortification of the kingdom, and his tutor in - mathematical arts; but upon the breaking out of the war he deserted the - Court, and was called by the State to several employments, in fortifying - the towns of Northampton, Newport, and Abingdon; and was also abroad - with the armies as Quartermaster-General of the Foot, and therein died. - Petitioner is left a disconsolate widow with ten children, the four - least of whom are now afflicted with sickness, and petitioner has - nothing left to support them. There are several considerable sums of - money due to the petitioner, as well from the King as the State. Prays - that she may have some relief amongst other widows. See L. J., VII. 6. - 657. - -[25]Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London, - 1820, p. 34. - -[26]The New Artificial Gauging Line or Rod: together with rules concerning - the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London, - 1633. The copy we have seen is in the Bodleian Library, Oxford. The book - is small sized and has 40 pages. - -[27]Oughtred, op. cit., p. 11. - -[28]S. J. Rigaud, Correspondence of Scientific Men of the 17th Century, - Oxford, Vol. I, 1841, p. 17. - -[29]Rigaud, loc. cit., p. 22. - -[30]Rigaud, loc. cit., pp. 30, 31. - -[31]Oughtred, An Addition vnto the Vse of the Instrument called the Circles - of Proportion, London, 1633, p. 63. - -[32]F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22, - Addenda, pp. vi-ix. - -[33]W. Leybourn, op. cit., 1673, Preface, and pp. 128-29. - -[34]Cajori op. cit., Addenda, p. ix. - -[35]William Leybourn, op. cit., 1673, p. 35. - -[36]See Cajori, op. cit., pp. 20, 28, Addenda, p. ix. - -[37]See F. Cajori, “A Note on the History of the Slide Rule,” Bibliotheca - mathematica, 3 F., Vol. 10, pp. 161-163. - -[38]John Atkinson, op. cit., 1694, p. 204. - -[39]Probably the oldest slide rule now in existence is owned by St. John’s - College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in - diameter. It was exhibited along with other instruments in May, 1919. - According to the Catalogue of a Loan Exhibition of Early Scientific - Instruments in Oxford, opened May 16, 1919, the instrument is inscribed - with the name of the maker (“Elias Allen fecit”) and with the name of - the donor, Georgius Barkham. It is dated 1635, which is only three years - after the first publication of Oughtred’s description of his circular - slide rule. It is stated in the Catalogue: “Unfortunately all the - movable parts but the base-plate and a couple of thumb-screws are - missing. The face of the instrument is engraved with Oughtred’s - Horizontal Instrument. The back is engraved with eleven Circles of - Proportion as described in Arthur Haughton’s book, a copy of which was - presented to St. John’s College by George Barkham, to explain the use of - the instrument.” As Arthur Haughton’s Oxford edition of Oughtred’s - Circles of Proportion did not appear until 1660, it would seem that the - instrument was probably not presented to the College before 1660. As far - as is known, the next oldest slide rule is of the year 1654, kept in the - South Kensington Museum, London, and is described in Nature of March 5, - 1914. It is a rectilinear rule, “of boxwood, well made, and bound - together with brass at the two ends. It is of the square type, a little - more than 2 ft. in length, and bears the logarithmic lines first - described by Edmund Gunter. Of these, the num, sin and tan lines are - arranged in pairs, identical and contiguous, one line in each pair being - on the fixed part, and the other on the slide.” The instrument is - inscribed, “Made by Robert Bissaker for T. W., 1654.” Nowhere else have - we seen reference to Robert Bissaker. His slide rule seems to antedate - the “Whites rule” mentioned above. [This foot-note was added on October - 15, 1919.] - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - - -Note.—The University of California Publications are offered in exchange for -the publication of learned societies and institutions, universities and -libraries. Complete lists of all the publications of the University will be -sent upon request. For sample copies, lists of publications or other -information, address the Manager of the University of California Press, -Berkeley, California, U. 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Current, Volume XXII. - - - - - Transcriber’s Notes - - ---Inserted subchapter headings from the Table of Contents into the - text. - ---Corrected two page numbers in the Table of Contents - ---Corrected a Greek quotation in footnote 6 by reference to the - original sources. - - - - - - - -End of the Project Gutenberg EBook of On the History of Gunter's Scale and -the Slide Rule during the Seventeent, by Florian Cajori - -*** END OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - -***** This file should be named 42216-0.txt or 42216-0.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/4/2/2/1/42216/ - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - -Updated editions will replace the previous one--the old editions -will be renamed. - -Creating the works from public domain print editions means that no -one owns a United States copyright in these works, so the Foundation -(and you!) can copy and distribute it in the United States without -permission and without paying copyright royalties. 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You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org - - -Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century - -Author: Florian Cajori - -Release Date: February 26, 2013 [EBook #42216] - -Language: English - -Character set encoding: ISO-8859-1 - -*** START OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - - - - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - IN - MATHEMATICS - - Vol. 1, No. 9, pp. 187-209 February 17, 1920 - - - - - ON THE HISTORY OF GUNTER'S SCALE AND - THE SLIDE RULE DURING THE - SEVENTEENTH CENTURY - - - BY - FLORIAN CAJORI - - - UNIVERSITY OF CALIFORNIA PRESS - BERKELEY - - - - - TABLE OF CONTENTS - - - PAGE - I. Introduction 187 - II. Innovations in Gunter's Scale 188 - Changes introduced by Edmund Wingate 188 - Changes introduced by Milbourn 189 - Changes introduced by Thomas Brown and John Brown 190 - Changes introduced by William Leybourn 192 - III. Richard Delamain's "Grammelogia" 192 - Different editions or impressions 194 - Description of Delamain's instrument of 1630 195 - Delamain's later designs, and directions for using his - instruments 197 - IV. Controversy between Oughtred and Delamain on the invention of - the circular slide rule 199 - V. Independence and priority of invention 203 - VI. Oughtred's "Gauging Line," 1633 206 - VII. Other seventeenth century slide rules 207 - - - - - I. INTRODUCTION - - - In my history of the slide rule[1], and my article on its invention[2] it -is shewn that William Oughtred and not Edmund Wingate is the inventor, that -Oughtred's circular rule was described in print in 1632, his rectilinear -rule in 1633. Richard Delamain is referred to as having tried to -appropriate the invention to himself[3] and as having written a scurrilous -pamphlet against Oughtred. All our information about Delamain was taken -from De Morgan,[4] who, however, gives no evidence of having read any of -Delamain's writings on the slide rule. Through Dr. Arthur Hutchinson of -Pembroke College, Cambridge, I learned that Delamain's writings on the -slide rule were available. In this article will be given: First, some -details of the changes introduced during the seventeenth century in the -design of Gunter's scale by Edmund Wingate, Milbourn, Thomas Brown, John -Brown and William Leybourn; second, an account of Delamain's book of 1630 -on the slide rule which antedates Oughtred's first publication (though -Oughtred's date of invention is earlier than the date of Delamain's alleged -invention) and of Delamain's later designs of slide rules; third, an -account of the controversy between Delamain and Oughtred; fourth, an -account of a later book on the slide rule written by William Oughtred, and -of other seventeenth century books on the slide rule. - - - - - II. INNOVATIONS IN GUNTER'S SCALE - - - Changes introduced by Wingate - - We begin with Anthony Wood's account of Wingate's introduction of -Gunter's scale into France.[5] - - In 1624 he transported into France the rule of proportion, having a - little before been invented by Edm. Gunter of Gresham Coll. and - communicated it to most of the chiefest mathematicians then residing in - Paris: who apprehending the great benefit that might accrue thereby, - importun'd him to express the use thereof in the French tongue. Which - being performed accordingly, he was advised by monsieur Alleawne the - King's chief engineer to dedicate his book to monsieur the King's only - brother, since duke of Orleans. Nevertheless the said work coming forth - as an abortive (the publishing thereof being somewhat hastened, by reason - an advocate of Dijon in Burgundy began to print some uses thereof, which - Wingate had in a friendly way communicated to him) especially in regard - Gunter himself had learnedly explained its use in a far larger volume.[6] - -Gunter's scale, which Wingate calls the "rule of proportion," contained, as -described in the French edition of 1624, four lines: (1) A single line of -numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot -in length, divided into 12 inches and tenths of inches, also a line, one -foot in length, divided into tenths and hundredths. - -The English editions of this book which appeared in 1623 and 1628 are -devoid of interest. The editions of 1645 and 1658 contain an important -innovation.[7] In the preface the reasons why this instrument has not been -used more are stated to be: (1) the difficulty of drawing the lines with -exactness, (2) the trouble of working thereupon by reason (sometimes) of -too large an extent of the compasses, (3) the fact that the instrument is -not readily portable. The drawing of Wingate's arrangement of the scale in -the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains -five parallel lines, about 66 cm. long, each having the divisions of one -line marked on one side and of another line on the other side. Thus each -line carries two graduations: (1) A single logarithmic line of numbers; (2) -a logarithmic line of numbers thrice repeated; (3) the first scale -repeated, but beginning with the graduations which are near the middle of -the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3; -(4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of -tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000 -equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of -inches; (10) a scale consisting of three kinds, viz., a gauge line, a line -of chords, and a foot measure, divided into 1000 equal parts. - -Important are the first and second scales, by which cube root extraction -was possible "by inspection only, without the aid of pen or compass;" -similarly the third and fourth scales, for square roots. This innovation is -due to Wingate. The 1645 edition announces that the instrument was made in -brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in -Hosier Lane. - - - Changes introduced by Milbourn - -William Leybourn, in his The Line of Proportion or Numbers, Commonly called -Gunter's Line, Made Easie, London, 1673, says in his preface "To the -Reader:" - - The Line of Proportion or Numbers, commonly called (by Artificers) - Gunter's Line, hath been discoursed of by several persons, and variously - applied to divers uses; for when Mr. Gunter had brought it from the - Tables to a Line, and written some Uses thereof, Mr. Wingate added divers - Lines of several lengths, thereby to extract the Square or Cube Roots, - without doubling or trebling the distance of the Compasses: After him Mr. - Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral - Line, thereby enlarging the divisions of the Line. - -On pages 127 and 128 Leybourn adds: - - Again, One T. Browne, a Maker of Mathematical Instruments, made it in a - Serpentine or Spiral Line, composed of divers Concentrick Circles, - thereby to enlarg the divisions, which was the contrivance of one Mr. - Milburn a Yorkshire Gentleman, who writ thereof, and communicated his - Uses to the aforesaid Brown, who (since his death) attributed it to - himself: But whoever was the contriver of it, it is not without - inconvenience; for it can in no wise be made portable; and besides - (instead of compasses) an opening Joynt with thirds [threads] must be - placed to move upon the Centre of the Instrument, without which no - proportion can be wrought. - -This Mr. Milburn is probably the person named in the diary of the -antiquarian, Elias Ashmole, on August 13 [1646?]; "I bought of Mr. Milbourn -all his Books and Mathematical Instruments."[8] Charles Hutton[9] says that -Milburne of Yorkshire designed the spiral form about 1650. This date is -doubtless wrong, for Thomas Browne who, according to Leybourn, got the -spiral form of line from Milbourn, is repeatedly mentioned by William -Oughtred in his Epistle[10] printed some time in 1632 or 1633. Oughtred -does not mention Milbourn, and says (page 4) that the spiral form "was -first hit upon by one Thomas Browne a Joyner, . . . the serpentine -revolution being but two true semicircles described on severall -centers."[11] - - - Changes introduced by Thomas Brown and John Brown - -Thomas Brown did not publish any description of his instrument, but his -son, John Brown, published in 1661 a small book,[12] in which he says -(preface) that he had done "as Mr. Oughtred with Gunter's Rule, to a -sliding and circular form; and as my father Thomas Brown into a Serpentine -form; or as Mr. Windgate in his Rule of Proportion." He says also that -"this brief touch of the Serpentine-line I made bold to assert, to see if I -could draw out a performance of that promise, that hath been so long -unperformed by the promisers thereof." Accordingly in Chapter XX he gives a -description of the serpentine line, "contrived in five (or rather 15) -turn." Whether this description, printed in 1661, exactly fits the -instrument as it was developed in 1632, we have no means of knowing. John -Brown says: - - 1. First next the center is two circles divided one into 60, the other - into 100 parts, for the reducing of minutes to 100 parts, and the - contrary. - - 2. You have in seven turnes two inpricks, and five in divisions, the - first Radius of the sines (or Tangents being neer the matter, alike to - the first three degrees,) ending at 5 degrees and 44 minutes. - - 3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in - three margents in divisions, and the line of versed sines in pricks, - under the line of Tangents, according to Mr. Gunter's cross-staff: the - sines and Tangents beginning at 5 degrees, and 44 minutes where the other - ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the - line of numbers beginning at 10, and proceeding to 100, being one entire - Radius, and graduated into as many divisions as the largeness of the - instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to - 100 into 20 parts in one unit of increase, but the Tangents are divided - into single minutes from the beginning to the end, both in the first, - second and third Radiusses, and the sines into minutes; also from 30 - minutes to 40 degrees, and from 40 to 60, into every two minutes, and - from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the - rest as many as can be well discovered. - - The versed sines are set after the manner of Mr. Gunter's Cross-staff, - and divided into every 10th minutes beginning at 0, and proceeding to 156 - going backwards under the line of Tangents. - - 4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is - the secants to 51 degrees, being nothing else but the sines reitterated - beyond 90. - - 5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85 - degrees, whereby all trouble of backward working is avoided. - - 6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and - also a line of natural sines, for finding of differences in sines, for - finding hour and Azimuth. - - 7. Seventhly, next the verge or outermost edge is a line of equal parts - to get the Logarithm of any number, or the Logarithm sine and Tangent of - any ark or angle to four figures besides the carracteristick. - - 8. Eightly and lastly, in the space place between the ending of the - middle five turnes, and one half of the circle are three prickt lines - fitted for reduction. The uppermost being for shillings, pence and - farthings. The next for pounds, and ounces, and quarters of small - Averdupoies weight. The last for pounds, shillings and pence, and to be - used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay - the hair or edge of one of the legs of the index on 16. 3 1/2 in the line - of 1. s. d. and the hair shall cut on the equal parts 81 16; and the - contrary, if you have a decimal fraction, and would reduce it to a proper - fraction, the like may you do for shillings, and pence, and pounds, and - ounces. - - The uses of the lines follow. - - As to the use of these lines, I shall in this place say but little, and - that for two reasons. First, because this instrument is so contrived, - that the use is sooner learned then any other, I speak as to the manner, - and way of using it, because by means of first second and third - radiusses, in sines and Tangents, the work is always right on, one way or - other, according to the Canon whatsoever it be, in any book that treats - of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in - Oughtred from page 64 to 107. - - Secondly, and more especially, because the more accurate, and large - handling thereof is more then promised, if not already performed by more - abler pens, and a large manuscript thereof by my Sires meanes, provided - many years ago, though to this day not extant in print; so for his sake I - claiming my interest therein, make bold to present you with these few - lines, in order to the use of them: And first note, - - 1. Which soever of the two legs is set to the first term in the question, - that I call the first leg always, and the other being set to the second - term, I call the second leg . . . - -The exact nature of the contrivance with the "two legs" is not described, -but it was probably a flat pair of compasses, attached to the metallic -surface on which the serpentine line was drawn. In that case the instrument -was a slide rule, rather than a form of Gunter's line. In his publication -of 1661, as also in later publications,[13] John Brown devoted more space -to Gunter's scales, requiring the use of a separate pair of compasses, than -to slide rules. - - - Changes introduced by William Leybourn - -The same remark applies to William Leybourn who, after speaking of Seth -Partridge's slide rule, returns to forms of Gunter's scale, saying:[14] - - There is yet another way of disposing of this Line of Proportion, by - having one Line of the full length of the Ruler, and another Line of the - same Radius broken in two parts between 3 and 4; so that in working your - Compasses never go off of the Line: This is one of the best contrivances, - but here Compasses must be used. These are all the Contrivances that I - have hitherto seen of these Lines: That which I here speak of, and will - shew how to use, is only two Lines of one and the same Radius, being set - upon a plain Ruler of any length (the larger the better) having the - beginning of one Line, at the end of the other, the divisions of each - Line being set so close together, that if you find any number upon one of - the Lines, you may easily see what number stands against it on the other - Line. This is all the Variation. . . . - - Example 1. If a Board be 1 Foot 64 parts broad, how much in length of - that Board will make a Foot Square? Look upon one of your Lines (it - matters not which) for 1 Foot 64 parts, and right against it on the other - Line you shall find 61; and so many parts of a Foot will make a Foot - square of that Board. - -This contrivance solves the equation 1.64x=1, yielding centesimal parts of -a foot. - -James Atkinson[15] speaks of "Gunter's scale" as "usually of Boxwood . . . -commonly 2 ft. long, 1 1/2 inch broad" and "of two kinds: long Gunter or -single Gunter, and the sliding Gunter. It appears that during the -seventeenth century (and long after) the Gunter's scale was a rival of the -slide rule. - - - - - III. RICHARD DELAMAIN'S GRAMMELOGIA - - -We begin with a brief statement of the relations between Oughtred and -Delamain. At one time Delamain, a teacher of mathematics in London, was -assisted by Oughtred in his mathematical studies. In 1630 Delamain -published the Grammelogia, a pamphlet describing a circular slide rule and -its use. In 1631 he published another tract, on the Horizontall -Quadrant.[16] In 1632 appeared Oughtred's Circles of Proportion[17] -translated into English from Oughtred's Latin manuscript by another pupil, -William Forster, in the preface of which Forster makes the charge (without -naming Delamain) that "another . . . went about to pre-ocupate" the new -invention. This led to verbal disputes and to the publication by Delamain -of several additions to the Grammelogia, describing further designs of -circular slide rules and also stating his side of the bitter controversy, -but without giving the name of his antagonist. Oughtred's Epistle was -published as a reply. Each combatant accuses the other of stealing the -invention of the circular slide rule and the horizontal quadrant. - -The two title-pages of the edition of the Grammelogia in the British Museum - in London which we have called "Grammelogia IV." - - - Different editions or impressions - -There are at least five different editions, or impressions, of the -Grammelogia which we designate, for convenience, as follows: - - Grammelogia I, 1630. One copy in the Cambridge University Library.[18] - - Grammelogia II, I have not seen a copy of this. - - Grammelogia III, One copy in the Cambridge University Library.[19] - - Grammelogia IV, One copy in the British Museum, another in the Bodleian - Library, Oxford.[20] - - Grammelogia V, One copy in the British Museum. - -In Grammelogia I the first three leaves and the last leaf are without -pagination. The first leaf contains the title-page; the second leaf, the -dedication to the King and the preface "To the Reader;" the third leaf, the -description of the Mathematical Ring. Then follow 22 numbered pages. -Counting the unnumbered pages, there are altogether 30 pages in the -pamphlet. Only the first three leaves of this pamphlet are omitted in -Grammelogia IV and V. - -In Grammelogia III the Appendix begins with a page numbered 52 and bears -the heading "Conclusion;" it ends with page 68, which contains the same two -poems on the mathematical ring that are given on the last page of -Grammelogia I but differs slightly in the spelling of some of the words. -The 51 pages which must originally have preceded page 52, we have not seen. -The edition containing these we have designated Grammelogia II. The reason -for the omission of these 51 pages can only be conjectured. In Oughtred's -Epistle (p. 24), it is stated that Delamain had given a copy of the -Grammelogia to Thomas Brown, and that two days later Delamain asked for the -return of the copy, "because he had found some things to be altered -therein" and "rent out all the middle part." Delamain labored "to recall -all the bookes he had given forth, (which were many) before the sight of -Brownes Lines." These spiral lines Oughtred claimed that Delamain had -stolen from Brown. The title-page and page 52 are the only parts of the -Appendix, as given in Grammelogia III, that are missing in the Grammelogia -IV and V. - -Grammelogia IV answers fully to the description of Delamain's pamphlet -contained in Oughtred's Epistle. It was brought out in 1632 or 1633, for -what appears to be the latest part of it contains a reference (page 99) to -the Grammelogia I (1630) as "being now more then two yeares past." -Moreover, it refers to Oughtred's Circles of Proportion, 1632, and -Oughtred's reply in the Epistle was bound in the Circles of Proportion -having the Addition of 1633. For convenience of reference we number the two -title-pages of Grammelogia IV, "page (1)" and "page (2)," as is done by -Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page -numbers which we assign will be placed in parentheses, to distinguish them -from the page numbers which are printed in Grammelogia IV. The pages -(44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the -same as the pages 53-68. Thus only thirty-eight pages have page numbers -printed on them. The pages (67) and (83) are identical in wording, except -for some printer's errors; they contain verses in praise of the Ring, and -have near the bottom the word "Finis." Also, pages (22) and (23) are -together identical in wording with page (113), which is set up in finer -type, containing an advertisement of a part of Grammelogia IV explaining -the mode of graduating the circular rules. There are altogether six parts -of Grammelogia IV which begin or end by an address to the reader, thus: "To -the Reader," "Courteous Reader," or "To the courteous and benevolent Reader -. . .," namely the pages (8), (22), (68), (89), (90), (108). In his Epistle -(page 2), Oughtred characterizes the make up of the book in the following -terms: - - In reading it . . . I met with such a patchery and confusion of - disjoynted stuffe, that I was striken with a new wonder, that any man - should be so simple, as to shame himselfe to the world with such a - hotch-potch. - -Grammelogia V differs from Grammelogia IV in having only the second -title-page. The first title-page may have been torn off from the copy I -have seen. A second difference is that the page with the printed numeral 22 -in Grammelogia IV has after the word "Finis" the following notice: - - This instrument is made in Silver, or Brasse for the Pocket, or at any - other bignesse, over against Saint Clements Church without Temple Barre, - by Elias Allen. - -This notice occurs also on page 22 of Grammelogia I and III, but is omitted -from page 22 of Grammelogia V. - - - Description of Delamain's instrument of 1630 - -In his address to King Charles I, in his Grammelogia I, Delamain emphasizes -the ease of operating with his slide rule by stating that it is "fit for -use . . . as well on Horse backe as on Foot." Speaking "To the Reader," he -states that he has "for many yeares taught the Mathematicks in this Towne," -and made efforts to improve Gunter's scale "by some Motion, so that the -whole body of Logarithmes might move proportionally the one to the other, -as occasion required. This conceit in February last [1629] I struke upon, -and so composed my Grammelogia or Mathematicall Ring; by which only with an -ocular inspection, there is had at one instant all proportionalls through -the said body of Numbers." He dates his preface "first of January, 1630." -The fifth and sixth pages contain his "Description of the Grammelogia," the -term Grammelogia being applied to the instrument, as well as to the book. -His description is as follows: - - The parts of the Instrument are two Circles, the one moveable, and the - other fixed; The moveable is that unto which is fastened a small pin to - move it by; the other Circle may be conceived to be fixed; The - circumference of the moveable Circle is divided into unequall parts, - charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures - doe represent themselves, or such numbers unto which a Cipher or Ciphers - are added, and are varied as the occasion falls out in the speech of - Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or - 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c. - -After elaborating this last point and explaining the decimal subdivisions -on the scales of the movable circle, he says that "the numbers and -divisions on the fixed Circle, are the very same that the moveable are, . -." There is no drawing of the slide rule in this publication. The -twenty-two numbered pages give explanations of the various uses to which -the instrument can be put: "How to performe the Golden Rule" (pp. 1-3), -"Further uses of the Golden Rule" (pp. 4-6), "Notions or Principles -touching the disposing or ordering of the Numbers in the Golden Rule in -their true places upon the Grammelogia" (pp. 7-11), "How to divide one -number by another" (pp. 12, 13), "to multiply one Number by another" (pp. -14, 15), "To find Numbers in continuall proportion" (pp. 16, 17), "How to -extract the Square Root," "How to extract the Cubicke Root" (pp. 18-21), -"How to performe the Golden Rule" (the rule of proportion) is explained -thus: - - Seeke the first number in the moveable, and bring it to the second number - in the fixed, so right against the third number in the moveable, is the - answer in the fixed. - - If the Interest of 100. li. be 8. li. in the yeare, what is the Interest - of 65. li. for the same time. - - Bring 100. in the moveable to 8. in the fixed, so right against 65. in - the moveable is 5.2. in the fixed, and so much is the Interest of 65. li. - for the yeare at 8. li. for 100. li. per annum. - - The Instrument not removed, you may at one instant right against any - summe of money in the moveable, see the Interest thereof in the fixed: - the reason of this is from the Definition of Logarithmes. - -These are the earliest known printed instructions on the use of a slide -rule. It will be noticed that the description of the instrument at the -opening makes no references to logarithmic lines for the trigonometric -functions; only the line of numbers is given. Yet the title-page promised -the "resolution of Plaine and Sphericall Triangles." Page 22 throws light -upon this matter: - - If there be composed three Circles of equal thicknesse, A.B.C. so that - the inner edge of D [should be B] and the outward edge of A bee - answerably graduated with Logarithmall signes [sines], and the outward - edge of B and the inner edge of A with Logarithmes; and then on the - backside be graduated the Logarithmall Tangents, and againe the - Logarithmall signes oppositly to the former graduations, it shall be - fitted for the resolution of Plaine and Sphericall Triangles. - -After twelve lines of further remarks on this point he adds: - - Hence from the forme, I have called it a Ring, and Grammelogia by - annoligie of a Lineary speech; which Ring, if it were projected in the - convex unto two yards Diameter, or thereabouts, and the line Decupled, it - would worke Trigonometrie unto seconds, and give proportionall numbers - unto six places only by an ocular inspection, which would compendiate - Astronomicall calculations, and be sufficient for the Prosthaphaeresis of - the Motions: But of this as God shall give life and ability to health and - time. - -The unnumbered page following page 22 contains the patent and copyright on -the instrument and book: - - Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs - an Instrument called Grammelogia, or The Mathematicall Ring, together - with a Booke so intituled, expressing the use thereof, being his owne - Invention; we of our Gracious and Princely favour have granted unto the - said Richard Delamain and his Assignes, Privilege, Licence, and - Authority, for the sole Making, Printing and Selling of the said - Instrument and Booke: straightly forbidding any other to Make, Imprint, - or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument - or Booke within any our Dominions, during the space of ten yeares next - ensuing the date hereof, upon paine of Our high displeasure. Given under - our hand and Signet at our Palace of Westminster, the fourth day of - January, in the sixth yeare of our Raigne. - - - Delamain's later designs, and directions for using his instruments - -In the Appendix of Grammelogia III, on page 52 is given a description of an -instrument promised near the end of Grammelogia I: - - That which I have formerly delivered hath been onely upon one of the - Circles of my Ring, simply concerning Arithmeticall Proportions, I will - by way of Conclusion touch upon some uses of the Circles, of Logarithmall - Sines, and Tangents, which are placed on the edge of both the moveable - and fixed Circles of the Ring in respect of Geometricall Proportions, but - first of the description of these Circles. - - First, upon the side that the Circle of Numbers is one, are graduated on - the edge of the moveable, and also on the edge of the fixed the - Logarithmall Sines, for if you bring 1. in the moveable amongst the - Numbers to 1. in the fixed, you may on the other edge of the moveable and - fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto - 6.6. and each degree subdivided, and then over the former divisions and - figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5. - 4. 3. 2. 1. each of those divided by small points. - - Secondly, (if the Ring is great) neere the outward edge of this side of - the fixed against the Numbers, are the usuall divisions of a Circle, and - the points of the Compasse: serving for observation in Astronomy, or - Geometry, and the sights belonging to those divisions, may be placed on - the moveable Circle. - - Thirdly, opposite to those Sines on the other side are the Logarithmall - Tangents, noted alike both in the moveable and fixed thus - 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or - divisions serve also for their Complements to 90. so 40 gr. stands for - 50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both - in the moveable and fixed is also divided into parts. As for the degrees - which are under 6. viz. 5.4.3.2.1. they are noted with small figures over - this divided Circle from 45.40.35.30.25. &c. and each of those degrees - divided into parts by small points both in the moveable and fixed. - - Fourthly, on the other edge of the moveable on the same side is another - graduation of Tangents, like that formerly described. And opposite unto - it, in the fixed is a Graduation of Logarithmall sines in every thing - answerable to the first descrition of Sines on the other side. - - Fifthly, on the edge of the Ring is graduated a parte of the quator, - numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees - of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree - both of the quator, and Meridian are subdivided into parts; these two - graduated Circles serve to resolve such Questions which concerne - Latitude, Longitude, Rumb, and Distance, in Nauticall operations. - - Sixthly, to the concave of the Ring may be added a Circle to be elevated - or depressed for any Latitude, representing the quator, and so divided - into houres and parts with an Axis, to shew both the houre, and Azimuth, - and within this Circle may be hanged a Box, and Needle with a Socket for - a staffe to slide into it, and this accommodated with scrue pines to - fasten it to the Ring and staffe, or to take it off at pleasure. - -The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and -V make no reference to the dispute with Oughtred and may, therefore, be -assumed to have been published before the appearance of Oughtred's Circles -of Proportion. On page 53, "To the Reader," he says: - - . . . you may make use of the Projection of the Circles of the Ring upon - a Plaine, having the feet of a paire of compasses (but so that they be - flat) to move on the Center of that Plaine, and those feet to open and - shut as a paire of Compasses . . . now if the feet bee opened to any two - termes or numbers in that Projection, then may you move the first foot to - the third number, and the other foot shall give the Answer; . . . it hath - pleased some to make use of this way. But in this there is a double - labour in respect to that of the Ring, the one in fitting those feet unto - the numbers assigned, and the other by moving them about, in which a man - can hardly accommodate the Instrument with one hand, and expresse the - Proportionals in writing with the other. By the Ring you need not but - bring one number to another, and right against any other number is the - Answer without any such motion. . . . upon that [the Ring] I write, - shewing some uses of those Circles amongst themselves, and conjoyned with - others . . . in Astronomy, Horolographie, in plaine Triangles applyed to - Dimensions, Navigation, Fortification, etc. . . . But before I come to - Construction, I have thought it convenient by way introduction, to - examine the truth of the graduation of those Circles . . . - -These are the words of a practical man, interested in the mechanical -development of his instrument. He considers not only questions of -convenience but also of accuracy. The instrument has, or may have now, also -lines of sines and tangents. To test the accuracy of the circles of -Numbers, "bring any number in the moveable to halfe of that number in the -fixed: so any number or part in the fixed shall give his double in the -moveable, and so may you trie of the thirds, fourths &c. of numbers, vel -contra," (p. 54). On page 55 are given two small drawings, labelled, "A -Type of the Ringe and Scheme of this Logarithmicall projection, the use -followeth. These Instruments are made in Silver or Brasse by John Allen -neare the Sauoy in the Strand." - - - IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE - CIRCULAR SLIDE RULE - -Delamain's publication of 1630 on the 'Mathematicall Ring' does not appear -at that time to have caused a rupture between him and Oughtred. When in -1631 Delamain brought out his Horizontall Quadrant, the invention of which -Delamain was afterwards charged to have stolen from Oughtred, Delamain was -still in close touch with Oughtred and was sending Oughtred in the Arundell -House, London, the sheets as they were printed. Oughtred's reference to -this in his Epistle (p. 20) written after the friendship was broken, is as -follows: - - While he was printing his tractate of the Horizontall quadrant, although - he could not but know that it was injurious to me in respect of my free - gift to Master Allen, and of William Forster, whose translation of my - rules was then about to come forth: yet such was my good nature, and his - shamelessnesse, that every day, as any sheet was printed, hee sent, or - brought the same to mee at my chamber in Arundell house to peruse which I - lovingly and ingenuously did, and gave him my judgment of it. - -Even after Forster's publication of Oughtred's Circles of Proportion, 1632, -Oughtred had a book, A canon of Sines Tangents and Secants, which he had -borrowed from Delamain and was then returning (Epistle, page (5)). The -attacks which Forster, in the preface to the Circles of Proportion, made -upon Delamain (though not naming Delamain) started the quarrel. Except for -Forster and other pupils of Oughtred who urged him on to castigate -Delamain, the controversy might never have arisen. Forster expressed -himself in part as follows: - - . . . being in the time of the long vacation 1630, in the Country, at the - house of the Reverend, and my most worthy friend, and Teacher, Mr. - William Oughtred (to whose instruction I owe both my initiation, and - whole progresse in these Sciences.) I vpon occasion of speech told him of - a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made - (such as it vsually called Mr. Gunter's Ruler) 6 feet long, to be vsed - with a payre of beame-compasses. "He answered that was a poore invention, - and the performance very troublesome: But, said he, seeing you are taken - with such mechanicall wayes of Instruments, I will shew you what deuises - I have had by mee these many yeares." And first, hee brought to mee two - Rulers of that sort, to be vsed by applying one to the other, without any - compasses: and after that hee shewed mee those lines cast into a circle - or Ring, with another moueable circle vpon it. I seeing the great - expeditenesse of both those wayes; but especially, of the latter, wherein - it farre excelleth any other Instrument which hath bin knowne; told him, - I wondered that hee could so many yeares conceale such vseful inuentions, - not onely from the world, but from my selfe, to whom in other parts and - mysteries of Art, he had bin so liberall. He answered, "That the true way - of Art is not by Instruments, but by Demonstration: and that it is a - preposterous course of vulgar Teachers, to begin with Instruments, and - not with the Sciences, and so in-stead of Artists, to make their - Schollers only doers of tricks, and as it were Iuglers: to the despite of - Art, losse of precious time, and betraying of willing and industrious - wits, vnto ignorance and idlenesse. That the vse of Instruments is indeed - excellent, if a man be an Artist: but contemptible, being set and opposed - to Art. And lastly, that he meant to commend to me, the skill of - Instruments, but first he would haue me well instructed in the Sciences. - He also shewed me many notes, and Rules for the vse of those circles, and - of his Horizontall Instrument, (which he had proiected about 30 yeares - before) the most part written in Latine. All which I obtained of him - leaue to translate into English, and make publique, for the vse, and - benefit of such as were studious, and louers of these excellent Sciences. - - Which thing while I with mature, and diligent care (as my occasions would - give me leaue) went about to doe: another to whom the Author in a louing - confidence discouered this intent, using more hast then good speed, went - about to preocupate; of which vntimely birth, and preuenting (if not - circumuenting) forwardnesse, I say no more: but aduise the studious - Reader, onely so farre to trust, as he shal be sure doth agree to truth & - Art. - -While in this dedication reference is made to a slide rule or "ring" with a -"moveable circle," the instrument actually described in the Circles of -Proportion consists of fixed circles "with an index to be opened after the -manner of a paire of Compasses." Delamain, as we have seen, had decided -preference for the moveable circle. To Oughtred, on the other hand, one -design was about as good as the other; he was more of a theorist and -repeatedly expressed his contempt for mathematical instruments. In his -Epistle (page (25)), he says he had not "the one halfe of my intentions -upon it" (the rule in his book), nor one with a "moveable circle and a -thread, but with an opening Index at the centre (if so be that bee cause -enough to make it to bee not the same, but another Instrument) for my part -I disclaime it: it may go seeke another Master: which for ought I know, -will prove to be Elias Allen himselfe: for at his request only I altered a -little my rules from the use of the moveable circle and the thread, to the -two armes of an Index." - -All parts of Delamain's Grammelogia IV, except pages 1-22 and 53-68 -considered above, were published after the Circles of Proportion, for they -contain references to the ill treatment that Delamain felt or made believe -that he felt, that he had received in the book published by Oughtred and -Forster. Oughtred's reference to teachers whose scholars are "doers of -tricks," "Iuglers," and Forster's allusion to "another to whom the Author -in a loving confidence" explained the instrument and who "went about to -preocupate" it, are repeatedly mentioned. Delamain says, (page (89)) that -at first he did not intend to express himself in print, "but sought peace -and my right by a private and friendly way." Oughtred's account of -Delamain's course is that of an "ill-natured man" with a "virulent tongue," -"sardonical laughter" and "malapert sawsiness." Contrasting Forster and -Delamain, he says that, of the former he "had the very first moulding" and -made him feel that "the way of Art" is "by demonstration." But Delamain was -"already corrupted with doing upon Instruments, and quite lost from ever -being made an Artist." (Epistle page (27)). Repeatedly does Oughtred assert -Delamain's ignorance of mathematics. The two men were evidently of wholly -different intellectual predilections. That Delamain loved instruments is -quite evident, and we proceed to describe his efforts to improve the -circular slide rule. - -The Grammelogia IV is dedicated to King Charles I. Delamain says: - - . . . Everything hath his beginning, and curious Arts seldome come to the - height at the first; It was my promise then to enlarge the invention by a - way of decuplating the Circles, which I now present unto your sacred - Majestie as the quintessence and excellencie there of . . . - -His enlarged circular rules are illustrated in the Bodleian Library copy of -Grammelogia IV by four diagrams, two of them being the two drawings on the -two title-pages at the beginning of the Grammelogia IV, 4 inches in -external diameter, and exhibiting eleven concentric circular lines carrying -graduations of different sorts. In the second of these designs all circles -are fixed. The other two drawings are each 10 3/4 inches in external -diameter and exhibit 18 concentric circular lines; the folded sheet of the -first of these drawings is inserted between pages (23) and (24), the second -folded sheet between pages (83) and (84). All circles of this second -instrument are fixed. Counting in the two small drawings in Grammelogia -III, there are in all six drawings of slide rules in the Bodleian -Grammelogia IV. On pages (24) to (43) Delamain explains the graduation of -slide rules. He takes first a rule which has one circle of equal parts, -divided into 1000 equal divisions. From a table of logarithms he gets log 2 -= 0.301; from the number 301 in the circle of equal parts he draws a line -to the center of the circle and marks the intersection with the circles of -numbers by the figure 2. Thus he proceeds with log 3, log 4, and so on; -also with log sin x and log tan x. For log sin x he uses two circles, the -first (see page (27)) for angles from 34' 24" to 5^o 44' 22", the second -circle from 5^o 44' 22" to 90^o. The drawings do not show the seconds. He -suggests many different designs of rules. On page (29) he says: - - For the single projection of the Circles of my Ring, and the dividing and - graduating of them: which may bee so inserted upon the edges of Circles - of mettle turned in the forme of a Ring, so that one Circle may moove - betweene two fixed, by helpe of two stayes, then may there be graduated - on the face of the Ring, upon the outer edge of the mooveable and inner - edge of the fixed, the Circle of Numbers, then upon the inner edge of - that mooveable Circle, and the outward edge of that inner fixed Circle - may be inserted the Circle of Sines, and so according to the description - of those that are usually made. - -In addition to these lines he proceeds to mention the circle giving the -ordinary division into degrees and minutes, and two circles of tangents on -the other side of the rule. - -Next Delamain explains an arrangement of all the graduation on one side of -the rule by means of "a small channell in the innermost fixed Circle, in -which may be placed a small single Index, which may have sufficient length -to reach from the innermost edge of the Mooveable Circle, unto the outmost -edge of the fixed Circle, which may be mooved to and fro at pleasure, in -the channell, which Index may serve to shew the opposition of Numbers" (p. -(31)). From this it is clear that the invention of the "runner" goes back -to the very first writers on the slide rule. - -After describing a modification of the above arrangement, he adds, "many -other formes might be deliverd, about this single projection" (p. (32)). - -Proceeding to the "enlarging" of the circles in the Ring, to, say, the -"Quadruple to that which is single, that is, foure times greater," the -"equall parts" are distributed over four circles instead of only one -circle, but the general method of graduation is the same as before (p. -(33)); there being now four circles carrying the logarithms of numbers, and -so on. Next he points out "severall wayes how the Circles of the -Mathematicall Ring (being inlarged) may be accommodated for practicall -use:" (1) The Circles are all fixed in a plain and movable flat compasses -(or better, a movable semicircle) are used for fixing any two positions; -(2) There is a "double projection" of each logarithmic line "inlarged on a -Plaine," one fixed, the other movable, as shown in his first figure on the -title-page, a single index only being used; (3) use of "my great Cylinder -which I have long proposed (in which all the Circles are of equall -greatnesse,) and it may be made of any magnitude or capacity, but for a -study (hee that will be at the charge) it may be of a yard diameter and of -such an indifferent length that it may containe 100 or more Circles fixed -parallel one to the other on the Cylinder, having a space betweene each of -them, so that there may bee as many mooveable Circles, as there are fixed -ones, and these of the mooveable linked, or fastened together, so that they -may all moove together by the fixed ones in these spaces, whose edges both -of the fixed, and mooveable being graduated by helpe of a single Index will -shew the proportionalls by opposition in this double Projection, or by a -double Index in a single Projection" (p. (36)). - -Next follows the detailed description of his Ring "on a Plaine, according -to the diagramme that was given the King (for a view of that projection) -and afterwards the Ring it selve." The diagram is the large one which we -mentioned as inserted between pages (23) and (24). The instrument has two -circles, one moveable, upon each of which are described 13 distinct -circular graduations. The lines on the fixed circle are: "The Circle of -degrees and calendar," E. "Circle of equall parts, and part of the Equator, -and Meridian," TT. "The Circle of Tangents," S. "The Circle of Sines," D. -"The Circle of Decimals," N. "The Circle of Numbers." The lines on the -movable circle are: N. "The Circle of Numbers," E. "The Circle of equated -figures, and bodies," S. "The Circle of Sines," TT. "The Circle of -Tangents," Y. "The Circle of time, yeares, and monethes." - -On pages (84)-(88) Delamain explains an enlargement of his Ring for -computations involving the sines of angles near to 90^o. On page (86) he -says: - - I have continued the Sines of the Projection unto two severall - revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr. - (being the last revolution of the decuplation of the former, or the - hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s. - and ends at 90.gr. (being the last of a ternary of decuplated - revolutions, or the thousand part of that Projection) and may bee thus - used. - -He explains the manner of using these extra graduations. Thus he claims to -have attained degrees of accuracy which enabled him to do what "some one" -had declared "could not bee done." It is hardly necessary to point out that -Delamain's Grammelogia IV suggests designs of slide rules which inventors -two hundred or more years later were endeavouring to produce. Which of -Delamain's designs of rules were actually made and used, he does not state -explicitly. He refers to a rule 18 inches in diameter as if it had been -actually constructed (pages (86), (88)). Oughtred showed no appreciation of -such study in designing and ridiculed Delamain's efforts, in his Epistle. - -Additional elucidations of his designs of rules, along with explanations of -the relations of his work to that of Gunter and Napier, and sallies -directed against Oughtred and Forster, are contained on pages (8)-(21) of -his Grammelogia IV. - - - - - V. INDEPENDENCE AND PRIORITY OF INVENTION - - -The question of independence and priority of invention is discussed by -Delamain more specifically on pages (89)-(113); Oughtred devotes his entire -Epistle to it. It is difficult to determine definitely which publication is -the later, Delamain's Grammelogia IV or Oughtred's Epistle. Each seems to -quote from the other. Probably the explanation is that the two publications -contain arguments which were previously passed from one antagonist to the -other by word of mouth or by private letter. Oughtred refers in his Epistle -(p. (12)) to a letter from Delamain. We believe that the Epistle came after -Delamain's Grammelogia IV. Delamain claims for himself the invention of the -circular slide rule. He says in his Grammelogia IV. (p. (99)), "when I had -a sight of it, which was in February, 1629 (as I specified in my Epistle) I -could not conceale it longer, envying my selfe, that others did not tast of -that which I found to carry with it so delightfull and pleasant a goate -[taste] . . ." Delamain asserts (without proof) that Oughtred "never saw it -as he now challengeth it to be his invention, untill it was so fitted to -his hand, and that he made all his practise on it after the publishing of -my Booke upon my Ring, and not before; so it was easie for him or some -other to write some uses of it in Latin after Christmas, 1630 and not the -Sommer before, as is falsely alledged by some one . . ." (p. (91)). -Delamain's accusation of theft on the part of Oughtred cannot be seriously -considered. Oughtred's reputation as a mathematician and his standing in -his community go against such a supposition. Moreover, William Forster is a -witness for Oughtred. The fact that Oughtred had the mastery of the -rectilinear slide rule as well, while Delamain in 1630 speaks only of the -circular rule, weighs in Oughtred's favour. - -Oughtred says he invented the slide rule "above twelve yeares agoe," that -is, about 1621, and "I with mine owne hand made me two such Circles, which -I have used ever since, as my occasions required," (Epistle p. (22)). On -the same page, he describes his mode of discovery thus: - - I found that it required many times too great a paire of Compasses [in - using Gunter's line], which would bee hard to open, apt to slip, and - troublesome for use. I therefore first devised to have another Ruler with - the former: and so by setting and applying one to the other, I did not - onely take away the use of Compasses, but also make the worke much more - easy and expedite: when I should not at all need the motion of my hand, - but onely the glancing at my sight: and with one position of the Rulers, - and view of mine eye, see not one onely, but the manifold proportions - incident unto the question intended. But yet this facility also wanted - not some difficulty especially in the line of tangents, when one arch was - in the former mediety of the quadrant, and the other in the latter: for - in this case it was needful that either one Ruler must bee as long againe - as the other; or else that I must use an inversion of the Ruler, and - regression. By this consideration I first of all saw that if those lines - upon both Rulers were inflected into two circles, that of the tangents - being in both doubled, and that those two Circles should move one upon - another; they with a small thread in the center to direct the sight, - would bee sufficient with incredible and wonderfull facility to worke all - questions of Trigonometry . . . - -Oughtred said that he had no desire to publish his invention, but in the -vacation of 1630 finally promised William Forster to let him bring out a -translation. Oughtred claims that Delamain got the invention from him at -Alhallontide [November 1], 1630, when they met in London. The accounts of -that meeting we proceed to give in double column. - - Delamain's Statement - Grammelogia IV, page (98) - - ". . . about Alhalontide 1630. (as our Authors reporteth) was the time he - was circumvented, and then his intent in a loving manner (as before) he - opened unto me, which particularly I will dismantle in the very naked - truth: for, wee being walking together some few weekes before Christmas, - upon Fishstreet hill, we discoursed upon sundry things Mathematicall, - both Theoreticall and Practicall, and of the excellent inventions and - helpes that in these dayes were produced, amongst which I was not a - little taken with that of the Logarythmes, commending greatly the - ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, - in the lines of proportion, extracted from these Logarythmes for ordinary - Practicall uses; He replyed unto me (in these very words) What will yov - say to an Invention that I have, which in a lesse extent of the Compasses - shall worke truer then that of Mr. Gunters Ruler, I asked him then of - what forme it was, he answered with some pause (which no doubt argued his - suspition of mee that I might conceive it) that it was Arching-wise, but - now hee sayes that hee told mee then, it was Circular (but were I put to - my oath to avoid the guilt of Conscience I would conclude in the former.) - At which immediately I answered, I had the like my selfe, and so we - discoursed not a word more touching that subject . . . Then after my - coming home I sent him a sight of my Projection drawne in Pastboard: Now - admit I had not the Invention of my Ring before I discoursed . . . it was - not so facil for mee . . . to raise and compose so complete, and absolute - an Instrument from so small a principle, or glimpse of light . . ." - - Oughtred's Statement - Epistle, page (23) - - "Shortly after my gift to Elias Allen, I chanced to meet with Richard - Delamain in the street (it was at Alhallontide) and as we walked together - I told him what an Instrument I had given to Master Allen, both of the - Logarithmes projected into circles, which being lesse then one foot - diameter would performe as much as one of Master Gunters Rulers of sixe - feet long: and also of the Prostaphaereses of the Plannets and second - motions. Such an invention have I said he: for now his intentions (that - is his ambition) beganne to worke: . . . But he saith, Then after my - comming home I sent him a sight of my projection drawne in past-board. - See how notoriously he jugleth without an Instrument. Then after: how - long after? a sight of my projection: of how much? More then seven weekes - after on December 23, he sent to mee the line of numbers onely set upon a - circle: . . . and so much onely he presented to his Majesty: but as for - Sine or tangent of his, there was not the least shew of any. Neither - could he give to Master Allen any direction for the composure of the - circles of his Ring, or for the division of them: as upon his oath Master - Allen will testify how hee misled him, and made him labour in vain above - three weeks together, until Master Allen himselfe found out his ignorance - and mistaking, which is more cleare then is possible with any impudence - to be outfaced." - -Oughtred makes a further statement (Epistle, p. (24)) as follows: - - Delamain hearing that Brown with his Serpentine had another line by which - he could worke to minutes in the 90 degree of sines . . . gave the [his] - booke to Browne: who in thankfulnesse could not but gratify Delamain with - his Lines also: and teach him the use of them, but especially of the - great Line: with this caution on both sides, that one should not meddle - with the others invention. Two dayes after Delamain . . . because he had - found some things to be altered therin, . . . asked for the booke . . . - but as soone as he had got it in his hands he rent out all the middle - part with the two Schemes & put them up in his pocket & went his way . . - . and . . . laboureth to recall all the bookes he had given forth . . . - And shortly after this he got a new Printer (who was ignorant of his - former Schemes) to print him new: giving him an especiall charge of the - outermost line newly graven in the Plate, which indeed is Brownes very - line: and then altering his book . . . - -This and other statements made by Oughtred seem damaging to Delamain's -reputation. But it is quite possible that Oughtred's guesses as to -Delamain's motives are wrong. Moreover, some of Oughtred's statements are -not first hand knowledge with him, but mere hearsay. One may accept his -first hand facts and still clear Delamain of wrong doing. There is always -danger that rival claimants of an invention or discovery will proceed on -the assumption that no one else could possibly have come independently upon -the same devices that they themselves did; the history of science proves -the opposite. Seldom is an invention of any note made by only one man. We -do not feel competent to judge Delamain's case. We know too little about -him as a man. We incline to the opinion that the hypothesis of independent -invention is the most plausible. At any rate, Delamain figures in the -history of the slide rule as the publisher of the earliest book thereon and -as an enthusiastic and skillful designer of slide rules. - -The effect of this controversy upon interested friends was probably small. -Doubtless few people read both sides. Oughtred says:[21] "this scandall . . -. hath with them, to whom I am not knowne, wrought me much prejudice and -disadvantage . ." Aubrey,[22] a friend of Oughtred, refers to Delamain "who -was so sawcy to write against him" and remembers having seen "many yeares -since, twenty or more good verses made" against Delamain. Another friend of -Oughtred, William Robinson, who had seen some of Delamain's publications, -but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before -the appearance of the latter's Epistle: - - I cannot but wonder at the indiscretion of Rich. Delamain, who being - conscious to himself that he is but the pickpurse of another man's wit, - would thus inconsiderately provoke and awake a sleeping lion . . . he - hath so weakly (though in my judgment, vaingloriously enough) commended - his own labour . . .[23] - -Delamain presented King Charles I with one of his sun-dials, also with a -manuscript and, later, with a printed copy of his book of 1630. A drawing -of his improved slide rule was sent to the King and the Grammelogia IV is -dedicated to him. The King must have been favorably impressed, for Delamain -was appointed tutor to the King in mathematics. His widow petitioned the -House of Lords in 1645 for relief; he had ten children.[24] - -Anthony Wood states that Charles I, on the day of his execution, commanded -his friend Thomas Herbert "to give his son the duke of York his large -ring-sundial of silver, a jewel his maj. much valued." Anthony Wood adds, -"it was invented and made by Rich. Delamaine a very able mathematician, who -projected it, and in a little printed book did shew its excellent use in -resolving many questions in arithmetic and other rare operations to be -wrought by it in the mathematics."[25] - - - - - VI. OUGHTRED'S GAUGING LINE, 1633 - - -It has not been generally known, hitherto, that Oughtred designed a -rectilinear slide rule for gauging and published a description thereof in -1633.[26] In his Circles of Proportion, chapter IX, Oughtred had offered a -closer approximation than that of Gunter for the capacity of casks. The -Gauger of London expostulated with Oughtred for presuming to question -anything that Gunter had written. The ensuing discussion led to an -invitation extended by the Company of Vintners to the instrument maker -Elias Allen to request Oughtred to design a gauging rod.[27] This he did, -and Allen received an order for "threescore" instruments. On page 19 -Oughtred describes his 'Gauging Rod:' - - It consisteth of two rulers of brasse about 32 ynches of length, which - also are halfe an ynch broad, and a quarter of an ynch thick . . . At one - end of both those rulers are two little sockets of brasse fastened on - strongly: by which the rulers are held together, and made to move one - upon another, and to bee drawne out unto any length, as occasion shall - require: and when you have them at the just length, there is upon one of - the sockets a long Scrue-pin to scrue them fast. - -There are graduations on three sides of the rulers, one graduation being -the logarithmic line of numbers. He says (p. 39), "the maner of computing -the Gauge-divisions I have concealed." W. Robinson, who was a friend of -Oughtred, wrote him as follows:[28] - - I have light upon your little book of artificial gauging, wherewith I am - much taken, but I want the rod, neither could I get a sight of one of - them at the time, because Mr. Allen had none left . . . I forgot to ask - Mr. Allen the price of one of them, which if not much I would have one of - them." Oughtred annotated this passage thus: "Or in wood, if any be made - in wood by Thompson or any other." - -Another of Oughtred's admirers, Sir Charles Cavendish, wrote, on February -11, 1635 thus:[29] - - I thank you for your little book, but especially for the way of - calculating the divisions of your gauging rod. I wish, both for their own - sakes and yours, that the citizens were as capable of the acuteness of - this invention, as they are commonly greedy of gain, and then I doubt not - but they would give you a better recompense than I doubt now they will. - -On April 20, 1638, we find Oughtred giving Elias Allen directions[30] -"about the making of the two rulers." As in 1633,[31] so now, Oughtred -takes one ruler longer than the other. This 1633 instrument was used also -as "a crosse-staffe to take the height of the Sunne, or any Starre above -the Horizon, and also their distances." The longer ruler was called staffe, -the shorter transversarie. While in 1633 he took the lengths of the two in -the ratio "almost 3 to 2," in 1638, he took "the transversary three -quarters of the staff's length, . . . that the divisions may be larger." - - - - - VII. OTHER SEVENTEENTH CENTURY SLIDE RULES - - -In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard, -Henry Coggeshall, W. Hunt and Sir Isaac Newton.[32] Of Partridge's Double -Scale of Proportion, London, I have examined a copy dated 1661, which is -the earliest date for this book that I have seen. As far as we know, 1661 -is the earliest date of publications on the slide rule, since Oughtred and -Delamain. But it would not be surprising if the intervening 28 years were -found not so barren as they seem at present. The 1661 and 1662 impressions -of Partridge are identical, except for the date on the title-page. William -Leybourn, who printed Partridge's book, speaks in high appreciation of it -in his own book.[33] - -In 1661 was published also John Brown's first book, Description and Use of -a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use -of "Mr. Whites rule" for the measuring of board and timber, round and -square. He calls this a "sliding rule." The existence, in 1661, of a -"Whites rule" indicates activities in designing of which we know as yet -very little. In his book of 1761, previously quoted, Brown gives a drawing -of "White's sliding rule" (p. 193); also a special contrivance of his own, -as indicated by him in these words: - - A further improvement of the Triangular Quadrant, as I have made it - several times, with a sliding Cover on the in-side, when made hollow, to - carry Ink, Pens, and Compasses; then on the sliding Cover, and Edges, is - put the Line of Numbers, according to Mr. White's first Contrivance for - manner of operation; but much augmented, and made easie, by John Brown. - -He gives no drawing of his "triangular quadrant," hence his account of it -is unsatisfactory. He explains the use of "gage-points." His placing -logarithmic lines on the edges of instrument boxes was outdone in oddity -later by Everard who placed them on tobacco-boxes.[34] In Brown's -publication of 1704 the White slide rule is given again, "being as neat and -ready a way as ever was used." He tells also of a "glasier's sliding rule." -William Leybourn explains in 1673 how Wingate's double and triple lines for -squaring and cubing, or square and cube root, can be used on slide -rules.[35] - -Beginning early in the history of the slide rule, when Oughtred designed -his "gauging rod," we notice the designing of rules intended for very -special purposes. Another such contrivance, which enjoyed long popularity, -was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677, -a booklet of 35 pages. Coggeshall says in his preface: - - For what can be more ready and easie, then having set twelve to the - length, to see the Content exactly against the Girt or Side of the - Square. Whereas on Mr. Partridge's Scale the Content is the Sixth Number, - which is far more troublesome then [even] with Compasses. - -One line on Coggeshall's rule begins with 4 and extends to 40, these -numbers being the "Girt" (a quarter of the circumference), which in -ordinary practice of measuring round timber lies between 4 inches and 40 -inches. This "Girt line" slides "against the line of Numbers in two -Lengths, to which it is exactly equal." A second edition, 1682, shows some -changes in the rule, as well as an enlargement and change of title of the -book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent, -London, 1682. In this, the description of the rule is given thus: - - There are four Lines on each flat of this Rule; two next the outward - edges, which are Lines of Measure; and two next the inward edges, which - are Lines of Proportion. On one flat, next the inward edges, is the - Square-line [Girt-line in round timber measurement] with the Line of - Numbers his fellow. Next the outward, a Line of Inches divided into - Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 - to 24 on the other. On the other flat, next the inward edges, is the - double Scale of Numbers [for solving proportions]. Next the outward on - one Rule a Line of Inches divided each into ten parts; and this for - gauging, etc. On the other a foot divided into 100 parts. - -Later further changes were introduced in Coggeshall's rule.[36] - -It is worthy of note that Coggeshall's slide rule book, The Art of -Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p. -473; hence Leupold's description[37] of the rectilinear slide rule in his -Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not -the earliest reference to the rectilinear rule found in German -publications. The above date is earlier even than Biler's reference to a -circular slide rule in his Descriptio instrumenti mathematici universalis -of 1696. - -Two noted slide rules for gauging were described by Tho. Everard, -Philomath, in his Stereometry made easie, London, 1684. He designates his -lines by the capital letters A, B, C, D, E. On the first instrument, A on -the rule, and B and C on the slide, have each two radiuses of numbers, D -has only one, while E has three. The second rule is described in an -Appendix; it is one foot long, with two slides enabling the rule to be -extended to 3 feet. - -Everard's instruments were made in London by Isaac Carver who, soon after, -himself wrote a sixteen-page Description and Use of a New Sliding Rule, -projected from the Tables in the Gauger's Magazine, London, 1687, which was -"printed for William Hunt" and bound in one volume with a book by Hunt, -called The Gauger's Magazine, London, 1687. This appears to be the same -William Hunt who later brought out descriptions of his own of slide rules. -The instrument described by Carver "consists of three pieces, two whereof -are moveable to be drawn out till the whole be 36 inches long." It has -several non-logarithmic graduations, together with logarithmic lines marked -A, B, C, D, of which A, B, C are "double lines," and D a "single line" used -for squares and square roots. It is designed for the determination of the -vacuity of a "spheroidal cask lying," a "spheroidal cask standing," and a -"parabolical cask lying." - -Another seventeenth century writer on the slide rule is John Atkinson, whom -we have mentioned earlier. He says:[38] "The Lines of Numbers, Sines and -Tangents, are set double, that is, one on each side, as the middle piece -slides: which middle piece is so contrived, to slip to and fro easily, to -slide out, and to be put in any side uppermost, in order to bring those -Lines together (or against one another) most proper for solving the -Question, wrought by Sliding-Gunter." - -The data presented in this article show that, while the earliest slide -rules were of the circular type, the later slide rules of the seventeenth -century were of the rectilinear type.[39] - - January 12, 1915. - - - - - Footnotes - - -[1]F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments, - New York, 1909, pp. 7-14, also Addenda i-vi. - -[2]F. Cajori, "On the Invention of the Slide Rule," in Colorado College - Publication, Engineering Series Vol. 1, 1910. An abstract of this is - given in Nature (London), Vol. 82, 1909, p. 267. - -[3]F. Cajori, History etc., p. 14. - -[4]Art. "Slide Rule" in the Penny Cyclopaedia and in the English - Cyclopaedia [Arts and Sciences]. - -[5]Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817, - p. 423. - -[6]The full title of the book which Wingate published on this subject in - Paris is as follows: - - L'Vsage | de la | Reigle de | Proportion | en l'Arithmetique & | - Geometrie. | Par Edmond Vvingate, | Gentil-homme Anglois. | - - Ean s philomaths, es polymaths. - - In tenui, sed no tenuis vsusve, laborne. | - - A Paris, | Chez Melchior Mondiere, | demeurant en l'Isle du Palais, | - la | ru de Harlay aux deux Viperes. | M. DC. XXIV. | Auec Priuilege du - Roy. | - - Back of the title page is the announcement: - - Notez que la Reigle de Proportion en toutes faons se vend Paris chez - Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces, - demeurant en l'Isle du Palais sur le Quay qui regarde la Megisserie - l'Espic d'or. - -[7]The title-page of the edition of 1658 is as follows: - - The Use of the Rule of Proportion in Arithmetick & Geometrie. First - published at Paris in the French tongue, and dedicated to Monsieur, the - then king's onely Brother (now Duke of Orleance). By Edm. Wingate, an - English Gent. And now translated into English by the Author. Whereinto - is now also inserted the Construction of the same Rule, & a farther use - thereof . . . 2nd edition inlarged and amended. London, 1658. - -[8]Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; - Drawn up by himself by way of Diary. With Appendix of original Letters. - Publish'd by Charles Burman, Esq., London, 1717, p. 23. - -[9]Mathematical Tables, 1811, p. 36, and art. "Gunter's Line" in his Phil. - and Math. Dictionary, London, 1815. - -[10]To the English Gentrie, and all others studious of the Mathematicks, - which shall bee readers hereof. The just Apologie of Wil: Ovghtred, - against the slaunderous insimulations of Richard Delamain, in a Pamphlet - called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum - projectio circularis. We shall refer to this document as Epistle. It was - published without date in 32 unnumbered pages of fine print, and was - bound in with Oughtred's Circles of Proportion, in the editions of 1633 - and 1639. In the 1633 edition it is inserted at the end of the volume - just after the Addition vnto the Vse of the Instrument etc., and in that - of 1639 immediately after the preface. It was omitted from the Oxford - edition of 1660. The Epistle was also published separately. There is a - separate copy in the British Museum, London. Aubrey, in his Brief Lives, - edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, "He - writt a stitch't pamphlet about 163(?4) against . . . Delamaine." - -[11]Thomas Browne is mentioned by Stone in his Mathematical Instruments, - London 1723, p. 16. See also Cajori, History of the Slide Rule, New - York, 1909, p. 15. - -[12]The Description and Use of a Joynt-Rule: . . . also the use of Mr. - White's Rule for measuring of Board and Timber, round and square; With - the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and - Versed Sines. By J. Brown, Philom., London, 1661. - -[13]A Collection of Centers and Useful Proportions on the Line of Numbers, - by John Brown, 1662(?), 16 pages; Description and Use of the Triangular - Quadrant, by John Brown, London, 1671; Wingate's Rule of Proportion in - Arithmetick and Geometry: or Gunter's Line. Newly rectified by Mr. Brown - and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The - Description and Use of the Carpenter's-Rule: Together with the Use of - the Line of Numbers commonly call'd Gunter's-Line, by John Brown, - London, 1704. - -[14]William Leybourn, op. cit., pp. 129, 130, 132, 133. - -[15]James Atkinson's edition of Andrew Wakely's The Mariners Compass - Rectified, London, 1694 [Wakely's preface dated 1664, Atkinson's - preface, 1693]. Atkinson adds An Appendix containing Use of Instruments - most useful in Navigation. Our quotation is from this Appendix, p. 199. - -[16]R. Delamain, The Making, Description, and Use of a small portable - Instrument . . . called a Horizontall Quadrant, etc., London, 1631. - -[17]Oughtred's description of his circular slide rule of 1632 and his - rectilinear slide rule of 1633, as well as a drawing of the circular - slide rule, are reproduced in Cajori's History of the Slide Rule, - Addenda, pp. ii-vi. - -[18]The full title of the Grammelogia I is as follows: - - Grammelogia | or, | The Mathematicall Ring. | Shewing (any reasonable - Capacity that hath | not Arithmeticke) how to resolve and worke | all - ordinary operations of Arithmeticke. | And those which are most - difficult with greatest | facilitie: The extraction of Roots, the - valuation of | Leases, &c. The measuring of Plaines | and Solids. | With - the resolution of Plaine and Sphericall | Triangles. | And that onely by - an Ocular Inspection, | and a Circular Motion. | Naturae secreta tempus - aperit. | London printed by John Haviland, 1630. - -[19]Grammelogia III is the same as Grammelogia I, except for the addition - of an appendix, entitled: - - De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit - nullo (praescripto tempore) vitae | vsuram nobis ingeniique Deus. | - London, | - - . . . The next line or two of this title-page which probably contained - the date of publication, were cut off by the binder in trimming the - edges of this and several other pamphlets for binding into one volume. - -[20]Grammelogia IV has two title pages. The first is Mirifica Logarithmoru' - Projectio Circularis. There follows a diagram of a circular slide rule, - with the inscription within the innermost ring: Nil Finis, Motvs, - Circvlvs vllvs Habet. The second title page is as follows: - - Grammelogia | Or, the Mathematicall Ring. | Extracted from the - Logarythmes, and projected Circular: Now published in the | inlargement - thereof unto any magnitude fit for use: shewing any reason- | able - capacity that hath not Arithmeticke how to resolve and worke, | all - ordinary operations of Arithmeticke: | And those that are most difficult - with greatest facilitie, the extracti- | on of Rootes, the valuation of - Leases, &c. the measuring of Plaines and Solids, | with the resolution - of Plaine and Sphericall Triangles applied to the | Practicall parts of - Geometrie, Horologographie, Geographie | Fortification, Navigation, - Astronomie, &c. | And that onely by an ocular inspection, and a Circular - motion, Invented and first published, by R. Delamain, Teacher, and - Student of the Mathematicks. | Naturae secreta tempus aperit. | - - There is no date. There follows the diagram of a second circular slide - rule, with the inscription within the innermost ring: Typus proiectionis - Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630 - promisi. There are numerous drawings in the Grammelogia, all of which, - excepting the drawings of slide rules on the engraved title-pages of - Grammelogia IV and V, were printed upon separate pieces of paper and - then inserted by hand into the vacant spaces on the printed pages - reserved for them. Some drawings are missing, so that the Bodleian - Grammelogia IV differs in this respect slightly from the copy in the - British Museum and from the British Museum copy of Grammelogia V. - -[21]Epistle, p. (8). - -[22]Aubrey, op. cit., Vol. II., p. 111. - -[23]Rigaud, Correspondence of Scientific Men during the 17th Century, Vol. - I, Oxford, 1841, p. 11. - -[24]Dictionary of National Biography, Art. "Delamain, Richard." See also - Rev. Charles J. Robinson, Taylors' School, from A.D. 1562 to 1874, Vol. - I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b; - Sixth Report of the Royal Commission on Historical Manuscripts, Part I, - Report and Appendix, London, 1877. In this Appendix, p. 82, we read the - following: - - Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. - Petitioner's husband was servant to the King, and one of His Majesty's - engineers for the fortification of the kingdom, and his tutor in - mathematical arts; but upon the breaking out of the war he deserted the - Court, and was called by the State to several employments, in fortifying - the towns of Northampton, Newport, and Abingdon; and was also abroad - with the armies as Quartermaster-General of the Foot, and therein died. - Petitioner is left a disconsolate widow with ten children, the four - least of whom are now afflicted with sickness, and petitioner has - nothing left to support them. There are several considerable sums of - money due to the petitioner, as well from the King as the State. Prays - that she may have some relief amongst other widows. See L. J., VII. 6. - 657. - -[25]Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London, - 1820, p. 34. - -[26]The New Artificial Gauging Line or Rod: together with rules concerning - the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London, - 1633. The copy we have seen is in the Bodleian Library, Oxford. The book - is small sized and has 40 pages. - -[27]Oughtred, op. cit., p. 11. - -[28]S. J. Rigaud, Correspondence of Scientific Men of the 17th Century, - Oxford, Vol. I, 1841, p. 17. - -[29]Rigaud, loc. cit., p. 22. - -[30]Rigaud, loc. cit., pp. 30, 31. - -[31]Oughtred, An Addition vnto the Vse of the Instrument called the Circles - of Proportion, London, 1633, p. 63. - -[32]F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22, - Addenda, pp. vi-ix. - -[33]W. Leybourn, op. cit., 1673, Preface, and pp. 128-29. - -[34]Cajori op. cit., Addenda, p. ix. - -[35]William Leybourn, op. cit., 1673, p. 35. - -[36]See Cajori, op. cit., pp. 20, 28, Addenda, p. ix. - -[37]See F. Cajori, "A Note on the History of the Slide Rule," Bibliotheca - mathematica, 3 F., Vol. 10, pp. 161-163. - -[38]John Atkinson, op. cit., 1694, p. 204. - -[39]Probably the oldest slide rule now in existence is owned by St. John's - College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in - diameter. It was exhibited along with other instruments in May, 1919. - According to the Catalogue of a Loan Exhibition of Early Scientific - Instruments in Oxford, opened May 16, 1919, the instrument is inscribed - with the name of the maker ("Elias Allen fecit") and with the name of - the donor, Georgius Barkham. It is dated 1635, which is only three years - after the first publication of Oughtred's description of his circular - slide rule. It is stated in the Catalogue: "Unfortunately all the - movable parts but the base-plate and a couple of thumb-screws are - missing. The face of the instrument is engraved with Oughtred's - Horizontal Instrument. The back is engraved with eleven Circles of - Proportion as described in Arthur Haughton's book, a copy of which was - presented to St. John's College by George Barkham, to explain the use of - the instrument." As Arthur Haughton's Oxford edition of Oughtred's - Circles of Proportion did not appear until 1660, it would seem that the - instrument was probably not presented to the College before 1660. As far - as is known, the next oldest slide rule is of the year 1654, kept in the - South Kensington Museum, London, and is described in Nature of March 5, - 1914. It is a rectilinear rule, "of boxwood, well made, and bound - together with brass at the two ends. It is of the square type, a little - more than 2 ft. in length, and bears the logarithmic lines first - described by Edmund Gunter. Of these, the num, sin and tan lines are - arranged in pairs, identical and contiguous, one line in each pair being - on the fixed part, and the other on the slide." The instrument is - inscribed, "Made by Robert Bissaker for T. W., 1654." Nowhere else have - we seen reference to Robert Bissaker. His slide rule seems to antedate - the "Whites rule" mentioned above. [This foot-note was added on October - 15, 1919.] - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - - -Note.--The University of California Publications are offered in exchange -for the publication of learned societies and institutions, universities and -libraries. Complete lists of all the publications of the University will be -sent upon request. For sample copies, lists of publications or other -information, address the Manager of the University of California Press, -Berkeley, California, U. 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Current, Volume XXII. - - - - - Transcriber's Notes - - ---Inserted subchapter headings from the Table of Contents into the - text. - ---Corrected two page numbers in the Table of Contents - ---Corrected a Greek quotation in footnote 6 by reference to the - original sources. - - - - - - - -End of the Project Gutenberg EBook of On the History of Gunter's Scale and -the Slide Rule during the Seventeenth Century, by Florian Cajori - -*** END OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - -***** This file should be named 42216-8.txt or 42216-8.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/4/2/2/1/42216/ - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - -Updated editions will replace the previous one--the old editions -will be renamed. - -Creating the works from public domain print editions means that no -one owns a United States copyright in these works, so the Foundation -(and you!) can copy and distribute it in the United States without -permission and without paying copyright royalties. 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You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org - - -Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century - -Author: Florian Cajori - -Release Date: February 26, 2013 [EBook #42216] - -Language: English - -Character set encoding: UTF-8 - -*** START OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - - - - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - - - - - -</pre> - -<div class="box"> -<p class="center"><span class="small">UNIVERSITY OF CALIFORNIA PUBLICATIONS</span> -<br /><span class="smaller">IN</span> -<br />MATHEMATICS</p> -<p class="center"><span class="small">Vol. 1, No. 9, pp. 187-209 <span class="hst">February 17, 1920</span></span></p> -<h2>ON THE HISTORY OF GUNTER’S SCALE AND THE SLIDE RULE DURING THE SEVENTEENTH CENTURY</h2> -<p class="center"><span class="small">BY</span> -<br />FLORIAN CAJORI</p> -<p class="tbcenter"><span class="smaller">UNIVERSITY OF CALIFORNIA PRESS -<br />BERKELEY</span></p> -</div> -<div class="pb" id="Page_187">[187]</div> -<h2>TABLE OF CONTENTS</h2> -<dl class="toc"> -<dt class="jr"><span class="small">PAGE</span></dt> -<dt><a href="#c1">I. Introduction</a> 187</dt> -<dt><a href="#c2">II. Innovations in Gunter’s Scale</a> 188</dt> -<dd><a href="#c3">Changes introduced by Edmund Wingate</a> 188</dd> -<dd><a href="#c4">Changes introduced by Milbourn</a> 189</dd> -<dd><a href="#c5">Changes introduced by Thomas Brown and John Brown</a> 190</dd> -<dd><a href="#c6">Changes introduced by William Leybourn</a> 192</dd> -<dt><a href="#c7">III. Richard Delamain’s “Grammelogia”</a> 192</dt> -<dd><a href="#c8">Different editions or impressions</a> 194</dd> -<dd><a href="#c9">Description of Delamain’s instrument of 1630</a> 195</dd> -<dd><a href="#c10">Delamain’s later designs, and directions for using his instruments</a> 197</dd> -<dt><a href="#c11">IV. Controversy between Oughtred and Delamain on the invention of the circular slide rule</a> 199</dt> -<dt><a href="#c12">V. Independence and priority of invention</a> 203</dt> -<dt><a href="#c13">VI. Oughtred’s “Gauging Line,” 1633</a> 206</dt> -<dt><a href="#c14">VII. Other seventeenth century slide rules</a> 207</dt> -</dl> -<h2 id="c1">I. INTRODUCTION</h2> -<p>In my history of the slide rule<a class="fn" id="fr_1" href="#fn_1">[1]</a>, and my article on its invention<a class="fn" id="fr_2" href="#fn_2">[2]</a> -it is shewn that William Oughtred and not Edmund Wingate is the -inventor, that Oughtred’s circular rule was described in print in 1632, -his rectilinear rule in 1633. Richard Delamain is referred to as having -tried to appropriate the invention to himself<a class="fn" id="fr_3" href="#fn_3">[3]</a> and as having written -a scurrilous pamphlet against Oughtred. All our information -<span class="pb" id="Page_188">[188]</span> -about Delamain was taken from De Morgan,<a class="fn" id="fr_4" href="#fn_4">[4]</a> who, however, gives no -evidence of having read any of Delamain’s writings on the slide rule. -Through Dr. Arthur Hutchinson of Pembroke College, Cambridge, I learned -that Delamain’s writings on the slide rule were available. In this -article will be given: First, some details of the changes introduced -during the seventeenth century in the design of Gunter’s scale by -Edmund Wingate, Milbourn, Thomas Brown, John Brown and William Leybourn; -second, an account of Delamain’s book of 1630 on the slide rule which antedates -Oughtred’s first <i>publication</i> (though Oughtred’s date of <i>invention</i> is earlier -than the date of Delamain’s alleged invention) and of Delamain’s later designs of -slide rules; third, an account of the controversy between Delamain and Oughtred; -fourth, an account of a later book on the slide rule written by William Oughtred, -and of other seventeenth century books on the slide rule.</p> -<h2 id="c2">II. INNOVATIONS IN GUNTER’S SCALE</h2> -<h3 id="c3">Changes introduced by Wingate</h3> -<p>We begin with Anthony Wood’s account of Wingate’s introduction of Gunter’s -scale into France.<a class="fn" id="fr_5" href="#fn_5">[5]</a></p> -<blockquote> -<p>In 1624 he transported into France the rule of proportion, having a little before been -invented by Edm. Gunter of Gresham Coll. and communicated it to most of the chiefest -mathematicians then residing in Paris: who apprehending the great benefit that might accrue -thereby, importun’d him to express the use thereof in the French tongue. Which being performed -accordingly, he was advised by monsieur Alleawne the King’s chief engineer to dedicate -his book to monsieur the King’s only brother, since duke of Orleans. Nevertheless the said -work coming forth as an abortive (the publishing thereof being somewhat hastened, by reason -an advocate of Dijon in Burgundy began to print some uses thereof, which Wingate had in a -friendly way communicated to him) especially in regard Gunter himself had learnedly explained -its use in a far larger volume.<a class="fn" id="fr_6" href="#fn_6">[6]</a></p> -</blockquote> -<p>Gunter’s scale, which Wingate calls the “rule of proportion,” contained, as -described in the French edition of 1624, four lines: (1) A single line of numbers; -(2) a line of tangents; (3) a line of sines; (4) a line, one foot in length, divided into -12 inches and tenths of inches, also a line, one foot in length, divided into tenths -and hundredths.</p> -<div class="pb" id="Page_189">[189]</div> -<p>The English editions of this book which appeared in 1623 and 1628 are devoid -of interest. The editions of 1645 and 1658 contain an important innovation.<a class="fn" id="fr_7" href="#fn_7">[7]</a> -In the preface the reasons why this instrument has not been used more are stated -to be: (1) the difficulty of drawing the lines with exactness, (2) the trouble of -working thereupon by reason (sometimes) of too large an extent of the compasses, -(3) the fact that the instrument is not readily portable. The drawing of Wingate’s -arrangement of the scale in the editions of 1645 and 1658 is about 66 cm. (26.5 in.) -long. It contains five parallel lines, about 66 cm. long, each having the divisions -of one line marked on one side and of another line on the other side. Thus each -line carries two graduations: (1) A single logarithmic line of numbers; (2) a logarithmic -line of numbers thrice repeated; (3) the first scale repeated, but beginning -with the graduations which are near the middle of the first scale, so that its graduation -reads 4, 5, 6, 7, 8, 9, 1, 2, 3; (4) a logarithmic line of numbers twice repeated; -(5) a logarithmic line of tangents; (6) a logarithmic line of sines; (7) the rule divided -into 1000 equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of -inches; (10) a scale consisting of three kinds, viz., a gauge line, a line of chords, -and a foot measure, divided into 1000 equal parts.</p> -<p>Important are the first and second scales, by which cube root extraction was -possible “by inspection only, without the aid of pen or compass;” similarly the -third and fourth scales, for square roots. This innovation is due to Wingate. -The 1645 edition announces that the instrument was made in brass by Elias Allen, -and in wood by John Thompson and Anthony Thompson in Hosier Lane.</p> -<h3 id="c4">Changes introduced by Milbourn</h3> -<p>William Leybourn, in his <i>The Line of Proportion or Numbers, Commonly called -Gunter’s Line, Made Easie</i>, London, 1673, says in his preface “To the Reader:”</p> -<blockquote> -<p>The Line of Proportion or Numbers, commonly called (by Artificers) Gunter’s Line, hath -been discoursed of by several persons, and variously applied to divers uses; for when Mr. -Gunter had brought it from the Tables to a Line, and written some Uses thereof, Mr. Wingate -added divers Lines of several lengths, thereby to extract the Square or Cube Roots, without -doubling or trebling the distance of the Compasses: After him Mr. Milbourn, a Yorkshire -Gentleman, disposed it in a Serpentine or Spiral Line, thereby enlarging the divisions of the -Line.</p> -</blockquote> -<p>On pages 127 and 128 Leybourn adds:</p> -<blockquote> -<p>Again, One T. Browne, a Maker of Mathematical Instruments, made it in a Serpentine -or Spiral Line, composed of divers Concentrick Circles, thereby to enlarg the divisions, -which was the contrivance of one Mr. Milburn a Yorkshire Gentleman, who writ thereof, and -communicated his Uses to the aforesaid Brown, who (since his death) attributed it to himself: -But whoever was the contriver of it, it is not without inconvenience; for it can in no wise be -made portable; and besides (instead of compasses) an opening Joynt with thirds [threads] -must be placed to move upon the Centre of the Instrument, without which no proportion -can be wrought.</p> -</blockquote> -<div class="pb" id="Page_190">[190]</div> -<p>This Mr. Milburn is probably the person named in the diary of the antiquarian, -Elias Ashmole, on August 13 [1646?]; “I bought of Mr. Milbourn all his Books -and Mathematical Instruments.”<a class="fn" id="fr_8" href="#fn_8">[8]</a> Charles Hutton<a class="fn" id="fr_9" href="#fn_9">[9]</a> says that Milburne of -Yorkshire designed the spiral form about 1650. This date is doubtless wrong, -for Thomas Browne who, according to Leybourn, got the spiral form of line from -Milbourn, is repeatedly mentioned by William Oughtred in his <i>Epistle</i><a class="fn" id="fr_10" href="#fn_10">[10]</a> printed -some time in 1632 or 1633. Oughtred does not mention Milbourn, and says -(page 4) that the spiral form “was first hit upon by one Thomas Browne a Joyner, -. . . the serpentine revolution being but two true semicircles described on severall -centers.”<a class="fn" id="fr_11" href="#fn_11">[11]</a></p> -<h3 id="c5">Changes introduced by Thomas Brown and John Brown</h3> -<p>Thomas Brown did not publish any description of his instrument, but his son, -John Brown, published in 1661 a small book,<a class="fn" id="fr_12" href="#fn_12">[12]</a> in which he says (preface) that he -had done “as Mr. Oughtred with Gunter’s Rule, to a sliding and circular form; -and as my father Thomas Brown into a Serpentine form; or as Mr. Windgate in -his <i>Rule of Proportion</i>.” He says also that “this brief touch of the Serpentine-line -I made bold to assert, to see if I could draw out a performance of that promise, -that hath been so long unperformed by the promisers thereof.” Accordingly in -Chapter XX he gives a description of the serpentine line, “contrived in five (or -rather 15) turn.” Whether this description, printed in 1661, exactly fits the instrument -as it was developed in 1632, we have no means of knowing. John Brown -says:</p> -<blockquote> -<p>1. First next the center is two circles divided one into 60, the other into 100 parts, for -the reducing of minutes to 100 parts, and the contrary.</p> -<p>2. You have in seven turnes two inpricks, and five in divisions, the first Radius of the -sines (or Tangents being neer the matter, alike to the first three degrees,) ending at 5 degrees -and 44 minutes.</p> -<p>3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in three margents -in divisions, and the line of versed sines in pricks, under the line of Tangents, according to -Mr. <i>Gunter’s</i> cross-staff: the sines and Tangents beginning at 5 degrees, and 44 minutes where -<span class="pb" id="Page_191">[191]</span> -the other ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the line -of numbers beginning at 10, and proceeding to 100, being one entire Radius, and graduated -into as many divisions as the largeness of the instrument will admit, being 10 to 10 50 into -50 parts, and from 50 to 100 into 20 parts in one unit of increase, but the Tangents are divided -into single minutes from the beginning to the end, both in the first, second and third Radiusses, -and the sines into minutes; also from 30 minutes to 40 degrees, and from 40 to 60, into every -two minutes, and from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the -rest as many as can be well discovered.</p> -<p>The versed sines are set after the manner of Mr. <i>Gunter’s</i> Cross-staff, and divided into every -10th minutes beginning at 0, and proceeding to 156 going backwards under the line of Tangents.</p> -<p>4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is the secants to -51 degrees, being nothing else but the sines reitterated beyond 90.</p> -<p>5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85 degrees, whereby -all trouble of backward working is avoided.</p> -<p>6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and also a line of natural -sines, for finding of differences in sines, for finding hour and Azimuth.</p> -<p>7. Seventhly, next the verge or outermost edge is a line of equal parts to get the Logarithm -of any number, or the Logarithm sine and Tangent of any ark or angle to four figures -besides the carracteristick.</p> -<p>8. Eightly and lastly, in the space place between the ending of the middle five turnes, -and one half of the circle are three prickt lines fitted for reduction. The uppermost being for -shillings, pence and farthings. The next for pounds, and ounces, and quarters of small -<i>Averdupoies</i> weight. The last for pounds, shillings and pence, and to be used thus: If you -would reduce 16s. 3d. 2q. to a decimal fraction, lay the hair or edge of one of the legs of the -index on 16. 3½ in the line of 1. s. d. and the hair shall cut on the equal parts 81 16; and the -contrary, if you have a decimal fraction, and would reduce it to a proper fraction, the like -may you do for shillings, and pence, and pounds, and ounces.</p> -</blockquote> -<p class="center"><i>The uses of the lines follow.</i></p> -<blockquote> -<p>As to the use of these lines, I shall in this place say but little, and that for two reasons. -First, because this instrument is so contrived, that the use is sooner learned then any other, -I speak as to the manner, and way of using it, because by means of first second and third -radiusses, in sines and Tangents, the work is always right on, one way or other, according to -the Canon whatsoever it be, in any book that treats of the Logarithms, as <i>Gunter</i>, <i>Wells</i>, -<i>Oughtred</i>, <i>Norwood</i>, or others, as in <i>Oughtred</i> from page 64 to 107.</p> -<p>Secondly, and more especially, because the more accurate, and large handling thereof is -more then promised, if not already performed by more abler pens, and a large manuscript -thereof by my <i>Sires</i> meanes, provided many years ago, though to this day not extant in print; -so for his sake I claiming my interest therein, make bold to present you with these few lines, -in order to the use of them: And first note,</p> -<p>1. Which soever of the two legs is set to the first term in the question, that I call the first -leg always, and the other being set to the second term, I call the second leg . . .</p> -</blockquote> -<p>The exact nature of the contrivance with the “two legs” is not described, but -it was probably a flat pair of compasses, attached to the metallic surface on which -the serpentine line was drawn. In that case the instrument was a slide rule, -rather than a form of Gunter’s line. In his publication of 1661, as also in later -<span class="pb" id="Page_192">[192]</span> -publications,<a class="fn" id="fr_13" href="#fn_13">[13]</a> John Brown devoted more space to Gunter’s scales, requiring the -use of a separate pair of compasses, than to slide rules.</p> -<h3 id="c6">Changes introduced by William Leybourn</h3> -<p>The same remark applies to William Leybourn who, after speaking of Seth -Partridge’s slide rule, returns to forms of Gunter’s scale, saying:<a class="fn" id="fr_14" href="#fn_14">[14]</a></p> -<blockquote> -<p>There is yet another way of disposing of this Line of Proportion, by having one Line of -the full length of the Ruler, and another Line of the same Radius broken in two parts between -3 and 4; so that in working your Compasses never go off of the Line: This is one of the best -contrivances, but here Compasses must be used. These are all the Contrivances that I have -hitherto seen of these Lines: That which I here speak of, and will shew how to use, is only -two Lines of one and the same Radius, being set upon a plain Ruler of any length (the larger -the better) having the beginning of one Line, at the end of the other, the divisions of each -Line being set so close together, that if you find any number upon one of the Lines, you may -easily see what number stands against it on the other Line. This is all the Variation. . . .</p> -<p><i>Example 1.</i> <i>If a Board be 1 Foot 64 parts broad, how much in length of that Board will make -a Foot Square?</i> Look upon one of your Lines (it matters not which) for 1 Foot 64 parts, -and right against it on the other Line you shall find 61; and so many parts of a Foot will make -a Foot square of that Board.</p> -</blockquote> -<p>This contrivance solves the equation 1.64<i>x</i>=1, yielding centesimal parts of -a foot.</p> -<p>James Atkinson<a class="fn" id="fr_15" href="#fn_15">[15]</a> speaks of “Gunter’s scale” as “usually of Boxwood . . . -commonly 2 ft. long, 1½ inch broad” and “of two kinds: <i>long Gunter</i> or <i>single -Gunter</i>, and the <i>sliding Gunter</i>. It appears that during the seventeenth century -(and long after) the Gunter’s scale was a rival of the slide rule.</p> -<h2 id="c7">III. RICHARD DELAMAIN’S GRAMMELOGIA</h2> -<p>We begin with a brief statement of the relations between Oughtred and Delamain. -At one time Delamain, a teacher of mathematics in London, was assisted -by Oughtred in his mathematical studies. In 1630 Delamain published the -<i>Grammelogia</i>, a pamphlet describing a circular slide rule and its use. In 1631 he -published another tract, on the <i>Horizontall Quadrant</i>.<a class="fn" id="fr_16" href="#fn_16">[16]</a> In 1632 appeared Oughtred’s -<i>Circles of Proportion</i><a class="fn" id="fr_17" href="#fn_17">[17]</a> translated into English from Oughtred’s Latin manuscript -by another pupil, William Forster, in the preface of which Forster makes -the charge (without naming Delamain) that “another . . . went about to pre-ocupate” -the new invention. This led to verbal disputes and to the publication -by Delamain of several additions to the <i>Grammelogia</i>, describing further designs -of circular slide rules and also stating his side of the bitter controversy, but without -giving the name of his antagonist. Oughtred’s <i>Epistle</i> was published as a reply. -Each combatant accuses the other of stealing the invention of the circular slide -rule and the horizontal quadrant.</p> -<div class="img"> -<img src="images/title1.jpg" alt="First title-page of “Grammelogia IV”" width="500" height="835" /> -</div> -<div class="img"> -<img src="images/title2.jpg" alt="Second title-page of “Grammelogia IV”" width="500" height="824" /> -</div> -<p class="center">The two title-pages of the edition of the <span class="sc">Grammelogia</span> in the British Museum in London which we have called “Grammelogia IV.”</p> -<div class="pb" id="Page_194">[194]</div> -<h3 id="c8">Different editions or impressions</h3> -<p>There are at least five different editions, or impressions, of the <i>Grammelogia</i> -which we designate, for convenience, as follows:</p> -<blockquote> -<p><i>Grammelogia I</i>, 1630. One copy in the Cambridge University Library.<a class="fn" id="fr_18" href="#fn_18">[18]</a></p> -<p><i>Grammelogia II</i>, I have not seen a copy of this.</p> -<p><i>Grammelogia III</i>, One copy in the Cambridge University Library.<a class="fn" id="fr_19" href="#fn_19">[19]</a></p> -<p><i>Grammelogia IV</i>, One copy in the British Museum, another in the Bodleian Library, Oxford.<a class="fn" id="fr_20" href="#fn_20">[20]</a></p> -<p><i>Grammelogia V</i>, One copy in the British Museum.</p> -</blockquote> -<p>In <i>Grammelogia I</i> the first three leaves and the last leaf are without pagination. -The first leaf contains the title-page; the second leaf, the dedication to the King -and the preface “To the Reader;” the third leaf, the description of the Mathematical -<span class="pb" id="Page_195">[195]</span> -Ring. Then follow 22 numbered pages. Counting the unnumbered pages, -there are altogether 30 pages in the pamphlet. Only the first three leaves of this -pamphlet are omitted in <i>Grammelogia IV</i> and <i>V</i>.</p> -<p>In <i>Grammelogia III</i> the <i>Appendix</i> begins with a page numbered 52 and bears -the heading “Conclusion;” it ends with page 68, which contains the same two poems -on the mathematical ring that are given on the last page of <i>Grammelogia I</i> but -differs slightly in the spelling of some of the words. The 51 pages which must -originally have preceded page 52, we have not seen. The edition containing these -we have designated <i>Grammelogia II</i>. The reason for the omission of these 51 -pages can only be conjectured. In Oughtred’s <i>Epistle</i> (p. 24), it is stated that -Delamain had given a copy of the <i>Grammelogia</i> to Thomas Brown, and that two -days later Delamain asked for the return of the copy, “because he had found -some things to be altered therein” and “rent out all the middle part.” Delamain -labored “to recall all the bookes he had given forth, (which were many) before the -sight of <i>Brownes Lines</i>.” These spiral lines Oughtred claimed that Delamain -had stolen from Brown. The title-page and page 52 are the only parts of the -<i>Appendix</i>, as given in <i>Grammelogia III</i>, that are missing in the <i>Grammelogia IV</i> -and <i>V</i>.</p> -<p><i>Grammelogia IV</i> answers fully to the description of Delamain’s pamphlet -contained in Oughtred’s <i>Epistle</i>. It was brought out in 1632 or 1633, for what -appears to be the latest part of it contains a reference (page 99) to the <i>Grammelogia -I</i> (1630) as “being now more then two yeares past.” Moreover, it refers to Oughtred’s -<i>Circles of Proportion</i>, 1632, and Oughtred’s reply in the <i>Epistle</i> was bound -in the <i>Circles of Proportion</i> having the <i>Addition</i> of 1633. For convenience of -reference we number the two title-pages of <i>Grammelogia IV</i>, “page (1)” and -“page (2),” as is done by Oughtred in his <i>Epistle</i>. <i>Grammelogia IV</i> contains, then, -113 pages. The page numbers which we assign will be placed in parentheses, to -distinguish them from the page numbers which are <i>printed</i> in <i>Grammelogia IV</i>. -The pages (44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are -the same as the pages 53-68. Thus only thirty-eight pages have page numbers -printed on them. The pages (67) and (83) are identical in wording, except for -some printer’s errors; they contain verses in praise of the <i>Ring</i>, and have near the -bottom the word “Finis.” Also, pages (22) and (23) are together identical in -wording with page (113), which is set up in finer type, containing an advertisement -of a part of <i>Grammelogia IV</i> explaining the mode of graduating the circular -rules. There are altogether six parts of <i>Grammelogia IV</i> which begin or end by -an address to the reader, thus: “To the Reader,” “Courteous Reader,” or “To -the courteous and benevolent Reader . . .,” namely the pages (8), (22), (68), (89), -(90), (108). In his <i>Epistle</i> (page 2), Oughtred characterizes the make up of the -book in the following terms:</p> -<blockquote> -<p>In reading it . . . I met with such a patchery and confusion of disjoynted stuffe, that -I was striken with a new wonder, that any man should be so simple, as to shame himselfe to -the world with such a hotch-potch.</p> -</blockquote> -<div class="pb" id="Page_196">[196]</div> -<p><i>Grammelogia V</i> differs from <i>Grammelogia IV</i> in having only the -second title-page. The first title-page may have been torn off from the -copy I have seen. A second difference is that the page with the printed -numeral 22 in <i>Grammelogia IV</i> has after the word “Finis” the -following notice:</p> -<blockquote> -<p>This instrument is made in Silver, or Brasse for the Pocket, or at any other bignesse, -over against Saint Clements Church without Temple Barre, by Elias Allen.</p> -</blockquote> -<p>This notice occurs also on page 22 of <i>Grammelogia I</i> and <i>III</i>, but is omitted from -page 22 of <i>Grammelogia V</i>.</p> -<h3 id="c9">Description of Delamain’s instrument of 1630</h3> -<p>In his address to King Charles I, in his <i>Grammelogia I</i>, Delamain emphasizes -the ease of operating with his slide rule by stating that it is “fit for use . . . as well -on Horse backe as on Foot.” Speaking “To the Reader,” he states that he has -“for many yeares taught the Mathematicks in this Towne,” and made efforts to -improve Gunter’s scale “by some Motion, so that the whole body of Logarithmes -might move proportionally the one to the other, as occasion required. This -conceit in February last [1629] I struke upon, and so composed my <i>Grammelogia</i> -or <i>Mathematicall Ring</i>; by which only with an <i>ocular inspection</i>, there is had at -one instant all proportionalls through the said body of Numbers.” He dates his -preface “first of January, 1630.” The fifth and sixth pages contain his “Description -of the Grammelogia,” the term <i>Grammelogia</i> being applied to the instrument, -as well as to the book. His description is as follows:</p> -<blockquote> -<p>The parts of the Instrument are two Circles, the one moveable, and the other fixed; The -moveable is that unto which is fastened a small pin to move it by; the other Circle may be -conceived to be fixed; The circumference of the moveable Circle is divided into unequall -parts, charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures doe represent themselves, -or such numbers unto which a Cipher or Ciphers are added, and are varied as the occasion -falls out in <i>the speech of Numbers</i>, so 1. stands for 1. or 10. or 100., &c. the 2. stands for -2. or 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c.</p> -</blockquote> -<p>After elaborating this last point and explaining the decimal subdivisions on -the scales of the movable circle, he says that “the numbers and divisions on the -fixed Circle, are the very same that the moveable are, . .” There is no drawing -of the slide rule in this publication. The twenty-two numbered pages give explanations -of the various uses to which the instrument can be put: “How to performe -the Golden Rule” (pp. 1-3), “Further uses of the Golden Rule” (pp. 4-6), -“Notions or Principles touching the disposing or ordering of the Numbers in the -Golden Rule in their true places upon the Grammelogia” (pp. 7-11), “How to -divide one number by another” (pp. 12, 13), “to multiply one Number by another” -(pp. 14, 15), “To find Numbers in continuall proportion” (pp. 16, 17), “How to -extract the Square Root,” “How to extract the Cubicke Root” (pp. 18-21), -“How to performe the Golden Rule” (the rule of proportion) is explained thus:</p> -<blockquote> -<p>Seeke the first number in the moveable, and bring it to the second number in the fixed, -so right against the third number in the moveable, is the answer in the fixed.</p> -<p>If the Interest of 100. li. be 8. li. in the yeare, what is the Interest of 65. li. for the same time.</p> -<p>Bring 100. in the moveable to 8. in the fixed, so right against 65. in the moveable is 5.2. -in the fixed, and so much is the Interest of 65. li. for the yeare at 8. li. for 100. li. <i>per annum</i>.</p> -<div class="pb" id="Page_197">[197]</div> -<p>The <i>Instrument</i> not removed, you may at one instant right against any summe of money -in the moveable, see the Interest thereof in the fixed: the reason of this is from the <i>Definition -of Logarithmes.</i></p> -</blockquote> -<p>These are the earliest known printed instructions on the use of a slide rule. -It will be noticed that the description of the instrument at the opening makes no -references to logarithmic lines for the trigonometric functions; only the line of -numbers is given. Yet the title-page promised the “resolution of Plaine and -Sphericall Triangles.” Page 22 throws light upon this matter:</p> -<blockquote> -<p>If there be composed three Circles of equal thicknesse, A.B.C. so that the inner edge of D -[should be B] and the outward edge of A bee answerably graduated with <i>Logarithmall signes</i> -[sines], and the outward edge of B and the inner edge of A with <i>Logarithmes</i>; and then on the -backside be graduated the <i>Logarithmall Tangents</i>, and againe the <i>Logarithmall signes</i> oppositly -to the former graduations, it shall be fitted for the resolution of <i>Plaine</i> and <i>Sphericall -Triangles</i>.</p> -</blockquote> -<p>After twelve lines of further remarks on this point he adds:</p> -<blockquote> -<p>Hence from the forme, I have called it a <i>Ring</i>, and <i>Grammelogia</i> by annoligie of a <i>Lineary -speech</i>; which <i>Ring</i>, if it were projected in the <i>convex</i> unto two yards <i>Diameter</i>, or thereabouts, -and the line <i>Decupled</i>, it would worke <i>Trigonometrie</i> unto seconds, and give <i>proportionall -numbers</i> unto six places only by an <i>ocular inspection</i>, which would compendiate <i>Astronomicall -calculations</i>, and be sufficient for the <i>Prosthaphaeresis</i> of the Motions: But of this as God shall -give life and ability to health and time.</p> -</blockquote> -<p>The unnumbered page following page 22 contains the patent and copyright -on the instrument and book:</p> -<blockquote> -<p>Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs an Instrument -called Grammelogia, or The Mathematicall Ring, together with a Booke so intituled, -expressing the use thereof, being his owne Invention; we of our Gracious and Princely favour -have granted unto the said Richard Delamain and his Assignes, Privilege, Licence, and Authority, -for the sole Making, Printing and Selling of the said Instrument and Booke: straightly -forbidding any other to Make, Imprint, or Sell, or cause to be Made, or Imprinted, or Sold, -the said Instrument or Booke within any our Dominions, during the space of ten yeares next -ensuing the date hereof, upon paine of Our high displeasure. Given under our hand and -Signet at our Palace of Westminster, the fourth day of January, in the sixth yeare of our -Raigne.</p> -</blockquote> -<h3 id="c10">Delamain’s later designs, and directions for using his instruments</h3> -<p>In the <i>Appendix</i> of <i>Grammelogia III</i>, on page 52 is given a description of an -instrument promised near the end of <i>Grammelogia I</i>:</p> -<blockquote> -<p>That which I have formerly delivered hath been onely upon one of the -<i>Circles</i> of my <i>Ring</i>, -simply concerning <i>Arithmeticall Proportions</i>, I will by way of -<i>Conclusion</i> touch upon some uses -of the <i>Circles</i>, of <i>Logarithmall Sines</i>, and <i>Tangents</i>, -which are placed on the edge of both the -moveable and fixed <i>Circles</i> of the <i>Ring</i> -in respect of <i>Geometricall Proportions</i>, but first of the -description of these <i>Circles</i>.</p> -<p><i>First</i>, upon the side that the <i>Circle of Numbers</i> is one, are graduated on the edge of the -moveable, and also on the edge of the fixed the <i>Logarithmall Sines</i>, for if you bring 1. in the -moveable amongst the <i>Numbers</i> to 1. in the fixed, you may on the other edge of the moveable -and fixed see the <i>sines</i> noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto 6.6. and each degree -subdivided, and then over the former divisions and figures 90. 90. 80. 80. 70. 70. &c. you have -the other degrees, viz. 5. 4. 3. 2. 1. each of those divided by small points.</p> -<div class="pb" id="Page_198">[198]</div> -<p><i>Secondly</i>, (if the <i>Ring</i> is great) neere the outward edge of this side of the fixed against the -<i>Numbers</i>, are the usuall divisions of a <i>Circle</i>, and the points of the <i>Compasse</i>: serving for observation -in <i>Astronomy</i>, or <i>Geometry</i>, and the sights belonging to those divisions, may be placed -on the moveable <i>Circle</i>.</p> -<p><i>Thirdly</i>, opposite to those <i>Sines</i> on the other side are the <i>Logarithmall Tangents</i>, noted -alike both in the moveable and fixed thus 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. -which numbers or divisions serve also for their <i>Complements</i> to 90. so 40 gr. stands for 50. -gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both in the moveable and fixed is -also divided into parts. As for the degrees which are under 6. viz. 5.4.3.2.1. they are noted -with small figures over this divided <i>Circle</i> from 45.40.35.30.25. &c. and each of those degrees -divided into parts by small points both in the moveable and fixed.</p> -<p><i>Fourthly</i>, on the other edge of the moveable on the same side is another graduation of -<i>Tangents</i>, like that formerly described. And opposite unto it, in the fixed is a Graduation -<i>of Logarithmall sines</i> in every thing answerable to the first descrition of Sines on the other side.</p> -<p><i>Fifthly</i>, on the edge of the <i>Ring</i> is graduated a parte of the <i>Æquator</i>, numbered thus 10 -20. 30. unto 100. and there unto is adjoyned the degrees of the <i>Meridian</i> inlarged, and numbered -thus 10 20.30 unto 70. each degree both of the <i>Æquator</i>, and <i>Meridian</i> are subdivided into -parts; these two graduated <i>Circles</i> serve to resolve such <i>Questions</i> which concerne <i>Latitude</i>, -<i>Longitude</i>, <i>Rumb</i>, and <i>Distance</i>, in <i>Nauticall</i> operations.</p> -<p><i>Sixthly</i>, to the concave of the <i>Ring</i> may be added a <i>Circle</i> to be elevated or depressed for -any <i>Latitude</i>, representing the <i>Æquator</i>, and so divided into houres and parts with an <i>Axis</i>, -to shew both the <i>houre</i>, and <i>Azimuth</i>, and within this <i>Circle</i> may be hanged a <i>Box</i>, and <i>Needle</i> -with a Socket for a <i>staffe</i> to slide into it, and this accommodated with <i>scrue pines</i> to fasten it -to the <i>Ring</i> and <i>staffe</i>, or to take it off at pleasure.</p> -</blockquote> -<p>The pages bearing the printed numbers 53-68 in the <i>Grammelogia III</i>, <i>IV</i> -and <i>V</i> make no reference to the dispute with Oughtred and may, therefore, be -assumed to have been published before the appearance of Oughtred’s <i>Circles of -Proportion</i>. On page 53, “To the Reader,” he says:</p> -<blockquote> -<p>. . . you may make use of the Projection of the <i>Circles</i> of the <i>Ring</i> upon a <i>Plaine</i>, having -the feet of a paire of <i>compasses</i> (but so that they be flat) to move on the <i>Center</i> of that <i>Plaine</i>, -and those feet to open and shut as a paire of <i>Compasses</i> . . . now if the feet bee opened to any -two termes or numbers in that <i>Projection</i>, then may you move the first foot to the third number, -and the other foot shall give the <i>Answer</i>; . . . it hath pleased some to make use of this way. -But in this there is a double labour in respect to that of the <i>Ring</i>, the one in fitting those feet -unto the numbers assigned, and the other by moving them about, in which a man can hardly -accommodate the <i>Instrument</i> with one hand, and expresse the <i>Proportionals</i> in writing with -the other. By the <i>Ring</i> you need not but bring one number to another, and right against -any other number is the <i>Answer</i> without any such motion. . . . upon that [the Ring] I write, -shewing some uses of those <i>Circles</i> amongst themselves, and conjoyned with others . . . in -<i>Astronomy</i>, <i>Horolographie</i>, in plaine <i>Triangles</i> applyed to <i>Dimensions</i>, <i>Navigation</i>, <i>Fortification</i>, -etc. . . . But before I come to <i>Construction</i>, I have thought it convenient by way -introduction, to examine the truth of the graduation of those <i>Circles</i> . . .</p> -</blockquote> -<p>These are the words of a practical man, interested in the mechanical development -of his instrument. He considers not only questions of convenience but also -of accuracy. The instrument has, or may have now, also lines of sines and tangents. -To test the accuracy of the circles of Numbers, “bring any number in the moveable -to halfe of that number in the fixed: so any number or part in the fixed shall give -his double in the moveable, and so may you trie of the thirds, fourths &c. of numbers, -<span class="pb" id="Page_199">[199]</span> -<i>vel contra</i>,” (p. 54). On page 55 are given two small drawings, labelled, -“A Type of the Ringe and Scheme of this Logarithmicall projection, the use followeth. -These Instruments are made in Silver or Brasse by John Allen neare -the Sauoy in the Strand.”</p> -<h3 id="c11">IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE CIRCULAR SLIDE RULE</h3> -<p>Delamain’s publication of 1630 on the ‘Mathematicall Ring’ does not appear -at that time to have caused a rupture between him and Oughtred. When in 1631 -Delamain brought out his <i>Horizontall Quadrant</i>, the invention of which Delamain -was afterwards charged to have stolen from Oughtred, Delamain was still in close -touch with Oughtred and was sending Oughtred in the Arundell House, London, -the sheets as they were printed. Oughtred’s reference to this in his <i>Epistle</i> (p. 20) -written after the friendship was broken, is as follows:</p> -<blockquote> -<p>While he was printing his tractate of the Horizontall quadrant, although he could not but -know that it was injurious to me in respect of my free gift to Master <i>Allen</i>, and of <i>William -Forster</i>, whose translation of my rules was then about to come forth: yet such was my good -nature, and his shamelessnesse, that every day, as any sheet was printed, hee sent, or brought -the same to mee at my chamber in Arundell house to peruse which I lovingly and ingenuously -did, and gave him my judgment of it.</p> -</blockquote> -<p>Even after Forster’s publication of Oughtred’s <i>Circles of Proportion</i>, 1632, -Oughtred had a book, <i>A canon of Sines Tangents and Secants</i>, which he had borrowed -from Delamain and was then returning (<i>Epistle</i>, page (5)). The attacks -which Forster, in the preface to the <i>Circles of Proportion</i>, made upon Delamain -(though not naming Delamain) started the quarrel. Except for Forster and other -pupils of Oughtred who urged him on to castigate Delamain, the controversy might -never have arisen. Forster expressed himself in part as follows:</p> -<blockquote> -<p>. . . being in the time of the long vacation 1630, in the Country, at the house of the Reverend, -and my most worthy friend, and Teacher, Mr. William Oughtred (to whose instruction I -owe both my initiation, and whole progresse in these Sciences.) I vpon occasion of speech -told him of a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made (such -as it vsually called Mr. Gunter’s Ruler) 6 feet long, to be vsed with a payre of beame-compasses. -“He answered that was a poore invention, and the performance very troublesome: But, said -he, seeing you are taken with such mechanicall wayes of Instruments, I will shew you what -deuises I have had by mee these many yeares.” And first, hee brought to mee two Rulers -of that sort, to be vsed by applying one to the other, without any compasses: and after that -hee shewed mee those lines cast into a circle or Ring, with another moueable circle vpon it. -I seeing the great expeditenesse of both those wayes; but especially, of the latter, wherein it -farre excelleth any other Instrument which hath bin knowne; told him, I wondered that hee -could so many yeares conceale such vseful inuentions, not onely from the world, but from my -selfe, to whom in other parts and mysteries of Art, he had bin so liberall. He answered, -“That the true way of Art is not by Instruments, but by Demonstration: and that it is a preposterous -course of vulgar Teachers, to begin with Instruments, and not with the Sciences, -and so in-stead of Artists, to make their Schollers only doers of tricks, and as it were Iuglers: -to the despite of Art, losse of precious time, and betraying of willing and industrious wits, -<span class="pb" id="Page_200">[200]</span> -vnto ignorance and idlenesse. That the vse of Instruments is indeed excellent, if a man be -an Artist: but contemptible, being set and opposed to Art. And lastly, that he meant to -commend to me, the skill of Instruments, but first he would haue me well instructed in the -Sciences. He also shewed me many notes, and Rules for the vse of those circles, and of his -Horizontall Instrument, (which he had proiected about 30 yeares before) the most part written -in Latine. All which I obtained of him leaue to translate into English, and make publique, -for the vse, and benefit of such as were studious, and louers of these excellent Sciences.</p> -<p>Which thing while I with mature, and diligent care (as my occasions would give me leaue) -went about to doe: another to whom the Author in a louing confidence discouered this intent, -using more hast then good speed, went about to preocupate; of which vntimely birth, and -preuenting (if not circumuenting) forwardnesse, I say no more: but aduise the studious Reader, -onely so farre to trust, as he shal be sure doth agree to truth & Art.</p> -</blockquote> -<p>While in this dedication reference is made to a slide rule or “ring” with a “moveable -circle,” the instrument actually described in the <i>Circles of Proportion</i> consists -of fixed circles “with an <i>index</i> to be opened after the manner of a paire of Compasses.” -Delamain, as we have seen, had decided preference for the moveable -circle. To Oughtred, on the other hand, one design was about as good as the -other; he was more of a theorist and repeatedly expressed his contempt for mathematical -instruments. In his <i>Epistle</i> (page (25)), he says he had not “the one halfe -of my intentions upon it” (the rule in his book), nor one with a “moveable circle -and a thread, but with an opening Index at the centre (if so be that bee cause -enough to make it to bee not the same, but another Instrument) for my part I -disclaime it: it may go seeke another Master: which for ought I know, will prove -to be <i>Elias Allen</i> himselfe: for at his request only I altered a little my rules from -the use of the moveable circle and the thread, to the two armes of an Index.”</p> -<p>All parts of Delamain’s <i>Grammelogia IV</i>, except pages 1-22 and 53-68 considered -above, were published after the <i>Circles of Proportion</i>, for they contain references -to the ill treatment that Delamain felt or made believe that he felt, that he had -received in the book published by Oughtred and Forster. Oughtred’s reference -to teachers whose scholars are “doers of tricks,” “Iuglers,” and Forster’s allusion -to “another to whom the Author in a loving confidence” explained the instrument -and who “went about to preocupate” it, are repeatedly mentioned. Delamain -says, (page (89)) that at first he did not intend to express himself in print, “but -sought peace and my right by a private and friendly way.” Oughtred’s account -of Delamain’s course is that of an “ill-natured man” with a “virulent tongue,” -“sardonical laughter” and “malapert sawsiness.” Contrasting Forster and -Delamain, he says that, of the former he “had the very first moulding” and made -him feel that “the way of Art” is “by demonstration.” But Delamain was -“already corrupted with doing upon Instruments, and quite lost from ever being -made an Artist.” (<i>Epistle</i> page (27)). Repeatedly does Oughtred assert Delamain’s -ignorance of mathematics. The two men were evidently of wholly different -intellectual predilections. That Delamain loved instruments is quite -evident, and we proceed to describe his efforts to improve the circular slide rule.</p> -<div class="pb" id="Page_201">[201]</div> -<p>The <i>Grammelogia IV</i> is dedicated to King Charles I. Delamain says:</p> -<blockquote> -<p>. . . Everything hath his beginning, and curious <i>Arts</i> seldome come to the height at the -first; It was my promise then to enlarge the <i>invention</i> by a way of <i>decuplating the Circles</i>, -which I now present unto your <i>sacred Majestie</i> as the <i>quintessence</i> and <i>excellencie</i> there of . . .</p> -</blockquote> -<p>His enlarged circular rules are illustrated in the Bodleian Library copy of -<i>Grammelogia IV</i> by four diagrams, two of them being the two drawings on the two -title-pages at the beginning of the <i>Grammelogia IV</i>, 4 inches in external diameter, -and exhibiting eleven concentric circular lines carrying graduations of different -sorts. In the second of these designs all circles are fixed. The other two drawings -are each 10¾ inches in external diameter and exhibit 18 concentric circular lines; -the folded sheet of the first of these drawings is inserted between pages (23) and -(24), the second folded sheet between pages (83) and (84). All circles of this -second instrument are fixed. Counting in the two small drawings in <i>Grammelogia -III</i>, there are in all six drawings of slide rules in the Bodleian <i>Grammelogia IV</i>. -On pages (24) to (43) Delamain explains the graduation of slide rules. He takes -first a rule which has one circle of equal parts, divided into 1000 equal divisions. -From a table of logarithms he gets log 2 = 0.301; from the number 301 in the circle -of equal parts he draws a line to the center of the circle and marks the intersection -with the circles of numbers by the figure 2. Thus he proceeds with log 3, log 4, -and so on; also with log sin <i>x</i> and log tan <i>x</i>. For log sin <i>x</i> he -uses two circles, the first (see page (27)) for angles from -34′ 24″ to 5° 44′ 22″, -the second circle from -5° 44′ 22″ to 90°. The drawings do not show the seconds. He suggests many -different designs of rules. On page (29) he says:</p> -<blockquote> -<p>For the single projection of the <i>Circles of my Ring</i>, and the dividing and graduating of -them: which may bee so inserted upon the edges of <i>Circles</i> of mettle turned in the forme of a -<i>Ring</i>, so that one <i>Circle</i> may moove betweene two fixed, by helpe of two stayes, then may -there be graduated on the <i>face of the Ring</i>, upon the outer edge of the mooveable and inner -edge of the fixed, the <i>Circle of Numbers</i>, then upon the inner edge of that mooveable <i>Circle</i>, -and the outward edge of that inner fixed Circle may be inserted the <i>Circle of Sines</i>, and so -according to the description of those that are usually made.</p> -</blockquote> -<p>In addition to these lines he proceeds to mention the circle giving the ordinary -division into degrees and minutes, and two circles of tangents on the other side of -the rule.</p> -<p>Next Delamain explains an arrangement of all the graduation on one side of -the rule by means of “a small channell in the innermost <i>fixed Circle</i>, in which may -be placed a small single Index, which may have sufficient length to reach from the -innermost edge of the <i>Mooveable Circle</i>, unto the outmost edge of the <i>fixed Circle</i>, -which may be mooved to and fro at pleasure, in the channell, which Index may -serve to shew the opposition of Numbers” (p. (31)). From this it is clear that the -invention of the “runner” goes back to the very first writers on the slide rule.</p> -<p>After describing a modification of the above arrangement, he adds, “many -other formes might be deliverd, about this single <i>projection</i>” (p. (32)).</p> -<div class="pb" id="Page_202">[202]</div> -<p>Proceeding to the “enlarging” of the circles in the Ring, to, say, the “<i>Quadruple</i> -to that which is single, that is, foure times greater,” the “equall parts” are -distributed over four circles instead of only one circle, but the general method of -graduation is the same as before (p. (33)); there being now four circles carrying -the logarithms of numbers, and so on. Next he points out “severall wayes how -the Circles of the Mathematicall Ring (being inlarged) may be accommodated -for practicall use:” (1) The Circles are all fixed in a plain and movable flat -compasses (or better, a movable semicircle) are used for fixing any two positions; -(2) There is a “double projection” of each logarithmic line “inlarged on a Plaine,” -one fixed, the other movable, as shown in his first figure on the title-page, a single -index only being used; (3) use of “my great <i>Cylinder</i> which I have long proposed -(in which all the Circles are of equall greatnesse,) and it may be made of any -magnitude or capacity, but for a study (hee that will be at the charge) it may be -of a yard diameter and of such an indifferent length that it may containe 100 or -more Circles fixed parallel one to the other on the <i>Cylinder</i>, having a space betweene -each of them, so that there may bee as many mooveable Circles, as there are fixed -ones, and these of the mooveable linked, or fastened together, so that they may -all moove together by the fixed ones in these spaces, whose edges both of the fixed, -and mooveable being graduated by helpe of a single Index will shew the proportionalls -by opposition in this double <i>Projection</i>, or by a double <i>Index</i> in a single <i>Projection</i>” -(p. (36)).</p> -<p>Next follows the detailed description of his Ring “on a Plaine, according to -the diagramme that was given the King (for a view of that projection) and afterwards -the Ring it selve.” The diagram is the large one which we mentioned as -inserted between pages (23) and (24). The instrument has two circles, one moveable, -upon each of which are described 13 distinct circular graduations. The -lines on the fixed circle are: “The Circle of degrees and calendar,” E. “Circle of -equall parts, and part of the Equator, and Meridian,” TT. “The Circle of Tangents,” -S. “The Circle of Sines,” D. “The Circle of Decimals,” N. “The -Circle of Numbers.” The lines on the movable circle are: N. “The Circle of Numbers,” -E. “The Circle of equated figures, and bodies,” S. “The Circle of Sines,” -TT. “The Circle of Tangents,” Y. “The Circle of time, yeares, and monethes.”</p> -<p>On pages (84)-(88) Delamain explains an enlargement of his Ring for computations -involving the sines of angles near to 90°. On page (86) he says:</p> -<blockquote> -<p>I have continued the <i>Sines</i> of the <i>Projection</i> unto two severall <i>revolutions</i>, the one beginning -at 77.gr. 45.m. 6.s. and ends at 90.gr. (being the last <i>revolution</i> of the <i>decuplation</i> of the former, -or the hundred part of that <i>Projection</i>) the other beginning at 86.gr. 6.m. 48.s. and ends at -90.gr. (being the last of a ternary of <i>decuplated revolutions</i>, or the thousand part of that <i>Projection</i>) -and may bee thus used.</p> -</blockquote> -<p>He explains the manner of using these extra graduations. Thus he claims to have -attained degrees of accuracy which enabled him to do what “some one” had declared -“could not bee done.” It is hardly necessary to point out that Delamain’s -<i>Grammelogia IV</i> suggests designs of slide rules which inventors two hundred or -more years later were endeavouring to produce. Which of Delamain’s designs -<span class="pb" id="Page_203">[203]</span> -of rules were actually made and used, he does not state explicitly. He refers to a -rule 18 inches in diameter as if it had been actually constructed (pages (86), (88)). -Oughtred showed no appreciation of such study in designing and ridiculed Delamain’s -efforts, in his <i>Epistle</i>.</p> -<p>Additional elucidations of his designs of rules, along with explanations of the -relations of his work to that of Gunter and Napier, and sallies directed against -Oughtred and Forster, are contained on pages (8)-(21) of his <i>Grammelogia IV</i>.</p> -<h2 id="c12">V. INDEPENDENCE AND PRIORITY OF INVENTION</h2> -<p>The question of independence and priority of invention is discussed by Delamain -more specifically on pages (89)-(113); Oughtred devotes his entire <i>Epistle</i> -to it. It is difficult to determine definitely which publication is the later, Delamain’s -<i>Grammelogia IV</i> or Oughtred’s <i>Epistle</i>. Each seems to quote from the -other. Probably the explanation is that the two publications contain arguments -which were previously passed from one antagonist to the other by word of mouth -or by private letter. Oughtred refers in his <i>Epistle</i> (p. (12)) to a letter from Delamain. -We believe that the <i>Epistle</i> came after Delamain’s <i>Grammelogia IV</i>. -Delamain claims for himself the invention of the circular slide rule. He says in -his <i>Grammelogia IV</i>. (p. (99)), “when I had a sight of it, which was in <i>February</i>, -1629 (as I specified in my <i>Epistle</i>) I could not conceale it longer, envying my -selfe, that others did not tast of that which I found to carry with it so delightfull -and pleasant a goate [taste] . . .” Delamain asserts (without proof) that Oughtred -“never saw it as he now challengeth it to be his invention, untill it was so fitted -to his hand, and that he made all his practise on it after the publishing of my -<i>Booke</i> upon my <i>Ring</i>, and not before; so it was easie for him or some other to write -some uses of it in Latin after Christmas, 1630 and not the <i>Sommer</i> before, as is -falsely alledged by some one . . .” (p. (91)). Delamain’s accusation of theft on -the part of Oughtred cannot be seriously considered. Oughtred’s reputation as a -mathematician and his standing in his community go against such a supposition. -Moreover, William Forster is a witness for Oughtred. The fact that Oughtred -had the mastery of the rectilinear slide rule as well, while Delamain in 1630 speaks -only of the circular rule, weighs in Oughtred’s favour.</p> -<p>Oughtred says he invented the slide rule “above twelve yeares agoe,” that is, -about 1621, and “I with mine owne hand made me two such Circles, which I have -used ever since, as my occasions required,” (<i>Epistle</i> p. (22)). On the same page, -he describes his mode of discovery thus:</p> -<blockquote> -<p>I found that it required many times too great a paire of Compasses [in using Gunter’s -line], which would bee hard to open, apt to slip, and troublesome for use. I therefore first -devised to have another Ruler with the former: and so by setting and applying one to the other, -I did not onely take away the use of Compasses, but also make the worke much more easy -and expedite: when I should not at all need the motion of my hand, but onely the glancing -at my sight: and with one position of the Rulers, and view of mine eye, see not one onely, -but the manifold proportions incident unto the question intended. But yet this facility also -<span class="pb" id="Page_204">[204]</span> -wanted not some difficulty especially in the line of tangents, when one arch was in the former -mediety of the quadrant, and the other in the latter: for in this case it was needful that either -one Ruler must bee as long againe as the other; or else that I must use an inversion of the Ruler, -and regression. By this consideration I first of all saw that if those lines upon both Rulers -were inflected into two circles, that of the tangents being in both doubled, and that those two -Circles should move one upon another; they with a small thread in the center to direct the -sight, would bee sufficient with incredible and wonderfull facility to worke all questions of -Trigonometry . . .</p> -</blockquote> -<p>Oughtred said that he had no desire to publish his invention, but in the vacation -of 1630 finally promised William Forster to let him bring out a translation. Oughtred -claims that Delamain got the invention from him at Alhallontide [November -1], 1630, when they met in London. The accounts of that meeting we proceed -to give in double column.</p> -<div class="lcol"> -<blockquote> -<p class="center"><span class="sc">Delamain’s Statement</span> -<br /><i>Grammelogia</i> IV, page (98)</p> -<p>“. . . about Alhalontide 1630. (as our <i>Authors</i> -reporteth) was the time he was <i>circumvented</i>, -and then <i>his intent in a loving manner (as -before) he opened unto me</i>, which particularly -I will dismantle in the very naked truth: for, -wee being walking together some few weekes -before <i>Christmas</i>, upon <i>Fishstreet hill</i>, we -discoursed upon sundry things <i>Mathematicall</i>, -both <i>Theoreticall</i> and Practicall, and of the -excellent inventions and helpes that in these -dayes were produced, amongst which I was -not a little taken with that of the <i>Logarythmes</i>, -commending greatly the ingenuitie -of Mr. Gunter in the <i>Projection</i>, and inventing -of his <i>Ruler</i>, in the lines of proportion, extracted -from these <i>Logarythmes</i> for ordinary -<i>Practicall uses</i>; He replyed unto me (in these -very words) What will yov say to an <i>Invention</i> -that I have, which in a lesse extent of the -<i>Compasses</i> shall worke truer then that of -Mr. <i>Gunters Ruler</i>, I asked him then of what -forme it was, he answered with some pause -(which no doubt argued his suspition of mee -that I might conceive it) that it was <i>Arching-wise</i>, -but now hee sayes that hee told mee -then, it was <i>Circular</i> (but were I put to my -oath to avoid the guilt of Conscience I -would conclude in the former.) At which -immediately I answered, I had the like my -selfe, and so we discoursed not a word more -touching that subject . . . Then after my -coming home I sent him a sight of my <i>Projection</i> -drawne in <i>Pastboard</i>: Now admit I -had not the <i>Invention</i> of my <i>Ring</i> before I -discoursed . . . it was not so facil for mee . . . -to raise and compose so complete, and -absolute an <i>Instrument</i> from so small a -principle, or glimpse of light . . .”</p> -</blockquote> -</div><div class="rcol"> -<blockquote> -<p class="center"><span class="sc">Oughtred’s Statement</span> -<br /><i>Epistle</i>, page (23)</p> -<p>“Shortly after my gift to <i>Elias Allen</i>, I -chanced to meet with <i>Richard Delamain</i> in -the street (it was at Alhallontide) and as -we walked together I told him what an Instrument -I had given to Master <i>Allen</i>, both -of the Logarithmes projected into circles, -which being lesse then one foot diameter -would performe as much as one of Master -<i>Gunters</i> Rulers of sixe feet long: and also of -the Prostaphaereses of the Plannets and -second motions. <i>Such an invention have I</i> -said he: for now his <i>intentions</i> (that is his -ambition) beganne to worke: . . . But he -saith, <i>Then after my comming home I sent -him a sight of my projection drawne in past-board</i>. -See how notoriously he jugleth without -an Instrument. <i>Then after</i>: how long -after? <i>a sight of my projection</i>: of how much? -More then seven weekes after on December -23, he sent to mee the line of numbers onely -set upon a circle: . . . and so much onely he -presented to his Majesty: but as for Sine or -tangent of his, there was not the least shew -of any. Neither could he give to Master -<i>Allen</i> any direction for the composure of -the circles of his Ring, or for the division of -them: as upon his oath Master <i>Allen</i> will -testify how hee misled him, and made him -labour in vain above three weeks together, -until Master <i>Allen</i> himselfe found out his -ignorance and mistaking, which is more -cleare then is possible with any impudence -to be outfaced.”</p> -</blockquote> -</div> -<div class="pb" id="Page_205">[205]</div> -<p class="clear">Oughtred makes a further statement (<i>Epistle</i>, p. (24)) as follows:</p> -<blockquote> -<p>Delamain hearing that Brown with his <i>Serpentine</i> had <i>another line</i> by which he could worke -to minutes in the 90 degree of sines . . . gave the [his] booke to Browne: who in thankfulnesse -could not but gratify Delamain with his <i>Lines</i> also: and teach him the use of them, but especially -of the <i>great Line</i>: with this caution on both sides, that one should not meddle with the -others invention. Two dayes after <i>Delamain</i> . . . because he had found some things to be -altered therin, . . . asked for the booke . . . but as soone as he had got it in his hands he rent -out all the middle part with the two Schemes & put them up in his pocket & went his way . . . -and . . . laboureth to recall all the bookes he had given forth . . . And shortly after this he -got a new Printer (who was ignorant of his former Schemes) to print him new: giving him an -especiall charge of the <i>outermost line newly graven</i> in the Plate, which indeed is <i>Brownes very -line</i>: and then altering his book . . .</p> -</blockquote> -<p>This and other statements made by Oughtred seem damaging to Delamain’s -reputation. But it is quite possible that Oughtred’s guesses as to Delamain’s -motives are wrong. Moreover, some of Oughtred’s statements are not first hand -knowledge with him, but mere hearsay. One may accept his first hand facts and -still clear Delamain of wrong doing. There is always danger that rival claimants -of an invention or discovery will proceed on the assumption that no one else could -possibly have come independently upon the same devices that they themselves -did; the history of science proves the opposite. Seldom is an invention of any note -made by only one man. We do not feel competent to judge Delamain’s case. -We know too little about him as a man. We incline to the opinion that the hypothesis -of independent invention is the most plausible. At any rate, Delamain -figures in the history of the slide rule as the publisher of the earliest book thereon -and as an enthusiastic and skillful designer of slide rules.</p> -<p>The effect of this controversy upon interested friends was probably small. -Doubtless few people read both sides. Oughtred says:<a class="fn" id="fr_21" href="#fn_21">[21]</a> “this scandall . . . hath -with them, to whom I am not knowne, wrought me much prejudice and disadvantage -. .” Aubrey,<a class="fn" id="fr_22" href="#fn_22">[22]</a> a friend of Oughtred, refers to Delamain “who was so sawcy -to write against him” and remembers having seen “many yeares since, twenty or -more good verses made” against Delamain. Another friend of Oughtred, William -Robinson, who had seen some of Delamain’s publications, but not his <i>Grammelogia -IV</i>, wrote in a letter to Oughtred, shortly before the appearance of the latter’s -<i>Epistle</i>:</p> -<blockquote> -<p>I cannot but wonder at the indiscretion of Rich. Delamain, who being conscious to himself -that he is but the pickpurse of another man’s wit, would thus inconsiderately provoke -and awake a sleeping lion . . . he hath so weakly (though in my judgment, vaingloriously -enough) commended his own labour . . .<a class="fn" id="fr_23" href="#fn_23">[23]</a></p> -</blockquote> -<p>Delamain presented King Charles I with one of his sun-dials, also with a manuscript -and, later, with a printed copy of his book of 1630. A drawing of his improved -slide rule was sent to the King and the <i>Grammelogia IV</i> is dedicated to him. -<span class="pb" id="Page_206">[206]</span> -The King must have been favorably impressed, for Delamain was appointed tutor -to the King in mathematics. His widow petitioned the House of Lords in 1645 -for relief; he had ten children.<a class="fn" id="fr_24" href="#fn_24">[24]</a></p> -<p>Anthony Wood states that Charles I, on the day of his execution, commanded -his friend Thomas Herbert “to give his son the duke of York his large ring-sundial -of silver, a jewel his maj. much valued.” Anthony Wood adds, “it was invented -and made by Rich. Delamaine a very able mathematician, who projected it, and -in a little printed book did shew its excellent use in resolving many questions in -arithmetic and other rare operations to be wrought by it in the mathematics.”<a class="fn" id="fr_25" href="#fn_25">[25]</a></p> -<h2 id="c13">VI. OUGHTRED’S GAUGING LINE, 1633</h2> -<p>It has not been generally known, hitherto, that Oughtred designed a rectilinear -slide rule for gauging and published a description thereof in 1633.<a class="fn" id="fr_26" href="#fn_26">[26]</a> In his <i>Circles -of Proportion</i>, chapter IX, Oughtred had offered a closer approximation than that -of Gunter for the capacity of casks. The Gauger of London expostulated with -Oughtred for presuming to question anything that Gunter had written. The -ensuing discussion led to an invitation extended by the Company of Vintners to -the instrument maker Elias Allen to request Oughtred to design a gauging rod.<a class="fn" id="fr_27" href="#fn_27">[27]</a> -This he did, and Allen received an order for “threescore” instruments. On page -19 Oughtred describes his ‘Gauging Rod:’</p> -<blockquote> -<p>It consisteth of <i>two rulers of brasse</i> about 32 ynches of length, which also are halfe an ynch -broad, and a quarter of an ynch thick . . . At one end of both those rulers are <i>two little sockets</i> -of brasse fastened on strongly: by which the rulers are held together, and made to move one -upon another, and to bee drawne out unto any length, as occasion shall require: and when -you have them at the just length, there is upon one of the sockets <i>a long Scrue-pin</i> to scrue -them fast.</p> -</blockquote> -<p>There are graduations on three sides of the rulers, one graduation being the logarithmic -line of numbers. He says (p. 39), “the maner of computing the <i>Gauge-divisions</i> -I have concealed.” W. Robinson, who was a friend of Oughtred, wrote -him as follows:<a class="fn" id="fr_28" href="#fn_28">[28]</a></p> -<blockquote> -<div class="pb" id="Page_207">[207]</div> -<p>I have light upon your little book of artificial gauging, wherewith I am much taken, but -I want the rod, neither could I get a sight of one of them at the time, because Mr. Allen had -none left . . . I forgot to ask Mr. Allen the price of one of them, which if not much I would -have one of them.” Oughtred annotated this passage thus: “Or in wood, if any be made in -wood by Thompson or any other.”</p> -</blockquote> -<p>Another of Oughtred’s admirers, Sir Charles Cavendish, wrote, on February 11, -1635 thus:<a class="fn" id="fr_29" href="#fn_29">[29]</a></p> -<blockquote> -<p>I thank you for your little book, but especially for the way of calculating the divisions of -your gauging rod. I wish, both for their own sakes and yours, that the citizens were as capable -of the acuteness of this invention, as they are commonly greedy of gain, and then I -doubt not but they would give you a better recompense than I doubt now they will.</p> -</blockquote> -<p>On April 20, 1638, we find Oughtred giving Elias Allen directions<a class="fn" id="fr_30" href="#fn_30">[30]</a> “about -the making of the two rulers.” As in 1633,<a class="fn" id="fr_31" href="#fn_31">[31]</a> so now, Oughtred takes one ruler -longer than the other. This 1633 instrument was used also as “a crosse-staffe to -take the height of the Sunne, or any Starre above the Horizon, and also their distances.” -The longer ruler was called <i>staffe</i>, the shorter <i>transversarie</i>. While in -1633 he took the lengths of the two in the ratio “almost 3 to 2,” in 1638, he took -“the transversary three quarters of the staff’s length, . . . that the divisions may -be larger.”</p> -<h2 id="c14">VII. OTHER SEVENTEENTH CENTURY SLIDE RULES</h2> -<p>In my <i>History of the Slide Rule</i> I treat of Seth Partridge, Thomas Everard, -Henry Coggeshall, W. Hunt and Sir Isaac Newton.<a class="fn" id="fr_32" href="#fn_32">[32]</a> Of Partridge’s <i>Double -Scale of Proportion</i>, London, I have examined a copy dated 1661, which is the -earliest date for this book that I have seen. As far as we know, 1661 is the earliest -date of publications on the slide rule, since Oughtred and Delamain. But it -would not be surprising if the intervening 28 years were found not so barren as -they seem at present. The 1661 and 1662 impressions of Partridge are identical, -except for the date on the title-page. William Leybourn, who printed Partridge’s -book, speaks in high appreciation of it in his own book.<a class="fn" id="fr_33" href="#fn_33">[33]</a></p> -<p>In 1661 was published also John Brown’s first book, <i>Description and Use of a -Joynt-Rule</i>, previously mentioned. In Chapter XVIII he describes the use of -“Mr. Whites rule” for the measuring of board and timber, round and square. He -calls this a “sliding rule.” The existence, in 1661, of a “Whites rule” indicates -activities in designing of which we know as yet very little. In his book of 1761, -previously quoted, Brown gives a drawing of “White’s sliding rule” (p. 193); -also a special contrivance of his own, as indicated by him in these words:</p> -<blockquote> -<p>A further improvement of the Triangular Quadrant, as I have made it several times, with -a sliding Cover on the in-side, when made hollow, to carry Ink, Pens, and Compasses; then -on the sliding Cover, and Edges, is put the Line of Numbers, according to Mr. White’s first -Contrivance for manner of operation; but much augmented, and made easie, by John Brown.</p> -</blockquote> -<div class="pb" id="Page_208">[208]</div> -<p>He gives no drawing of his “triangular quadrant,” hence his account of it is unsatisfactory. -He explains the use of “gage-points.” His placing logarithmic lines -on the edges of instrument boxes was outdone in oddity later by Everard who -placed them on tobacco-boxes.<a class="fn" id="fr_34" href="#fn_34">[34]</a> In Brown’s publication of 1704 the White slide -rule is given again, “being as neat and ready a way as ever was used.” He tells -also of a “glasier’s sliding rule.” William Leybourn explains in 1673 how Wingate’s -double and triple lines for squaring and cubing, or square and cube root, can be -used on slide rules.<a class="fn" id="fr_35" href="#fn_35">[35]</a></p> -<p>Beginning early in the history of the slide rule, when Oughtred designed his -“gauging rod,” we notice the designing of rules intended for very special purposes. -Another such contrivance, which enjoyed long popularity, was the <i>Timber Measure -by a Line</i>, by Hen. Coggeshall, Gent., London, 1677, a booklet of 35 pages. Coggeshall -says in his preface:</p> -<blockquote> -<p>For what can be more ready and easie, then having set twelve to the length, to see the -Content exactly against the Girt or Side of the Square. Whereas on Mr. Partridge’s Scale -the Content is the Sixth Number, which is far more troublesome then [even] with Compasses.</p> -</blockquote> -<p>One line on Coggeshall’s rule begins with 4 and extends to 40, these numbers being -the “Girt” (a quarter of the circumference), which in ordinary practice of measuring -round timber lies between 4 inches and 40 inches. This “Girt line” slides “against -the line of Numbers in two Lengths, to which it is exactly equal.” A second edition, -1682, shows some changes in the rule, as well as an enlargement and change of -title of the book itself: <i>A Treatise of Measures, by a Two-foot Rule</i>, by H. C. Gent, -London, 1682. In this, the description of the rule is given thus:</p> -<blockquote> -<p>There are four Lines on each flat of this Rule; two next the outward edges, which are -Lines of Measure; and two next the inward edges, which are Lines of Proportion. On one -flat, next the inward edges, is the Square-line [Girt-line in round timber measurement] with -the Line of Numbers his fellow. Next the outward, a Line of Inches divided into Halfs, -Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 to 24 on the other. On -the other flat, next the inward edges, is the double Scale of Numbers [for solving proportions]. -Next the outward on one Rule a Line of Inches divided each into ten parts; and this for gauging, -etc. On the other a foot divided into 100 parts.</p> -</blockquote> -<p>Later further changes were introduced in Coggeshall’s rule.<a class="fn" id="fr_36" href="#fn_36">[36]</a></p> -<p>It is worthy of note that Coggeshall’s slide rule book, <i>The Art of Practical -Measuring</i>, was reviewed in the <i>Acta eruditorum</i>, anno 1691, p. 473; hence Leupold’s -description<a class="fn" id="fr_37" href="#fn_37">[37]</a> of the rectilinear slide rule in his <i>Theatrum arithmetico-geometricum</i>, -Leipzig, 1727, Cap. XIII, p. 71, is not the earliest reference to the rectilinear rule -found in German publications. The above date is earlier even than Biler’s reference -to a circular slide rule in his <i>Descriptio instrumenti mathematici universalis</i> -of 1696.</p> -<div class="pb" id="Page_209">[209]</div> -<p>Two noted slide rules for gauging were described by Tho. Everard, -Philomath, in his <i>Stereometry made easie</i>, London, 1684. He -designates his lines by the capital letters A, B, C, D, E. On the -first instrument, <i>A</i> on the rule, and <i>B</i> and <i>C</i> -on the slide, have each two radiuses of numbers, <i>D</i> has only -one, while <i>E</i> has three. The second rule is described in an -<i>Appendix</i>; it is one foot long, with two slides enabling -the rule to be extended to 3 feet.</p> -<p>Everard’s instruments were made in London by Isaac Carver who, soon after, -himself wrote a sixteen-page <i>Description and Use of a New Sliding Rule, projected -from the Tables in the Gauger’s Magazine</i>, London, 1687, which was “printed for -William Hunt” and bound in one volume with a book by Hunt, called <i>The Gauger’s -Magazine</i>, London, 1687. This appears to be the same William Hunt who later -brought out descriptions of his own of slide rules. The instrument described by -Carver “consists of three pieces, two whereof are moveable to be drawn out till -the whole be 36 inches long.” It has several non-logarithmic graduations, together -with logarithmic lines marked A, B, C, D, of which A, B, C are “double -lines,” and D a “single line” used for squares and square roots. It is designed -for the determination of the vacuity of a “spheroidal cask lying,” a “spheroidal -cask standing,” and a “parabolical cask lying.”</p> -<p>Another seventeenth century writer on the slide rule is John Atkinson, whom we -have mentioned earlier. He says:<a class="fn" id="fr_38" href="#fn_38">[38]</a> “The Lines of Numbers, Sines and Tangents, -are set double, that is, one on each side, as the middle piece slides: which middle -piece is so contrived, to slip to and fro easily, to slide out, and to be put in any side -uppermost, in order to bring those Lines together (or against one another) most -proper for solving the Question, wrought by <i>Sliding-Gunter</i>.”</p> -<p>The data presented in this article show that, while the earliest -slide rules were of the circular type, the later slide rules of the -seventeenth century were of the rectilinear type.<a class="fn" id="fr_39" href="#fn_39">[39]</a></p> -<p class="jr1"><i>January 12, 1915.</i></p> -<h2 id="c15">Footnotes</h2> -<div class="fnblock"><div class="fndef"><a class="fn" id="fn_1" href="#fr_1">[1]</a>F. Cajori, <i>History of the Logarithmic Slide Rule and Allied Instruments</i>, New York, 1909, -pp. 7-14, also Addenda i-vi. -</div><div class="fndef"><a class="fn" id="fn_2" href="#fr_2">[2]</a>F. Cajori, “On the Invention of the Slide Rule,” in <i>Colorado College Publication</i>, Engineering -Series Vol. 1, 1910. An abstract of this is given in <i>Nature</i> (London), Vol. 82, 1909, p. 267. -</div><div class="fndef"><a class="fn" id="fn_3" href="#fr_3">[3]</a>F. Cajori, <i>History</i> etc., p. 14. -</div><div class="fndef"><a class="fn" id="fn_4" href="#fr_4">[4]</a>Art. “Slide Rule” in the <i>Penny Cyclopaedia</i> and in the <i>English Cyclopaedia</i> [Arts and -Sciences]. -</div><div class="fndef"><a class="fn" id="fn_5" href="#fr_5">[5]</a>Anthony Wood, <i>Athenae oxonienses</i> (Ed. P. Bliss), London, Vol. III, 1817, p. 423. -</div><div class="fndef"><a class="fn" id="fn_6" href="#fr_6">[6]</a>The full title of the book -which Wingate published on this subject in Paris is as follows: -<p class="fncont">L’Vsage | de la | Reigle de | Proportion | en l’Arithmetique & | -Geometrie. | Par Edmond Vvingate, | Gentil-homme Anglois. |</p> -<p class="fncont"><span class="greek" title="Ean ês philomathês, esê polymathês.">Εἂν -ἧς -φιλεμαθὴς, -ἕση ἥση -πολυμαθὴς.</span></p> -<p class="fnq">In tenui, sed nõ tenuis vsusve, laborne. |</p> -<p class="fnq">A Paris, | Chez Melchior Mondiere, | demeurant en l’Isle du Palais, | à la | ruë de Harlay aux -deux Viperes. | M. DC. XXIV. | Auec Priuilege du Roy. |</p> -<p class="fncont">Back of the title page is the announcement:</p> -<p class="fnq">Notez que la Reigle de Proportion en toutes -façons se vend à Paris chez Melchior Tauernier, Graueur -& Imprimeur du Roy pour les Tailles douces, demeurant en l’Isle du -Palais sur le Quay qui regarde la Megisserie à l’Espic d’or.</p> -</div><div class="fndef"><a class="fn" id="fn_7" href="#fr_7">[7]</a>The title-page of the edition of 1658 is as follows: -<p class="fnq">The Use of the Rule of Proportion in Arithmetick -& Geometrie. First published at -Paris in the French tongue, and dedicated to Monsieur, the then king’s onely Brother (now -Duke of Orleance). By Edm. Wingate, an English Gent. And now translated into -English by the Author. Whereinto is now also inserted the Construction of the same Rule, -& a farther use thereof . . . 2nd edition inlarged and amended. -London, 1658.</p> -</div><div class="fndef"><a class="fn" id="fn_8" href="#fr_8">[8]</a><i>Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; Drawn up by himself by -way of Diary. With Appendix of original Letters.</i> Publish’d by Charles Burman, Esq., London, -1717, p. 23. -</div><div class="fndef"><a class="fn" id="fn_9" href="#fr_9">[9]</a><i>Mathematical Tables</i>, 1811, p. 36, and art. “Gunter’s Line” in his <i>Phil. and Math. Dictionary</i>, -London, 1815. -</div><div class="fndef"><a class="fn" id="fn_10" href="#fr_10">[10]</a><i>To the English Gentrie, and all others studious of the Mathematicks, which shall bee readers -hereof. The just Apologie of Wil: Ovghtred, against the slaunderous insimulations of Richard -Delamain, in a Pamphlet called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum -projectio circularis.</i> We shall refer to this document as <i>Epistle</i>. It was published without date -in 32 unnumbered pages of fine print, and was bound in with Oughtred’s <i>Circles of Proportion</i>, -in the editions of 1633 and 1639. In the 1633 edition it is inserted at the end of the volume just -after the <i>Addition vnto the Vse of the Instrument etc.</i>, and in that of 1639 immediately after the -preface. It was omitted from the Oxford edition of 1660. The <i>Epistle</i> was also published -separately. There is a separate copy in the British Museum, London. Aubrey, in his -<i>Brief Lives</i>, edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, “He writt a stitch’t -pamphlet about 163(?4) against . . . Delamaine.” -</div><div class="fndef"><a class="fn" id="fn_11" href="#fr_11">[11]</a>Thomas Browne is mentioned by Stone in his <i>Mathematical Instruments</i>, London 1723, p. 16. -See also Cajori, <i>History of the Slide Rule</i>, New York, 1909, p. 15. -</div><div class="fndef"><a class="fn" id="fn_12" href="#fr_12">[12]</a><i>The Description and Use of a Joynt-Rule: . . . also the use of Mr. White’s Rule for measuring -of Board and Timber, round and square; With the manner of Vsing the Serpentine-line of Numbers, -Sines, Tangents, and Versed Sines.</i> By J. Brown, Philom., London, 1661. -</div><div class="fndef"><a class="fn" id="fn_13" href="#fr_13">[13]</a><i>A Collection of Centers and Useful Proportions on the Line of Numbers</i>, by John Brown, -1662(?), 16 pages; <i>Description and Use of the Triangular Quadrant</i>, by John Brown, London, 1671; -<i>Wingate’s Rule of Proportion in Arithmetick and Geometry: or Gunter’s Line</i>. <i>Newly rectified by -Mr. Brown and Mr. Atkinson, Teachers of the Mathematicks</i>, London, 1683; <i>The Description and -Use of the Carpenter’s-Rule: Together with the Use of the Line of Numbers commonly call’d Gunter’s-Line</i>, -by John Brown, London, 1704. -</div><div class="fndef"><a class="fn" id="fn_14" href="#fr_14">[14]</a>William Leybourn, <i>op. cit.</i>, pp. 129, 130, 132, 133. -</div><div class="fndef"><a class="fn" id="fn_15" href="#fr_15">[15]</a>James Atkinson’s edition of Andrew Wakely’s <i>The Mariners Compass Rectified</i>, London, -1694 [Wakely’s preface dated 1664, Atkinson’s preface, 1693]. Atkinson adds <i>An Appendix -containing Use of Instruments most useful in Navigation</i>. Our quotation is from this <i>Appendix</i>, -p. 199. -</div><div class="fndef"><a class="fn" id="fn_16" href="#fr_16">[16]</a>R. Delamain, <i>The Making, Description, and Use of a small portable Instrument . . . called a -Horizontall Quadrant</i>, etc., London, 1631. -</div><div class="fndef"><a class="fn" id="fn_17" href="#fr_17">[17]</a>Oughtred’s description of his circular slide rule of 1632 and his rectilinear slide rule of 1633, -as well as a drawing of the circular slide rule, are reproduced in Cajori’s <i>History of the Slide Rule</i>, -Addenda, pp. ii-vi. -</div><div class="fndef"><a class="fn" id="fn_18" href="#fr_18">[18]</a>The full title of the <i>Grammelogia I</i> is as follows: -<p class="fnq">Gram̄elogia | or, | The Mathematicall Ring. | Shewing (any reasonable Capacity that hath | -not Arithmeticke) how to resolve and worke | all ordinary operations of Arithmeticke. | -And those which are most difficult with greatest | facilitie: The extraction of Roots, the -valuation of | Leases, &c. The measuring of Plaines | and Solids. | With the resolution of -Plaine and Sphericall | Triangles. | And that onely by an Ocular Inspection, | and a Circular -Motion. | Naturae secreta tempus aperit. | London printed by John Haviland, 1630.</p> -</div><div class="fndef"><a class="fn" id="fn_19" href="#fr_19">[19]</a><i>Grammelogia III</i> is the same as <i>Grammelogia I</i>, -except for the addition of an appendix, entitled: -<p class="fnq">De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit nullo (praescripto -tempore) vitae | vsuram nobis ingeniique Deus. | London, |</p> -<p class="fncont">. . . The next line or two of this title-page which probably contained the date of publication, were -cut off by the binder in trimming the edges of this and several other pamphlets for binding into -one volume.</p> -</div><div class="fndef"><a class="fn" id="fn_20" href="#fr_20">[20]</a><i>Grammelogia IV</i> has two title pages. The first is <i>Mirifica Logarithmoru’ Projectio Circularis</i>. -There follows a diagram of a circular slide rule, with the inscription within the innermost -ring: <i>Nil Finis, Motvs, Circvlvs vllvs Habet</i>. The second title page is as follows: -<p class="fnq">Grammelogia | Or, the Mathematicall Ring. | Extracted from the Logarythmes, and projected -Circular: Now published in the | inlargement thereof unto any magnitude fit for use: -shewing any reason- | able capacity that hath not Arithmeticke how to resolve and worke, | -all ordinary operations of Arithmeticke: | And those that are most difficult with greatest -facilitie, the extracti- | on of Rootes, the valuation of Leases, &c. the measuring of Plaines -and Solids, | with the resolution of Plaine and Sphericall Triangles applied to the | Practicall -parts of Geometrie, Horologographie, Geographie | Fortification, Navigation, Astronomie, -&c. | And that onely by an ocular inspection, and a Circular motion, Invented and first -published, by R. Delamain, Teacher, and Student of the Mathematicks. | Naturae secreta -tempus aperit. |</p> -<p class="fncont">There is no date. There follows the diagram of a second circular slide rule, with the inscription -within the innermost ring: <i>Typus proiectionis Annuli adaucti vt in Conslusione Lybri praelo -commissi, Anno 1630 promisi</i>. There are numerous drawings in the <i>Grammelogia</i>, all of which, -excepting the drawings of slide rules on the engraved title-pages of <i>Grammelogia IV</i> and <i>V</i>, were -printed upon separate pieces of paper and then inserted by hand into the vacant spaces on the -printed pages reserved for them. Some drawings are missing, so that the Bodleian <i>Grammelogia -IV</i> differs in this respect slightly from the copy in the British Museum and from the British -Museum copy of <i>Grammelogia V</i>.</p> -</div><div class="fndef"><a class="fn" id="fn_21" href="#fr_21">[21]</a><i>Epistle</i>, p. (8). -</div><div class="fndef"><a class="fn" id="fn_22" href="#fr_22">[22]</a>Aubrey, <i>op. cit.</i>, Vol. II., p. 111. -</div><div class="fndef"><a class="fn" id="fn_23" href="#fr_23">[23]</a>Rigaud, <i>Correspondence of Scientific Men during the 17th Century</i>, Vol. I, Oxford, 1841, p. 11. -</div><div class="fndef"><a class="fn" id="fn_24" href="#fr_24">[24]</a><i>Dictionary of National Biography</i>, Art. “Delamain, Richard.” See also Rev. Charles J. -Robinson, <i>Taylors’ School, from A.D. 1562 to 1874</i>, Vol. I, 1882, p. 151; <i>Journal of the House of -Commons</i>, Vol. IV., p. 197<i>b</i>; <i>Sixth Report of the Royal Commission on Historical Manuscripts</i>, -Part I, Report and Appendix, London, 1877. In this <i>Appendix</i>, p. 82, we read the following: -<p class="fnq">Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. Petitioner’s -husband was servant to the King, and one of His Majesty’s engineers for the fortification -of the kingdom, and his tutor in mathematical arts; but upon the breaking out of the war -he deserted the Court, and was called by the State to several employments, in fortifying -the towns of Northampton, Newport, and Abingdon; and was also abroad with the armies -as Quartermaster-General of the Foot, and therein died. Petitioner is left a disconsolate -widow with ten children, the four least of whom are now afflicted with sickness, and -petitioner has nothing left to support them. There are several considerable sums of -money due to the petitioner, as well from the King as the State. Prays that she may -have some relief amongst other widows. See L. J., VII. 6. 657.</p> -</div><div class="fndef"><a class="fn" id="fn_25" href="#fr_25">[25]</a>Anthony Wood, <i>Athenae Oxonienses</i> (Edition Bliss) Vol. IV., London, 1820, p. 34. -</div><div class="fndef"><a class="fn" id="fn_26" href="#fr_26">[26]</a><i>The New Artificial Gauging Line or Rod: together with rules concerning the use thereof: Invented -and written by WILLIAM OUGHTRED</i>, etc., London, 1633. The copy we have seen is -in the Bodleian Library, Oxford. The book is small sized and has 40 pages. -</div><div class="fndef"><a class="fn" id="fn_27" href="#fr_27">[27]</a>Oughtred, <i>op. cit.</i>, p. 11. -</div><div class="fndef"><a class="fn" id="fn_28" href="#fr_28">[28]</a>S. J. Rigaud, <i>Correspondence of Scientific Men of the 17th Century</i>, Oxford, Vol. I, 1841, p. 17. -</div><div class="fndef"><a class="fn" id="fn_29" href="#fr_29">[29]</a>Rigaud, <i>loc. cit.</i>, p. 22. -</div><div class="fndef"><a class="fn" id="fn_30" href="#fr_30">[30]</a>Rigaud, <i>loc. cit.</i>, pp. 30, 31. -</div><div class="fndef"><a class="fn" id="fn_31" href="#fr_31">[31]</a>Oughtred, <i>An Addition vnto the Vse of the Instrument called the Circles of Proportion</i>, London, -1633, p. 63. -</div><div class="fndef"><a class="fn" id="fn_32" href="#fr_32">[32]</a>F. Cajori, <i>History of the Slide Rule</i>, New York, 1909, pp. 16-22, Addenda, pp. vi-ix. -</div><div class="fndef"><a class="fn" id="fn_33" href="#fr_33">[33]</a>W. Leybourn, <i>op. cit.</i>, 1673, Preface, and pp. 128-29. -</div><div class="fndef"><a class="fn" id="fn_34" href="#fr_34">[34]</a>Cajori <i>op. cit.</i>, Addenda, p. ix. -</div><div class="fndef"><a class="fn" id="fn_35" href="#fr_35">[35]</a>William Leybourn, <i>op. cit.</i>, 1673, p. 35. -</div><div class="fndef"><a class="fn" id="fn_36" href="#fr_36">[36]</a>See Cajori, <i>op. cit.</i>, pp. 20, 28, Addenda, p. ix. -</div><div class="fndef"><a class="fn" id="fn_37" href="#fr_37">[37]</a>See F. Cajori, “A Note on the History of the Slide Rule,” <i>Bibliotheca mathematica</i>, 3 F., -Vol. 10, pp. 161-163. -</div><div class="fndef"><a class="fn" id="fn_38" href="#fr_38">[38]</a>John Atkinson, <i>op. cit.</i>, 1694, p. 204. -</div><div class="fndef"><a class="fn" id="fn_39" href="#fr_39">[39]</a>Probably the oldest slide rule now in existence is owned by St. John’s College, Oxford, and -is in the form of a brass disc, 1 ft. 6 in. in diameter. It was exhibited along with other instruments -in May, 1919. According to the <i>Catalogue of a Loan Exhibition of Early Scientific Instruments</i> -in Oxford, opened May 16, 1919, the instrument is inscribed with the name of the maker (“<i>Elias -Allen fecit</i>”) and with the name of the donor, Georgius Barkham. It is dated 1635, which is only -three years after the first publication of Oughtred’s description of his circular slide rule. It -is stated in the <i>Catalogue</i>: “Unfortunately all the movable parts but the base-plate and a couple -of thumb-screws are missing. The face of the instrument is engraved with Oughtred’s <i>Horizontal -Instrument</i>. The back is engraved with eleven Circles of Proportion as described in Arthur -Haughton’s book, a copy of which was presented to St. John’s College by George Barkham, to -explain the use of the instrument.” As Arthur Haughton’s Oxford edition of Oughtred’s <i>Circles -of Proportion</i> did not appear until 1660, it would seem that the instrument was probably not presented -to the College before 1660. As far as is known, the next oldest slide rule is of the year -1654, kept in the South Kensington Museum, London, and is described in <i>Nature</i> of March 5, -1914. It is a rectilinear rule, “of boxwood, well made, and bound together with brass at the two -ends. It is of the square type, a little more than 2 ft. in length, and bears the logarithmic lines -first described by Edmund Gunter. Of these, the <i>num</i>, <i>sin</i> and <i>tan</i> lines are arranged in pairs, -identical and contiguous, one line in each pair being on the fixed part, and the other on the slide.” -The instrument is inscribed, “Made by Robert Bissaker for T. W., 1654.” Nowhere else have -we seen reference to Robert Bissaker. His slide rule seems to antedate the “Whites rule” -mentioned above. [This foot-note was added on October 15, 1919.] -</div> -</div> -<div class="pb" id="Page_211">[211]</div> -<h2 id="c16">UNIVERSITY OF CALIFORNIA PUBLICATIONS</h2> -<p>Note.—The University of California Publications are offered in -exchange for the publication of learned societies and institutions, -universities and libraries. Complete lists of all the publications -of the University will be sent upon request. For sample copies, lists -of publications or other information, address the Manager of the -University of California Press, Berkeley, California, U. S. A. 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Current, -Volume XXII.</p> -<h2 id="c17">Transcriber’s Notes</h2> -<ul><li>Inserted subchapter headings from the Table of Contents into the text.</li> -<li>Corrected two page numbers in the Table of Contents</li> -<li>Corrected a Greek quotation in footnote 6 by reference to the original sources.</li></ul> - - - - - - - -<pre> - - - - - -End of the Project Gutenberg EBook of On the History of Gunter's Scale and -the Slide Rule during the Seventeent, by Florian Cajori - -*** END OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - -***** This file should be named 42216-h.htm or 42216-h.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/4/2/2/1/42216/ - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - -Updated editions will replace the previous one--the old editions -will be renamed. - -Creating the works from public domain print editions means that no -one owns a United States copyright in these works, so the Foundation -(and you!) can copy and distribute it in the United States without -permission and without paying copyright royalties. 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You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org - - -Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century - -Author: Florian Cajori - -Release Date: February 26, 2013 [EBook #42216] - -Language: English - -Character set encoding: ASCII - -*** START OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - - - - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - IN - MATHEMATICS - - Vol. 1, No. 9, pp. 187-209 February 17, 1920 - - - - - ON THE HISTORY OF GUNTER'S SCALE AND - THE SLIDE RULE DURING THE - SEVENTEENTH CENTURY - - - BY - FLORIAN CAJORI - - - UNIVERSITY OF CALIFORNIA PRESS - BERKELEY - - - - - TABLE OF CONTENTS - - - PAGE - I. Introduction 187 - II. Innovations in Gunter's Scale 188 - Changes introduced by Edmund Wingate 188 - Changes introduced by Milbourn 189 - Changes introduced by Thomas Brown and John Brown 190 - Changes introduced by William Leybourn 192 - III. Richard Delamain's "Grammelogia" 192 - Different editions or impressions 194 - Description of Delamain's instrument of 1630 195 - Delamain's later designs, and directions for using his - instruments 197 - IV. Controversy between Oughtred and Delamain on the invention of - the circular slide rule 199 - V. Independence and priority of invention 203 - VI. Oughtred's "Gauging Line," 1633 206 - VII. Other seventeenth century slide rules 207 - - - - - I. INTRODUCTION - - - In my history of the slide rule[1], and my article on its invention[2] it -is shewn that William Oughtred and not Edmund Wingate is the inventor, that -Oughtred's circular rule was described in print in 1632, his rectilinear -rule in 1633. Richard Delamain is referred to as having tried to -appropriate the invention to himself[3] and as having written a scurrilous -pamphlet against Oughtred. All our information about Delamain was taken -from De Morgan,[4] who, however, gives no evidence of having read any of -Delamain's writings on the slide rule. Through Dr. Arthur Hutchinson of -Pembroke College, Cambridge, I learned that Delamain's writings on the -slide rule were available. In this article will be given: First, some -details of the changes introduced during the seventeenth century in the -design of Gunter's scale by Edmund Wingate, Milbourn, Thomas Brown, John -Brown and William Leybourn; second, an account of Delamain's book of 1630 -on the slide rule which antedates Oughtred's first publication (though -Oughtred's date of invention is earlier than the date of Delamain's alleged -invention) and of Delamain's later designs of slide rules; third, an -account of the controversy between Delamain and Oughtred; fourth, an -account of a later book on the slide rule written by William Oughtred, and -of other seventeenth century books on the slide rule. - - - - - II. INNOVATIONS IN GUNTER'S SCALE - - - Changes introduced by Wingate - - We begin with Anthony Wood's account of Wingate's introduction of -Gunter's scale into France.[5] - - In 1624 he transported into France the rule of proportion, having a - little before been invented by Edm. Gunter of Gresham Coll. and - communicated it to most of the chiefest mathematicians then residing in - Paris: who apprehending the great benefit that might accrue thereby, - importun'd him to express the use thereof in the French tongue. Which - being performed accordingly, he was advised by monsieur Alleawne the - King's chief engineer to dedicate his book to monsieur the King's only - brother, since duke of Orleans. Nevertheless the said work coming forth - as an abortive (the publishing thereof being somewhat hastened, by reason - an advocate of Dijon in Burgundy began to print some uses thereof, which - Wingate had in a friendly way communicated to him) especially in regard - Gunter himself had learnedly explained its use in a far larger volume.[6] - -Gunter's scale, which Wingate calls the "rule of proportion," contained, as -described in the French edition of 1624, four lines: (1) A single line of -numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot -in length, divided into 12 inches and tenths of inches, also a line, one -foot in length, divided into tenths and hundredths. - -The English editions of this book which appeared in 1623 and 1628 are -devoid of interest. The editions of 1645 and 1658 contain an important -innovation.[7] In the preface the reasons why this instrument has not been -used more are stated to be: (1) the difficulty of drawing the lines with -exactness, (2) the trouble of working thereupon by reason (sometimes) of -too large an extent of the compasses, (3) the fact that the instrument is -not readily portable. The drawing of Wingate's arrangement of the scale in -the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains -five parallel lines, about 66 cm. long, each having the divisions of one -line marked on one side and of another line on the other side. Thus each -line carries two graduations: (1) A single logarithmic line of numbers; (2) -a logarithmic line of numbers thrice repeated; (3) the first scale -repeated, but beginning with the graduations which are near the middle of -the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3; -(4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of -tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000 -equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of -inches; (10) a scale consisting of three kinds, viz., a gauge line, a line -of chords, and a foot measure, divided into 1000 equal parts. - -Important are the first and second scales, by which cube root extraction -was possible "by inspection only, without the aid of pen or compass;" -similarly the third and fourth scales, for square roots. This innovation is -due to Wingate. The 1645 edition announces that the instrument was made in -brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in -Hosier Lane. - - - Changes introduced by Milbourn - -William Leybourn, in his The Line of Proportion or Numbers, Commonly called -Gunter's Line, Made Easie, London, 1673, says in his preface "To the -Reader:" - - The Line of Proportion or Numbers, commonly called (by Artificers) - Gunter's Line, hath been discoursed of by several persons, and variously - applied to divers uses; for when Mr. Gunter had brought it from the - Tables to a Line, and written some Uses thereof, Mr. Wingate added divers - Lines of several lengths, thereby to extract the Square or Cube Roots, - without doubling or trebling the distance of the Compasses: After him Mr. - Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral - Line, thereby enlarging the divisions of the Line. - -On pages 127 and 128 Leybourn adds: - - Again, One T. Browne, a Maker of Mathematical Instruments, made it in a - Serpentine or Spiral Line, composed of divers Concentrick Circles, - thereby to enlarg the divisions, which was the contrivance of one Mr. - Milburn a Yorkshire Gentleman, who writ thereof, and communicated his - Uses to the aforesaid Brown, who (since his death) attributed it to - himself: But whoever was the contriver of it, it is not without - inconvenience; for it can in no wise be made portable; and besides - (instead of compasses) an opening Joynt with thirds [threads] must be - placed to move upon the Centre of the Instrument, without which no - proportion can be wrought. - -This Mr. Milburn is probably the person named in the diary of the -antiquarian, Elias Ashmole, on August 13 [1646?]; "I bought of Mr. Milbourn -all his Books and Mathematical Instruments."[8] Charles Hutton[9] says that -Milburne of Yorkshire designed the spiral form about 1650. This date is -doubtless wrong, for Thomas Browne who, according to Leybourn, got the -spiral form of line from Milbourn, is repeatedly mentioned by William -Oughtred in his Epistle[10] printed some time in 1632 or 1633. Oughtred -does not mention Milbourn, and says (page 4) that the spiral form "was -first hit upon by one Thomas Browne a Joyner, . . . the serpentine -revolution being but two true semicircles described on severall -centers."[11] - - - Changes introduced by Thomas Brown and John Brown - -Thomas Brown did not publish any description of his instrument, but his -son, John Brown, published in 1661 a small book,[12] in which he says -(preface) that he had done "as Mr. Oughtred with Gunter's Rule, to a -sliding and circular form; and as my father Thomas Brown into a Serpentine -form; or as Mr. Windgate in his Rule of Proportion." He says also that -"this brief touch of the Serpentine-line I made bold to assert, to see if I -could draw out a performance of that promise, that hath been so long -unperformed by the promisers thereof." Accordingly in Chapter XX he gives a -description of the serpentine line, "contrived in five (or rather 15) -turn." Whether this description, printed in 1661, exactly fits the -instrument as it was developed in 1632, we have no means of knowing. John -Brown says: - - 1. First next the center is two circles divided one into 60, the other - into 100 parts, for the reducing of minutes to 100 parts, and the - contrary. - - 2. You have in seven turnes two inpricks, and five in divisions, the - first Radius of the sines (or Tangents being neer the matter, alike to - the first three degrees,) ending at 5 degrees and 44 minutes. - - 3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in - three margents in divisions, and the line of versed sines in pricks, - under the line of Tangents, according to Mr. Gunter's cross-staff: the - sines and Tangents beginning at 5 degrees, and 44 minutes where the other - ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the - line of numbers beginning at 10, and proceeding to 100, being one entire - Radius, and graduated into as many divisions as the largeness of the - instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to - 100 into 20 parts in one unit of increase, but the Tangents are divided - into single minutes from the beginning to the end, both in the first, - second and third Radiusses, and the sines into minutes; also from 30 - minutes to 40 degrees, and from 40 to 60, into every two minutes, and - from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the - rest as many as can be well discovered. - - The versed sines are set after the manner of Mr. Gunter's Cross-staff, - and divided into every 10th minutes beginning at 0, and proceeding to 156 - going backwards under the line of Tangents. - - 4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is - the secants to 51 degrees, being nothing else but the sines reitterated - beyond 90. - - 5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85 - degrees, whereby all trouble of backward working is avoided. - - 6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and - also a line of natural sines, for finding of differences in sines, for - finding hour and Azimuth. - - 7. Seventhly, next the verge or outermost edge is a line of equal parts - to get the Logarithm of any number, or the Logarithm sine and Tangent of - any ark or angle to four figures besides the carracteristick. - - 8. Eightly and lastly, in the space place between the ending of the - middle five turnes, and one half of the circle are three prickt lines - fitted for reduction. The uppermost being for shillings, pence and - farthings. The next for pounds, and ounces, and quarters of small - Averdupoies weight. The last for pounds, shillings and pence, and to be - used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay - the hair or edge of one of the legs of the index on 16. 3 1/2 in the line - of 1. s. d. and the hair shall cut on the equal parts 81 16; and the - contrary, if you have a decimal fraction, and would reduce it to a proper - fraction, the like may you do for shillings, and pence, and pounds, and - ounces. - - The uses of the lines follow. - - As to the use of these lines, I shall in this place say but little, and - that for two reasons. First, because this instrument is so contrived, - that the use is sooner learned then any other, I speak as to the manner, - and way of using it, because by means of first second and third - radiusses, in sines and Tangents, the work is always right on, one way or - other, according to the Canon whatsoever it be, in any book that treats - of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in - Oughtred from page 64 to 107. - - Secondly, and more especially, because the more accurate, and large - handling thereof is more then promised, if not already performed by more - abler pens, and a large manuscript thereof by my Sires meanes, provided - many years ago, though to this day not extant in print; so for his sake I - claiming my interest therein, make bold to present you with these few - lines, in order to the use of them: And first note, - - 1. Which soever of the two legs is set to the first term in the question, - that I call the first leg always, and the other being set to the second - term, I call the second leg . . . - -The exact nature of the contrivance with the "two legs" is not described, -but it was probably a flat pair of compasses, attached to the metallic -surface on which the serpentine line was drawn. In that case the instrument -was a slide rule, rather than a form of Gunter's line. In his publication -of 1661, as also in later publications,[13] John Brown devoted more space -to Gunter's scales, requiring the use of a separate pair of compasses, than -to slide rules. - - - Changes introduced by William Leybourn - -The same remark applies to William Leybourn who, after speaking of Seth -Partridge's slide rule, returns to forms of Gunter's scale, saying:[14] - - There is yet another way of disposing of this Line of Proportion, by - having one Line of the full length of the Ruler, and another Line of the - same Radius broken in two parts between 3 and 4; so that in working your - Compasses never go off of the Line: This is one of the best contrivances, - but here Compasses must be used. These are all the Contrivances that I - have hitherto seen of these Lines: That which I here speak of, and will - shew how to use, is only two Lines of one and the same Radius, being set - upon a plain Ruler of any length (the larger the better) having the - beginning of one Line, at the end of the other, the divisions of each - Line being set so close together, that if you find any number upon one of - the Lines, you may easily see what number stands against it on the other - Line. This is all the Variation. . . . - - Example 1. If a Board be 1 Foot 64 parts broad, how much in length of - that Board will make a Foot Square? Look upon one of your Lines (it - matters not which) for 1 Foot 64 parts, and right against it on the other - Line you shall find 61; and so many parts of a Foot will make a Foot - square of that Board. - -This contrivance solves the equation 1.64x=1, yielding centesimal parts of -a foot. - -James Atkinson[15] speaks of "Gunter's scale" as "usually of Boxwood . . . -commonly 2 ft. long, 1 1/2 inch broad" and "of two kinds: long Gunter or -single Gunter, and the sliding Gunter. It appears that during the -seventeenth century (and long after) the Gunter's scale was a rival of the -slide rule. - - - - - III. RICHARD DELAMAIN'S GRAMMELOGIA - - -We begin with a brief statement of the relations between Oughtred and -Delamain. At one time Delamain, a teacher of mathematics in London, was -assisted by Oughtred in his mathematical studies. In 1630 Delamain -published the Grammelogia, a pamphlet describing a circular slide rule and -its use. In 1631 he published another tract, on the Horizontall -Quadrant.[16] In 1632 appeared Oughtred's Circles of Proportion[17] -translated into English from Oughtred's Latin manuscript by another pupil, -William Forster, in the preface of which Forster makes the charge (without -naming Delamain) that "another . . . went about to pre-ocupate" the new -invention. This led to verbal disputes and to the publication by Delamain -of several additions to the Grammelogia, describing further designs of -circular slide rules and also stating his side of the bitter controversy, -but without giving the name of his antagonist. Oughtred's Epistle was -published as a reply. Each combatant accuses the other of stealing the -invention of the circular slide rule and the horizontal quadrant. - -The two title-pages of the edition of the Grammelogia in the British Museum - in London which we have called "Grammelogia IV." - - - Different editions or impressions - -There are at least five different editions, or impressions, of the -Grammelogia which we designate, for convenience, as follows: - - Grammelogia I, 1630. One copy in the Cambridge University Library.[18] - - Grammelogia II, I have not seen a copy of this. - - Grammelogia III, One copy in the Cambridge University Library.[19] - - Grammelogia IV, One copy in the British Museum, another in the Bodleian - Library, Oxford.[20] - - Grammelogia V, One copy in the British Museum. - -In Grammelogia I the first three leaves and the last leaf are without -pagination. The first leaf contains the title-page; the second leaf, the -dedication to the King and the preface "To the Reader;" the third leaf, the -description of the Mathematical Ring. Then follow 22 numbered pages. -Counting the unnumbered pages, there are altogether 30 pages in the -pamphlet. Only the first three leaves of this pamphlet are omitted in -Grammelogia IV and V. - -In Grammelogia III the Appendix begins with a page numbered 52 and bears -the heading "Conclusion;" it ends with page 68, which contains the same two -poems on the mathematical ring that are given on the last page of -Grammelogia I but differs slightly in the spelling of some of the words. -The 51 pages which must originally have preceded page 52, we have not seen. -The edition containing these we have designated Grammelogia II. The reason -for the omission of these 51 pages can only be conjectured. In Oughtred's -Epistle (p. 24), it is stated that Delamain had given a copy of the -Grammelogia to Thomas Brown, and that two days later Delamain asked for the -return of the copy, "because he had found some things to be altered -therein" and "rent out all the middle part." Delamain labored "to recall -all the bookes he had given forth, (which were many) before the sight of -Brownes Lines." These spiral lines Oughtred claimed that Delamain had -stolen from Brown. The title-page and page 52 are the only parts of the -Appendix, as given in Grammelogia III, that are missing in the Grammelogia -IV and V. - -Grammelogia IV answers fully to the description of Delamain's pamphlet -contained in Oughtred's Epistle. It was brought out in 1632 or 1633, for -what appears to be the latest part of it contains a reference (page 99) to -the Grammelogia I (1630) as "being now more then two yeares past." -Moreover, it refers to Oughtred's Circles of Proportion, 1632, and -Oughtred's reply in the Epistle was bound in the Circles of Proportion -having the Addition of 1633. For convenience of reference we number the two -title-pages of Grammelogia IV, "page (1)" and "page (2)," as is done by -Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page -numbers which we assign will be placed in parentheses, to distinguish them -from the page numbers which are printed in Grammelogia IV. The pages -(44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the -same as the pages 53-68. Thus only thirty-eight pages have page numbers -printed on them. The pages (67) and (83) are identical in wording, except -for some printer's errors; they contain verses in praise of the Ring, and -have near the bottom the word "Finis." Also, pages (22) and (23) are -together identical in wording with page (113), which is set up in finer -type, containing an advertisement of a part of Grammelogia IV explaining -the mode of graduating the circular rules. There are altogether six parts -of Grammelogia IV which begin or end by an address to the reader, thus: "To -the Reader," "Courteous Reader," or "To the courteous and benevolent Reader -. . .," namely the pages (8), (22), (68), (89), (90), (108). In his Epistle -(page 2), Oughtred characterizes the make up of the book in the following -terms: - - In reading it . . . I met with such a patchery and confusion of - disjoynted stuffe, that I was striken with a new wonder, that any man - should be so simple, as to shame himselfe to the world with such a - hotch-potch. - -Grammelogia V differs from Grammelogia IV in having only the second -title-page. The first title-page may have been torn off from the copy I -have seen. A second difference is that the page with the printed numeral 22 -in Grammelogia IV has after the word "Finis" the following notice: - - This instrument is made in Silver, or Brasse for the Pocket, or at any - other bignesse, over against Saint Clements Church without Temple Barre, - by Elias Allen. - -This notice occurs also on page 22 of Grammelogia I and III, but is omitted -from page 22 of Grammelogia V. - - - Description of Delamain's instrument of 1630 - -In his address to King Charles I, in his Grammelogia I, Delamain emphasizes -the ease of operating with his slide rule by stating that it is "fit for -use . . . as well on Horse backe as on Foot." Speaking "To the Reader," he -states that he has "for many yeares taught the Mathematicks in this Towne," -and made efforts to improve Gunter's scale "by some Motion, so that the -whole body of Logarithmes might move proportionally the one to the other, -as occasion required. This conceit in February last [1629] I struke upon, -and so composed my Grammelogia or Mathematicall Ring; by which only with an -ocular inspection, there is had at one instant all proportionalls through -the said body of Numbers." He dates his preface "first of January, 1630." -The fifth and sixth pages contain his "Description of the Grammelogia," the -term Grammelogia being applied to the instrument, as well as to the book. -His description is as follows: - - The parts of the Instrument are two Circles, the one moveable, and the - other fixed; The moveable is that unto which is fastened a small pin to - move it by; the other Circle may be conceived to be fixed; The - circumference of the moveable Circle is divided into unequall parts, - charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures - doe represent themselves, or such numbers unto which a Cipher or Ciphers - are added, and are varied as the occasion falls out in the speech of - Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or - 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c. - -After elaborating this last point and explaining the decimal subdivisions -on the scales of the movable circle, he says that "the numbers and -divisions on the fixed Circle, are the very same that the moveable are, . -." There is no drawing of the slide rule in this publication. The -twenty-two numbered pages give explanations of the various uses to which -the instrument can be put: "How to performe the Golden Rule" (pp. 1-3), -"Further uses of the Golden Rule" (pp. 4-6), "Notions or Principles -touching the disposing or ordering of the Numbers in the Golden Rule in -their true places upon the Grammelogia" (pp. 7-11), "How to divide one -number by another" (pp. 12, 13), "to multiply one Number by another" (pp. -14, 15), "To find Numbers in continuall proportion" (pp. 16, 17), "How to -extract the Square Root," "How to extract the Cubicke Root" (pp. 18-21), -"How to performe the Golden Rule" (the rule of proportion) is explained -thus: - - Seeke the first number in the moveable, and bring it to the second number - in the fixed, so right against the third number in the moveable, is the - answer in the fixed. - - If the Interest of 100. li. be 8. li. in the yeare, what is the Interest - of 65. li. for the same time. - - Bring 100. in the moveable to 8. in the fixed, so right against 65. in - the moveable is 5.2. in the fixed, and so much is the Interest of 65. li. - for the yeare at 8. li. for 100. li. per annum. - - The Instrument not removed, you may at one instant right against any - summe of money in the moveable, see the Interest thereof in the fixed: - the reason of this is from the Definition of Logarithmes. - -These are the earliest known printed instructions on the use of a slide -rule. It will be noticed that the description of the instrument at the -opening makes no references to logarithmic lines for the trigonometric -functions; only the line of numbers is given. Yet the title-page promised -the "resolution of Plaine and Sphericall Triangles." Page 22 throws light -upon this matter: - - If there be composed three Circles of equal thicknesse, A.B.C. so that - the inner edge of D [should be B] and the outward edge of A bee - answerably graduated with Logarithmall signes [sines], and the outward - edge of B and the inner edge of A with Logarithmes; and then on the - backside be graduated the Logarithmall Tangents, and againe the - Logarithmall signes oppositly to the former graduations, it shall be - fitted for the resolution of Plaine and Sphericall Triangles. - -After twelve lines of further remarks on this point he adds: - - Hence from the forme, I have called it a Ring, and Grammelogia by - annoligie of a Lineary speech; which Ring, if it were projected in the - convex unto two yards Diameter, or thereabouts, and the line Decupled, it - would worke Trigonometrie unto seconds, and give proportionall numbers - unto six places only by an ocular inspection, which would compendiate - Astronomicall calculations, and be sufficient for the Prosthaphaeresis of - the Motions: But of this as God shall give life and ability to health and - time. - -The unnumbered page following page 22 contains the patent and copyright on -the instrument and book: - - Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs - an Instrument called Grammelogia, or The Mathematicall Ring, together - with a Booke so intituled, expressing the use thereof, being his owne - Invention; we of our Gracious and Princely favour have granted unto the - said Richard Delamain and his Assignes, Privilege, Licence, and - Authority, for the sole Making, Printing and Selling of the said - Instrument and Booke: straightly forbidding any other to Make, Imprint, - or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument - or Booke within any our Dominions, during the space of ten yeares next - ensuing the date hereof, upon paine of Our high displeasure. Given under - our hand and Signet at our Palace of Westminster, the fourth day of - January, in the sixth yeare of our Raigne. - - - Delamain's later designs, and directions for using his instruments - -In the Appendix of Grammelogia III, on page 52 is given a description of an -instrument promised near the end of Grammelogia I: - - That which I have formerly delivered hath been onely upon one of the - Circles of my Ring, simply concerning Arithmeticall Proportions, I will - by way of Conclusion touch upon some uses of the Circles, of Logarithmall - Sines, and Tangents, which are placed on the edge of both the moveable - and fixed Circles of the Ring in respect of Geometricall Proportions, but - first of the description of these Circles. - - First, upon the side that the Circle of Numbers is one, are graduated on - the edge of the moveable, and also on the edge of the fixed the - Logarithmall Sines, for if you bring 1. in the moveable amongst the - Numbers to 1. in the fixed, you may on the other edge of the moveable and - fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto - 6.6. and each degree subdivided, and then over the former divisions and - figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5. - 4. 3. 2. 1. each of those divided by small points. - - Secondly, (if the Ring is great) neere the outward edge of this side of - the fixed against the Numbers, are the usuall divisions of a Circle, and - the points of the Compasse: serving for observation in Astronomy, or - Geometry, and the sights belonging to those divisions, may be placed on - the moveable Circle. - - Thirdly, opposite to those Sines on the other side are the Logarithmall - Tangents, noted alike both in the moveable and fixed thus - 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or - divisions serve also for their Complements to 90. so 40 gr. stands for - 50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both - in the moveable and fixed is also divided into parts. As for the degrees - which are under 6. viz. 5.4.3.2.1. they are noted with small figures over - this divided Circle from 45.40.35.30.25. &c. and each of those degrees - divided into parts by small points both in the moveable and fixed. - - Fourthly, on the other edge of the moveable on the same side is another - graduation of Tangents, like that formerly described. And opposite unto - it, in the fixed is a Graduation of Logarithmall sines in every thing - answerable to the first descrition of Sines on the other side. - - Fifthly, on the edge of the Ring is graduated a parte of the AEquator, - numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees - of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree - both of the AEquator, and Meridian are subdivided into parts; these two - graduated Circles serve to resolve such Questions which concerne - Latitude, Longitude, Rumb, and Distance, in Nauticall operations. - - Sixthly, to the concave of the Ring may be added a Circle to be elevated - or depressed for any Latitude, representing the AEquator, and so divided - into houres and parts with an Axis, to shew both the houre, and Azimuth, - and within this Circle may be hanged a Box, and Needle with a Socket for - a staffe to slide into it, and this accommodated with scrue pines to - fasten it to the Ring and staffe, or to take it off at pleasure. - -The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and -V make no reference to the dispute with Oughtred and may, therefore, be -assumed to have been published before the appearance of Oughtred's Circles -of Proportion. On page 53, "To the Reader," he says: - - . . . you may make use of the Projection of the Circles of the Ring upon - a Plaine, having the feet of a paire of compasses (but so that they be - flat) to move on the Center of that Plaine, and those feet to open and - shut as a paire of Compasses . . . now if the feet bee opened to any two - termes or numbers in that Projection, then may you move the first foot to - the third number, and the other foot shall give the Answer; . . . it hath - pleased some to make use of this way. But in this there is a double - labour in respect to that of the Ring, the one in fitting those feet unto - the numbers assigned, and the other by moving them about, in which a man - can hardly accommodate the Instrument with one hand, and expresse the - Proportionals in writing with the other. By the Ring you need not but - bring one number to another, and right against any other number is the - Answer without any such motion. . . . upon that [the Ring] I write, - shewing some uses of those Circles amongst themselves, and conjoyned with - others . . . in Astronomy, Horolographie, in plaine Triangles applyed to - Dimensions, Navigation, Fortification, etc. . . . But before I come to - Construction, I have thought it convenient by way introduction, to - examine the truth of the graduation of those Circles . . . - -These are the words of a practical man, interested in the mechanical -development of his instrument. He considers not only questions of -convenience but also of accuracy. The instrument has, or may have now, also -lines of sines and tangents. To test the accuracy of the circles of -Numbers, "bring any number in the moveable to halfe of that number in the -fixed: so any number or part in the fixed shall give his double in the -moveable, and so may you trie of the thirds, fourths &c. of numbers, vel -contra," (p. 54). On page 55 are given two small drawings, labelled, "A -Type of the Ringe and Scheme of this Logarithmicall projection, the use -followeth. These Instruments are made in Silver or Brasse by John Allen -neare the Sauoy in the Strand." - - - IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE - CIRCULAR SLIDE RULE - -Delamain's publication of 1630 on the 'Mathematicall Ring' does not appear -at that time to have caused a rupture between him and Oughtred. When in -1631 Delamain brought out his Horizontall Quadrant, the invention of which -Delamain was afterwards charged to have stolen from Oughtred, Delamain was -still in close touch with Oughtred and was sending Oughtred in the Arundell -House, London, the sheets as they were printed. Oughtred's reference to -this in his Epistle (p. 20) written after the friendship was broken, is as -follows: - - While he was printing his tractate of the Horizontall quadrant, although - he could not but know that it was injurious to me in respect of my free - gift to Master Allen, and of William Forster, whose translation of my - rules was then about to come forth: yet such was my good nature, and his - shamelessnesse, that every day, as any sheet was printed, hee sent, or - brought the same to mee at my chamber in Arundell house to peruse which I - lovingly and ingenuously did, and gave him my judgment of it. - -Even after Forster's publication of Oughtred's Circles of Proportion, 1632, -Oughtred had a book, A canon of Sines Tangents and Secants, which he had -borrowed from Delamain and was then returning (Epistle, page (5)). The -attacks which Forster, in the preface to the Circles of Proportion, made -upon Delamain (though not naming Delamain) started the quarrel. Except for -Forster and other pupils of Oughtred who urged him on to castigate -Delamain, the controversy might never have arisen. Forster expressed -himself in part as follows: - - . . . being in the time of the long vacation 1630, in the Country, at the - house of the Reverend, and my most worthy friend, and Teacher, Mr. - William Oughtred (to whose instruction I owe both my initiation, and - whole progresse in these Sciences.) I vpon occasion of speech told him of - a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made - (such as it vsually called Mr. Gunter's Ruler) 6 feet long, to be vsed - with a payre of beame-compasses. "He answered that was a poore invention, - and the performance very troublesome: But, said he, seeing you are taken - with such mechanicall wayes of Instruments, I will shew you what deuises - I have had by mee these many yeares." And first, hee brought to mee two - Rulers of that sort, to be vsed by applying one to the other, without any - compasses: and after that hee shewed mee those lines cast into a circle - or Ring, with another moueable circle vpon it. I seeing the great - expeditenesse of both those wayes; but especially, of the latter, wherein - it farre excelleth any other Instrument which hath bin knowne; told him, - I wondered that hee could so many yeares conceale such vseful inuentions, - not onely from the world, but from my selfe, to whom in other parts and - mysteries of Art, he had bin so liberall. He answered, "That the true way - of Art is not by Instruments, but by Demonstration: and that it is a - preposterous course of vulgar Teachers, to begin with Instruments, and - not with the Sciences, and so in-stead of Artists, to make their - Schollers only doers of tricks, and as it were Iuglers: to the despite of - Art, losse of precious time, and betraying of willing and industrious - wits, vnto ignorance and idlenesse. That the vse of Instruments is indeed - excellent, if a man be an Artist: but contemptible, being set and opposed - to Art. And lastly, that he meant to commend to me, the skill of - Instruments, but first he would haue me well instructed in the Sciences. - He also shewed me many notes, and Rules for the vse of those circles, and - of his Horizontall Instrument, (which he had proiected about 30 yeares - before) the most part written in Latine. All which I obtained of him - leaue to translate into English, and make publique, for the vse, and - benefit of such as were studious, and louers of these excellent Sciences. - - Which thing while I with mature, and diligent care (as my occasions would - give me leaue) went about to doe: another to whom the Author in a louing - confidence discouered this intent, using more hast then good speed, went - about to preocupate; of which vntimely birth, and preuenting (if not - circumuenting) forwardnesse, I say no more: but aduise the studious - Reader, onely so farre to trust, as he shal be sure doth agree to truth & - Art. - -While in this dedication reference is made to a slide rule or "ring" with a -"moveable circle," the instrument actually described in the Circles of -Proportion consists of fixed circles "with an index to be opened after the -manner of a paire of Compasses." Delamain, as we have seen, had decided -preference for the moveable circle. To Oughtred, on the other hand, one -design was about as good as the other; he was more of a theorist and -repeatedly expressed his contempt for mathematical instruments. In his -Epistle (page (25)), he says he had not "the one halfe of my intentions -upon it" (the rule in his book), nor one with a "moveable circle and a -thread, but with an opening Index at the centre (if so be that bee cause -enough to make it to bee not the same, but another Instrument) for my part -I disclaime it: it may go seeke another Master: which for ought I know, -will prove to be Elias Allen himselfe: for at his request only I altered a -little my rules from the use of the moveable circle and the thread, to the -two armes of an Index." - -All parts of Delamain's Grammelogia IV, except pages 1-22 and 53-68 -considered above, were published after the Circles of Proportion, for they -contain references to the ill treatment that Delamain felt or made believe -that he felt, that he had received in the book published by Oughtred and -Forster. Oughtred's reference to teachers whose scholars are "doers of -tricks," "Iuglers," and Forster's allusion to "another to whom the Author -in a loving confidence" explained the instrument and who "went about to -preocupate" it, are repeatedly mentioned. Delamain says, (page (89)) that -at first he did not intend to express himself in print, "but sought peace -and my right by a private and friendly way." Oughtred's account of -Delamain's course is that of an "ill-natured man" with a "virulent tongue," -"sardonical laughter" and "malapert sawsiness." Contrasting Forster and -Delamain, he says that, of the former he "had the very first moulding" and -made him feel that "the way of Art" is "by demonstration." But Delamain was -"already corrupted with doing upon Instruments, and quite lost from ever -being made an Artist." (Epistle page (27)). Repeatedly does Oughtred assert -Delamain's ignorance of mathematics. The two men were evidently of wholly -different intellectual predilections. That Delamain loved instruments is -quite evident, and we proceed to describe his efforts to improve the -circular slide rule. - -The Grammelogia IV is dedicated to King Charles I. Delamain says: - - . . . Everything hath his beginning, and curious Arts seldome come to the - height at the first; It was my promise then to enlarge the invention by a - way of decuplating the Circles, which I now present unto your sacred - Majestie as the quintessence and excellencie there of . . . - -His enlarged circular rules are illustrated in the Bodleian Library copy of -Grammelogia IV by four diagrams, two of them being the two drawings on the -two title-pages at the beginning of the Grammelogia IV, 4 inches in -external diameter, and exhibiting eleven concentric circular lines carrying -graduations of different sorts. In the second of these designs all circles -are fixed. The other two drawings are each 10 3/4 inches in external -diameter and exhibit 18 concentric circular lines; the folded sheet of the -first of these drawings is inserted between pages (23) and (24), the second -folded sheet between pages (83) and (84). All circles of this second -instrument are fixed. Counting in the two small drawings in Grammelogia -III, there are in all six drawings of slide rules in the Bodleian -Grammelogia IV. On pages (24) to (43) Delamain explains the graduation of -slide rules. He takes first a rule which has one circle of equal parts, -divided into 1000 equal divisions. From a table of logarithms he gets log 2 -= 0.301; from the number 301 in the circle of equal parts he draws a line -to the center of the circle and marks the intersection with the circles of -numbers by the figure 2. Thus he proceeds with log 3, log 4, and so on; -also with log sin x and log tan x. For log sin x he uses two circles, the -first (see page (27)) for angles from 34' 24" to 5^o 44' 22", the second -circle from 5^o 44' 22" to 90^o. The drawings do not show the seconds. He -suggests many different designs of rules. On page (29) he says: - - For the single projection of the Circles of my Ring, and the dividing and - graduating of them: which may bee so inserted upon the edges of Circles - of mettle turned in the forme of a Ring, so that one Circle may moove - betweene two fixed, by helpe of two stayes, then may there be graduated - on the face of the Ring, upon the outer edge of the mooveable and inner - edge of the fixed, the Circle of Numbers, then upon the inner edge of - that mooveable Circle, and the outward edge of that inner fixed Circle - may be inserted the Circle of Sines, and so according to the description - of those that are usually made. - -In addition to these lines he proceeds to mention the circle giving the -ordinary division into degrees and minutes, and two circles of tangents on -the other side of the rule. - -Next Delamain explains an arrangement of all the graduation on one side of -the rule by means of "a small channell in the innermost fixed Circle, in -which may be placed a small single Index, which may have sufficient length -to reach from the innermost edge of the Mooveable Circle, unto the outmost -edge of the fixed Circle, which may be mooved to and fro at pleasure, in -the channell, which Index may serve to shew the opposition of Numbers" (p. -(31)). From this it is clear that the invention of the "runner" goes back -to the very first writers on the slide rule. - -After describing a modification of the above arrangement, he adds, "many -other formes might be deliverd, about this single projection" (p. (32)). - -Proceeding to the "enlarging" of the circles in the Ring, to, say, the -"Quadruple to that which is single, that is, foure times greater," the -"equall parts" are distributed over four circles instead of only one -circle, but the general method of graduation is the same as before (p. -(33)); there being now four circles carrying the logarithms of numbers, and -so on. Next he points out "severall wayes how the Circles of the -Mathematicall Ring (being inlarged) may be accommodated for practicall -use:" (1) The Circles are all fixed in a plain and movable flat compasses -(or better, a movable semicircle) are used for fixing any two positions; -(2) There is a "double projection" of each logarithmic line "inlarged on a -Plaine," one fixed, the other movable, as shown in his first figure on the -title-page, a single index only being used; (3) use of "my great Cylinder -which I have long proposed (in which all the Circles are of equall -greatnesse,) and it may be made of any magnitude or capacity, but for a -study (hee that will be at the charge) it may be of a yard diameter and of -such an indifferent length that it may containe 100 or more Circles fixed -parallel one to the other on the Cylinder, having a space betweene each of -them, so that there may bee as many mooveable Circles, as there are fixed -ones, and these of the mooveable linked, or fastened together, so that they -may all moove together by the fixed ones in these spaces, whose edges both -of the fixed, and mooveable being graduated by helpe of a single Index will -shew the proportionalls by opposition in this double Projection, or by a -double Index in a single Projection" (p. (36)). - -Next follows the detailed description of his Ring "on a Plaine, according -to the diagramme that was given the King (for a view of that projection) -and afterwards the Ring it selve." The diagram is the large one which we -mentioned as inserted between pages (23) and (24). The instrument has two -circles, one moveable, upon each of which are described 13 distinct -circular graduations. The lines on the fixed circle are: "The Circle of -degrees and calendar," E. "Circle of equall parts, and part of the Equator, -and Meridian," TT. "The Circle of Tangents," S. "The Circle of Sines," D. -"The Circle of Decimals," N. "The Circle of Numbers." The lines on the -movable circle are: N. "The Circle of Numbers," E. "The Circle of equated -figures, and bodies," S. "The Circle of Sines," TT. "The Circle of -Tangents," Y. "The Circle of time, yeares, and monethes." - -On pages (84)-(88) Delamain explains an enlargement of his Ring for -computations involving the sines of angles near to 90^o. On page (86) he -says: - - I have continued the Sines of the Projection unto two severall - revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr. - (being the last revolution of the decuplation of the former, or the - hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s. - and ends at 90.gr. (being the last of a ternary of decuplated - revolutions, or the thousand part of that Projection) and may bee thus - used. - -He explains the manner of using these extra graduations. Thus he claims to -have attained degrees of accuracy which enabled him to do what "some one" -had declared "could not bee done." It is hardly necessary to point out that -Delamain's Grammelogia IV suggests designs of slide rules which inventors -two hundred or more years later were endeavouring to produce. Which of -Delamain's designs of rules were actually made and used, he does not state -explicitly. He refers to a rule 18 inches in diameter as if it had been -actually constructed (pages (86), (88)). Oughtred showed no appreciation of -such study in designing and ridiculed Delamain's efforts, in his Epistle. - -Additional elucidations of his designs of rules, along with explanations of -the relations of his work to that of Gunter and Napier, and sallies -directed against Oughtred and Forster, are contained on pages (8)-(21) of -his Grammelogia IV. - - - - - V. INDEPENDENCE AND PRIORITY OF INVENTION - - -The question of independence and priority of invention is discussed by -Delamain more specifically on pages (89)-(113); Oughtred devotes his entire -Epistle to it. It is difficult to determine definitely which publication is -the later, Delamain's Grammelogia IV or Oughtred's Epistle. Each seems to -quote from the other. Probably the explanation is that the two publications -contain arguments which were previously passed from one antagonist to the -other by word of mouth or by private letter. Oughtred refers in his Epistle -(p. (12)) to a letter from Delamain. We believe that the Epistle came after -Delamain's Grammelogia IV. Delamain claims for himself the invention of the -circular slide rule. He says in his Grammelogia IV. (p. (99)), "when I had -a sight of it, which was in February, 1629 (as I specified in my Epistle) I -could not conceale it longer, envying my selfe, that others did not tast of -that which I found to carry with it so delightfull and pleasant a goate -[taste] . . ." Delamain asserts (without proof) that Oughtred "never saw it -as he now challengeth it to be his invention, untill it was so fitted to -his hand, and that he made all his practise on it after the publishing of -my Booke upon my Ring, and not before; so it was easie for him or some -other to write some uses of it in Latin after Christmas, 1630 and not the -Sommer before, as is falsely alledged by some one . . ." (p. (91)). -Delamain's accusation of theft on the part of Oughtred cannot be seriously -considered. Oughtred's reputation as a mathematician and his standing in -his community go against such a supposition. Moreover, William Forster is a -witness for Oughtred. The fact that Oughtred had the mastery of the -rectilinear slide rule as well, while Delamain in 1630 speaks only of the -circular rule, weighs in Oughtred's favour. - -Oughtred says he invented the slide rule "above twelve yeares agoe," that -is, about 1621, and "I with mine owne hand made me two such Circles, which -I have used ever since, as my occasions required," (Epistle p. (22)). On -the same page, he describes his mode of discovery thus: - - I found that it required many times too great a paire of Compasses [in - using Gunter's line], which would bee hard to open, apt to slip, and - troublesome for use. I therefore first devised to have another Ruler with - the former: and so by setting and applying one to the other, I did not - onely take away the use of Compasses, but also make the worke much more - easy and expedite: when I should not at all need the motion of my hand, - but onely the glancing at my sight: and with one position of the Rulers, - and view of mine eye, see not one onely, but the manifold proportions - incident unto the question intended. But yet this facility also wanted - not some difficulty especially in the line of tangents, when one arch was - in the former mediety of the quadrant, and the other in the latter: for - in this case it was needful that either one Ruler must bee as long againe - as the other; or else that I must use an inversion of the Ruler, and - regression. By this consideration I first of all saw that if those lines - upon both Rulers were inflected into two circles, that of the tangents - being in both doubled, and that those two Circles should move one upon - another; they with a small thread in the center to direct the sight, - would bee sufficient with incredible and wonderfull facility to worke all - questions of Trigonometry . . . - -Oughtred said that he had no desire to publish his invention, but in the -vacation of 1630 finally promised William Forster to let him bring out a -translation. Oughtred claims that Delamain got the invention from him at -Alhallontide [November 1], 1630, when they met in London. The accounts of -that meeting we proceed to give in double column. - - Delamain's Statement - Grammelogia IV, page (98) - - ". . . about Alhalontide 1630. (as our Authors reporteth) was the time he - was circumvented, and then his intent in a loving manner (as before) he - opened unto me, which particularly I will dismantle in the very naked - truth: for, wee being walking together some few weekes before Christmas, - upon Fishstreet hill, we discoursed upon sundry things Mathematicall, - both Theoreticall and Practicall, and of the excellent inventions and - helpes that in these dayes were produced, amongst which I was not a - little taken with that of the Logarythmes, commending greatly the - ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, - in the lines of proportion, extracted from these Logarythmes for ordinary - Practicall uses; He replyed unto me (in these very words) What will yov - say to an Invention that I have, which in a lesse extent of the Compasses - shall worke truer then that of Mr. Gunters Ruler, I asked him then of - what forme it was, he answered with some pause (which no doubt argued his - suspition of mee that I might conceive it) that it was Arching-wise, but - now hee sayes that hee told mee then, it was Circular (but were I put to - my oath to avoid the guilt of Conscience I would conclude in the former.) - At which immediately I answered, I had the like my selfe, and so we - discoursed not a word more touching that subject . . . Then after my - coming home I sent him a sight of my Projection drawne in Pastboard: Now - admit I had not the Invention of my Ring before I discoursed . . . it was - not so facil for mee . . . to raise and compose so complete, and absolute - an Instrument from so small a principle, or glimpse of light . . ." - - Oughtred's Statement - Epistle, page (23) - - "Shortly after my gift to Elias Allen, I chanced to meet with Richard - Delamain in the street (it was at Alhallontide) and as we walked together - I told him what an Instrument I had given to Master Allen, both of the - Logarithmes projected into circles, which being lesse then one foot - diameter would performe as much as one of Master Gunters Rulers of sixe - feet long: and also of the Prostaphaereses of the Plannets and second - motions. Such an invention have I said he: for now his intentions (that - is his ambition) beganne to worke: . . . But he saith, Then after my - comming home I sent him a sight of my projection drawne in past-board. - See how notoriously he jugleth without an Instrument. Then after: how - long after? a sight of my projection: of how much? More then seven weekes - after on December 23, he sent to mee the line of numbers onely set upon a - circle: . . . and so much onely he presented to his Majesty: but as for - Sine or tangent of his, there was not the least shew of any. Neither - could he give to Master Allen any direction for the composure of the - circles of his Ring, or for the division of them: as upon his oath Master - Allen will testify how hee misled him, and made him labour in vain above - three weeks together, until Master Allen himselfe found out his ignorance - and mistaking, which is more cleare then is possible with any impudence - to be outfaced." - -Oughtred makes a further statement (Epistle, p. (24)) as follows: - - Delamain hearing that Brown with his Serpentine had another line by which - he could worke to minutes in the 90 degree of sines . . . gave the [his] - booke to Browne: who in thankfulnesse could not but gratify Delamain with - his Lines also: and teach him the use of them, but especially of the - great Line: with this caution on both sides, that one should not meddle - with the others invention. Two dayes after Delamain . . . because he had - found some things to be altered therin, . . . asked for the booke . . . - but as soone as he had got it in his hands he rent out all the middle - part with the two Schemes & put them up in his pocket & went his way . . - . and . . . laboureth to recall all the bookes he had given forth . . . - And shortly after this he got a new Printer (who was ignorant of his - former Schemes) to print him new: giving him an especiall charge of the - outermost line newly graven in the Plate, which indeed is Brownes very - line: and then altering his book . . . - -This and other statements made by Oughtred seem damaging to Delamain's -reputation. But it is quite possible that Oughtred's guesses as to -Delamain's motives are wrong. Moreover, some of Oughtred's statements are -not first hand knowledge with him, but mere hearsay. One may accept his -first hand facts and still clear Delamain of wrong doing. There is always -danger that rival claimants of an invention or discovery will proceed on -the assumption that no one else could possibly have come independently upon -the same devices that they themselves did; the history of science proves -the opposite. Seldom is an invention of any note made by only one man. We -do not feel competent to judge Delamain's case. We know too little about -him as a man. We incline to the opinion that the hypothesis of independent -invention is the most plausible. At any rate, Delamain figures in the -history of the slide rule as the publisher of the earliest book thereon and -as an enthusiastic and skillful designer of slide rules. - -The effect of this controversy upon interested friends was probably small. -Doubtless few people read both sides. Oughtred says:[21] "this scandall . . -. hath with them, to whom I am not knowne, wrought me much prejudice and -disadvantage . ." Aubrey,[22] a friend of Oughtred, refers to Delamain "who -was so sawcy to write against him" and remembers having seen "many yeares -since, twenty or more good verses made" against Delamain. Another friend of -Oughtred, William Robinson, who had seen some of Delamain's publications, -but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before -the appearance of the latter's Epistle: - - I cannot but wonder at the indiscretion of Rich. Delamain, who being - conscious to himself that he is but the pickpurse of another man's wit, - would thus inconsiderately provoke and awake a sleeping lion . . . he - hath so weakly (though in my judgment, vaingloriously enough) commended - his own labour . . .[23] - -Delamain presented King Charles I with one of his sun-dials, also with a -manuscript and, later, with a printed copy of his book of 1630. A drawing -of his improved slide rule was sent to the King and the Grammelogia IV is -dedicated to him. The King must have been favorably impressed, for Delamain -was appointed tutor to the King in mathematics. His widow petitioned the -House of Lords in 1645 for relief; he had ten children.[24] - -Anthony Wood states that Charles I, on the day of his execution, commanded -his friend Thomas Herbert "to give his son the duke of York his large -ring-sundial of silver, a jewel his maj. much valued." Anthony Wood adds, -"it was invented and made by Rich. Delamaine a very able mathematician, who -projected it, and in a little printed book did shew its excellent use in -resolving many questions in arithmetic and other rare operations to be -wrought by it in the mathematics."[25] - - - - - VI. OUGHTRED'S GAUGING LINE, 1633 - - -It has not been generally known, hitherto, that Oughtred designed a -rectilinear slide rule for gauging and published a description thereof in -1633.[26] In his Circles of Proportion, chapter IX, Oughtred had offered a -closer approximation than that of Gunter for the capacity of casks. The -Gauger of London expostulated with Oughtred for presuming to question -anything that Gunter had written. The ensuing discussion led to an -invitation extended by the Company of Vintners to the instrument maker -Elias Allen to request Oughtred to design a gauging rod.[27] This he did, -and Allen received an order for "threescore" instruments. On page 19 -Oughtred describes his 'Gauging Rod:' - - It consisteth of two rulers of brasse about 32 ynches of length, which - also are halfe an ynch broad, and a quarter of an ynch thick . . . At one - end of both those rulers are two little sockets of brasse fastened on - strongly: by which the rulers are held together, and made to move one - upon another, and to bee drawne out unto any length, as occasion shall - require: and when you have them at the just length, there is upon one of - the sockets a long Scrue-pin to scrue them fast. - -There are graduations on three sides of the rulers, one graduation being -the logarithmic line of numbers. He says (p. 39), "the maner of computing -the Gauge-divisions I have concealed." W. Robinson, who was a friend of -Oughtred, wrote him as follows:[28] - - I have light upon your little book of artificial gauging, wherewith I am - much taken, but I want the rod, neither could I get a sight of one of - them at the time, because Mr. Allen had none left . . . I forgot to ask - Mr. Allen the price of one of them, which if not much I would have one of - them." Oughtred annotated this passage thus: "Or in wood, if any be made - in wood by Thompson or any other." - -Another of Oughtred's admirers, Sir Charles Cavendish, wrote, on February -11, 1635 thus:[29] - - I thank you for your little book, but especially for the way of - calculating the divisions of your gauging rod. I wish, both for their own - sakes and yours, that the citizens were as capable of the acuteness of - this invention, as they are commonly greedy of gain, and then I doubt not - but they would give you a better recompense than I doubt now they will. - -On April 20, 1638, we find Oughtred giving Elias Allen directions[30] -"about the making of the two rulers." As in 1633,[31] so now, Oughtred -takes one ruler longer than the other. This 1633 instrument was used also -as "a crosse-staffe to take the height of the Sunne, or any Starre above -the Horizon, and also their distances." The longer ruler was called staffe, -the shorter transversarie. While in 1633 he took the lengths of the two in -the ratio "almost 3 to 2," in 1638, he took "the transversary three -quarters of the staff's length, . . . that the divisions may be larger." - - - - - VII. OTHER SEVENTEENTH CENTURY SLIDE RULES - - -In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard, -Henry Coggeshall, W. Hunt and Sir Isaac Newton.[32] Of Partridge's Double -Scale of Proportion, London, I have examined a copy dated 1661, which is -the earliest date for this book that I have seen. As far as we know, 1661 -is the earliest date of publications on the slide rule, since Oughtred and -Delamain. But it would not be surprising if the intervening 28 years were -found not so barren as they seem at present. The 1661 and 1662 impressions -of Partridge are identical, except for the date on the title-page. William -Leybourn, who printed Partridge's book, speaks in high appreciation of it -in his own book.[33] - -In 1661 was published also John Brown's first book, Description and Use of -a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use -of "Mr. Whites rule" for the measuring of board and timber, round and -square. He calls this a "sliding rule." The existence, in 1661, of a -"Whites rule" indicates activities in designing of which we know as yet -very little. In his book of 1761, previously quoted, Brown gives a drawing -of "White's sliding rule" (p. 193); also a special contrivance of his own, -as indicated by him in these words: - - A further improvement of the Triangular Quadrant, as I have made it - several times, with a sliding Cover on the in-side, when made hollow, to - carry Ink, Pens, and Compasses; then on the sliding Cover, and Edges, is - put the Line of Numbers, according to Mr. White's first Contrivance for - manner of operation; but much augmented, and made easie, by John Brown. - -He gives no drawing of his "triangular quadrant," hence his account of it -is unsatisfactory. He explains the use of "gage-points." His placing -logarithmic lines on the edges of instrument boxes was outdone in oddity -later by Everard who placed them on tobacco-boxes.[34] In Brown's -publication of 1704 the White slide rule is given again, "being as neat and -ready a way as ever was used." He tells also of a "glasier's sliding rule." -William Leybourn explains in 1673 how Wingate's double and triple lines for -squaring and cubing, or square and cube root, can be used on slide -rules.[35] - -Beginning early in the history of the slide rule, when Oughtred designed -his "gauging rod," we notice the designing of rules intended for very -special purposes. Another such contrivance, which enjoyed long popularity, -was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677, -a booklet of 35 pages. Coggeshall says in his preface: - - For what can be more ready and easie, then having set twelve to the - length, to see the Content exactly against the Girt or Side of the - Square. Whereas on Mr. Partridge's Scale the Content is the Sixth Number, - which is far more troublesome then [even] with Compasses. - -One line on Coggeshall's rule begins with 4 and extends to 40, these -numbers being the "Girt" (a quarter of the circumference), which in -ordinary practice of measuring round timber lies between 4 inches and 40 -inches. This "Girt line" slides "against the line of Numbers in two -Lengths, to which it is exactly equal." A second edition, 1682, shows some -changes in the rule, as well as an enlargement and change of title of the -book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent, -London, 1682. In this, the description of the rule is given thus: - - There are four Lines on each flat of this Rule; two next the outward - edges, which are Lines of Measure; and two next the inward edges, which - are Lines of Proportion. On one flat, next the inward edges, is the - Square-line [Girt-line in round timber measurement] with the Line of - Numbers his fellow. Next the outward, a Line of Inches divided into - Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 - to 24 on the other. On the other flat, next the inward edges, is the - double Scale of Numbers [for solving proportions]. Next the outward on - one Rule a Line of Inches divided each into ten parts; and this for - gauging, etc. On the other a foot divided into 100 parts. - -Later further changes were introduced in Coggeshall's rule.[36] - -It is worthy of note that Coggeshall's slide rule book, The Art of -Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p. -473; hence Leupold's description[37] of the rectilinear slide rule in his -Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not -the earliest reference to the rectilinear rule found in German -publications. The above date is earlier even than Biler's reference to a -circular slide rule in his Descriptio instrumenti mathematici universalis -of 1696. - -Two noted slide rules for gauging were described by Tho. Everard, -Philomath, in his Stereometry made easie, London, 1684. He designates his -lines by the capital letters A, B, C, D, E. On the first instrument, A on -the rule, and B and C on the slide, have each two radiuses of numbers, D -has only one, while E has three. The second rule is described in an -Appendix; it is one foot long, with two slides enabling the rule to be -extended to 3 feet. - -Everard's instruments were made in London by Isaac Carver who, soon after, -himself wrote a sixteen-page Description and Use of a New Sliding Rule, -projected from the Tables in the Gauger's Magazine, London, 1687, which was -"printed for William Hunt" and bound in one volume with a book by Hunt, -called The Gauger's Magazine, London, 1687. This appears to be the same -William Hunt who later brought out descriptions of his own of slide rules. -The instrument described by Carver "consists of three pieces, two whereof -are moveable to be drawn out till the whole be 36 inches long." It has -several non-logarithmic graduations, together with logarithmic lines marked -A, B, C, D, of which A, B, C are "double lines," and D a "single line" used -for squares and square roots. It is designed for the determination of the -vacuity of a "spheroidal cask lying," a "spheroidal cask standing," and a -"parabolical cask lying." - -Another seventeenth century writer on the slide rule is John Atkinson, whom -we have mentioned earlier. He says:[38] "The Lines of Numbers, Sines and -Tangents, are set double, that is, one on each side, as the middle piece -slides: which middle piece is so contrived, to slip to and fro easily, to -slide out, and to be put in any side uppermost, in order to bring those -Lines together (or against one another) most proper for solving the -Question, wrought by Sliding-Gunter." - -The data presented in this article show that, while the earliest slide -rules were of the circular type, the later slide rules of the seventeenth -century were of the rectilinear type.[39] - - January 12, 1915. - - - - - Footnotes - - -[1]F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments, - New York, 1909, pp. 7-14, also Addenda i-vi. - -[2]F. Cajori, "On the Invention of the Slide Rule," in Colorado College - Publication, Engineering Series Vol. 1, 1910. An abstract of this is - given in Nature (London), Vol. 82, 1909, p. 267. - -[3]F. Cajori, History etc., p. 14. - -[4]Art. "Slide Rule" in the Penny Cyclopaedia and in the English - Cyclopaedia [Arts and Sciences]. - -[5]Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817, - p. 423. - -[6]The full title of the book which Wingate published on this subject in - Paris is as follows: - - L'Vsage | de la | Reigle de | Proportion | en l'Arithmetique & | - Geometrie. | Par Edmond Vvingate, | Gentil-homme Anglois. | - - Ean es philomathes, ese polymathes. - - In tenui, sed no tenuis vsusve, laborne. | - - A Paris, | Chez Melchior Mondiere, | demeurant en l'Isle du Palais, | a - la | rue de Harlay aux deux Viperes. | M. DC. XXIV. | Auec Priuilege du - Roy. | - - Back of the title page is the announcement: - - Notez que la Reigle de Proportion en toutes facons se vend a Paris chez - Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces, - demeurant en l'Isle du Palais sur le Quay qui regarde la Megisserie a - l'Espic d'or. - -[7]The title-page of the edition of 1658 is as follows: - - The Use of the Rule of Proportion in Arithmetick & Geometrie. First - published at Paris in the French tongue, and dedicated to Monsieur, the - then king's onely Brother (now Duke of Orleance). By Edm. Wingate, an - English Gent. And now translated into English by the Author. Whereinto - is now also inserted the Construction of the same Rule, & a farther use - thereof . . . 2nd edition inlarged and amended. London, 1658. - -[8]Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; - Drawn up by himself by way of Diary. With Appendix of original Letters. - Publish'd by Charles Burman, Esq., London, 1717, p. 23. - -[9]Mathematical Tables, 1811, p. 36, and art. "Gunter's Line" in his Phil. - and Math. Dictionary, London, 1815. - -[10]To the English Gentrie, and all others studious of the Mathematicks, - which shall bee readers hereof. The just Apologie of Wil: Ovghtred, - against the slaunderous insimulations of Richard Delamain, in a Pamphlet - called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum - projectio circularis. We shall refer to this document as Epistle. It was - published without date in 32 unnumbered pages of fine print, and was - bound in with Oughtred's Circles of Proportion, in the editions of 1633 - and 1639. In the 1633 edition it is inserted at the end of the volume - just after the Addition vnto the Vse of the Instrument etc., and in that - of 1639 immediately after the preface. It was omitted from the Oxford - edition of 1660. The Epistle was also published separately. There is a - separate copy in the British Museum, London. Aubrey, in his Brief Lives, - edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, "He - writt a stitch't pamphlet about 163(?4) against . . . Delamaine." - -[11]Thomas Browne is mentioned by Stone in his Mathematical Instruments, - London 1723, p. 16. See also Cajori, History of the Slide Rule, New - York, 1909, p. 15. - -[12]The Description and Use of a Joynt-Rule: . . . also the use of Mr. - White's Rule for measuring of Board and Timber, round and square; With - the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and - Versed Sines. By J. Brown, Philom., London, 1661. - -[13]A Collection of Centers and Useful Proportions on the Line of Numbers, - by John Brown, 1662(?), 16 pages; Description and Use of the Triangular - Quadrant, by John Brown, London, 1671; Wingate's Rule of Proportion in - Arithmetick and Geometry: or Gunter's Line. Newly rectified by Mr. Brown - and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The - Description and Use of the Carpenter's-Rule: Together with the Use of - the Line of Numbers commonly call'd Gunter's-Line, by John Brown, - London, 1704. - -[14]William Leybourn, op. cit., pp. 129, 130, 132, 133. - -[15]James Atkinson's edition of Andrew Wakely's The Mariners Compass - Rectified, London, 1694 [Wakely's preface dated 1664, Atkinson's - preface, 1693]. Atkinson adds An Appendix containing Use of Instruments - most useful in Navigation. Our quotation is from this Appendix, p. 199. - -[16]R. Delamain, The Making, Description, and Use of a small portable - Instrument . . . called a Horizontall Quadrant, etc., London, 1631. - -[17]Oughtred's description of his circular slide rule of 1632 and his - rectilinear slide rule of 1633, as well as a drawing of the circular - slide rule, are reproduced in Cajori's History of the Slide Rule, - Addenda, pp. ii-vi. - -[18]The full title of the Grammelogia I is as follows: - - Grammelogia | or, | The Mathematicall Ring. | Shewing (any reasonable - Capacity that hath | not Arithmeticke) how to resolve and worke | all - ordinary operations of Arithmeticke. | And those which are most - difficult with greatest | facilitie: The extraction of Roots, the - valuation of | Leases, &c. The measuring of Plaines | and Solids. | With - the resolution of Plaine and Sphericall | Triangles. | And that onely by - an Ocular Inspection, | and a Circular Motion. | Naturae secreta tempus - aperit. | London printed by John Haviland, 1630. - -[19]Grammelogia III is the same as Grammelogia I, except for the addition - of an appendix, entitled: - - De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit - nullo (praescripto tempore) vitae | vsuram nobis ingeniique Deus. | - London, | - - . . . The next line or two of this title-page which probably contained - the date of publication, were cut off by the binder in trimming the - edges of this and several other pamphlets for binding into one volume. - -[20]Grammelogia IV has two title pages. The first is Mirifica Logarithmoru' - Projectio Circularis. There follows a diagram of a circular slide rule, - with the inscription within the innermost ring: Nil Finis, Motvs, - Circvlvs vllvs Habet. The second title page is as follows: - - Grammelogia | Or, the Mathematicall Ring. | Extracted from the - Logarythmes, and projected Circular: Now published in the | inlargement - thereof unto any magnitude fit for use: shewing any reason- | able - capacity that hath not Arithmeticke how to resolve and worke, | all - ordinary operations of Arithmeticke: | And those that are most difficult - with greatest facilitie, the extracti- | on of Rootes, the valuation of - Leases, &c. the measuring of Plaines and Solids, | with the resolution - of Plaine and Sphericall Triangles applied to the | Practicall parts of - Geometrie, Horologographie, Geographie | Fortification, Navigation, - Astronomie, &c. | And that onely by an ocular inspection, and a Circular - motion, Invented and first published, by R. Delamain, Teacher, and - Student of the Mathematicks. | Naturae secreta tempus aperit. | - - There is no date. There follows the diagram of a second circular slide - rule, with the inscription within the innermost ring: Typus proiectionis - Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630 - promisi. There are numerous drawings in the Grammelogia, all of which, - excepting the drawings of slide rules on the engraved title-pages of - Grammelogia IV and V, were printed upon separate pieces of paper and - then inserted by hand into the vacant spaces on the printed pages - reserved for them. Some drawings are missing, so that the Bodleian - Grammelogia IV differs in this respect slightly from the copy in the - British Museum and from the British Museum copy of Grammelogia V. - -[21]Epistle, p. (8). - -[22]Aubrey, op. cit., Vol. II., p. 111. - -[23]Rigaud, Correspondence of Scientific Men during the 17th Century, Vol. - I, Oxford, 1841, p. 11. - -[24]Dictionary of National Biography, Art. "Delamain, Richard." See also - Rev. Charles J. Robinson, Taylors' School, from A.D. 1562 to 1874, Vol. - I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b; - Sixth Report of the Royal Commission on Historical Manuscripts, Part I, - Report and Appendix, London, 1877. In this Appendix, p. 82, we read the - following: - - Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. - Petitioner's husband was servant to the King, and one of His Majesty's - engineers for the fortification of the kingdom, and his tutor in - mathematical arts; but upon the breaking out of the war he deserted the - Court, and was called by the State to several employments, in fortifying - the towns of Northampton, Newport, and Abingdon; and was also abroad - with the armies as Quartermaster-General of the Foot, and therein died. - Petitioner is left a disconsolate widow with ten children, the four - least of whom are now afflicted with sickness, and petitioner has - nothing left to support them. There are several considerable sums of - money due to the petitioner, as well from the King as the State. Prays - that she may have some relief amongst other widows. See L. J., VII. 6. - 657. - -[25]Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London, - 1820, p. 34. - -[26]The New Artificial Gauging Line or Rod: together with rules concerning - the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London, - 1633. The copy we have seen is in the Bodleian Library, Oxford. The book - is small sized and has 40 pages. - -[27]Oughtred, op. cit., p. 11. - -[28]S. J. Rigaud, Correspondence of Scientific Men of the 17th Century, - Oxford, Vol. I, 1841, p. 17. - -[29]Rigaud, loc. cit., p. 22. - -[30]Rigaud, loc. cit., pp. 30, 31. - -[31]Oughtred, An Addition vnto the Vse of the Instrument called the Circles - of Proportion, London, 1633, p. 63. - -[32]F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22, - Addenda, pp. vi-ix. - -[33]W. Leybourn, op. cit., 1673, Preface, and pp. 128-29. - -[34]Cajori op. cit., Addenda, p. ix. - -[35]William Leybourn, op. cit., 1673, p. 35. - -[36]See Cajori, op. cit., pp. 20, 28, Addenda, p. ix. - -[37]See F. Cajori, "A Note on the History of the Slide Rule," Bibliotheca - mathematica, 3 F., Vol. 10, pp. 161-163. - -[38]John Atkinson, op. cit., 1694, p. 204. - -[39]Probably the oldest slide rule now in existence is owned by St. John's - College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in - diameter. It was exhibited along with other instruments in May, 1919. - According to the Catalogue of a Loan Exhibition of Early Scientific - Instruments in Oxford, opened May 16, 1919, the instrument is inscribed - with the name of the maker ("Elias Allen fecit") and with the name of - the donor, Georgius Barkham. It is dated 1635, which is only three years - after the first publication of Oughtred's description of his circular - slide rule. It is stated in the Catalogue: "Unfortunately all the - movable parts but the base-plate and a couple of thumb-screws are - missing. The face of the instrument is engraved with Oughtred's - Horizontal Instrument. The back is engraved with eleven Circles of - Proportion as described in Arthur Haughton's book, a copy of which was - presented to St. John's College by George Barkham, to explain the use of - the instrument." As Arthur Haughton's Oxford edition of Oughtred's - Circles of Proportion did not appear until 1660, it would seem that the - instrument was probably not presented to the College before 1660. As far - as is known, the next oldest slide rule is of the year 1654, kept in the - South Kensington Museum, London, and is described in Nature of March 5, - 1914. It is a rectilinear rule, "of boxwood, well made, and bound - together with brass at the two ends. It is of the square type, a little - more than 2 ft. in length, and bears the logarithmic lines first - described by Edmund Gunter. Of these, the num, sin and tan lines are - arranged in pairs, identical and contiguous, one line in each pair being - on the fixed part, and the other on the slide." The instrument is - inscribed, "Made by Robert Bissaker for T. W., 1654." Nowhere else have - we seen reference to Robert Bissaker. His slide rule seems to antedate - the "Whites rule" mentioned above. [This foot-note was added on October - 15, 1919.] - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - - -Note.--The University of California Publications are offered in exchange -for the publication of learned societies and institutions, universities and -libraries. Complete lists of all the publications of the University will be -sent upon request. For sample copies, lists of publications or other -information, address the Manager of the University of California Press, -Berkeley, California, U. 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Current, Volume XXII. - - - - - Transcriber's Notes - - ---Inserted subchapter headings from the Table of Contents into the - text. - ---Corrected two page numbers in the Table of Contents - ---Corrected a Greek quotation in footnote 6 by reference to the - original sources. - - - - - - - -End of the Project Gutenberg EBook of On the History of Gunter's Scale and -the Slide Rule during the Seventeenth Century, by Florian Cajori - -*** END OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - -***** This file should be named 42216.txt or 42216.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/4/2/2/1/42216/ - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - -Updated editions will replace the previous one--the old editions -will be renamed. - -Creating the works from public domain print editions means that no -one owns a United States copyright in these works, so the Foundation -(and you!) can copy and distribute it in the United States without -permission and without paying copyright royalties. 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Hart was the originator of the Project Gutenberg-tm -concept of a library of electronic works that could be freely shared -with anyone. For forty years, he produced and distributed Project -Gutenberg-tm eBooks with only a loose network of volunteer support. - -Project Gutenberg-tm eBooks are often created from several printed -editions, all of which are confirmed as Public Domain in the U.S. -unless a copyright notice is included. Thus, we do not necessarily -keep eBooks in compliance with any particular paper edition. - -Most people start at our Web site which has the main PG search facility: - - www.gutenberg.org - -This Web site includes information about Project Gutenberg-tm, -including how to make donations to the Project Gutenberg Literary -Archive Foundation, how to help produce our new eBooks, and how to -subscribe to our email newsletter to hear about new eBooks. diff --git a/old/old/42216-0.txt b/old/old/42216-0.txt deleted file mode 100644 index 7b784d0..0000000 --- a/old/old/42216-0.txt +++ /dev/null @@ -1,2002 +0,0 @@ -The Project Gutenberg EBook of On the History of Gunter's Scale and the -Slide Rule during the Seventeenth Centur, by Florian Cajori - -This eBook is for the use of anyone anywhere at no cost and with -almost no restrictions whatsoever. You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org - - -Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century - -Author: Florian Cajori - -Release Date: February 26, 2013 [EBook #42216] - -Language: English - -Character set encoding: UTF-8 - -*** START OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - - - - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - IN - MATHEMATICS - - Vol. 1, No. 9, pp. 187-209 February 17, 1920 - - - - - ON THE HISTORY OF GUNTER’S SCALE AND - THE SLIDE RULE DURING THE - SEVENTEENTH CENTURY - - - BY - FLORIAN CAJORI - - - UNIVERSITY OF CALIFORNIA PRESS - BERKELEY - - - - - TABLE OF CONTENTS - - - PAGE - I. Introduction 187 - II. Innovations in Gunter’s Scale 188 - Changes introduced by Edmund Wingate 188 - Changes introduced by Milbourn 189 - Changes introduced by Thomas Brown and John Brown 190 - Changes introduced by William Leybourn 192 - III. Richard Delamain’s “Grammelogia” 192 - Different editions or impressions 194 - Description of Delamain’s instrument of 1630 195 - Delamain’s later designs, and directions for using his - instruments 197 - IV. Controversy between Oughtred and Delamain on the invention of - the circular slide rule 199 - V. Independence and priority of invention 203 - VI. Oughtred’s “Gauging Line,” 1633 206 - VII. Other seventeenth century slide rules 207 - - - - - I. INTRODUCTION - - - In my history of the slide rule[1], and my article on its invention[2] it -is shewn that William Oughtred and not Edmund Wingate is the inventor, that -Oughtred’s circular rule was described in print in 1632, his rectilinear -rule in 1633. Richard Delamain is referred to as having tried to -appropriate the invention to himself[3] and as having written a scurrilous -pamphlet against Oughtred. All our information about Delamain was taken -from De Morgan,[4] who, however, gives no evidence of having read any of -Delamain’s writings on the slide rule. Through Dr. Arthur Hutchinson of -Pembroke College, Cambridge, I learned that Delamain’s writings on the -slide rule were available. In this article will be given: First, some -details of the changes introduced during the seventeenth century in the -design of Gunter’s scale by Edmund Wingate, Milbourn, Thomas Brown, John -Brown and William Leybourn; second, an account of Delamain’s book of 1630 -on the slide rule which antedates Oughtred’s first publication (though -Oughtred’s date of invention is earlier than the date of Delamain’s alleged -invention) and of Delamain’s later designs of slide rules; third, an -account of the controversy between Delamain and Oughtred; fourth, an -account of a later book on the slide rule written by William Oughtred, and -of other seventeenth century books on the slide rule. - - - - - II. INNOVATIONS IN GUNTER’S SCALE - - - Changes introduced by Wingate - - We begin with Anthony Wood’s account of Wingate’s introduction of -Gunter’s scale into France.[5] - - In 1624 he transported into France the rule of proportion, having a - little before been invented by Edm. Gunter of Gresham Coll. and - communicated it to most of the chiefest mathematicians then residing in - Paris: who apprehending the great benefit that might accrue thereby, - importun’d him to express the use thereof in the French tongue. Which - being performed accordingly, he was advised by monsieur Alleawne the - King’s chief engineer to dedicate his book to monsieur the King’s only - brother, since duke of Orleans. Nevertheless the said work coming forth - as an abortive (the publishing thereof being somewhat hastened, by reason - an advocate of Dijon in Burgundy began to print some uses thereof, which - Wingate had in a friendly way communicated to him) especially in regard - Gunter himself had learnedly explained its use in a far larger volume.[6] - -Gunter’s scale, which Wingate calls the “rule of proportion,” contained, as -described in the French edition of 1624, four lines: (1) A single line of -numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot -in length, divided into 12 inches and tenths of inches, also a line, one -foot in length, divided into tenths and hundredths. - -The English editions of this book which appeared in 1623 and 1628 are -devoid of interest. The editions of 1645 and 1658 contain an important -innovation.[7] In the preface the reasons why this instrument has not been -used more are stated to be: (1) the difficulty of drawing the lines with -exactness, (2) the trouble of working thereupon by reason (sometimes) of -too large an extent of the compasses, (3) the fact that the instrument is -not readily portable. The drawing of Wingate’s arrangement of the scale in -the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains -five parallel lines, about 66 cm. long, each having the divisions of one -line marked on one side and of another line on the other side. Thus each -line carries two graduations: (1) A single logarithmic line of numbers; (2) -a logarithmic line of numbers thrice repeated; (3) the first scale -repeated, but beginning with the graduations which are near the middle of -the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3; -(4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of -tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000 -equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of -inches; (10) a scale consisting of three kinds, viz., a gauge line, a line -of chords, and a foot measure, divided into 1000 equal parts. - -Important are the first and second scales, by which cube root extraction -was possible “by inspection only, without the aid of pen or compass;” -similarly the third and fourth scales, for square roots. This innovation is -due to Wingate. The 1645 edition announces that the instrument was made in -brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in -Hosier Lane. - - - Changes introduced by Milbourn - -William Leybourn, in his The Line of Proportion or Numbers, Commonly called -Gunter’s Line, Made Easie, London, 1673, says in his preface “To the -Reader:” - - The Line of Proportion or Numbers, commonly called (by Artificers) - Gunter’s Line, hath been discoursed of by several persons, and variously - applied to divers uses; for when Mr. Gunter had brought it from the - Tables to a Line, and written some Uses thereof, Mr. Wingate added divers - Lines of several lengths, thereby to extract the Square or Cube Roots, - without doubling or trebling the distance of the Compasses: After him Mr. - Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral - Line, thereby enlarging the divisions of the Line. - -On pages 127 and 128 Leybourn adds: - - Again, One T. Browne, a Maker of Mathematical Instruments, made it in a - Serpentine or Spiral Line, composed of divers Concentrick Circles, - thereby to enlarg the divisions, which was the contrivance of one Mr. - Milburn a Yorkshire Gentleman, who writ thereof, and communicated his - Uses to the aforesaid Brown, who (since his death) attributed it to - himself: But whoever was the contriver of it, it is not without - inconvenience; for it can in no wise be made portable; and besides - (instead of compasses) an opening Joynt with thirds [threads] must be - placed to move upon the Centre of the Instrument, without which no - proportion can be wrought. - -This Mr. Milburn is probably the person named in the diary of the -antiquarian, Elias Ashmole, on August 13 [1646?]; “I bought of Mr. Milbourn -all his Books and Mathematical Instruments.”[8] Charles Hutton[9] says that -Milburne of Yorkshire designed the spiral form about 1650. This date is -doubtless wrong, for Thomas Browne who, according to Leybourn, got the -spiral form of line from Milbourn, is repeatedly mentioned by William -Oughtred in his Epistle[10] printed some time in 1632 or 1633. Oughtred -does not mention Milbourn, and says (page 4) that the spiral form “was -first hit upon by one Thomas Browne a Joyner, . . . the serpentine -revolution being but two true semicircles described on severall -centers.”[11] - - - Changes introduced by Thomas Brown and John Brown - -Thomas Brown did not publish any description of his instrument, but his -son, John Brown, published in 1661 a small book,[12] in which he says -(preface) that he had done “as Mr. Oughtred with Gunter’s Rule, to a -sliding and circular form; and as my father Thomas Brown into a Serpentine -form; or as Mr. Windgate in his Rule of Proportion.” He says also that -“this brief touch of the Serpentine-line I made bold to assert, to see if I -could draw out a performance of that promise, that hath been so long -unperformed by the promisers thereof.” Accordingly in Chapter XX he gives a -description of the serpentine line, “contrived in five (or rather 15) -turn.” Whether this description, printed in 1661, exactly fits the -instrument as it was developed in 1632, we have no means of knowing. John -Brown says: - - 1. First next the center is two circles divided one into 60, the other - into 100 parts, for the reducing of minutes to 100 parts, and the - contrary. - - 2. You have in seven turnes two inpricks, and five in divisions, the - first Radius of the sines (or Tangents being neer the matter, alike to - the first three degrees,) ending at 5 degrees and 44 minutes. - - 3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in - three margents in divisions, and the line of versed sines in pricks, - under the line of Tangents, according to Mr. Gunter’s cross-staff: the - sines and Tangents beginning at 5 degrees, and 44 minutes where the other - ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the - line of numbers beginning at 10, and proceeding to 100, being one entire - Radius, and graduated into as many divisions as the largeness of the - instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to - 100 into 20 parts in one unit of increase, but the Tangents are divided - into single minutes from the beginning to the end, both in the first, - second and third Radiusses, and the sines into minutes; also from 30 - minutes to 40 degrees, and from 40 to 60, into every two minutes, and - from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the - rest as many as can be well discovered. - - The versed sines are set after the manner of Mr. Gunter’s Cross-staff, - and divided into every 10th minutes beginning at 0, and proceeding to 156 - going backwards under the line of Tangents. - - 4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is - the secants to 51 degrees, being nothing else but the sines reitterated - beyond 90. - - 5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85 - degrees, whereby all trouble of backward working is avoided. - - 6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and - also a line of natural sines, for finding of differences in sines, for - finding hour and Azimuth. - - 7. Seventhly, next the verge or outermost edge is a line of equal parts - to get the Logarithm of any number, or the Logarithm sine and Tangent of - any ark or angle to four figures besides the carracteristick. - - 8. Eightly and lastly, in the space place between the ending of the - middle five turnes, and one half of the circle are three prickt lines - fitted for reduction. The uppermost being for shillings, pence and - farthings. The next for pounds, and ounces, and quarters of small - Averdupoies weight. The last for pounds, shillings and pence, and to be - used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay - the hair or edge of one of the legs of the index on 16. 3½ in the line of - 1. s. d. and the hair shall cut on the equal parts 81 16; and the - contrary, if you have a decimal fraction, and would reduce it to a proper - fraction, the like may you do for shillings, and pence, and pounds, and - ounces. - - The uses of the lines follow. - - As to the use of these lines, I shall in this place say but little, and - that for two reasons. First, because this instrument is so contrived, - that the use is sooner learned then any other, I speak as to the manner, - and way of using it, because by means of first second and third - radiusses, in sines and Tangents, the work is always right on, one way or - other, according to the Canon whatsoever it be, in any book that treats - of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in - Oughtred from page 64 to 107. - - Secondly, and more especially, because the more accurate, and large - handling thereof is more then promised, if not already performed by more - abler pens, and a large manuscript thereof by my Sires meanes, provided - many years ago, though to this day not extant in print; so for his sake I - claiming my interest therein, make bold to present you with these few - lines, in order to the use of them: And first note, - - 1. Which soever of the two legs is set to the first term in the question, - that I call the first leg always, and the other being set to the second - term, I call the second leg . . . - -The exact nature of the contrivance with the “two legs” is not described, -but it was probably a flat pair of compasses, attached to the metallic -surface on which the serpentine line was drawn. In that case the instrument -was a slide rule, rather than a form of Gunter’s line. In his publication -of 1661, as also in later publications,[13] John Brown devoted more space -to Gunter’s scales, requiring the use of a separate pair of compasses, than -to slide rules. - - - Changes introduced by William Leybourn - -The same remark applies to William Leybourn who, after speaking of Seth -Partridge’s slide rule, returns to forms of Gunter’s scale, saying:[14] - - There is yet another way of disposing of this Line of Proportion, by - having one Line of the full length of the Ruler, and another Line of the - same Radius broken in two parts between 3 and 4; so that in working your - Compasses never go off of the Line: This is one of the best contrivances, - but here Compasses must be used. These are all the Contrivances that I - have hitherto seen of these Lines: That which I here speak of, and will - shew how to use, is only two Lines of one and the same Radius, being set - upon a plain Ruler of any length (the larger the better) having the - beginning of one Line, at the end of the other, the divisions of each - Line being set so close together, that if you find any number upon one of - the Lines, you may easily see what number stands against it on the other - Line. This is all the Variation. . . . - - Example 1. If a Board be 1 Foot 64 parts broad, how much in length of - that Board will make a Foot Square? Look upon one of your Lines (it - matters not which) for 1 Foot 64 parts, and right against it on the other - Line you shall find 61; and so many parts of a Foot will make a Foot - square of that Board. - -This contrivance solves the equation 1.64x=1, yielding centesimal parts of -a foot. - -James Atkinson[15] speaks of “Gunter’s scale” as “usually of Boxwood . . . -commonly 2 ft. long, 1½ inch broad” and “of two kinds: long Gunter or -single Gunter, and the sliding Gunter. It appears that during the -seventeenth century (and long after) the Gunter’s scale was a rival of the -slide rule. - - - - - III. RICHARD DELAMAIN’S GRAMMELOGIA - - -We begin with a brief statement of the relations between Oughtred and -Delamain. At one time Delamain, a teacher of mathematics in London, was -assisted by Oughtred in his mathematical studies. In 1630 Delamain -published the Grammelogia, a pamphlet describing a circular slide rule and -its use. In 1631 he published another tract, on the Horizontall -Quadrant.[16] In 1632 appeared Oughtred’s Circles of Proportion[17] -translated into English from Oughtred’s Latin manuscript by another pupil, -William Forster, in the preface of which Forster makes the charge (without -naming Delamain) that “another . . . went about to pre-ocupate” the new -invention. This led to verbal disputes and to the publication by Delamain -of several additions to the Grammelogia, describing further designs of -circular slide rules and also stating his side of the bitter controversy, -but without giving the name of his antagonist. Oughtred’s Epistle was -published as a reply. Each combatant accuses the other of stealing the -invention of the circular slide rule and the horizontal quadrant. - -The two title-pages of the edition of the Grammelogia in the British Museum - in London which we have called “Grammelogia IV.” - - - Different editions or impressions - -There are at least five different editions, or impressions, of the -Grammelogia which we designate, for convenience, as follows: - - Grammelogia I, 1630. One copy in the Cambridge University Library.[18] - - Grammelogia II, I have not seen a copy of this. - - Grammelogia III, One copy in the Cambridge University Library.[19] - - Grammelogia IV, One copy in the British Museum, another in the Bodleian - Library, Oxford.[20] - - Grammelogia V, One copy in the British Museum. - -In Grammelogia I the first three leaves and the last leaf are without -pagination. The first leaf contains the title-page; the second leaf, the -dedication to the King and the preface “To the Reader;” the third leaf, the -description of the Mathematical Ring. Then follow 22 numbered pages. -Counting the unnumbered pages, there are altogether 30 pages in the -pamphlet. Only the first three leaves of this pamphlet are omitted in -Grammelogia IV and V. - -In Grammelogia III the Appendix begins with a page numbered 52 and bears -the heading “Conclusion;” it ends with page 68, which contains the same two -poems on the mathematical ring that are given on the last page of -Grammelogia I but differs slightly in the spelling of some of the words. -The 51 pages which must originally have preceded page 52, we have not seen. -The edition containing these we have designated Grammelogia II. The reason -for the omission of these 51 pages can only be conjectured. In Oughtred’s -Epistle (p. 24), it is stated that Delamain had given a copy of the -Grammelogia to Thomas Brown, and that two days later Delamain asked for the -return of the copy, “because he had found some things to be altered -therein” and “rent out all the middle part.” Delamain labored “to recall -all the bookes he had given forth, (which were many) before the sight of -Brownes Lines.” These spiral lines Oughtred claimed that Delamain had -stolen from Brown. The title-page and page 52 are the only parts of the -Appendix, as given in Grammelogia III, that are missing in the Grammelogia -IV and V. - -Grammelogia IV answers fully to the description of Delamain’s pamphlet -contained in Oughtred’s Epistle. It was brought out in 1632 or 1633, for -what appears to be the latest part of it contains a reference (page 99) to -the Grammelogia I (1630) as “being now more then two yeares past.” -Moreover, it refers to Oughtred’s Circles of Proportion, 1632, and -Oughtred’s reply in the Epistle was bound in the Circles of Proportion -having the Addition of 1633. For convenience of reference we number the two -title-pages of Grammelogia IV, “page (1)” and “page (2),” as is done by -Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page -numbers which we assign will be placed in parentheses, to distinguish them -from the page numbers which are printed in Grammelogia IV. The pages -(44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the -same as the pages 53-68. Thus only thirty-eight pages have page numbers -printed on them. The pages (67) and (83) are identical in wording, except -for some printer’s errors; they contain verses in praise of the Ring, and -have near the bottom the word “Finis.” Also, pages (22) and (23) are -together identical in wording with page (113), which is set up in finer -type, containing an advertisement of a part of Grammelogia IV explaining -the mode of graduating the circular rules. There are altogether six parts -of Grammelogia IV which begin or end by an address to the reader, thus: “To -the Reader,” “Courteous Reader,” or “To the courteous and benevolent Reader -. . .,” namely the pages (8), (22), (68), (89), (90), (108). In his Epistle -(page 2), Oughtred characterizes the make up of the book in the following -terms: - - In reading it . . . I met with such a patchery and confusion of - disjoynted stuffe, that I was striken with a new wonder, that any man - should be so simple, as to shame himselfe to the world with such a - hotch-potch. - -Grammelogia V differs from Grammelogia IV in having only the second -title-page. The first title-page may have been torn off from the copy I -have seen. A second difference is that the page with the printed numeral 22 -in Grammelogia IV has after the word “Finis” the following notice: - - This instrument is made in Silver, or Brasse for the Pocket, or at any - other bignesse, over against Saint Clements Church without Temple Barre, - by Elias Allen. - -This notice occurs also on page 22 of Grammelogia I and III, but is omitted -from page 22 of Grammelogia V. - - - Description of Delamain’s instrument of 1630 - -In his address to King Charles I, in his Grammelogia I, Delamain emphasizes -the ease of operating with his slide rule by stating that it is “fit for -use . . . as well on Horse backe as on Foot.” Speaking “To the Reader,” he -states that he has “for many yeares taught the Mathematicks in this Towne,” -and made efforts to improve Gunter’s scale “by some Motion, so that the -whole body of Logarithmes might move proportionally the one to the other, -as occasion required. This conceit in February last [1629] I struke upon, -and so composed my Grammelogia or Mathematicall Ring; by which only with an -ocular inspection, there is had at one instant all proportionalls through -the said body of Numbers.” He dates his preface “first of January, 1630.” -The fifth and sixth pages contain his “Description of the Grammelogia,” the -term Grammelogia being applied to the instrument, as well as to the book. -His description is as follows: - - The parts of the Instrument are two Circles, the one moveable, and the - other fixed; The moveable is that unto which is fastened a small pin to - move it by; the other Circle may be conceived to be fixed; The - circumference of the moveable Circle is divided into unequall parts, - charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures - doe represent themselves, or such numbers unto which a Cipher or Ciphers - are added, and are varied as the occasion falls out in the speech of - Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or - 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c. - -After elaborating this last point and explaining the decimal subdivisions -on the scales of the movable circle, he says that “the numbers and -divisions on the fixed Circle, are the very same that the moveable are, . -.” There is no drawing of the slide rule in this publication. The -twenty-two numbered pages give explanations of the various uses to which -the instrument can be put: “How to performe the Golden Rule” (pp. 1-3), -“Further uses of the Golden Rule” (pp. 4-6), “Notions or Principles -touching the disposing or ordering of the Numbers in the Golden Rule in -their true places upon the Grammelogia” (pp. 7-11), “How to divide one -number by another” (pp. 12, 13), “to multiply one Number by another” (pp. -14, 15), “To find Numbers in continuall proportion” (pp. 16, 17), “How to -extract the Square Root,” “How to extract the Cubicke Root” (pp. 18-21), -“How to performe the Golden Rule” (the rule of proportion) is explained -thus: - - Seeke the first number in the moveable, and bring it to the second number - in the fixed, so right against the third number in the moveable, is the - answer in the fixed. - - If the Interest of 100. li. be 8. li. in the yeare, what is the Interest - of 65. li. for the same time. - - Bring 100. in the moveable to 8. in the fixed, so right against 65. in - the moveable is 5.2. in the fixed, and so much is the Interest of 65. li. - for the yeare at 8. li. for 100. li. per annum. - - The Instrument not removed, you may at one instant right against any - summe of money in the moveable, see the Interest thereof in the fixed: - the reason of this is from the Definition of Logarithmes. - -These are the earliest known printed instructions on the use of a slide -rule. It will be noticed that the description of the instrument at the -opening makes no references to logarithmic lines for the trigonometric -functions; only the line of numbers is given. Yet the title-page promised -the “resolution of Plaine and Sphericall Triangles.” Page 22 throws light -upon this matter: - - If there be composed three Circles of equal thicknesse, A.B.C. so that - the inner edge of D [should be B] and the outward edge of A bee - answerably graduated with Logarithmall signes [sines], and the outward - edge of B and the inner edge of A with Logarithmes; and then on the - backside be graduated the Logarithmall Tangents, and againe the - Logarithmall signes oppositly to the former graduations, it shall be - fitted for the resolution of Plaine and Sphericall Triangles. - -After twelve lines of further remarks on this point he adds: - - Hence from the forme, I have called it a Ring, and Grammelogia by - annoligie of a Lineary speech; which Ring, if it were projected in the - convex unto two yards Diameter, or thereabouts, and the line Decupled, it - would worke Trigonometrie unto seconds, and give proportionall numbers - unto six places only by an ocular inspection, which would compendiate - Astronomicall calculations, and be sufficient for the Prosthaphaeresis of - the Motions: But of this as God shall give life and ability to health and - time. - -The unnumbered page following page 22 contains the patent and copyright on -the instrument and book: - - Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs - an Instrument called Grammelogia, or The Mathematicall Ring, together - with a Booke so intituled, expressing the use thereof, being his owne - Invention; we of our Gracious and Princely favour have granted unto the - said Richard Delamain and his Assignes, Privilege, Licence, and - Authority, for the sole Making, Printing and Selling of the said - Instrument and Booke: straightly forbidding any other to Make, Imprint, - or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument - or Booke within any our Dominions, during the space of ten yeares next - ensuing the date hereof, upon paine of Our high displeasure. Given under - our hand and Signet at our Palace of Westminster, the fourth day of - January, in the sixth yeare of our Raigne. - - - Delamain’s later designs, and directions for using his instruments - -In the Appendix of Grammelogia III, on page 52 is given a description of an -instrument promised near the end of Grammelogia I: - - That which I have formerly delivered hath been onely upon one of the - Circles of my Ring, simply concerning Arithmeticall Proportions, I will - by way of Conclusion touch upon some uses of the Circles, of Logarithmall - Sines, and Tangents, which are placed on the edge of both the moveable - and fixed Circles of the Ring in respect of Geometricall Proportions, but - first of the description of these Circles. - - First, upon the side that the Circle of Numbers is one, are graduated on - the edge of the moveable, and also on the edge of the fixed the - Logarithmall Sines, for if you bring 1. in the moveable amongst the - Numbers to 1. in the fixed, you may on the other edge of the moveable and - fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto - 6.6. and each degree subdivided, and then over the former divisions and - figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5. - 4. 3. 2. 1. each of those divided by small points. - - Secondly, (if the Ring is great) neere the outward edge of this side of - the fixed against the Numbers, are the usuall divisions of a Circle, and - the points of the Compasse: serving for observation in Astronomy, or - Geometry, and the sights belonging to those divisions, may be placed on - the moveable Circle. - - Thirdly, opposite to those Sines on the other side are the Logarithmall - Tangents, noted alike both in the moveable and fixed thus - 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or - divisions serve also for their Complements to 90. so 40 gr. stands for - 50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both - in the moveable and fixed is also divided into parts. As for the degrees - which are under 6. viz. 5.4.3.2.1. they are noted with small figures over - this divided Circle from 45.40.35.30.25. &c. and each of those degrees - divided into parts by small points both in the moveable and fixed. - - Fourthly, on the other edge of the moveable on the same side is another - graduation of Tangents, like that formerly described. And opposite unto - it, in the fixed is a Graduation of Logarithmall sines in every thing - answerable to the first descrition of Sines on the other side. - - Fifthly, on the edge of the Ring is graduated a parte of the Æquator, - numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees - of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree - both of the Æquator, and Meridian are subdivided into parts; these two - graduated Circles serve to resolve such Questions which concerne - Latitude, Longitude, Rumb, and Distance, in Nauticall operations. - - Sixthly, to the concave of the Ring may be added a Circle to be elevated - or depressed for any Latitude, representing the Æquator, and so divided - into houres and parts with an Axis, to shew both the houre, and Azimuth, - and within this Circle may be hanged a Box, and Needle with a Socket for - a staffe to slide into it, and this accommodated with scrue pines to - fasten it to the Ring and staffe, or to take it off at pleasure. - -The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and -V make no reference to the dispute with Oughtred and may, therefore, be -assumed to have been published before the appearance of Oughtred’s Circles -of Proportion. On page 53, “To the Reader,” he says: - - . . . you may make use of the Projection of the Circles of the Ring upon - a Plaine, having the feet of a paire of compasses (but so that they be - flat) to move on the Center of that Plaine, and those feet to open and - shut as a paire of Compasses . . . now if the feet bee opened to any two - termes or numbers in that Projection, then may you move the first foot to - the third number, and the other foot shall give the Answer; . . . it hath - pleased some to make use of this way. But in this there is a double - labour in respect to that of the Ring, the one in fitting those feet unto - the numbers assigned, and the other by moving them about, in which a man - can hardly accommodate the Instrument with one hand, and expresse the - Proportionals in writing with the other. By the Ring you need not but - bring one number to another, and right against any other number is the - Answer without any such motion. . . . upon that [the Ring] I write, - shewing some uses of those Circles amongst themselves, and conjoyned with - others . . . in Astronomy, Horolographie, in plaine Triangles applyed to - Dimensions, Navigation, Fortification, etc. . . . But before I come to - Construction, I have thought it convenient by way introduction, to - examine the truth of the graduation of those Circles . . . - -These are the words of a practical man, interested in the mechanical -development of his instrument. He considers not only questions of -convenience but also of accuracy. The instrument has, or may have now, also -lines of sines and tangents. To test the accuracy of the circles of -Numbers, “bring any number in the moveable to halfe of that number in the -fixed: so any number or part in the fixed shall give his double in the -moveable, and so may you trie of the thirds, fourths &c. of numbers, vel -contra,” (p. 54). On page 55 are given two small drawings, labelled, “A -Type of the Ringe and Scheme of this Logarithmicall projection, the use -followeth. These Instruments are made in Silver or Brasse by John Allen -neare the Sauoy in the Strand.” - - - IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE - CIRCULAR SLIDE RULE - -Delamain’s publication of 1630 on the ‘Mathematicall Ring’ does not appear -at that time to have caused a rupture between him and Oughtred. When in -1631 Delamain brought out his Horizontall Quadrant, the invention of which -Delamain was afterwards charged to have stolen from Oughtred, Delamain was -still in close touch with Oughtred and was sending Oughtred in the Arundell -House, London, the sheets as they were printed. Oughtred’s reference to -this in his Epistle (p. 20) written after the friendship was broken, is as -follows: - - While he was printing his tractate of the Horizontall quadrant, although - he could not but know that it was injurious to me in respect of my free - gift to Master Allen, and of William Forster, whose translation of my - rules was then about to come forth: yet such was my good nature, and his - shamelessnesse, that every day, as any sheet was printed, hee sent, or - brought the same to mee at my chamber in Arundell house to peruse which I - lovingly and ingenuously did, and gave him my judgment of it. - -Even after Forster’s publication of Oughtred’s Circles of Proportion, 1632, -Oughtred had a book, A canon of Sines Tangents and Secants, which he had -borrowed from Delamain and was then returning (Epistle, page (5)). The -attacks which Forster, in the preface to the Circles of Proportion, made -upon Delamain (though not naming Delamain) started the quarrel. Except for -Forster and other pupils of Oughtred who urged him on to castigate -Delamain, the controversy might never have arisen. Forster expressed -himself in part as follows: - - . . . being in the time of the long vacation 1630, in the Country, at the - house of the Reverend, and my most worthy friend, and Teacher, Mr. - William Oughtred (to whose instruction I owe both my initiation, and - whole progresse in these Sciences.) I vpon occasion of speech told him of - a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made - (such as it vsually called Mr. Gunter’s Ruler) 6 feet long, to be vsed - with a payre of beame-compasses. “He answered that was a poore invention, - and the performance very troublesome: But, said he, seeing you are taken - with such mechanicall wayes of Instruments, I will shew you what deuises - I have had by mee these many yeares.” And first, hee brought to mee two - Rulers of that sort, to be vsed by applying one to the other, without any - compasses: and after that hee shewed mee those lines cast into a circle - or Ring, with another moueable circle vpon it. I seeing the great - expeditenesse of both those wayes; but especially, of the latter, wherein - it farre excelleth any other Instrument which hath bin knowne; told him, - I wondered that hee could so many yeares conceale such vseful inuentions, - not onely from the world, but from my selfe, to whom in other parts and - mysteries of Art, he had bin so liberall. He answered, “That the true way - of Art is not by Instruments, but by Demonstration: and that it is a - preposterous course of vulgar Teachers, to begin with Instruments, and - not with the Sciences, and so in-stead of Artists, to make their - Schollers only doers of tricks, and as it were Iuglers: to the despite of - Art, losse of precious time, and betraying of willing and industrious - wits, vnto ignorance and idlenesse. That the vse of Instruments is indeed - excellent, if a man be an Artist: but contemptible, being set and opposed - to Art. And lastly, that he meant to commend to me, the skill of - Instruments, but first he would haue me well instructed in the Sciences. - He also shewed me many notes, and Rules for the vse of those circles, and - of his Horizontall Instrument, (which he had proiected about 30 yeares - before) the most part written in Latine. All which I obtained of him - leaue to translate into English, and make publique, for the vse, and - benefit of such as were studious, and louers of these excellent Sciences. - - Which thing while I with mature, and diligent care (as my occasions would - give me leaue) went about to doe: another to whom the Author in a louing - confidence discouered this intent, using more hast then good speed, went - about to preocupate; of which vntimely birth, and preuenting (if not - circumuenting) forwardnesse, I say no more: but aduise the studious - Reader, onely so farre to trust, as he shal be sure doth agree to truth & - Art. - -While in this dedication reference is made to a slide rule or “ring” with a -“moveable circle,” the instrument actually described in the Circles of -Proportion consists of fixed circles “with an index to be opened after the -manner of a paire of Compasses.” Delamain, as we have seen, had decided -preference for the moveable circle. To Oughtred, on the other hand, one -design was about as good as the other; he was more of a theorist and -repeatedly expressed his contempt for mathematical instruments. In his -Epistle (page (25)), he says he had not “the one halfe of my intentions -upon it” (the rule in his book), nor one with a “moveable circle and a -thread, but with an opening Index at the centre (if so be that bee cause -enough to make it to bee not the same, but another Instrument) for my part -I disclaime it: it may go seeke another Master: which for ought I know, -will prove to be Elias Allen himselfe: for at his request only I altered a -little my rules from the use of the moveable circle and the thread, to the -two armes of an Index.” - -All parts of Delamain’s Grammelogia IV, except pages 1-22 and 53-68 -considered above, were published after the Circles of Proportion, for they -contain references to the ill treatment that Delamain felt or made believe -that he felt, that he had received in the book published by Oughtred and -Forster. Oughtred’s reference to teachers whose scholars are “doers of -tricks,” “Iuglers,” and Forster’s allusion to “another to whom the Author -in a loving confidence” explained the instrument and who “went about to -preocupate” it, are repeatedly mentioned. Delamain says, (page (89)) that -at first he did not intend to express himself in print, “but sought peace -and my right by a private and friendly way.” Oughtred’s account of -Delamain’s course is that of an “ill-natured man” with a “virulent tongue,” -“sardonical laughter” and “malapert sawsiness.” Contrasting Forster and -Delamain, he says that, of the former he “had the very first moulding” and -made him feel that “the way of Art” is “by demonstration.” But Delamain was -“already corrupted with doing upon Instruments, and quite lost from ever -being made an Artist.” (Epistle page (27)). Repeatedly does Oughtred assert -Delamain’s ignorance of mathematics. The two men were evidently of wholly -different intellectual predilections. That Delamain loved instruments is -quite evident, and we proceed to describe his efforts to improve the -circular slide rule. - -The Grammelogia IV is dedicated to King Charles I. Delamain says: - - . . . Everything hath his beginning, and curious Arts seldome come to the - height at the first; It was my promise then to enlarge the invention by a - way of decuplating the Circles, which I now present unto your sacred - Majestie as the quintessence and excellencie there of . . . - -His enlarged circular rules are illustrated in the Bodleian Library copy of -Grammelogia IV by four diagrams, two of them being the two drawings on the -two title-pages at the beginning of the Grammelogia IV, 4 inches in -external diameter, and exhibiting eleven concentric circular lines carrying -graduations of different sorts. In the second of these designs all circles -are fixed. The other two drawings are each 10¾ inches in external diameter -and exhibit 18 concentric circular lines; the folded sheet of the first of -these drawings is inserted between pages (23) and (24), the second folded -sheet between pages (83) and (84). All circles of this second instrument -are fixed. Counting in the two small drawings in Grammelogia III, there are -in all six drawings of slide rules in the Bodleian Grammelogia IV. On pages -(24) to (43) Delamain explains the graduation of slide rules. He takes -first a rule which has one circle of equal parts, divided into 1000 equal -divisions. From a table of logarithms he gets log 2 = 0.301; from the -number 301 in the circle of equal parts he draws a line to the center of -the circle and marks the intersection with the circles of numbers by the -figure 2. Thus he proceeds with log 3, log 4, and so on; also with log sin -x and log tan x. For log sin x he uses two circles, the first (see page -(27)) for angles from 34′ 24″ to 5° 44′ 22″, the second circle from 5° 44′ -22″ to 90°. The drawings do not show the seconds. He suggests many -different designs of rules. On page (29) he says: - - For the single projection of the Circles of my Ring, and the dividing and - graduating of them: which may bee so inserted upon the edges of Circles - of mettle turned in the forme of a Ring, so that one Circle may moove - betweene two fixed, by helpe of two stayes, then may there be graduated - on the face of the Ring, upon the outer edge of the mooveable and inner - edge of the fixed, the Circle of Numbers, then upon the inner edge of - that mooveable Circle, and the outward edge of that inner fixed Circle - may be inserted the Circle of Sines, and so according to the description - of those that are usually made. - -In addition to these lines he proceeds to mention the circle giving the -ordinary division into degrees and minutes, and two circles of tangents on -the other side of the rule. - -Next Delamain explains an arrangement of all the graduation on one side of -the rule by means of “a small channell in the innermost fixed Circle, in -which may be placed a small single Index, which may have sufficient length -to reach from the innermost edge of the Mooveable Circle, unto the outmost -edge of the fixed Circle, which may be mooved to and fro at pleasure, in -the channell, which Index may serve to shew the opposition of Numbers” (p. -(31)). From this it is clear that the invention of the “runner” goes back -to the very first writers on the slide rule. - -After describing a modification of the above arrangement, he adds, “many -other formes might be deliverd, about this single projection” (p. (32)). - -Proceeding to the “enlarging” of the circles in the Ring, to, say, the -“Quadruple to that which is single, that is, foure times greater,” the -“equall parts” are distributed over four circles instead of only one -circle, but the general method of graduation is the same as before (p. -(33)); there being now four circles carrying the logarithms of numbers, and -so on. Next he points out “severall wayes how the Circles of the -Mathematicall Ring (being inlarged) may be accommodated for practicall -use:” (1) The Circles are all fixed in a plain and movable flat compasses -(or better, a movable semicircle) are used for fixing any two positions; -(2) There is a “double projection” of each logarithmic line “inlarged on a -Plaine,” one fixed, the other movable, as shown in his first figure on the -title-page, a single index only being used; (3) use of “my great Cylinder -which I have long proposed (in which all the Circles are of equall -greatnesse,) and it may be made of any magnitude or capacity, but for a -study (hee that will be at the charge) it may be of a yard diameter and of -such an indifferent length that it may containe 100 or more Circles fixed -parallel one to the other on the Cylinder, having a space betweene each of -them, so that there may bee as many mooveable Circles, as there are fixed -ones, and these of the mooveable linked, or fastened together, so that they -may all moove together by the fixed ones in these spaces, whose edges both -of the fixed, and mooveable being graduated by helpe of a single Index will -shew the proportionalls by opposition in this double Projection, or by a -double Index in a single Projection” (p. (36)). - -Next follows the detailed description of his Ring “on a Plaine, according -to the diagramme that was given the King (for a view of that projection) -and afterwards the Ring it selve.” The diagram is the large one which we -mentioned as inserted between pages (23) and (24). The instrument has two -circles, one moveable, upon each of which are described 13 distinct -circular graduations. The lines on the fixed circle are: “The Circle of -degrees and calendar,” E. “Circle of equall parts, and part of the Equator, -and Meridian,” TT. “The Circle of Tangents,” S. “The Circle of Sines,” D. -“The Circle of Decimals,” N. “The Circle of Numbers.” The lines on the -movable circle are: N. “The Circle of Numbers,” E. “The Circle of equated -figures, and bodies,” S. “The Circle of Sines,” TT. “The Circle of -Tangents,” Y. “The Circle of time, yeares, and monethes.” - -On pages (84)-(88) Delamain explains an enlargement of his Ring for -computations involving the sines of angles near to 90°. On page (86) he -says: - - I have continued the Sines of the Projection unto two severall - revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr. - (being the last revolution of the decuplation of the former, or the - hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s. - and ends at 90.gr. (being the last of a ternary of decuplated - revolutions, or the thousand part of that Projection) and may bee thus - used. - -He explains the manner of using these extra graduations. Thus he claims to -have attained degrees of accuracy which enabled him to do what “some one” -had declared “could not bee done.” It is hardly necessary to point out that -Delamain’s Grammelogia IV suggests designs of slide rules which inventors -two hundred or more years later were endeavouring to produce. Which of -Delamain’s designs of rules were actually made and used, he does not state -explicitly. He refers to a rule 18 inches in diameter as if it had been -actually constructed (pages (86), (88)). Oughtred showed no appreciation of -such study in designing and ridiculed Delamain’s efforts, in his Epistle. - -Additional elucidations of his designs of rules, along with explanations of -the relations of his work to that of Gunter and Napier, and sallies -directed against Oughtred and Forster, are contained on pages (8)-(21) of -his Grammelogia IV. - - - - - V. INDEPENDENCE AND PRIORITY OF INVENTION - - -The question of independence and priority of invention is discussed by -Delamain more specifically on pages (89)-(113); Oughtred devotes his entire -Epistle to it. It is difficult to determine definitely which publication is -the later, Delamain’s Grammelogia IV or Oughtred’s Epistle. Each seems to -quote from the other. Probably the explanation is that the two publications -contain arguments which were previously passed from one antagonist to the -other by word of mouth or by private letter. Oughtred refers in his Epistle -(p. (12)) to a letter from Delamain. We believe that the Epistle came after -Delamain’s Grammelogia IV. Delamain claims for himself the invention of the -circular slide rule. He says in his Grammelogia IV. (p. (99)), “when I had -a sight of it, which was in February, 1629 (as I specified in my Epistle) I -could not conceale it longer, envying my selfe, that others did not tast of -that which I found to carry with it so delightfull and pleasant a goate -[taste] . . .” Delamain asserts (without proof) that Oughtred “never saw it -as he now challengeth it to be his invention, untill it was so fitted to -his hand, and that he made all his practise on it after the publishing of -my Booke upon my Ring, and not before; so it was easie for him or some -other to write some uses of it in Latin after Christmas, 1630 and not the -Sommer before, as is falsely alledged by some one . . .” (p. (91)). -Delamain’s accusation of theft on the part of Oughtred cannot be seriously -considered. Oughtred’s reputation as a mathematician and his standing in -his community go against such a supposition. Moreover, William Forster is a -witness for Oughtred. The fact that Oughtred had the mastery of the -rectilinear slide rule as well, while Delamain in 1630 speaks only of the -circular rule, weighs in Oughtred’s favour. - -Oughtred says he invented the slide rule “above twelve yeares agoe,” that -is, about 1621, and “I with mine owne hand made me two such Circles, which -I have used ever since, as my occasions required,” (Epistle p. (22)). On -the same page, he describes his mode of discovery thus: - - I found that it required many times too great a paire of Compasses [in - using Gunter’s line], which would bee hard to open, apt to slip, and - troublesome for use. I therefore first devised to have another Ruler with - the former: and so by setting and applying one to the other, I did not - onely take away the use of Compasses, but also make the worke much more - easy and expedite: when I should not at all need the motion of my hand, - but onely the glancing at my sight: and with one position of the Rulers, - and view of mine eye, see not one onely, but the manifold proportions - incident unto the question intended. But yet this facility also wanted - not some difficulty especially in the line of tangents, when one arch was - in the former mediety of the quadrant, and the other in the latter: for - in this case it was needful that either one Ruler must bee as long againe - as the other; or else that I must use an inversion of the Ruler, and - regression. By this consideration I first of all saw that if those lines - upon both Rulers were inflected into two circles, that of the tangents - being in both doubled, and that those two Circles should move one upon - another; they with a small thread in the center to direct the sight, - would bee sufficient with incredible and wonderfull facility to worke all - questions of Trigonometry . . . - -Oughtred said that he had no desire to publish his invention, but in the -vacation of 1630 finally promised William Forster to let him bring out a -translation. Oughtred claims that Delamain got the invention from him at -Alhallontide [November 1], 1630, when they met in London. The accounts of -that meeting we proceed to give in double column. - - Delamain’s Statement - Grammelogia IV, page (98) - - “. . . about Alhalontide 1630. (as our Authors reporteth) was the time he - was circumvented, and then his intent in a loving manner (as before) he - opened unto me, which particularly I will dismantle in the very naked - truth: for, wee being walking together some few weekes before Christmas, - upon Fishstreet hill, we discoursed upon sundry things Mathematicall, - both Theoreticall and Practicall, and of the excellent inventions and - helpes that in these dayes were produced, amongst which I was not a - little taken with that of the Logarythmes, commending greatly the - ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, - in the lines of proportion, extracted from these Logarythmes for ordinary - Practicall uses; He replyed unto me (in these very words) What will yov - say to an Invention that I have, which in a lesse extent of the Compasses - shall worke truer then that of Mr. Gunters Ruler, I asked him then of - what forme it was, he answered with some pause (which no doubt argued his - suspition of mee that I might conceive it) that it was Arching-wise, but - now hee sayes that hee told mee then, it was Circular (but were I put to - my oath to avoid the guilt of Conscience I would conclude in the former.) - At which immediately I answered, I had the like my selfe, and so we - discoursed not a word more touching that subject . . . Then after my - coming home I sent him a sight of my Projection drawne in Pastboard: Now - admit I had not the Invention of my Ring before I discoursed . . . it was - not so facil for mee . . . to raise and compose so complete, and absolute - an Instrument from so small a principle, or glimpse of light . . .” - - Oughtred’s Statement - Epistle, page (23) - - “Shortly after my gift to Elias Allen, I chanced to meet with Richard - Delamain in the street (it was at Alhallontide) and as we walked together - I told him what an Instrument I had given to Master Allen, both of the - Logarithmes projected into circles, which being lesse then one foot - diameter would performe as much as one of Master Gunters Rulers of sixe - feet long: and also of the Prostaphaereses of the Plannets and second - motions. Such an invention have I said he: for now his intentions (that - is his ambition) beganne to worke: . . . But he saith, Then after my - comming home I sent him a sight of my projection drawne in past-board. - See how notoriously he jugleth without an Instrument. Then after: how - long after? a sight of my projection: of how much? More then seven weekes - after on December 23, he sent to mee the line of numbers onely set upon a - circle: . . . and so much onely he presented to his Majesty: but as for - Sine or tangent of his, there was not the least shew of any. Neither - could he give to Master Allen any direction for the composure of the - circles of his Ring, or for the division of them: as upon his oath Master - Allen will testify how hee misled him, and made him labour in vain above - three weeks together, until Master Allen himselfe found out his ignorance - and mistaking, which is more cleare then is possible with any impudence - to be outfaced.” - -Oughtred makes a further statement (Epistle, p. (24)) as follows: - - Delamain hearing that Brown with his Serpentine had another line by which - he could worke to minutes in the 90 degree of sines . . . gave the [his] - booke to Browne: who in thankfulnesse could not but gratify Delamain with - his Lines also: and teach him the use of them, but especially of the - great Line: with this caution on both sides, that one should not meddle - with the others invention. Two dayes after Delamain . . . because he had - found some things to be altered therin, . . . asked for the booke . . . - but as soone as he had got it in his hands he rent out all the middle - part with the two Schemes & put them up in his pocket & went his way . . - . and . . . laboureth to recall all the bookes he had given forth . . . - And shortly after this he got a new Printer (who was ignorant of his - former Schemes) to print him new: giving him an especiall charge of the - outermost line newly graven in the Plate, which indeed is Brownes very - line: and then altering his book . . . - -This and other statements made by Oughtred seem damaging to Delamain’s -reputation. But it is quite possible that Oughtred’s guesses as to -Delamain’s motives are wrong. Moreover, some of Oughtred’s statements are -not first hand knowledge with him, but mere hearsay. One may accept his -first hand facts and still clear Delamain of wrong doing. There is always -danger that rival claimants of an invention or discovery will proceed on -the assumption that no one else could possibly have come independently upon -the same devices that they themselves did; the history of science proves -the opposite. Seldom is an invention of any note made by only one man. We -do not feel competent to judge Delamain’s case. We know too little about -him as a man. We incline to the opinion that the hypothesis of independent -invention is the most plausible. At any rate, Delamain figures in the -history of the slide rule as the publisher of the earliest book thereon and -as an enthusiastic and skillful designer of slide rules. - -The effect of this controversy upon interested friends was probably small. -Doubtless few people read both sides. Oughtred says:[21] “this scandall . . -. hath with them, to whom I am not knowne, wrought me much prejudice and -disadvantage . .” Aubrey,[22] a friend of Oughtred, refers to Delamain “who -was so sawcy to write against him” and remembers having seen “many yeares -since, twenty or more good verses made” against Delamain. Another friend of -Oughtred, William Robinson, who had seen some of Delamain’s publications, -but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before -the appearance of the latter’s Epistle: - - I cannot but wonder at the indiscretion of Rich. Delamain, who being - conscious to himself that he is but the pickpurse of another man’s wit, - would thus inconsiderately provoke and awake a sleeping lion . . . he - hath so weakly (though in my judgment, vaingloriously enough) commended - his own labour . . .[23] - -Delamain presented King Charles I with one of his sun-dials, also with a -manuscript and, later, with a printed copy of his book of 1630. A drawing -of his improved slide rule was sent to the King and the Grammelogia IV is -dedicated to him. The King must have been favorably impressed, for Delamain -was appointed tutor to the King in mathematics. His widow petitioned the -House of Lords in 1645 for relief; he had ten children.[24] - -Anthony Wood states that Charles I, on the day of his execution, commanded -his friend Thomas Herbert “to give his son the duke of York his large -ring-sundial of silver, a jewel his maj. much valued.” Anthony Wood adds, -“it was invented and made by Rich. Delamaine a very able mathematician, who -projected it, and in a little printed book did shew its excellent use in -resolving many questions in arithmetic and other rare operations to be -wrought by it in the mathematics.”[25] - - - - - VI. OUGHTRED’S GAUGING LINE, 1633 - - -It has not been generally known, hitherto, that Oughtred designed a -rectilinear slide rule for gauging and published a description thereof in -1633.[26] In his Circles of Proportion, chapter IX, Oughtred had offered a -closer approximation than that of Gunter for the capacity of casks. The -Gauger of London expostulated with Oughtred for presuming to question -anything that Gunter had written. The ensuing discussion led to an -invitation extended by the Company of Vintners to the instrument maker -Elias Allen to request Oughtred to design a gauging rod.[27] This he did, -and Allen received an order for “threescore” instruments. On page 19 -Oughtred describes his ‘Gauging Rod:’ - - It consisteth of two rulers of brasse about 32 ynches of length, which - also are halfe an ynch broad, and a quarter of an ynch thick . . . At one - end of both those rulers are two little sockets of brasse fastened on - strongly: by which the rulers are held together, and made to move one - upon another, and to bee drawne out unto any length, as occasion shall - require: and when you have them at the just length, there is upon one of - the sockets a long Scrue-pin to scrue them fast. - -There are graduations on three sides of the rulers, one graduation being -the logarithmic line of numbers. He says (p. 39), “the maner of computing -the Gauge-divisions I have concealed.” W. Robinson, who was a friend of -Oughtred, wrote him as follows:[28] - - I have light upon your little book of artificial gauging, wherewith I am - much taken, but I want the rod, neither could I get a sight of one of - them at the time, because Mr. Allen had none left . . . I forgot to ask - Mr. Allen the price of one of them, which if not much I would have one of - them.” Oughtred annotated this passage thus: “Or in wood, if any be made - in wood by Thompson or any other.” - -Another of Oughtred’s admirers, Sir Charles Cavendish, wrote, on February -11, 1635 thus:[29] - - I thank you for your little book, but especially for the way of - calculating the divisions of your gauging rod. I wish, both for their own - sakes and yours, that the citizens were as capable of the acuteness of - this invention, as they are commonly greedy of gain, and then I doubt not - but they would give you a better recompense than I doubt now they will. - -On April 20, 1638, we find Oughtred giving Elias Allen directions[30] -“about the making of the two rulers.” As in 1633,[31] so now, Oughtred -takes one ruler longer than the other. This 1633 instrument was used also -as “a crosse-staffe to take the height of the Sunne, or any Starre above -the Horizon, and also their distances.” The longer ruler was called staffe, -the shorter transversarie. While in 1633 he took the lengths of the two in -the ratio “almost 3 to 2,” in 1638, he took “the transversary three -quarters of the staff’s length, . . . that the divisions may be larger.” - - - - - VII. OTHER SEVENTEENTH CENTURY SLIDE RULES - - -In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard, -Henry Coggeshall, W. Hunt and Sir Isaac Newton.[32] Of Partridge’s Double -Scale of Proportion, London, I have examined a copy dated 1661, which is -the earliest date for this book that I have seen. As far as we know, 1661 -is the earliest date of publications on the slide rule, since Oughtred and -Delamain. But it would not be surprising if the intervening 28 years were -found not so barren as they seem at present. The 1661 and 1662 impressions -of Partridge are identical, except for the date on the title-page. William -Leybourn, who printed Partridge’s book, speaks in high appreciation of it -in his own book.[33] - -In 1661 was published also John Brown’s first book, Description and Use of -a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use -of “Mr. Whites rule” for the measuring of board and timber, round and -square. He calls this a “sliding rule.” The existence, in 1661, of a -“Whites rule” indicates activities in designing of which we know as yet -very little. In his book of 1761, previously quoted, Brown gives a drawing -of “White’s sliding rule” (p. 193); also a special contrivance of his own, -as indicated by him in these words: - - A further improvement of the Triangular Quadrant, as I have made it - several times, with a sliding Cover on the in-side, when made hollow, to - carry Ink, Pens, and Compasses; then on the sliding Cover, and Edges, is - put the Line of Numbers, according to Mr. White’s first Contrivance for - manner of operation; but much augmented, and made easie, by John Brown. - -He gives no drawing of his “triangular quadrant,” hence his account of it -is unsatisfactory. He explains the use of “gage-points.” His placing -logarithmic lines on the edges of instrument boxes was outdone in oddity -later by Everard who placed them on tobacco-boxes.[34] In Brown’s -publication of 1704 the White slide rule is given again, “being as neat and -ready a way as ever was used.” He tells also of a “glasier’s sliding rule.” -William Leybourn explains in 1673 how Wingate’s double and triple lines for -squaring and cubing, or square and cube root, can be used on slide -rules.[35] - -Beginning early in the history of the slide rule, when Oughtred designed -his “gauging rod,” we notice the designing of rules intended for very -special purposes. Another such contrivance, which enjoyed long popularity, -was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677, -a booklet of 35 pages. Coggeshall says in his preface: - - For what can be more ready and easie, then having set twelve to the - length, to see the Content exactly against the Girt or Side of the - Square. Whereas on Mr. Partridge’s Scale the Content is the Sixth Number, - which is far more troublesome then [even] with Compasses. - -One line on Coggeshall’s rule begins with 4 and extends to 40, these -numbers being the “Girt” (a quarter of the circumference), which in -ordinary practice of measuring round timber lies between 4 inches and 40 -inches. This “Girt line” slides “against the line of Numbers in two -Lengths, to which it is exactly equal.” A second edition, 1682, shows some -changes in the rule, as well as an enlargement and change of title of the -book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent, -London, 1682. In this, the description of the rule is given thus: - - There are four Lines on each flat of this Rule; two next the outward - edges, which are Lines of Measure; and two next the inward edges, which - are Lines of Proportion. On one flat, next the inward edges, is the - Square-line [Girt-line in round timber measurement] with the Line of - Numbers his fellow. Next the outward, a Line of Inches divided into - Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 - to 24 on the other. On the other flat, next the inward edges, is the - double Scale of Numbers [for solving proportions]. Next the outward on - one Rule a Line of Inches divided each into ten parts; and this for - gauging, etc. On the other a foot divided into 100 parts. - -Later further changes were introduced in Coggeshall’s rule.[36] - -It is worthy of note that Coggeshall’s slide rule book, The Art of -Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p. -473; hence Leupold’s description[37] of the rectilinear slide rule in his -Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not -the earliest reference to the rectilinear rule found in German -publications. The above date is earlier even than Biler’s reference to a -circular slide rule in his Descriptio instrumenti mathematici universalis -of 1696. - -Two noted slide rules for gauging were described by Tho. Everard, -Philomath, in his Stereometry made easie, London, 1684. He designates his -lines by the capital letters A, B, C, D, E. On the first instrument, A on -the rule, and B and C on the slide, have each two radiuses of numbers, D -has only one, while E has three. The second rule is described in an -Appendix; it is one foot long, with two slides enabling the rule to be -extended to 3 feet. - -Everard’s instruments were made in London by Isaac Carver who, soon after, -himself wrote a sixteen-page Description and Use of a New Sliding Rule, -projected from the Tables in the Gauger’s Magazine, London, 1687, which was -“printed for William Hunt” and bound in one volume with a book by Hunt, -called The Gauger’s Magazine, London, 1687. This appears to be the same -William Hunt who later brought out descriptions of his own of slide rules. -The instrument described by Carver “consists of three pieces, two whereof -are moveable to be drawn out till the whole be 36 inches long.” It has -several non-logarithmic graduations, together with logarithmic lines marked -A, B, C, D, of which A, B, C are “double lines,” and D a “single line” used -for squares and square roots. It is designed for the determination of the -vacuity of a “spheroidal cask lying,” a “spheroidal cask standing,” and a -“parabolical cask lying.” - -Another seventeenth century writer on the slide rule is John Atkinson, whom -we have mentioned earlier. He says:[38] “The Lines of Numbers, Sines and -Tangents, are set double, that is, one on each side, as the middle piece -slides: which middle piece is so contrived, to slip to and fro easily, to -slide out, and to be put in any side uppermost, in order to bring those -Lines together (or against one another) most proper for solving the -Question, wrought by Sliding-Gunter.” - -The data presented in this article show that, while the earliest slide -rules were of the circular type, the later slide rules of the seventeenth -century were of the rectilinear type.[39] - - January 12, 1915. - - - - - Footnotes - - -[1]F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments, - New York, 1909, pp. 7-14, also Addenda i-vi. - -[2]F. Cajori, “On the Invention of the Slide Rule,” in Colorado College - Publication, Engineering Series Vol. 1, 1910. An abstract of this is - given in Nature (London), Vol. 82, 1909, p. 267. - -[3]F. Cajori, History etc., p. 14. - -[4]Art. “Slide Rule” in the Penny Cyclopaedia and in the English - Cyclopaedia [Arts and Sciences]. - -[5]Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817, - p. 423. - -[6]The full title of the book which Wingate published on this subject in - Paris is as follows: - - L’Vsage | de la | Reigle de | Proportion | en l’Arithmetique & | - Geometrie. | Par Edmond Vvingate, | Gentil-homme Anglois. | - - Εἂν ἧς φιλεµαθὴς, ἕση ἥση πολυµαθὴς. - - In tenui, sed nõ tenuis vsusve, laborne. | - - A Paris, | Chez Melchior Mondiere, | demeurant en l’Isle du Palais, | à - la | ruë de Harlay aux deux Viperes. | M. DC. XXIV. | Auec Priuilege du - Roy. | - - Back of the title page is the announcement: - - Notez que la Reigle de Proportion en toutes façons se vend à Paris chez - Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces, - demeurant en l’Isle du Palais sur le Quay qui regarde la Megisserie à - l’Espic d’or. - -[7]The title-page of the edition of 1658 is as follows: - - The Use of the Rule of Proportion in Arithmetick & Geometrie. First - published at Paris in the French tongue, and dedicated to Monsieur, the - then king’s onely Brother (now Duke of Orleance). By Edm. Wingate, an - English Gent. And now translated into English by the Author. Whereinto - is now also inserted the Construction of the same Rule, & a farther use - thereof . . . 2nd edition inlarged and amended. London, 1658. - -[8]Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; - Drawn up by himself by way of Diary. With Appendix of original Letters. - Publish’d by Charles Burman, Esq., London, 1717, p. 23. - -[9]Mathematical Tables, 1811, p. 36, and art. “Gunter’s Line” in his Phil. - and Math. Dictionary, London, 1815. - -[10]To the English Gentrie, and all others studious of the Mathematicks, - which shall bee readers hereof. The just Apologie of Wil: Ovghtred, - against the slaunderous insimulations of Richard Delamain, in a Pamphlet - called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum - projectio circularis. We shall refer to this document as Epistle. It was - published without date in 32 unnumbered pages of fine print, and was - bound in with Oughtred’s Circles of Proportion, in the editions of 1633 - and 1639. In the 1633 edition it is inserted at the end of the volume - just after the Addition vnto the Vse of the Instrument etc., and in that - of 1639 immediately after the preface. It was omitted from the Oxford - edition of 1660. The Epistle was also published separately. There is a - separate copy in the British Museum, London. Aubrey, in his Brief Lives, - edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, “He - writt a stitch’t pamphlet about 163(?4) against . . . Delamaine.” - -[11]Thomas Browne is mentioned by Stone in his Mathematical Instruments, - London 1723, p. 16. See also Cajori, History of the Slide Rule, New - York, 1909, p. 15. - -[12]The Description and Use of a Joynt-Rule: . . . also the use of Mr. - White’s Rule for measuring of Board and Timber, round and square; With - the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and - Versed Sines. By J. Brown, Philom., London, 1661. - -[13]A Collection of Centers and Useful Proportions on the Line of Numbers, - by John Brown, 1662(?), 16 pages; Description and Use of the Triangular - Quadrant, by John Brown, London, 1671; Wingate’s Rule of Proportion in - Arithmetick and Geometry: or Gunter’s Line. Newly rectified by Mr. Brown - and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The - Description and Use of the Carpenter’s-Rule: Together with the Use of - the Line of Numbers commonly call’d Gunter’s-Line, by John Brown, - London, 1704. - -[14]William Leybourn, op. cit., pp. 129, 130, 132, 133. - -[15]James Atkinson’s edition of Andrew Wakely’s The Mariners Compass - Rectified, London, 1694 [Wakely’s preface dated 1664, Atkinson’s - preface, 1693]. Atkinson adds An Appendix containing Use of Instruments - most useful in Navigation. Our quotation is from this Appendix, p. 199. - -[16]R. Delamain, The Making, Description, and Use of a small portable - Instrument . . . called a Horizontall Quadrant, etc., London, 1631. - -[17]Oughtred’s description of his circular slide rule of 1632 and his - rectilinear slide rule of 1633, as well as a drawing of the circular - slide rule, are reproduced in Cajori’s History of the Slide Rule, - Addenda, pp. ii-vi. - -[18]The full title of the Grammelogia I is as follows: - - Gram̄elogia | or, | The Mathematicall Ring. | Shewing (any - reasonable Capacity that hath | not Arithmeticke) how to resolve and - worke | all ordinary operations of Arithmeticke. | And those which are - most difficult with greatest | facilitie: The extraction of Roots, the - valuation of | Leases, &c. The measuring of Plaines | and Solids. | With - the resolution of Plaine and Sphericall | Triangles. | And that onely by - an Ocular Inspection, | and a Circular Motion. | Naturae secreta tempus - aperit. | London printed by John Haviland, 1630. - -[19]Grammelogia III is the same as Grammelogia I, except for the addition - of an appendix, entitled: - - De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit - nullo (praescripto tempore) vitae | vsuram nobis ingeniique Deus. | - London, | - - . . . The next line or two of this title-page which probably contained - the date of publication, were cut off by the binder in trimming the - edges of this and several other pamphlets for binding into one volume. - -[20]Grammelogia IV has two title pages. The first is Mirifica Logarithmoru’ - Projectio Circularis. There follows a diagram of a circular slide rule, - with the inscription within the innermost ring: Nil Finis, Motvs, - Circvlvs vllvs Habet. The second title page is as follows: - - Grammelogia | Or, the Mathematicall Ring. | Extracted from the - Logarythmes, and projected Circular: Now published in the | inlargement - thereof unto any magnitude fit for use: shewing any reason- | able - capacity that hath not Arithmeticke how to resolve and worke, | all - ordinary operations of Arithmeticke: | And those that are most difficult - with greatest facilitie, the extracti- | on of Rootes, the valuation of - Leases, &c. the measuring of Plaines and Solids, | with the resolution - of Plaine and Sphericall Triangles applied to the | Practicall parts of - Geometrie, Horologographie, Geographie | Fortification, Navigation, - Astronomie, &c. | And that onely by an ocular inspection, and a Circular - motion, Invented and first published, by R. Delamain, Teacher, and - Student of the Mathematicks. | Naturae secreta tempus aperit. | - - There is no date. There follows the diagram of a second circular slide - rule, with the inscription within the innermost ring: Typus proiectionis - Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630 - promisi. There are numerous drawings in the Grammelogia, all of which, - excepting the drawings of slide rules on the engraved title-pages of - Grammelogia IV and V, were printed upon separate pieces of paper and - then inserted by hand into the vacant spaces on the printed pages - reserved for them. Some drawings are missing, so that the Bodleian - Grammelogia IV differs in this respect slightly from the copy in the - British Museum and from the British Museum copy of Grammelogia V. - -[21]Epistle, p. (8). - -[22]Aubrey, op. cit., Vol. II., p. 111. - -[23]Rigaud, Correspondence of Scientific Men during the 17th Century, Vol. - I, Oxford, 1841, p. 11. - -[24]Dictionary of National Biography, Art. “Delamain, Richard.” See also - Rev. Charles J. Robinson, Taylors’ School, from A.D. 1562 to 1874, Vol. - I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b; - Sixth Report of the Royal Commission on Historical Manuscripts, Part I, - Report and Appendix, London, 1877. In this Appendix, p. 82, we read the - following: - - Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. - Petitioner’s husband was servant to the King, and one of His Majesty’s - engineers for the fortification of the kingdom, and his tutor in - mathematical arts; but upon the breaking out of the war he deserted the - Court, and was called by the State to several employments, in fortifying - the towns of Northampton, Newport, and Abingdon; and was also abroad - with the armies as Quartermaster-General of the Foot, and therein died. - Petitioner is left a disconsolate widow with ten children, the four - least of whom are now afflicted with sickness, and petitioner has - nothing left to support them. There are several considerable sums of - money due to the petitioner, as well from the King as the State. Prays - that she may have some relief amongst other widows. See L. J., VII. 6. - 657. - -[25]Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London, - 1820, p. 34. - -[26]The New Artificial Gauging Line or Rod: together with rules concerning - the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London, - 1633. The copy we have seen is in the Bodleian Library, Oxford. The book - is small sized and has 40 pages. - -[27]Oughtred, op. cit., p. 11. - -[28]S. J. Rigaud, Correspondence of Scientific Men of the 17th Century, - Oxford, Vol. I, 1841, p. 17. - -[29]Rigaud, loc. cit., p. 22. - -[30]Rigaud, loc. cit., pp. 30, 31. - -[31]Oughtred, An Addition vnto the Vse of the Instrument called the Circles - of Proportion, London, 1633, p. 63. - -[32]F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22, - Addenda, pp. vi-ix. - -[33]W. Leybourn, op. cit., 1673, Preface, and pp. 128-29. - -[34]Cajori op. cit., Addenda, p. ix. - -[35]William Leybourn, op. cit., 1673, p. 35. - -[36]See Cajori, op. cit., pp. 20, 28, Addenda, p. ix. - -[37]See F. Cajori, “A Note on the History of the Slide Rule,” Bibliotheca - mathematica, 3 F., Vol. 10, pp. 161-163. - -[38]John Atkinson, op. cit., 1694, p. 204. - -[39]Probably the oldest slide rule now in existence is owned by St. John’s - College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in - diameter. It was exhibited along with other instruments in May, 1919. - According to the Catalogue of a Loan Exhibition of Early Scientific - Instruments in Oxford, opened May 16, 1919, the instrument is inscribed - with the name of the maker (“Elias Allen fecit”) and with the name of - the donor, Georgius Barkham. It is dated 1635, which is only three years - after the first publication of Oughtred’s description of his circular - slide rule. It is stated in the Catalogue: “Unfortunately all the - movable parts but the base-plate and a couple of thumb-screws are - missing. The face of the instrument is engraved with Oughtred’s - Horizontal Instrument. The back is engraved with eleven Circles of - Proportion as described in Arthur Haughton’s book, a copy of which was - presented to St. John’s College by George Barkham, to explain the use of - the instrument.” As Arthur Haughton’s Oxford edition of Oughtred’s - Circles of Proportion did not appear until 1660, it would seem that the - instrument was probably not presented to the College before 1660. As far - as is known, the next oldest slide rule is of the year 1654, kept in the - South Kensington Museum, London, and is described in Nature of March 5, - 1914. It is a rectilinear rule, “of boxwood, well made, and bound - together with brass at the two ends. It is of the square type, a little - more than 2 ft. in length, and bears the logarithmic lines first - described by Edmund Gunter. Of these, the num, sin and tan lines are - arranged in pairs, identical and contiguous, one line in each pair being - on the fixed part, and the other on the slide.” The instrument is - inscribed, “Made by Robert Bissaker for T. W., 1654.” Nowhere else have - we seen reference to Robert Bissaker. His slide rule seems to antedate - the “Whites rule” mentioned above. [This foot-note was added on October - 15, 1919.] - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - - -Note.—The University of California Publications are offered in exchange for -the publication of learned societies and institutions, universities and -libraries. Complete lists of all the publications of the University will be -sent upon request. For sample copies, lists of publications or other -information, address the Manager of the University of California Press, -Berkeley, California, U. 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Current, Volume XXII. - - - - - Transcriber’s Notes - - ---Inserted subchapter headings from the Table of Contents into the - text. - ---Corrected two page numbers in the Table of Contents - ---Corrected a Greek quotation in footnote 6 by reference to the - original sources. - - - - - - - -End of the Project Gutenberg EBook of On the History of Gunter's Scale and -the Slide Rule during the Seventeent, by Florian Cajori - -*** END OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - -***** This file should be named 42216-0.txt or 42216-0.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/4/2/2/1/42216/ - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - -Updated editions will replace the previous one--the old editions -will be renamed. - -Creating the works from public domain print editions means that no -one owns a United States copyright in these works, so the Foundation -(and you!) can copy and distribute it in the United States without -permission and without paying copyright royalties. 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You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org - - -Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century - -Author: Florian Cajori - -Release Date: February 26, 2013 [EBook #42216] - -Language: English - -Character set encoding: ISO-8859-1 - -*** START OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - - - - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - IN - MATHEMATICS - - Vol. 1, No. 9, pp. 187-209 February 17, 1920 - - - - - ON THE HISTORY OF GUNTER'S SCALE AND - THE SLIDE RULE DURING THE - SEVENTEENTH CENTURY - - - BY - FLORIAN CAJORI - - - UNIVERSITY OF CALIFORNIA PRESS - BERKELEY - - - - - TABLE OF CONTENTS - - - PAGE - I. Introduction 187 - II. Innovations in Gunter's Scale 188 - Changes introduced by Edmund Wingate 188 - Changes introduced by Milbourn 189 - Changes introduced by Thomas Brown and John Brown 190 - Changes introduced by William Leybourn 192 - III. Richard Delamain's "Grammelogia" 192 - Different editions or impressions 194 - Description of Delamain's instrument of 1630 195 - Delamain's later designs, and directions for using his - instruments 197 - IV. Controversy between Oughtred and Delamain on the invention of - the circular slide rule 199 - V. Independence and priority of invention 203 - VI. Oughtred's "Gauging Line," 1633 206 - VII. Other seventeenth century slide rules 207 - - - - - I. INTRODUCTION - - - In my history of the slide rule[1], and my article on its invention[2] it -is shewn that William Oughtred and not Edmund Wingate is the inventor, that -Oughtred's circular rule was described in print in 1632, his rectilinear -rule in 1633. Richard Delamain is referred to as having tried to -appropriate the invention to himself[3] and as having written a scurrilous -pamphlet against Oughtred. All our information about Delamain was taken -from De Morgan,[4] who, however, gives no evidence of having read any of -Delamain's writings on the slide rule. Through Dr. Arthur Hutchinson of -Pembroke College, Cambridge, I learned that Delamain's writings on the -slide rule were available. In this article will be given: First, some -details of the changes introduced during the seventeenth century in the -design of Gunter's scale by Edmund Wingate, Milbourn, Thomas Brown, John -Brown and William Leybourn; second, an account of Delamain's book of 1630 -on the slide rule which antedates Oughtred's first publication (though -Oughtred's date of invention is earlier than the date of Delamain's alleged -invention) and of Delamain's later designs of slide rules; third, an -account of the controversy between Delamain and Oughtred; fourth, an -account of a later book on the slide rule written by William Oughtred, and -of other seventeenth century books on the slide rule. - - - - - II. INNOVATIONS IN GUNTER'S SCALE - - - Changes introduced by Wingate - - We begin with Anthony Wood's account of Wingate's introduction of -Gunter's scale into France.[5] - - In 1624 he transported into France the rule of proportion, having a - little before been invented by Edm. Gunter of Gresham Coll. and - communicated it to most of the chiefest mathematicians then residing in - Paris: who apprehending the great benefit that might accrue thereby, - importun'd him to express the use thereof in the French tongue. Which - being performed accordingly, he was advised by monsieur Alleawne the - King's chief engineer to dedicate his book to monsieur the King's only - brother, since duke of Orleans. Nevertheless the said work coming forth - as an abortive (the publishing thereof being somewhat hastened, by reason - an advocate of Dijon in Burgundy began to print some uses thereof, which - Wingate had in a friendly way communicated to him) especially in regard - Gunter himself had learnedly explained its use in a far larger volume.[6] - -Gunter's scale, which Wingate calls the "rule of proportion," contained, as -described in the French edition of 1624, four lines: (1) A single line of -numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot -in length, divided into 12 inches and tenths of inches, also a line, one -foot in length, divided into tenths and hundredths. - -The English editions of this book which appeared in 1623 and 1628 are -devoid of interest. The editions of 1645 and 1658 contain an important -innovation.[7] In the preface the reasons why this instrument has not been -used more are stated to be: (1) the difficulty of drawing the lines with -exactness, (2) the trouble of working thereupon by reason (sometimes) of -too large an extent of the compasses, (3) the fact that the instrument is -not readily portable. The drawing of Wingate's arrangement of the scale in -the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains -five parallel lines, about 66 cm. long, each having the divisions of one -line marked on one side and of another line on the other side. Thus each -line carries two graduations: (1) A single logarithmic line of numbers; (2) -a logarithmic line of numbers thrice repeated; (3) the first scale -repeated, but beginning with the graduations which are near the middle of -the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3; -(4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of -tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000 -equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of -inches; (10) a scale consisting of three kinds, viz., a gauge line, a line -of chords, and a foot measure, divided into 1000 equal parts. - -Important are the first and second scales, by which cube root extraction -was possible "by inspection only, without the aid of pen or compass;" -similarly the third and fourth scales, for square roots. This innovation is -due to Wingate. The 1645 edition announces that the instrument was made in -brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in -Hosier Lane. - - - Changes introduced by Milbourn - -William Leybourn, in his The Line of Proportion or Numbers, Commonly called -Gunter's Line, Made Easie, London, 1673, says in his preface "To the -Reader:" - - The Line of Proportion or Numbers, commonly called (by Artificers) - Gunter's Line, hath been discoursed of by several persons, and variously - applied to divers uses; for when Mr. Gunter had brought it from the - Tables to a Line, and written some Uses thereof, Mr. Wingate added divers - Lines of several lengths, thereby to extract the Square or Cube Roots, - without doubling or trebling the distance of the Compasses: After him Mr. - Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral - Line, thereby enlarging the divisions of the Line. - -On pages 127 and 128 Leybourn adds: - - Again, One T. Browne, a Maker of Mathematical Instruments, made it in a - Serpentine or Spiral Line, composed of divers Concentrick Circles, - thereby to enlarg the divisions, which was the contrivance of one Mr. - Milburn a Yorkshire Gentleman, who writ thereof, and communicated his - Uses to the aforesaid Brown, who (since his death) attributed it to - himself: But whoever was the contriver of it, it is not without - inconvenience; for it can in no wise be made portable; and besides - (instead of compasses) an opening Joynt with thirds [threads] must be - placed to move upon the Centre of the Instrument, without which no - proportion can be wrought. - -This Mr. Milburn is probably the person named in the diary of the -antiquarian, Elias Ashmole, on August 13 [1646?]; "I bought of Mr. Milbourn -all his Books and Mathematical Instruments."[8] Charles Hutton[9] says that -Milburne of Yorkshire designed the spiral form about 1650. This date is -doubtless wrong, for Thomas Browne who, according to Leybourn, got the -spiral form of line from Milbourn, is repeatedly mentioned by William -Oughtred in his Epistle[10] printed some time in 1632 or 1633. Oughtred -does not mention Milbourn, and says (page 4) that the spiral form "was -first hit upon by one Thomas Browne a Joyner, . . . the serpentine -revolution being but two true semicircles described on severall -centers."[11] - - - Changes introduced by Thomas Brown and John Brown - -Thomas Brown did not publish any description of his instrument, but his -son, John Brown, published in 1661 a small book,[12] in which he says -(preface) that he had done "as Mr. Oughtred with Gunter's Rule, to a -sliding and circular form; and as my father Thomas Brown into a Serpentine -form; or as Mr. Windgate in his Rule of Proportion." He says also that -"this brief touch of the Serpentine-line I made bold to assert, to see if I -could draw out a performance of that promise, that hath been so long -unperformed by the promisers thereof." Accordingly in Chapter XX he gives a -description of the serpentine line, "contrived in five (or rather 15) -turn." Whether this description, printed in 1661, exactly fits the -instrument as it was developed in 1632, we have no means of knowing. John -Brown says: - - 1. First next the center is two circles divided one into 60, the other - into 100 parts, for the reducing of minutes to 100 parts, and the - contrary. - - 2. You have in seven turnes two inpricks, and five in divisions, the - first Radius of the sines (or Tangents being neer the matter, alike to - the first three degrees,) ending at 5 degrees and 44 minutes. - - 3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in - three margents in divisions, and the line of versed sines in pricks, - under the line of Tangents, according to Mr. Gunter's cross-staff: the - sines and Tangents beginning at 5 degrees, and 44 minutes where the other - ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the - line of numbers beginning at 10, and proceeding to 100, being one entire - Radius, and graduated into as many divisions as the largeness of the - instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to - 100 into 20 parts in one unit of increase, but the Tangents are divided - into single minutes from the beginning to the end, both in the first, - second and third Radiusses, and the sines into minutes; also from 30 - minutes to 40 degrees, and from 40 to 60, into every two minutes, and - from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the - rest as many as can be well discovered. - - The versed sines are set after the manner of Mr. Gunter's Cross-staff, - and divided into every 10th minutes beginning at 0, and proceeding to 156 - going backwards under the line of Tangents. - - 4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is - the secants to 51 degrees, being nothing else but the sines reitterated - beyond 90. - - 5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85 - degrees, whereby all trouble of backward working is avoided. - - 6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and - also a line of natural sines, for finding of differences in sines, for - finding hour and Azimuth. - - 7. Seventhly, next the verge or outermost edge is a line of equal parts - to get the Logarithm of any number, or the Logarithm sine and Tangent of - any ark or angle to four figures besides the carracteristick. - - 8. Eightly and lastly, in the space place between the ending of the - middle five turnes, and one half of the circle are three prickt lines - fitted for reduction. The uppermost being for shillings, pence and - farthings. The next for pounds, and ounces, and quarters of small - Averdupoies weight. The last for pounds, shillings and pence, and to be - used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay - the hair or edge of one of the legs of the index on 16. 3 1/2 in the line - of 1. s. d. and the hair shall cut on the equal parts 81 16; and the - contrary, if you have a decimal fraction, and would reduce it to a proper - fraction, the like may you do for shillings, and pence, and pounds, and - ounces. - - The uses of the lines follow. - - As to the use of these lines, I shall in this place say but little, and - that for two reasons. First, because this instrument is so contrived, - that the use is sooner learned then any other, I speak as to the manner, - and way of using it, because by means of first second and third - radiusses, in sines and Tangents, the work is always right on, one way or - other, according to the Canon whatsoever it be, in any book that treats - of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in - Oughtred from page 64 to 107. - - Secondly, and more especially, because the more accurate, and large - handling thereof is more then promised, if not already performed by more - abler pens, and a large manuscript thereof by my Sires meanes, provided - many years ago, though to this day not extant in print; so for his sake I - claiming my interest therein, make bold to present you with these few - lines, in order to the use of them: And first note, - - 1. Which soever of the two legs is set to the first term in the question, - that I call the first leg always, and the other being set to the second - term, I call the second leg . . . - -The exact nature of the contrivance with the "two legs" is not described, -but it was probably a flat pair of compasses, attached to the metallic -surface on which the serpentine line was drawn. In that case the instrument -was a slide rule, rather than a form of Gunter's line. In his publication -of 1661, as also in later publications,[13] John Brown devoted more space -to Gunter's scales, requiring the use of a separate pair of compasses, than -to slide rules. - - - Changes introduced by William Leybourn - -The same remark applies to William Leybourn who, after speaking of Seth -Partridge's slide rule, returns to forms of Gunter's scale, saying:[14] - - There is yet another way of disposing of this Line of Proportion, by - having one Line of the full length of the Ruler, and another Line of the - same Radius broken in two parts between 3 and 4; so that in working your - Compasses never go off of the Line: This is one of the best contrivances, - but here Compasses must be used. These are all the Contrivances that I - have hitherto seen of these Lines: That which I here speak of, and will - shew how to use, is only two Lines of one and the same Radius, being set - upon a plain Ruler of any length (the larger the better) having the - beginning of one Line, at the end of the other, the divisions of each - Line being set so close together, that if you find any number upon one of - the Lines, you may easily see what number stands against it on the other - Line. This is all the Variation. . . . - - Example 1. If a Board be 1 Foot 64 parts broad, how much in length of - that Board will make a Foot Square? Look upon one of your Lines (it - matters not which) for 1 Foot 64 parts, and right against it on the other - Line you shall find 61; and so many parts of a Foot will make a Foot - square of that Board. - -This contrivance solves the equation 1.64x=1, yielding centesimal parts of -a foot. - -James Atkinson[15] speaks of "Gunter's scale" as "usually of Boxwood . . . -commonly 2 ft. long, 1 1/2 inch broad" and "of two kinds: long Gunter or -single Gunter, and the sliding Gunter. It appears that during the -seventeenth century (and long after) the Gunter's scale was a rival of the -slide rule. - - - - - III. RICHARD DELAMAIN'S GRAMMELOGIA - - -We begin with a brief statement of the relations between Oughtred and -Delamain. At one time Delamain, a teacher of mathematics in London, was -assisted by Oughtred in his mathematical studies. In 1630 Delamain -published the Grammelogia, a pamphlet describing a circular slide rule and -its use. In 1631 he published another tract, on the Horizontall -Quadrant.[16] In 1632 appeared Oughtred's Circles of Proportion[17] -translated into English from Oughtred's Latin manuscript by another pupil, -William Forster, in the preface of which Forster makes the charge (without -naming Delamain) that "another . . . went about to pre-ocupate" the new -invention. This led to verbal disputes and to the publication by Delamain -of several additions to the Grammelogia, describing further designs of -circular slide rules and also stating his side of the bitter controversy, -but without giving the name of his antagonist. Oughtred's Epistle was -published as a reply. Each combatant accuses the other of stealing the -invention of the circular slide rule and the horizontal quadrant. - -The two title-pages of the edition of the Grammelogia in the British Museum - in London which we have called "Grammelogia IV." - - - Different editions or impressions - -There are at least five different editions, or impressions, of the -Grammelogia which we designate, for convenience, as follows: - - Grammelogia I, 1630. One copy in the Cambridge University Library.[18] - - Grammelogia II, I have not seen a copy of this. - - Grammelogia III, One copy in the Cambridge University Library.[19] - - Grammelogia IV, One copy in the British Museum, another in the Bodleian - Library, Oxford.[20] - - Grammelogia V, One copy in the British Museum. - -In Grammelogia I the first three leaves and the last leaf are without -pagination. The first leaf contains the title-page; the second leaf, the -dedication to the King and the preface "To the Reader;" the third leaf, the -description of the Mathematical Ring. Then follow 22 numbered pages. -Counting the unnumbered pages, there are altogether 30 pages in the -pamphlet. Only the first three leaves of this pamphlet are omitted in -Grammelogia IV and V. - -In Grammelogia III the Appendix begins with a page numbered 52 and bears -the heading "Conclusion;" it ends with page 68, which contains the same two -poems on the mathematical ring that are given on the last page of -Grammelogia I but differs slightly in the spelling of some of the words. -The 51 pages which must originally have preceded page 52, we have not seen. -The edition containing these we have designated Grammelogia II. The reason -for the omission of these 51 pages can only be conjectured. In Oughtred's -Epistle (p. 24), it is stated that Delamain had given a copy of the -Grammelogia to Thomas Brown, and that two days later Delamain asked for the -return of the copy, "because he had found some things to be altered -therein" and "rent out all the middle part." Delamain labored "to recall -all the bookes he had given forth, (which were many) before the sight of -Brownes Lines." These spiral lines Oughtred claimed that Delamain had -stolen from Brown. The title-page and page 52 are the only parts of the -Appendix, as given in Grammelogia III, that are missing in the Grammelogia -IV and V. - -Grammelogia IV answers fully to the description of Delamain's pamphlet -contained in Oughtred's Epistle. It was brought out in 1632 or 1633, for -what appears to be the latest part of it contains a reference (page 99) to -the Grammelogia I (1630) as "being now more then two yeares past." -Moreover, it refers to Oughtred's Circles of Proportion, 1632, and -Oughtred's reply in the Epistle was bound in the Circles of Proportion -having the Addition of 1633. For convenience of reference we number the two -title-pages of Grammelogia IV, "page (1)" and "page (2)," as is done by -Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page -numbers which we assign will be placed in parentheses, to distinguish them -from the page numbers which are printed in Grammelogia IV. The pages -(44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the -same as the pages 53-68. Thus only thirty-eight pages have page numbers -printed on them. The pages (67) and (83) are identical in wording, except -for some printer's errors; they contain verses in praise of the Ring, and -have near the bottom the word "Finis." Also, pages (22) and (23) are -together identical in wording with page (113), which is set up in finer -type, containing an advertisement of a part of Grammelogia IV explaining -the mode of graduating the circular rules. There are altogether six parts -of Grammelogia IV which begin or end by an address to the reader, thus: "To -the Reader," "Courteous Reader," or "To the courteous and benevolent Reader -. . .," namely the pages (8), (22), (68), (89), (90), (108). In his Epistle -(page 2), Oughtred characterizes the make up of the book in the following -terms: - - In reading it . . . I met with such a patchery and confusion of - disjoynted stuffe, that I was striken with a new wonder, that any man - should be so simple, as to shame himselfe to the world with such a - hotch-potch. - -Grammelogia V differs from Grammelogia IV in having only the second -title-page. The first title-page may have been torn off from the copy I -have seen. A second difference is that the page with the printed numeral 22 -in Grammelogia IV has after the word "Finis" the following notice: - - This instrument is made in Silver, or Brasse for the Pocket, or at any - other bignesse, over against Saint Clements Church without Temple Barre, - by Elias Allen. - -This notice occurs also on page 22 of Grammelogia I and III, but is omitted -from page 22 of Grammelogia V. - - - Description of Delamain's instrument of 1630 - -In his address to King Charles I, in his Grammelogia I, Delamain emphasizes -the ease of operating with his slide rule by stating that it is "fit for -use . . . as well on Horse backe as on Foot." Speaking "To the Reader," he -states that he has "for many yeares taught the Mathematicks in this Towne," -and made efforts to improve Gunter's scale "by some Motion, so that the -whole body of Logarithmes might move proportionally the one to the other, -as occasion required. This conceit in February last [1629] I struke upon, -and so composed my Grammelogia or Mathematicall Ring; by which only with an -ocular inspection, there is had at one instant all proportionalls through -the said body of Numbers." He dates his preface "first of January, 1630." -The fifth and sixth pages contain his "Description of the Grammelogia," the -term Grammelogia being applied to the instrument, as well as to the book. -His description is as follows: - - The parts of the Instrument are two Circles, the one moveable, and the - other fixed; The moveable is that unto which is fastened a small pin to - move it by; the other Circle may be conceived to be fixed; The - circumference of the moveable Circle is divided into unequall parts, - charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures - doe represent themselves, or such numbers unto which a Cipher or Ciphers - are added, and are varied as the occasion falls out in the speech of - Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or - 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c. - -After elaborating this last point and explaining the decimal subdivisions -on the scales of the movable circle, he says that "the numbers and -divisions on the fixed Circle, are the very same that the moveable are, . -." There is no drawing of the slide rule in this publication. The -twenty-two numbered pages give explanations of the various uses to which -the instrument can be put: "How to performe the Golden Rule" (pp. 1-3), -"Further uses of the Golden Rule" (pp. 4-6), "Notions or Principles -touching the disposing or ordering of the Numbers in the Golden Rule in -their true places upon the Grammelogia" (pp. 7-11), "How to divide one -number by another" (pp. 12, 13), "to multiply one Number by another" (pp. -14, 15), "To find Numbers in continuall proportion" (pp. 16, 17), "How to -extract the Square Root," "How to extract the Cubicke Root" (pp. 18-21), -"How to performe the Golden Rule" (the rule of proportion) is explained -thus: - - Seeke the first number in the moveable, and bring it to the second number - in the fixed, so right against the third number in the moveable, is the - answer in the fixed. - - If the Interest of 100. li. be 8. li. in the yeare, what is the Interest - of 65. li. for the same time. - - Bring 100. in the moveable to 8. in the fixed, so right against 65. in - the moveable is 5.2. in the fixed, and so much is the Interest of 65. li. - for the yeare at 8. li. for 100. li. per annum. - - The Instrument not removed, you may at one instant right against any - summe of money in the moveable, see the Interest thereof in the fixed: - the reason of this is from the Definition of Logarithmes. - -These are the earliest known printed instructions on the use of a slide -rule. It will be noticed that the description of the instrument at the -opening makes no references to logarithmic lines for the trigonometric -functions; only the line of numbers is given. Yet the title-page promised -the "resolution of Plaine and Sphericall Triangles." Page 22 throws light -upon this matter: - - If there be composed three Circles of equal thicknesse, A.B.C. so that - the inner edge of D [should be B] and the outward edge of A bee - answerably graduated with Logarithmall signes [sines], and the outward - edge of B and the inner edge of A with Logarithmes; and then on the - backside be graduated the Logarithmall Tangents, and againe the - Logarithmall signes oppositly to the former graduations, it shall be - fitted for the resolution of Plaine and Sphericall Triangles. - -After twelve lines of further remarks on this point he adds: - - Hence from the forme, I have called it a Ring, and Grammelogia by - annoligie of a Lineary speech; which Ring, if it were projected in the - convex unto two yards Diameter, or thereabouts, and the line Decupled, it - would worke Trigonometrie unto seconds, and give proportionall numbers - unto six places only by an ocular inspection, which would compendiate - Astronomicall calculations, and be sufficient for the Prosthaphaeresis of - the Motions: But of this as God shall give life and ability to health and - time. - -The unnumbered page following page 22 contains the patent and copyright on -the instrument and book: - - Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs - an Instrument called Grammelogia, or The Mathematicall Ring, together - with a Booke so intituled, expressing the use thereof, being his owne - Invention; we of our Gracious and Princely favour have granted unto the - said Richard Delamain and his Assignes, Privilege, Licence, and - Authority, for the sole Making, Printing and Selling of the said - Instrument and Booke: straightly forbidding any other to Make, Imprint, - or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument - or Booke within any our Dominions, during the space of ten yeares next - ensuing the date hereof, upon paine of Our high displeasure. Given under - our hand and Signet at our Palace of Westminster, the fourth day of - January, in the sixth yeare of our Raigne. - - - Delamain's later designs, and directions for using his instruments - -In the Appendix of Grammelogia III, on page 52 is given a description of an -instrument promised near the end of Grammelogia I: - - That which I have formerly delivered hath been onely upon one of the - Circles of my Ring, simply concerning Arithmeticall Proportions, I will - by way of Conclusion touch upon some uses of the Circles, of Logarithmall - Sines, and Tangents, which are placed on the edge of both the moveable - and fixed Circles of the Ring in respect of Geometricall Proportions, but - first of the description of these Circles. - - First, upon the side that the Circle of Numbers is one, are graduated on - the edge of the moveable, and also on the edge of the fixed the - Logarithmall Sines, for if you bring 1. in the moveable amongst the - Numbers to 1. in the fixed, you may on the other edge of the moveable and - fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto - 6.6. and each degree subdivided, and then over the former divisions and - figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5. - 4. 3. 2. 1. each of those divided by small points. - - Secondly, (if the Ring is great) neere the outward edge of this side of - the fixed against the Numbers, are the usuall divisions of a Circle, and - the points of the Compasse: serving for observation in Astronomy, or - Geometry, and the sights belonging to those divisions, may be placed on - the moveable Circle. - - Thirdly, opposite to those Sines on the other side are the Logarithmall - Tangents, noted alike both in the moveable and fixed thus - 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or - divisions serve also for their Complements to 90. so 40 gr. stands for - 50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both - in the moveable and fixed is also divided into parts. As for the degrees - which are under 6. viz. 5.4.3.2.1. they are noted with small figures over - this divided Circle from 45.40.35.30.25. &c. and each of those degrees - divided into parts by small points both in the moveable and fixed. - - Fourthly, on the other edge of the moveable on the same side is another - graduation of Tangents, like that formerly described. And opposite unto - it, in the fixed is a Graduation of Logarithmall sines in every thing - answerable to the first descrition of Sines on the other side. - - Fifthly, on the edge of the Ring is graduated a parte of the quator, - numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees - of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree - both of the quator, and Meridian are subdivided into parts; these two - graduated Circles serve to resolve such Questions which concerne - Latitude, Longitude, Rumb, and Distance, in Nauticall operations. - - Sixthly, to the concave of the Ring may be added a Circle to be elevated - or depressed for any Latitude, representing the quator, and so divided - into houres and parts with an Axis, to shew both the houre, and Azimuth, - and within this Circle may be hanged a Box, and Needle with a Socket for - a staffe to slide into it, and this accommodated with scrue pines to - fasten it to the Ring and staffe, or to take it off at pleasure. - -The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and -V make no reference to the dispute with Oughtred and may, therefore, be -assumed to have been published before the appearance of Oughtred's Circles -of Proportion. On page 53, "To the Reader," he says: - - . . . you may make use of the Projection of the Circles of the Ring upon - a Plaine, having the feet of a paire of compasses (but so that they be - flat) to move on the Center of that Plaine, and those feet to open and - shut as a paire of Compasses . . . now if the feet bee opened to any two - termes or numbers in that Projection, then may you move the first foot to - the third number, and the other foot shall give the Answer; . . . it hath - pleased some to make use of this way. But in this there is a double - labour in respect to that of the Ring, the one in fitting those feet unto - the numbers assigned, and the other by moving them about, in which a man - can hardly accommodate the Instrument with one hand, and expresse the - Proportionals in writing with the other. By the Ring you need not but - bring one number to another, and right against any other number is the - Answer without any such motion. . . . upon that [the Ring] I write, - shewing some uses of those Circles amongst themselves, and conjoyned with - others . . . in Astronomy, Horolographie, in plaine Triangles applyed to - Dimensions, Navigation, Fortification, etc. . . . But before I come to - Construction, I have thought it convenient by way introduction, to - examine the truth of the graduation of those Circles . . . - -These are the words of a practical man, interested in the mechanical -development of his instrument. He considers not only questions of -convenience but also of accuracy. The instrument has, or may have now, also -lines of sines and tangents. To test the accuracy of the circles of -Numbers, "bring any number in the moveable to halfe of that number in the -fixed: so any number or part in the fixed shall give his double in the -moveable, and so may you trie of the thirds, fourths &c. of numbers, vel -contra," (p. 54). On page 55 are given two small drawings, labelled, "A -Type of the Ringe and Scheme of this Logarithmicall projection, the use -followeth. These Instruments are made in Silver or Brasse by John Allen -neare the Sauoy in the Strand." - - - IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE - CIRCULAR SLIDE RULE - -Delamain's publication of 1630 on the 'Mathematicall Ring' does not appear -at that time to have caused a rupture between him and Oughtred. When in -1631 Delamain brought out his Horizontall Quadrant, the invention of which -Delamain was afterwards charged to have stolen from Oughtred, Delamain was -still in close touch with Oughtred and was sending Oughtred in the Arundell -House, London, the sheets as they were printed. Oughtred's reference to -this in his Epistle (p. 20) written after the friendship was broken, is as -follows: - - While he was printing his tractate of the Horizontall quadrant, although - he could not but know that it was injurious to me in respect of my free - gift to Master Allen, and of William Forster, whose translation of my - rules was then about to come forth: yet such was my good nature, and his - shamelessnesse, that every day, as any sheet was printed, hee sent, or - brought the same to mee at my chamber in Arundell house to peruse which I - lovingly and ingenuously did, and gave him my judgment of it. - -Even after Forster's publication of Oughtred's Circles of Proportion, 1632, -Oughtred had a book, A canon of Sines Tangents and Secants, which he had -borrowed from Delamain and was then returning (Epistle, page (5)). The -attacks which Forster, in the preface to the Circles of Proportion, made -upon Delamain (though not naming Delamain) started the quarrel. Except for -Forster and other pupils of Oughtred who urged him on to castigate -Delamain, the controversy might never have arisen. Forster expressed -himself in part as follows: - - . . . being in the time of the long vacation 1630, in the Country, at the - house of the Reverend, and my most worthy friend, and Teacher, Mr. - William Oughtred (to whose instruction I owe both my initiation, and - whole progresse in these Sciences.) I vpon occasion of speech told him of - a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made - (such as it vsually called Mr. Gunter's Ruler) 6 feet long, to be vsed - with a payre of beame-compasses. "He answered that was a poore invention, - and the performance very troublesome: But, said he, seeing you are taken - with such mechanicall wayes of Instruments, I will shew you what deuises - I have had by mee these many yeares." And first, hee brought to mee two - Rulers of that sort, to be vsed by applying one to the other, without any - compasses: and after that hee shewed mee those lines cast into a circle - or Ring, with another moueable circle vpon it. I seeing the great - expeditenesse of both those wayes; but especially, of the latter, wherein - it farre excelleth any other Instrument which hath bin knowne; told him, - I wondered that hee could so many yeares conceale such vseful inuentions, - not onely from the world, but from my selfe, to whom in other parts and - mysteries of Art, he had bin so liberall. He answered, "That the true way - of Art is not by Instruments, but by Demonstration: and that it is a - preposterous course of vulgar Teachers, to begin with Instruments, and - not with the Sciences, and so in-stead of Artists, to make their - Schollers only doers of tricks, and as it were Iuglers: to the despite of - Art, losse of precious time, and betraying of willing and industrious - wits, vnto ignorance and idlenesse. That the vse of Instruments is indeed - excellent, if a man be an Artist: but contemptible, being set and opposed - to Art. And lastly, that he meant to commend to me, the skill of - Instruments, but first he would haue me well instructed in the Sciences. - He also shewed me many notes, and Rules for the vse of those circles, and - of his Horizontall Instrument, (which he had proiected about 30 yeares - before) the most part written in Latine. All which I obtained of him - leaue to translate into English, and make publique, for the vse, and - benefit of such as were studious, and louers of these excellent Sciences. - - Which thing while I with mature, and diligent care (as my occasions would - give me leaue) went about to doe: another to whom the Author in a louing - confidence discouered this intent, using more hast then good speed, went - about to preocupate; of which vntimely birth, and preuenting (if not - circumuenting) forwardnesse, I say no more: but aduise the studious - Reader, onely so farre to trust, as he shal be sure doth agree to truth & - Art. - -While in this dedication reference is made to a slide rule or "ring" with a -"moveable circle," the instrument actually described in the Circles of -Proportion consists of fixed circles "with an index to be opened after the -manner of a paire of Compasses." Delamain, as we have seen, had decided -preference for the moveable circle. To Oughtred, on the other hand, one -design was about as good as the other; he was more of a theorist and -repeatedly expressed his contempt for mathematical instruments. In his -Epistle (page (25)), he says he had not "the one halfe of my intentions -upon it" (the rule in his book), nor one with a "moveable circle and a -thread, but with an opening Index at the centre (if so be that bee cause -enough to make it to bee not the same, but another Instrument) for my part -I disclaime it: it may go seeke another Master: which for ought I know, -will prove to be Elias Allen himselfe: for at his request only I altered a -little my rules from the use of the moveable circle and the thread, to the -two armes of an Index." - -All parts of Delamain's Grammelogia IV, except pages 1-22 and 53-68 -considered above, were published after the Circles of Proportion, for they -contain references to the ill treatment that Delamain felt or made believe -that he felt, that he had received in the book published by Oughtred and -Forster. Oughtred's reference to teachers whose scholars are "doers of -tricks," "Iuglers," and Forster's allusion to "another to whom the Author -in a loving confidence" explained the instrument and who "went about to -preocupate" it, are repeatedly mentioned. Delamain says, (page (89)) that -at first he did not intend to express himself in print, "but sought peace -and my right by a private and friendly way." Oughtred's account of -Delamain's course is that of an "ill-natured man" with a "virulent tongue," -"sardonical laughter" and "malapert sawsiness." Contrasting Forster and -Delamain, he says that, of the former he "had the very first moulding" and -made him feel that "the way of Art" is "by demonstration." But Delamain was -"already corrupted with doing upon Instruments, and quite lost from ever -being made an Artist." (Epistle page (27)). Repeatedly does Oughtred assert -Delamain's ignorance of mathematics. The two men were evidently of wholly -different intellectual predilections. That Delamain loved instruments is -quite evident, and we proceed to describe his efforts to improve the -circular slide rule. - -The Grammelogia IV is dedicated to King Charles I. Delamain says: - - . . . Everything hath his beginning, and curious Arts seldome come to the - height at the first; It was my promise then to enlarge the invention by a - way of decuplating the Circles, which I now present unto your sacred - Majestie as the quintessence and excellencie there of . . . - -His enlarged circular rules are illustrated in the Bodleian Library copy of -Grammelogia IV by four diagrams, two of them being the two drawings on the -two title-pages at the beginning of the Grammelogia IV, 4 inches in -external diameter, and exhibiting eleven concentric circular lines carrying -graduations of different sorts. In the second of these designs all circles -are fixed. The other two drawings are each 10 3/4 inches in external -diameter and exhibit 18 concentric circular lines; the folded sheet of the -first of these drawings is inserted between pages (23) and (24), the second -folded sheet between pages (83) and (84). All circles of this second -instrument are fixed. Counting in the two small drawings in Grammelogia -III, there are in all six drawings of slide rules in the Bodleian -Grammelogia IV. On pages (24) to (43) Delamain explains the graduation of -slide rules. He takes first a rule which has one circle of equal parts, -divided into 1000 equal divisions. From a table of logarithms he gets log 2 -= 0.301; from the number 301 in the circle of equal parts he draws a line -to the center of the circle and marks the intersection with the circles of -numbers by the figure 2. Thus he proceeds with log 3, log 4, and so on; -also with log sin x and log tan x. For log sin x he uses two circles, the -first (see page (27)) for angles from 34' 24" to 5^o 44' 22", the second -circle from 5^o 44' 22" to 90^o. The drawings do not show the seconds. He -suggests many different designs of rules. On page (29) he says: - - For the single projection of the Circles of my Ring, and the dividing and - graduating of them: which may bee so inserted upon the edges of Circles - of mettle turned in the forme of a Ring, so that one Circle may moove - betweene two fixed, by helpe of two stayes, then may there be graduated - on the face of the Ring, upon the outer edge of the mooveable and inner - edge of the fixed, the Circle of Numbers, then upon the inner edge of - that mooveable Circle, and the outward edge of that inner fixed Circle - may be inserted the Circle of Sines, and so according to the description - of those that are usually made. - -In addition to these lines he proceeds to mention the circle giving the -ordinary division into degrees and minutes, and two circles of tangents on -the other side of the rule. - -Next Delamain explains an arrangement of all the graduation on one side of -the rule by means of "a small channell in the innermost fixed Circle, in -which may be placed a small single Index, which may have sufficient length -to reach from the innermost edge of the Mooveable Circle, unto the outmost -edge of the fixed Circle, which may be mooved to and fro at pleasure, in -the channell, which Index may serve to shew the opposition of Numbers" (p. -(31)). From this it is clear that the invention of the "runner" goes back -to the very first writers on the slide rule. - -After describing a modification of the above arrangement, he adds, "many -other formes might be deliverd, about this single projection" (p. (32)). - -Proceeding to the "enlarging" of the circles in the Ring, to, say, the -"Quadruple to that which is single, that is, foure times greater," the -"equall parts" are distributed over four circles instead of only one -circle, but the general method of graduation is the same as before (p. -(33)); there being now four circles carrying the logarithms of numbers, and -so on. Next he points out "severall wayes how the Circles of the -Mathematicall Ring (being inlarged) may be accommodated for practicall -use:" (1) The Circles are all fixed in a plain and movable flat compasses -(or better, a movable semicircle) are used for fixing any two positions; -(2) There is a "double projection" of each logarithmic line "inlarged on a -Plaine," one fixed, the other movable, as shown in his first figure on the -title-page, a single index only being used; (3) use of "my great Cylinder -which I have long proposed (in which all the Circles are of equall -greatnesse,) and it may be made of any magnitude or capacity, but for a -study (hee that will be at the charge) it may be of a yard diameter and of -such an indifferent length that it may containe 100 or more Circles fixed -parallel one to the other on the Cylinder, having a space betweene each of -them, so that there may bee as many mooveable Circles, as there are fixed -ones, and these of the mooveable linked, or fastened together, so that they -may all moove together by the fixed ones in these spaces, whose edges both -of the fixed, and mooveable being graduated by helpe of a single Index will -shew the proportionalls by opposition in this double Projection, or by a -double Index in a single Projection" (p. (36)). - -Next follows the detailed description of his Ring "on a Plaine, according -to the diagramme that was given the King (for a view of that projection) -and afterwards the Ring it selve." The diagram is the large one which we -mentioned as inserted between pages (23) and (24). The instrument has two -circles, one moveable, upon each of which are described 13 distinct -circular graduations. The lines on the fixed circle are: "The Circle of -degrees and calendar," E. "Circle of equall parts, and part of the Equator, -and Meridian," TT. "The Circle of Tangents," S. "The Circle of Sines," D. -"The Circle of Decimals," N. "The Circle of Numbers." The lines on the -movable circle are: N. "The Circle of Numbers," E. "The Circle of equated -figures, and bodies," S. "The Circle of Sines," TT. "The Circle of -Tangents," Y. "The Circle of time, yeares, and monethes." - -On pages (84)-(88) Delamain explains an enlargement of his Ring for -computations involving the sines of angles near to 90^o. On page (86) he -says: - - I have continued the Sines of the Projection unto two severall - revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr. - (being the last revolution of the decuplation of the former, or the - hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s. - and ends at 90.gr. (being the last of a ternary of decuplated - revolutions, or the thousand part of that Projection) and may bee thus - used. - -He explains the manner of using these extra graduations. Thus he claims to -have attained degrees of accuracy which enabled him to do what "some one" -had declared "could not bee done." It is hardly necessary to point out that -Delamain's Grammelogia IV suggests designs of slide rules which inventors -two hundred or more years later were endeavouring to produce. Which of -Delamain's designs of rules were actually made and used, he does not state -explicitly. He refers to a rule 18 inches in diameter as if it had been -actually constructed (pages (86), (88)). Oughtred showed no appreciation of -such study in designing and ridiculed Delamain's efforts, in his Epistle. - -Additional elucidations of his designs of rules, along with explanations of -the relations of his work to that of Gunter and Napier, and sallies -directed against Oughtred and Forster, are contained on pages (8)-(21) of -his Grammelogia IV. - - - - - V. INDEPENDENCE AND PRIORITY OF INVENTION - - -The question of independence and priority of invention is discussed by -Delamain more specifically on pages (89)-(113); Oughtred devotes his entire -Epistle to it. It is difficult to determine definitely which publication is -the later, Delamain's Grammelogia IV or Oughtred's Epistle. Each seems to -quote from the other. Probably the explanation is that the two publications -contain arguments which were previously passed from one antagonist to the -other by word of mouth or by private letter. Oughtred refers in his Epistle -(p. (12)) to a letter from Delamain. We believe that the Epistle came after -Delamain's Grammelogia IV. Delamain claims for himself the invention of the -circular slide rule. He says in his Grammelogia IV. (p. (99)), "when I had -a sight of it, which was in February, 1629 (as I specified in my Epistle) I -could not conceale it longer, envying my selfe, that others did not tast of -that which I found to carry with it so delightfull and pleasant a goate -[taste] . . ." Delamain asserts (without proof) that Oughtred "never saw it -as he now challengeth it to be his invention, untill it was so fitted to -his hand, and that he made all his practise on it after the publishing of -my Booke upon my Ring, and not before; so it was easie for him or some -other to write some uses of it in Latin after Christmas, 1630 and not the -Sommer before, as is falsely alledged by some one . . ." (p. (91)). -Delamain's accusation of theft on the part of Oughtred cannot be seriously -considered. Oughtred's reputation as a mathematician and his standing in -his community go against such a supposition. Moreover, William Forster is a -witness for Oughtred. The fact that Oughtred had the mastery of the -rectilinear slide rule as well, while Delamain in 1630 speaks only of the -circular rule, weighs in Oughtred's favour. - -Oughtred says he invented the slide rule "above twelve yeares agoe," that -is, about 1621, and "I with mine owne hand made me two such Circles, which -I have used ever since, as my occasions required," (Epistle p. (22)). On -the same page, he describes his mode of discovery thus: - - I found that it required many times too great a paire of Compasses [in - using Gunter's line], which would bee hard to open, apt to slip, and - troublesome for use. I therefore first devised to have another Ruler with - the former: and so by setting and applying one to the other, I did not - onely take away the use of Compasses, but also make the worke much more - easy and expedite: when I should not at all need the motion of my hand, - but onely the glancing at my sight: and with one position of the Rulers, - and view of mine eye, see not one onely, but the manifold proportions - incident unto the question intended. But yet this facility also wanted - not some difficulty especially in the line of tangents, when one arch was - in the former mediety of the quadrant, and the other in the latter: for - in this case it was needful that either one Ruler must bee as long againe - as the other; or else that I must use an inversion of the Ruler, and - regression. By this consideration I first of all saw that if those lines - upon both Rulers were inflected into two circles, that of the tangents - being in both doubled, and that those two Circles should move one upon - another; they with a small thread in the center to direct the sight, - would bee sufficient with incredible and wonderfull facility to worke all - questions of Trigonometry . . . - -Oughtred said that he had no desire to publish his invention, but in the -vacation of 1630 finally promised William Forster to let him bring out a -translation. Oughtred claims that Delamain got the invention from him at -Alhallontide [November 1], 1630, when they met in London. The accounts of -that meeting we proceed to give in double column. - - Delamain's Statement - Grammelogia IV, page (98) - - ". . . about Alhalontide 1630. (as our Authors reporteth) was the time he - was circumvented, and then his intent in a loving manner (as before) he - opened unto me, which particularly I will dismantle in the very naked - truth: for, wee being walking together some few weekes before Christmas, - upon Fishstreet hill, we discoursed upon sundry things Mathematicall, - both Theoreticall and Practicall, and of the excellent inventions and - helpes that in these dayes were produced, amongst which I was not a - little taken with that of the Logarythmes, commending greatly the - ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, - in the lines of proportion, extracted from these Logarythmes for ordinary - Practicall uses; He replyed unto me (in these very words) What will yov - say to an Invention that I have, which in a lesse extent of the Compasses - shall worke truer then that of Mr. Gunters Ruler, I asked him then of - what forme it was, he answered with some pause (which no doubt argued his - suspition of mee that I might conceive it) that it was Arching-wise, but - now hee sayes that hee told mee then, it was Circular (but were I put to - my oath to avoid the guilt of Conscience I would conclude in the former.) - At which immediately I answered, I had the like my selfe, and so we - discoursed not a word more touching that subject . . . Then after my - coming home I sent him a sight of my Projection drawne in Pastboard: Now - admit I had not the Invention of my Ring before I discoursed . . . it was - not so facil for mee . . . to raise and compose so complete, and absolute - an Instrument from so small a principle, or glimpse of light . . ." - - Oughtred's Statement - Epistle, page (23) - - "Shortly after my gift to Elias Allen, I chanced to meet with Richard - Delamain in the street (it was at Alhallontide) and as we walked together - I told him what an Instrument I had given to Master Allen, both of the - Logarithmes projected into circles, which being lesse then one foot - diameter would performe as much as one of Master Gunters Rulers of sixe - feet long: and also of the Prostaphaereses of the Plannets and second - motions. Such an invention have I said he: for now his intentions (that - is his ambition) beganne to worke: . . . But he saith, Then after my - comming home I sent him a sight of my projection drawne in past-board. - See how notoriously he jugleth without an Instrument. Then after: how - long after? a sight of my projection: of how much? More then seven weekes - after on December 23, he sent to mee the line of numbers onely set upon a - circle: . . . and so much onely he presented to his Majesty: but as for - Sine or tangent of his, there was not the least shew of any. Neither - could he give to Master Allen any direction for the composure of the - circles of his Ring, or for the division of them: as upon his oath Master - Allen will testify how hee misled him, and made him labour in vain above - three weeks together, until Master Allen himselfe found out his ignorance - and mistaking, which is more cleare then is possible with any impudence - to be outfaced." - -Oughtred makes a further statement (Epistle, p. (24)) as follows: - - Delamain hearing that Brown with his Serpentine had another line by which - he could worke to minutes in the 90 degree of sines . . . gave the [his] - booke to Browne: who in thankfulnesse could not but gratify Delamain with - his Lines also: and teach him the use of them, but especially of the - great Line: with this caution on both sides, that one should not meddle - with the others invention. Two dayes after Delamain . . . because he had - found some things to be altered therin, . . . asked for the booke . . . - but as soone as he had got it in his hands he rent out all the middle - part with the two Schemes & put them up in his pocket & went his way . . - . and . . . laboureth to recall all the bookes he had given forth . . . - And shortly after this he got a new Printer (who was ignorant of his - former Schemes) to print him new: giving him an especiall charge of the - outermost line newly graven in the Plate, which indeed is Brownes very - line: and then altering his book . . . - -This and other statements made by Oughtred seem damaging to Delamain's -reputation. But it is quite possible that Oughtred's guesses as to -Delamain's motives are wrong. Moreover, some of Oughtred's statements are -not first hand knowledge with him, but mere hearsay. One may accept his -first hand facts and still clear Delamain of wrong doing. There is always -danger that rival claimants of an invention or discovery will proceed on -the assumption that no one else could possibly have come independently upon -the same devices that they themselves did; the history of science proves -the opposite. Seldom is an invention of any note made by only one man. We -do not feel competent to judge Delamain's case. We know too little about -him as a man. We incline to the opinion that the hypothesis of independent -invention is the most plausible. At any rate, Delamain figures in the -history of the slide rule as the publisher of the earliest book thereon and -as an enthusiastic and skillful designer of slide rules. - -The effect of this controversy upon interested friends was probably small. -Doubtless few people read both sides. Oughtred says:[21] "this scandall . . -. hath with them, to whom I am not knowne, wrought me much prejudice and -disadvantage . ." Aubrey,[22] a friend of Oughtred, refers to Delamain "who -was so sawcy to write against him" and remembers having seen "many yeares -since, twenty or more good verses made" against Delamain. Another friend of -Oughtred, William Robinson, who had seen some of Delamain's publications, -but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before -the appearance of the latter's Epistle: - - I cannot but wonder at the indiscretion of Rich. Delamain, who being - conscious to himself that he is but the pickpurse of another man's wit, - would thus inconsiderately provoke and awake a sleeping lion . . . he - hath so weakly (though in my judgment, vaingloriously enough) commended - his own labour . . .[23] - -Delamain presented King Charles I with one of his sun-dials, also with a -manuscript and, later, with a printed copy of his book of 1630. A drawing -of his improved slide rule was sent to the King and the Grammelogia IV is -dedicated to him. The King must have been favorably impressed, for Delamain -was appointed tutor to the King in mathematics. His widow petitioned the -House of Lords in 1645 for relief; he had ten children.[24] - -Anthony Wood states that Charles I, on the day of his execution, commanded -his friend Thomas Herbert "to give his son the duke of York his large -ring-sundial of silver, a jewel his maj. much valued." Anthony Wood adds, -"it was invented and made by Rich. Delamaine a very able mathematician, who -projected it, and in a little printed book did shew its excellent use in -resolving many questions in arithmetic and other rare operations to be -wrought by it in the mathematics."[25] - - - - - VI. OUGHTRED'S GAUGING LINE, 1633 - - -It has not been generally known, hitherto, that Oughtred designed a -rectilinear slide rule for gauging and published a description thereof in -1633.[26] In his Circles of Proportion, chapter IX, Oughtred had offered a -closer approximation than that of Gunter for the capacity of casks. The -Gauger of London expostulated with Oughtred for presuming to question -anything that Gunter had written. The ensuing discussion led to an -invitation extended by the Company of Vintners to the instrument maker -Elias Allen to request Oughtred to design a gauging rod.[27] This he did, -and Allen received an order for "threescore" instruments. On page 19 -Oughtred describes his 'Gauging Rod:' - - It consisteth of two rulers of brasse about 32 ynches of length, which - also are halfe an ynch broad, and a quarter of an ynch thick . . . At one - end of both those rulers are two little sockets of brasse fastened on - strongly: by which the rulers are held together, and made to move one - upon another, and to bee drawne out unto any length, as occasion shall - require: and when you have them at the just length, there is upon one of - the sockets a long Scrue-pin to scrue them fast. - -There are graduations on three sides of the rulers, one graduation being -the logarithmic line of numbers. He says (p. 39), "the maner of computing -the Gauge-divisions I have concealed." W. Robinson, who was a friend of -Oughtred, wrote him as follows:[28] - - I have light upon your little book of artificial gauging, wherewith I am - much taken, but I want the rod, neither could I get a sight of one of - them at the time, because Mr. Allen had none left . . . I forgot to ask - Mr. Allen the price of one of them, which if not much I would have one of - them." Oughtred annotated this passage thus: "Or in wood, if any be made - in wood by Thompson or any other." - -Another of Oughtred's admirers, Sir Charles Cavendish, wrote, on February -11, 1635 thus:[29] - - I thank you for your little book, but especially for the way of - calculating the divisions of your gauging rod. I wish, both for their own - sakes and yours, that the citizens were as capable of the acuteness of - this invention, as they are commonly greedy of gain, and then I doubt not - but they would give you a better recompense than I doubt now they will. - -On April 20, 1638, we find Oughtred giving Elias Allen directions[30] -"about the making of the two rulers." As in 1633,[31] so now, Oughtred -takes one ruler longer than the other. This 1633 instrument was used also -as "a crosse-staffe to take the height of the Sunne, or any Starre above -the Horizon, and also their distances." The longer ruler was called staffe, -the shorter transversarie. While in 1633 he took the lengths of the two in -the ratio "almost 3 to 2," in 1638, he took "the transversary three -quarters of the staff's length, . . . that the divisions may be larger." - - - - - VII. OTHER SEVENTEENTH CENTURY SLIDE RULES - - -In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard, -Henry Coggeshall, W. Hunt and Sir Isaac Newton.[32] Of Partridge's Double -Scale of Proportion, London, I have examined a copy dated 1661, which is -the earliest date for this book that I have seen. As far as we know, 1661 -is the earliest date of publications on the slide rule, since Oughtred and -Delamain. But it would not be surprising if the intervening 28 years were -found not so barren as they seem at present. The 1661 and 1662 impressions -of Partridge are identical, except for the date on the title-page. William -Leybourn, who printed Partridge's book, speaks in high appreciation of it -in his own book.[33] - -In 1661 was published also John Brown's first book, Description and Use of -a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use -of "Mr. Whites rule" for the measuring of board and timber, round and -square. He calls this a "sliding rule." The existence, in 1661, of a -"Whites rule" indicates activities in designing of which we know as yet -very little. In his book of 1761, previously quoted, Brown gives a drawing -of "White's sliding rule" (p. 193); also a special contrivance of his own, -as indicated by him in these words: - - A further improvement of the Triangular Quadrant, as I have made it - several times, with a sliding Cover on the in-side, when made hollow, to - carry Ink, Pens, and Compasses; then on the sliding Cover, and Edges, is - put the Line of Numbers, according to Mr. White's first Contrivance for - manner of operation; but much augmented, and made easie, by John Brown. - -He gives no drawing of his "triangular quadrant," hence his account of it -is unsatisfactory. He explains the use of "gage-points." His placing -logarithmic lines on the edges of instrument boxes was outdone in oddity -later by Everard who placed them on tobacco-boxes.[34] In Brown's -publication of 1704 the White slide rule is given again, "being as neat and -ready a way as ever was used." He tells also of a "glasier's sliding rule." -William Leybourn explains in 1673 how Wingate's double and triple lines for -squaring and cubing, or square and cube root, can be used on slide -rules.[35] - -Beginning early in the history of the slide rule, when Oughtred designed -his "gauging rod," we notice the designing of rules intended for very -special purposes. Another such contrivance, which enjoyed long popularity, -was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677, -a booklet of 35 pages. Coggeshall says in his preface: - - For what can be more ready and easie, then having set twelve to the - length, to see the Content exactly against the Girt or Side of the - Square. Whereas on Mr. Partridge's Scale the Content is the Sixth Number, - which is far more troublesome then [even] with Compasses. - -One line on Coggeshall's rule begins with 4 and extends to 40, these -numbers being the "Girt" (a quarter of the circumference), which in -ordinary practice of measuring round timber lies between 4 inches and 40 -inches. This "Girt line" slides "against the line of Numbers in two -Lengths, to which it is exactly equal." A second edition, 1682, shows some -changes in the rule, as well as an enlargement and change of title of the -book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent, -London, 1682. In this, the description of the rule is given thus: - - There are four Lines on each flat of this Rule; two next the outward - edges, which are Lines of Measure; and two next the inward edges, which - are Lines of Proportion. On one flat, next the inward edges, is the - Square-line [Girt-line in round timber measurement] with the Line of - Numbers his fellow. Next the outward, a Line of Inches divided into - Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 - to 24 on the other. On the other flat, next the inward edges, is the - double Scale of Numbers [for solving proportions]. Next the outward on - one Rule a Line of Inches divided each into ten parts; and this for - gauging, etc. On the other a foot divided into 100 parts. - -Later further changes were introduced in Coggeshall's rule.[36] - -It is worthy of note that Coggeshall's slide rule book, The Art of -Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p. -473; hence Leupold's description[37] of the rectilinear slide rule in his -Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not -the earliest reference to the rectilinear rule found in German -publications. The above date is earlier even than Biler's reference to a -circular slide rule in his Descriptio instrumenti mathematici universalis -of 1696. - -Two noted slide rules for gauging were described by Tho. Everard, -Philomath, in his Stereometry made easie, London, 1684. He designates his -lines by the capital letters A, B, C, D, E. On the first instrument, A on -the rule, and B and C on the slide, have each two radiuses of numbers, D -has only one, while E has three. The second rule is described in an -Appendix; it is one foot long, with two slides enabling the rule to be -extended to 3 feet. - -Everard's instruments were made in London by Isaac Carver who, soon after, -himself wrote a sixteen-page Description and Use of a New Sliding Rule, -projected from the Tables in the Gauger's Magazine, London, 1687, which was -"printed for William Hunt" and bound in one volume with a book by Hunt, -called The Gauger's Magazine, London, 1687. This appears to be the same -William Hunt who later brought out descriptions of his own of slide rules. -The instrument described by Carver "consists of three pieces, two whereof -are moveable to be drawn out till the whole be 36 inches long." It has -several non-logarithmic graduations, together with logarithmic lines marked -A, B, C, D, of which A, B, C are "double lines," and D a "single line" used -for squares and square roots. It is designed for the determination of the -vacuity of a "spheroidal cask lying," a "spheroidal cask standing," and a -"parabolical cask lying." - -Another seventeenth century writer on the slide rule is John Atkinson, whom -we have mentioned earlier. He says:[38] "The Lines of Numbers, Sines and -Tangents, are set double, that is, one on each side, as the middle piece -slides: which middle piece is so contrived, to slip to and fro easily, to -slide out, and to be put in any side uppermost, in order to bring those -Lines together (or against one another) most proper for solving the -Question, wrought by Sliding-Gunter." - -The data presented in this article show that, while the earliest slide -rules were of the circular type, the later slide rules of the seventeenth -century were of the rectilinear type.[39] - - January 12, 1915. - - - - - Footnotes - - -[1]F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments, - New York, 1909, pp. 7-14, also Addenda i-vi. - -[2]F. Cajori, "On the Invention of the Slide Rule," in Colorado College - Publication, Engineering Series Vol. 1, 1910. An abstract of this is - given in Nature (London), Vol. 82, 1909, p. 267. - -[3]F. Cajori, History etc., p. 14. - -[4]Art. "Slide Rule" in the Penny Cyclopaedia and in the English - Cyclopaedia [Arts and Sciences]. - -[5]Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817, - p. 423. - -[6]The full title of the book which Wingate published on this subject in - Paris is as follows: - - L'Vsage | de la | Reigle de | Proportion | en l'Arithmetique & | - Geometrie. | Par Edmond Vvingate, | Gentil-homme Anglois. | - - Ean s philomaths, es polymaths. - - In tenui, sed no tenuis vsusve, laborne. | - - A Paris, | Chez Melchior Mondiere, | demeurant en l'Isle du Palais, | - la | ru de Harlay aux deux Viperes. | M. DC. XXIV. | Auec Priuilege du - Roy. | - - Back of the title page is the announcement: - - Notez que la Reigle de Proportion en toutes faons se vend Paris chez - Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces, - demeurant en l'Isle du Palais sur le Quay qui regarde la Megisserie - l'Espic d'or. - -[7]The title-page of the edition of 1658 is as follows: - - The Use of the Rule of Proportion in Arithmetick & Geometrie. First - published at Paris in the French tongue, and dedicated to Monsieur, the - then king's onely Brother (now Duke of Orleance). By Edm. Wingate, an - English Gent. And now translated into English by the Author. Whereinto - is now also inserted the Construction of the same Rule, & a farther use - thereof . . . 2nd edition inlarged and amended. London, 1658. - -[8]Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; - Drawn up by himself by way of Diary. With Appendix of original Letters. - Publish'd by Charles Burman, Esq., London, 1717, p. 23. - -[9]Mathematical Tables, 1811, p. 36, and art. "Gunter's Line" in his Phil. - and Math. Dictionary, London, 1815. - -[10]To the English Gentrie, and all others studious of the Mathematicks, - which shall bee readers hereof. The just Apologie of Wil: Ovghtred, - against the slaunderous insimulations of Richard Delamain, in a Pamphlet - called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum - projectio circularis. We shall refer to this document as Epistle. It was - published without date in 32 unnumbered pages of fine print, and was - bound in with Oughtred's Circles of Proportion, in the editions of 1633 - and 1639. In the 1633 edition it is inserted at the end of the volume - just after the Addition vnto the Vse of the Instrument etc., and in that - of 1639 immediately after the preface. It was omitted from the Oxford - edition of 1660. The Epistle was also published separately. There is a - separate copy in the British Museum, London. Aubrey, in his Brief Lives, - edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, "He - writt a stitch't pamphlet about 163(?4) against . . . Delamaine." - -[11]Thomas Browne is mentioned by Stone in his Mathematical Instruments, - London 1723, p. 16. See also Cajori, History of the Slide Rule, New - York, 1909, p. 15. - -[12]The Description and Use of a Joynt-Rule: . . . also the use of Mr. - White's Rule for measuring of Board and Timber, round and square; With - the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and - Versed Sines. By J. Brown, Philom., London, 1661. - -[13]A Collection of Centers and Useful Proportions on the Line of Numbers, - by John Brown, 1662(?), 16 pages; Description and Use of the Triangular - Quadrant, by John Brown, London, 1671; Wingate's Rule of Proportion in - Arithmetick and Geometry: or Gunter's Line. Newly rectified by Mr. Brown - and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The - Description and Use of the Carpenter's-Rule: Together with the Use of - the Line of Numbers commonly call'd Gunter's-Line, by John Brown, - London, 1704. - -[14]William Leybourn, op. cit., pp. 129, 130, 132, 133. - -[15]James Atkinson's edition of Andrew Wakely's The Mariners Compass - Rectified, London, 1694 [Wakely's preface dated 1664, Atkinson's - preface, 1693]. Atkinson adds An Appendix containing Use of Instruments - most useful in Navigation. Our quotation is from this Appendix, p. 199. - -[16]R. Delamain, The Making, Description, and Use of a small portable - Instrument . . . called a Horizontall Quadrant, etc., London, 1631. - -[17]Oughtred's description of his circular slide rule of 1632 and his - rectilinear slide rule of 1633, as well as a drawing of the circular - slide rule, are reproduced in Cajori's History of the Slide Rule, - Addenda, pp. ii-vi. - -[18]The full title of the Grammelogia I is as follows: - - Grammelogia | or, | The Mathematicall Ring. | Shewing (any reasonable - Capacity that hath | not Arithmeticke) how to resolve and worke | all - ordinary operations of Arithmeticke. | And those which are most - difficult with greatest | facilitie: The extraction of Roots, the - valuation of | Leases, &c. The measuring of Plaines | and Solids. | With - the resolution of Plaine and Sphericall | Triangles. | And that onely by - an Ocular Inspection, | and a Circular Motion. | Naturae secreta tempus - aperit. | London printed by John Haviland, 1630. - -[19]Grammelogia III is the same as Grammelogia I, except for the addition - of an appendix, entitled: - - De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit - nullo (praescripto tempore) vitae | vsuram nobis ingeniique Deus. | - London, | - - . . . The next line or two of this title-page which probably contained - the date of publication, were cut off by the binder in trimming the - edges of this and several other pamphlets for binding into one volume. - -[20]Grammelogia IV has two title pages. The first is Mirifica Logarithmoru' - Projectio Circularis. There follows a diagram of a circular slide rule, - with the inscription within the innermost ring: Nil Finis, Motvs, - Circvlvs vllvs Habet. The second title page is as follows: - - Grammelogia | Or, the Mathematicall Ring. | Extracted from the - Logarythmes, and projected Circular: Now published in the | inlargement - thereof unto any magnitude fit for use: shewing any reason- | able - capacity that hath not Arithmeticke how to resolve and worke, | all - ordinary operations of Arithmeticke: | And those that are most difficult - with greatest facilitie, the extracti- | on of Rootes, the valuation of - Leases, &c. the measuring of Plaines and Solids, | with the resolution - of Plaine and Sphericall Triangles applied to the | Practicall parts of - Geometrie, Horologographie, Geographie | Fortification, Navigation, - Astronomie, &c. | And that onely by an ocular inspection, and a Circular - motion, Invented and first published, by R. Delamain, Teacher, and - Student of the Mathematicks. | Naturae secreta tempus aperit. | - - There is no date. There follows the diagram of a second circular slide - rule, with the inscription within the innermost ring: Typus proiectionis - Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630 - promisi. There are numerous drawings in the Grammelogia, all of which, - excepting the drawings of slide rules on the engraved title-pages of - Grammelogia IV and V, were printed upon separate pieces of paper and - then inserted by hand into the vacant spaces on the printed pages - reserved for them. Some drawings are missing, so that the Bodleian - Grammelogia IV differs in this respect slightly from the copy in the - British Museum and from the British Museum copy of Grammelogia V. - -[21]Epistle, p. (8). - -[22]Aubrey, op. cit., Vol. II., p. 111. - -[23]Rigaud, Correspondence of Scientific Men during the 17th Century, Vol. - I, Oxford, 1841, p. 11. - -[24]Dictionary of National Biography, Art. "Delamain, Richard." See also - Rev. Charles J. Robinson, Taylors' School, from A.D. 1562 to 1874, Vol. - I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b; - Sixth Report of the Royal Commission on Historical Manuscripts, Part I, - Report and Appendix, London, 1877. In this Appendix, p. 82, we read the - following: - - Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. - Petitioner's husband was servant to the King, and one of His Majesty's - engineers for the fortification of the kingdom, and his tutor in - mathematical arts; but upon the breaking out of the war he deserted the - Court, and was called by the State to several employments, in fortifying - the towns of Northampton, Newport, and Abingdon; and was also abroad - with the armies as Quartermaster-General of the Foot, and therein died. - Petitioner is left a disconsolate widow with ten children, the four - least of whom are now afflicted with sickness, and petitioner has - nothing left to support them. There are several considerable sums of - money due to the petitioner, as well from the King as the State. Prays - that she may have some relief amongst other widows. See L. J., VII. 6. - 657. - -[25]Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London, - 1820, p. 34. - -[26]The New Artificial Gauging Line or Rod: together with rules concerning - the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London, - 1633. The copy we have seen is in the Bodleian Library, Oxford. The book - is small sized and has 40 pages. - -[27]Oughtred, op. cit., p. 11. - -[28]S. J. Rigaud, Correspondence of Scientific Men of the 17th Century, - Oxford, Vol. I, 1841, p. 17. - -[29]Rigaud, loc. cit., p. 22. - -[30]Rigaud, loc. cit., pp. 30, 31. - -[31]Oughtred, An Addition vnto the Vse of the Instrument called the Circles - of Proportion, London, 1633, p. 63. - -[32]F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22, - Addenda, pp. vi-ix. - -[33]W. Leybourn, op. cit., 1673, Preface, and pp. 128-29. - -[34]Cajori op. cit., Addenda, p. ix. - -[35]William Leybourn, op. cit., 1673, p. 35. - -[36]See Cajori, op. cit., pp. 20, 28, Addenda, p. ix. - -[37]See F. Cajori, "A Note on the History of the Slide Rule," Bibliotheca - mathematica, 3 F., Vol. 10, pp. 161-163. - -[38]John Atkinson, op. cit., 1694, p. 204. - -[39]Probably the oldest slide rule now in existence is owned by St. John's - College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in - diameter. It was exhibited along with other instruments in May, 1919. - According to the Catalogue of a Loan Exhibition of Early Scientific - Instruments in Oxford, opened May 16, 1919, the instrument is inscribed - with the name of the maker ("Elias Allen fecit") and with the name of - the donor, Georgius Barkham. It is dated 1635, which is only three years - after the first publication of Oughtred's description of his circular - slide rule. It is stated in the Catalogue: "Unfortunately all the - movable parts but the base-plate and a couple of thumb-screws are - missing. The face of the instrument is engraved with Oughtred's - Horizontal Instrument. The back is engraved with eleven Circles of - Proportion as described in Arthur Haughton's book, a copy of which was - presented to St. John's College by George Barkham, to explain the use of - the instrument." As Arthur Haughton's Oxford edition of Oughtred's - Circles of Proportion did not appear until 1660, it would seem that the - instrument was probably not presented to the College before 1660. As far - as is known, the next oldest slide rule is of the year 1654, kept in the - South Kensington Museum, London, and is described in Nature of March 5, - 1914. It is a rectilinear rule, "of boxwood, well made, and bound - together with brass at the two ends. It is of the square type, a little - more than 2 ft. in length, and bears the logarithmic lines first - described by Edmund Gunter. Of these, the num, sin and tan lines are - arranged in pairs, identical and contiguous, one line in each pair being - on the fixed part, and the other on the slide." The instrument is - inscribed, "Made by Robert Bissaker for T. W., 1654." Nowhere else have - we seen reference to Robert Bissaker. His slide rule seems to antedate - the "Whites rule" mentioned above. [This foot-note was added on October - 15, 1919.] - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - - -Note.--The University of California Publications are offered in exchange -for the publication of learned societies and institutions, universities and -libraries. Complete lists of all the publications of the University will be -sent upon request. For sample copies, lists of publications or other -information, address the Manager of the University of California Press, -Berkeley, California, U. 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Current, Volume XXII. - - - - - Transcriber's Notes - - ---Inserted subchapter headings from the Table of Contents into the - text. - ---Corrected two page numbers in the Table of Contents - ---Corrected a Greek quotation in footnote 6 by reference to the - original sources. - - - - - - - -End of the Project Gutenberg EBook of On the History of Gunter's Scale and -the Slide Rule during the Seventeenth Century, by Florian Cajori - -*** END OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - -***** This file should be named 42216-8.txt or 42216-8.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/4/2/2/1/42216/ - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - -Updated editions will replace the previous one--the old editions -will be renamed. - -Creating the works from public domain print editions means that no -one owns a United States copyright in these works, so the Foundation -(and you!) can copy and distribute it in the United States without -permission and without paying copyright royalties. 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You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org - - -Title: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century - -Author: Florian Cajori - -Release Date: February 26, 2013 [EBook #42216] - -Language: English - -Character set encoding: ASCII - -*** START OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - - - - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - IN - MATHEMATICS - - Vol. 1, No. 9, pp. 187-209 February 17, 1920 - - - - - ON THE HISTORY OF GUNTER'S SCALE AND - THE SLIDE RULE DURING THE - SEVENTEENTH CENTURY - - - BY - FLORIAN CAJORI - - - UNIVERSITY OF CALIFORNIA PRESS - BERKELEY - - - - - TABLE OF CONTENTS - - - PAGE - I. Introduction 187 - II. Innovations in Gunter's Scale 188 - Changes introduced by Edmund Wingate 188 - Changes introduced by Milbourn 189 - Changes introduced by Thomas Brown and John Brown 190 - Changes introduced by William Leybourn 192 - III. Richard Delamain's "Grammelogia" 192 - Different editions or impressions 194 - Description of Delamain's instrument of 1630 195 - Delamain's later designs, and directions for using his - instruments 197 - IV. Controversy between Oughtred and Delamain on the invention of - the circular slide rule 199 - V. Independence and priority of invention 203 - VI. Oughtred's "Gauging Line," 1633 206 - VII. Other seventeenth century slide rules 207 - - - - - I. INTRODUCTION - - - In my history of the slide rule[1], and my article on its invention[2] it -is shewn that William Oughtred and not Edmund Wingate is the inventor, that -Oughtred's circular rule was described in print in 1632, his rectilinear -rule in 1633. Richard Delamain is referred to as having tried to -appropriate the invention to himself[3] and as having written a scurrilous -pamphlet against Oughtred. All our information about Delamain was taken -from De Morgan,[4] who, however, gives no evidence of having read any of -Delamain's writings on the slide rule. Through Dr. Arthur Hutchinson of -Pembroke College, Cambridge, I learned that Delamain's writings on the -slide rule were available. In this article will be given: First, some -details of the changes introduced during the seventeenth century in the -design of Gunter's scale by Edmund Wingate, Milbourn, Thomas Brown, John -Brown and William Leybourn; second, an account of Delamain's book of 1630 -on the slide rule which antedates Oughtred's first publication (though -Oughtred's date of invention is earlier than the date of Delamain's alleged -invention) and of Delamain's later designs of slide rules; third, an -account of the controversy between Delamain and Oughtred; fourth, an -account of a later book on the slide rule written by William Oughtred, and -of other seventeenth century books on the slide rule. - - - - - II. INNOVATIONS IN GUNTER'S SCALE - - - Changes introduced by Wingate - - We begin with Anthony Wood's account of Wingate's introduction of -Gunter's scale into France.[5] - - In 1624 he transported into France the rule of proportion, having a - little before been invented by Edm. Gunter of Gresham Coll. and - communicated it to most of the chiefest mathematicians then residing in - Paris: who apprehending the great benefit that might accrue thereby, - importun'd him to express the use thereof in the French tongue. Which - being performed accordingly, he was advised by monsieur Alleawne the - King's chief engineer to dedicate his book to monsieur the King's only - brother, since duke of Orleans. Nevertheless the said work coming forth - as an abortive (the publishing thereof being somewhat hastened, by reason - an advocate of Dijon in Burgundy began to print some uses thereof, which - Wingate had in a friendly way communicated to him) especially in regard - Gunter himself had learnedly explained its use in a far larger volume.[6] - -Gunter's scale, which Wingate calls the "rule of proportion," contained, as -described in the French edition of 1624, four lines: (1) A single line of -numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot -in length, divided into 12 inches and tenths of inches, also a line, one -foot in length, divided into tenths and hundredths. - -The English editions of this book which appeared in 1623 and 1628 are -devoid of interest. The editions of 1645 and 1658 contain an important -innovation.[7] In the preface the reasons why this instrument has not been -used more are stated to be: (1) the difficulty of drawing the lines with -exactness, (2) the trouble of working thereupon by reason (sometimes) of -too large an extent of the compasses, (3) the fact that the instrument is -not readily portable. The drawing of Wingate's arrangement of the scale in -the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains -five parallel lines, about 66 cm. long, each having the divisions of one -line marked on one side and of another line on the other side. Thus each -line carries two graduations: (1) A single logarithmic line of numbers; (2) -a logarithmic line of numbers thrice repeated; (3) the first scale -repeated, but beginning with the graduations which are near the middle of -the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3; -(4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of -tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000 -equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of -inches; (10) a scale consisting of three kinds, viz., a gauge line, a line -of chords, and a foot measure, divided into 1000 equal parts. - -Important are the first and second scales, by which cube root extraction -was possible "by inspection only, without the aid of pen or compass;" -similarly the third and fourth scales, for square roots. This innovation is -due to Wingate. The 1645 edition announces that the instrument was made in -brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in -Hosier Lane. - - - Changes introduced by Milbourn - -William Leybourn, in his The Line of Proportion or Numbers, Commonly called -Gunter's Line, Made Easie, London, 1673, says in his preface "To the -Reader:" - - The Line of Proportion or Numbers, commonly called (by Artificers) - Gunter's Line, hath been discoursed of by several persons, and variously - applied to divers uses; for when Mr. Gunter had brought it from the - Tables to a Line, and written some Uses thereof, Mr. Wingate added divers - Lines of several lengths, thereby to extract the Square or Cube Roots, - without doubling or trebling the distance of the Compasses: After him Mr. - Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral - Line, thereby enlarging the divisions of the Line. - -On pages 127 and 128 Leybourn adds: - - Again, One T. Browne, a Maker of Mathematical Instruments, made it in a - Serpentine or Spiral Line, composed of divers Concentrick Circles, - thereby to enlarg the divisions, which was the contrivance of one Mr. - Milburn a Yorkshire Gentleman, who writ thereof, and communicated his - Uses to the aforesaid Brown, who (since his death) attributed it to - himself: But whoever was the contriver of it, it is not without - inconvenience; for it can in no wise be made portable; and besides - (instead of compasses) an opening Joynt with thirds [threads] must be - placed to move upon the Centre of the Instrument, without which no - proportion can be wrought. - -This Mr. Milburn is probably the person named in the diary of the -antiquarian, Elias Ashmole, on August 13 [1646?]; "I bought of Mr. Milbourn -all his Books and Mathematical Instruments."[8] Charles Hutton[9] says that -Milburne of Yorkshire designed the spiral form about 1650. This date is -doubtless wrong, for Thomas Browne who, according to Leybourn, got the -spiral form of line from Milbourn, is repeatedly mentioned by William -Oughtred in his Epistle[10] printed some time in 1632 or 1633. Oughtred -does not mention Milbourn, and says (page 4) that the spiral form "was -first hit upon by one Thomas Browne a Joyner, . . . the serpentine -revolution being but two true semicircles described on severall -centers."[11] - - - Changes introduced by Thomas Brown and John Brown - -Thomas Brown did not publish any description of his instrument, but his -son, John Brown, published in 1661 a small book,[12] in which he says -(preface) that he had done "as Mr. Oughtred with Gunter's Rule, to a -sliding and circular form; and as my father Thomas Brown into a Serpentine -form; or as Mr. Windgate in his Rule of Proportion." He says also that -"this brief touch of the Serpentine-line I made bold to assert, to see if I -could draw out a performance of that promise, that hath been so long -unperformed by the promisers thereof." Accordingly in Chapter XX he gives a -description of the serpentine line, "contrived in five (or rather 15) -turn." Whether this description, printed in 1661, exactly fits the -instrument as it was developed in 1632, we have no means of knowing. John -Brown says: - - 1. First next the center is two circles divided one into 60, the other - into 100 parts, for the reducing of minutes to 100 parts, and the - contrary. - - 2. You have in seven turnes two inpricks, and five in divisions, the - first Radius of the sines (or Tangents being neer the matter, alike to - the first three degrees,) ending at 5 degrees and 44 minutes. - - 3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in - three margents in divisions, and the line of versed sines in pricks, - under the line of Tangents, according to Mr. Gunter's cross-staff: the - sines and Tangents beginning at 5 degrees, and 44 minutes where the other - ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the - line of numbers beginning at 10, and proceeding to 100, being one entire - Radius, and graduated into as many divisions as the largeness of the - instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to - 100 into 20 parts in one unit of increase, but the Tangents are divided - into single minutes from the beginning to the end, both in the first, - second and third Radiusses, and the sines into minutes; also from 30 - minutes to 40 degrees, and from 40 to 60, into every two minutes, and - from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the - rest as many as can be well discovered. - - The versed sines are set after the manner of Mr. Gunter's Cross-staff, - and divided into every 10th minutes beginning at 0, and proceeding to 156 - going backwards under the line of Tangents. - - 4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is - the secants to 51 degrees, being nothing else but the sines reitterated - beyond 90. - - 5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85 - degrees, whereby all trouble of backward working is avoided. - - 6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and - also a line of natural sines, for finding of differences in sines, for - finding hour and Azimuth. - - 7. Seventhly, next the verge or outermost edge is a line of equal parts - to get the Logarithm of any number, or the Logarithm sine and Tangent of - any ark or angle to four figures besides the carracteristick. - - 8. Eightly and lastly, in the space place between the ending of the - middle five turnes, and one half of the circle are three prickt lines - fitted for reduction. The uppermost being for shillings, pence and - farthings. The next for pounds, and ounces, and quarters of small - Averdupoies weight. The last for pounds, shillings and pence, and to be - used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay - the hair or edge of one of the legs of the index on 16. 3 1/2 in the line - of 1. s. d. and the hair shall cut on the equal parts 81 16; and the - contrary, if you have a decimal fraction, and would reduce it to a proper - fraction, the like may you do for shillings, and pence, and pounds, and - ounces. - - The uses of the lines follow. - - As to the use of these lines, I shall in this place say but little, and - that for two reasons. First, because this instrument is so contrived, - that the use is sooner learned then any other, I speak as to the manner, - and way of using it, because by means of first second and third - radiusses, in sines and Tangents, the work is always right on, one way or - other, according to the Canon whatsoever it be, in any book that treats - of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in - Oughtred from page 64 to 107. - - Secondly, and more especially, because the more accurate, and large - handling thereof is more then promised, if not already performed by more - abler pens, and a large manuscript thereof by my Sires meanes, provided - many years ago, though to this day not extant in print; so for his sake I - claiming my interest therein, make bold to present you with these few - lines, in order to the use of them: And first note, - - 1. Which soever of the two legs is set to the first term in the question, - that I call the first leg always, and the other being set to the second - term, I call the second leg . . . - -The exact nature of the contrivance with the "two legs" is not described, -but it was probably a flat pair of compasses, attached to the metallic -surface on which the serpentine line was drawn. In that case the instrument -was a slide rule, rather than a form of Gunter's line. In his publication -of 1661, as also in later publications,[13] John Brown devoted more space -to Gunter's scales, requiring the use of a separate pair of compasses, than -to slide rules. - - - Changes introduced by William Leybourn - -The same remark applies to William Leybourn who, after speaking of Seth -Partridge's slide rule, returns to forms of Gunter's scale, saying:[14] - - There is yet another way of disposing of this Line of Proportion, by - having one Line of the full length of the Ruler, and another Line of the - same Radius broken in two parts between 3 and 4; so that in working your - Compasses never go off of the Line: This is one of the best contrivances, - but here Compasses must be used. These are all the Contrivances that I - have hitherto seen of these Lines: That which I here speak of, and will - shew how to use, is only two Lines of one and the same Radius, being set - upon a plain Ruler of any length (the larger the better) having the - beginning of one Line, at the end of the other, the divisions of each - Line being set so close together, that if you find any number upon one of - the Lines, you may easily see what number stands against it on the other - Line. This is all the Variation. . . . - - Example 1. If a Board be 1 Foot 64 parts broad, how much in length of - that Board will make a Foot Square? Look upon one of your Lines (it - matters not which) for 1 Foot 64 parts, and right against it on the other - Line you shall find 61; and so many parts of a Foot will make a Foot - square of that Board. - -This contrivance solves the equation 1.64x=1, yielding centesimal parts of -a foot. - -James Atkinson[15] speaks of "Gunter's scale" as "usually of Boxwood . . . -commonly 2 ft. long, 1 1/2 inch broad" and "of two kinds: long Gunter or -single Gunter, and the sliding Gunter. It appears that during the -seventeenth century (and long after) the Gunter's scale was a rival of the -slide rule. - - - - - III. RICHARD DELAMAIN'S GRAMMELOGIA - - -We begin with a brief statement of the relations between Oughtred and -Delamain. At one time Delamain, a teacher of mathematics in London, was -assisted by Oughtred in his mathematical studies. In 1630 Delamain -published the Grammelogia, a pamphlet describing a circular slide rule and -its use. In 1631 he published another tract, on the Horizontall -Quadrant.[16] In 1632 appeared Oughtred's Circles of Proportion[17] -translated into English from Oughtred's Latin manuscript by another pupil, -William Forster, in the preface of which Forster makes the charge (without -naming Delamain) that "another . . . went about to pre-ocupate" the new -invention. This led to verbal disputes and to the publication by Delamain -of several additions to the Grammelogia, describing further designs of -circular slide rules and also stating his side of the bitter controversy, -but without giving the name of his antagonist. Oughtred's Epistle was -published as a reply. Each combatant accuses the other of stealing the -invention of the circular slide rule and the horizontal quadrant. - -The two title-pages of the edition of the Grammelogia in the British Museum - in London which we have called "Grammelogia IV." - - - Different editions or impressions - -There are at least five different editions, or impressions, of the -Grammelogia which we designate, for convenience, as follows: - - Grammelogia I, 1630. One copy in the Cambridge University Library.[18] - - Grammelogia II, I have not seen a copy of this. - - Grammelogia III, One copy in the Cambridge University Library.[19] - - Grammelogia IV, One copy in the British Museum, another in the Bodleian - Library, Oxford.[20] - - Grammelogia V, One copy in the British Museum. - -In Grammelogia I the first three leaves and the last leaf are without -pagination. The first leaf contains the title-page; the second leaf, the -dedication to the King and the preface "To the Reader;" the third leaf, the -description of the Mathematical Ring. Then follow 22 numbered pages. -Counting the unnumbered pages, there are altogether 30 pages in the -pamphlet. Only the first three leaves of this pamphlet are omitted in -Grammelogia IV and V. - -In Grammelogia III the Appendix begins with a page numbered 52 and bears -the heading "Conclusion;" it ends with page 68, which contains the same two -poems on the mathematical ring that are given on the last page of -Grammelogia I but differs slightly in the spelling of some of the words. -The 51 pages which must originally have preceded page 52, we have not seen. -The edition containing these we have designated Grammelogia II. The reason -for the omission of these 51 pages can only be conjectured. In Oughtred's -Epistle (p. 24), it is stated that Delamain had given a copy of the -Grammelogia to Thomas Brown, and that two days later Delamain asked for the -return of the copy, "because he had found some things to be altered -therein" and "rent out all the middle part." Delamain labored "to recall -all the bookes he had given forth, (which were many) before the sight of -Brownes Lines." These spiral lines Oughtred claimed that Delamain had -stolen from Brown. The title-page and page 52 are the only parts of the -Appendix, as given in Grammelogia III, that are missing in the Grammelogia -IV and V. - -Grammelogia IV answers fully to the description of Delamain's pamphlet -contained in Oughtred's Epistle. It was brought out in 1632 or 1633, for -what appears to be the latest part of it contains a reference (page 99) to -the Grammelogia I (1630) as "being now more then two yeares past." -Moreover, it refers to Oughtred's Circles of Proportion, 1632, and -Oughtred's reply in the Epistle was bound in the Circles of Proportion -having the Addition of 1633. For convenience of reference we number the two -title-pages of Grammelogia IV, "page (1)" and "page (2)," as is done by -Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page -numbers which we assign will be placed in parentheses, to distinguish them -from the page numbers which are printed in Grammelogia IV. The pages -(44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the -same as the pages 53-68. Thus only thirty-eight pages have page numbers -printed on them. The pages (67) and (83) are identical in wording, except -for some printer's errors; they contain verses in praise of the Ring, and -have near the bottom the word "Finis." Also, pages (22) and (23) are -together identical in wording with page (113), which is set up in finer -type, containing an advertisement of a part of Grammelogia IV explaining -the mode of graduating the circular rules. There are altogether six parts -of Grammelogia IV which begin or end by an address to the reader, thus: "To -the Reader," "Courteous Reader," or "To the courteous and benevolent Reader -. . .," namely the pages (8), (22), (68), (89), (90), (108). In his Epistle -(page 2), Oughtred characterizes the make up of the book in the following -terms: - - In reading it . . . I met with such a patchery and confusion of - disjoynted stuffe, that I was striken with a new wonder, that any man - should be so simple, as to shame himselfe to the world with such a - hotch-potch. - -Grammelogia V differs from Grammelogia IV in having only the second -title-page. The first title-page may have been torn off from the copy I -have seen. A second difference is that the page with the printed numeral 22 -in Grammelogia IV has after the word "Finis" the following notice: - - This instrument is made in Silver, or Brasse for the Pocket, or at any - other bignesse, over against Saint Clements Church without Temple Barre, - by Elias Allen. - -This notice occurs also on page 22 of Grammelogia I and III, but is omitted -from page 22 of Grammelogia V. - - - Description of Delamain's instrument of 1630 - -In his address to King Charles I, in his Grammelogia I, Delamain emphasizes -the ease of operating with his slide rule by stating that it is "fit for -use . . . as well on Horse backe as on Foot." Speaking "To the Reader," he -states that he has "for many yeares taught the Mathematicks in this Towne," -and made efforts to improve Gunter's scale "by some Motion, so that the -whole body of Logarithmes might move proportionally the one to the other, -as occasion required. This conceit in February last [1629] I struke upon, -and so composed my Grammelogia or Mathematicall Ring; by which only with an -ocular inspection, there is had at one instant all proportionalls through -the said body of Numbers." He dates his preface "first of January, 1630." -The fifth and sixth pages contain his "Description of the Grammelogia," the -term Grammelogia being applied to the instrument, as well as to the book. -His description is as follows: - - The parts of the Instrument are two Circles, the one moveable, and the - other fixed; The moveable is that unto which is fastened a small pin to - move it by; the other Circle may be conceived to be fixed; The - circumference of the moveable Circle is divided into unequall parts, - charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures - doe represent themselves, or such numbers unto which a Cipher or Ciphers - are added, and are varied as the occasion falls out in the speech of - Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or - 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c. - -After elaborating this last point and explaining the decimal subdivisions -on the scales of the movable circle, he says that "the numbers and -divisions on the fixed Circle, are the very same that the moveable are, . -." There is no drawing of the slide rule in this publication. The -twenty-two numbered pages give explanations of the various uses to which -the instrument can be put: "How to performe the Golden Rule" (pp. 1-3), -"Further uses of the Golden Rule" (pp. 4-6), "Notions or Principles -touching the disposing or ordering of the Numbers in the Golden Rule in -their true places upon the Grammelogia" (pp. 7-11), "How to divide one -number by another" (pp. 12, 13), "to multiply one Number by another" (pp. -14, 15), "To find Numbers in continuall proportion" (pp. 16, 17), "How to -extract the Square Root," "How to extract the Cubicke Root" (pp. 18-21), -"How to performe the Golden Rule" (the rule of proportion) is explained -thus: - - Seeke the first number in the moveable, and bring it to the second number - in the fixed, so right against the third number in the moveable, is the - answer in the fixed. - - If the Interest of 100. li. be 8. li. in the yeare, what is the Interest - of 65. li. for the same time. - - Bring 100. in the moveable to 8. in the fixed, so right against 65. in - the moveable is 5.2. in the fixed, and so much is the Interest of 65. li. - for the yeare at 8. li. for 100. li. per annum. - - The Instrument not removed, you may at one instant right against any - summe of money in the moveable, see the Interest thereof in the fixed: - the reason of this is from the Definition of Logarithmes. - -These are the earliest known printed instructions on the use of a slide -rule. It will be noticed that the description of the instrument at the -opening makes no references to logarithmic lines for the trigonometric -functions; only the line of numbers is given. Yet the title-page promised -the "resolution of Plaine and Sphericall Triangles." Page 22 throws light -upon this matter: - - If there be composed three Circles of equal thicknesse, A.B.C. so that - the inner edge of D [should be B] and the outward edge of A bee - answerably graduated with Logarithmall signes [sines], and the outward - edge of B and the inner edge of A with Logarithmes; and then on the - backside be graduated the Logarithmall Tangents, and againe the - Logarithmall signes oppositly to the former graduations, it shall be - fitted for the resolution of Plaine and Sphericall Triangles. - -After twelve lines of further remarks on this point he adds: - - Hence from the forme, I have called it a Ring, and Grammelogia by - annoligie of a Lineary speech; which Ring, if it were projected in the - convex unto two yards Diameter, or thereabouts, and the line Decupled, it - would worke Trigonometrie unto seconds, and give proportionall numbers - unto six places only by an ocular inspection, which would compendiate - Astronomicall calculations, and be sufficient for the Prosthaphaeresis of - the Motions: But of this as God shall give life and ability to health and - time. - -The unnumbered page following page 22 contains the patent and copyright on -the instrument and book: - - Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs - an Instrument called Grammelogia, or The Mathematicall Ring, together - with a Booke so intituled, expressing the use thereof, being his owne - Invention; we of our Gracious and Princely favour have granted unto the - said Richard Delamain and his Assignes, Privilege, Licence, and - Authority, for the sole Making, Printing and Selling of the said - Instrument and Booke: straightly forbidding any other to Make, Imprint, - or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument - or Booke within any our Dominions, during the space of ten yeares next - ensuing the date hereof, upon paine of Our high displeasure. Given under - our hand and Signet at our Palace of Westminster, the fourth day of - January, in the sixth yeare of our Raigne. - - - Delamain's later designs, and directions for using his instruments - -In the Appendix of Grammelogia III, on page 52 is given a description of an -instrument promised near the end of Grammelogia I: - - That which I have formerly delivered hath been onely upon one of the - Circles of my Ring, simply concerning Arithmeticall Proportions, I will - by way of Conclusion touch upon some uses of the Circles, of Logarithmall - Sines, and Tangents, which are placed on the edge of both the moveable - and fixed Circles of the Ring in respect of Geometricall Proportions, but - first of the description of these Circles. - - First, upon the side that the Circle of Numbers is one, are graduated on - the edge of the moveable, and also on the edge of the fixed the - Logarithmall Sines, for if you bring 1. in the moveable amongst the - Numbers to 1. in the fixed, you may on the other edge of the moveable and - fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto - 6.6. and each degree subdivided, and then over the former divisions and - figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5. - 4. 3. 2. 1. each of those divided by small points. - - Secondly, (if the Ring is great) neere the outward edge of this side of - the fixed against the Numbers, are the usuall divisions of a Circle, and - the points of the Compasse: serving for observation in Astronomy, or - Geometry, and the sights belonging to those divisions, may be placed on - the moveable Circle. - - Thirdly, opposite to those Sines on the other side are the Logarithmall - Tangents, noted alike both in the moveable and fixed thus - 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or - divisions serve also for their Complements to 90. so 40 gr. stands for - 50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both - in the moveable and fixed is also divided into parts. As for the degrees - which are under 6. viz. 5.4.3.2.1. they are noted with small figures over - this divided Circle from 45.40.35.30.25. &c. and each of those degrees - divided into parts by small points both in the moveable and fixed. - - Fourthly, on the other edge of the moveable on the same side is another - graduation of Tangents, like that formerly described. And opposite unto - it, in the fixed is a Graduation of Logarithmall sines in every thing - answerable to the first descrition of Sines on the other side. - - Fifthly, on the edge of the Ring is graduated a parte of the AEquator, - numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees - of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree - both of the AEquator, and Meridian are subdivided into parts; these two - graduated Circles serve to resolve such Questions which concerne - Latitude, Longitude, Rumb, and Distance, in Nauticall operations. - - Sixthly, to the concave of the Ring may be added a Circle to be elevated - or depressed for any Latitude, representing the AEquator, and so divided - into houres and parts with an Axis, to shew both the houre, and Azimuth, - and within this Circle may be hanged a Box, and Needle with a Socket for - a staffe to slide into it, and this accommodated with scrue pines to - fasten it to the Ring and staffe, or to take it off at pleasure. - -The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and -V make no reference to the dispute with Oughtred and may, therefore, be -assumed to have been published before the appearance of Oughtred's Circles -of Proportion. On page 53, "To the Reader," he says: - - . . . you may make use of the Projection of the Circles of the Ring upon - a Plaine, having the feet of a paire of compasses (but so that they be - flat) to move on the Center of that Plaine, and those feet to open and - shut as a paire of Compasses . . . now if the feet bee opened to any two - termes or numbers in that Projection, then may you move the first foot to - the third number, and the other foot shall give the Answer; . . . it hath - pleased some to make use of this way. But in this there is a double - labour in respect to that of the Ring, the one in fitting those feet unto - the numbers assigned, and the other by moving them about, in which a man - can hardly accommodate the Instrument with one hand, and expresse the - Proportionals in writing with the other. By the Ring you need not but - bring one number to another, and right against any other number is the - Answer without any such motion. . . . upon that [the Ring] I write, - shewing some uses of those Circles amongst themselves, and conjoyned with - others . . . in Astronomy, Horolographie, in plaine Triangles applyed to - Dimensions, Navigation, Fortification, etc. . . . But before I come to - Construction, I have thought it convenient by way introduction, to - examine the truth of the graduation of those Circles . . . - -These are the words of a practical man, interested in the mechanical -development of his instrument. He considers not only questions of -convenience but also of accuracy. The instrument has, or may have now, also -lines of sines and tangents. To test the accuracy of the circles of -Numbers, "bring any number in the moveable to halfe of that number in the -fixed: so any number or part in the fixed shall give his double in the -moveable, and so may you trie of the thirds, fourths &c. of numbers, vel -contra," (p. 54). On page 55 are given two small drawings, labelled, "A -Type of the Ringe and Scheme of this Logarithmicall projection, the use -followeth. These Instruments are made in Silver or Brasse by John Allen -neare the Sauoy in the Strand." - - - IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE - CIRCULAR SLIDE RULE - -Delamain's publication of 1630 on the 'Mathematicall Ring' does not appear -at that time to have caused a rupture between him and Oughtred. When in -1631 Delamain brought out his Horizontall Quadrant, the invention of which -Delamain was afterwards charged to have stolen from Oughtred, Delamain was -still in close touch with Oughtred and was sending Oughtred in the Arundell -House, London, the sheets as they were printed. Oughtred's reference to -this in his Epistle (p. 20) written after the friendship was broken, is as -follows: - - While he was printing his tractate of the Horizontall quadrant, although - he could not but know that it was injurious to me in respect of my free - gift to Master Allen, and of William Forster, whose translation of my - rules was then about to come forth: yet such was my good nature, and his - shamelessnesse, that every day, as any sheet was printed, hee sent, or - brought the same to mee at my chamber in Arundell house to peruse which I - lovingly and ingenuously did, and gave him my judgment of it. - -Even after Forster's publication of Oughtred's Circles of Proportion, 1632, -Oughtred had a book, A canon of Sines Tangents and Secants, which he had -borrowed from Delamain and was then returning (Epistle, page (5)). The -attacks which Forster, in the preface to the Circles of Proportion, made -upon Delamain (though not naming Delamain) started the quarrel. Except for -Forster and other pupils of Oughtred who urged him on to castigate -Delamain, the controversy might never have arisen. Forster expressed -himself in part as follows: - - . . . being in the time of the long vacation 1630, in the Country, at the - house of the Reverend, and my most worthy friend, and Teacher, Mr. - William Oughtred (to whose instruction I owe both my initiation, and - whole progresse in these Sciences.) I vpon occasion of speech told him of - a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made - (such as it vsually called Mr. Gunter's Ruler) 6 feet long, to be vsed - with a payre of beame-compasses. "He answered that was a poore invention, - and the performance very troublesome: But, said he, seeing you are taken - with such mechanicall wayes of Instruments, I will shew you what deuises - I have had by mee these many yeares." And first, hee brought to mee two - Rulers of that sort, to be vsed by applying one to the other, without any - compasses: and after that hee shewed mee those lines cast into a circle - or Ring, with another moueable circle vpon it. I seeing the great - expeditenesse of both those wayes; but especially, of the latter, wherein - it farre excelleth any other Instrument which hath bin knowne; told him, - I wondered that hee could so many yeares conceale such vseful inuentions, - not onely from the world, but from my selfe, to whom in other parts and - mysteries of Art, he had bin so liberall. He answered, "That the true way - of Art is not by Instruments, but by Demonstration: and that it is a - preposterous course of vulgar Teachers, to begin with Instruments, and - not with the Sciences, and so in-stead of Artists, to make their - Schollers only doers of tricks, and as it were Iuglers: to the despite of - Art, losse of precious time, and betraying of willing and industrious - wits, vnto ignorance and idlenesse. That the vse of Instruments is indeed - excellent, if a man be an Artist: but contemptible, being set and opposed - to Art. And lastly, that he meant to commend to me, the skill of - Instruments, but first he would haue me well instructed in the Sciences. - He also shewed me many notes, and Rules for the vse of those circles, and - of his Horizontall Instrument, (which he had proiected about 30 yeares - before) the most part written in Latine. All which I obtained of him - leaue to translate into English, and make publique, for the vse, and - benefit of such as were studious, and louers of these excellent Sciences. - - Which thing while I with mature, and diligent care (as my occasions would - give me leaue) went about to doe: another to whom the Author in a louing - confidence discouered this intent, using more hast then good speed, went - about to preocupate; of which vntimely birth, and preuenting (if not - circumuenting) forwardnesse, I say no more: but aduise the studious - Reader, onely so farre to trust, as he shal be sure doth agree to truth & - Art. - -While in this dedication reference is made to a slide rule or "ring" with a -"moveable circle," the instrument actually described in the Circles of -Proportion consists of fixed circles "with an index to be opened after the -manner of a paire of Compasses." Delamain, as we have seen, had decided -preference for the moveable circle. To Oughtred, on the other hand, one -design was about as good as the other; he was more of a theorist and -repeatedly expressed his contempt for mathematical instruments. In his -Epistle (page (25)), he says he had not "the one halfe of my intentions -upon it" (the rule in his book), nor one with a "moveable circle and a -thread, but with an opening Index at the centre (if so be that bee cause -enough to make it to bee not the same, but another Instrument) for my part -I disclaime it: it may go seeke another Master: which for ought I know, -will prove to be Elias Allen himselfe: for at his request only I altered a -little my rules from the use of the moveable circle and the thread, to the -two armes of an Index." - -All parts of Delamain's Grammelogia IV, except pages 1-22 and 53-68 -considered above, were published after the Circles of Proportion, for they -contain references to the ill treatment that Delamain felt or made believe -that he felt, that he had received in the book published by Oughtred and -Forster. Oughtred's reference to teachers whose scholars are "doers of -tricks," "Iuglers," and Forster's allusion to "another to whom the Author -in a loving confidence" explained the instrument and who "went about to -preocupate" it, are repeatedly mentioned. Delamain says, (page (89)) that -at first he did not intend to express himself in print, "but sought peace -and my right by a private and friendly way." Oughtred's account of -Delamain's course is that of an "ill-natured man" with a "virulent tongue," -"sardonical laughter" and "malapert sawsiness." Contrasting Forster and -Delamain, he says that, of the former he "had the very first moulding" and -made him feel that "the way of Art" is "by demonstration." But Delamain was -"already corrupted with doing upon Instruments, and quite lost from ever -being made an Artist." (Epistle page (27)). Repeatedly does Oughtred assert -Delamain's ignorance of mathematics. The two men were evidently of wholly -different intellectual predilections. That Delamain loved instruments is -quite evident, and we proceed to describe his efforts to improve the -circular slide rule. - -The Grammelogia IV is dedicated to King Charles I. Delamain says: - - . . . Everything hath his beginning, and curious Arts seldome come to the - height at the first; It was my promise then to enlarge the invention by a - way of decuplating the Circles, which I now present unto your sacred - Majestie as the quintessence and excellencie there of . . . - -His enlarged circular rules are illustrated in the Bodleian Library copy of -Grammelogia IV by four diagrams, two of them being the two drawings on the -two title-pages at the beginning of the Grammelogia IV, 4 inches in -external diameter, and exhibiting eleven concentric circular lines carrying -graduations of different sorts. In the second of these designs all circles -are fixed. The other two drawings are each 10 3/4 inches in external -diameter and exhibit 18 concentric circular lines; the folded sheet of the -first of these drawings is inserted between pages (23) and (24), the second -folded sheet between pages (83) and (84). All circles of this second -instrument are fixed. Counting in the two small drawings in Grammelogia -III, there are in all six drawings of slide rules in the Bodleian -Grammelogia IV. On pages (24) to (43) Delamain explains the graduation of -slide rules. He takes first a rule which has one circle of equal parts, -divided into 1000 equal divisions. From a table of logarithms he gets log 2 -= 0.301; from the number 301 in the circle of equal parts he draws a line -to the center of the circle and marks the intersection with the circles of -numbers by the figure 2. Thus he proceeds with log 3, log 4, and so on; -also with log sin x and log tan x. For log sin x he uses two circles, the -first (see page (27)) for angles from 34' 24" to 5^o 44' 22", the second -circle from 5^o 44' 22" to 90^o. The drawings do not show the seconds. He -suggests many different designs of rules. On page (29) he says: - - For the single projection of the Circles of my Ring, and the dividing and - graduating of them: which may bee so inserted upon the edges of Circles - of mettle turned in the forme of a Ring, so that one Circle may moove - betweene two fixed, by helpe of two stayes, then may there be graduated - on the face of the Ring, upon the outer edge of the mooveable and inner - edge of the fixed, the Circle of Numbers, then upon the inner edge of - that mooveable Circle, and the outward edge of that inner fixed Circle - may be inserted the Circle of Sines, and so according to the description - of those that are usually made. - -In addition to these lines he proceeds to mention the circle giving the -ordinary division into degrees and minutes, and two circles of tangents on -the other side of the rule. - -Next Delamain explains an arrangement of all the graduation on one side of -the rule by means of "a small channell in the innermost fixed Circle, in -which may be placed a small single Index, which may have sufficient length -to reach from the innermost edge of the Mooveable Circle, unto the outmost -edge of the fixed Circle, which may be mooved to and fro at pleasure, in -the channell, which Index may serve to shew the opposition of Numbers" (p. -(31)). From this it is clear that the invention of the "runner" goes back -to the very first writers on the slide rule. - -After describing a modification of the above arrangement, he adds, "many -other formes might be deliverd, about this single projection" (p. (32)). - -Proceeding to the "enlarging" of the circles in the Ring, to, say, the -"Quadruple to that which is single, that is, foure times greater," the -"equall parts" are distributed over four circles instead of only one -circle, but the general method of graduation is the same as before (p. -(33)); there being now four circles carrying the logarithms of numbers, and -so on. Next he points out "severall wayes how the Circles of the -Mathematicall Ring (being inlarged) may be accommodated for practicall -use:" (1) The Circles are all fixed in a plain and movable flat compasses -(or better, a movable semicircle) are used for fixing any two positions; -(2) There is a "double projection" of each logarithmic line "inlarged on a -Plaine," one fixed, the other movable, as shown in his first figure on the -title-page, a single index only being used; (3) use of "my great Cylinder -which I have long proposed (in which all the Circles are of equall -greatnesse,) and it may be made of any magnitude or capacity, but for a -study (hee that will be at the charge) it may be of a yard diameter and of -such an indifferent length that it may containe 100 or more Circles fixed -parallel one to the other on the Cylinder, having a space betweene each of -them, so that there may bee as many mooveable Circles, as there are fixed -ones, and these of the mooveable linked, or fastened together, so that they -may all moove together by the fixed ones in these spaces, whose edges both -of the fixed, and mooveable being graduated by helpe of a single Index will -shew the proportionalls by opposition in this double Projection, or by a -double Index in a single Projection" (p. (36)). - -Next follows the detailed description of his Ring "on a Plaine, according -to the diagramme that was given the King (for a view of that projection) -and afterwards the Ring it selve." The diagram is the large one which we -mentioned as inserted between pages (23) and (24). The instrument has two -circles, one moveable, upon each of which are described 13 distinct -circular graduations. The lines on the fixed circle are: "The Circle of -degrees and calendar," E. "Circle of equall parts, and part of the Equator, -and Meridian," TT. "The Circle of Tangents," S. "The Circle of Sines," D. -"The Circle of Decimals," N. "The Circle of Numbers." The lines on the -movable circle are: N. "The Circle of Numbers," E. "The Circle of equated -figures, and bodies," S. "The Circle of Sines," TT. "The Circle of -Tangents," Y. "The Circle of time, yeares, and monethes." - -On pages (84)-(88) Delamain explains an enlargement of his Ring for -computations involving the sines of angles near to 90^o. On page (86) he -says: - - I have continued the Sines of the Projection unto two severall - revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr. - (being the last revolution of the decuplation of the former, or the - hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s. - and ends at 90.gr. (being the last of a ternary of decuplated - revolutions, or the thousand part of that Projection) and may bee thus - used. - -He explains the manner of using these extra graduations. Thus he claims to -have attained degrees of accuracy which enabled him to do what "some one" -had declared "could not bee done." It is hardly necessary to point out that -Delamain's Grammelogia IV suggests designs of slide rules which inventors -two hundred or more years later were endeavouring to produce. Which of -Delamain's designs of rules were actually made and used, he does not state -explicitly. He refers to a rule 18 inches in diameter as if it had been -actually constructed (pages (86), (88)). Oughtred showed no appreciation of -such study in designing and ridiculed Delamain's efforts, in his Epistle. - -Additional elucidations of his designs of rules, along with explanations of -the relations of his work to that of Gunter and Napier, and sallies -directed against Oughtred and Forster, are contained on pages (8)-(21) of -his Grammelogia IV. - - - - - V. INDEPENDENCE AND PRIORITY OF INVENTION - - -The question of independence and priority of invention is discussed by -Delamain more specifically on pages (89)-(113); Oughtred devotes his entire -Epistle to it. It is difficult to determine definitely which publication is -the later, Delamain's Grammelogia IV or Oughtred's Epistle. Each seems to -quote from the other. Probably the explanation is that the two publications -contain arguments which were previously passed from one antagonist to the -other by word of mouth or by private letter. Oughtred refers in his Epistle -(p. (12)) to a letter from Delamain. We believe that the Epistle came after -Delamain's Grammelogia IV. Delamain claims for himself the invention of the -circular slide rule. He says in his Grammelogia IV. (p. (99)), "when I had -a sight of it, which was in February, 1629 (as I specified in my Epistle) I -could not conceale it longer, envying my selfe, that others did not tast of -that which I found to carry with it so delightfull and pleasant a goate -[taste] . . ." Delamain asserts (without proof) that Oughtred "never saw it -as he now challengeth it to be his invention, untill it was so fitted to -his hand, and that he made all his practise on it after the publishing of -my Booke upon my Ring, and not before; so it was easie for him or some -other to write some uses of it in Latin after Christmas, 1630 and not the -Sommer before, as is falsely alledged by some one . . ." (p. (91)). -Delamain's accusation of theft on the part of Oughtred cannot be seriously -considered. Oughtred's reputation as a mathematician and his standing in -his community go against such a supposition. Moreover, William Forster is a -witness for Oughtred. The fact that Oughtred had the mastery of the -rectilinear slide rule as well, while Delamain in 1630 speaks only of the -circular rule, weighs in Oughtred's favour. - -Oughtred says he invented the slide rule "above twelve yeares agoe," that -is, about 1621, and "I with mine owne hand made me two such Circles, which -I have used ever since, as my occasions required," (Epistle p. (22)). On -the same page, he describes his mode of discovery thus: - - I found that it required many times too great a paire of Compasses [in - using Gunter's line], which would bee hard to open, apt to slip, and - troublesome for use. I therefore first devised to have another Ruler with - the former: and so by setting and applying one to the other, I did not - onely take away the use of Compasses, but also make the worke much more - easy and expedite: when I should not at all need the motion of my hand, - but onely the glancing at my sight: and with one position of the Rulers, - and view of mine eye, see not one onely, but the manifold proportions - incident unto the question intended. But yet this facility also wanted - not some difficulty especially in the line of tangents, when one arch was - in the former mediety of the quadrant, and the other in the latter: for - in this case it was needful that either one Ruler must bee as long againe - as the other; or else that I must use an inversion of the Ruler, and - regression. By this consideration I first of all saw that if those lines - upon both Rulers were inflected into two circles, that of the tangents - being in both doubled, and that those two Circles should move one upon - another; they with a small thread in the center to direct the sight, - would bee sufficient with incredible and wonderfull facility to worke all - questions of Trigonometry . . . - -Oughtred said that he had no desire to publish his invention, but in the -vacation of 1630 finally promised William Forster to let him bring out a -translation. Oughtred claims that Delamain got the invention from him at -Alhallontide [November 1], 1630, when they met in London. The accounts of -that meeting we proceed to give in double column. - - Delamain's Statement - Grammelogia IV, page (98) - - ". . . about Alhalontide 1630. (as our Authors reporteth) was the time he - was circumvented, and then his intent in a loving manner (as before) he - opened unto me, which particularly I will dismantle in the very naked - truth: for, wee being walking together some few weekes before Christmas, - upon Fishstreet hill, we discoursed upon sundry things Mathematicall, - both Theoreticall and Practicall, and of the excellent inventions and - helpes that in these dayes were produced, amongst which I was not a - little taken with that of the Logarythmes, commending greatly the - ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, - in the lines of proportion, extracted from these Logarythmes for ordinary - Practicall uses; He replyed unto me (in these very words) What will yov - say to an Invention that I have, which in a lesse extent of the Compasses - shall worke truer then that of Mr. Gunters Ruler, I asked him then of - what forme it was, he answered with some pause (which no doubt argued his - suspition of mee that I might conceive it) that it was Arching-wise, but - now hee sayes that hee told mee then, it was Circular (but were I put to - my oath to avoid the guilt of Conscience I would conclude in the former.) - At which immediately I answered, I had the like my selfe, and so we - discoursed not a word more touching that subject . . . Then after my - coming home I sent him a sight of my Projection drawne in Pastboard: Now - admit I had not the Invention of my Ring before I discoursed . . . it was - not so facil for mee . . . to raise and compose so complete, and absolute - an Instrument from so small a principle, or glimpse of light . . ." - - Oughtred's Statement - Epistle, page (23) - - "Shortly after my gift to Elias Allen, I chanced to meet with Richard - Delamain in the street (it was at Alhallontide) and as we walked together - I told him what an Instrument I had given to Master Allen, both of the - Logarithmes projected into circles, which being lesse then one foot - diameter would performe as much as one of Master Gunters Rulers of sixe - feet long: and also of the Prostaphaereses of the Plannets and second - motions. Such an invention have I said he: for now his intentions (that - is his ambition) beganne to worke: . . . But he saith, Then after my - comming home I sent him a sight of my projection drawne in past-board. - See how notoriously he jugleth without an Instrument. Then after: how - long after? a sight of my projection: of how much? More then seven weekes - after on December 23, he sent to mee the line of numbers onely set upon a - circle: . . . and so much onely he presented to his Majesty: but as for - Sine or tangent of his, there was not the least shew of any. Neither - could he give to Master Allen any direction for the composure of the - circles of his Ring, or for the division of them: as upon his oath Master - Allen will testify how hee misled him, and made him labour in vain above - three weeks together, until Master Allen himselfe found out his ignorance - and mistaking, which is more cleare then is possible with any impudence - to be outfaced." - -Oughtred makes a further statement (Epistle, p. (24)) as follows: - - Delamain hearing that Brown with his Serpentine had another line by which - he could worke to minutes in the 90 degree of sines . . . gave the [his] - booke to Browne: who in thankfulnesse could not but gratify Delamain with - his Lines also: and teach him the use of them, but especially of the - great Line: with this caution on both sides, that one should not meddle - with the others invention. Two dayes after Delamain . . . because he had - found some things to be altered therin, . . . asked for the booke . . . - but as soone as he had got it in his hands he rent out all the middle - part with the two Schemes & put them up in his pocket & went his way . . - . and . . . laboureth to recall all the bookes he had given forth . . . - And shortly after this he got a new Printer (who was ignorant of his - former Schemes) to print him new: giving him an especiall charge of the - outermost line newly graven in the Plate, which indeed is Brownes very - line: and then altering his book . . . - -This and other statements made by Oughtred seem damaging to Delamain's -reputation. But it is quite possible that Oughtred's guesses as to -Delamain's motives are wrong. Moreover, some of Oughtred's statements are -not first hand knowledge with him, but mere hearsay. One may accept his -first hand facts and still clear Delamain of wrong doing. There is always -danger that rival claimants of an invention or discovery will proceed on -the assumption that no one else could possibly have come independently upon -the same devices that they themselves did; the history of science proves -the opposite. Seldom is an invention of any note made by only one man. We -do not feel competent to judge Delamain's case. We know too little about -him as a man. We incline to the opinion that the hypothesis of independent -invention is the most plausible. At any rate, Delamain figures in the -history of the slide rule as the publisher of the earliest book thereon and -as an enthusiastic and skillful designer of slide rules. - -The effect of this controversy upon interested friends was probably small. -Doubtless few people read both sides. Oughtred says:[21] "this scandall . . -. hath with them, to whom I am not knowne, wrought me much prejudice and -disadvantage . ." Aubrey,[22] a friend of Oughtred, refers to Delamain "who -was so sawcy to write against him" and remembers having seen "many yeares -since, twenty or more good verses made" against Delamain. Another friend of -Oughtred, William Robinson, who had seen some of Delamain's publications, -but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before -the appearance of the latter's Epistle: - - I cannot but wonder at the indiscretion of Rich. Delamain, who being - conscious to himself that he is but the pickpurse of another man's wit, - would thus inconsiderately provoke and awake a sleeping lion . . . he - hath so weakly (though in my judgment, vaingloriously enough) commended - his own labour . . .[23] - -Delamain presented King Charles I with one of his sun-dials, also with a -manuscript and, later, with a printed copy of his book of 1630. A drawing -of his improved slide rule was sent to the King and the Grammelogia IV is -dedicated to him. The King must have been favorably impressed, for Delamain -was appointed tutor to the King in mathematics. His widow petitioned the -House of Lords in 1645 for relief; he had ten children.[24] - -Anthony Wood states that Charles I, on the day of his execution, commanded -his friend Thomas Herbert "to give his son the duke of York his large -ring-sundial of silver, a jewel his maj. much valued." Anthony Wood adds, -"it was invented and made by Rich. Delamaine a very able mathematician, who -projected it, and in a little printed book did shew its excellent use in -resolving many questions in arithmetic and other rare operations to be -wrought by it in the mathematics."[25] - - - - - VI. OUGHTRED'S GAUGING LINE, 1633 - - -It has not been generally known, hitherto, that Oughtred designed a -rectilinear slide rule for gauging and published a description thereof in -1633.[26] In his Circles of Proportion, chapter IX, Oughtred had offered a -closer approximation than that of Gunter for the capacity of casks. The -Gauger of London expostulated with Oughtred for presuming to question -anything that Gunter had written. The ensuing discussion led to an -invitation extended by the Company of Vintners to the instrument maker -Elias Allen to request Oughtred to design a gauging rod.[27] This he did, -and Allen received an order for "threescore" instruments. On page 19 -Oughtred describes his 'Gauging Rod:' - - It consisteth of two rulers of brasse about 32 ynches of length, which - also are halfe an ynch broad, and a quarter of an ynch thick . . . At one - end of both those rulers are two little sockets of brasse fastened on - strongly: by which the rulers are held together, and made to move one - upon another, and to bee drawne out unto any length, as occasion shall - require: and when you have them at the just length, there is upon one of - the sockets a long Scrue-pin to scrue them fast. - -There are graduations on three sides of the rulers, one graduation being -the logarithmic line of numbers. He says (p. 39), "the maner of computing -the Gauge-divisions I have concealed." W. Robinson, who was a friend of -Oughtred, wrote him as follows:[28] - - I have light upon your little book of artificial gauging, wherewith I am - much taken, but I want the rod, neither could I get a sight of one of - them at the time, because Mr. Allen had none left . . . I forgot to ask - Mr. Allen the price of one of them, which if not much I would have one of - them." Oughtred annotated this passage thus: "Or in wood, if any be made - in wood by Thompson or any other." - -Another of Oughtred's admirers, Sir Charles Cavendish, wrote, on February -11, 1635 thus:[29] - - I thank you for your little book, but especially for the way of - calculating the divisions of your gauging rod. I wish, both for their own - sakes and yours, that the citizens were as capable of the acuteness of - this invention, as they are commonly greedy of gain, and then I doubt not - but they would give you a better recompense than I doubt now they will. - -On April 20, 1638, we find Oughtred giving Elias Allen directions[30] -"about the making of the two rulers." As in 1633,[31] so now, Oughtred -takes one ruler longer than the other. This 1633 instrument was used also -as "a crosse-staffe to take the height of the Sunne, or any Starre above -the Horizon, and also their distances." The longer ruler was called staffe, -the shorter transversarie. While in 1633 he took the lengths of the two in -the ratio "almost 3 to 2," in 1638, he took "the transversary three -quarters of the staff's length, . . . that the divisions may be larger." - - - - - VII. OTHER SEVENTEENTH CENTURY SLIDE RULES - - -In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard, -Henry Coggeshall, W. Hunt and Sir Isaac Newton.[32] Of Partridge's Double -Scale of Proportion, London, I have examined a copy dated 1661, which is -the earliest date for this book that I have seen. As far as we know, 1661 -is the earliest date of publications on the slide rule, since Oughtred and -Delamain. But it would not be surprising if the intervening 28 years were -found not so barren as they seem at present. The 1661 and 1662 impressions -of Partridge are identical, except for the date on the title-page. William -Leybourn, who printed Partridge's book, speaks in high appreciation of it -in his own book.[33] - -In 1661 was published also John Brown's first book, Description and Use of -a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use -of "Mr. Whites rule" for the measuring of board and timber, round and -square. He calls this a "sliding rule." The existence, in 1661, of a -"Whites rule" indicates activities in designing of which we know as yet -very little. In his book of 1761, previously quoted, Brown gives a drawing -of "White's sliding rule" (p. 193); also a special contrivance of his own, -as indicated by him in these words: - - A further improvement of the Triangular Quadrant, as I have made it - several times, with a sliding Cover on the in-side, when made hollow, to - carry Ink, Pens, and Compasses; then on the sliding Cover, and Edges, is - put the Line of Numbers, according to Mr. White's first Contrivance for - manner of operation; but much augmented, and made easie, by John Brown. - -He gives no drawing of his "triangular quadrant," hence his account of it -is unsatisfactory. He explains the use of "gage-points." His placing -logarithmic lines on the edges of instrument boxes was outdone in oddity -later by Everard who placed them on tobacco-boxes.[34] In Brown's -publication of 1704 the White slide rule is given again, "being as neat and -ready a way as ever was used." He tells also of a "glasier's sliding rule." -William Leybourn explains in 1673 how Wingate's double and triple lines for -squaring and cubing, or square and cube root, can be used on slide -rules.[35] - -Beginning early in the history of the slide rule, when Oughtred designed -his "gauging rod," we notice the designing of rules intended for very -special purposes. Another such contrivance, which enjoyed long popularity, -was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677, -a booklet of 35 pages. Coggeshall says in his preface: - - For what can be more ready and easie, then having set twelve to the - length, to see the Content exactly against the Girt or Side of the - Square. Whereas on Mr. Partridge's Scale the Content is the Sixth Number, - which is far more troublesome then [even] with Compasses. - -One line on Coggeshall's rule begins with 4 and extends to 40, these -numbers being the "Girt" (a quarter of the circumference), which in -ordinary practice of measuring round timber lies between 4 inches and 40 -inches. This "Girt line" slides "against the line of Numbers in two -Lengths, to which it is exactly equal." A second edition, 1682, shows some -changes in the rule, as well as an enlargement and change of title of the -book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent, -London, 1682. In this, the description of the rule is given thus: - - There are four Lines on each flat of this Rule; two next the outward - edges, which are Lines of Measure; and two next the inward edges, which - are Lines of Proportion. On one flat, next the inward edges, is the - Square-line [Girt-line in round timber measurement] with the Line of - Numbers his fellow. Next the outward, a Line of Inches divided into - Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 - to 24 on the other. On the other flat, next the inward edges, is the - double Scale of Numbers [for solving proportions]. Next the outward on - one Rule a Line of Inches divided each into ten parts; and this for - gauging, etc. On the other a foot divided into 100 parts. - -Later further changes were introduced in Coggeshall's rule.[36] - -It is worthy of note that Coggeshall's slide rule book, The Art of -Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p. -473; hence Leupold's description[37] of the rectilinear slide rule in his -Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not -the earliest reference to the rectilinear rule found in German -publications. The above date is earlier even than Biler's reference to a -circular slide rule in his Descriptio instrumenti mathematici universalis -of 1696. - -Two noted slide rules for gauging were described by Tho. Everard, -Philomath, in his Stereometry made easie, London, 1684. He designates his -lines by the capital letters A, B, C, D, E. On the first instrument, A on -the rule, and B and C on the slide, have each two radiuses of numbers, D -has only one, while E has three. The second rule is described in an -Appendix; it is one foot long, with two slides enabling the rule to be -extended to 3 feet. - -Everard's instruments were made in London by Isaac Carver who, soon after, -himself wrote a sixteen-page Description and Use of a New Sliding Rule, -projected from the Tables in the Gauger's Magazine, London, 1687, which was -"printed for William Hunt" and bound in one volume with a book by Hunt, -called The Gauger's Magazine, London, 1687. This appears to be the same -William Hunt who later brought out descriptions of his own of slide rules. -The instrument described by Carver "consists of three pieces, two whereof -are moveable to be drawn out till the whole be 36 inches long." It has -several non-logarithmic graduations, together with logarithmic lines marked -A, B, C, D, of which A, B, C are "double lines," and D a "single line" used -for squares and square roots. It is designed for the determination of the -vacuity of a "spheroidal cask lying," a "spheroidal cask standing," and a -"parabolical cask lying." - -Another seventeenth century writer on the slide rule is John Atkinson, whom -we have mentioned earlier. He says:[38] "The Lines of Numbers, Sines and -Tangents, are set double, that is, one on each side, as the middle piece -slides: which middle piece is so contrived, to slip to and fro easily, to -slide out, and to be put in any side uppermost, in order to bring those -Lines together (or against one another) most proper for solving the -Question, wrought by Sliding-Gunter." - -The data presented in this article show that, while the earliest slide -rules were of the circular type, the later slide rules of the seventeenth -century were of the rectilinear type.[39] - - January 12, 1915. - - - - - Footnotes - - -[1]F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments, - New York, 1909, pp. 7-14, also Addenda i-vi. - -[2]F. Cajori, "On the Invention of the Slide Rule," in Colorado College - Publication, Engineering Series Vol. 1, 1910. An abstract of this is - given in Nature (London), Vol. 82, 1909, p. 267. - -[3]F. Cajori, History etc., p. 14. - -[4]Art. "Slide Rule" in the Penny Cyclopaedia and in the English - Cyclopaedia [Arts and Sciences]. - -[5]Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817, - p. 423. - -[6]The full title of the book which Wingate published on this subject in - Paris is as follows: - - L'Vsage | de la | Reigle de | Proportion | en l'Arithmetique & | - Geometrie. | Par Edmond Vvingate, | Gentil-homme Anglois. | - - Ean es philomathes, ese polymathes. - - In tenui, sed no tenuis vsusve, laborne. | - - A Paris, | Chez Melchior Mondiere, | demeurant en l'Isle du Palais, | a - la | rue de Harlay aux deux Viperes. | M. DC. XXIV. | Auec Priuilege du - Roy. | - - Back of the title page is the announcement: - - Notez que la Reigle de Proportion en toutes facons se vend a Paris chez - Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces, - demeurant en l'Isle du Palais sur le Quay qui regarde la Megisserie a - l'Espic d'or. - -[7]The title-page of the edition of 1658 is as follows: - - The Use of the Rule of Proportion in Arithmetick & Geometrie. First - published at Paris in the French tongue, and dedicated to Monsieur, the - then king's onely Brother (now Duke of Orleance). By Edm. Wingate, an - English Gent. And now translated into English by the Author. Whereinto - is now also inserted the Construction of the same Rule, & a farther use - thereof . . . 2nd edition inlarged and amended. London, 1658. - -[8]Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; - Drawn up by himself by way of Diary. With Appendix of original Letters. - Publish'd by Charles Burman, Esq., London, 1717, p. 23. - -[9]Mathematical Tables, 1811, p. 36, and art. "Gunter's Line" in his Phil. - and Math. Dictionary, London, 1815. - -[10]To the English Gentrie, and all others studious of the Mathematicks, - which shall bee readers hereof. The just Apologie of Wil: Ovghtred, - against the slaunderous insimulations of Richard Delamain, in a Pamphlet - called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum - projectio circularis. We shall refer to this document as Epistle. It was - published without date in 32 unnumbered pages of fine print, and was - bound in with Oughtred's Circles of Proportion, in the editions of 1633 - and 1639. In the 1633 edition it is inserted at the end of the volume - just after the Addition vnto the Vse of the Instrument etc., and in that - of 1639 immediately after the preface. It was omitted from the Oxford - edition of 1660. The Epistle was also published separately. There is a - separate copy in the British Museum, London. Aubrey, in his Brief Lives, - edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, "He - writt a stitch't pamphlet about 163(?4) against . . . Delamaine." - -[11]Thomas Browne is mentioned by Stone in his Mathematical Instruments, - London 1723, p. 16. See also Cajori, History of the Slide Rule, New - York, 1909, p. 15. - -[12]The Description and Use of a Joynt-Rule: . . . also the use of Mr. - White's Rule for measuring of Board and Timber, round and square; With - the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and - Versed Sines. By J. Brown, Philom., London, 1661. - -[13]A Collection of Centers and Useful Proportions on the Line of Numbers, - by John Brown, 1662(?), 16 pages; Description and Use of the Triangular - Quadrant, by John Brown, London, 1671; Wingate's Rule of Proportion in - Arithmetick and Geometry: or Gunter's Line. Newly rectified by Mr. Brown - and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The - Description and Use of the Carpenter's-Rule: Together with the Use of - the Line of Numbers commonly call'd Gunter's-Line, by John Brown, - London, 1704. - -[14]William Leybourn, op. cit., pp. 129, 130, 132, 133. - -[15]James Atkinson's edition of Andrew Wakely's The Mariners Compass - Rectified, London, 1694 [Wakely's preface dated 1664, Atkinson's - preface, 1693]. Atkinson adds An Appendix containing Use of Instruments - most useful in Navigation. Our quotation is from this Appendix, p. 199. - -[16]R. Delamain, The Making, Description, and Use of a small portable - Instrument . . . called a Horizontall Quadrant, etc., London, 1631. - -[17]Oughtred's description of his circular slide rule of 1632 and his - rectilinear slide rule of 1633, as well as a drawing of the circular - slide rule, are reproduced in Cajori's History of the Slide Rule, - Addenda, pp. ii-vi. - -[18]The full title of the Grammelogia I is as follows: - - Grammelogia | or, | The Mathematicall Ring. | Shewing (any reasonable - Capacity that hath | not Arithmeticke) how to resolve and worke | all - ordinary operations of Arithmeticke. | And those which are most - difficult with greatest | facilitie: The extraction of Roots, the - valuation of | Leases, &c. The measuring of Plaines | and Solids. | With - the resolution of Plaine and Sphericall | Triangles. | And that onely by - an Ocular Inspection, | and a Circular Motion. | Naturae secreta tempus - aperit. | London printed by John Haviland, 1630. - -[19]Grammelogia III is the same as Grammelogia I, except for the addition - of an appendix, entitled: - - De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit - nullo (praescripto tempore) vitae | vsuram nobis ingeniique Deus. | - London, | - - . . . The next line or two of this title-page which probably contained - the date of publication, were cut off by the binder in trimming the - edges of this and several other pamphlets for binding into one volume. - -[20]Grammelogia IV has two title pages. The first is Mirifica Logarithmoru' - Projectio Circularis. There follows a diagram of a circular slide rule, - with the inscription within the innermost ring: Nil Finis, Motvs, - Circvlvs vllvs Habet. The second title page is as follows: - - Grammelogia | Or, the Mathematicall Ring. | Extracted from the - Logarythmes, and projected Circular: Now published in the | inlargement - thereof unto any magnitude fit for use: shewing any reason- | able - capacity that hath not Arithmeticke how to resolve and worke, | all - ordinary operations of Arithmeticke: | And those that are most difficult - with greatest facilitie, the extracti- | on of Rootes, the valuation of - Leases, &c. the measuring of Plaines and Solids, | with the resolution - of Plaine and Sphericall Triangles applied to the | Practicall parts of - Geometrie, Horologographie, Geographie | Fortification, Navigation, - Astronomie, &c. | And that onely by an ocular inspection, and a Circular - motion, Invented and first published, by R. Delamain, Teacher, and - Student of the Mathematicks. | Naturae secreta tempus aperit. | - - There is no date. There follows the diagram of a second circular slide - rule, with the inscription within the innermost ring: Typus proiectionis - Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630 - promisi. There are numerous drawings in the Grammelogia, all of which, - excepting the drawings of slide rules on the engraved title-pages of - Grammelogia IV and V, were printed upon separate pieces of paper and - then inserted by hand into the vacant spaces on the printed pages - reserved for them. Some drawings are missing, so that the Bodleian - Grammelogia IV differs in this respect slightly from the copy in the - British Museum and from the British Museum copy of Grammelogia V. - -[21]Epistle, p. (8). - -[22]Aubrey, op. cit., Vol. II., p. 111. - -[23]Rigaud, Correspondence of Scientific Men during the 17th Century, Vol. - I, Oxford, 1841, p. 11. - -[24]Dictionary of National Biography, Art. "Delamain, Richard." See also - Rev. Charles J. Robinson, Taylors' School, from A.D. 1562 to 1874, Vol. - I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b; - Sixth Report of the Royal Commission on Historical Manuscripts, Part I, - Report and Appendix, London, 1877. In this Appendix, p. 82, we read the - following: - - Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. - Petitioner's husband was servant to the King, and one of His Majesty's - engineers for the fortification of the kingdom, and his tutor in - mathematical arts; but upon the breaking out of the war he deserted the - Court, and was called by the State to several employments, in fortifying - the towns of Northampton, Newport, and Abingdon; and was also abroad - with the armies as Quartermaster-General of the Foot, and therein died. - Petitioner is left a disconsolate widow with ten children, the four - least of whom are now afflicted with sickness, and petitioner has - nothing left to support them. There are several considerable sums of - money due to the petitioner, as well from the King as the State. Prays - that she may have some relief amongst other widows. See L. J., VII. 6. - 657. - -[25]Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London, - 1820, p. 34. - -[26]The New Artificial Gauging Line or Rod: together with rules concerning - the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London, - 1633. The copy we have seen is in the Bodleian Library, Oxford. The book - is small sized and has 40 pages. - -[27]Oughtred, op. cit., p. 11. - -[28]S. J. Rigaud, Correspondence of Scientific Men of the 17th Century, - Oxford, Vol. I, 1841, p. 17. - -[29]Rigaud, loc. cit., p. 22. - -[30]Rigaud, loc. cit., pp. 30, 31. - -[31]Oughtred, An Addition vnto the Vse of the Instrument called the Circles - of Proportion, London, 1633, p. 63. - -[32]F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22, - Addenda, pp. vi-ix. - -[33]W. Leybourn, op. cit., 1673, Preface, and pp. 128-29. - -[34]Cajori op. cit., Addenda, p. ix. - -[35]William Leybourn, op. cit., 1673, p. 35. - -[36]See Cajori, op. cit., pp. 20, 28, Addenda, p. ix. - -[37]See F. Cajori, "A Note on the History of the Slide Rule," Bibliotheca - mathematica, 3 F., Vol. 10, pp. 161-163. - -[38]John Atkinson, op. cit., 1694, p. 204. - -[39]Probably the oldest slide rule now in existence is owned by St. John's - College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in - diameter. It was exhibited along with other instruments in May, 1919. - According to the Catalogue of a Loan Exhibition of Early Scientific - Instruments in Oxford, opened May 16, 1919, the instrument is inscribed - with the name of the maker ("Elias Allen fecit") and with the name of - the donor, Georgius Barkham. It is dated 1635, which is only three years - after the first publication of Oughtred's description of his circular - slide rule. It is stated in the Catalogue: "Unfortunately all the - movable parts but the base-plate and a couple of thumb-screws are - missing. The face of the instrument is engraved with Oughtred's - Horizontal Instrument. The back is engraved with eleven Circles of - Proportion as described in Arthur Haughton's book, a copy of which was - presented to St. John's College by George Barkham, to explain the use of - the instrument." As Arthur Haughton's Oxford edition of Oughtred's - Circles of Proportion did not appear until 1660, it would seem that the - instrument was probably not presented to the College before 1660. As far - as is known, the next oldest slide rule is of the year 1654, kept in the - South Kensington Museum, London, and is described in Nature of March 5, - 1914. It is a rectilinear rule, "of boxwood, well made, and bound - together with brass at the two ends. It is of the square type, a little - more than 2 ft. in length, and bears the logarithmic lines first - described by Edmund Gunter. Of these, the num, sin and tan lines are - arranged in pairs, identical and contiguous, one line in each pair being - on the fixed part, and the other on the slide." The instrument is - inscribed, "Made by Robert Bissaker for T. W., 1654." Nowhere else have - we seen reference to Robert Bissaker. His slide rule seems to antedate - the "Whites rule" mentioned above. [This foot-note was added on October - 15, 1919.] - - - - - UNIVERSITY OF CALIFORNIA PUBLICATIONS - - -Note.--The University of California Publications are offered in exchange -for the publication of learned societies and institutions, universities and -libraries. Complete lists of all the publications of the University will be -sent upon request. For sample copies, lists of publications or other -information, address the Manager of the University of California Press, -Berkeley, California, U. 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Current, Volume XXII. - - - - - Transcriber's Notes - - ---Inserted subchapter headings from the Table of Contents into the - text. - ---Corrected two page numbers in the Table of Contents - ---Corrected a Greek quotation in footnote 6 by reference to the - original sources. - - - - - - - -End of the Project Gutenberg EBook of On the History of Gunter's Scale and -the Slide Rule during the Seventeenth Century, by Florian Cajori - -*** END OF THIS PROJECT GUTENBERG EBOOK HISTORY OF GUNTER'S SCALE, SLIDE RULE *** - -***** This file should be named 42216.txt or 42216.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/4/2/2/1/42216/ - -Produced by Brenda Lewis, Stephen Hutcheson and the Online -Distributed Proofreading Canada Team at -http://www.pgdpcanada.net (This file was produced from -images generously made available by The Internet -Archive/American Libraries.) - - -Updated editions will replace the previous one--the old editions -will be renamed. - -Creating the works from public domain print editions means that no -one owns a United States copyright in these works, so the Foundation -(and you!) can copy and distribute it in the United States without -permission and without paying copyright royalties. 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