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diff --git a/22062-0.txt b/22062-0.txt new file mode 100644 index 0000000..b334bd6 --- /dev/null +++ b/22062-0.txt @@ -0,0 +1,4295 @@ +The Project Gutenberg eBook of The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara, by John Dee + +This eBook is for the use of anyone anywhere in the United States and +most other parts of the world 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. If you are not located in the United States, you +will have to check the laws of the country where you are located before +using this eBook. + +Title: The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara + +Author: John Dee + +Release Date: July 13, 2007 [eBook #22062] +[Most recently updated: June 28, 2021] + +Language: English + +Character set encoding: UTF-8 + +Produced by: Louise Hope, David Starner, Suzanne Lybarger and the Online Distributed Proofreading Team + +*** START OF THE PROJECT GUTENBERG EBOOK THE MATHEMATICALL PRAEFACE *** + + + + +[Transcriber’s Note: + +This e-text includes characters that will only display in UTF-8 +(Unicode) text readers, including a few words of Greek: + + Τακτικὴ [Taktikê] + ã ẽ õ ũ [overline or tilde to show following -n or -m] + ❧ ☞ [leaf symbol; pointing-finger symbol] + ‡ [double-ended dagger, used in size notations (below)] + +If any of these characters do not display properly-- in particular, if +the diacritic does not appear directly above the letter-- or if the +quotation marks in this paragraph appear as garbage, make sure your text +reader’s “character set” or “file encoding” is set to Unicode (UTF-8). +You may also need to change the default font. As a last resort, use the +ASCII version of this file instead. + +Some aspects of the original book had to be modified for all versions +of this plain-text file. + +Superscript letters are shown with ^: y^e, y^t. + +Marginal quotation marks are shown inline as “ and ”, approximating the +beginning and end of the marked passage. In the original text, no +quotation marks were printed inline. + +Paragraphs are broken up for sidenotes, with blank lines before and +after. Original paragraph breaks are shown as two blank lines. Brackets +within the body text are in the original. + +All sidenotes except the one beginning “This noble Earle” were printed +in italics; markup has been omitted to reduce visual clutter. + +At least four sizes of text were used, often in combination with +_italics_. The variants are shown here as: + + +‡very large‡+ + +larger+ + =smaller= + +Further errors and anomalies are listed at the end of the text, along +with those Euclid citations identified by number.] + + * * * * * + * * * * + * * * * * + + + THE ELEMENTS + OF GEOMETRIE + + of the most auncient + Philosopher + _EVCLIDE_ + of Megara. + + _Faithfully (now first) translated + into the Englishe toung, by + _H. Billingsley_, Citizen of London_. + + _Whereunto are annexed certaine + Scholies, Annotations, and Inuentions, + of the best Mathematiciens, + both of time past, and + in this our age._ + + + _With a very fruitfull Præface made by _M. I. Dee_, + specifying the chiefe Mathematicall Sciẽces, what they are, + and wherunto commodious: where, also, are disclosed + certaine new Secrets Mathematicall and Mechanicall, + vntill these our daies, greatly missed._ + + + Imprinted at London by _Iohn Daye_. + + + + +❧ The Translator to the Reader. + + +_There is (gentle Reader) nothing (the word of God onely set +apart) which so much beautifieth and adorneth the soule and +minde of mã, as doth the knowledge of good artes and sciences: +as the knowledge of naturall and morall Philosophie. The one +setteth before our eyes, the creatures of God, both in the +heauens aboue, and in the earth beneath: in which as in a +glasse, we beholde the exceding maiestie and wisedome of God, +in adorning and beautifying them as we see: in geuing vnto them +such wonderfull and manifolde proprieties, and naturall +workinges, and that so diuersly and in such varietie: farther in +maintaining and conseruing them continually, whereby to praise +and adore him, as by S. Paule we are taught. The other teacheth +vs rules and preceptes of vertue, how, in common life amongest +men, we ought to walke vprightly: what dueties pertaine to our +selues, what pertaine to the gouernment or good order both of an +housholde, and also of a citie or common wealth. The reading +likewise of histories, conduceth not a litle, to the adorning of +the soule & minde of man, a studie of all men cõmended: by it +are seene and knowen the artes and doinges of infinite wise men +gone before vs. In histories are contained infinite examples of +heroicall vertues to be of vs followed, and horrible examples of +vices to be of vs eschewed. Many other artes also there are +which beautifie the minde of man: but of all other none do more +garnishe & beautifie it, then those artes which are called +Mathematicall. Unto the knowledge of which no man can attaine, +without the perfecte knowledge and instruction of the +principles, groundes, and Elementes of Geometrie. But perfectly +to be instructed in them, requireth diligent studie and reading +of olde auncient authors. Amongest which, none for a beginner is +to be preferred before the most auncient Philosopher _Euclide_ +of _Megara_. For of all others he hath in a true methode and +iuste order, gathered together whatsoeuer any before him had of +these Elementes written: inuenting also and adding many thinges +of his owne: wherby he hath in due forme accomplished the arte: +first geuing definitions, principles, & groundes, wherof he +deduceth his Propositions or conclusions, in such wonderfull +wise, that that which goeth before, is of necessitie required to +the proufe of that which followeth. So that without the diligent +studie of _Euclides_ Elementes, it is impossible to attaine vnto +the perfecte knowledge of Geometrie, and consequently of any of +the other Mathematicall sciences. Wherefore considering the want +& lacke of such good authors hitherto in our Englishe tounge, +lamenting also the negligence, and lacke of zeale to their +countrey in those of our nation, to whom God hath geuen both +knowledge, & also abilitie to translate into our tounge, and +to publishe abroad such good authors, and bookes (the chiefe +instrumentes of all learninges): seing moreouer that many good +wittes both of gentlemen and of others of all degrees, much +desirous and studious of these artes, and seeking for them as +much as they can, sparing no paines, and yet frustrate of their +intent, by no meanes attaining to that which they seeke: I haue +for their sakes, with some charge & great trauaile, faithfully +translated into our vulgare toũge, & set abroad in Print, this +booke of _Euclide_. Whereunto I haue added easie and plaine +declarations and examples by figures, of the definitions. In +which booke also ye shall in due place finde manifolde +additions, Scholies, Annotations, and Inuentions: which I haue +gathered out of many of the most famous & chiefe Mathematiciẽs, +both of old time, and in our age: as by diligent reading it in +course, ye shall well perceaue. The fruite and gaine which I +require for these my paines and trauaile, shall be nothing els, +but onely that thou gentle reader, will gratefully accept the +same: and that thou mayest thereby receaue some profite: and +moreouer to excite and stirre vp others learned, to do the like, +& to take paines in that behalfe. By meanes wherof, our Englishe +tounge shall no lesse be enriched with good Authors, then are +other straunge tounges: as the Dutch, French, Italian, and +Spanishe: in which are red all good authors in a maner, found +amongest the Grekes or Latines. Which is the chiefest cause, +that amongest thẽ do florishe so many cunning and skilfull men, +in the inuentions of straunge and wonderfull thinges, as in +these our daies we see there do. Which fruite and gaine if I +attaine vnto, it shall encourage me hereafter, in such like sort +to translate, and set abroad some other good authors, both +pertaining to religion (as partly I haue already done) and also +pertaining to the Mathematicall Artes. Thus gentle +reader farewell._ (?¿) + + + [Decoration] + + + + + ❧ TO THE VNFAINED LOVERS + of truthe, and constant Studentes of Noble + _Sciences, _IOHN DEE_ of London, hartily_ + wisheth grace from heauen, and most prosperous + _successe in all their honest attemptes and_ + exercises. + + +Diuine _Plato_, the great Master of many worthy Philosophers, and the +constant auoucher, and pithy perswader of _Vnum_, _Bonum_, and _Ens_: in +his Schole and Academie, sundry times (besides his ordinary Scholers) +was visited of a certaine kinde of men, allured by the noble fame of +_Plato_, and the great commendation of hys profound and profitable +doctrine. But when such Hearers, after long harkening to him, perceaued, +that the drift of his discourses issued out, to conclude, this _Vnum_, +_Bonum_, and _Ens_, to be Spirituall, Infinite, Æternall, Omnipotent, +&c. Nothyng beyng alledged or expressed, How, worldly goods: how, +worldly dignitie: how, health, Strẽgth or lustines of body: nor yet the +meanes, how a merueilous sensible and bodyly blysse and felicitie +hereafter, might be atteyned: Straightway, the fantasies of those +hearers, were dampt: their opinion of _Plato_, was clene chaunged: yea +his doctrine was by them despised: and his schole, no more of them +visited. Which thing, his Scholer, _Aristotle_, narrowly cõsidering, +founde the cause therof, to be, “For that they had no forwarnyng and +information, in generall,” whereto his doctrine tended. For, so, might +they haue had occasion, either to haue forborne his schole hauntyng: (if +they, then, had misliked his Scope and purpose) or constantly to haue +continued therin: to their full satisfaction: if such his finall scope & +intent, had ben to their desire. Wherfore, _Aristotle_, euer, after +that, vsed in brief, to forewarne his owne Scholers and hearers, “both +of what matter, and also to what ende, he tooke in hand to speake, or +teach.” While I consider the diuerse trades of these two excellent +Philosophers (and am most sure, both, that _Plato_ right well, otherwise +could teach: and that _Aristotle_ mought boldely, with his hearers, haue +dealt in like sorte as _Plato_ did) I am in no little pang of +perplexitie: Bycause, that, which I mislike, is most easy for me to +performe (and to haue _Plato_ for my exãple.) And that, which I know to +be most commendable: and (in this first bringyng, into common handling, +the _Artes Mathematicall_) to be most necessary: is full of great +difficultie and sundry daungers. Yet, neither do I think it mete, for so +straunge matter (as now is ment to be published) and to so straunge an +audience, to be bluntly, at first, put forth, without a peculiar +Preface: Nor (Imitatyng _Aristotle_) well can I hope, that accordyng to +the amplenes and dignitie of the _State Mathematicall_, I am able, +either playnly to prescribe the materiall boundes: or precisely to +expresse the chief purposes, and most wonderfull applications therof. +And though I am sure, that such as did shrinke from _Plato_ his schole, +after they had perceiued his finall conclusion, would in these thinges +haue ben his most diligent hearers (so infinitely mought their desires, +in fine and at length, by our _Artes Mathematicall_ be satisfied) yet, +by this my Præface & forewarnyng, Aswell all such, may (to their great +behofe) the soner, hither be allured: as also the _Pythagoricall_, and +_Platonicall_ perfect scholer, and the constant profound Philosopher, +with more ease and spede, may (like the Bee,) gather, hereby, both wax +and hony. + + + [The intent of this Preface.] + +Wherfore, seyng I finde great occasion (for the causes alleged, and +farder, in respect of my _Art Mathematike generall_) to vse “a certaine +forewarnyng and Præface, whose content shalbe, that mighty, most +plesaunt, and frutefull _Mathematicall Tree_, with his chief armes and +second (grifted) braunches: Both, what euery one is, and also, what +commodity, in generall, is to be looked for, aswell of griff as stocke: +And forasmuch as this enterprise is so great, that, to this our tyme, it +neuer was (to my knowledge) by any achieued: And also it is most hard, +in these our drery dayes, to such rare and straunge Artes, to wyn due +and common credit:” Neuertheles, if, for my sincere endeuour to satisfie +your honest expectation, you will but lend me your thãkefull mynde a +while: and, to such matter as, for this time, my penne (with spede) is +hable to deliuer, apply your eye or eare attentifely: perchaunce, at +once, and for the first salutyng, this Preface you will finde a lesson +long enough. And either you will, for a second (by this) be made much +the apter: or shortly become, well hable your selues, of the lyons claw, +to coniecture his royall symmetrie, and farder propertie. Now then, +gentle, my frendes, and countrey men, Turne your eyes, and bend your +myndes to that doctrine, which for our present purpose, my simple talent +is hable to yeld you. + + +All thinges which are, & haue beyng, are found vnder a triple diuersitie +generall. For, either, they are demed Supernaturall, Naturall, or, of a +third being. Thinges Supernaturall, are immateriall, simple, +indiuisible, incorruptible, & vnchangeable. Things Naturall, are +materiall, compounded, diuisible, corruptible, and chaungeable. Thinges +Supernaturall, are, of the minde onely, comprehended: Things Naturall, +of the sense exterior, ar hable to be perceiued. In thinges Naturall, +probabilitie and coniecture hath place: But in things Supernaturall, +chief demõstration, & most sure Science is to be had. By which +properties & comparasons of these two, more easily may be described, the +state, condition, nature and property of those thinges, which, we before +termed of a third being: which, by a peculier name also, are called +_Thynges Mathematicall_. For, these, beyng (in a maner) middle, betwene +thinges supernaturall and naturall: are not so absolute and excellent, +as thinges supernatural: Nor yet so base and grosse, as things naturall: +But are thinges immateriall: and neuerthelesse, by materiall things +hable somewhat to be signified. And though their particular Images, by +Art, are aggregable and diuisible: yet the generall _Formes_, +notwithstandyng, are constant, vnchaungeable, vntrãsformable, and +incorruptible. Neither of the sense, can they, at any tyme, be perceiued +or iudged. Nor yet, for all that, in the royall mynde of man, first +conceiued. But, surmountyng the imperfectiõ of coniecture, weenyng and +opinion: and commyng short of high intellectuall cõceptiõ, are the +Mercurial fruite of _Dianœticall_ discourse, in perfect imagination +subsistyng. A meruaylous newtralitie haue these thinges _Mathematicall_, +and also a straunge participatiõ betwene thinges supernaturall, +immortall, intellectual, simple and indiuisible: and thynges naturall, +mortall, sensible, compounded and diuisible. Probabilitie and sensible +prose, may well serue in thinges naturall: and is commendable: In +Mathematicall reasoninges, a probable Argument, is nothyng regarded: nor +yet the testimony of sense, any whit credited: But onely a perfect +demonstration, of truthes certaine, necessary, and inuincible: +vniuersally and necessaryly concluded: is allowed as sufficient for “an +Argument exactly and purely Mathematical.” + + + [Note the worde, Vnit, to expresse the Greke Monas, + & not Vnitie: as we haue all, commonly, till now, vsed.] + +Of _Mathematicall_ thinges, are two principall kindes: namely, _Number_, +and _Magnitude_. + + [Number.] + +_Number_, we define, to be, a certayne Mathematicall Sũme, of _Vnits_. +And, an _Vnit_, is that thing Mathematicall, Indiuisible, by +participation of some likenes of whose property, any thing, which is in +deede, or is counted One, may resonably be called One. We account an +_Vnit_, a thing _Mathematicall_, though it be no Number, and also +indiuisible: because, of it, materially, Number doth consist: which, +principally, is a thing _Mathematicall_. + + [Magnitude.] + +_Magnitude_ is a thing _Mathematicall_, by participation of some likenes +of whose nature, any thing is iudged long, broade, or thicke. “A thicke +_Magnitude_ we call a _Solide_, or a _Body_. What _Magnitude_ so euer, +is Solide or Thicke, is also broade, & long. A broade magnitude, we call +a _Superficies_ or a Plaine. Euery playne magnitude, hath also length. +A long magnitude, we terme a _Line_. A _Line_ is neither thicke nor +broade, but onely long: Euery certayne Line, hath two endes: + + [A point.] + +The endes of a line, are _Pointes_ called. A _Point_, is a thing +_Mathematicall_, indiuisible, which may haue a certayne determined +situation.” If a Poynt moue from a determined situation, the way wherein +it moued, is also a _Line_: mathematically produced, whereupon, of the +auncient Mathematiciens, + + [A Line.] + +a _Line_ is called the race or course of a _Point_. A Poynt we define, +by the name of a thing Mathematicall: though it be no Magnitude, and +indiuisible: because it is the propre ende, and bound of a Line: which +is a true _Magnitude_. + + [Magnitude.] + +And _Magnitude_ we may define to be that thing _Mathematicall_, which is +diuisible for euer, in partes diuisible, long, broade or thicke. +Therefore though a Poynt be no _Magnitude_, yet _Terminatiuely_, we +recken it a thing _Mathematicall_ (as I sayd) by reason it is properly +the end, and bound of a line. Neither _Number_, nor _Magnitude_, haue +any Materialitie. First, we will consider of _Number_, and of the +Science _Mathematicall_, to it appropriate, called _Arithmetike_: and +afterward of _Magnitude_, and his Science, called _Geometrie_. But that +name contenteth me not: whereof a word or two hereafter shall be sayd. +How Immateriall and free from all matter, _Number_ is, who doth not +perceaue? yea, who doth not wonderfully wõder at it? For, neither pure +_Element_, nor _Aristoteles, Quinta Essentia_, is hable to serue for +Number, as his propre matter. Nor yet the puritie and simplenes of +Substance Spirituall or Angelicall, will be found propre enough thereto. +And therefore the great & godly Philosopher _Anitius Boetius_, sayd: +_Omnia quæcun[que] a primæua rerum natura constructa sunt, Numerorum +videntur ratione formata. Hoc enim fuit principale in animo Conditoris +Exemplar_. That is: +_All thinges (which from the very first originall +being of thinges, haue bene framed and made) do appeare to be Formed by +the reason of Numbers. For this was the principall example or patterne +in the minde of the Creator_.+ O comfortable allurement, O rauishing +perswasion, to deale with a Science, whose Subiect, is so Auncient, so +pure, so excellent, so surmounting all creatures, so vsed of the +Almighty and incomprehensible wisdome of the Creator, in the distinct +creation of all creatures: in all their distinct partes, properties, +natures, and vertues, by order, and most absolute number, brought, from +_Nothing_, to the _Formalitie_ of their being and state. By _Numbers_ +propertie therefore, of vs, by all possible meanes, (to the perfection +of the Science) learned, we may both winde and draw our selues into the +inward and deepe search and vew, of all creatures distinct vertues, +natures, properties, and _Formes_: And also, farder, arise, clime, +ascend, and mount vp (with Speculatiue winges) in spirit, to behold in +the Glas of Creation, the _Forme of Formes_, the _Exemplar Number_ of +all thinges _Numerable_: both visible and inuisible, mortall and +immortall, Corporall and Spirituall. Part of this profound and diuine +Science, had _Ioachim_ the Prophesier atteyned vnto: by _Numbers +Formall, Naturall_, and _Rationall_, forseyng, concludyng, and +forshewyng great particular euents, long before their comming. His +bookes yet remainyng, hereof, are good profe: And the noble Earle of +_Mirandula_, (besides that,) a sufficient witnesse: that _Ioachim, in +his prophesies, proceded by no other way, then by Numbers Formall_. And +this Earle hym selfe, in Rome, + + [Ano. 1488.] + +* set vp 900. Conclusions, in all kinde of Sciences, openly to be +disputed of: and among the rest, in his Conclusions _Mathematicall_, (in +the eleuenth Conclusion) hath in Latin, this English sentence. _By +Numbers, a way is had, to the searchyng out, and vnderstandyng of euery +thyng, hable to be knowen. For the verifying of which Conclusion, +I promise to aunswere to the 74. Questions, vnder written, by the way of +Numbers_. Which Cõclusions, I omit here to rehearse: aswell auoidyng +superfluous prolixitie: as, bycause _Ioannes Picus, workes_, are +commonly had. But, in any case, I would wish that those Conclusions were +red diligently, and perceiued of such, as are earnest Obseruers and +Considerers of the constant law of nũbers: which is planted in thyngs +Naturall and Supernaturall: and is prescribed to all Creatures, +inuiolably to be kept. For, so, besides many other thinges, in those +Conclusions to be marked, it would apeare, how sincerely, & within my +boundes, I disclose the wonderfull mysteries, by numbers, to be atteyned +vnto. + + +Of my former wordes, easy it is to be gathered, that _Number_ hath a +treble state: One, in the Creator: an other in euery Creature (in +respect of his complete constitution:) and the third, in Spirituall and +Angelicall Myndes, and in the Soule of mã. In the first and third state, +_Number_, is termed _Number Numbryng_. But in all Creatures, otherwise, +_Number_, is termed _Nũber Numbred_. And in our Soule, Nũber beareth +such a swaye, and hath such an affinitie therwith: that some of the old +_Philosophers_ taught, _Mans Soule, to be a Number mouyng it selfe_. And +in dede, in vs, though it be a very Accident: yet such an Accident it +is, that before all Creatures it had perfect beyng, in the Creator, +Sempiternally. _Number Numbryng_ therfore, is the discretion discerning, +and distincting of thinges. But in God the Creator, This discretion, in +the beginnyng, produced orderly and distinctly all thinges. For his +_Numbryng_, then, was his Creatyng of all thinges. And his Continuall +_Numbryng_, of all thinges, is the Conseruation of them in being: And, +where and when he will lacke an _Vnit_: there and then, that particular +thyng shalbe _Discreated_. Here I stay. But our Seuerallyng, +distinctyng, and _Numbryng_, createth nothyng: but of Multitude +considered, maketh certaine and distinct determination. And albeit these +thynges be waighty and truthes of great importance, yet (by the infinite +goodnes of the Almighty _Ternarie_,) Artificiall Methods and easy wayes +are made, by which the zelous Philosopher, may wyn nere this Riuerish +_Ida_, this Mountayne of Contemplation: and more then Contemplation. And +also, though _Number_, be a thyng so Immateriall, so diuine, and +æternall: yet by degrees, by litle and litle, stretchyng forth, and +applying some likenes of it, as first, to thinges Spirituall: and then, +bryngyng it lower, to thynges sensibly perceiued: as of a momentanye +sounde iterated: then to the least thynges that may be seen, numerable: +And at length, (most grossely,) to a multitude of any corporall thynges +seen, or felt: and so, of these grosse and sensible thynges, we are +trayned to learne a certaine Image or likenes of numbers: and to vse +Arte in them to our pleasure and proffit. So grosse is our conuersation, +and dull is our apprehension: while mortall Sense, in vs, ruleth the +common wealth of our litle world. Hereby we say, Three Lyons, are three: +or a _Ternarie_. Three Egles, are three, or a _Ternarie_. + + [☞] + +Which * _Ternaries_, are eche, the _Vnion_, _knot_, and _Vniformitie_, +of three discrete and distinct _Vnits_. That is, we may in eche +_Ternarie_, thrise, seuerally pointe, and shew a part, _One_, _One_, and +_One_. Where, in Numbryng, we say One, two, Three. But how farre, these +visible Ones, do differre from our Indiuisible Vnits (in pure +_Arithmetike_, principally considered) no man is ignorant. Yet from +these grosse and materiall thynges, may we be led vpward, by degrees, +so, informyng our rude Imagination, toward the cõceiuyng of _Numbers_, +absolutely (:Not supposing, nor admixtyng any thyng created, Corporall +or Spirituall, to support, conteyne, or represent those _Numbers_ +imagined:) that at length, we may be hable, to finde the number of our +owne name, gloriously exemplified and registred in the booke of the +_Trinitie_ most blessed and æternall. + + +But farder vnderstand, that vulgar Practisers, haue Numbers, otherwise, +in sundry Considerations: and extend their name farder, then to Numbers, +whose least part is an _Vnit_. For the common Logist, Reckenmaster, or +Arithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lesse +partes: and calleth them _Fractions_. As of an _Vnit_, he maketh an +halfe, and thus noteth it, ½. and so of other, (infinitely diuerse) +partes of an _Vnit_. Yea and farder, hath, _Fractions of Fractions. &c_. +And, forasmuch, as, _Addition_, _Substraction_, _Multiplication_, +_Diuision_ and _Extraction of Rotes_, are the chief, and sufficient +partes of _Arithmetike_: + + [Arithmetike.] + +which is, the _Science that demonstrateth the properties, of Numbers, +and all operatiõs, in numbers to be performed_: + + [Note.] + +“How often, therfore, these fiue sundry sortes of Operations, do, for +the most part, of their execution, differre from the fiue operations of +like generall property and name, in our Whole numbers practisable, So +often, (for a more distinct doctrine) we, vulgarly account and name it, +an other kynde of _Arithmetike_.” And by this reason: + + [1.] + +the Consideration, doctrine, and working, in whole numbers onely: where, +of an _Vnit_, is no lesse part to be allowed: is named (as it were) an +_Arithmetike_ by it selfe. And so of the _Arithmetike of Fractions_. + + [2.] + +In lyke sorte, the necessary, wonderfull and Secret doctrine of +Proportion, and proportionalytie hath purchased vnto it selfe a peculier +maner of handlyng and workyng: and so may seme an other forme of +_Arithmetike_. + + [3.] + +Moreouer, the _Astronomers_, for spede and more commodious calculation, +haue deuised a peculier maner of orderyng nũbers, about theyr circular +motions, by Sexagenes, and Sexagesmes. By Signes, Degrees and Minutes +&c. which commonly is called the _Arithmetike_ of _Astronomical_ or +_Phisicall Fractions_. That, haue I briefly noted, by the name of +_Arithmetike Circular_. Bycause it is also vsed in circles, not +_Astronomicall. &c._ + + [4.] + +Practise hath led _Numbers_ farder, and hath framed them, to take vpon +them, the shew of _Magnitudes_ propertie: Which is _Incommensurabilitie_ +and _Irrationalitie_. (For in pure _Arithmetike_, an _Vnit_, is the +common Measure of all Numbers.) And, here, Nũbers are become, as Lynes, +Playnes and Solides: some tymes _Rationall_, some tymes _Irrationall_. +And haue propre and peculier characters, (as ²√. ³√. and so of other. +Which is to signifie _Rote Square, Rote Cubik: and so forth_:) & propre +and peculier fashions in the fiue principall partes: Wherfore the +practiser, estemeth this, a diuerse _Arithmetike_ from the other. +Practise bryngeth in, here, diuerse compoundyng of Numbers: as some +tyme, two, three, foure (or more) _Radicall_ nũbers, diuersly knit, by +signes, of More & Lesse: as thus ²√12 + ³√15. Or thus ⁴√19 + ³√12 - ²√2. +&c. And some tyme with whole numbers, or fractions of whole Number, amõg +them: as 20 + ²√24. ³√16 + 33 - ²√10. ⁴√44 + 12¼ + ³√9. And so, +infinitely, may hap the varietie. After this: Both the one and the other +hath fractions incident: and so is this _Arithmetike_ greately enlarged, +by diuerse exhibityng and vse of Compositions and mixtynges. Consider +how, I (beyng desirous to deliuer the student from error and +Cauillation) do giue to this _Practise_, the name of the _Arithmetike of +Radicall numbers_: Not, of _Irrationall_ or _Surd Numbers_: which other +while, are Rationall: though they haue the Signe of a Rote before them, +which, _Arithmetike_ of whole Numbers most vsuall, would say they had no +such Roote: and so account them _Surd Numbers_: which, generally spokẽ, +is vntrue: as _Euclides_ tenth booke may teach you. Therfore to call +them, generally, _Radicall Numbers_, (by reason of the signe √. +prefixed,) is a sure way: and a sufficient generall distinction from all +other ordryng and vsing of Numbers: And yet (beside all this) Consider: +the infinite desire of knowledge, and incredible power of mans Search +and Capacitye: how, they, ioyntly haue waded farder (by mixtyng of +speculation and practise) and haue found out, and atteyned to the very +chief perfection (almost) of _Numbers_ Practicall vse. Which thing, is +well to be perceiued in that great Arithmeticall Arte of _Æquation_: +commonly called the _Rule of Coss._ or _Algebra_. The Latines termed it, +_Regulam Rei & Census_, that is, the +_Rule of the thyng and his +value_+. With an apt name: comprehendyng the first and last pointes of +the worke. And the vulgar names, both in Italian, Frenche and Spanish, +depend (in namyng it,) vpon the signification of the Latin word, _Res_: ++_A thing_+: vnleast they vse the name of _Algebra_. And therin +(commonly) is a dubble error. The one, of them, which thinke it to be of +_Geber_ his inuentyng: the other of such as call it _Algebra_. For, +first, though _Geber_ for his great skill in Numbers, Geometry, +Astronomy, and other maruailous Artes, mought haue semed hable to haue +first deuised the sayd Rule: and also the name carryeth with it a very +nere likenes of _Geber_ his name: yet true it is, that a _Greke_ +Philosopher and Mathematicien, named _Diophantus_, before _Geber_ his +tyme, wrote 13. bookes therof (of which, six are yet extant: and I had +them to *vse, + + [* Anno. 1550.] + +of the famous Mathematicien, and my great frende, _Petrus Montaureus_:) +And secondly, the very name, is _Algiebar_, and not _Algebra_: as by the +Arabien _Auicen_, may be proued: who hath these precise wordes in +Latine, by _Andreas Alpagus_ (most perfect in the Arabik tung) so +translated. _Scientia faciendi Algiebar & Almachabel. i. Scientia +inueniendi numerum ignotum, per additionem Numeri, & diuisionem & +æquationem_. Which is to say: +_The Science of workyng Algiebar and +Almachabel_+, that is, the +_Science of findyng an vnknowen number, by +Addyng of a Number, & Diuision & æquation_+. Here haue you the name: and +also the principall partes of the Rule, touched. To name it, _The rule, +or Art of Æquation_, doth signifie the middle part and the State of the +Rule. This Rule, hath his peculier Characters: + + [5.] + +and the principal partes of _Arithmetike_, to it appertayning, do +differre from the other _Arithmeticall operations_. This _Arithmetike, +hath Nũbers_ Simple, Cõpound, Mixt: and Fractions, accordingly. This +Rule, and _Arithmetike of Algiebar_, is so profound, so generall and so +(in maner) conteyneth the whole power of Numbers Application practicall: +that mans witt, can deale with nothyng, more proffitable about numbers: +nor match, with a thyng, more mete for the diuine force of the Soule, +(in humane Studies, affaires, or exercises) to be tryed in. Perchaunce +you looked for, (long ere now,) to haue had some particular profe, or +euident testimony of the vse, proffit and Commodity of Arithmetike +vulgar, in the Common lyfe and trade of men. Therto, then, I will now +frame my selfe: But herein great care I haue, least length of sundry +profes, might make you deme, that either I did misdoute your zelous +mynde to vertues schole: or els mistrust your hable witts, by some, to +gesse much more. A profe then, foure, fiue, or six, such, will I bryng, +as any reasonable man, therwith may be persuaded, to loue & honor, yea +learne and exercise the excellent Science of _Arithmetike_. + + +And first: who, nerer at hand, can be a better witnesse of the frute +receiued by _Arithmetike_, then all kynde of Marchants? Though not all, +alike, either nede it, or vse it. How could they forbeare the vse and +helpe of the Rule, called the Golden Rule? Simple and Compounde: both +forward and backward? How might they misse _Arithmeticall_ helpe in the +Rules of Felowshyp: either without tyme, or with tyme? and betwene the +Marchant & his Factor? The Rules of Bartering in wares onely: or part in +wares, and part in money, would they gladly want? Our Marchant +venturers, and Trauaylers ouer Sea, how could they order their doynges +iustly and without losse, vnleast certaine and generall Rules for +Exchaũge of money, and Rechaunge, were, for their vse, deuised? The Rule +of Alligation, in how sundry cases, doth it conclude for them, such +precise verities, as neither by naturall witt, nor other experience, +they, were hable, els, to know? And (with the Marchant then to make an +end) how ample & wonderfull is the Rule of False positions? especially +as it is now, by two excellent Mathematiciens (of my familier +acquayntance in their life time) enlarged? I meane _Gemma Frisius_, and +_Simon Iacob_. Who can either in brief conclude, the generall and +Capitall Rules? or who can Imagine the Myriades of sundry Cases, and +particular examples, in Act and earnest, continually wrought, tried and +concluded by the forenamed Rules, onely? How sundry other _Arithmeticall +practises_, are commonly in Marchantes handes, and knowledge: They them +selues, can, at large, testifie. + + +The Mintmaster, and Goldsmith, in their Mixture of Metals, either of +diuerse kindes, or diuerse values: how are they, or may they, exactly be +directed, and meruailously pleasured, if _Arithmetike_ be their guide? +And the honorable Phisiciãs, will gladly confesse them selues, much +beholding to the Science of _Arithmetike_, and that sundry wayes: But +chiefly in their Art of Graduation, and compounde Medicines. And though +_Galenus_, _Auerrois_, _Arnoldus_, _Lullus_, and other haue published +their positions, aswell in the quantities of the Degrees aboue +Temperament, as in the Rules, concluding the new _Forme_ resulting: yet +a more precise, commodious, and easy _Method_, is extant: by a +Countreyman of ours + + [R. B.] + +(aboue 200. yeares ago) inuented. And forasmuch as I am vncertaine, who +hath the same: or when that litle Latin treatise, (as the Author writ +it,) shall come to be Printed: (Both to declare the desire I haue to +pleasure my Countrey, wherin I may: and also, for very good profe of +Numbers vse, in this most subtile and frutefull, Philosophicall +Conclusion,) I entend in the meane while, most briefly, and with my +farder helpe, to communicate the pith therof vnto you. + + +First describe a circle: whose diameter let be an inch. Diuide the +Circumference into foure equall partes. Frõ the Center, by those 4. +sections, extend 4. right lines: eche of 4. inches and a halfe long: or +of as many as you liste, aboue 4. without the circumference of the +circle: So that they shall be of 4. inches long (at the least) without +the Circle. Make good euident markes, at euery inches end. If you list, +you may subdiuide the inches againe into 10. or 12. smaller partes, +equall. At the endes of the lines, write the names of the 4. principall +elementall Qualities. _Hote_ and _Colde_, one against the other. And +likewise _Moyst_ and _Dry_, one against the other. And in the Circle +write _Temperate_. Which _Temperature_ hath a good Latitude: as +appeareth by the Complexion of man. And therefore we haue allowed vnto +it, the foresayd Circle: and not a point Mathematicall or Physicall. + + + [* Take some part of Lullus counsayle in his booke + de Q. Essentia.] + +Now, when you haue two thinges Miscible, whose degrees are * truely +knowen: Of necessitie, either they are of one Quantitie and waight, or +of diuerse. If they be of one Quantitie and waight: whether their +formes, be Contrary Qualities, or of one kinde (but of diuerse +intentions and degrees) or a _Temperate_, and a Contrary, _The forme +resulting of their Mixture, is in the Middle betwene the degrees of the +formes mixt_. As for example, let _A_, be _Moist_ in the first degree: +and _B_, _Dry_ in the third degree. Adde 1. and 3. that maketh 4: the +halfe or middle of 4. is 2. This 2. is the middle, equally distant from +_A_ and _B_ + + [* Note.] + +(for the * _Temperament_ is counted none. And for it, you must put a +Ciphre, if at any time, it be in mixture). + + HOTE + +C + | + | + + + | + | + + + | + | + +E + | + MOIST A TEMPERATE B DRYE + +------+------+------+------+------+------+------+------+ + |D + | + + + | + | + + + | + | + + + | + | + + + COLD + +Counting then from _B_, 2. degrees, toward _A_: you finde it to be _Dry_ +in the first degree: So is the _Forme resulting_ of the Mixture of _A_, +and _B_, in our example. I will geue you an other example. Suppose, you +haue two thinges, as _C_, and _D_: and of _C_, the Heate to be in the 4. +degree: and of _D_, the Colde, to be remisse, euen vnto the +_Temperament_. Now, for _C_, you take 4: and for _D_, you take a Ciphre: +which, added vnto 4, yeldeth onely 4. The middle, or halfe, whereof, is +2. Wherefore the _Forme resulting_ of _C_, and _D_, is Hote in the +second degree: for, 2. degrees, accounted from _C_, toward _D_, ende +iuste in the 2. degree of heate. Of the third maner, I will geue also an +example: which let be this: + + [Note.] + +I haue a liquid Medicine whose Qualitie of heate is in the 4. degree +exalted: as was _C_, in the example foregoing: and an other liquid +Medicine I haue: whose Qualitie, is heate, in the first degree. Of eche +of these, I mixt a like quantitie: Subtract here, the lesse frõ the +more: and the residue diuide into two equall partes: whereof, the one +part, either added to the lesse, or subtracted from the higher degree, +doth produce the degree of the Forme resulting, by this mixture of _C_, +and _E_. As, if from 4. ye abate 1. there resteth 3. the halfe of 3. is +1½: Adde to 1. this 1½: you haue 2½. Or subtract from 4. this 1½: you +haue likewise 2½ remayning. Which declareth, the _Forme resulting_, to +be _Heate_, in the middle of the third degree. + + + [The Second Rule.] + +“But if the Quantities of two thinges Commixt, be diuerse, and the +Intensions (of their Formes Miscible) be in diuerse degrees, and +heigthes. (Whether those Formes be of one kinde, or of Contrary kindes, +or of a Temperate and a Contrary, _What proportion is of the lesse +quantitie to the greater, the same shall be of the difference, which is +betwene the degree of the Forme resulting, and the degree of the greater +quantitie of the thing miscible, to the difference, which is betwene the +same degree of the Forme resulting, and the degree of the lesse +quantitie_. As for example. Let two pound of Liquor be geuen, hote in +the 4. degree: & one pound of Liquor be geuen, hote in the third +degree.” I would gladly know the Forme resulting, in the Mixture of +these two Liquors. Set downe your nũbers in order, thus. + ___________________________ + | | | + | {P}. _2._ | _Hote. 4._ | + | | | + | {P}. _1._ | _Hote. 3._ | + |____________|______________| + +Now by the rule of Algiebar, haue I deuised a very easie, briefe, and +generall maner of working in this case. Let vs first, suppose that +_Middle Forme resulting_, to be 1{x}: as that Rule teacheth. And because +(by our Rule, here geuen) as the waight of 1. is to 2: So is the +difference betwene 4. (the degree of the greater quantitie) and 1{x}: to +the difference betwene 1{x} and 3: (the degree of the thing, in lesse +quãtitie. And with all, 1{x}, being alwayes in a certaine middell, +betwene the two heigthes or degrees). For the first difference, I set +4-1{x}: and for the second, I set 1{x}-3. And, now againe, I say, as 1. +is to 2. so is 4-1{x} to 1{x}-3. Wherfore, of these foure proportionall +numbers, the first and the fourth Multiplied, one by the other, do make +as much, as the second and the third Multiplied the one by the other. +Let these Multiplications be made accordingly. And of the first and the +fourth, we haue 1{x}-3. and of the second & the third, 8-2{x}. Wherfore, +our Æquation is betwene 1{x}-3: and 8-2{x}. Which may be reduced, +according to the Arte of Algiebar: as, here, adding 3. to eche part, +geueth the Æquation, thus, 1{x}=11-2{x}. And yet againe, contracting, or +Reducing it: Adde to eche part, 2{x}: Then haue you 3{x} æquall to 11: +thus represented 3{x}=11. Wherefore, diuiding 11. by 3: the Quotient is +3⅔: the _Valew_ of our 1{x}, _Coss_, or _Thing_, first supposed. And +that is the heigth, or Intension of the _Forme resulting:_ which is, +_Heate_, in two thirdes of the fourth degree: And here I set the shew of +the worke in conclusion, thus. The proufe hereof is easie: by +subtracting 3. from 3⅔, resteth ⅔. Subtracte the same heigth of the +Forme resulting, (which is 3⅔) frõ 4: then resteth ⅓: You see, that ⅔ is +double to ⅓: as 2.{P}. is double to 1.{P}. So should it be: by the rule +here geuen. Note. As you added to eche part of the Æquation, 3: so if ye +first added to eche part 2{x}, it would stand, 3{x}-3=8. And now adding +to eche part 3: you haue (as afore) 3{x}=11. + _________________________ + | | | _ + | {P}. _2._ | _Hote. 4._ | ⅓ _ _The forme_ + | | | _ _3⅔ resulting._ + | {P}. _1._ | _Hote. 3._ | _ ⅔ + |___________|_____________| + + +And though I, here, speake onely of two thyngs Miscible: and most +commonly mo then three, foure, fiue or six, (&c.) are to be Mixed: (and +in one Compound to be reduced: & the Forme resultyng of the same, to +serue the turne) yet these Rules are sufficient: duely repeated and +iterated. + + [Note.] + +In procedyng first, with any two: and then, with the Forme Resulting, +and an other: & so forth: For, the last worke, concludeth the Forme +resultyng of them all: I nede nothing to speake, of the Mixture (here +supposed) what it is. Common Philosophie hath defined it, saying, +_Mixtio est miscibilium, alteratorum, per minima coniunctorum, Vnio_. +Euery word in the definition, is of great importance. I nede not also +spend any time, to shew, how, the other manner of distributing of +degrees, doth agree to these Rules. Neither nede I of the farder vse +belonging to the Crosse of Graduation (before described) in this place +declare, vnto such as are capable of that, which I haue all ready sayd. +Neither yet with examples specifie the Manifold varieties, by the +foresayd two generall Rules, to be ordered. The witty and Studious, +here, haue sufficient: And they which are not hable to atteine to this, +without liuely teaching, and more in particular: would haue larger +discoursing, then is mete in this place to be dealt withall: And other +(perchaunce) with a proude snuffe will disdaine this litle: and would be +vnthankefull for much more. I, therfore conclude: and wish such as haue +modest and earnest Philosophicall mindes, to laude God highly for this: +and to Meruayle, that the profoundest and subtilest point, concerning +_Mixture of Formes and Qualities Naturall_, is so Matcht and maryed with +the most simple, easie, and short way of the noble Rule of _Algiebar_. +Who can remaine, therfore vnpersuaded, to loue, alow, and honor the +excellent Science of _Arithmetike_? For, here, you may perceiue that the +litle finger of _Arithmetike_, is of more might and contriuing, then a +hunderd thousand mens wittes, of the middle sorte, are hable to +perfourme, or truely to conclude, with out helpe thereof. + + +Now will we farder, by the wise and valiant Capitaine, be certified, +what helpe he hath, by the Rules of _Arithmetike_: in one of the Artes +to him appertaining: And of the Grekes named + + [Τακτικὴ.] + +Τακτικὴ. “That is, the Skill of Ordring Souldiers in Battell ray after +the best maner to all purposes.” This Art so much dependeth vppon +Numbers vse, and the Mathematicals, that _Ælianus_ (the best writer +therof,) in his worke, to the _Emperour Hadrianus_, by his perfection, +in the Mathematicals, (beyng greater, then other before him had,) +thinketh his booke to passe all other the excellent workes, written of +that Art, vnto his dayes. For, of it, had written _Æneas_: _Cyneas_ of +_Thessaly_: _Pyrrhus Epirota_: and _Alexander_ his sonne: _Clearchus_: +_Pausanias_: _Euangelus_: _Polybius_, familier frende to _Scipio_: +_Eupolemus_: _Iphicrates_, _Possidonius_: and very many other worthy +Capitaines, Philosophers and Princes of Immortall fame and memory: Whose +fayrest floure of their garland (in this feat) was _Arithmetike_: and a +litle perceiuerance, in _Geometricall_ Figures. But in many other cases +doth _Arithmetike_ stand the Capitaine in great stede. As in +proportionyng of vittayles, for the Army, either remaining at a stay: or +suddenly to be encreased with a certaine number of Souldiers: and for a +certain tyme. Or by good Art to diminish his company, to make the +victuals, longer to serue the remanent, & for a certaine determined +tyme: if nede so require. And so in sundry his other accountes, +Reckeninges, Measurynges, and proportionynges, the wise, expert, and +Circumspect Capitaine will affirme the Science of _Arithmetike_, to be +one of his chief Counsaylors, directers and aiders. Which thing (by good +meanes) was euident to the Noble, the Couragious, the loyall, and +Curteous + + [☞] + +_Iohn_, late Earle of Warwicke. Who was a yong Gentleman, throughly +knowne to very few. Albeit his lusty valiantnes, force, and Skill in +Chiualrous feates and exercises: his humblenes, and frendelynes to all +men, were thinges, openly, of the world perceiued. But what rotes +(otherwise,) vertue had fastened in his brest, what Rules of godly and +honorable life he had framed to him selfe: what vices, (in some then +liuing) notable, he tooke great care to eschew: what manly vertues, in +other noble men, (florishing before his eyes,) he Sythingly aspired +after: what prowesses he purposed and ment to achieue: with what feats +and Artes, he began to furnish and fraught him selfe, for the better +seruice of his Kyng and Countrey, both in peace & warre. These (I say) +his Heroicall Meditations, forecastinges and determinations, no twayne, +(I thinke) beside my selfe, can so perfectly, and truely report. And +therfore, in Conscience, I count it my part, for the honor, preferment, +& procuring of vertue (thus, briefly) to haue put his Name, in the +Register of _Fame Immortall_. + + +To our purpose. This _Iohn_, by one of his actes (besides many other: +both in England and Fraunce, by me, in him noted.) did disclose his +harty loue to vertuous Sciences: and his noble intent, to excell in +Martiall prowesse: When he, with humble request, and instant +Solliciting: got the best Rules (either in time past by Greke or +Romaine, or in our time vsed: and new Stratagemes therin deuised) for +ordring of all Companies, summes and Numbers of mẽ, (Many, or few) with +one kinde of weapon, or mo, appointed: with Artillery, or without: on +horsebacke, or on fote: to giue, or take onset: to seem many, being few: +to seem few, being many. To marche in battaile or Iornay: with many such +feates, to Foughten field, Skarmoush, or Ambushe appartaining: + + [This noble Earle, dyed Anno. 1554. skarse of 24. yeares + of age: hauing no issue by his wife: Daughter to the Duke + of Somerset.] + +And of all these, liuely designementes (most curiously) to be in velame +parchement described: with Notes & peculier markes, as the Arte +requireth: and all these Rules, and descriptions Arithmeticall, inclosed +in a riche Case of Gold, he vsed to weare about his necke: as his Iuell +most precious, and Counsaylour most trusty. Thus, _Arithmetike_, of him, +was shryned in gold: Of _Numbers_ frute, he had good hope. Now, Numbers +therfore innumerable, in _Numbers_ prayse, his shryne shall finde. + + +What nede I, (for farder profe to you) of the Scholemasters of Iustice, +to require testimony: how nedefull, how frutefull, how skillfull a thing +_Arithmetike_ is? I meane, the Lawyers of all sortes. Vndoubtedly, the +Ciuilians, can meruaylously declare: how, neither the Auncient Romaine +lawes, without good knowledge of _Numbers art_, can be perceiued: Nor +(Iustice in infinite Cases) without due proportion, (narrowly +considered,) is hable to be executed. How Iustly, & with great knowledge +of Arte, did _Papinianus_ institute a law of partition, and allowance, +betwene man and wife after a diuorce? But how _Accursius_, _Baldus_, +_Bartolus_, _Iason_, _Alexander_, and finally _Alciatus_, (being +otherwise, notably well learned) do iumble, gesse, and erre, from the +æquity, art and Intent of the lawmaker: _Arithmetike_ can detect, and +conuince: and clerely, make the truth to shine. Good _Bartolus_, tyred +in the examining & proportioning of the matter: and with _Accursius_ +Glosse, much cumbred: burst out, and sayd: _Nulla est in toto libro, hac +glossa difficilior: Cuius computationem nec Scholastici nec Doctores +intelligunt. &c._ That is: +_In the whole booke, there is no Glosse +harder then this: Whose accoumpt or reckenyng, neither the Scholers, nor +the Doctours vnderstand. &c._+ What can they say of _Iulianus_ law, _Si +ita Scriptum. &c._ Of the Testators will iustly performing, betwene the +wife, Sonne and daughter? How can they perceiue the æquitie of +_Aphricanus_, _Arithmeticall_ Reckening, where he treateth of _Lex +Falcidia_? How can they deliuer him, from his Reprouers: and their +maintainers: as _Ioannes_, _Accursius Hypolitus_ and _Alciatus_? How +Iustly and artificially, was _Africanus_ reckening made? Proportionating +to the Sommes bequeathed, the Contributions of eche part? Namely, for +the hundred presently receiued, 17-1/7. And for the hundred, receiued +after ten monethes, 12-6/7: which make the 30: which were to be +cõtributed by the legataries to the heire. For, what proportion, 100 +hath to 75: the same hath 17-1/7 to 12-6/7: Which is Sesquitertia: that +is, as 4, to 3. which make 7. Wonderfull many places, in the Ciuile law, +require an expert _Arithmeticien_, to vnderstand the deepe Iudgemẽt, & +Iust determinatiõ of the Auncient Romaine Lawmakers. But much more +expert ought he to be, who should be hable, to decide with æquitie, the +infinite varietie of Cases, which do, or may happen, vnder euery one of +those lawes and ordinances Ciuile. Hereby, easely, ye may now +coniecture: that in the Canon law: and in the lawes of the Realme (which +with vs, beare the chief Authoritie), Iustice and equity might be +greately preferred, and skilfully executed, through due skill of +Arithmetike, and proportions appertainyng. The worthy Philosophers, and +prudent lawmakers (who haue written many bookes _De Republica:_ How the +best state of Common wealthes might be procured and mainteined,) haue +very well determined of Iustice: (which, not onely, is the Base and +foundacion of Common weales: but also the totall perfection of all our +workes, words, and thoughtes:) defining it, + + [Iustice.] + +“to be that vertue, by which, to euery one, is rendred, that to him +appertaineth.” God challengeth this at our handes, to be honored as God: +to be loued, as a father: to be feared as a Lord & master. Our +neighbours proportiõ, is also prescribed of the Almighty lawmaker: which +is, to do to other, euen as we would be done vnto. These proportions, +are in Iustice necessary: in duety, commendable: and of Common wealthes, +the life, strength, stay and florishing. _Aristotle_ in his _Ethikes_ +(to fatch the sede of Iustice, and light of direction, to vse and +execute the same) was fayne to fly to the perfection, and power of +Numbers: for proportions Arithmeticall and Geometricall. _Plato_ in his +booke called _Epinomis_ (which boke, is the Threasury of all his +doctrine) where, his purpose is, to seke a Science, which, when a man +had it, perfectly: he might seme, and so be, in dede, _Wise_. He, +briefly, of other Sciences discoursing, findeth them, not hable to bring +it to passe: But of the Science of Numbers, he sayth. _Illa, quæ numerum +mortalium generi dedit, id profecto efficiet. Deum autem aliquem, magis +quam fortunam, ad salutem nostram, hoc munus nobis arbitror contulisse. +&c. Nam ipsum bonorum omnium Authorem, cur non maximi boni, Prudentiæ +dico, causam arbitramur? +That Science, verely, which hath taught +mankynde number, shall be able to bryng it to passe. And, I thinke, +a certaine God, rather then fortune, to haue giuen vs this gift, for our +blisse. For, why should we not Iudge him, who is the Author of all good +things, to be also the cause of the greatest good thyng, namely, +Wisedome?+_ There, at length, he proueth _Wisedome_ to be atteyned, by +good Skill of _Numbers_. With which great Testimony, and the manifold +profes, and reasons, before expressed, you may be sufficiently and fully +persuaded: of the perfect Science of _Arithmetike_, to make this +accounte: That + + [☞] + +of all Sciences, next to _Theologie_, it is most diuine, most pure, most +ample and generall, most profounde, most subtile, most commodious and +most necessary. Whose next Sister, is the Absolute Science of +_Magnitudes_: of which (by the Direction and aide of him, whose +_Magnitude_ is Infinite, and of vs Incomprehensible) I now entend, so to +write, that both with the _Multitude_, and also with the _Magnitude_ of +Meruaylous and frutefull verities, you (my frendes and Countreymen) may +be stird vp, and awaked, to behold what certaine Artes and Sciences, (to +our vnspeakable behofe) our heauenly father, hath for vs prepared, and +reuealed, by sundry _Philosophers_ and _Mathematiciens_. + + +Both, _Number_ and _Magnitude_, haue a certaine Originall sede, (as it +were) of an incredible property: and of man, neuer hable, Fully, to be +declared. Of _Number_, an Vnit, and of _Magnitude_, a Poynte, doo seeme +to be much like Originall causes: But the diuersitie neuerthelesse, is +great. We defined an _Vnit_, to be a thing Mathematicall Indiuisible: +A Point, likewise, we sayd to be a Mathematicall thing Indiuisible. And +farder, that a Point may haue a certaine determined Situation: that is, +that we may assigne, and prescribe a Point, to be here, there, yonder. +&c. Herein, (behold) our Vnit is free, and can abyde no bondage, or to +be tyed to any place, or seat: diuisible or indiuisible. Agayne, by +reason, a Point may haue a Situation limited to him: a certaine motion, +therfore (to a place, and from a place) is to a Point incident and +appertainyng. But an _Vnit_, can not be imagined to haue any motion. +A Point, by his motion, produceth, Mathematically, a line: (as we sayd +before) which is the first kinde of Magnitudes, and most simple: An +_Vnit_, can not produce any number. A Line, though it be produced of a +Point moued, yet, it doth not consist of pointes: Number, though it be +not produced of an _Vnit_, yet doth it Consist of vnits, as a materiall +cause. But formally, + + [Number.] + +Number, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting, +is the workemanship of our minde: which, of distinct and discrete Vnits, +maketh a Number: by vniformitie, resulting of a certaine multitude of +Vnits. And so, euery number, may haue his least part, giuen: namely, an +Vnit: But not of a Magnitude, (no, not of a Lyne,) the least part can be +giuẽ: by cause, infinitly, diuision therof, may be conceiued. All +Magnitude, is either a Line, a Plaine, or a Solid. Which Line, Plaine, +or Solid, of no Sense, can be perceiued, nor exactly by hãd (any way) +represented: nor of Nature produced: But, as (by degrees) Number did +come to our perceiuerance: So, by visible formes, we are holpen to +imagine, what our Line Mathematicall, is. What our Point, is. So +precise, are our Magnitudes, that one Line is no broader then an other: +for they haue no bredth: Nor our Plaines haue any thicknes. Nor yet our +Bodies, any weight: be they neuer so large of dimensiõ. Our Bodyes, we +can haue Smaller, then either Arte or Nature can produce any: and +Greater also, then all the world can comprehend. Our least Magnitudes, +can be diuided into so many partes, as the greatest. As, a Line of an +inch long, (with vs) may be diuided into as many partes, as may the +diameter of the whole world, from East to West: or any way extended: +What priuiledges, aboue all manual Arte, and Natures might, haue our two +Sciences Mathematicall? to exhibite, and to deale with thinges of such +power, liberty, simplicity, puritie, and perfection? And in them, so +certainly, so orderly, so precisely to procede: as, excellent is that +workemã Mechanicall Iudged, who nerest can approche to the representing +of workes, Mathematically demonstrated? + + [☞] + +And our two Sciences, remaining pure, and absolute, in their proper +termes, and in their owne Matter: to haue, and allowe, onely such +Demonstrations, as are plaine, certaine, vniuersall, and of an æternall +veritye? + + [Geometrie.] + +This Science of _Magnitude_, his properties, conditions, and +appertenances: commonly, now is, and from the beginnyng, hath of all +Philosophers, ben called _Geometrie_. But, veryly, with a name to base +and scant, for a Science of such dignitie and amplenes. And, perchaunce, +that name, by cõmon and secret consent, of all wisemen, hitherto hath +ben suffred to remayne: that it might carry with it a perpetuall +memorye, of the first and notablest benefite, by that Science, to common +people shewed: Which was, when Boundes and meres of land and ground were +lost, and confounded (as in _Egypt_, yearely, with the ouerflowyng of +_Nilus_, the greatest and longest riuer in the world) or, that ground +bequeathed, were to be assigned: or, ground sold, were to be layd out: +or (when disorder preuailed) that Commõs were distributed into +seueralties. For, where, vpon these & such like occasiõs, Some by +ignorãce, some by negligẽce, Some by fraude, and some by violence, did +wrongfully limite, measure, encroach, or challenge (by pretence of iust +content, and measure) those landes and groundes: great losse, +disquietnes, murder, and warre did (full oft) ensue: Till, by Gods +mercy, and mans Industrie, The perfect Science of Lines, Plaines, and +Solides (like a diuine Iusticier,) gaue vnto euery man, his owne. The +people then, by this art pleasured, and greatly relieued, in their +landes iust measuring: & other Philosophers, writing Rules for land +measuring: betwene them both, thus, confirmed the name of _Geometria_, +that is, (according to the very etimologie of the word) Land measuring. +Wherin, the people knew no farder, of Magnitudes vse, but in Plaines: +and the Philosophers, of thẽ, had no feet hearers, or Scholers: farder +to disclose vnto, then of flat, plaine _Geometrie_. And though, these +Philosophers, knew of farder vse, and best vnderstode the etymologye of +the worde, yet this name _Geometria_, was of them applyed generally to +all sortes of Magnitudes: vnleast, otherwhile, of _Plato_, and +_Pythagoras_: When they would precisely declare their owne doctrine. +Then, was + + [* Plato. 7. de Rep.] + +* _Geometria_, with them, _Studium quod circa planum versatur_. But, +well you may perceiue by _Euclides Elementes_, that more ample is our +Science, then to measure Plaines: and nothyng lesse therin is tought (of +purpose) then how to measure Land. An other name, therfore, must nedes +be had, for our Mathematicall Science of Magnitudes: which regardeth +neither clod, nor turff: neither hill, nor dale: neither earth nor +heauen: but is absolute _Megethologia_: not creping on ground, and +dasseling the eye, with pole perche, rod or lyne: but “liftyng the hart +aboue the heauens, by inuisible lines, and + + [☞] + +immortall beames meteth with the reflexions, of the light +incomprehensible: and so procureth Ioye, and perfection vnspeakable.” Of +which true vse of our _Megethica_, or _Megethologia_, _Diuine Plato_ +seemed to haue good taste, and iudgement: and (by the name of +_Geometrie_) so noted it: and warned his Scholers therof: as, in hys +seuenth _Dialog_, of the Common wealth, may euidently be sene. Where (in +Latin) thus it is: right well translated: _Profecto, nobis hoc non +negabunt, Quicun[que] vel paululum quid Geometriæ gustârunt, quin hæc +Scientia, contrà, omnino se habeat, quàm de ea loquuntur, qui in ipsa +versantur._ In English, thus. +_Verely_+ (sayth _Plato_) +_whosoeuer +haue, (but euen very litle) tasted of Geometrie, will not denye vnto vs, +this: but that this Science, is of an other condicion, quite contrary to +that, which they that are exercised in it, do speake of it._+ And there +it followeth, of our _Geometrie_, _Quòd quæritur cognoscendi illius +gratia, quod semper est, non & eius quod oritur quando[que] & interit. +Geometria, eius quod est semper, Cognitio est. Attollet igitur +(ô Generose vir) ad Veritatem, animum: at[que] ita, ad Philosophandum +preparabit cogitationem, vt ad supera conuertamus: quæ, nunc, contra +quàm decet, ad inferiora deijcimus. &c. Quàm maximè igitur præcipiendum +est, vt qui præclarissimam hanc habitãt Civitatem, nullo modo, +Geometriam spernant. Nam & quæ præter ipsius propositum, quodam modo +esse videntur, haud exigua sunt. &c._ It must nedes be confessed (saith +_Plato_) +_That =[Geometrie]= is learned, for the knowyng of that, which +is euer: and not of that, which, in tyme, both is bred and is brought to +an ende. &c. Geometrie is the knowledge of that which is euerlastyng. It +will lift vp therfore (O Gentle Syr) our mynde to the Veritie: and by +that meanes, it will prepare the Thought, to the Philosophicall loue of +wisdome: that we may turne or conuert, toward heauenly thinges =[both +mynde and thought]= which now, otherwise then becommeth vs, we cast down +on base or inferior things. &c. Chiefly, therfore, Commaundement must be +giuen, that such as do inhabit this most honorable Citie, by no meanes, +despise Geometrie. For euen those thinges =[done by it]= which, in +manner, seame to be, beside the purpose of Geometrie: are of no small +importance. &c._+ And besides the manifold vses of _Geometrie_, in +matters appertainyng to warre, he addeth more, of second vnpurposed +frute, and commoditye, arrising by _Geometrie_: saying: _Scimus quin +etiam, ad Disciplinas omnes facilius per discendas, interesse omnino, +attigerit ne Geometriam aliquis, an non. &c. Hanc ergo Doctrinam, +secundo loco discendam Iuuenibus statuamus._ That is. +_But, also, we +know, that for the more easy learnyng of all Artes, it importeth much, +whether one haue any knowledge in Geometrie, or no. &c. Let vs therfore +make an ordinance or decree, that this Science, of young men shall be +learned in the second place._+ This was _Diuine Plato_ his Iudgement, +both of the purposed, chief, and perfect vse of _Geometrie_: and of his +second, dependyng, deriuatiue commodities. And for vs, Christen men, +a thousand thousand mo occasions are, to haue nede of the helpe of * + + [I. D. + * Herein, I would gladly shake of, the earthly name, + of Geometrie.] + +_Megethologicall_ Contemplations: wherby, to trayne our Imaginations and +Myndes, by litle and litle, to forsake and abandon, the grosse and +corruptible Obiectes, of our vtward senses: and to apprehend, by sure +doctrine demonstratiue, Things Mathematicall. And by them, readily to be +holpen and conducted to conceiue, discourse, and conclude of things +Intellectual, Spirituall, æternall, and such as concerne our Blisse +euerlasting: which, otherwise (without Speciall priuiledge of +Illumination, or Reuelation frõ heauen) No mortall mans wyt (naturally) +is hable to reach vnto, or to Compasse. And, veryly, by my small Talent +(from aboue) I am hable to proue and testifie, that the litterall Text, +and order of our diuine Law, Oracles, and Mysteries, require more skill +in Numbers, and Magnitudes: then (commonly) the expositors haue vttered: +but rather onely (at the most) so warned: & shewed their own want +therin. (To name any, is nedeles: and to note the places, is, here, no +place: But if I be duely asked, my answere is ready.) And without the +litterall, Grammaticall, Mathematicall or Naturall verities of such +places, by good and certaine Arte, perceiued, no Spirituall sense +(propre to those places, by Absolute _Theologie_) will thereon depend. + + [☞] + +“No man, therfore, can doute, but toward the atteyning of knowledge +incomparable, and Heauenly Wisedome: Mathematicall Speculations, both of +Numbers and Magnitudes: are meanes, aydes, and guides: ready, certaine, +and necessary.” From henceforth, in this my Preface, will I frame my +talke, to _Plato_ his fugitiue Scholers: or, rather, to such, who well +can, (and also wil,) vse their vtward senses, to the glory of God, the +benefite of their Countrey, and their owne secret contentation, or +honest preferment, on this earthly Scaffold. To them, I will orderly +recite, describe & declare a great Number of Artes, from our two +Mathematicall fountaines, deriued into the fieldes of _Nature_. Wherby, +such Sedes, and Rotes, as lye depe hyd in the groũd of _Nature_, are +refreshed, quickened, and prouoked to grow, shote vp, floure, and giue +frute, infinite, and incredible. And these Artes, shalbe such, as vpon +Magnitudes properties do depende, more, then vpon Number. And by good +reason we may call them Artes, and Artes Mathematicall Deriuatiue: for +(at this tyme) I Define + + [An Arte.] + ++An Arte, to be a Methodicall cõplete Doctrine, hauing abundancy of +sufficient, and peculier matter to deale with, by the allowance of the +Metaphisicall Philosopher: the knowledge whereof, to humaine state is +necessarye.+ And that I account, + + [Art Mathematicall Deriuatiue.] + ++An Art Mathematicall deriuatiue, which by Mathematicall demonstratiue +Method, in Nũbers, or Magnitudes, ordreth and confirmeth his doctrine, +as much & as perfectly, as the matter subiect will admit.+ And for that, +I entend to vse the name and propertie of a + + [A Mechanitien.] + +_Mechanicien_, otherwise, then (hitherto) it hath ben vsed, I thinke it +good, (for distinction sake) to giue you also a brief description, what +I meane therby. +A Mechanicien, or a Mechanicall workman is he, whose +skill is, without knowledge of Mathematicall demonstration, perfectly to +worke and finishe any sensible worke, by the Mathematicien principall or +deriuatiue, demonstrated or demonstrable.+ Full well I know, that he +which inuenteth, or maketh these demonstrations, is generally called _A +speculatiue Mechanicien_: which differreth nothyng from a _Mechanicall +Mathematicien_. So, in respect of diuerse actions, one man may haue the +name of sundry artes: as, some tyme, of a Logicien, some tymes (in the +same matter otherwise handled) of a Rethoricien. Of these trifles, +I make, (as now, in respect of my Preface,) small account: to fyle thẽ +for the fine handlyng of subtile curious disputers. In other places, +they may commaunde me, to giue good reason: and yet, here, I will not be +vnreasonable. + + + [+1.+] + +First, then, from the puritie, absolutenes, and Immaterialitie of +Principall _Geometrie_, is that kinde of _Geometrie_ deriued, which +vulgarly is counted _Geometrie_: and is the +Arte of Measuring sensible +magnitudes, their iust quãtities and contentes.+ + + [Geometrie vulgar.] + +This, teacheth to measure, either at hand: and the practiser, to be by +the thing Measured: and so, by due applying of Cumpase, Rule, Squire, +Yarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) to +the Length, Plaine, or Solide measured, + + [1.] + +* to be certified, either of the length, perimetry, or distance lineall: +and this is called, _Mecometrie_. Or + + [2.] + +* to be certified of the content of any plaine Superficies: whether it +be in ground Surueyed, Borde, or Glasse measured, or such like thing: +which measuring, is named _Embadometrie_. + + [3.] + +* Or els to vnderstand the Soliditie, and content of any bodily thing: +as of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels, +small & great, of all fashions. Where, of Wine, Oyle, Beere, or Ale +vessells, &c, the Measuring, commonly, hath a peculier name: and is +called _Gaging_. And the generall name of these Solide measures, is +_Stereometrie_. + + [+2.+] + +Or els, this _vulgar Geometrie_, hath consideration to teach the +practiser, how to measure things, with good distance betwene him and the +thing measured: and to vnderstand thereby, either + + [1.] + +* how Farre, a thing seene (on land or water) is from the measurer: and +this may be called _Apomecometrie_: + + [2.] + +Or, how High or depe, aboue or vnder the leuel of the measurers stãding, +any thing is, which is sene on land or water, called _Hypsometrie_. + + [3.] + +* Or, it informeth the measurer, how Broad any thing is, which is in the +measurers vew: so it be on Land or Water, situated: and may be called +_Platometrie_. Though I vse here to condition, the thing measured, to be +on Land, or Water Situated: + + [Note.] + +yet, know for certaine, that the sundry heigthe of Cloudes, blasing +Starres, and of the Mone, may (by these meanes) haue their distances +from the earth: and, of the blasing Starres and Mone, the Soliditie +(aswell as distances) to be measured: But because, neither these things +are vulgarly taught: nor of a common practiser so ready to be executed: +I, rather, let such measures be reckened incident to some of our other +Artes, dealing with thinges on high, more purposely, then this vulgar +Land measuring Geometrie doth: as in _Perspectiue_ and _Astronomie, &c._ + + +Of these Feates (farther applied) is Sprong the Feate of _Geodesie_, or +Land Measuring: more cunningly to measure & Suruey Land, Woods, and +Waters, a farre of. More cunningly, I say: But God knoweth (hitherto) in +these Realmes of England and Ireland (whether through ignorance or +fraude, I can not tell, in euery particular) + + [Note.] + +how great wrong and iniurie hath (in my time) bene committed by vntrue +measuring and surueying of Land or Woods, any way. And, this I am sure: +that the Value of the difference, betwene the truth and such Surueyes, +would haue bene hable to haue foũd (for euer) in eche of our two +Vniuersities, an excellent Mathematicall Reader: to eche, allowing +(yearly) a hundred Markes of lawfull money of this realme: which, in +dede, would seme requisit, here, to be had (though by other wayes +prouided for) as well, as, the famous Vniuersitie of Paris, hath two +Mathematicall Readers: and eche, two hundreth French Crownes yearly, of +the French Kinges magnificent liberalitie onely. Now, againe, to our +purpose returning: Moreouer, of the former knowledge Geometricall, are +growen the Skills of _Geographie_, _Chorographie_, _Hydrographie_, and +_Stratarithmetrie_. + + +“+‡Geographie‡+ teacheth wayes, by which, in sũdry formes, (as +_Sphærike_, _Plaine_ or other), the Situation of Cities, Townes, +Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, & +such other things, vpõ the outface of the earthly Globe (either in the +whole, or in some principall mẽber and portion therof cõtayned) may be +described and designed, in cõmensurations Analogicall to Nature and +veritie: and most aptly to our vew, may be represented.” Of this Arte +how great pleasure, and how manifolde commodities do come vnto vs, daily +and hourely: of most men, is perceaued. While, some, to beautifie their +Halls, Parlers, Chambers, Galeries, Studies, or Libraries with: other +some, for thinges past, as battels fought, earthquakes, heauenly +fyringes, & such occurentes, in histories mentioned: therby liuely, as +it were, to vewe the place, the region adioyning, the distance from vs: +and such other circumstances. Some other, presently to vewe the large +dominion of the Turke: the wide Empire of the Moschouite: and the litle +morsell of ground, where Christendome (by profession) is certainly +knowen. Litle, I say, in respecte of the rest. &c. Some, either for +their owne iorneyes directing into farre landes: or to vnderstand of +other mens trauailes. To conclude, some, for one purpose: and some, for +an other, liketh, loueth, getteth, and vseth, Mappes, Chartes, & +Geographicall Globes. Of whose vse, to speake sufficiently, would +require a booke peculier. + + ++‡Chorographie‡+ seemeth to be an vnderling, and a twig, of +_Geographie_: and yet neuerthelesse, is in practise manifolde, and in +vse very ample. “This teacheth Analogically to describe a small portion +or circuite of ground, with the contentes: not regarding what +commensuration it hath to the whole, or any parcell, without it, +contained. But in the territory or parcell of ground which it taketh in +hand to make description of, it leaueth out (or vndescribed) no notable, +or odde thing, aboue the ground visible. Yea and sometimes, of thinges +vnder ground, geueth some peculier marke: or warning: as of Mettall +mines, Cole pittes, Stone quarries. &c.” Thus, a Dukedome, a Shiere, +a Lordship, or lesse, may be described distinctly. But marueilous +pleasant, and profitable it is, in the exhibiting to our eye, and +commensuration, the plat of a Citie, Towne, Forte, or Pallace, in true +Symmetry: not approching to any of them: and out of Gunne shot. &c. +Hereby, the _Architect_ may furnishe him selfe, with store of what +patterns he liketh: to his great instruction: euen in those thinges +which outwardly are proportioned: either simply in them selues: or +respectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also, +terme this particular description of places, _Topographie_. + + +“+‡Hydrographie‡+, deliuereth to our knowledge, on Globe or in Plaine, +the perfect Analogicall description of the Ocean Sea coastes, through +the whole world: or in the chiefe and principall partes thereof:” with +the Iles and chiefe particular places of daungers, conteyned within the +boundes, and Sea coastes described: as, of Quicksandes, Bankes, Pittes, +Rockes, Races, Countertides, Whorlepooles. &c. This, dealeth with the +Element of the water chiefly: as _Geographie_ did principally take the +Element of the Earthes description (with his appertenances) to taske. +And besides thys, _Hydrographie_, requireth a particular Register of +certaine Landmarkes (where markes may be had) from the sea, well hable +to be skried, in what point of the Seacumpase they appeare, and what +apparent forme, Situation, and bignes they haue, in respecte of any +daungerous place in the sea, or nere vnto it, assigned: And in all +Coastes, what Mone, maketh full Sea: and what way, the Tides and Ebbes, +come and go, the _Hydrographer_ ought to recorde. The Soundinges +likewise: and the Chanels wayes: their number, and depthes ordinarily, +at ebbe and flud, ought the _Hydrographer_, by obseruation and diligence +of _Measuring_, to haue certainly knowen. And many other pointes, are +belonging to perfecte _Hydrographie_, and for to make a _Rutter_, by: of +which, I nede not here speake: as of the describing, in any place, vpon +Globe or Plaine, the 32. pointes of the Compase, truely: (wherof, +scarsly foure, in England, haue right knowledge: bycause, the lines +therof, are no straight lines, nor Circles.) Of making due proiection of +a Sphere in plaine. Of the Variacion of the Compas, from true Northe: +And such like matters (of great importance, all) I leaue to speake of, +in this place: bycause, I may seame (al ready) to haue enlarged the +boundes, and duety of an Hydographer, much more, then any man (to this +day) hath noted, or prescribed. Yet am I well hable to proue, all these +thinges, to appertaine, and also to be proper to the Hydrographer. The +chief vse and ende of this Art, is the Art of Nauigation: but it hath +other diuerse vses: euen by them to be enioyed, that neuer lacke sight +of land. + + ++‡Stratarithmetrie‡+, is the Skill, (appertainyng to the warre,) by +which a man can set in figure, analogicall to any _Geometricall_ figure +appointed, any certaine number or summe of men: of such a figure +capable: (by reason of the vsuall spaces betwene Souldiers allowed: and +for that, of men, can be made no Fractions. Yet, neuertheles, he can +order the giuen summe of men, for the greatest such figure, that of +them, cã be ordred) and certifie, of the ouerplus: (if any be) and of +the next certaine summe, which, with the ouerplus, will admit a figure +exactly proportionall to the figure assigned. By which Skill, also, of +any army or company of men: (the figure & sides of whose orderly +standing, or array, is knowen) he is able to expresse the iust number of +men, within that figure conteined: or (orderly) able to be conteined. + + [* Note.] + +* And this figure, and sides therof, he is hable to know: either beyng +by, and at hand: or a farre of. Thus farre, stretcheth the description +and property of _Stratarithmetrie_: sufficient for this tyme and place. + + [The difference betwene Stratarithmetrie and Tacticie.] + +“It differreth from the Feate _Tacticall_, _De aciebus instruendis._ +bycause, there, is necessary the wisedome and foresight, to what purpose +he so ordreth the men: and Skillfull hability, also, for any occasion, +or purpose, to deuise and vse the aptest and most necessary order, array +and figure of his Company and Summe of men.” By figure, I meane: as, +either of a _Perfect Square_, _Triangle_, _Circle_, _Ouale_, _long +square_, (of the Grekes it is called _Eteromekes_) _Rhombe_, _Rhomboïd_, +_Lunular_, _Ryng_, _Serpentine_, and such other Geometricall figures: +Which, in warres, haue ben, and are to be vsed: for commodiousnes, +necessity, and auauntage &c. And no small skill ought he to haue, that +should make true report, or nere the truth, of the numbers and Summes, +of footemen or horsemen, in the Enemyes ordring. A farre of, to make an +estimate, betwene nere termes of More and Lesse, is not a thyng very +rife, among those that gladly would do it. + + [I. D. + Frende, you will finde it hard, to performe my description + of this Feate. But by Chorographie, you may helpe your selfe + some what: where the Figures knowne (in Sides and Angles) + are not Regular: And where, Resolution into Triangles can + serue. &c. And yet you will finde it strange to deale thus + generally with Arithmeticall figures: and, that for Battayle + ray. Their contentes, differ so much from like Geometricall + Figures.] + +Great pollicy may be vsed of the Capitaines, (at tymes fete, and in +places conuenient) as to vse Figures, which make greatest shew, of so +many as he hath: and vsing the aduauntage of the three kindes of vsuall +spaces: (betwene footemen or horsemen) to take the largest: or when he +would seme to haue few, (beyng many:) contrarywise, in Figure, and +space. The Herald, Purseuant, Sergeant Royall, Capitaine, or who soeuer +is carefull to come nere the truth herein, besides the Iudgement of his +expert eye, his skill of Ordering _Tacticall_, the helpe of his +Geometricall instrument: Ring, or Staffe Astronomicall: (commodiously +framed for cariage and vse) He may wonderfully helpe him selfe, by +perspectiue Glasses. In which, (I trust) our posterity will proue more +skillfull and expert, and to greater purposes, then in these dayes, can +(almost) be credited to be possible. + + +Thus haue I lightly passed ouer the Artificiall Feates, chiefly +dependyng vpon vulgar _Geometrie_: & commonly and generally reckened +vnder the name of _Geometrie_. But there are other (very many) +_Methodicall Artes_, which, declyning from the purity, simplicitie, and +Immateriality, of our Principall Science of _Magnitudes_: do yet +neuertheles vse the great ayde, direction, and Method of the sayd +principall Science, and haue propre names, and distinct: both from the +Science of _Geometrie_, (from which they are deriued) and one from the +other. As +Perspectiue, Astronomie, Musike, Cosmographie, Astrologie, +Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie, +Menadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture, +Nauigation, Thaumaturgike+ and +Archemastrie+. I thinke it necessary, +orderly, of these to giue some peculier descriptions: and withall, to +touch some of their commodious vses, and so to make this Preface, to be +a little swete, pleasant Nosegaye for you: to comfort your Spirites, +beyng almost out of courage, and in despayre, (through brutish brute) +Weenyng that _Geometrie_, had but serued for buildyng of an house, or a +curious bridge, or the roufe of Westminster hall, or some witty pretty +deuise, or engyn, appropriate to a Carpenter, or a Ioyner &c. That the +thing is farre otherwise, then the world, (commonly) to this day, hath +demed, by worde and worke, good profe wilbe made. + + +Among these Artes, by good reason, +‡Perspectiue‡+ ought to be had, ere +of _Astronomicall Apparences_, perfect knowledge can be atteyned. And +bycause of the prerogatiue of _Light_, beyng the first of _Gods +Creatures_: and the eye, the light of our body, and his Sense most +mighty, and his organ most Artificiall and _Geometricall_: At +_Perspectiue_, we will begyn therfore. +Perspectiue, is an Art +Mathematicall, which demonstrateth the maner, and properties, of all +Radiations Direct, Broken, and Reflected.+ This Description, or +Notation, is brief: but it reacheth so farre, as the world is wyde. It +concerneth all Creatures, all Actions, and passions, by Emanation of +beames perfourmed. Beames, or naturall lines, (here) I meane, not of +light onely, or of colour (though they, to eye, giue shew, witnes, and +profe, wherby to ground the Arte vpon) but also of other _Formes_, both +_Substantiall_, and _Accidentall_, the certaine and determined actiue +Radiall emanations. By this Art (omitting to speake of the highest +pointes) we may vse our eyes, and the light, with greater pleasure: and +perfecter Iudgement: both of things, in light seen, & of other: which by +like order of Lightes Radiations, worke and produce their effectes. We +may be ashamed to be ignorant of the cause, why so sundry wayes our eye +is deceiued, and abused: as, while the eye weeneth a roũd Globe or +Sphere (beyng farre of) to be a flat and plaine Circle, and so likewise +iudgeth a plaine Square, to be roũd: supposeth walles parallels, to +approche, a farre of: rofe and floure parallels, the one to bend +downward, the other to rise vpward, at a little distance from you. +Againe, of thinges being in like swiftnes of mouing, to thinke the +nerer, to moue faster: and the farder, much slower. Nay, of two thinges, +wherof the one (incomparably) doth moue swifter then the other, to deme +the slower to moue very swift, & the other to stand: what an error is +this, of our eye? Of the Raynbow, both of his Colours, of the order of +the colours, of the bignes of it, the place and heith of it, (&c) to +know the causes demonstratiue, is it not pleasant, is it not necessary? +of two or three Sonnes appearing: of Blasing Sterres: and such like +thinges: by naturall causes, brought to passe, (and yet neuertheles, of +farder matter, Significatiue) is it not commodious for man to know the +very true cause, & occasion Naturall? Yea, rather, is it not, greatly, +against the Souerainty of Mans nature, to be so ouershot and abused, +with thinges (at hand) before his eyes? as with a Pecockes tayle, and a +Doues necke: or a whole ore, in water, holden, to seme broken. Thynges, +farre of, to seeme nere: and nere, to seme farre of. Small thinges, to +seme great: and great, to seme small. One man, to seme an Army. Or a man +to be curstly affrayed of his owne shaddow. Yea, so much, to feare, +that, if you, being (alone) nere a certaine glasse, and proffer, with +dagger or sword, to foyne at the glasse, you shall suddenly be moued to +giue backe (in maner) by reason of an Image, + + [☞ A marueilous Glasse.] + +appearing in the ayre, betwene you & the glasse, with like hand, sword +or dagger, & with like quicknes, foyning at your very eye, likewise as +you do at the Glasse. Straunge, this is, to heare of: but more +meruailous to behold, then these my wordes can signifie. And +neuerthelesse by demonstration Opticall, the order and cause therof, is +certified: euen so, as the effect is consequent. Yea, thus much more, +dare I take vpon me, toward the satisfying of the noble courrage, that +longeth ardently for the wisedome of Causes Naturall: as to let him +vnderstand, that, in London, he may with his owne eyes, haue profe of +that, which I haue sayd herein. A Gentleman, (which, for his good +seruice, done to his Countrey, is famous and honorable: + + [S. W. P.] + +and for skill in the Mathematicall Sciences, and Languages, is the Od +man of this land. &c.) euen he, is hable: and (I am sure) will, very +willingly, let the Glasse, and profe be sene: and so I (here) request +him: for the encrease of wisedome, in the honorable: and for the +stopping of the mouthes malicious: and repressing the arrogancy of the +ignorant. Ye may easily gesse, what I meane. This Art of _Perspectiue_, +is of that excellency, and may be led, to the certifying, and executing +of such thinges, as no man would easily beleue: without Actuall profe +perceiued. I speake nothing of _Naturall Philosophie_, which, without +_Perspectiue_, can not be fully vnderstanded, nor perfectly atteined +vnto. Nor, of _Astronomie_: which, without _Perspectiue_, can not well +be grounded: Nor _Astrologie_, naturally Verified, and auouched. That +part hereof, which dealeth with Glasses (which name, Glasse, is a +generall name, in this Arte, for any thing, from which, a Beame +reboundeth) is called _Catoptrike_: and hath so many vses, both +merueilous, and proffitable: that, both, it would hold me to long, to +note therin the principall conclusions, all ready knowne: And also +(perchaunce) some thinges, might lacke due credite with you: And I, +therby, to leese my labor: and you, to slip into light Iudgement, + + [* ☞] + +Before you haue learned sufficiently the powre of Nature and Arte. + + +Now, to procede: +‡Astronomie‡, is an Arte Mathematicall, which +demonstrateth the distance, magnitudes, and all naturall motions, +apparences, and passions propre to the Planets and fixed Sterres: for +any time past, present and to come: in respect of a certaine Horizon, or +without respect of any Horizon.+ By this Arte we are certified of the +distance of the Starry Skye, and of eche _Planete_ from the Centre of +the Earth: and of the greatnes of any Fixed starre sene, or _Planete_, +in respect of the Earthes greatnes. As, we are sure (by this Arte) that +the Solidity, Massines and Body of the _Sonne_, conteineth the quantitie +of the whole Earth and Sea, a hundred thre score and two times, lesse by +⅛ one eight parte of the earth. But the Body of the whole earthly globe +and Sea, is bigger then the body of the Mone, three and forty times +lesse by ⅛ of the Mone. Wherfore the _Sonne_ is bigger then the _Mone_, +7000 times, lesse, by 59 39/64 that is, precisely 6940 25/64 bigger then +the _Mone_. And yet the vnskillfull man, would iudge them a like bigge. +Wherfore, of Necessity, the one is much farder from vs, then the other. +The _Sonne_, when he is fardest from the earth (which, now, in our age, +is, when he is in the 8. degree, of Cancer) is, 1179 Semidiameters of +the Earth, distante. And the _Mone_ when she is fardest from the earth, +is 68 Semidiameters of the earth and ⅓ The nerest, that the _Mone_ +commeth to the earth, is Semidiameters 52¼ The distance of the Starry +Skye is, frõ vs, in Semidiameters of the earth 20081½ Twenty thousand +fourescore, one, and almost a halfe. Subtract from this, the _Mones_ +nerest distance, from the Earth: and therof remaineth Semidiameters of +the earth 20029¼ Twenty thousand nine and twenty and a quarter. + + [Note.] + +So thicke is the heauenly Palace, that the _Planetes_ haue all their +exercise in, and most meruailously perfourme the Commaũdement and Charge +to them giuen by the omnipotent Maiestie of the king of kings. This is +that, which in _Genesis_ is called _Ha Rakia_. Consider it well. The +Semidiameter of the earth, cõteineth of our common miles 3436 4/11 three +thousand, foure hundred thirty six and foure eleuenth partes of one +myle: Such as the whole earth and Sea, round about, is 21600. One and +twenty thousand six hundred of our myles. Allowyng for euery degree of +the greatest circle, thre score myles. Now if you way well with your +selfe but this litle parcell of frute _Astronomicall_, as concerning the +bignesse, Distances of _Sonne_, _Mone_, _Sterry Sky_, and the huge +massines of _Ha Rakia_, will you not finde your Consciences moued, with +the kingly Prophet, to sing the confession of Gods Glory, and say, +_The +Heauens declare the glory of God, and the Firmament =[Ha Rakia]= sheweth +forth the workes of his handes_+. And so forth, for those fiue first +staues, of that kingly Psalme. Well, well, It is time for some to lay +hold on wisedome, and to Iudge truly of thinges: and notso to expound +the Holy word, all by Allegories: as to Neglect the wisedome, powre and +Goodnes of God, in, and by his Creatures, and Creation to be seen and +learned. By parables and Analogies of whose natures and properties, the +course of the Holy Scripture, also, declareth to vs very many Mysteries. +The whole Frame of Gods Creatures, (which is the whole world,) is to vs, +a bright glasse: from which, by reflexion, reboundeth to our knowledge +and perceiuerance, Beames, and Radiations: representing the Image of his +Infinite goodnes, Omnipotẽcy, and wisedome. And we therby, are taught +and persuaded to Glorifie our Creator, as God: and be thankefull +therfore. Could the Heathenistes finde these vses, of these most pure, +beawtifull, and Mighty Corporall Creatures: and shall we, after that the +true _Sonne_ of rightwisenesse is risen aboue the _Horizon_, of our +temporall _Hemisphærie_, and hath so abundantly streamed into our +hartes, the direct beames of his goodnes, mercy, and grace: Whose heat +All Creatures feele: Spirituall and Corporall: Visible and Inuisible. +Shall we (I say) looke vpon the _Heauen_, _Sterres_, and _Planets_, as +an Oxe and an Asse doth: no furder carefull or inquisitiue, what they +are: why were they Created, How do they execute that they were Created +for? Seing, All Creatures, were for our sake created: and both we, and +they, Created, chiefly to glorifie the Almighty Creator: and that, by +all meanes, to vs possible. _Nolite ignorare_ (saith _Plato in +Epinomis_) _Astronomiam, Sapientissimũ quiddam esse._ +_Be ye not +ignorant, Astronomie to be a thyng of excellent wisedome._+ +_Astronomie_, was to vs, from the beginning commended, and in maner +commaunded by God him selfe. In asmuch as he made the _Sonne_, _Mone_, +and _Sterres_, to be to vs, for _Signes_, and knowledge of Seasons, and +for Distinctions of Dayes, and yeares. Many wordes nede not. But I wish, +euery man should way this word, _Signes_. And besides that, conferre it +also with the tenth Chapter of _Hieremie_. And though Some thinke, that +there, they haue found a rod: Yet Modest Reason, will be indifferent +Iudge, who ought to be beaten therwith, in respect of our purpose. +Leauing that: I pray you vnderstand this: that without great diligence +of Obseruation, examination and Calculation, their periods and courses +(wherby _Distinction_ of Seasons, yeares, and New Mones might precisely +be knowne) could not exactely be certified. Which thing to performe, is +that _Art_, which we here haue Defined to be _Astronomie_. Wherby, we +may haue the distinct Course of Times, dayes, yeares, and Ages: aswell +for Consideratiõ of Sacred Prophesies, accomplished in due time, +foretold: as for high Mysticall Solemnities holding: And for all other +humaine affaires, Conditions, and couenantes, vpon certaine time, +betwene man and man: with many other great vses: Wherin, (verely), would +be great incertainty, Confusion, vntruth, and brutish Barbarousnes: +without the wonderfull diligence and skill of this Arte: continually +learning, and determining Times, and periodes of Time, by the Record of +the heauenly booke, wherin all times are written: and to be read with an +_Astronomicall staffe_, in stede of a festue. + + ++‡Musike‡+, of Motion, hath his Originall cause: Therfore, after the +motions most swift, and most Slow, which are in the Firmament, of Nature +performed: and vnder the _Astronomers Consideration_: now I will Speake +of an other kinde of _Motion_, producing sound, audible, and of Man +numerable. _Musike_ I call here that _Science_, which of the Grekes is +called _Harmonice_. Not medling with the Controuersie betwene the +auncient _Harmonistes_, and _Canonistes_. +Musike is a Mathematicall +Science, which teacheth, by sense and reason, perfectly to iudge, and +order the diuersities of soundes, hye and low.+ _Astronomie_ and +_Musike_ are Sisters, saith _Plato_. As, for _Astronomie_, the eyes: So, +for _Harmonious Motion_, the eares were made. But as _Astronomie_ hath a +more diuine Contemplation, and cõmodity, then mortall eye can perceiue: +So, is _Musike_ to be considered, + + [1.] + +that the * Minde may be preferred, before the eare. And from audible +sound, we ought to ascende, to the examination: which numbers are +_Harmonious_, and which not. And why, either, the one are: or the other +are not. I could at large, + + [2.] + +in the heauenly * motions and distances, describe a meruailous Harmonie, +of _Pythagoras_ Harpe + + [3.] + +with eight stringes. Also, somwhat might be sayd of _Mercurius_ * two +Harpes, + + [4.] + +eche of foure Stringes Elementall. And very straunge matter, might be +alledged of the _Harmonie_, + + [5.] + +to our * Spirituall part appropriate. As in _Ptolomaus_ third boke, in +the fourth and sixth Chapters may appeare. * + + [6.] + +And what is the cause of the apt bonde, and frendly felowship, of the +Intellectuall and Mentall part of vs, with our grosse & corruptible +body: but a certaine Meane, and _Harmonious Spiritualitie, with both +participatyng, & of both (in a maner) resultynge In + + [7.] + +the * Tune of Mans voyce, and also + + [8.] + +* the sound of Instrument_, what might be sayd, of _Harmonie_: No common +Musicien would lightly beleue. + + [I. D. + Read in Aristotle his 8. booke of Politikes: the 5, 6, and 7. + chapters. Where you shall haue some occasion farder to thinke + of Musike, than commonly is thought.] + +But of the sundry Mixture (as I may terme it) and concurse, diuerse +collation, and Application of these _Harmonies_: as of thre, foure, +fiue, or mo: Maruailous haue the effectes ben: and yet may be founde, +and produced the like: with some proportionall consideration for our +time, and being: in respect of the State, of the thinges then: in which, +and by which, the wondrous effectes were wrought. _Democritus_ and +_Theophrastus_ affirmed, that, by _Musike_, griefes and diseases of the +Minde, and body might be cured, or inferred. And we finde in Recorde, +that _Terpander_, _Arion_, _Ismenias_, _Orpheus_, _Amphion_, _Dauid_, +_Pythagoras_, _Empedocles_, _Asclepiades_ and _Timotheus_, by +_Harmonicall_ Consonãcy, haue done, and brought to pas, thinges, more +then meruailous, to here of. Of them then, making no farder discourse, +in this place: Sure I am, that Common _Musike_, commonly vsed, is found +to the _Musiciens_ and Hearers, to be so Commodious and pleasant, That +if I would say and dispute, but thus much: That it were to be otherwise +vsed, then it is, I should finde more repreeuers, then I could finde +priuy, or skilfull of my meaning. In thinges therfore euident, and +better knowen, then I can expresse: and so allowed and liked of, (as I +would wish, some other thinges, had the like hap) I will spare to +enlarge my lines any farder, but consequently follow my purpose. + + ++‡Of Cosmographie‡+, I appointed briefly in this place, to geue you some +intelligence. +Cosmographie, is the whole and perfect description of the +heauenly, and also elementall parte of the world, and their homologall +application, and mutuall collation necessarie.+ This Art, requireth +_Astronomie_, _Geographie_, _Hydrographie_ and _Musike_. Therfore, it is +no small Arte, nor so simple, as in common practise, it is (slightly) +considered. This matcheth Heauen, and the Earth, in one frame, and aptly +applieth parts Correspõdent: So, as, the Heauenly Globe, may (in +practise) be duely described vpon the Geographicall, and Hydrographicall +Globe. And there, for vs to consider an _Æquonoctiall Circle_, _an +Ecliptike line_, _Colures_, _Poles_, _Sterres_ in their true Longitudes, +Latitudes, Declinations, and Verticalitie: also Climes, and Parallels: +and by an _Horizon_ annexed, and reuolution of the earthly Globe (as the +Heauen, is, by the _Primouant_, caried about in 24. æquall Houres) to +learne the Risinges and Settinges of Sterres (of _Virgill_ in his +_Georgikes_: of _Hesiod_: of _Hippocrates_ in his _Medicinall Sphære_, +to Perdicca King of the Macedonians: of _Diocles_, to King _Antigonus_, +and of other famous _Philosophers_ prescribed) a thing necessary, for +due manuring of the earth, for _Nauigation_, for the Alteration of mans +body: being, whole, Sicke, wounded, or brused. By the Reuolution, also, +or mouing of the Globe Cosmographicall, the Rising and Setting of the +Sonne: the Lengthes, of dayes and nightes: the Houres and times (both +night and day) are knowne: with very many other pleasant and necessary +vses: Wherof, some are knowne: but better remaine, for such to know and +vse: + + [☞] + +who of a sparke of true fire, can make a wonderfull bonfire, by applying +of due matter, duely. + + ++‡Of Astrologie‡+, here I make an Arte, seuerall from _Astronomie_: not +by new deuise, but by good reason and authoritie: for, +Astrologie, is +an Arte Mathematicall, which reasonably demonstrateth the operations and +effectes, of the naturall beames, of light, and secrete influence: of +the Sterres and Planets: in euery element and elementall body: at all +times, in any Horizon assigned.+ This Arte is furnished with many other +great Artes and experiences: As with perfecte _Perspectiue_, +_Astronomie_, _Cosmographie_, _Naturall Philosophie_ of the 4. +Elementes, the Arte of Graduation, and some good vnderstãding in +_Musike_: and yet moreouer, with an other great Arte, hereafter +following, though I, here, set this before, for some considerations me +mouing. Sufficient (you see) is the stuffe, to make this rare and +secrete Arte, of: and hard enough to frame to the Conclusion +Syllogisticall. Yet both the manifolde and continuall trauailes of the +most auncient and wise Philosophers, for the atteyning of this Arte: and +by examples of effectes, to confirme the same: hath left vnto vs +sufficient proufe and witnesse: and we, also, daily may perceaue, That +mans body, and all other Elementall bodies, are altered, disposed, +ordred, pleasured, and displeasured, by the Influentiall working of the +_Sunne_, _Mone_, and the other Starres and Planets. And therfore, sayth +_Aristotle_, in the first of his _Meteorologicall_ bookes, in the second +Chapter: _Est autem necessariò Mundus iste, supernis lationibus ferè +continuus. Vt, inde, vis eius vniuersa regatur. Ea siquidem Causà prima +putanda omnibus est, vnde motus principium existit._ That is: +_This +=[Elementall]= World is of necessitie, almost, next adioyning, to the +heauenly motions: That, from thence, all his vertue or force may be +gouerned. For, that is to be thought the first Cause vnto all: from +which, the beginning of motion, is._+ And againe, in the tenth Chapter. +_Oportet igitur & horum principia sumamus, & causas omnium similiter. +Principium igitur vt mouens, præcipuum[que] & omnium primum, Circulus +ille est, in quo manifeste Solis latio, &c._ And so forth. His +_Meteorologicall_ bookes, are full of argumentes, and effectuall +demonstrations, of the vertue, operation, and power of the heauenly +bodies, in and vpon the fower Elementes, and other bodies, of them +(either perfectly, or vnperfectly) composed. And in his second booke, +_De Generatione & Corruptione_, in the tenth Chapter. _Quocirca & prima +latio, Ortus & Interitus causa non est: Sed obliqui Circuli latio: ea +nam[que] & continua est, & duobus motibus fit:_ In Englishe, thus. ++_Wherefore the vppermost motion, is not the cause of Generation and +Corruption, but the motion of the Zodiake: for, that, both, is +continuall, and is caused of two mouinges._+ And in his second booke, +and second Chapter of hys _Physikes_. _Homo nam[que] generat hominem, +at[que] Sol._ +_For Man (sayth he) and the Sonne, are cause of mans +generation._+ Authorities may be brought, very many: both of 1000. 2000. +yea and 3000. yeares Antiquitie: of great _Philosophers_, _Expert_, +_Wise_, and godly men, for that Conclusion: which, daily and hourely, we +men, may discerne and perceaue by sense and reason: All beastes do +feele, and simply shew, by their actions and passions, outward and +inward: All Plants, Herbes, Trees, Flowers, and Fruites. And finally, +the Elementes, and all thinges of the Elementes composed, do geue +Testimonie (as _Aristotle_ sayd) that theyr +_Whole Dispositions, +vertues, and naturall motions, depend of the Actiuitie of the heauenly +motions and Influences. Whereby, beside the specificall order and forme, +due to euery seede: and beside the Nature, propre to the Indiuiduall +Matrix, of the thing produced: What shall be the heauenly Impression, +the perfect and circumspecte Astrologien hath to Conclude._+ Not onely +(by _Apotelesmes_) τὸ ὁτὶ]. but by Naturall and Mathematicall +demonstration τὸ διότι. Whereunto, what Sciences are requisite (without +exception) I partly haue here warned: And in my _Propædeumes_ (besides +other matter there disclosed) I haue Mathematically furnished vp the +whole Method: To this our age, not so carefully handled by any, that +euer I saw, or heard of. I was, + + [* Anno. 1548 and 1549. in Louayn.] + +(for * 21. yeares ago) by certaine earnest disputations, of the Learned +_Gerardus Mercator_, and _Antonius Gogaua_, (and other,) therto so +prouoked: and (by my constant and inuincible zeale to the veritie) in +obseruations of Heauenly Influencies (to the Minute of time,) than, so +diligent: And chiefly by the Supernaturall influence, from the Starre of +Iacob, so directed: That any Modest and Sober Student, carefully and +diligently seking for the Truth, will both finde & cõfesse, therin, to +be the Veritie, of these my wordes: And also become a Reasonable +Reformer, of three Sortes of people: about these Influentiall +Operations, greatly erring from the truth. + + [Note.] + +Wherof, the one, is +Light Beleuers+, the other, +Light Despisers+, and +the third +Light Practisers+. The first, & most cõmon Sort, thinke the +Heauen and Sterres, to be answerable to any their doutes or desires: + + [1.] + +which is not so: and, in dede, they, to much, ouer reache. The Second +sorte thinke no Influentiall vertue (frõ the heauenly bodies) to beare +any Sway in Generation + + [2.] + +and Corruption, in this Elementall world. And to the _Sunne_, _Mone_ and +_Sterres_ (being so many, so pure, so bright, so wonderfull bigge, so +farre in distance, so manifold in their motions, so constant in their +periodes. &c.) they assigne a sleight, simple office or two, and so +allow vnto thẽ (according to their capacities) as much vertue, and power +Influentiall, as to the Signe of the _Sunne_, _Mone_, and seuen Sterres, +hanged vp (for Signes) in London, for distinction of houses, & such +grosse helpes, in our worldly affaires: And they vnderstand not (or will +not vnderstand) of the other workinges, and vertues of the Heauenly +_Sunne_, _Mone_, and _Sterres_: not so much, as the Mariner, or Husband +man: no, not so much, as the _Elephant_ doth, as the _Cynocephalus_, as +the Porpentine doth: nor will allow these perfect, and incorruptible +mighty bodies, so much vertuall Radiation, & Force, as they see in a +litle peece of a _Magnes stone_: which, at great distance, sheweth his +operation. And perchaunce they thinke, the Sea & Riuers (as the Thames) +to be some quicke thing, and so to ebbe, and flow, run in and out, of +them selues, at their owne fantasies. God helpe, God helpe. Surely, +these men, come to short: and either are to dull: or willfully blind: +or, perhaps, to malicious. The third man, is the common and vulgare +_Astrologien_, or Practiser: who, being not duely, artificially, and +perfectly + + [3.] + +furnished: yet, either for vaine glory, or gayne: or like a simple dolt, +& blinde Bayard, both in matter and maner, erreth: to the discredit of +the _Wary_, and modest _Astrologien_: and to the robbing of those most +noble corporall Creatures, of their Naturall Vertue: being most mighty: +most beneficiall to all elementall Generation, Corruption and the +appartenances: and most Harmonious in their Monarchie: For which +thinges, being knowen, and modestly vsed: we might highly, and +continually glorifie God, with the princely Prophet, saying. +_The +Heauens declare the Glorie of God: who made the Heauẽs in his wisedome: +who made the Sonne, for to haue dominion of the day: the Mone and +Sterres to haue dominion of the nyght: whereby, Day to day vttereth +talke: and night, to night declareth knowledge. Prayse him, all ye +Sterres, and Light. Amen._+ + + +In order, now foloweth, of +‡Statike‡+, somewhat to say, what we meane +by that name: and what commodity, doth, on such Art, depend. +Statike, +is an Arte Mathematicall, which demonstrateth the causes of heauynes, +and lightnes of all thynges: and of motions and properties, to heauynes +and lightnes, belonging.+ And for asmuch as, by the Bilanx, or Balance +(as the chief sensible Instrument,) Experience of these demonstrations +may be had: we call this Art, _Statike:_ that is, _the Experimentes of +the Balance_. Oh, that men wist, what proffit, (all maner of wayes) by +this Arte might grow, to the hable examiner, and diligent practiser. +“Thou onely, knowest all thinges precisely (O God) who hast made weight +and Balance, thy Iudgement: who hast created all thinges in _Number, +Waight, and Measure_: and hast wayed the mountaines and hils in a +Balance: who hast peysed in thy hand, both Heauen and earth. We therfore +warned by the Sacred word, to Consider thy Creatures: and by that +consideration, to wynne a glyms (as it were,) or shaddow of +perceiuerance, that thy wisedome, might, and goodnes is infinite, and +vnspeakable, in thy Creatures declared: And being farder aduertised, by +thy mercifull goodnes, that, three principall wayes, were, of the, vsed +in Creation of all thy Creatures, namely, _Number_, _Waight_ and +_Measure_, And for as much as, of _Number_ and _Measure_, the two Artes +(auncient, famous, and to humaine vses most necessary,) are, all ready, +sufficiently knowen and extant: This third key, we beseche thee (through +thy accustomed goodnes,) that it may come to the nedefull and sufficient +knowledge, of such thy Seruauntes, as in thy workemanship, would gladly +finde, thy true occasions (purposely of the vsed) whereby we should +glorifie thy name, and shew forth (to the weaklinges in faith) thy +wondrous wisedome and Goodnes. Amen.” + + +Meruaile nothing at this pang (godly frend, you Gentle and zelous +Student.) An other day, perchaunce, you will perceiue, what occasion +moued me. Here, as now, I will giue you some ground, and withall some +shew, of certaine commodities, by this Arte arising. And bycause this +Arte is rare, my wordes and practises might be to darke: vnleast you had +some light, holden before the matter: and that, best will be, in giuing +you, out of _Archimedes_ demonstrations, a few principal Conclusions, as +foloweth. + + +1.+ + + +The Superficies of euery Liquor, by it selfe consistyng, and in + quyet, is Sphæricall: the centre whereof, is the same, which is the + centre of the Earth.+ + + +2.+ + + +If Solide Magnitudes, being of the same bignes, or quãtitie, that + any Liquor is, and hauyng also the same Waight: be let downe into + the same Liquor, they will settle downeward, so, that no parte of + them, shall be aboue the Superficies of the Liquor: and yet + neuertheles, they will not sinke vtterly downe, or drowne.+ + + +3.+ + + +If any Solide Magnitude beyng Lighter then a Liquor, be let downe + into the same Liquor, it will settle downe, so farre into the same + Liquor, that so great a quantitie of that Liquor, as is the parte of + the Solid Magnitude, settled downe into the same Liquor: is in + Waight, æquall, to the waight of the whole Solid Magnitude.+ + + +4.+ + + +Any Solide Magnitude, Lighter then a Liquor, forced downe into the + same Liquor, will moue vpward, with so great a power, by how much, + the Liquor hauyng æquall quantitie to the whole Magnitude, is + heauyer then the same Magnitude.+ + + +5.+ + + +Any Solid Magnitude, heauyer then a Liquor, beyng let downe into + the same Liquor, will sinke downe vtterly: And wilbe in that Liquor, + Lighter by so much, as is the waight or heauynes of the Liquor, + hauing bygnes or quantitie, æquall to the Solid Magnitude.+ + + +6.+ + + [I. D. + The Cutting of a Sphære according to any proportion assigned + may by this proposition be done Mechanically by tempering + Liquor to a certayne waight in respect of the waight of the + Sphære therein Swymming.] + + +If any Solide Magnitude, Lighter then a Liquor, be let downe into + the same Liquor, the waight of the same Magnitude, will be, to the + Waight of the Liquor. (Which is æquall in quantitie to the whole + Magnitude,) in that proportion, that the parte, of the Magnitude + settled downe, is to the whole Magnitude.+ + + +By these verities, great Errors may be reformed, in Opinion of the +Naturall Motion of thinges, Light and Heauy. Which errors, are in +Naturall Philosophie (almost) of all mẽ allowed: to much trusting to +Authority: and false Suppositions. As, +Of any two bodyes, the heauyer, +to moue downward faster then the lighter.+ + + [A common error, noted.] + +This error, is not first by me, Noted: but by one _Iohn Baptist de +Benedictis_. The chief of his propositions, is this: which seemeth a +Paradox. + + ++If there be two bodyes of one forme, and of one kynde, æquall in +quantitie or vnæquall, + + [A paradox.] + +they will moue by æquall space, in æquall tyme: So that both theyr +mouynges be in ayre, or both in water: or in any one Middle.+ + + +Hereupon, in the feate of +Gunnyng+, + + [N. T.] + +certaine good discourses (otherwise) may receiue great amendement, and +furderance. + + [The wonderfull vse of these Propositions.] + +In the entended purpose, also, allowing somwhat to the imperfection of +Nature: not aunswerable to the precisenes of demonstration. Moreouer, by +the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth, +the Fire, may be nerely, knowen, how light or heauy they are (Naturally) +in their assigned partes: or in the whole. And then, to thinges +Elementall, turning your practise: you may deale for the proportion of +the Elementes, in the thinges Compounded. Then, to the proportions of +the Humours in Man: their waightes: and the waight of his bones, and +flesh. &c. Than, by waight, to haue consideration of the Force of man, +any maner of way: in whole or in part. Then, may you, of Ships water +drawing, diuersly, in the Sea and in fresh water, haue pleasant +consideration: and of waying vp of any thing, sonken in Sea or in fresh +water &c. And (to lift vp your head a loft:) by waight, you may, as +precisely, as by any instrument els, measure the Diameters of _Sonne_ +and _Mone. &c._ Frende, I pray you, way these thinges, with the iust +Balance of Reason. And you will finde Meruailes vpon Meruailes: And +esteme one Drop of Truth (yea in Naturall Philosophie) more worth, then +whole Libraries of Opinions, vndemonstrated: or not aunswering to +Natures Law, and your experience. Leauing these thinges, thus: I will +giue you two or three, light practises, to great purpose: and so finish +my Annotation _Staticall_. In Mathematicall matters, by the Mechaniciens +ayde, we will behold, here, the Commodity of waight. + + [The practise Staticall, to know the proportion, betwene + the Cube, and the Sphære.] + +Make a Cube, of any one Vniforme: and through like heauy stuffe: of the +same Stuffe, make a Sphære or Globe, precisely, of a Diameter æquall to +the Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne, +Lead, Siluer. &c. (being, as I sayd, of like nature, condition, and like +waight throughout.) And you may, by Say Balance, haue prepared a great +number of the smallest waightes: which, by those Balance can be +discerned or tryed: and so, haue proceded to make you a perfect Pyle, +company & Number of waightes: to the waight of six, eight, or twelue +pound waight: most diligently tryed, all. And of euery one, the Content +knowen, in your least waight, that is wayable. [They that can not haue +these waightes of precisenes: may, by Sand, Vniforme, and well dusted, +make them a number of waightes, somewhat nere precisenes: by halfing +euer the Sand: they shall, at length, come to a least common waight. +Therein, I leaue the farder matter, to their discretion, whom nede shall +pinche.] The _Venetians_ consideration of waight, may seme precise +enough: by eight descentes progressionall, * halfing, from a grayne. + + [I. D. + * For, so, haue you .256. partes of a Graine.] + +Your Cube, Sphære, apt Balance, and conuenient waightes, being ready: +fall to worke.❉. First, way your Cube. Note the Number of the waight. +Way, after that, your Sphære. Note likewise, the Nũber of the waight. If +you now find the waight of your Cube, to be to the waight of the Sphære, +as 21. is to 11: Then you see, how the Mechanicien and _Experimenter_, +without Geometrie and Demonstration, are (as nerely in effect) tought +the proportion of the Cube to the Sphere: as I haue demonstrated it, in +the end of the twelfth boke of _Euclide_. Often, try with the same Cube +and Sphære. Then, chaunge, your Sphære and Cube, to an other matter: or +to an other bignes: till you haue made a perfect vniuersall Experience +of it. Possible it is, that you shall wynne to nerer termes, in the +proportion. + + +When you haue found this one certaine Drop of Naturall veritie, procede +on, to Inferre, and duely to make assay, of matter depending. As, +bycause it is well demonstrated, that a Cylinder, whose heith, and +Diameter of his base, is æquall to the Diameter of the Sphære, is +Sesquialter to the same Sphære (that is, as 3. to 2:) To the number of +the waight of the Sphære, adde halfe so much, as it is: and so haue you +the number of the waight of that Cylinder. Which is also Comprehended of +our former Cube: So, that the base of that Cylinder, is a Circle +described in the Square, which is the base of our Cube. But the Cube and +the Cylinder, being both of one heith, haue their Bases in the same +proportion, in the which, they are, one to an other, in their Massines +or Soliditie. But, before, we haue two numbers, expressing their +Massines, Solidities, and Quantities, by waight: wherfore, + + [* =The proportion of the Square to the Circle inscribed.=] + +we haue * the proportion of the Square, to the Circle, inscribed in the +same Square. And so are we fallen into the knowledge sensible, and +Experimentall of _Archimedes_ great Secret: of him, by great trauaile of +minde, sought and found. Wherfore, to any Circle giuen, you can giue a +Square æquall: + + [* =The Squaring of the Circle, Mechanically.=] + +* as I haue taught, in my Annotation, vpon the first proposition of the +twelfth boke, And likewise, to any Square giuen, you may giue a Circle +æquall: + + [* =To any Square geuen, to geue a Circle, equall.=] + +* If you describe a Circle, which shall be in that proportion, to your +Circle inscribed, as the Square is to the same Circle: This, you may do, +by my Annotations, vpon the second proposition of the twelfth boke of +_Euclide_, in my third Probleme there. Your diligence may come to a +proportion, of the Square to the Circle inscribed, nerer the truth, then +is the proportion of 14. to 11. And consider, that you may begyn at the +Circle and Square, and so come to conclude of the Sphære, & the Cube, +what their proportion is: as now, you came from the Sphære to the +Circle. For, of Siluer, or Gold, or Latton Lamyns or plates (thorough +one hole drawẽ, as the maner is) if you make a Square figure & way it: +and then, describing theron, the Circle inscribed: & cut of, & file +away, precisely (to the Circle) the ouerplus of the Square: you shall +then, waying your Circle, see, whether the waight of the Square, be to +your Circle, as 14. to 11. As I haue Noted, in the beginning of +_Euclides_ twelfth boke. &c. after this resort to my last proposition, +vpon the last of the twelfth. And there, helpe your selfe, to the end. +And, here, Note this, by the way. + + [Note Squaring of the Circle without knowledge of the + proportion betwene Circumference and Diameter.] + +That we may Square the Circle, without hauing knowledge of the +proportion, of the Circumference to the Diameter: as you haue here +perceiued. And otherwayes also, I can demonstrate it. So that, many haue +cumberd them selues superfluously, by trauailing in that point first, +which was not of necessitie, first: and also very intricate. And easily, +you may, (and that diuersly) come to the knowledge of the Circumference: +the Circles Quantitie, being first knowen. Which thing, I leaue to your +consideration: making hast to despatch an other Magistrall Probleme: and +to bring it, nerer to your knowledge, and readier dealing with, then the +world (before this day,) had it for you, that I can tell of. And that +is, _A Mechanicall Dubblyng of the Cube: &c._ Which may, thus, be done: + + [To Dubble the Cube redily: by Art Mechanicall: depending + vppon Demonstration Mathematicall.] + ++Make of Copper plates, or Tyn plates, a foursquare vpright Pyramis, or +a Cone: perfectly fashioned in the holow, within. Wherin, let great +diligence be vsed, to approche (as nere as may be) to the Mathematicall +perfection of those figures. At their bases, let them be all open: euery +where, els, most close, and iust to. From the vertex, to the +Circumference of the base of the Cone: & to the sides of the base of the +Pyramis:+ + + [=I. D.= + =The 4. sides of this Pyramis must be 4. Isosceles + Triangles alike and æquall.=] + ++Let 4. straight lines be drawen, in the inside of the Cone and Pyramis: +makyng at their fall, on the perimeters of the bases, equall angles on +both sides them selues, with the sayd perimeters. These 4. lines (in the +Pyramis: and as many, in the Cone) diuide: one, in 12. æquall partes: +and an other, in 24. an other, in 60, and an other, in 100. (reckenyng +vp from the vertex.) Or vse other numbers of diuision, as experience +shall teach you.+ + + [=I. D.= + =* In all workinges with this Pyramis or Cone, Let their + Situations be in all Pointes and Conditions, alike, + or all one: while you are about one Worke. Els you + will erre.=] + ++Then, * set your Cone or Pyramis, with the vertex downward, +perpendicularly, in respect of the Base. (Though it be otherwayes, it +hindreth nothyng.) So let thẽ most stedily be stayed.+ Now, if there be +a Cube, which you wold haue Dubbled. Make you a prety Cube of Copper, +Siluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, or +Cubik coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so +proportiõ in respect of your Pyramis or Cone, that the Pyramis or Cone, +will be hable to conteine the waight of them, in water, 3. or 4. times: +at the least: what stuff so euer they be made of. Let not your Solid +angle, at the vertex, be to sharpe: but that the water may come with +ease, to the very vertex, of your hollow Cone or Pyramis. Put one of +your Solid Cubes in a Balance apt: take the waight therof exactly in +water. Powre that water, (without losse) into the hollow Pyramis or +Cone, quietly. Marke in your lines, what numbers the water Cutteth: Take +the waight of the same Cube againe: in the same kinde of water, which +you had before: + + [=I. D.= + =* Consider well whan you must put your waters togyther: + and whan, you must empty your first water, out of your + Pyramis or Cone. Els you will erre.=] + +put that* also, into the Pyramis or Cone, where you did put the first. +Marke now againe, in what number or place of the lines, the water +Cutteth them. Two wayes you may conclude your purpose: it is to wete, +either by numbers or lines. By numbers: as, if you diuide the side of +your Fundamentall Cube into so many æquall partes, as it is capable of, +conueniently, with your ease, and precisenes of the diuision. For, as +the number of your first and lesse line (in your hollow Pyramis or +Cone,) is to the second or greater (both being counted from the vertex) +so shall the number of the side of your Fundamentall Cube, be to the +nũber belonging to the Radicall side, of the Cube, dubble to your +Fundamentall Cube: Which being multiplied Cubik wise, will sone shew it +selfe, whether it be dubble or no, to the Cubik number of your +Fundamentall Cube. By lines, thus: As your lesse and first line, (in +your hollow Pyramis or Cone,) is to the second or greater, so let the +Radical side of your Fundamẽtall Cube, be to a fourth proportionall +line, by the 12. proposition, of the sixth boke of _Euclide_. Which +fourth line, shall be the Rote Cubik, or Radicall side of the Cube, +dubble to your Fundamentall Cube: which is the thing we desired. + + [☞ God be thanked for this Inuention, & the fruite ensuing.] + +For this, may I (with ioy) say, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ: thanking the +holy and glorious Trinity: hauing greater cause therto, then + + [* Vitruuius. Lib. 9. Cap. 3.] + +* _Archimedes_ had (for finding the fraude vsed in the Kinges Crowne, of +Gold): as all men may easily Iudge: by the diuersitie of the frute +following of the one, and the other. Where I spake before, of a hollow +Cubik Coffen: the like vse, is of it: and without waight. Thus. Fill it +with water, precisely full, and poure that water into your Pyramis or +Cone. And here note the lines cutting in your Pyramis or Cone. Againe, +fill your coffen, like as you did before. Put that Water, also, to the +first. Marke the second cutting of your lines. Now, as you proceded +before, so must you here procede. + + [* Note.] + +* And if the Cube, which you should Double, be neuer so great: you haue, +thus, the proportion (in small) betwene your two litle Cubes: And then, +the side, of that great Cube (to be doubled) being the third, will haue +the fourth, found, to it proportionall: by the 12. of the sixth of +Euclide. + + + [Note, as concerning the Sphæricall Superficies of the Water.] + +Note, that all this while, I forget not my first Proposition Staticall, +here rehearsed: that, the Superficies of the water, is Sphæricall. +Wherein, vse your discretion: to the first line, adding a small heare +breadth, more: and to the second, halfe a heare breadth more, to his +length. For, you will easily perceaue, that the difference can be no +greater, in any Pyramis or Cone, of you to be handled. Which you shall +thus trye. _For finding the swelling of the water aboue leuell._ + + [☞] + +“Square the Semidiameter, from the Centre of the earth, to your first +Waters Superficies. Square then, halfe the Subtendent of that watry +Superficies (which Subtendent must haue the equall partes of his +measure, all one, with those of the Semidiameter of the earth to your +watry Superficies): Subtracte this square, from the first: Of the +residue, take the Rote Square. That Rote, Subtracte from your first +Semidiameter of the earth to your watry Superficies: that, which +remaineth, is the heith of the water, in the middle, aboue the leuell.” +Which, you will finde, to be a thing insensible. And though it were +greatly sensible, * + + [* Note.] + +yet, by helpe of my sixt Theoreme vpon the last Proposition of Euclides +twelfth booke, noted: you may reduce all, to a true Leuell. But, farther +diligence, of you is to be vsed, against accidentall causes of the +waters swelling: as by hauing (somwhat) with a moyst Sponge, before, +made moyst your hollow Pyramis or Cone, will preuent an accidentall +cause of Swelling, &c. Experience will teach you abundantly: with great +ease, pleasure, and cõmoditie. + + +Thus, may you Double the Cube Mechanically, Treble it, and so forth, in +any proportion. + + [Note this Abridgement of Dubbling the Cube. &c.] + +Now will I Abridge your paine, cost, and Care herein. Without all +preparing of your Fundamentall Cubes: you may (alike) worke this +Conclusion. For, that, was rather a kinde of Experimentall demõstration, +then the shortest way: and all, vpon one Mathematicall Demonstration +depending. “Take water (as much as conueniently will serue your turne: +as I warned before of your Fundamentall Cubes bignes) Way it precisely. +Put that water, into your Pyramis or Cone. Of the same kinde of water, +then take againe, the same waight you had before: put that likewise into +the Pyramis or Cone. For, in eche time, your marking of the lines, how +the Water doth cut them, shall geue you the proportion betwen the +Radicall sides, of any two Cubes, wherof the one is Double to the other: +working as before I haue taught you: + + [* ☞ Note.] + +* sauing that for you Fundamentall Cube his Radicall side: here, you may +take a right line, at pleasure.” + + +Yet farther proceding with our droppe of Naturall truth: + + [To giue Cubes one to the other in any proportion, + Rationall or Irrationall.] + ++you may (now) geue Cubes, one to the other, in any proportiõ geuẽ: +Rationall or Irrationall+: on this maner. Make a hollow Parallelipipedon +of Copper or Tinne: with one Base wãting, or open: as in our Cubike +Coffen. Frõ the bottome of that Parallelipipedon, raise vp, many +perpendiculars, in euery of his fower sides. Now if any proportion be +assigned you, in right lines: Cut one of your perpendiculars (or a line +equall to it, or lesse then it) likewise: by the 10. of the sixth of +Euclide. And those two partes, set in two sundry lines of those +perpendiculars (or you may set them both, in one line) making their +beginninges, to be, at the base: and so their lengthes to extend vpward. +Now, set your hollow Parallelipipedon, vpright, perpendicularly, +steadie. Poure in water, handsomly, to the heith of your shorter line. +Poure that water, into the hollow Pyramis or Cone. Marke the place of +the rising. Settle your hollow Parallelipipedon againe. Poure water into +it: vnto the heith of the second line, exactly. + + [* Emptying the first.] + +Poure that water * duely into the hollow Pyramis or Cone: Marke now +againe, where the water cutteth the same line which you marked before. +For, there, as the first marked line, is to the second: So shall the two +Radicall sides be, one to the other, of any two Cubes: which, in their +Soliditie, shall haue the same proportion, which was at the first +assigned: were it Rationall or Irrationall. + + +Thus, in sundry waies you may furnishe your selfe with such straunge and +profitable matter: which, long hath bene wished for. And though it be +Naturally done and Mechanically: yet hath it a good Demonstration +Mathematicall. + + [=The demonstrations of this Dubbling of the Cube, and of the +rest.=] + +Which is this: Alwaies, you haue two Like Pyramids: or two Like Cones, +in the proportions assigned: and like Pyramids or Cones, are in +proportion, one to the other, in the proportion of their Homologall +sides (or lines) tripled. Wherefore, if to the first, and second lines, +found in your hollow Pyramis or Cone, you ioyne a third and a fourth, in +continuall proportion: that fourth line, shall be to the first, as the +greater Pyramis or Cone, is to the lesse: by the 33. of the eleuenth of +Euclide. If Pyramis to Pyramis, or Cone to Cone, be double, + + [I. D. + = * Hereby, helpe your self to become a præcise practiser. + And so consider, how, nothing at all, you are hindred + (sensibly) by the Conuexitie of the water.=] + +then shall * Line to Line, be also double, &c. But, as our first line, +is to the second, so is the Radicall side of our Fundamentall Cube, to +the Radicall side of the Cube to be made, or to be doubled: and +therefore, to those twaine also, a third and a fourth line, in +continuall proportion, ioyned: will geue the fourth line in that +proportion to the first, as our fourth Pyramidall, or Conike line, was +to his first: but that was double, or treble, &c. as the Pyramids or +Cones were, one to an other (as we haue proued) therfore, this fourth, +shalbe also double or treble to the first, as the Pyramids or Cones were +one to an other: But our made Cube, is described of the second in +proportion, of the fower proportionall lines: + + [= * By the 33. of the eleuenth booke of Euclide.=] + +therfore * as the fourth line, is to the first, so is that Cube, to the +first Cube: and we haue proued the fourth line, to be to the first, as +the Pyramis or Cone, is to the Pyramis or Cone: Wherefore the Cube is to +the Cube, as Pyramis is to Pyramis, or Cone is to Cone. + + [I. D. + = * And your diligence in practise, can so (in waight of + water) performe it: Therefore, now, you are able to geue + good reason of your whole doing.=] + +But we * Suppose Pyramis to Pyramis, or Cone to Cone, to be double or +treble. &c. Therfore Cube, is to Cube, double, or treble, &c. Which was +to be demonstrated. And of the Parallelipipedõ, it is euidẽt, that the +water Solide Parallelipipedons, are one to the other, as their heithes +are, seing they haue one base. Wherfore the Pyramids or Cones, made of +those water Parallelipipedons, are one to the other, as the lines are +(one to the other) betwene which, our proportion was assigned. But the +Cubes made of lines, after the proportiõ of the Pyramidal or Conik +_homologall_ lines, are one to the other, as the Pyramides or Cones are, +one to the other (as we before did proue) therfore, the Cubes made, +shalbe one to the other, as the lines assigned, are one to the other: +Which was to be demonstrated. Note. + + [* _Note this Corollary._] + +* This, my Demonstratiõ is more generall, then onely in Square Pyramis +or Cone: Consider well. Thus, haue I, both Mathematically and +Mechanically, ben very long in wordes: yet (I trust) nothing tedious to +them, who, to these thinges, are well affected. And verily I am forced +(auoiding prolixitie) to omit sundry such things, easie to be practised: +which to the Mathematicien, would be a great Threasure: and to the +Mechanicien, no small gaine. + + [* The great Commodities following of these new Inuentions.] + +* Now may you, +Betwene two lines giuen, finde two middle proportionals, +in Continuall proportion: by the hollow Parallelipipedon, and the hollow +Pyramis, or Cone.+ Now, any Parallelipipedon rectangle being giuen: thre +right lines may be found, proportionall in any proportion assigned, of +which, shal be produced a Parallelipipedon, æquall to the +Parallelipipedon giuen. Hereof, I noted somwhat, vpon the 36. +proposition, of the 11. boke of _Euclide_. Now, all those thinges, which +_Vitruuius_ in his Architecture, specified hable to be done, by dubbling +of the Cube: Or, by finding of two middle proportionall lines, betwene +two lines giuen, may easely be performed. Now, that Probleme, which I +noted vnto you, in the end of my Addition, vpon the 34. of the 11. boke +of _Euclide_, is proued possible. Now, may any regular body, be +Transformed into an other, &c. Now, any regular body: any Sphere, yea +any Mixt Solid: and (that more is) Irregular Solides, may be made (in +any proportiõ assigned) like vnto the body, first giuen. Thus, of a +_Manneken_, (as the _Dutch_ Painters terme it) in the same _Symmetrie_, +may a Giant be made: and that, with any gesture, by the Manneken vsed: +and contrarywise. Now, may you, of any Mould, or Modell of a Ship, make +one, of the same Mould (in any assigned proportion) bigger or lesser. + + [* ☞] + +Now, may you, of any * Gunne, or little peece of ordinaũce, make an +other, with the same _Symmetrie_ (in all pointes) as great, and as +little, as you will. Marke that: and thinke on it. Infinitely, +may you +apply this, so long sought for, and now so easily concluded: and +withall, so willingly and frankly communicated to such, as faithfully +deale with vertuous studies.+ + + [Such is the Fruite of the Mathematicall Sciences and Artes.] + +Thus, can the Mathematicall minde, deale Speculatiuely in his own Arte: +and by good meanes, Mount aboue the cloudes and sterres: And thirdly, he +can, by order, Descend, to frame Naturall thinges, to wonderfull vses: +and when he list, retire home into his owne Centre: and there, prepare +more Meanes, to Ascend or Descend by: and, all, to the glory of God, and +our honest delectation in earth. + + +Although, the Printer, hath looked for this Præface, a day or two, yet +could I not bring my pen from the paper, before I had giuen you +comfortable warning, and brief instructions, of some of the Commodities, +by _Statike_, hable to be reaped: In the rest, I will therfore, be as +brief, as it is possible: and with all, describing them, somwhat +accordingly. And that, you shall perceiue, by this, which in order +commeth next. For, wheras, it is so ample and wonderfull, that, an whole +yeare long, one might finde fruitfull matter therin, to speake of: and +also in practise, is a Threasure endeles: yet will I glanse ouer it, +with wordes very few. + + +This do I call +‡Anthropographie‡+. Which is an Art restored, and of my +preferment to your Seruice. I pray you, thinke of it, as of one of the +chief pointes, of Humane knowledge. Although it be, but now, first +Cõfirmed, with this new name: yet the matter, hath from the beginning, +ben in consideration of all perfect Philosophers. +Anthropographie, is +the description of the Number, Measure, Waight, figure, Situation, and +colour of euery diuerse thing, conteyned in the perfect body of MAN: +with certain knowledge of the Symmetrie, figure, waight, +Characterization, and due locall motion, of any parcell of the sayd +body, assigned: and of Nũbers, to the sayd parcell appertainyng.+ This, +is the one part of the Definition, mete for this place: Sufficient to +notifie, the particularitie, and excellency of the Arte: and why it is, +here, ascribed to the Mathematicals. Yf the description of the heauenly +part of the world, had a peculier Art, called _Astronomie:_ If the +description of the earthly Globe, hath his peculier arte, called +_Geographie_. If the Matching of both, hath his peculier Arte, called +_Cosmographie:_ Which is the Descriptiõ of the whole, and vniuersall +frame of the world: Why should not the description of + + [MAN is the Lesse World.] + +him, who is the Lesse world: and, frõ the beginning, called +_Microcosmus_ (that is. _The Lesse World._) And for whose sake, and +seruice, all bodily creatures els, were created: Who, also, +participateth with Spirites, and Angels: and is made to the Image and +similitude of _God_: haue his peculier Art? and be called the _Arte of +Artes_: rather, then, either to want a name, or to haue to base and +impropre a name? You must of sundry professions, borow or challenge +home, peculier partes hereof: and farder procede: as, God, Nature, +Reason and Experience shall informe you. The Anatomistes will restore to +you, some part: The Physiognomistes, some: The Chyromantistes some. The +Metaposcopistes, some: The excellent, _Albert Durer_, a good part: the +Arte of Perspectiue, will somwhat, for the Eye, helpe forward: +_Pythagoras_, _Hipocrates_, _Plato_, _Galenus_, _Meletius_, & many other +(in certaine thinges) will be Contributaries. And farder, the Heauen, +the Earth, and all other Creatures, will eche shew, and offer their +Harmonious seruice, to fill vp, that, which wanteth hereof: and with +your own Experience, concluding: you may Methodically register the +whole, for the posteritie: Whereby, good profe will be had, of our +Harmonious, and + + [Micro Cosmus.] + +Microcosmicall constitution. + + [* ☞] + +The outward Image, and vew hereof: to the Art of _Zographie_ and +Painting, to Sculpture, and Architecture: (for Church, House, Fort, or +Ship) is most necessary and profitable: for that, it is the chiefe base +and foundation of them. + + [* Lib. 3. Cap. 1.] + +Looke in * _Vitruuius_, whether I deale sincerely for your behoufe, or +no. Looke in _Albertus Durerus_, _De Symmetria humani Corporis_. Looke +in the 27. and 28. Chapters, of the second booke, _De occulta +Philosophia_. Consider the _Arke_ of _Noe_. And by that, wade farther. +Remember the _Delphicall Oracle NOSCE TEIPSVM_ +_(Knowe thy selfe)_+ so +long agoe pronounced: of so many a Philosopher repeated: and of the +_Wisest_ attempted: And then, you will perceaue, how long agoe, you haue +bene called to the Schole, where this Arte might be learned. Well. I am +nothing affrayde, of the disdayne of some such, as thinke Sciences and +Artes, to be but Seuen. Perhaps, those Such, may, with ignorance, and +shame enough, come short of them Seuen also: and yet neuerthelesse they +can not prescribe a certaine number of Artes: and in eche, certaine +vnpassable boundes, to God, Nature, and mans Industrie. New Artes, dayly +rise vp: and there was no such order taken, that, + + [☞] + +All Artes, should in one age, or in one land, or of one man, be made +knowen to the world. Let vs embrace the giftes of God, and wayes to +wisedome, in this time of grace, from aboue, continually bestowed on +them, who thankefully will receiue them: _Et bonis Omnia Cooperabuntur +in bonum._ + + ++‡Trochilike,‡ is that Art Mathematicall, which demonstrateth the +properties of all Circular motions, Simple and Compounde.+ And bycause +the frute hereof, vulgarly receiued, is in Wheles, it hath the name of +_Trochilike:_ as a man would say, _Whele Art_. By this art, a Whele may +be geuen which shall moue ones about, in any tyme assigned. Two Wheles +may be giuen, whose turnynges about in one and the same tyme, (or equall +tymes), shall haue, one to the other, any proportion appointed. By +Wheles, may a straight line be described: Likewise, a Spirall line in +plaine, Conicall Section lines, and other Irregular lines, at pleasure, +may be drawen. These, and such like, are principall Conclusions of this +Arte: and helpe forward many pleasant and profitable Mechanicall workes: + + [Saw Milles.] + +As Milles, to Saw great and very long Deale bordes, no man being by. +Such haue I seene in Germany: and in the Citie of Prage: in the kingdome +of Bohemia: Coyning Milles, Hand Milles for Corne grinding: And all +maner of Milles, and Whele worke: By Winde, Smoke, Water, Waight, +Spring, Man or Beast, moued. Take in your hand, _Agricola De re +Metallica:_ and then shall you (in all Mines) perceaue, how great nede +is, of Whele worke. By Wheles, straunge workes and incredible, are done: +as will, in other Artes hereafter, appeare. A wonderfull example of +farther possibilitie, and present commoditie, was sene in my time, in a +certaine Instrument: which by the Inuenter and Artificer (before) was +solde for xx. Talentes of Golde: and then had (by misfortune) receaued +some iniurie and hurt: And one _Ianellus_ of _Cremona_ did mend the +same, and presented it vnto the Emperour _Charles_ the fifth. +_Hieronymus Cardanus_, can be my witnesse, that therein, was one Whele, +which moued, and that, in such rate, that, in 7000. yeares onely, his +owne periode should be finished. A thing almost incredible: But how +farre, I keepe me within my boundes: very many men (yet aliue) can tell. + + ++‡Helicosophie‡+, is nere Sister to _Trochilike:_ and is, +An Arte +Mathematicall, which demonstrateth the designing of all Spirall lines in +Plaine, on Cylinder, Cone, Sphære, Conoid, and Sphæroid, and their +properties appertayning.+ The vse hereof, in _Architecture_, and diuerse +Instrumentes and Engines, is most necessary. For, in many thinges, the +Skrue worketh the feate, which, els, could not be performed. By helpe +hereof, + + [* Atheneus Lib. 5. cap. 8.] + +it is * recorded, that, where all the power of the Citie of Syracusa, +was not hable to moue a certaine Ship (being on ground) mightie +_Archimedes_, setting to, his Skruish Engine, caused _Hiero_ the king, +by him self, at ease, to remoue her, as he would. + + [Proclus. Pag. 18.] + +Wherat, the King wondring: Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει +λέγοντι πιϛευτεόν. _From this day, forward_ (said the King) _Credit +ought to be giuen to Archimedes, what soeuer he sayth._ + + ++‡Pneumatithmie‡ demonstrateth by close hollow Geometricall Figures, +(regular and irregular) the straunge properties (in motion or stay) of +the Water, Ayre, Smoke, and Fire, in theyr cõtinuitie, and as they are +ioyned to the Elementes next them.+ This Arte, to the Naturall +Philosopher, is very proffitable: to proue, that _Vacuum_, or _Emptines_ +is not in the world. And that, all Nature, abhorreth it so much: that, +contrary to ordinary law, the Elementes will moue or stand. As, Water to +ascend: rather then betwene him and Ayre, Space or place should be left, +more then (naturally) that quãtitie of Ayre requireth, or can fill. +Againe, Water to hang, and not descend: rather then by descending, to +leaue Emptines at his backe. The like, is of Fire and Ayre: they will +descend: when, either, their Cõtinuitie should be dissolued: or their +next Element forced from them. And as they will not be extended, to +discontinuitie: So, will they not, nor yet of mans force, can be prest +or pent, in space, not sufficient and aunswerable to their bodily +substance. Great force and violence will they vse, to enioy their +naturall right and libertie. + + [To go to the bottom of the Sea without daunger.] + +Hereupon, two or three men together, by keping Ayre vnder a great +Cauldron, and forcyng the same downe, orderly, may without harme descend +to the Sea bottome: and continue there a tyme &c. Where, Note, how the +thicker Element (as the Water) giueth place to the thynner (as, is the +ayre:) and receiueth violence of the thinner, in maner. &c. Pumps and +all maner of Bellowes, haue their ground of this Art: and many other +straunge deuises. As, _Hydraulica_, Organes goyng by water. &c. Of this +Feat, (called commonly _Pneumatica_,) goodly workes are extant, both in +Greke, and Latin. With old and learned Schole men, it is called +_Scientia de pleno & vacuo._ + + ++‡Menadrie‡, is an Arte Mathematicall, which demonstrateth, how, aboue +Natures vertue and power simple: Vertue and force may be multiplied: and +so, to direct, to lift, to pull to, and to put or cast fro, any +multiplied or simple, determined Vertue, Waight or Force: naturally, +not, so, directible or moueable.+ Very much is this Art furdred by other +Artes: as, in some pointes, by _Perspectiue_: in some, by _Statike_: in +some, by _Trochilike_: and in other, by _Helicosophie_: and +_Pneumatithmie_. By this Art, all Cranes, Gybbettes, & Ingines to lift +vp, or to force any thing, any maner way, are ordred: and the certaine +cause of their force, is knowne: As, the force which one man hath with +the Duche waghen Racke: therwith, to set vp agayne, a mighty waghen +laden, being ouerthrowne. The force of the Crossebow Racke, is +certainly, here, demonstrated. The reason, why one mã, doth with a +leauer, lift that, which Sixe men, with their handes onely, could not, +so easily do. By this Arte, in our common Cranes in London, where powre +is to Crane vp, the waight of 2000. pound: by two Wheles more (by good +order added) Arte concludeth, that there may be Craned vp 200000. pound +waight &c. So well knew _Archimedes_ this Arte: that he alone, with his +deuises and engynes, (twise or thrise) spoyled and discomfited the whole +Army and Hoste of the Romaines, besieging _Syracusa_, + + [=Plutarchus in Marco Marcello.=] + +_Marcus Marcellus the Consul_, being their Generall Capitaine. + + [=Synesius in Epistolis.=] + +Such huge Stones, so many, with such force, and so farre, did he with +his engynes hayle among them, out of the Citie. + + [=Polybius.=] + + [=Plinius.=] + + [=Quintilianus.=] + + [=T. Liuius.=] + +And by Sea likewise: though their Ships might come to the walls of +_Syracusa_, yet hee vtterly confounded the Romaine Nauye. What with his +mighty Stones hurlyng: + + [=* Athenæus.=] + +what with Pikes of * 18 fote long, made like shaftes: which he forced +almost a quarter of a myle. What, with his catchyng hold of their Shyps, +and hoysing them vp aboue the water, and suddenly letting them fall into +the Sea againe: + + [= * Galenus.=] + + [=Anthemius.=] + +what with his * Burning Glasses: by which he fired their other Shippes a +far-of: what, with his other pollicies, deuises, and engines, he so +manfully acquit him selfe: that all the Force, courage, and pollicie of +the Romaines (for a great season) could nothing preuaile, for the +winning of Syracusa. Wherupon, the Romanes named _Archimedes_, +_Briareus_, and _Centimanus_. _Zonaras_ maketh mention of one _Proclus_, +who so well had perceiued _Archimedes_ Arte of _Menadrie_, and had so +well inuented of his owne, that with his Burning Glasses, + + [Burning Glasses.] + +being placed vpon the walles of Bysance, he multiplied so the heate of +the Sunne, and directed the beames of the same against his enemies Nauie +with such force, and so sodeinly (like lightening) that he burned and +destroyed both man and ship. And _Dion_specifieth of _Priscus_, +a _Geometricien_ in Bysance, who inuented and vsed sondry Engins, of +Force multiplied: Which was cause, that the _Emperour Seuerus_ pardoned +him, his life, after he had wonne Bysance: Bycause he honored the Arte, +wytt, and rare industrie of _Priscus_. But nothing inferior to the +inuention of these engines of Force, was the inuention of Gunnes. + + [Gunnes.] + +Which, from an English man, had the occasion and order of first +inuenting: though in an other land, and by other men, it was first +executed. And they that should see the record, where the occasion and +order generall, of Gunning, is first discoursed of, would thinke: that, +“small thinges, slight, and cõmon: comming to wise mens consideration, +and industrious mens handling, may grow to be of force incredible.” + + ++‡Hypogeiodie‡, is an Arte Mathematicall, demonstratyng, how, vnder the +Sphæricall Superficies of the earth, at any depth, to any perpendicular +line assigned (whose distance from the perpendicular of the entrance: +and the Azimuth, likewise, in respect of the said entrance, is knowen) +certaine way may be præscribed and gone: And how, any way aboue the +Superficies of the earth designed, may vnder earth, at any depth +limited, be kept: goyng alwayes, perpendicularly, vnder the way, on +earth designed: And, contrarywise, Any way, (straight or croked,) vnder +the earth, beyng giuen: vppon the vtface, or Superficies of the earth, +to Lyne out the same: So, as, from the Centre of the earth, +perpendiculars drawen to the Sphæricall Superficies of the earth, shall +precisely fall in the Correspondent pointes of those two wayes. This, +with all other Cases and circumstances herein, and appertenances, this +Arte demonstrateth.+ This Arte, is very ample in varietie of +Conclusions: and very profitable sundry wayes to the Common Wealth. The +occasion of my Inuenting this Arte, was at the request of two Gentlemen, +who had a certaine worke (of gaine) vnder ground: and their groundes did +ioyne ouer the worke: and by reason of the crokednes, diuers depthes, +and heithes of the way vnder ground, they were in doubt, and at +controuersie, vnder whose ground, as then, the worke was. The name onely +(before this) was of me published, _De Itinere Subterraneo_: The rest, +be at Gods will. For Pioners, Miners, Diggers for Mettalls, Stone, Cole, +and for secrete passages vnder ground, betwene place and place (as this +land hath diuerse) and for other purposes, any man may easily perceaue, +both the great fruite of this Arte, and also in this Arte, the great +aide of Geometrie. + + ++‡Hydragogie‡, demonstrateth the possible leading of Water, by Natures +lawe, and by artificiall helpe, from any head (being a Spring, standing, +or running Water) to any other place assigned.+ Long, hath this Arte +bene in vse: and much thereof written: and very marueilous workes +therein, performed: as may yet appeare, in Italy: by the Ruynes +remaining of the Aqueductes. In other places, of Riuers leading through +the Maine land, Nauigable many a Mile. And in other places, of the +marueilous forcinges of Water to Ascend. which all, declare the great +skill, to be required of him, who should in this Arte be perfecte, for +all occasions of waters possible leading. To speake of the allowance of +the Fall, for euery hundred foote: or of the Ventills (if the waters +labour be farre, and great) I neede not: Seing, at hand (about vs) many +expert men can sufficiently testifie, in effecte, the order: though the +Demonstration of the Necessitie thereof, they know not: Nor yet, if they +should be led, vp and downe, and about Mountaines, from the head of the +Spring: and then, a place being assigned: and of them, to be demaunded, +how low or high, that last place is, in respecte of the head, from which +(so crokedly, and vp and downe) they be come: Perhaps, they would not, +or could not, very redily, or nerely assoyle that question. _Geometrie_ +therefore, is necessary to _Hydragogie_. Of the sundry wayes to force +water to ascend, eyther by _Tympane_, _Kettell mills_, _Skrue_, +_Ctesibike_, or such like: in _Vitruuius_, _Agricola_, (and other,) +fully, the maner may appeare. And so, thereby, also be most euident, how +the Artes, of _Pneumatithmie_, _Helicosophie_, _Statike_, _Trochilike_, +and _Menadrie_, come to the furniture of this, in Speculation, and to +the Commoditie of the Common Wealth, in practise. + + ++‡Horometrie‡, is an Arte Mathematicall, which demõstrateth, how, at all +times appointed, the precise vsuall denominatiõ of time, may be knowen, +for any place assigned.+ These wordes, are smoth and plaine easie +Englishe, but the reach of their meaning, is farther, then you woulde +lightly imagine. Some part of this Arte, was called in olde time, +_Gnomonice_: and of late, _Horologiographia_: and in Englishe, may be +termed, _Dialling_. Auncient is the vse, and more auncient, is the +Inuention. The vse, doth well appeare to haue bene (at the least) aboue +two thousand and three hundred yeare agoe: + + [4. Reg. 20.] + +in * King _Achaz_ Diall, then, by the Sunne, shewing the distinction of +time. By Sunne, Mone, and Sterres, this Dialling may be performed, and +the precise Time of day or night knowen. But the demonstratiue +delineation of these Dialls, of all sortes, requireth good skill, both +of _Astronomie_, and _Geometrie_ Elementall, Sphæricall, Phænomenall, +and Conikall. Then, to vse the groundes of the Arte, for any regular +Superficies, in any place offred: and (in any possible apt position +therof) theron, to describe (all maner of wayes) how, vsuall howers, may +be (by the _Sunnes_ shadow) truely determined: will be found no sleight +Painters worke. So to Paint, and prescribe the Sunnes Motion, to the +breadth of a heare. In this Feate (in my youth) I Inuented a way, +How +in any Horizontall, Murall, or Æquinoctiall Diall, &c. At all howers +(the Sunne shining) the Signe and Degree ascendent, may be knowen.+ +Which is a thing very necessary for the Rising of those fixed Sterres: +whose Operation in the Ayre, is of great might, euidently. I speake no +further, of the vse hereof. Bur forasmuch as, Mans affaires require +knowledge of Times & Momentes, when, neither Sunne, Mone, or Sterre, can +be sene: Therefore, by Industrie Mechanicall, was inuented, first, how, +by Water, running orderly, the Time and howers might be knowen: whereof, +the famous _Ctesibius_, was Inuentor: a man, of _Vitruuius_, to the Skie +(iustly) extolled. Then, after that, by Sand running, were howers +measured: Then, by _Trochilike_ with waight: And of late time, by +_Trochilike_ with Spring: without waight. All these, by Sunne or Sterres +direction (in certaine time) require ouersight and reformation, +according to the heauenly Æquinoctiall Motion: besides the inæqualitie +of their owne Operation. There remayneth (without parabolicall meaning +herein) among the Philosophers, + + [A perpetuall Motion.] + +a more excellent, more commodious, and more marueilous way, then all +these: of hauing the motion of the Primouant (or first æquinoctiall +motion,) by Nature and Arte, Imitated: which you shall (by furder search +in waightier studyes) hereafter, vnderstand more of. And so, it is tyme +to finish this Annotation, of Tymes distinction, vsed in our common, and +priuate affaires: The commoditie wherof, no man would want, that can +tell, how to bestow his tyme. + + ++‡Zographie‡, is an Arte Mathematicall, which teacheth and +demonstrateth, how, the Intersection of all visuall Pyramides, made by +any playne assigned, (the Centre, distance, and lightes, beyng +determined) may be, by lynes, and due propre colours, represented.+ A +notable Arte, is this: and would require a whole Volume, to declare the +property thereof: and the Commodities ensuyng. Great skill of +_Geometrie_, _Arithmetike_, _Perspectiue_, and _Anthropographie_, with +many other particular Artes, hath the _Zographer_, nede of, for his +perfection. For, the most excellent Painter, (who is but the propre +Mechanicien, & Imitator sensible, of the Zographer) hath atteined to +such perfection, that Sense of Man and beast, haue iudged thinges +painted, to be things naturall, and not artificiall: aliue, and not +dead. This Mechanicall Zographer (commonly called the Painter) is +meruailous in his skill: and seemeth to haue a certaine diuine power: +As, of frendes absent, to make a frendly, present comfort: yea, and of +frendes dead, to giue a continuall, silent presence: not onely with vs, +but with our posteritie, for many Ages. And so procedyng, Consider, How, +in Winter, he can shew you, the liuely vew of Sommers Ioy, and riches: +and in Sommer, exhibite the countenance of Winters dolefull State, and +nakednes. Cities, Townes, Fortes, Woodes, Armyes, yea whole Kingdomes +(be they neuer so farre, or greate) can he, with ease, bring with him, +home (to any mans Iudgement) as Paternes liuely, of the thinges +rehearsed. In one little house, can he, enclose (with great pleasure of +the beholders,) the portrayture liuely, of all visible Creatures, either +on earth, or in the earth, liuing: or in the waters lying, Creping, +slyding, or swimming: or of any foule, or fly, in the ayre flying. Nay, +in respect of the Starres, the Skie, the Cloudes: yea, in the shew of +the very light it selfe (that Diuine Creature) can he match our eyes +Iudgement, most nerely. What a thing is this? thinges not yet being, he +can represent so, as, at their being, the Picture shall seame (in maner) +to haue Created them. To what Artificer, is not Picture, a great +pleasure and Commoditie? Which of them all, will refuse the Direction +and ayde of Picture? The Architect, the Goldsmith, and the Arras Weauer: +of Picture, make great account. Our liuely Herbals, our portraitures of +birdes, beastes, and fishes: and our curious Anatomies, which way, are +they most perfectly made, or with most pleasure, of vs beholden? Is it +not, by Picture onely? And if Picture, by the Industry of the Painter, +be thus commodious and meruailous: what shall be thought of _Zographie_, +the Scholemaster of Picture, and chief gouernor? Though I mencion not +_Sculpture_, in my Table of Artes Mathematicall: yet may all men +perceiue, How, that _Picture_ and _Sculpture_, are Sisters germaine: and +both, right profitable, in a Commõ wealth. and of _Sculpture_, aswell as +of Picture, excellent Artificers haue written great bokes in +commendation. Witnesse I take, of _Georgio Vasari_, _Pittore Aretino_: +of _Pomponius Gauricus_: and other. To these two Artes, (with other,) is +a certaine od Arte, called _Althalmasat_, much beholdyng: more, then the +common _Sculptor_, _Entayler_, _Keruer_, _Cutter_, _Grauer_, _Founder_, +or _Paynter (&c)_ know their Arte, to be commodious. + + + [An objection.] + ++‡Architecture‡+, to many may seme not worthy, or not mete, to be +reckned among the _Artes Mathematicall_. To whom, I thinke good, to giue +some account of my so doyng. Not worthy, (will they say,) bycause it is +but for building, of a house, Pallace, Church, Forte, or such like, +grosse workes. And you, also, defined the _Artes Mathematicall_, to be +such, as dealed with no Materiall or corruptible thing: and also did +demonstratiuely procede in their faculty, by Number or Magnitude. First, + + [The Answer.] + +you see, that I count, here, _Architecture_, among those _Artes +Mathematicall_, which are Deriued from the Principals: and you know, +that such, may deale with Naturall thinges, and sensible matter. Of +which, “some draw nerer, to the Simple and absolute Mathematicall +Speculation, then other do. + + [☞] + +And though, the _Architect_ procureth, enformeth, & directeth, the +_Mechanicien_, to handworke, & the building actuall, of house, Castell, +or Pallace, and is chief Iudge of the same: yet, with him selfe (as +chief _Master_ and _Architect_,) remaineth the Demonstratiue reason and +cause, of the Mechaniciens worke: in Lyne, plaine, and Solid: by +_Geometricall_, _Arithmeticall_, _Opticall_, _Musicall_, +_Astronomicall_, _Cosmographicall_” (& to be brief) by all the former +Deriued _Artes Mathematicall_, and other Naturall Artes, hable to be +confirmed and stablished. If this be so: then, may you thinke, that +_Architecture_, hath good and due allowance, in this honest Company of +_Artes Mathematicall_ Deriuatiue. I will, herein, craue Iudgement of two +most perfect _Architectes_: the one, being _Vitruuius_, the Romaine: who +did write ten bookes thereof, to the Emperour _Augustus_ (in whose daies +our Heauenly Archemaster, was borne): and the other, _Leo Baptista +Albertus_, a Florentine: who also published ten bookes therof. +_Architectura_ (sayth _Vitruuius_) _est Scientia pluribus disciplinis & +varijs eruditionibus ornata: cuius Iudicio probantur omnia, quæ ab +cæteris Artificibus perficiuntur opera._ That is. +Architecture, is a +Science garnished with many doctrines & diuerse instructions: by whose +Iudgement, all workes, by other workmen finished, are Iudged.+ It +followeth. _Ea nascitur ex Fabrica, & Ratiocinatione. &c. Ratiocinatio +autem est, quæ, res fabricatas, Solertia ac ratione proportionis, +demonstrare at[que] explicare potest. +Architecture, groweth of Framing, +and Reasoning. &c. Reasoning, is that, which of thinges framed, with +forecast, and proportion: can make demonstration, and manifest +declaration.+_ Againe. _Cùm, in omnibus enim rebus, tùm maximè etiam in +Architectura, hæc duo insunt: quod significatur, & quod significat. +Significatur proposita res, de qua dicitur: hanc autem Significat +Demonstratio, rationibus doctrinarum explicata. +Forasmuch as, in all +thinges: therefore chiefly in Architecture, these two thinges are: the +thing signified: and that which signifieth. The thing propounded, +whereof we speake, is the thing Signified. But Demonstration, expressed +with the reasons of diuerse doctrines, doth signifie the same thing.+_ +After that. _Vt literatus sit, peritus Graphidos, eruditus Geometriæ, & +Optices non ignarus: instructus Arithmetica: historias complures +nouerit, Philosophos diligenter audiuerit: Musicam sciuerit: Medicinæ +non sit ignarus, responsa Iurisperitorũ nouerit: Astrologiam, Cæli[que] +rationes cognitas habeat. +An Architect+_ (sayth he) +_ought to +vnderstand Languages, to be skilfull of Painting, well instructed in +Geometrie, not ignorant of Perspectiue, furnished with Arithmetike, haue +knowledge of many histories, and diligently haue heard Philosophers, +haue skill of Musike, not ignorant of Physike, know the aunsweres of +Lawyers, and haue Astronomie, and the courses Cælestiall, in good +knowledge._+ He geueth reason, orderly, wherefore all these Artes, +Doctrines, and Instructions, are requisite in an excellent _Architect_. +And (for breuitie) omitting the Latin text, thus he hath. +_Secondly, it +is behofefull for an Architect to haue the knowledge of Painting: that +he may the more easilie fashion out, in patternes painted, the forme of +what worke he liketh. And Geometrie, geueth to Architecture many helpes: +and first teacheth the Vse of the Rule, and the Cumpasse: wherby +(chiefly and easilie) the descriptions of Buildinges, are despatched in +Groundplats: and the directions of Squires, Leuells, and Lines. +Likewise, by Perspectiue, the Lightes of the heauen, are well led, in +the buildinges: from certaine quarters of the world. By Arithmetike, the +charges of Buildinges are summed together: the measures are expressed, +and the hard questions of Symmetries, are by Geometricall Meanes and +Methods discoursed on. &c. Besides this, of the Nature of thinges (which +in Greke is called φυσιολογία) Philosophie doth make declaration. Which, +it is necessary, for an Architect, with diligence to haue learned: +because it hath many and diuers naturall questions: as specially, in +Aqueductes. For in their courses, leadinges about, in the leuell ground, +and in the mountinges, the naturall Spirites or breathes are ingendred +diuers wayes: The hindrances, which they cause, no man can helpe, but +he, which out of Philosophie, hath learned the originall causes of +thinges. Likewise, who soeuer shall read Ctesibius, or Archimedes +bookes, (and of others, who haue written such Rules) can not thinke, as +they do: vnlesse he shall haue receaued of Philosophers, instructions in +these thinges. And Musike he must nedes know: that he may haue +vnderstanding, both of Regular and Mathematicall Musike: that he may +temper well his Balistes, Catapultes, and Scorpions. &c. Moreouer, the +Brasen Vessels, which in Theatres, are placed by Mathematicall order, in +ambries, vnder the steppes: and the diuersities of the soundes (which +y^e Grecians call ηχεῖα) are ordred according to Musicall Symphonies & +Harmonies: being distributed in y^e Circuites, by Diatessaron, Diapente, +and Diapason. That the conuenient voyce, of the players sound, whẽ it +came to these preparations, made in order, there being increased: with +y^t increasing, might come more cleare & pleasant, to y^e eares of the +lokers on. &c. And of Astronomie, is knowẽ y^e East, West, South, and +North. The fashion of the heauen, the Æquinox, the Solsticie, and the +course of the sterres. Which thinges, vnleast one know: he can not +perceiue, any thyng at all, the reason of Horologies. Seyng therfore +this ample Science, is garnished, beautified and stored, with so many +and sundry skils and knowledges: I thinke, that none can iustly account +them selues Architectes, of the suddeyne. But they onely, who from their +childes yeares, ascendyng by these degrees of knowledges, beyng fostered +vp with the atteynyng of many Languages and Artes, haue wonne to the +high Tabernacle of Architecture. &c. And to whom Nature hath giuen such +quicke Circumspection, sharpnes of witt, and Memorie, that they may be +very absolutely skillfull in Geometrie, Astronomie, Musike, and the rest +of the Artes Mathematicall: Such, surmount and passe the callyng, and +state, of Architectes: + + [A Mathematicien.] + +and are become Mathematiciens. &c. And they are found, seldome. As, in +tymes past, was Aristarchus Samius: Philolaus, and Archytas, Tarentynes: +Apollonius Pergęus: Eratosthenes Cyreneus: Archimedes, and Scopas, +Syracusians. Who also, left to theyr posteritie, many Engines and +Gnomonicall workes: by numbers and naturall meanes, inuented and +declared._+ + + +Thus much, and the same wordes (in sense) in one onely Chapter of this +Incõparable _Architect Vitruuius_, shall you finde. And if you should, +but take his boke in your hand, and slightly loke thorough it, you would +say straight way: + + [Vitruuius.] + +This is _Geometrie_, _Arithmetike_, _Astronomie_, _Musike_, +_Anthropographie_, _Hydragogie_, _Horometrie_. _&c_. and (to cõclude) +the Storehouse of all workmãship. Now, let vs listen to our other Iudge, +our Florentine, _Leo Baptista_: and narrowly consider, how he doth +determine of _Architecture_. _Sed ante[que] vltra progrediar. &c. +But +before I procede any further +_(sayth he) +_I thinke, that I ought to +expresse, what man I would haue to bee allowed an Architect. For, I will +not bryng in place a Carpenter: as though you might Compare him to the +Chief Masters of other Artes. For the hand of the Carpenter, is the +Architectes Instrument._+ + + [VVho is an Architect.] + ++_But I will appoint the Architect to be “that man, who hath the skill, +(by a certaine and meruailous meanes and way,) both in minde and +Imagination to determine and also in worke to finish: what workes so +euer, by motion of waight, and cuppling and framyng together of bodyes, +may most aptly be Commodious for the worthiest Vses of Man.” And that he +may be able to performe these thinges, he hath nede of atteynyng and +knowledge of the best, and most worthy thynges. &c. The whole Feate of +Architecture in buildyng, consisteth in Lineamentes, and in Framyng. And +the whole power and skill of Lineamentes, tendeth to this: that the +right and absolute way may be had, of Coaptyng and ioyning Lines and +angles: by which, the face of the buildyng or frame, may be comprehended +and concluded. And it is the property of Lineamentes, to prescribe vnto +buildynges, and euery part of them, an apt place, & certaine nũber: +a worthy maner, and a semely order: that, so, y^e whole forme and figure +of the buildyng, may rest in the very Lineamentes. &c. And we may +prescribe in mynde and imagination the whole formes, * + + [* The Immaterialitie of perfect Architecture.] + +all material stuffe beyng secluded. Which point we shall atteyne, by +Notyng and forepointyng the angles, and lines, by a sure and certaine +direction and connexion. Seyng then, these thinges, are thus:_+ + + [What, Lineament is.] + ++_Lineamente, shalbe the certaine and constant prescribyng, conceiued in +mynde: made in lines and angles: and finished with a learned minde and +wyt._+ “We thanke you Master _Baptist_, that you haue so aptly brought +your Arte, and phrase therof, to haue some Mathematicall perfection: + + [Note.] + +by certaine order, nũber, forme, figure, and _Symmetrie_ mentall:” all +naturall & sensible stuffe set a part. Now, then, it is euident, (Gentle +reader) how aptely and worthely, I haue preferred _Architecture_, to be +bred and fostered vp in the Dominion of the pereles _Princesse_, +_Mathematica_: and to be a naturall Subiect of hers. And the name of +_Architecture_, is of the principalitie, which this Science hath, aboue +all other Artes. And _Plato_ affirmeth, the _Architect_ to be _Master_ +ouer all, that make any worke. Wherupon, he is neither Smith, nor +Builder: nor, separately, any Artificer: but the Hed, the Prouost, the +Directer, and Iudge of all Artificiall workes, and all Artificers. For, +the true _Architect_, is hable to teach, Demonstrate, distribute, +describe, and Iudge all workes wrought. And he, onely, searcheth out the +causes and reasons of all Artificiall thynges. Thus excellent, is +_Architecture_: though few (in our dayes) atteyne thereto: yet may not +the Arte, be otherwise thought on, then in very dede it is worthy. Nor +we may not, of auncient Artes, make new and imperfect Definitions in our +dayes: for scarsitie of Artificers: No more, than we may pynche in, the +Definitions of _Wisedome_, or _Honestie_, or of _Frendeshyp_ or of +_Iustice_. No more will I consent, to Diminish any whit, of the +perfection and dignitie, (by iust cause) allowed to absolute +_Architecture_. Vnder the Direction of this Arte, are thre principall, +necessary _Mechanicall Artes_. Namely, _Howsing_, _Fortification_, and +_Naupegie_. _Howsing_, I vnderstand, both for Diuine Seruice, and Mans +common vsage: publike, and priuate. Of _Fortification_ and _Naupegie_, +straunge matter might be told you: But perchaunce, some will be tyred, +with this Bederoll, all ready rehearsed: and other some, will nycely nip +my grosse and homely discoursing with you: made in post hast: for feare +you should wante this true and frendly warnyng, and tast giuyng, of the +_Power Mathematicall_. Lyfe is short, and vncertaine: Tymes are +perilouse: &c. And still the Printer awayting, for my pen staying: All +these thinges, with farder matter of Ingratefulnes, giue me occasion to +passe away, to the other Artes remainyng, with all spede possible. + + ++The Arte of ‡Nauigation‡, demonstrateth how, by the shortest good way, +by the aptest Directiõ, & in the shortest time, a sufficient Ship, +betwene any two places (in passage Nauigable,) assigned: may be +cõducted: and in all stormes, & naturall disturbances chauncyng, how, to +vse the best possible meanes, whereby to recouer the place first +assigned.+ What nede, the _Master Pilote_, hath of other Artes, here +before recited, it is easie to know: as, of _Hydrographie_, +_Astronomie_, _Astrologie_, and _Horometrie_. Presupposing continually, +the common Base, and foundacion of all: namely _Arithmetike_ and +_Geometrie_. So that, he be hable to vnderstand, and Iudge his own +necessary Instrumentes, and furniture Necessary: Whether they be +perfectly made or no: and also can, (if nede be) make them, hym selfe. +As Quadrantes, The Astronomers Ryng, The Astronomers staffe, The +Astrolabe vniuersall. An Hydrographicall Globe. Charts Hydrographicall, +true, (not with parallell Meridians). The Common Sea Compas: The Compas +of variacion: The Proportionall, and Paradoxall Compasses + + [Anno. 1559.] + +(of me Inuented, for our two Moscouy Master Pilotes, at the request of +the Company) Clockes with spryng: houre, halfe houre, and three houre +Sandglasses: & sundry other Instrumẽtes: And also, be hable, on Globe, +or Playne to describe the Paradoxall Compasse: and duely to vse the +same, to all maner of purposes, whereto it was inuented. And also, be +hable to Calculate the Planetes places for all tymes. + + +Moreouer, with Sonne Mone or Sterre (or without) be hable to define the +Longitude & Latitude of the place, which he is in: So that, the +Longitude & Latitude of the place, from which he sayled, be giuen: or by +him, be knowne. whereto, appertayneth expert meanes, to be certified +euer, of the Ships way. &c. And by foreseing the Rising, Settyng, +Nonestedyng, or Midnightyng of certaine tempestuous fixed Sterres: or +their Coniunctions, and Anglynges with the Planetes, &c. he ought to +haue expert coniecture of Stormes, Tempestes, and Spoutes: and such lyke +Meteorologicall effectes, daungerous on Sea. For (as _Plato_ sayth,) +_Mutationes, opportunitates[que] temporum presentire, non minus rei +militari, quàm Agriculturæ, Nauigationi[que] conuenit. +To foresee the +alterations and opportunities of tymes, is conuenient, no lesse to the +Art of Warre, then to Husbandry and Nauigation.+_ And besides such +cunnyng meanes, more euident tokens in Sonne and Mone, ought of hym to +be knowen: such as (the Philosophicall Poëte) _Virgilius_ teacheth, in +hys _Georgikes_. Where he sayth, + + [Sidenote: Georgic. 1.] + + _Sol quo[que] & exoriens & quum se condet in vndas, + Signa dabit, Solem certissima signa sequuntur. &c. + -------- Nam sæpe videmus, + Ipsius in vultu varios errare colores. + Cæruleus, pluuiam denunciat, igneus Euros. + Sin maculæ incipient rutilo immiscerier igni, + Omnia tum pariter vento, nimbis[que] videbis + Feruere: non illa quisquam me nocte per altum + Ire, ne[que] a terra moueat conuellere funem. &c. + Sol tibi signa dabit. Solem quis dicere falsum + Audeat? -------- &c._ + +And so of Mone, Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, and +Beastes, and of many thynges els, a certaine Sympathicall forewarnyng +may be had: sometymes to great pleasure and proffit, both on Sea and +Land. Sufficiently, for my present purpose, it doth appeare, by the +premisses, how _Mathematicall_, the _Arte_ of _Nauigation_, is: and how +it nedeth and also vseth other _Mathematicall Artes_: And now, if I +would go about to speake of the manifold Commodities, commyng to this +Land, and others, by Shypps and _Nauigation_, you might thinke, that I +catch at occasions, to vse many wordes, where no nede is. + + +Yet, this one thyng may I, (iustly) say. In _Nauigation_, none ought to +haue greater care, to be skillfull, then our English Pylotes. And +perchaunce, Some, would more attempt: And other Some, more willingly +would be aydyng, it they wist certainely, What Priuiledge, God had +endued this Iland with, by reason of Situation, most commodious for +_Nauigation_, to Places most Famous & Riche. And though, + + [* Anno. 1567 S. H. G.] + +(of * Late) a young Gentleman, a Courragious Capitaine, was in a great +readynes, with good hope, and great causes of persuasion, to haue +ventured, for a Discouerye, (either _Westerly_, by _Cape de Paramantia_: +or _Esterly_, aboue _Noua Zemla_, and the _Cyremisses_) and was, at the +very nere tyme of Attemptyng, called and employed otherwise (both then, +and since,) in great good seruice to his Countrey, as the Irish Rebels +haue * tasted: + + [* Anno. 1569] + +Yet, I say, (though the same Gentleman, doo not hereafter, deale +therewith) Some one, or other, should listen to the Matter: and by good +aduise, and discrete Circumspection, by little, and little, wynne to the +sufficient knowledge of that +Trade+ and +Voyage+: Which, now, I would +be sory, (through Carelesnesse, want of Skill, and Courrage,) should +remayne Vnknowne and vnheard of. Seyng, also, we are herein, halfe +Challenged, by the learned, by halfe request, published. Therof, verely, +might grow Commoditye, to this Land chiefly, and to the rest of the +Christen Common wealth, farre passing all riches and worldly Threasure. + + ++‡Thaumaturgike‡, is that Art Mathematicall, which giueth certaine order +to make straunge workes, of the sense to be perceiued, and of men +greatly to be wondred at.+ By sundry meanes, this _Wonder-worke_ is +wrought. Some, by _Pneumatithmie_. As the workes of _Ctesibius_ and +_Hero_, Some by waight. wherof _Timæus_ speaketh. Some, by Stringes +strayned, or Springs, therwith Imitating liuely Motions. Some, by other +meanes, as the Images of Mercurie: and the brasen hed, made by _Albertus +Magnus_, which dyd seme to speake. _Boethius_ was excellent in these +feates. To whom, _Cassiodorus_ writyng, sayth. +_Your purpose is to know +profound thynges: and to shew meruayles. By the disposition of your +Arte, Metals do low: Diomedes of brasse, doth blow a Trumpet loude: +a brasen Serpent hisseth: byrdes made, sing swetely. Small thynges we +rehearse of you, who can Imitate the heauen. &c._+ Of the straunge +Selfmouyng, which, at Saint Denys, by Paris, + + [* Anno. 1551] + +* I saw, ones or twise (_Orontius_ beyng then with me, in Company) it +were to straunge to tell. But some haue written it. And yet, (I hope) it +is there, of other to be sene. And by _Perspectiue_ also straunge +thinges, are done. As partly (before) I gaue you to vnderstand in +_Perspectiue_. As, to see in the Ayre, a loft, the lyuely Image of an +other man, either walkyng to and fro: or standyng still. Likewise, to +come into an house, and there to see the liuely shew of Gold, Siluer or +precious stones: and commyng to take them in your hand, to finde nought +but Ayre. Hereby, haue some men (in all other matters counted wise) +fouly ouershot thẽ selues: misdeaming of the meanes. Therfore sayd +_Claudius Cælestinus_. + + [De his quæ Mundo mirabiliter eueniunt. cap. 8.] + +_Hodie magnæ literaturæ viros & magna reputationis videmus, opera quedam +quasi miranda, supra Naturã putare: de quibus in Perspectiua doctus +causam faciliter reddidisset._ That is. +_Now a dayes, we see some men, +yea of great learnyng and reputation, to Iudge certain workes as +meruaylous, aboue the power of Nature: Of which workes, one that were +skillfull in Perspectiue might easely haue giuen the Cause._+ Of +_Archimedes Sphære_, _Cicero_ witnesseth. + + [Tusc. 1.] + +Which is very straunge to thinke on. +_For when Archimedes_+ (sayth he) ++_did fasten in a Sphære, the mouynges of the Sonne, Mone, and of the +fiue other Planets, he did, as the God, which (in Timæus of Plato) did +make the world. That, one turnyng, should rule motions most vnlike in +slownes, and swiftnes._+ But a greater cause of meruayling we haue by +_Claudianus_ report hereof. Who affirmeth this _Archimedes worke_, to +haue ben of Glasse. And discourseth of it more at large: which I omit. +The Doue of wood, which the _Mathematicien Archytas_ did make to flye, +is by _Agellius_ spoken of. Of _Dædalus_ straunge Images, _Plato_ +reporteth. _Homere_ of _Vulcans Selfmouers_, (by secret wheles) leaueth +in writyng. _Aristotle_, in hys _Politikes_, of both, maketh mention. +Meruaylous was the workemanshyp, of late dayes, performed by good skill +of _Trochilike. &c._ For in Noremberge, A flye of Iern, beyng let out of +the Artificers hand, did (as it were) fly about by the gestes, at the +table, and at length, as though it were weary, retourne to his masters +hand agayne. Moreouer, an Artificiall Egle, was ordred, to fly out of +the same Towne, a mighty way, and that a loft in the Ayre, toward the +Emperour comming thether: and followed hym, beyng come to the gate of +the towne. * + + [* ☞] + +Thus, you see, what, Arte Mathematicall can performe, when Skill, will, +Industry, and Hability, are duely applyed to profe. + + + [A Digression.] + +And for these, and such like marueilous Actes and Feates, Naturally, +Mathematically, and Mechanically, wrought and contriued: + + [Apologeticall.] + +ought any honest Student, and Modest Christian Philosopher, be counted, +& called a +Coniurer+? Shall the folly of Idiotes, and the Mallice of +the Scornfull, so much preuaile, that He, who seeketh no worldly gaine +or glory at their handes: But onely, of God, the threasor of heauenly +wisedome, & knowledge of pure veritie: Shall he (I say) in the meane +space, be robbed and spoiled of his honest name and fame? He that seketh +(by S. Paules aduertisement) in the Creatures Properties, and wonderfull +vertues, to finde iuste cause, to glorifie the Æternall, and Almightie +Creator by: Shall that man, be (in hugger mugger) condemned, as a +Companion of the Helhoundes, and a Caller, and Coniurer of wicked and +damned Spirites? He that bewaileth his great want of time, sufficient +(to his contentation) for learning of Godly wisdome, and Godly Verities +in: and onely therin setteth all his delight: Will that mã leese and +abuse his time, in dealing with the Chiefe enemie of Christ our Redemer: +the deadly foe of all mankinde: the subtile and impudent peruerter of +Godly Veritie: the Hypocriticall Crocodile: the Enuious Basiliske, +continually desirous, in the twinke of an eye, to destroy all Mankinde, +both in Body and Soule, æternally? Surely (for my part, somewhat to say +herein) I haue not learned to make so brutish, and so wicked a Bargaine. +Should I, for my xx. or xxv. yeares Studie: for two or three thousand +Markes spending: seuen or eight thousand Miles going and trauailing, +onely for good learninges sake: And that, in all maner of wethers: in +all maner of waies and passages: both early and late: in daunger of +violence by man: in daunger of destruction by wilde beastes: in hunger: +in thirst: in perilous heates by day, with toyle on foote: in daungerous +dampes of colde, by night, almost bereuing life: (as God knoweth): with +lodginges, oft times, to small ease: and somtime to lesse securitie. And +for much more (then all this) done & suffred, for Learning and attaining +of Wisedome: Should I (I pray you) for all this, no otherwise, nor more +warily: or (by Gods mercifulnes) no more luckily, haue fished, with so +large, and costly, a Nette, so long time in drawing (and that with the +helpe and aduise of Lady Philosophie, & Queene Theologie): but at +length, to haue catched, and drawen vp, * a Frog? + + [* A prouerb. Fayre fisht, and caught a Frog.] + +Nay, a Deuill? For, so, doth the Common peuish Pratler Imagine and +Iangle: And, so, doth the Malicious skorner, secretly wishe, & brauely +and boldly face down, behinde my backe. Ah, what a miserable thing, is +this kinde of Men? How great is the blindnes & boldnes, of the +Multitude, in thinges aboue their Capacitie? What a Land: what a People: +what Maners: what Times are these? Are they become Deuils, them selues: +and, by false witnesse bearing against their Neighbour, would they also, +become Murderers? Doth God, so long geue them respite, to reclaime them +selues in, from this horrible slaundering of the giltlesse: contrary to +their owne Consciences: and yet will they not cease? Doth the Innocent, +forbeare the calling of them, Iuridically to aunswere him, according to +the rigour of the Lawes: and will they despise his Charitable pacience? +As they, against him, by name, do forge, fable, rage, and raise +slaunder, by Worde & Print: Will they prouoke him, by worde and Print, +likewise, to Note their Names to the World: with their particular +deuises, fables, beastly Imaginations, and vnchristen-like slaunders? +Well: Well. O (you such) my vnkinde Countrey men. O vnnaturall Countrey +men. O vnthankfull Countrey men. O Brainsicke, Rashe, Spitefull, and +Disdainfull Countrey men. Why oppresse you me, thus violently, with your +slaundering of me: Contrary to Veritie: and contrary to your owne +Consciences? And I, to this hower, neither by worde, deede, or thought, +haue bene, any way, hurtfull, damageable, or iniurious to you, or yours? +Haue I, so long, so dearly, so farre, so carefully, so painfully, so +daungerously sought & trauailed for the learning of Wisedome, & +atteyning of Vertue: And in the end (in your iudgemẽt) am I become, +worse, then when I begã? Worse, thẽ a Mad man? A dangerous Member in the +Common Wealth: and no Member of the Church of Christ? Call you this, to +be Learned? Call you this, to be a Philosopher? and a louer of Wisedome? +To forsake the straight heauenly way: and to wallow in the broad way of +damnation? To forsake the light of heauenly Wisedome: and to lurke in +the dungeon of the Prince of darkenesse? To forsake the Veritie of God, +& his Creatures: and to fawne vpon the Impudent, Craftie, Obstinate +Lier, and continuall disgracer of Gods Veritie, to the vttermost of his +power? To forsake the Life & Blisse Æternall: and to cleaue vnto the +Author of Death euerlasting? that Murderous Tyrant, most gredily +awaiting the Pray of Mans Soule? Well: I thanke God and our Lorde Iesus +Christ, for the Comfort which I haue by the Examples of other men, +before my time: To whom, neither in godlines of life, nor in perfection +of learning, I am worthy to be compared: and yet, they sustained the +very like Iniuries, that I do: or rather, greater. Pacient _Socrates_, +his _Apologie_ will testifie: _Apuleius_ his _Apologies_, will declare +the Brutishnesse of the Multitude. _Ioannes Picus_, Earle of Mirandula, +his _Apologie_ will teach you, of the Raging slaunder of the Malicious +Ignorant against him. _Ioannes Trithemius_, his _Apologie_ will +specifie, how he had occasion to make publike Protestation: as well by +reason of the Rude Simple: as also, in respect of such, as were counted +to be of the wisest sort of men. “Many could I recite: But I deferre the +precise and determined handling of this matter: being loth to detect the +Folly & Mallice of my Natiue Countrey men. * + + [* ☞] + +Who, so hardly, can disgest or like any extraordinary course of +Philosophicall Studies: not falling within the Cumpasse of their +Capacitie: or where they are not made priuie of the true and secrete +cause, of such wonderfull Philosophicall Feates.” These men, are of +fower sortes, chiefly. The first, I may name, _Vaine pratling busie +bodies_: The second, _Fond Frendes_: The third, _Imperfectly zelous_: +and the fourth, _Malicious Ignorant_. To eche of these (briefly, and in +charitie) I will say a word or two, and so returne to my Præface. + + [1.] + +_Vaine pratling busie bodies_, vse your idle assemblies, and +conferences, otherwise, then in talke of matter, either aboue your +Capacities, for hardnesse: or contrary to your Consciences, in Veritie. + + [2.] + +_Fonde Frendes_, leaue of, so to commend your vnacquainted frend, vpon +blinde affection: As, because he knoweth more, then the common Student: +that, therfore, he must needes be skilfull, and a doer, in such matter +and maner, as you terme _Coniuring_. Weening, thereby, you aduaunce his +fame: and that you make other men, great marueilers of your hap, to haue +such a learned frend. Cease to ascribe Impietie, where you pretend +Amitie. For, if your tounges were true, then were that your frend, +_Vntrue_, both to God, and his Soueraigne. Such _Frendes_ and +_Fondlinges_, I shake of, and renounce you: Shake you of, your Folly. + + [3.] + +_Imperfectly zelous_, to you, do I say: that (perhaps) well, do you +Meane: But farre you misse the Marke: If a Lambe you will kill, to feede +the flocke with his bloud. Sheepe, with Lambes bloud, haue no naturall +sustenaunce: No more, is Christes flocke, with horrible slaunders, duely +ædified. Nor your faire pretense, by such rashe ragged Rhetorike, any +whit, well graced. But such, as so vse me, will finde a fowle Cracke in +their Credite. Speake that you know: And know, as you ought: Know not, +by Heare say, when life lieth in daunger. Search to the quicke, & let +Charitie be your guide. + + [4.] + +_Malicious Ignorant_, what shall I say to thee? _Prohibe linguam tuam a +malo. A detractione parcite linguæ. +Cause thy toung to refraine frõ +euill. Refraine your toung from slaunder.+_ Though your tounges be +sharpned, Serpent like, & Adders poyson lye in your lippes: + + [Psal. 140.] + +yet take heede, and thinke, betimes, with your selfe, _Vir linguosus non +stabilietur in terra. Virum violentum venabitur malum, donec +præcipitetur._ For, sure I am, _Quia faciet Dominus Iudicium afflicti: +& vindictam pauperum._ + + +Thus, I require you, my assured frendes, and Countrey men (you +Mathematiciens, Mechaniciens, and Philosophers, Charitable and discrete) +to deale in my behalf, with the light & vntrue tounged, my enuious +Aduersaries, or Fond frends. And farther, I would wishe, that at leysor, +you would consider, how _Basilius Magnus_, layeth _Moses_ and _Daniel_, +before the eyes of those, which count all such Studies Philosophicall +(as mine hath bene) to be vngodly, or vnprofitable. Waye well +_S. Stephen_ his witnesse of _Moses_. + + [Act. 7. C.] + +_Eruditus est Moses omni Sapientia Ægyptiorũ: & erat potens in verbis & +operibus suis. +Moses was instructed in all maner of wisedome of the +Ægyptians: and he was of power both in his wordes, and workes.+_ You see +this Philosophicall Power & Wisedome, which _Moses_ had, to be nothing +misliked of the Holy Ghost. Yet _Plinius_ hath recorded, _Moses_ to be a +wicked _Magicien_. And that (of force) must be, either for this +Philosophicall wisedome, learned, before his calling to the leading of +the Children of _Israel_: or for those his wonders, wrought before King +_Pharao_, after he had the conducting of the _Israelites_. As concerning +the first, you perceaue, how _S. Stephen_, at his Martyrdome (being full +of the Holy Ghost) in his Recapitulation of the olde Testament, hath +made mention of _Moses_ Philosophie: with good liking of it: And +_Basilius Magnus_ also, auoucheth it, to haue bene to _Moses_ profitable +(and therefore, I say, to the Church of God, necessary). But as +cõcerning _Moses_ wonders, done before King _Pharao_: God, him selfe, +sayd: _Vide vt omnia ostenta, quæ posui in manu tua, facias coram +Pharaone. +See that thou do all those wonders before Pharao, which I +haue put in thy hand.+_ Thus, you euidently perceaue, how rashly, +_Plinius_ hath slaundered _Moses_, + + [Lib. 30. Cap. 1.] + +of vayne fraudulent _Magike_, saying: _Est & alia Magices Factio, +a Mose, Iamne, & Iotape, Iudæis pendens: sed multis millibus annorum +post Zoroastrem. &c._ + + [1.] + +Let all such, therefore, who, in Iudgement and Skill of Philosophie, are +farre Inferior to _Plinie_, “take good heede, least they ouershoote them +selues rashly,” in + + [☞] + +Iudging of _Philosophers straunge Actes_: and the Meanes, how they are +done. + + [2.] + +But, much more, ought they to beware of forging, deuising, and imagining +monstrous feates, and wonderfull workes, when and where, no such were +done: no, not any sparke or likelihode, of such, as they, without all +shame, do report. + + [3.] + +And (to conclude) most of all, let them be ashamed of Man, and afraide +of the dreadfull and Iuste Iudge: both Folishly or Maliciously to +deuise: and then, deuilishly to father their new fond Monsters on me: +Innocent, in hand and hart: for trespacing either against the lawe of +God, or Man, in any my Studies or Exercises, Philosophicall, or +Mathematicall: As in due time, I hope, will be more manifest. + + +Now end I, with +‡Archemastrie‡+. Which name, is not so new, as this +Arte is rare. For an other Arte, vnder this, a degree (for skill and +power) hath bene indued with this English name before. And yet, this, +may serue for our purpose, sufficiently, at this present. +This Arte, +teacheth to bryng to actuall experience sensible, all worthy conclusions +by all the Artes Mathematicall purposed, & by true Naturall Philosophie +concluded: & both addeth to them a farder scope, in the termes of the +same Artes, & also by hys propre Method, and in peculier termes, +procedeth, with helpe of the foresayd Artes, to the performance of +complet Experiẽces, which of no particular Art, are hable (Formally) to +be challenged.+ If you remember, how we considered _Architecture_, in +respect of all common handworkes: some light may you haue, therby, to +vnderstand the Souerainty and propertie of this Science. _Science_ I may +call it, rather, then an Arte: for the excellency and Mastershyp it +hath, ouer so many, and so mighty Artes and Sciences. And bycause it +procedeth by _Experiences_, and searcheth forth the causes of +Conclusions, by _Experiences_: and also putteth the Conclusions them +selues, in _Experience_, it is named of some, _Scientia Experimentalis_. +The +_Experimentall Science_+. _Nicolaus Cusanus_ termeth it so, in hys +_Experimentes Statikall_, And an other _Philosopher_, + + [R. B.] + +of this land Natiue (the floure of whose worthy fame, can neuer dye nor +wither) did write therof largely, at the request of _Clement the sixt_. +The Arte carrieth with it, a wonderfull Credit: By reason, it +certefieth, sensibly, fully, and completely to the vtmost power of +Nature, and Arte. This Arte, certifieth by _Experience_ complete and +absolute: and other Artes, with their Argumentes, and Demonstrations, +persuade: and in wordes, proue very well their Conclusions. * + + [☞] + +But wordes, and Argumentes, are no sensible certifying: nor the full and +finall frute of Sciences practisable. And though some Artes, haue in +them, _Experiences_, yet they are not complete, and brought to the +vttermost, they may be stretched vnto, and applyed sensibly. As for +example: the Naturall Philosopher disputeth and maketh goodly shew of +reason: And the Astronomer, and the Opticall Mechanicien, put some +thynges in _Experience_: but neither, all, that they may: nor yet +sufficiently, and to the vtmost, those, which they do, There, then, the +_Archemaster_ steppeth in, and leadeth forth on, the _Experiences_, by +order of his doctrine _Experimentall_, to the chief and finall power of +Naturall and Mathematicall Artes. Of two or three men, in whom, this +Description of _Archemastry_ was _Experimentally_, verified, I haue read +and hard: and good record, is of their such perfection. So that, this +Art, is no fantasticall Imagination: as some Sophister, might, _Cum suis +Insolubilibus_, make a florish: and dassell your Imagination: and dash +your honest desire and Courage, from beleuing these thinges, so vnheard +of, so meruaylous, & of such Importance. Well: as you will. I haue +forewarned you. I haue done the part of a frende: I haue discharged my +Duety toward God: for my small Talent, at hys most mercyfull handes +receiued. To this Science, doth the _Science Alnirangiat_, great +Seruice. Muse nothyng of this name. I chaunge not the name, so vsed, and +in Print published by other: beyng a name, propre to the Science. Vnder +this, commeth _Ars Sintrillia_, by _Artephius_, briefly written. But the +chief Science, of the Archemaster, (in this world) as yet knowen, is an +other (as it were) OPTICAL Science: wherof, the name shall be told (God +willyng) when I shall haue some, (more iust) occasion, therof, to +Discourse. + + +Here, I must end, thus abruptly (Gentle frende, and vnfayned louer of +honest and necessary verities.) For, they, who haue (for your sake, and +vertues cause) requested me, (an old forworne Mathematicien) to take pen +in hand: (through the confidence they reposed in my long experience: and +tryed sincerity) for the declaryng and reportyng somewhat, of the frute +and commodity, by the +Artes Mathematicall, to be atteyned vnto+: euen +they, Sore agaynst their willes, are forced, for sundry causes, to +satisfie the workemans request, in endyng forthwith: He, so feareth +this, so new an attempt, & so costly: And in matter so slenderly +(hetherto) among the common Sorte of Studentes, considered or estemed. + + +And where I was willed, somewhat to alledge, why, in our vulgare Speche, +this part of the Principall Science of _Geometrie_, called _Euclides +Geometricall Elementes_, is published, to your handlyng: being vnlatined +people, and not Vniuersitie Scholers: Verily, I thinke it nedelesse. + + + [1.] + +For, the Honour, and Estimation of the +Vniuersities, and Graduates+, +is, hereby, nothing diminished. Seing, from, and by their Nurse +Children, you receaue all this Benefite: how great soeuer it be. + + + [2.] + +Neither are their Studies, hereby, any whit hindred. No more, then the +Italian _Vniuersities_, as _Academia Bononiensis_, _Ferrariensis_, +_Florentina_, _Mediolanensis_, _Patauina_, _Papiensis_, _Perusina_, +_Pisana_, _Romana_, _Senensis_, or any one of them, finde them selues, +any deale, disgraced, or their Studies any thing hindred, by _Frater +Lucas de Burgo_, or by _Nicolaus Tartalea_, who in vulgar Italian +language, haue published, not onely _Euclides Geometrie_, but of +_Archimedes_ somewhat: and in Arithmetike and Practicall Geometrie, very +large volumes, all in their vulgar speche. Nor in Germany haue the +famous _Vniuersities_, any thing bene discontent with _Albertus +Durerus_, his Geometricall Institutions in Dutch: or with _Gulielmus +Xylander_, his learned translation of the first sixe bookes of +_Euclide_, out of the Greke into the high Dutch. Nor with _Gualterus H. +Riffius_, his Geometricall Volume: very diligently translated into the +high Dutch tounge, and published. Nor yet the _Vniuersities_ of Spaine, +or Portugall, thinke their reputation to be decayed: or suppose any +their Studies to be hindred by the Excellent _P. Nonnius_, his +Mathematicall workes, in vulgare speche by him put forth. Haue you not, +likewise, in the French tounge, the whole Mathematicall Quadriuie? and +yet neither Paris, Orleance, or any of the other Vniuersities of +Fraunce, at any time, with the Translaters, or Publishers offended: or +any mans Studie thereby hindred? + + + [3.] + +And surely, the Common and Vulgar Scholer (much more, the Gramarian) +before his comming to the _Vniuersitie_, shall (or may) be, now +(according to _Plato_ his Counsell) sufficiently instructed in +_Arithmetike_ and _Geometrie_, for the better and easier learning of all +maner of _Philosophie_, _Academicall_, or _Peripateticall_. And by that +meanes, goe more cherefully, more skilfully, and spedily forwarde, in +his Studies, there to be learned. And, so, in lesse time, profite more, +then (otherwise) he should, or could do. + + + [4.] + +Also many good and pregnant Englishe wittes, of young Gentlemen, and of +other, who neuer intend to meddle with the profound search and Studie of +Philosophie (in the _Vniuersities_ to be learned) may neuerthelesse, +now, with more ease and libertie, haue good occasion, vertuously to +occupie the sharpnesse of their wittes: where, els (perchance) +otherwise, they would in fond exercises, spend (or rather leese) their +time: neither seruing God: nor furdering the Weale, common or priuate. + + + [5.] + +And great Comfort, with good hope, may the _Vniuersities_ haue, by +reason of this _Englishe_ +Geometrie, and Mathematicall Præface+, that +they (hereafter) shall be the more regarded, esteemed, and resorted +vnto. For, when it shall be knowen and reported, that of the +_Mathematicall Sciences_ onely, such great Commodities are ensuing (as I +haue specified): and that in dede, some of you vnlatined Studentes, can +be good witnesse, of such rare fruite by you enioyed (thereby): as +either, before this, was not heard of: or els, not so fully credited: +“Well, may all men coniecture, that farre greater ayde, and better +furniture, to winne to the Perfection of all Philosophie, + + [Vniuersities.] + +may in the Vniuersities be had: being the Storehouses & Threasory of all +Sciences, + + [☞] + +and all Artes, necessary for the best, and most noble State of Common +Wealthes.” + + [6.] + +Besides this, how many a Common Artificer, is there, in these Realmes of +England and Ireland, that dealeth with Numbers, Rule, & Cumpasse: Who, +with their owne Skill and experience, already had, will be hable (by +these good helpes and informations) to finde out, and deuise, new +workes, straunge Engines, and Instrumentes: for sundry purposes in the +Common Wealth? or for priuate pleasure? and for the better maintayning +of their owne estate? I will not (therefore) fight against myne owne +shadowe. For, no man (I am sure) will open his mouth against this +Enterprise. No mã (I say) who either hath Charitie toward his brother +(and would be glad of his furtherance in vertuous knowledge): or that +hath any care & zeale for the bettering of the Cõmon state of this +Realme. Neither any, that make accompt, what the wiser sort of men (Sage +and Stayed) do thinke of them. To none (therefore) will I make any +_Apologie,_ for a vertuous acte doing: and for cõmending, or setting +forth, Profitable Artes to English men, in the English toung. “But, vnto +God our Creator, let vs all be thankefull: for that, +_As he, of his +Goodnes, by his Powre, and in his wisedome, + + [☞] + +hath Created all thynges, in Number, Waight, and Measure_+: So, to vs, +of hys great Mercy, he hath reuealed Meanes, whereby, to atteyne the +sufficient and necessary knowledge of the foresayd hys three principall +Instrumentes: Which Meanes, I haue abundantly proued vnto you, to be the +_Sciences_ and _Artes Mathematicall_.” + + +And though I haue ben pinched with straightnes of tyme: that, no way, +I could so pen downe the matter (in my Mynde) as I determined: hopyng of +conuenient laysure: Yet. if vertuous zeale, and honest Intent prouoke +and bryng you to the readyng and examinyng of this Compendious treatise, +I do not doute, but, as the veritie therof (accordyng to our purpose) +will be euident vnto you: So the pith and force therof, will persuade +you: and the wonderfull frute therof, highly pleasure you. And that you +may the easier perceiue, and better remember, the principall pointes, +whereof my Preface treateth, + + [The Ground platt of this Præface in a Table.] + +I will giue you the +Groundplatt+ of my whole discourse, in a Table +annexed: from the first to the last, somewhat Methodically contriued. + + +If Hast, hath caused my poore pen, any where, to stumble: You will, + (I am sure) in part of recompence, (for my earnest and sincere + good will to pleasure you), Consider the rockish huge + mountaines, and the perilous vnbeaten wayes, which + (both night and day, for the while) it hath + toyled and labored through, to bryng you + this good Newes, and Comfortable + profe, of Vertues frute. + + So, I Commit you vnto Gods Mercyfull direction, for the rest: + hartely besechyng hym, to prosper your Studyes, and + honest Intentes: to his Glory, & the Commodity + of our Countrey. _Amen_. + + + _Written at my poore House + At Mortlake._ + + _Anno. 1570. February. 9._ + + + [Decoration] + + + + + [Transcriber’s Note: + + The “Groundplat” was printed in the form of a stemma, or tree, on an + oversized fold-out page. The layout was impossible to reproduce for + this e-text, so the information has been rearranged in nested-list + form. Size markings (see note at beginning of e-text) are relative + within each paragraph.] + + + _J. DEE_ + + +‡Here haue you (according to my promisse) the Groundplat of‡+ + +my MATHEMATICALL Præface: annexed to _Euclide_ (now first)+ + published in our Englishe tounge. An. 1570. Febr. 3. + + ++‡Sciences, and Artes Mathematicall,‡ are, either+ + + +‡Principall,‡ which are two, onely,+ + + +Arithmetike.+ + + +‡Simple‡+, Which dealeth with Numbers onely: and demonstrateth + all their properties and appertenances: where, an Vnit, is + Indiuisible. + +‡Mixt‡+, Which with aide of Geometrie principall, demonstrateth + some Arithmeticall Conclusion, or Purpose. + + +Geometrie.+ + + +‡Simple‡+, Which dealeth with Magnitudes, onely: and + demonstrateth all their properties, passions, and + appertenances: whose Point, is Indiuisible. + +‡Mixt‡+, Which with aide of Arithmetike principall, + demonstrateth some Geometricall purpose, as +EVCLIDES + ELEMENTES+. + + +‡The vse‡ whereof, is either,+ + + In thinges Supernaturall, æternall, & Diuine: By Application, + _Ascending_. + In thinges Mathematicall: without farther Application. + In thinges Naturall: both Substãtiall, & Accidentall, Visible, + & Inuisible. &c. By Application: _Descending_. + + The like Vses and Applications are, (though in a degree lower) in + the +Artes Mathematicall Deriuatiue+. + + +‡Deriuatiue‡ frõ the Principalls: of which, some haue+ + + +‡The names of‡ the Principalls: as,+ + + +_Arithmetike_, vulgar: which considereth+ + + --Arithmetike of most vsuall whole numbers: And of Fractions to + them appertaining. + --Arithmetike of Proportions. + --Arithmetike Circular. + --Arithmetike of Radicall Nũbers: Simple, Compound, Mixt: And of + their Fractions. + --Arithmetike of Cossike Nũbers: with their Fractions: And the + great Arte of Algiebar. + + +_Geometrie_, vulgar: which teacheth Measuring+ + + +‡At hand‡+ + + All Lengthes.--+Mecometrie.+ + All Plaines: As, Land, Borde, Glasse, &c.--+Embadometrie.+ + All Solids: As, Timber, Stone, Vessels, &c.--+Stereometrie.+ + + +‡With distãce‡+ from the thing Measured, as, + + +‡How farre‡+, from the Measurer, any thing is: of him + sene, on Land or Water: called +Apomecometrie+. + +‡How high or deepe‡+, from the leuell of the Measurers + standing, any thing is: Seene of hym, on Land or Water: + called +Hypsometrie+. + +‡How broad‡+, a thing is, which is in the Measurers view: + so it be situated on Land or Water: called + +Platometrie+. + + +‡Of which‡ are growen the Feates & Artes of+ + + +Geodesie+: more cunningly to Measure and Suruey Landes, + Woods, Waters. &c. + +Geographie.+ + +Chorographie.+ + +Hydrographie.+ + +Stratarithmetrie.+ + + +‡Propre names‡ as+, + + +Perspectiue,+--Which demonstrateth the maners and properties + of all Radiations: Directe, Broken, and Reflected. + + +Astronomie,+--Which demonstrateth the Distances, Magnitudes, + and all Naturall motions, Apparences, and Passions, proper to + the Planets and fixed Starres: for any time, past, present, and + to come: in respecte of a certaine Horizon, or without respecte + of any Horizon. + + +Musike,+--Which demonstrateth by reason, and teacheth by + sense, perfectly to iudge and order the diuersitie of Soundes, + hie or low. + + +Cosmographie,+--Which, wholy and perfectly maketh description + of the Heauenlym and also Elementall part of the World: and of + these partes, maketh homologall application, and mutuall + collation necessary. + + +Astrologie,+--Which reasonably demonstrateth the operations + and effectes of the naturall beames of light, and secrete + Influence of the Planets, and fixed Starres, in euery Element + and Elementall body: at all times, in any Horizon assigned. + + +Statike,+--Which demonstrateth the causes of heauines and + lightnes of all thinges: and of the motions and properties to + heauines and lightnes belonging. + + +Anthropographie,+ Which describeth the Nũber, Measure, Waight, + Figure, Situation, and colour of euery diuers thing contained in + the perfecte body of MAN: and geueth certaine knowledge of the + Figure, Symmetrie, Waight, Characterization, & due Locall motion + of any percell of the said body assigned: and of numbers to the + said percell appertaining. + + +Trochilike,+--Which demonstrateth the properties of all + Circular motions: Simple and Compound. + + +Helicosophie,+--Which demonstrateth the designing of all + Spirall lines: in Plaine, on Cylinder, Cone, Sphære, Conoïd, and + Sphæroid: and their properties. + + +Pneumatithmie,+--Which demonstrateth by close hollow + Geometricall figures (Regular and Irregular) the straunge + properties (in motion or stay) of the Water, Ayre, Smoke, and + Fire, in their Continuitie, and as they are ioyned to the + Elementes next them. + + +Menadrie,+--Which demonstrateth, how, aboue Natures Vertue, + and power simple: Vertue and force, may be multiplied: and so + to directe, to lift, to pull to, and to put or cast fro, any + multiplied, or simple determined Vertue, Waight, or Force: + naturally, not, so, directible, or moueable. + + +Hypogeiodie,+--Which demonstrateth, how, vnder the Sphæricall + Superficies of the Earth, at any depth, to any perpendicular + line assigned (whose distance from the perpendicular of the + entrance: and the Azimuth likewise, in respecte of the sayd + entrance, is knowen) certaine way, may be prescribed and gone, + &c. + + +Hydragogie,+--Which demonstrateth the possible leading of + water by Natures law, and by artificiall helpe, from any head + (being Spring, standing, or running water) to any other place + assigned. + + +Horometrie,+--Which demonstrateth, how, at all times + appointed, the precise, vsuall denomination of time, may be + knowen, for any place assigned. + + +Zographie,+--Which demonstrateth and teacheth, how, the + Intersection of all visuall Pyramids, made by any plaine + assigned (the Center, distance, and lightes being determined) + may be, by lines, and proper colours represented. + + +Architecture,+--Which is a Science garnished with many + doctrines, and diuers Instructions: by whose iudgement, all + workes by other workmen finished, are iudged. + + +Nauigation,+--Which demonstrateth, how, by the Shortest good + way, by the aptest direction, and in the shortest time: + a sufficient Shippe, betwene any two places (in passage + nauigable) assigned, may be conducted: and in all stormes and + naturall disturbances chauncing, how to vse the best possible + meanes, to recouer the place first assigned. + + +Thaumaturgike,+--Which geueth certaine order to make straunge + workes, of the sense to be perceiued: and of men greatly to be + wondred at. + + +Archemastrie,+--Which teacheth to bring to actuall experience + sensible, all worthy conclusions, by all the Artes Mathematicall + purposed: and by true Naturall philosophie, concluded: And both + addeth to them a farder Scope, in the termes of the same Artes: + and also, by his proper Method, and in peculiar termes, + procedeth, with helpe of the forsayd Artes, to the performance + of complete Experiences: which, of no particular Arte, are hable + (Formally) to be challenged. + + ++¶ Imprinted by _Iohn Day_.+ + +An. 1570. Feb. 25. + + * * * * * + * * * * + * * * * * + +Errors and Anomalies: + +Unless otherwise noted, spelling and punctuation are unchanged. +Errors are listed below, with the original form, if changed, shown in +[brackets]. Unusual words include “fatch” (probably used as a variant +of “fetch”) and the mathematical terms “sexagene” and “sexagesme”. + + How, worldly goods: how, worldly dignitie + [_“o” in second “worldly” invisible_] + his most diligent hearers (so infinitely mought [hearers) so] + the boundes, and duety of an Hydrographer [Hydographer] + of the Grekes it is called _Eteromekes_ + [_text unchanged: correct form is “Heteromekes”_] + τὸ ὁτὶ [_accent unchanged_] + in our worldly affaires [wordly] + fall to worke.❉. + [_Some text readers may not display the oversized-asterisk symbol._] + _Emptying the first._ [Emptyting] + Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει λέγοντι πιϛευτεόν + [ἡμήρας ... πιϛευτέομ] + of the suddeyne [snddeyne] + that the right and absolute way may be had [he had] + Georgic I: [_The quoted segments, each ending in “&c.”, are + 438-439; 451-457; 463-464._] + +Additional Notes: + + The Greek letter η (eta) was consistently printed as if it were the + ou-ligature ȣ. + The Latin “-que” was written as an abbreviation resembling “-q´;”. + It is shown here as [que]. + + Mathematical symbols seen in the section accompanying the diagrams + could not be reproduced. The following substitutions were made: + --The curly “P” used for “Pounds” is shown as {P}. + --The “potestas” symbol, used to represent “x” (the unknown), + is shown as {x}. + --All roots were expressed as the “root” sign √ combined with + symbols for the power of 2 (doubled for power of 4, or fourth root) + and 3. They are shown as ²√ ³√ ⁴√. + +Euclid: + +The following Propositions were identified by number. + +6.12: (How) to find a fourth (line) proportional to three given straight +lines. + +11.34: In equal parallelepipedal solids the bases are reciprocally +proportional to the heights; and those parallelepipedal solids in which +the bases are reciprocally proportional to the heights are equal. + +11.36: If three straight lines are proportional, then the +parallelepipedal solid formed out of the three equals the +parallelepipedal solid on the mean which is equilateral, but equiangular +with the aforesaid solid. + +12.1: Similar polygons inscribed in circles are to one another as the +squares on their diameters. + +12.2: Circles are to one another as the squares on their diameters. + +12.18 (“last”): Spheres are to one another in triplicate ratio of their +respective diameters. + + + + +*** END OF THE PROJECT GUTENBERG EBOOK THE MATHEMATICALL PRAEFACE *** + +Updated editions will replace the previous one--the old editions will +be renamed. + +Creating the works from print editions not protected by U.S. copyright +law 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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