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+
+*** START OF THE PROJECT GUTENBERG EBOOK 75904 ***
+
+
+
+
+
+                                  THE
+                              SLIDE RULE:
+                           A PRACTICAL MANUAL
+
+
+                                   BY
+
+                          CHARLES N. PICKWORTH
+
+     WHITWORTH SCHOLAR; EDITOR OF “THE MECHANICAL WORLD”; AUTHOR OF
+    “LOGARITHMS FOR BEGINNERS”; “THE INDICATOR: ITS CONSTRUCTION AND
+  APPLICATION”; “THE INDICATOR DIAGRAM: ITS ANALYSIS AND CALCULATION,”
+                                  ETC.
+
+                         _SEVENTEENTH EDITION_
+
+                              MANCHESTER:
+                        EMMOTT AND CO., LIMITED,
+                            65 KING STREET;
+
+                               NEW YORK:
+                          D. VAN NOSTRAND CO.,
+                            8 WARREN STREET.
+
+                                LONDON:
+                        EMMOTT AND CO., LIMITED,
+                        20 BEDFORD STREET, W.C.
+
+                                  AND
+                       PITMAN AND SONS, LIMITED,
+                     PARKER ST., KINGSWAY, W.C. 2.
+
+                  [_Three Shillings and Sixpence net_]
+
+
+
+
+                         _All rights reserved._
+
+
+
+
+                   PREFACE TO THE FIFTEENTH EDITION.
+
+
+Several new slide rules for special calculations are described in this
+edition, and the contents further extended to include a section dealing
+with screw-cutting gear calculations by the slide rule—an application of
+the instrument to which attention has been given recently.
+
+Mention should be made of the fact that some of the special slide rules
+described in previous editions are no longer obtainable. As, however,
+the descriptive notes may be of service to those possessing the
+instruments, and are, in some measure, of general interest, they have
+been allowed to remain in the present issue.
+
+The author tenders his thanks to the many who have evinced their
+appreciation of his efforts to popularise the subject; also for the many
+kind hints and suggestions which he has received from time to time, and
+with a continuance of which he trusts to be favoured in the future.
+
+                                                                C. N. P.
+
+WITHINGTON, MANCHESTER, _November 1917_.
+
+
+                  PREFACE TO THE SEVENTEENTH EDITION.
+
+The sustained demand for this very successful work having resulted in
+the early call for a new edition, the opportunity has been taken to
+introduce descriptions of new slide rules and to effect some slight
+revisions.
+
+                                                                C. N. P.
+
+WITHINGTON, MANCHESTER, _December 1920_.
+
+
+
+
+                               CONTENTS.
+
+
+                                                                    PAGE
+ Introductory                                                          5
+ The Mathematical Principle of the Slide Rule                          6
+ Notation by Powers of 10                                              8
+ The Mechanical Principle of the Slide Rule                            9
+ The Primitive Slide Rule                                             10
+ The Modern Slide Rule                                                12
+ The Notation of the Slide Rule                                       14
+ The Cursor or Runner                                                 17
+ Multiplication                                                       19
+ Division                                                             24
+ The Use of the Upper Scales for Multiplication and Division          26
+ Reciprocals                                                          27
+ Continued Multiplication and Division                                28
+ Multiplication and Division with the Slide Inverted                  30
+ Proportion                                                           31
+ General Hints on the Elementary Uses of the Slide Rule               36
+ Squares and Square Roots                                             37
+ Cubes and Cube Roots                                                 40
+ Miscellaneous Powers and Roots                                       45
+ Power and Roots by Logarithms                                        45
+ Other Methods of Obtaining Powers and Roots                          47
+ Combined Operations                                                  49
+ Hints on Evaluating Expressions                                      52
+ Gauge Points                                                         53
+ Examples in Technical Calculations                                   56
+ Trigonometrical Application                                          74
+ Slide Rules with Log-log Scales                                      84
+ Special Types of Slide Rules                                         92
+ Long-Scale Slide Rules                                               96
+ Circular Calculators                                                101
+ Slide Rules for Special Calculations                                109
+ Construction Improvements in Slide Rules                            110
+ The Accuracy of Slide Rule Results                                  111
+ Appendix:—
+   New Slide Rules                                                   113
+   The Solution of Algebraic Equations                               122
+   Screw-Cutting Gear Calculations                                   124
+   Gauge Points and Signs on Slide Rules                             126
+   Tables and Data                                                   128
+   Slide Rule Data Slips                                             133
+
+
+
+
+                            THE SLIDE RULE.
+
+
+
+
+                             INTRODUCTORY.
+
+
+The slide rule may be defined as an instrument for mechanically
+effecting calculations by logarithms. Those familiar with logarithms and
+their use will recognise that the slide rule provides what is in effect
+a concisely arranged table of logarithms, together with a simple and
+convenient means for adding and subtracting any selected values. Those,
+however, who have no acquaintance with logarithms will find that only an
+elementary knowledge of the subject is necessary to enable them to make
+full use of the slide rule. It is true that for simple slide-rule
+operations, as multiplication and division, a knowledge of logarithms is
+unnecessary; indeed, many who have no conscious understanding of
+logarithms make good use of the instrument. But this involves a blind
+reliance upon rules without an appreciation of their origin or
+limitations, and this, in turn, engenders a want of confidence in the
+results of any but the simplest operations, and prevents the fullest use
+being made of the instrument. For this reason a brief, but probably
+sufficient _résumé_ of the principles of logarithmic calculation will be
+given. Those desiring a more detailed explanation are referred to the
+writer’s “Logarithms for Beginners.”
+
+The slide rule enables various arithmetical, algebraical and
+trigonometrical processes to be performed with ease and rapidity, and
+with sufficient accuracy for most practical purposes. A grasp of the
+simple fundamental principles which underlie its operation, together
+with a little patient practice, are all that are necessary to acquire
+facility in using the instrument, and few who have become proficient in
+this system of calculating would willingly revert to the laborious
+arithmetical processes.
+
+
+
+
+             THE MATHEMATICAL PRINCIPLE OF THE SLIDE RULE.
+
+
+Logarithms may be defined as a series of numbers in _arithmetical_
+progression, as 0, 1, 2, 3, 4, etc., which bear a definite relationship
+to another series of numbers in _geometrical_ progression, as 1, 2, 4,
+8, 16, etc. A more precise definition is:—The logarithm of a number to
+any base, is the _index of the power_ to which the base must be raised
+to equal the given number. In the logarithms in general use, known as
+_common logarithms_, and with which we are alone concerned, 10 is the
+base selected. The general definition may therefore be stated in the
+following modified form:—_The common logarithm of a number is the index
+of the power to which 10 must be raised to equal the given number._
+Applying this rule to a simple case, as 100 = 10^2, we see that the base
+10 must be squared (_i.e._, raised to the 2nd power) in order to equal
+100, the number selected. Therefore, as 2 is the index of the power to
+which 10 must be raised to equal 100, it follows from our definition
+that 2 is the common logarithm of 100. Similarly the common logarithm of
+1000 will be 3, while proceeding in the opposite direction the common
+log. of 10 must equal 1. Tabulating these results and extending, we
+have:—
+
+             Numbers    10,000  1000   100     10     1
+             Logarithms   4      3      2      1      0
+
+It will now be evident that for numbers
+
+       between    1 and     10 the logs. will be between 0 and 1
+          „      10  „     100       „          „        1  „  2
+          „     100  „    1000       „          „        2  „  3
+          „    1000  „  10,000       „          „        3  „  4
+
+In other words, the logarithms of numbers between 1 and 10 will be
+wholly fractional (_i.e._, decimal); the logs. of numbers between 10 and
+100 will be 1 _followed by a decimal quantity_; the logs. of numbers
+between 100 and 1000 will be 2 followed by a decimal quantity, and so
+on. These decimal quantities for numbers from 1 to 10 (which are the
+logarithms of this particular series) are as follows:—
+
+ Numbers      1     2     3     4     5     6     7     8     9    10
+ Logarithms   0   0·301 0·477 0·602 0·699 0·778 0·845 0·903 0·954 1·000
+
+Combining the two tables, we can complete the logarithms. Thus for 3
+multiplied successively by 10, we have:—
+
+          Numbers      3      30    300    3000  30,000  etc.
+          Logarithms 0·477  1·477  2·477  3·477  4·477    „
+
+We see from this that for numbers having the _same significant figure_
+(or figures), 3 in this case, the decimal part or _mantissa_ of the
+logarithm is the same, but that the integral part or _characteristic_ is
+always _one less than the number of figures before the decimal point_.
+
+For numbers less than 1 the same plan is followed. Thus extending our
+first table downwards, we have:—
+
+          Numbers      1     0·1    0·01  0·001  0·0001  etc.
+          Logarithms   0      −1     −2     −3     −4     „
+
+so that for 3 divided successively by 10, we have:—
+
+          Numbers      3     0·3    0·03  0·003  0·0003  etc.
+          Logarithms 0·477  ̅1·477  ̅2·477  ̅3·477  ̅4·477    „
+
+Here again we see that with the same significant figures in the numbers,
+the mantissa of the logarithm has always the same (_positive_) value,
+but the characteristic is _one more_ than the _number of 0’s immediately
+following the decimal point_, and is _negative_, as indicated by the
+minus sign written over it. Only the decimal parts of the logarithms of
+numbers between 1 and 10 are given in the usual tables, for, as shown
+above, the logarithms of all tenfold multiples or submultiples of a
+number can be obtained at once by modifying the characteristic in
+accordance with the rules given.
+
+An examination of the two rows of figures giving the logarithms of
+numbers from 1 to 10 will reveal some striking peculiarities, and at the
+same time serve to illustrate the principle of logarithmic calculation.
+First, it will be noticed that the addition of any two of the logarithms
+gives the logarithm of the _product_ of these two numbers. Thus, the
+addition of log. 2 and log. 4 = 0·301 + 0·602 = 0·903, and this is seen
+to be the logarithm of 8, that is, of 2 × 4. Conversely, the difference
+of the logarithms of two numbers gives the logarithm of the _quotient_
+resulting from the division of these two numbers. Thus, log. 8 − log. 2
+= 0·903 − 0·301 = 0·602, which is the log. of 4, or of 8 ÷ 2.
+
+One other important point is to be noted. If the logarithm of any number
+is _multiplied_ by 2, 3, or any other quantity, whole or fractional, the
+result is the logarithm of the original number, raised to the 2nd, 3rd,
+or other power respectively. Thus, multiplying the log. of 3 by 2, we
+obtain 0·477 × 2 = 0·954, and this is seen to be the log. of 9, that is,
+of 3 raised to the 2nd power, or 3 _squared_. Again, log. 2 multiplied
+by 3 = 0·903—that is, the log. of 8, or of 2 raised to the 3rd power, or
+2 _cubed_. Conversely, dividing the logarithm of any original number by
+any number _n_, we obtain the logarithm of the _n_th root of the
+original number. Thus, log. 8 ÷ 3 = 0·903 ÷ 3 = 0·301, and is therefore
+equal to log. 2 or to the log. of the _cube root_ of 8.
+
+Only simple logs. have been taken in these examples, but the student
+will understand that the same reasoning applies, whatever the number.
+Thus for 20^3 we prefix the characteristic (1 in this case) to log. 2,
+giving 1·301. Multiplying by 3, we have 3·903 as the resulting
+logarithm, and as its characteristic is 3, we know that it corresponds
+to the number 8000. Hence 20^3 = 8000.
+
+In this brief explanation is included all that need now be said with
+regard to the properties of logarithms. The main facts to be borne
+clearly in mind are:—(1.) That to find the _product_ of two numbers, the
+logarithms of the numbers are to be _added_ together, the result being
+the logarithm of the product required, the value of which can then be
+determined. (2.) That in finding the _quotient_ resulting from the
+division of one number by another, _the difference_ of the logarithms of
+the numbers gives the logarithm of the quotient, from which the value of
+the latter can be ascertained. (3.) That to find the result of _raising
+a number to the nth power_, we _multiply_ the logarithm of the number by
+_n_, thus obtaining the logarithm, and hence the value, of the desired
+result. And (4.) That to find the n_th root of a number_, we _divide_
+the logarithm of the number by _n_, this giving the logarithm of the
+result, from which its value may be determined.
+
+
+
+
+                       NOTATION BY POWERS OF 10.
+
+
+A convenient method of representing an arithmetical quantity is to split
+it up into two factors, of which the first is the original number, with
+the decimal point moved so as to immediately follow the first
+significant figure, and the second, 10^{_n_} where _n_ is the number of
+places the decimal point has been moved, this index being _positive_ for
+numbers greater than 1, and _negative_ for numbers less than 1.[1] In
+this system, therefore, we regard 3,610,000 as 3·61 × 1,000,000, and
+write it as 3·61 × 10^6. Similarly 361 = 3·61 x 10^2; 0·0361 (=
+(3·61)/(100)) = 3·61 × 10^{−2}; 0·0000361 = 3·61 × 10^{−5}, etc. To
+restore a number to its original form, we have only to move the decimal
+point through the number of places indicated by the index, moving to the
+right if the index is positive and to the left (prefixing 0’s) if
+negative. This method, which should be cultivated for ordinary
+arithmetical work, is substantially that followed in calculating by the
+slide rule. Thus with the slide rule the multiplication of 63,200 by
+0·0035 virtually resolves itself into 6·32 × 10^4 × 3·5 × 10^{−3} or
+6·32 × 3·5 × 10^{4–3} = 22·12 x 10^1 = 221·2. It will be seen later,
+however, that the result can be arrived at by a more direct, if less
+systematic, method of working.
+
+
+
+
+              THE MECHANICAL PRINCIPLE OF THE SLIDE RULE.
+
+
+[Illustration: FIG. 1.]
+
+The mechanical principle involved in the slide rule is of a very simple
+character. In Fig. 1, A and B represent two rules divided into 10 equal
+parts, the division lines being numbered consecutively as shown. If the
+rule B is moved to the right until 0 on B is opposite 3 on A, it is seen
+that any number on A is equal to the coinciding number on B, plus 3.
+Thus opposite 4 on B is 7 on A. The reason is obvious. By moving B to
+the right, we add to a length 0·3, another length 0·4, the result read
+off on A being 7. Evidently, the same result would have been obtained if
+a length 0·4 had been added, by means of a pair of dividers, to the
+length 0·3 on the scale A. By means of the slide B, however, the
+addition is more readily effected, and, what is of much greater
+importance, the result of adding 3 to _any one of the numbers_ within
+range, on the lower scale, is _immediately_ seen by reading the adjacent
+number on A.
+
+Of course, subtraction can be quite as readily performed. Thus, to
+subtract 4 from 7, we require to deduct from 0·7 on the A scale, a
+length 0·4 on B. We do this by placing 4 on B under 7 on A, when over 0
+on B we find 3, on A. It is here evident that the _difference_ of any
+pair of coinciding numbers on the scales is constantly equal to 3.
+
+[Illustration: FIG. 2.]
+
+An important modification results if the slide-scale B is inverted as in
+Fig. 2. In this case, to find the sum of 4 and 3 we require to place the
+4 of the A scale to 3 on the B scale, and the result is read on A over 0
+on B. Here it will be noted, the _sum_ of any pair of coinciding numbers
+on the scales is constant and equal to 7. This case, therefore,
+resembles that of the immediately preceding one, except that the _sum_,
+instead of the _difference_, of any pair of coinciding numbers is
+constant.
+
+To find the difference of two factors, the converse operation is
+necessary. Thus, to subtract 4 from 7, 0 on B is placed opposite 7 on A,
+and over 4 on B is found 3 on A.
+
+From these examples it will be seen that with the slide _inverted_ the
+methods of operation are the reverse of those used when the slide is in
+its normal position.
+
+It will be understood that although we have only considered the primary
+divisions of the scales, the remarks apply equally to any subdivisions
+into which the primary spaces of the scales might be divided. Further,
+we note that the length of scale taken to represent a unit is quite
+arbitrary.
+
+
+
+
+                       THE PRIMITIVE SLIDE RULE.
+
+
+The application of the foregoing principles to the slide rule can be
+shown most conveniently by describing the construction of a simple form
+of slide rule:—Take a strip of card about 11 in. long and 2 in. wide;
+draw a line down the centre of its width, and mark off two points, 10
+in. apart. Draw cross lines at these points and figure them 1 and 10 on
+each side, as in Fig. 3. Next mark off lengths of 3·01, 4·77, 6·02,
+6·99, 7·78, 8·45, 9·03 and 9·54 inches, from the line marked 1. Draw
+cross lines as before, and figure these lines, 2, 3, 4, 5, 6, 7, 8 and
+9. To fill in the intermediate divisions of the scale, take the logs, of
+1·1, 1·2, 1·3, etc. (from a table), multiply each by 10, and thus obtain
+the distances from 1, at which the several subdivisions are to be
+placed. Mark these 1·2, 1·3, 1·4, etc., and complete the scale, making
+the interpolated division marks shorter to facilitate reading, as with
+an ordinary measuring rule. Cutting the card cleanly down the centre
+line, we have the essentials of the slide rule.
+
+[Illustration: FIG. 3.]
+
+The fundamental principle of the slide rule is now evident:—Each scale
+is graduated in such a manner that the _distance of any number from 1 is
+proportional to the logarithm of that number_.
+
+[Illustration: FIG. 4.]
+
+“We know that to find the product of 2 × 3 by logarithms, we add 0·301,
+or log. 2, to 0·477, the log. of 3, obtaining 0·778, or log. 6. With our
+primitive slide rule we place 1 on the lower scale to 3·01 in. (which we
+have marked 2) on the upper scale (Fig. 4). Then over 4·77 in. on the
+lower scale (which we marked 3), we have 7·78 in. (which we marked 6) on
+the upper scale. Conversely, to divide 6 by 3, we place 3 on the lower
+scale in agreement with 6 on the upper, and over 1 on the lower scale
+read 2 on the upper scale. This method of adding and subtracting scale
+lengths will be seen to be identical with that used in the simple case
+shown in Fig. 1.
+
+
+
+
+                         THE MODERN SLIDE RULE.
+
+
+The modern form of slide rule, variously styled the Gravêt, the
+Tavernier-Gravêt, and the Mannheim rule, is frequently made of boxwood,
+but all the leading instrument makers now supply rules made of boxwood
+or mahogany, and faced with celluloid, the white surface of which brings
+out the graduations much more distinctly than lines engraved on a
+boxwood surface. The celluloid facings should not be polished, as a dull
+surface is much less fatiguing to the eyes. The most generally used, and
+on the whole the most convenient size of rule, is about 10½in. long,
+1¼in. wide, and about ⅜in. thick; but 5 in., 8 in., 15 in., 20 in., 24
+in. and 40 in. rules are also made. In the centre of the stock of the
+rule a movable slip is fitted, which constitutes the slide, and
+corresponds to the lower of the two rules of our rudimentary examples.
+
+[Illustration: FIG. 5.]
+
+From Fig. 5, which is a representation of the face of a Gravêt or
+Mannheim slide rule, it will be seen that four series of logarithmic
+graduations or scale-lines are employed, the upper and lower being
+engraved on the stock or body of the rule, while the other two are
+engraved upon the slide. The two upper sets of graduations are exactly
+alike in every particular, and the lower sets are also similar. It is
+usual to identify the two upper scale-lines by the letters A and B, and
+the two lower by the letters C and D, as indicated in the figure at the
+left-hand extremities of the scales.
+
+Referring to the scales C and D, these will each be seen to be a
+development of the elementary scales of Fig. 3, but in this case each
+principal space is subdivided, more or less minutely. The principle,
+however, is exactly the same, so that by moving the slide (carrying
+scale C), multiplication and division can be mechanically performed in
+the manner described.
+
+The upper scale-line A consists of two exactly similar scales, placed
+end to end, the first lying between IL and IC, and the second between IC
+and IR. The first of these scales will be designated the _left-hand A
+scale_, and the second the _right-hand A scale_. Similarly the
+coinciding scales on the slide are the _left-hand B scale_ and the
+_right-hand B scale_. Each of these four scales is divided (as finely as
+convenient) as in the case of the C and D scales, but, of course, they
+are exactly one half the length of the latter.
+
+The two end graduations of both the C and D scales are known as the
+_left-_ and _right-hand indices_ of these scales. Sometimes they are
+figured 1 and 10 respectively; sometimes both are marked 1. Similarly IL
+and IR are the left- and right-hand indices of the A and B lines, while
+IC is the centre index of these scales. Other division lines usually
+found on the face of the rule are one on the left-hand A and B scales,
+indicating the ratio of the circumference of a circle to its diameter, π
+= 3·1416; and a line on the right-hand B scale marking the position of
+(π)/(4) = 0·7854, used in calculating the areas of circles. Reference
+will be made hereafter to the scales on the under-side of the slide, and
+we need now only add that one of the edges of the rule, usually
+bevelled, is generally graduated in millimetres, while the other edge
+has engraved on it a scale of inches divided into eighths or tenths. On
+the bottom face inside the groove of the rule either one or the other of
+these scales is continued in such a manner that by drawing the slide out
+to the right and using the scale inside the rule, in conjunction with
+the corresponding scale on the edge, it is possible to measure 20 inches
+in the one case, or nearly 500 millimetres in the other. On the back of
+the rule there is usually a collection of data, for which the slips
+given at the end of this work may often be substituted with advantage.
+
+
+
+
+                    THE NOTATION OF THE SLIDE RULE.
+
+
+Hitherto our attention has been confined to a consideration of the
+primary divisions of the scales. The same principle of graduation is,
+however, used throughout; and after what has been said, this part of the
+subject need not be further enlarged upon. Some explanation of the
+method of reading the scales is necessary, as facility in using the
+instrument depends in a very great measure upon the dexterity of the
+operator in assigning the correct value to each division on the rule. By
+reference to Fig. 5, it will be seen that each of the primary spacings
+in the several scales is invariably subdivided into ten; but since the
+lengths of the successive primary divisions rapidly diminish, it is
+impossible to subdivide each main space into the same number of parts
+that the space 1–2 can be subdivided. This variable spacing of the
+scales is at first confusing to the student, but with a little practice
+the difficulty is soon overcome.
+
+With the C or D scale, it will be noticed that the length of the
+interval 1–2 is sufficient to allow each of the 10 subdivisions to be
+again divided into 10 parts, so that the whole interval 1–2 is divided
+into 100. The shorter main space 2–3, and the still shorter one 3–4,
+only allow of the 10 subdivisions of each being divided into five parts.
+Each of these main spaces is therefore divided into 50 parts. For the
+remainder of the scale each of the 10 subdivisions of each main space is
+divided into two parts only; so that from the main division 4 to the end
+of the scale the primary spaces are divided into 20 parts only.
+
+In the upper scales A or B, it will be found that—as the space 1–2 is of
+only half the length of the corresponding space on C or D—the 10
+subdivisions of this interval are divided into five parts only.
+Similarly each of the 10 subdivisions of the intervals 2–3, 3–4, and 4–5
+are further divided into two parts only, while for the remainder of the
+scale only the 10 subdivisions are possible, owing to the rapidly
+diminishing lengths of the primary spacings.
+
+The values actually given on the rule run from 1 to 10 on the lower
+scales and from 1 to 100 on the upper scales, and, as explained on page
+9, all factors are brought within these ranges of values by multiplying
+or dividing them by powers of 10. By following this plan, we virtually
+regard each factor as merely a series of significant figures, and make
+the necessary modification due to the “powers of 10” when fixing the
+position of the decimal point in the answer.
+
+Many, however, find it convenient in practice to regard the values on
+the rule as multiplied or divided by such powers of 10 as may be
+necessary to suit the factors entering into the calculation. If this
+plan is adopted, the values given to each graduation of the scales will
+depend on that given to the left index figure (1) of the lower scales,
+this being any multiple or submultiple of 10. Thus IL on the D scale may
+be regarded as 1, 10, 100, 1000, etc., or as 0·1, 0·01, 0·001, 0·0001,
+etc.; but once the initial value is assigned to the index, the ratio of
+value must be maintained throughout the whole scale. For example, if 1
+on C is taken to represent 10, the main divisions 2, 3, 4, etc., will be
+read as 20, 30, 40, etc. On the other hand, if the fourth main division
+is read as 0·004, then the left index figure of the scale will be read
+as 0·001. The figured subdivisions of the main space 1–2 are to be read
+as 11, 12, 13, 14, 15, 16, 17, 18 and 19—if the index represents 10,—and
+as corresponding multiples for any other value of the index.
+
+Independently considered, these remarks apply equally to the A or B
+scale, but in this case the notation is continued through the second
+half of the scale, the figures of which are to be read as tenfold values
+of the corresponding figures in the first half of the scale.
+
+The reading of the intermediate divisions will, of course, be determined
+by the values assigned to the main divisions. Thus, if IL on D is read
+as 1, then each of the smallest subdivisions of the space 1–2 will be
+read as 0·01, and each of the smallest subdivisions of the spaces 2–3 or
+3–4 as 0·02, while for the remainder of the scale the smallest
+subdivisions are read as 0·05. In the A or B scale the subdivisions of
+the space 1–2 of the first half of the scale are (if IL = 1) read as
+0·02, 0·04, etc.; for the divisions 2–3, 3–4, and 4–5, the smallest
+intervals are read as 0·05 of the primary spaces, and from 5 to the
+centre index of the scale the divisions represent 0·1 of each main
+interval. Passing the centre index, which is, now read as 10, the
+smallest subdivisions immediately following are read 10·2, 10·4, etc.,
+until 20·0 is reached; then we read 20·5, 21·0, 21·5 22·0, etc., until
+the figured main division 5 is reached. The remainder of the scale is
+read 51, 52, 53, etc., up to 100, the right-hand index.
+
+Further subdivision of any of the spaces of the rule can be effected by
+the eye, and after a little practice the operator will become quite
+expert in estimating any intermediate value. It affords good practice to
+set 1 on C to 1·04, 1·09, etc. on D, and to read the values on D, under
+4, 6, 8, etc. on C. As the exact results are easily calculated mentally,
+the student, by this means, will receive better instruction in
+estimating intermediate results than can be given by any diagram.
+
+Some rules will be found figured as shown in Fig. 5; in others, the
+right-hand upper scales are marked 10, 20, 30, etc. Again, others are
+marked decimally, the lower scales and the left-hand upper scales being
+figured 1, 1·1, 1·2, 1·3 ... 2·5, etc. The latter form has advantages
+from the point of view of the beginner.
+
+The method of reading the A and B scales, just given, applies only when
+these scales are regarded as altogether independent of the lower pair of
+scales C and D. Some operators prefer to use the A and B scales, and
+some the C and D scales, for the ordinary operations of proportion,
+multiplication, and division. Each method has its advantages, as will be
+shown, but in the more complex calculations, as involution and
+evolution, etc., the relation of the upper scales to the lower scales
+becomes a very important factor.
+
+The distance 1–10 on the upper scales is one-half of the distance 1–10
+on the lower scales. Hence any distance from 1, taken on the upper
+scales, represents _twice the logarithm_ which the same distance
+represents on the lower scales. In other words, the length which
+represents log. N on D, would represent 2 log. N on A; and, conversely,
+the length which represents log. N on A, would represent (log. N)/(2) on
+D.
+
+Now we have seen (page 8) that multiplying the log. of a number by 2
+gives the log. of the square of the number. Hence, above any number on D
+we find its _square_ on A, or, conversely, below any number on A, we
+find its _square root_ on D. Thus, above 2 we find 4; under 49, we find
+7 and so on. Obviously the same relation exists between the B and C
+scales.
+
+
+
+
+                         THE CURSOR OR RUNNER.
+
+
+All modern slide rules are now fitted with a _cursor_ or _runner_, which
+usually consists of a light metal frame moving under spring control in
+grooves in the edges of the stock of the rule. This frame carries a
+piece of glass, mica or transparent celluloid, about 1 in. square,
+across the centre of which a fine reference line is drawn exactly at
+right angles to the line of scales. To “set the cursor” to any value on
+the scales of the rule, the frame is taken between the thumb and
+forefinger and adjusted in position until the line falls exactly upon
+the graduation, or upon an estimated value, between a pair of
+graduations, as the case may be. Having fixed one number in this way,
+another value on either of the scales on the slide may be similarly
+adjusted in reference to the cursor line. The cursor will be found very
+convenient in making such settings, especially when either or both of
+the numbers are located by eye estimation. It also finds a very
+important use in referring the readings of the upper scale to those of
+the lower, or _vice versa_, while as an aid in continued multiplication
+and division and complex calculations generally, its value is
+inestimable.
+
+_Multiple Line Cursors._—Cursors can be obtained with _two_ lines, the
+distance between them being that between 7·854 and 10 on the A scale.
+The use of this cursor is explained on page 57. Another multiple line
+cursor has short lines engraved on it, corresponding to the main
+graduations from 95 to 105 on the respective scales. This is useful for
+adding or deducting small percentages.
+
+_The Broken Line Cursor._—To facilitate setting, broken line cursors are
+made, in which the hair-line is not continued across the scales, but has
+two gaps, as shown in Fig. 6.
+
+_The Pointed Cursor_ has an index or pointer, extending over the
+bevelled edge of the rule, on which is a scale of inches. It is useful
+for summing the lengths of the ordinates of indicator diagrams, and also
+for plotting lengths representing the logarithms of numbers, sometimes
+required in graphic calculations.
+
+_The Goulding Cursor._—It has been pointed out that in order to obtain
+the third or fourth figure of a reading on the 10 in. slide rule, it is
+frequently necessary to depend upon the operator’s ability to mentally
+subdivide the space within which the reading falls. This subdivision can
+be mechanically effected by the aid of the Goulding Cursor (Fig. 7),
+which consists of a frame fitting into the usual grooves in the rule,
+and carrying a metal plate faced with celluloid, upon which is engraved
+a triangular scale A B C. The portion carrying the chisel edges E is not
+fixed to the cursor proper, but slides on the latter, so that the index
+marks on the projecting prongs can be moved slightly along the scales of
+the rule, this movement being effected by the short end of the bent
+lever F working in the slot as shown. D is a pointer which can be moved
+along F under spring control. As illustrating the method of use, we will
+assume that 1 on C is placed to 155 on D, and that we require to read
+the value on D under 27 on C. This is seen to lie between 4150 and 4200,
+so setting the pointer D to the line B C—always the first operation—we
+move the whole along the rule until the index line on the lower prong
+agrees with 4200. We then move F across the scale until the index line
+agrees with 4100, set the pointer D to the line A C, and move the lever
+back until the index line agrees with 27 on the slide. It will then be
+found that the pointer D gives 85 on A B as the value of the
+supplementary figures, and hence the complete reading is 4185.
+
+[Illustration: FIG. 6.]
+
+[Illustration: FIG. 8.]
+
+[Illustration: FIG. 7.]
+
+[Illustration: FIG. 9.]
+
+_Magnifying Cursors_ are of assistance in reading the scales, and in a
+good and direct light are very helpful. In one form an ordinary lens is
+carried by two light arms hinged to the upper and lower edges of the
+cursor, so that it can be folded down to the face of the rule when not
+in use. A more compact form, shown in Fig. 8, consists of a strip of
+plano-convex glass, on the under-side of which is the hair-line. In a
+cursor made by Nestler of Lahr, the plano-convex strip is fixed on the
+ordinary cursor. The magnifying power is about 2, so that a 5 in. rule,
+having the same number of graduations as a 10 in. rule, can be read with
+equal facility, by the aid of this cursor.
+
+The Digit-registering Cursor, supplied by Mr. A. W. Faber, London, and
+shown in Fig. 9, has a semicircular scale running from 0 at the centre
+upward to −6 and downward to +6. A small finger enables the operator to
+register the number of digits to be added or subtracted at the end of a
+lengthy operation, as explained at page 28.
+
+
+
+
+                            MULTIPLICATION.
+
+
+In the preliminary notes it was shown that by mechanically adding two
+lengths representing the logarithms of two numbers, we can obtain the
+_product_ of these numbers; while by subtracting one log. length from
+another, the number represented by the latter is divided by the number
+represented by the former. Hence, using the C and D scales, we have the
+
+RULE FOR MULTIPLICATION.—_Set the index of the C scale to one of the
+factors on D, and under the other factor on C, find the product on D._
+
+[Illustration: FIG. 10.]
+
+Thus, to find the product of 2 × 4, the slide is moved to the right
+until the left index (1) of C is brought over 2 on D, when under the
+other factor (4) on C, is found the required product (8) on D. Following
+along the slide, to the right, we find that beyond 5 on C (giving 10 on
+D), we have no scale below the projecting slide (Fig. 10). If we imagine
+the D scale prolonged to the right, we should have a repetition of the
+earlier portion, but, as with the two parts of the A scales, the
+repeated portion would be of tenfold value, and 10 on C would agree with
+20 on the prolonged D scale. We turn this fact to account by moving the
+slide to the left until 10 on C agrees with 2 on D, and we can then read
+off such results as 2 × 6 = 12; 2 × 8 = 16, etc., remembering that as
+the scale is now of tenfold value, there will be two figures in the
+result. Hence, for those who prefer rules, we have the
+
+RULE FOR THE NUMBER OF DIGITS IN A PRODUCT.—_If the product is read with
+the slide projecting to the_ LEFT, ADD THE NUMBER OF THE DIGITS IN THE
+TWO FACTORS; _if read with the slide to the_ RIGHT, _deduct 1 from this
+sum_.
+
+
+  EX.—25 × 70 = 1750.
+
+  The product is found with the slide projecting to the _left_, so the
+  number of digits in the product = 2 + 2 = 4.
+
+  EX.—3·6 × 25 = 90.
+
+  The slide projects to the _right_, and the number of digits in the
+  product is therefore 1 + 2 − 1 = 2.
+
+  EX.—0·025 × 0·7 = 0·0175.
+
+  The product is obtained with the slide projecting to the _left_, and
+  the number of digits is therefore −1 + 0 = −1.
+
+  EX.—0·000184 × 0·005 = 0·00000092.
+
+  The sum of the number of digits in the two factors = −3 + (−2) = −5,
+  but as the slide projects to the _right_, the number of digits will be
+  −5 − 1 = −6.
+
+
+From the last two examples it will be seen that when the first
+significant figure of a decimal factor does not immediately follow the
+decimal point, the minus sign is to be prefixed to the number of digits,
+counting as many digits _minus_ as there are 0’s following the decimal
+point. Thus, 0·03 has −1 digit, 0·0035 has −2 digits, and so on. Some
+little care is necessary to ensure these minus values being correctly
+taken into account in determining the number of digits in the answer.
+For this reason many prefer to treat decimal factors as whole numbers,
+and to locate the decimal point according to the usual rules for the
+multiplication of decimals. Thus, in the last example we take 184 × 5 =
+920, but as by the usual rule the product must contain 6 + 3 = 9 decimal
+places, we prefix six cyphers, obtaining 0·00000092. When both factors
+consist of integers as well as decimals, the number of digits in the
+product, and therefore the position of the decimal point, will be
+determined by the usual rule for whole numbers.
+
+Another method of determining the number of digits in a product deserves
+mention, which, not being dependent upon the position of the slide, is
+applicable to all calculating instruments.
+
+GENERAL RULE FOR NUMBER OF DIGITS IN A PRODUCT.—_When the first
+significant figure in the product is smaller than in_ EITHER _of the
+factors, the number of digits in the product is equal to the_ SUM _of
+the digits in the two factors. When the contrary is the case, the number
+of digits is 1_ LESS _than the sum of the digits in the two factors.
+When the first figures are the same, those following must be compared._
+
+_Estimation of the Figures in a Product._—We have given rules for those
+who prefer to decide the number of figures by this means, but experience
+will show that to make the best use of the instrument, the result, as
+read on the rule, should be regarded merely as the _significant figures
+of the answer_, the position of the decimal point, if not obvious, being
+decided by a very rough mental calculation. In very many instances, the
+magnitude of the result will be evident from the conditions of the
+problem—_e.g._, whether the answer should be 0·3 in., 3 in., or 30 in.;
+or 10 tons, 0·1 ton, 100 tons, etc. In those cases where the magnitude
+of the answer cannot be estimated, and the factors contain many figures,
+or have a number of 0’s following the decimal point, the use of notation
+by powers of 10 (page 8) is of considerable assistance; but more usually
+it will be found, that a very rough calculation will settle the point
+with comparatively little trouble. Considerable practice is needed to
+work rapidly and with certainty, when using rules. Moreover, the
+experience thus acquired is confined to slide-rule work. The same time
+spent in practising the “rough approximation” method will enable
+reliable results to be obtained rapidly, with the advantage that the
+method is applicable to calculations generally. However, the choice of
+methods is a matter of personal preference. Both methods will be given,
+but whichever plan is followed, the student is strongly advised to
+cultivate the habit of forming an idea of the magnitude of the result.
+
+
+    EX.—33·6 × 236 = 7930.
+
+    Setting 1 on C to 33·6 on D, we read under 236 on D and find 793 on
+    D, as the significant figures of the answer. A rough calculation, as
+    30 × 200 = 6000, indicates that the result will consist of 4
+    figures, and is therefore to be read as 7930.
+
+    EX.—17,300 × 3780 = 65,400,000.
+
+    By factorising with powers of 10
+
+              1·73 × 10^4 × 3·78 × 10^3 = 1·73 × 3·78 × 10^7.
+
+    Setting 1 on C to 1·73 on D, we read, under 3·78 on C, the result of
+    the simple multiplication, as 6·54. Multiplying by 10^7 moves the
+    decimal point 7 places to the right, and the answer is 65,400,000.
+
+
+If it is required to find a series of products of which one of the
+factors is _constant_, set 1 on C to the constant factor on D and read
+the several products on D, under the respective variable factors.
+
+If the factors are required which will give a constant _product_ (really
+a case of division), set the cursor to the constant product on D. Then
+obviously, as the slide is moved along, any pair of factors found
+simultaneously under the cursor line on C, and on D under index of C,
+will give the product. A better method of working will be explained when
+we deal with the inversion of the slide.
+
+It is sometimes useful to remember that although we usually set the
+slide to the rule, we can obtain the result equally well by setting the
+rule to the slide. Thus, bringing 1 (or 10) on D to 2 on C, we find on
+C, _over_ any other factor, _n_ on D, the product of 2 × _n_. But note
+that the slide and rule have now changed places, and if we use rules for
+the number of digits in the result, we must now deduct 1 from the sum of
+the digits in the factors, when the _rule projects_ to the _right of the
+slide_.
+
+With the ordinary 10 in. rule it will be found in general that the
+extent to which the C and D scales are subdivided is such as to enable
+not more than three figures in either factor being dealt with. For the
+same reason it is impossible to directly read more than the first three
+figures of any product, although it is often possible—by mentally
+dividing the smallest space involved in the reading—to correctly
+determine the fourth figure of a product. Necessarily this method is
+only reliable when used in the earlier parts of the C and D scales.
+However, the last numeral of a three-figure, and in some cases the last
+of a four-figure, product can be readily ascertained by an inspection of
+the factors.
+
+EX.—19 × 27 = 513. Placing the L.H. index of C to 19 on D, we find
+opposite 27 on C, the product, which lies between 510 and 515. A glance
+at the factors, however, is sufficient to decide that the third figure
+must be 3, since the product of 9 and 7 is 63, and the last figure of
+this product must be the last figure in the answer.
+
+EX.—79 × 91 = 7189.
+
+In this case the division line 91 on C indicates on D that the answer
+lies between 7180 and 7190. As the last figure must be 9, it is at once
+inferred that the last two figures are 89.
+
+When there are more than three figures in either or both of the factors,
+the fourth and following figures to the right must be neglected. It is
+well to note, however, that if the first neglected figure is 5, or
+greater than 5, it will generally be advisable to increase by 1 the
+third figure of the factor employed. Generally it will suffice to make
+this increase in one of the two factors only, but it is obvious that in
+some cases greater accuracy will be obtained by increasing both factors
+in this way.
+
+CONTINUED MULTIPLICATION.—To find the product of more than two factors,
+we make use of the cursor to mark the position of successive products
+(the value of which does not concern us) as the several factors are
+taken into the calculation. Setting the index of C to the 1st factor on
+D, we bring the line of the cursor to the 2nd factor on C, then the
+index of C to the cursor, the cursor to the 3rd factor, index of C to
+cursor, and so on, reading the final product on D under the last factor
+on C. (Note that the 1st factor and the result are read on D; all
+intermediate readings are taken on C.)
+
+If the rule for the number of digits in a product is used, it is
+necessary to note the number of times multiplication is effected with
+the slide projecting to the right. This number, deducted from the sum of
+the digits of the several factors, gives the number of digits in the
+product. Ingenious devices have been adopted to record the number of
+times the slide projects to the right, but some of these are very
+inconvenient. The author’s method is to record each time the slide so
+projects, by a minus mark, thus −. These can be noted down in any
+convenient manner, and the sum of the marks so obtained deducted from
+the sum of the digits in the several factors, gives the number of digits
+in the product as before explained.
+
+EX.—42 × 71 × 1·5 × 0·32 × 121 = 173,200.
+
+The product given, which is that read on the rule, is obtained as
+follows:—Set R.H. index of C to 42 on D, and bring the cursor to 71 on
+C. Next bring the L.H. index of C to the cursor, and the latter to 1·5
+on C. This multiplication is effected with the slide to the right, and a
+memorandum of this fact is kept by making a mark −. Bring the R.H. index
+of C to the cursor and the latter to 0·32 on C. Then set the L.H. index
+of C to the cursor and read the result, 1732, on D under 121 on C, while
+as a slide again projects to the right, a second − memo-mark is
+recorded. There are 2 + 2 + 1 + 0 + 3 = 8 digits in the factors, and as
+there were 2 − marks recorded during the operation, there will be 8 − 2
+= 6 digits in the product, which will therefore read 173,200
+(173,194·56).
+
+For a very rough evaluation of the result, we note that 1·5 × 0·3 is
+about 0·5; hence, as a clue to the number of figures we have
+
+                  40 × 70 × 60 = 3000 × 60 = 180,000.
+
+
+
+
+                               DIVISION.
+
+
+The instructions for multiplication having been given in some detail, a
+full discussion of the inverse process of division will be unnecessary.
+
+RULE FOR DIVISION.—_Place the divisor on C, opposite the dividend on D,
+and read the quotient on D under the index of C._
+
+EX.—225 ÷ 18 = 12·5.
+
+Bringing 18 on C to 225 on D, we find 12·5 under the L.H. index of C.
+
+As in multiplication, the factors are treated as whole numbers, and the
+position of the decimal point afterwards decided according to the
+following rule, which, as will be seen, is the reverse of that for
+multiplication:—
+
+RULE FOR THE NUMBER OF DIGITS IN A QUOTIENT.—_If the quotient is read
+with the slide projecting to the_ LEFT, _subtract the number of digits
+in the divisor from those in the dividend; but if read with the slide to
+the_ RIGHT, ADD _1 to this difference_.[2]
+
+In the above example the quotient is read off with the slide to the
+right, so the number of digits in the answer = 3 − 2 + 1 = 2.
+
+EX.—0·000221 ÷ 0·017 = 0·013.
+
+Here the number of digits in the dividend is −3, and in the divisor −1.
+The difference is −2; but as the result is obtained with the slide to
+the right, this result must be increased by 1, so that the number of
+digits in the quotient is −2 + 1 = −1, giving the answer as 0·013.
+
+If preferred, the result can be obtained in the manner referred to when
+considering the multiplication of decimals. Thus, treating the above as
+whole numbers, we find that the result of dividing 221 by 17 = 13, since
+the difference in the number of digits in the factors, which is 1, is,
+owing to the position of the slide, increased by 1, giving 2 as the
+number of digits in the answer. Then by the rules for the division of
+decimals we know that the number of decimal places in the quotient is
+equal to 6 − 3 = 3, showing that a cypher is to be prefixed to the
+result read on the rule.
+
+As in multiplication, so in division, we have a
+
+GENERAL RULE FOR NUMBER OF DIGITS IN A QUOTIENT.—_When the first
+significant figure in the_ DIVISOR _is greater than that in the_
+DIVIDEND_, the number of digits in the quotient is found by subtracting
+the digits in the divisor from those in the dividend. When the contrary
+is the case, 1_ IS TO BE ADDED _to this difference. When the first
+figures are the same, those following must be compared._
+
+ESTIMATION OF THE FIGURES IN A QUOTIENT.—The method of roughly
+estimating the number of figures in a quotient needs little explanation.
+
+EX.—3·95 ÷ 5340 = 0·00074.
+
+
+  Setting 534 on C to 3·95 on D we read under the (R.H.) index of C, the
+  significant figures on D, which are 74. Then 3·9 ÷ 5 is about 0·8 and
+  0·8 ÷ 1000 gives 0·0008 as a rough estimate.
+
+
+EX.—0·00000285 ÷ 0·000197 = 0·01446.
+
+
+  Regarding this as 2·85 × 10^{−6} ÷ 1·97 × 10^{−4} we divide 2·85 by
+  1·97 and obtain 1·446. Dividing the powers of 10 we have 10^{−6} ÷
+  10^{−4} = 10^{−2}, so the decimal point is to be moved two places to
+  the left and the answer is read as 0·01446.
+
+
+Another method of dividing deserves mention as of special service when
+dividing a number of quantities by a _constant divisor_:—Set the index
+of C to the divisor on D and over any dividend on D, read the quotient
+on C.
+
+For the division of a _constant dividend_ by a variable divisor, set the
+cursor to the dividend on D and bring the divisor on C successively to
+the cursor, reading the corresponding quotients on D under the index of
+C. Another method which avoids moving the slide is explained in the
+section on “Multiplication and Division with the Slide Inverted.”
+
+CONTINUED DIVISION, if we can so call such an expression as
+
+             (3·14)/(785 × 0·00021 × 4·3 × 64·4) = 0·0688,
+
+may be worked by repeating as follows:—Set 7·85 on C to 3·14 on D, bring
+cursor to index of C, 2·1 on C to cursor, cursor to index, 4·3 to
+cursor, cursor to index, 6·44 to cursor, and under index of C read 688
+on D as the significant figures of the answer.
+
+For the number of figures in the result, we deduct the sum of the number
+of digits in the several factors and add 1 for each time the slide
+projects to the right, which in this case occurs once. There are 3 +
+(−3) + 1 + 2 = 3 denominator digits, 1 numerator digit, and 1 is to be
+added to the difference. Therefore there are 1 − 3 + 1 = −1 digits in
+the answer, which is therefore 0·0688. The foregoing method of working
+may confuse the beginner, who is apt to fall into the process of
+continued multiplication. For this reason, until familiarity with
+combined methods has been acquired, the product of the several
+denominators should be first found by the continued multiplication
+process, and the figures in this product determined. Then divide the
+numerator by this product to obtain the result.
+
+As the denominator product will be read on D, we may avoid resetting the
+slide by bringing the numerator on C to this product and reading the
+result on C _over_ the index of D. The slide and rule have here changed
+places; hence if rules are followed for the number of figures in the
+result, 1 must be added to the difference of digits, when the _rule
+projects_ to the _right of the slide_.
+
+The author’s method of recording the number of times division is
+performed with the slide to the right is by vertical memorandum marks,
+thus |. The full significance of these memo-marks will appear in the
+following section.
+
+For a rough calculation to fix the decimal point, in this example we
+move the decimal points in the factors, obtaining
+
+               (3)/(0·8 × 2 × 4 × 6) = (3)/(40) = 0·075.
+
+
+
+
+      THE USE OF THE UPPER SCALES FOR MULTIPLICATION AND DIVISION.
+
+
+Many prefer to use the upper scales A and B, in preference to C and D.
+The disadvantage is that as the scales are only one-half the length of C
+or D, the graduation does not permit of the same degree of accuracy
+being obtained as when working with the lower scales. But the result can
+always be read directly from the rule without ever having to change the
+position of the slide after it has been initially set. Hence, it
+obviates the uncertainty as to the direction in which the slide is to be
+moved in making a setting.
+
+When the A and B scales are employed, it is understood that the
+left-hand pair of scales are to be used in the same manner as C and D,
+and so far the rules relating to the latter are entirely applicable. But
+in this case the slide is always moved to the right, so that in
+multiplication the product is found either upon the left or right scales
+of A. If it is found on the left A scale, the rule for the number of
+digits in the product is found as for the C and D scales, and is equal
+to the _sum of the digits in the two factors, minus 1_; but if found on
+the right-hand A scale, the number of digits in the product is equal to
+the sum of the digits in the two factors.
+
+In division, similar modifications are necessary. If when moving the
+slide to the right the division can be completely effected by using the
+L.H. scale of A, the quotient (read on A above the L.H. of index B) has
+a number of digits equal to the number in the dividend, less the number
+in the divisor, _plus 1_. But if the division necessitates the use of
+both the A scales, the number of digits in the quotient equals the
+number in the dividend, less the number in the divisor.
+
+
+
+
+                              RECIPROCALS.
+
+
+A special case of division to be considered is the determination of the
+_reciprocal_ of a number _n_, or (1)/(_n_). Following the ordinary rule
+for division, it is evident that setting _n_ on C to 1 on D, gives
+(1)/(_n_) on D under 1 on C. It is more important to observe that by
+inverting the operation—setting 1 (or 10) on C to _n_ on D—we can read
+(1)/(_n_) on C over 1 (or 10) on D. Hence whenever a result is read on D
+under an index of C, we can also read its reciprocal on C over whichever
+index of D is available.
+
+_The Number of Digits in a Reciprocal_ is obvious when _n_ = 10, 100, or
+any power (_p_) of 10. Thus (1)/(10) = 0·1; (1)/(100) = 0·01;
+(1)/(10^{_p_}) = 1 preceded by _p_ − 1 cyphers. For all other cases we
+have the rule:—_Subtract from 1 the number of digits in the number._
+
+EX.—(1)/(339) = 0·00295.
+
+There are 3 digits in the number; hence, there are 1 − 3 = −2 digits in
+the answer.
+
+EX.—(1)/(0·0000156) = 64,100.
+
+There are −4 digits in the number; hence, there are 1 − (−4) = 5 digits
+in the result.
+
+
+
+
+                 CONTINUED MULTIPLICATION AND DIVISION.
+
+
+By combining the rules for multiplication and division, we can readily
+evaluate expressions of the form (_a_)/(_b_) × (_c_)/(_d_) × (_e_)/(_f_)
+× (_g_)/(_h_) = _x_. The simplest case, (_a_ × _c_)/(_b_) can be solved
+by one setting of the slide.[3] Take as an example, (14·45 × 60)/(8·5) =
+102. Setting 8·5 on C to 14·45 on D, we can, if desired, read 1·7 on D
+under 1 on C, as the quotient. However, we are not concerned with this,
+but require its multiplication by 60, and the slide being already set
+for this operation, we at once read under 60 on C the result, 102, on D.
+The figures in the answer are obvious.
+
+When there are more factors to take into account, we place the cursor
+over 102 on D, bring the next divisor on C to the cursor, move the
+cursor to the next multiplier on C, bring the next divisor on C to the
+cursor, and so on, until all the factors have been dealt with. Note that
+only the first factor and the result are read on D; also _that the
+cursor is moved for multiplying and the slide for dividing_.
+
+_Number of Digits in Result in Combined Multiplication and
+Division._—For those who use rules the author’s method of determining
+the decimal point in combined multiplication and division may be used.
+Each time _multiplication_ is performed with the slide projecting to the
+_right_, make a − mark; each time _division_ is effected with the slide
+to the right, make a | mark; _but allow the_ | _marks to cancel the_ −
+_marks as far as they will_. Subtract the sum of the digits in the
+denominator from the sum of digits in the numerator, and to this
+difference _add_ any uncancelled memo-marks, if of | character, or
+_subtract_ them if of − character.
+
+EX.—(43·5 × 29·4 × 51 × 32)/(27 × 3·83 × 10·5 × 1·31) = 1468.
+
+[Sidenote: ⵜ
+           ⵜ
+           ⵏ
+           ⵏ]
+
+Set 27 on C to 43·5 on D, and as with this _division_ the slide is to
+the right, make the first ⵏ mark. Bring cursor to 29·4 on C, and as in
+this _multiplication_ the slide is to the right, make the first − mark,
+cancelling as shown. Setting 3·83 on C to the cursor, requires the
+second ⵏ mark, which, however, is cancelled in turn by the
+multiplication by 51. The division by 10·5 requires the third ⵏ mark,
+and after multiplying by 32 (requiring no mark) the final division by
+1·31 requires the fourth ⵏ mark. Then, as there are 8 numerator digits,
+6 denominator, and 2 uncancelled memo-marks (which, being 1, are
+additive) we have
+
+              Number of digits in result = 8 − 6 + 2 = 4.
+
+Had the uncancelled marks been − in character, the number of digits
+would have been 8 − 6 − 2 = 0.
+
+For quantities less than 0·1 the digit place numbers will be _negative_.
+The troublesome addition of these may be avoided by transferring them to
+the opposite side and treating them as positive.
+
+                                   _2_        _4_
+                         0·00356 × 27·1  × 0·08375
+                Thus:—   ───────────────────────── = 288
+                         0·1426  × 9·85  × 0·00002
+                            _2_     _1_       _1_
+
+The first numerator, 0·00356, has −2 digits. Note this by placing 2
+_below the lower line_ as shown. 27·1 has 2 digits; place 2 over it.
+0·08375 has −1 digit; hence place 1 _below the lower line_. The first
+denominator has no digits; the second, 9·85, has 1 digit; hence place 1
+under it. 0·00002 has −4 digits; place 4 _above the upper line_. The sum
+of the top series is 2 + 4 = 6; of the bottom series 2 + 1 + 1 = 4.
+Subtracting the bottom from the top, we have 6 − 4 = 2 digits, to which
+1 has to be added for an uncancelled memo-mark, and the result is read
+as 288.
+
+Moving the decimal point often facilitates matters. Thus, (32·4 × 0·98 ×
+432 × 0·0217)/(4·71 × 0·175 × 0·00000621 × 412000) is much more
+conveniently dealt with when re-arranged as (32·4 × 9·8 × 432 ×
+2·17)/(4·71 × 17·5 × 6·21 × 4·12) = 141.
+
+To determine the number of figures in the result by rough cancelling and
+mental calculation, we note that 4·71 enters 432 about 100 times; 9·8
+enters 17·5 about 2; 6·21 into 32·4 about 5; and 2·17 into 4·12 about 2.
+This gives (500)/(4) = 125, showing that the result contains 3 digits.
+From the slide rule we read 141, which is therefore the result sought.
+
+The occasional traversing of the slide through the rule, to interchange
+the indices—a contingency which the use of the C and D scales always
+involves—may often be avoided by a very simple expedient. Such an
+example as (6·19 × 31·2 × 422)/(1120 × 8·86 × 2.09) = 3·93 is sometimes
+cited as a particularly difficult case. Working through the expression
+as given, two traversings of the slide are necessary; but by taking the
+factors in the slightly different order, (6·19 × 31·2 × 422)/(8·86 ×
+2·09 × 1120), _so that the significant figures of each pair are more
+nearly alike_, we not only avoid any traversing the slide, but we also
+reduce the extent to which the slide is moved to effect the several
+divisions.
+
+Such cases as (_a_ × _b_)/(_c_ × _d_ × _e_ × _f_ × _g_) or (_a_ × _b_ ×
+_c_ × _d_ × _e_)/(_f_ × _g_) really resolve themselves into (_a_ × _b_ ×
+1 × 1 × 1)/(_c_ × _d_ × _e_ × _f_ × _g_) and (_a_ × _b_ × _c_ × _d_ ×
+_e_)/(_f_ × _g_ × 1 × 1 × 1), but, of course, if rules are used to
+locate the decimal point, the 1’s so (mentally) introduced are not to be
+counted as additional figures in the factors.
+
+
+
+
+          MULTIPLICATION AND DIVISION WITH THE SLIDE INVERTED.
+
+
+If the slide be inverted in the rule but with the same face uppermost,
+so that the Ɔ scale lies adjacent to the A scale, and the right and left
+indices of the slide and rule are placed in coincidence, we find the
+product of any number on D by the coincident number on Ɔ (readily
+referred to each other by the cursor) is always 10. Hence, by reading
+the numbers on Ɔ as decimals, we have over any unit number on D, its
+_reciprocal_ on Ɔ. Thus 2 on D is found opposite 0·5 on Ɔ; 3 on D
+opposite to 0·333; while opposite 8 on Ɔ is 0·125 on D, etc. The reason
+of this is that the sum of the lengths of the slide and rule
+corresponding to the factors, is always equal to the length
+corresponding to the product—in this case, 10.
+
+It will be seen that if we attempt to apply the ordinary rule for
+multiplication, with the slide inverted, we shall actually be
+multiplying the one factor taken on D by the _reciprocal_ of the other
+taken on Ɔ. But multiplying by the _reciprocal of a number_ is
+equivalent to _dividing_ by that number, and _dividing_ a factor by the
+_reciprocal_ of a number is equivalent to _multiplying_ by that number.
+It follows that with the slide inverted the operations of multiplication
+and division are reversed, as are also the rules for the number of
+digits in the product and the position of the decimal point. Hence, in
+multiplying with the slide inverted, we place (by the aid of the cursor)
+one factor on Ɔ opposite the other factor on D, and read the result on D
+under either index of Ɔ. It follows that with the slide thus set, any
+pair of coinciding factors on Ɔ and D will give the same constant
+product found on D under the index of Ɔ. One useful application of this
+fact is found in selecting the scantlings of rectangular sections of
+given areas or in deciding upon the dimensions of rectangular sheets,
+plates, cisterns, etc. Thus by placing the index of Ɔ to 72 on D, it is
+readily seen that a plate having an area of 72 sq. ft. may have sides 8
+by 9 ft., 6 by 12, 5 by 14·4, 4 by 18, 3 by 24, 2 by 36, with
+innumerable intermediate values. Many other useful applications of a
+similar character will suggest themselves.
+
+
+
+
+                              PROPORTION.
+
+
+With the slide in the ordinary position and with the indices of the C
+and D scales in exact agreement, the _ratio_ of the corresponding
+divisions of these scales is 1. If the slide is moved so that 1 on C
+agrees with 2 on D, we know that under any number _n_ on C is _n_ × 2 on
+D, so that if we read numerators on C and denominators on D we have
+
+                 C      1     1·5     2      3      4
+               ─────────────────────────────────────────
+                 D1     2      3      4      6      8.
+
+In other words, the numbers on D bear to the coinciding numbers on C a
+ratio of 2 to 1. Obviously the same condition will obtain no matter in
+what position the slide may be placed. The rule for proportion, which is
+apparent from the foregoing, may be expressed as follows:—
+
+RULE FOR PROPORTION.—_Set the first term of a proportion on the C scale
+to the second term on the D scale, and opposite the third term on the C
+scale read the fourth term on the D scale._
+
+
+  EX.—Find the 4th term in the proportion of 20 ∶ 27 ∷ 70 ∶ _x_. Set 20
+  on C to 27 on D, and opposite 70 on C read 94·5 on D. Thus
+
+                             C    20    70
+                           ─────────────────
+                             D    27   94·5.
+
+
+It will be evident that this is merely a case of combined multiplication
+and division of the form, (20 × 70)/(27) = 94·5. Hence, given any three
+terms of a proportion, we set the 1st to the 2nd, or the 3rd to the 4th,
+as the case may be, and opposite the other given term read the term
+required.[4]
+
+Thus, in reducing vulgar fractions to decimals, the decimal equivalent
+of (3)/(16) is determined by placing 3 on C to 16 on D, when over the
+index or 1 of D we read 0·1875 on C. In this case the terms are
+3 ∶ 16 ∷ _x_ ∶ 1. For the inverse operation—to find a vulgar fraction
+equivalent to a given decimal—the given decimal fraction on C is set to
+the index of D, and then opposite any denominator on D is the
+corresponding numerator of the fraction on C.
+
+If the index of C be placed to agree with 3·1416 on D, it will be clear
+from what has been said that this ratio exists throughout between the
+numbers of the two scales. Therefore, against any _diameter_ of a circle
+on C will be found the corresponding _circumference_ on D. In the same
+way, by setting 1 on C to the appropriate conversion factor on D, we can
+convert a series of values in one denomination to their equivalents in
+another denomination. In this connection the following table of
+conversion factors will be found of service. If the A and B scales are
+used instead of the C and D scales, a complete set of conversions will
+be at once obtained. In this case, however, the left-hand A and B scales
+should be used for the initial setting, any values read on the
+right-hand A or B scales being read as of tenfold value. With the C and
+D scales a portion of the one scale will project beyond the other. To
+read this portion of the scale, the cursor or runner is brought to
+whichever index of the C scale falls within the rule, and the slide
+moved until the other index of the C scale coincides with the cursor,
+when the remainder of the equivalent values can then be read off. It
+must be remembered that if the slide is moved in the direction of
+notation (to the _right_), the values read thereon have a tenfold
+_greater_ value; if the slide is moved to the _left_, the readings
+thereon are _decreased_ in a tenfold degree. Although preferred by many,
+in the form given, the case is obviously one of multiplication, and is
+so treated in the Data Slips at the end of the book.
+
+                     TABLE OF CONVERSION FACTORS.
+    ───────────────────────────────────────────────────────────────
+                       GEOMETRICAL EQUIVALENTS.
+    ──────────────────────────┬──────────────────────────┬─────────
+             SCALE C.         │         SCALE D.         │If C = 1,
+                              │                          │   D =
+    ──────────────────────────┼──────────────────────────┼─────────
+    Diameter of circle        │Circumference of circle   │3·1416
+    „         „               │Side of inscribed square  │0·707
+    „         „               │„    equal square         │0·886
+    „         „               │„      „   equilateral    │
+                              │  triangle                │1·346
+    Circum. of circle         │„    inscribed square     │0·225
+    „         „               │„    equal square         │0·282
+    Side of square            │Diagonal of square        │1·414
+    Square inch               │Circular inch             │1·273
+    Area of circle            │Area of inscribed square  │0·636
+    ──────────────────────────┴──────────────────────────┴─────────
+                          MEASURES OF LENGTH.
+    ──────────────────────────┬──────────────────────────┬─────────
+    Inches                    │Millimetres               │25·40
+    „                         │Centimetres               │2·54
+    8ths of an inch           │Millimetres               │3·175
+    16ths „      „            │„                         │1·587
+    32nds „      „            │„                         │0·794
+    64ths „      „            │„                         │0·397
+    Feet                      │Metres                    │0·3048
+    Yards                     │„                         │0·9144
+    Chains                    │„                         │20·116
+    Miles                     │Kilometres                │1·609
+    ──────────────────────────┴──────────────────────────┴─────────
+                           MEASURES OF AREA.
+    ──────────────────────────┬──────────────────────────┬─────────
+    Square inches             │Square centimetres        │6·46
+    Circular  „               │„        „                │5·067
+    Square feet               │„    metres               │0·0929
+    „   yards                 │„      „                  │0·836
+    „   miles                 │„    kilometres           │2·59
+    „     „                   │Hectares                  │259·00
+    Acres                     │„                         │0·4046
+    ──────────────────────────┴──────────────────────────┴─────────
+                         MEASURES OF CAPACITY.
+    ──────────────────────────┬──────────────────────────┬─────────
+    Cubic inches              │Cubic centimetres         │16·38
+    „     „                   │Imperial gallons          │0·00360
+    „     „                   │U.S. gallons              │0·00432
+    „     „                   │Litres                    │0·01638
+    Cubic feet                │Cubic metres              │0·0283
+    „    „                    │Imperial gallons          │6·23
+    „    „                    │U.S. gallons              │7·48
+    „    „                    │Litres                    │28·37
+    „   yards                 │Cubic metres              │0·764
+    Imperial gallons          │Litres                    │4·54
+    „        „                │U.S. gallons              │1·200
+    Bushels                   │Cubic metres              │0·0363
+    „                         │„   feet                  │1·283
+    ──────────────────────────┴──────────────────────────┴─────────
+                          MEASURES OF WEIGHT.
+    ──────────────────────────┬──────────────────────────┬─────────
+    Grains                    │Grammes                   │0·0648
+    Ounces (Troy)             │„                         │31·103
+    „    (Avoird.)            │„                         │28·35
+    „       „                 │Kilogrammes               │0·02835
+    Pounds (Troy)             │„                         │0·3732
+    „    (Avoird.)            │„                         │0·4536
+    Hundredweights            │„                         │50·802
+    Tons                      │„                         │1016·4
+    „                         │Metric tonnes             │1·016
+    ──────────────────────────┴──────────────────────────┴─────────
+                     COMPOUND FACTORS—VELOCITIES.
+    ──────────────────────────┬──────────────────────────┬─────────
+    Feet per second           │Metres per second         │0·3048
+    „       „                 │„    minute               │18·288
+    „       „                 │Miles per hour            │0.682
+    „      minute             │Meters per second         │0·00508
+    „       „                 │„    minute               │0·3048
+    „       „                 │Miles per hour            │0·01136
+    Yards per „               │„      „                  │0·0341
+    Miles per hour            │Metres per minute         │26·82
+    Knots                     │„      „                  │30·88
+    „                         │Miles per hour            │1·151
+    ──────────────────────────┴──────────────────────────┴─────────
+                      COMPOUND FACTORS—PRESSURES.
+    ──────────────────────────┬──────────────────────────┬─────────
+    Pounds per sq. inch       │Grammes per sq. mm.       │0·7031
+    „         „               │Kilos. per sq. centimetre │0·0703
+    „         „               │Atmospheres               │0·068
+    „         „               │Head of water in inches   │27·71
+    „         „               │„        „   feet         │2·309
+    „         „               │„        „   metres       │0·757
+    „         „               │Inches of Mercury         │2·04
+    Inches of water           │Pounds per square inch    │0·0361
+    „       „                 │Inches of mercury         │0·0714
+    „       „                 │Pounds per square foot    │5·20
+    Inches of mercury         │Atmospheres               │0·0333
+    Atmospheres               │Metres of water           │10·34
+    „                         │Kilos. per sq. cm.        │1·033
+    Feet of water             │Pounds per square foot    │62·35
+    „     „                   │Atmospheres               │0·0294
+    „     „                   │Inches of mercury         │0·883
+    Pounds per sq. foot       │„       „                 │0·01417
+    „        „                │Kilos. per square metre   │4·883
+    „        „                │Atmospheres               │0·000472
+    Pounds per sq. yard       │Kilos. per square metre   │0·5425
+    Tons per sq. inch         │„       square mm.        │1·575
+    „    sq. foot             │Tonnes per square metre   │10·936
+    ──────────────────────────┴──────────────────────────┴─────────
+              COMPOUND FACTORS—WEIGHTS, CAPACITIES, ETC.
+    ──────────────────────────┬──────────────────────────┬─────────
+    Pounds per lineal ft.     │Kilos. per lineal metre   │1·488
+    „ per lineal yd.          │„        „      „         │0·496
+    „ per lineal mile         │Kilos. per kilometre      │0·2818
+    Tons     „       „        │Tonnes         „          │0·6313
+    Feet     „       „        │Metres         „          │1·894
+    Pounds per cubic in.      │Grammes per cubic cm.     │27·68
+    „  per cubic ft.          │Kilos. per cubic metre    │16·02
+    „  per cubic yd.          │„        „     „          │0·593
+    Tons per cubic yard       │Tonnes      „     „       │1·329
+    Cubic yds. per pound      │Cubic metres per kilo.    │1·685
+    „      per ton            │„    „    per tonne       │0·7525
+    Cubic inch of water       │Weight in pounds          │0·03608
+    Cubic feet of water       │„         „               │62·35
+    „          „              │„      kilos              │28·23
+    „          „              │Imperial gallons          │6·235
+    „          „              │U.S. gallons              │7·48
+    Litre of water            │Cubic inches              │61·025
+    Gallons of water          │Weight in kilos           │4·54
+    Pounds of fresh water     │Pounds of sea water       │1·026
+    Grains per gallon         │Grammes per litre         │0·01426
+    Pounds per gallon         │Kilos. per litre          │0·0998
+    „  per U.S. gal.          │„       „                 │0·115
+    ──────────────────────────┴──────────────────────────┴─────────
+                  COMPOUND FACTORS—POWER UNITS, ETC.
+    ──────────────────────────┬──────────────────────────┬─────────
+    British Ther. Units.      │Kilogrammetres.           │108
+    „          „              │Joules                    │1058
+    „          „              │Calories (Fr. Ther. units)│0·252
+    „    „   per sq. ft.      │„     per square metre    │2·713
+    „    „   per pound        │„     per kilogramme      │0·555
+    Pounds per sq. ft.        │Dynes, per sq. cm.        │479
+    Foot-pounds               │Kilogrammetres            │0·1382
+    „    „                    │Joules                    │1·356
+    „    „                    │Thermal Units             │0·00129
+    „    „                    │Calorie                   │0·000324
+    Foot-tons                 │Tonne-metres              │0·333
+    Horse-power               │Force decheval  (Fr.H.P.) │1·014
+    „    „                    │Kilowatts                 │0·746
+    Pounds per H.P.           │Kilos. per cheval         │0·447
+    Square feet per H. P.     │Square metres per cheval  │0·0196
+    Cubic     „      „        │Cubic    „          „     │0·0279
+    Watts                     │Ther. Units per hour      │3·44
+    „                         │Foot-pounds per second    │0·73
+    „                         │„      per minute         │44·24
+    Watt-hours                │Kilogrammetres            │367
+    „    „                    │Joules                    │3600
+    Kilogrammetres            │„                         │9·806
+    ──────────────────────────┴──────────────────────────┴─────────
+
+_Inverse Proportion._—If “more” requires “less,” or “less” requires
+“more,” the case is one of _inverse_ proportion, and although it will be
+seen that this form of proportion is quite readily dealt with by the
+preceding method, the working is simplified to some extent by inverting
+the slide so that the C scale is adjacent to the A scale. By the aid of
+the cursor, the values on the inverted C (or Ɔ) scale, and on the D
+scale, can be then read off. These will now constitute a series of
+inverse ratios. For example, in the proportion
+
+                              ───────────
+                               Ɔ   8   4
+                              ───────────
+                               D  1·5  3
+
+the 4 on the Ɔ scale is brought opposite 3 on D, when under 8 on Ɔ is
+found 1·5 on D.[5]
+
+
+        GENERAL HINTS ON THE ELEMENTARY USES OF THE SLIDE RULE.
+
+Before the more complex operations of involution, evolution, etc., are
+considered, a few general hints on the use of the slide rule for
+elementary operations may be of service, especially as these will serve
+to enforce some of the more important points brought out in the
+preceding sections.
+
+Always use the slide rule in as _direct_ a light as possible.
+
+Study the manner in which the scales are divided. Follow the graduations
+of the C and D scales from 1 to 10, noting the values given by each
+successive graduation and how these values change as we follow along to
+the right. Do the same with the two halves of the A and B scales and
+note the difference in the value of the subdivisions, due to the shorter
+scale-lengths.
+
+Practise reading values by setting 1 on C to some value on D and reading
+under 2, 3, 4, etc., on C, checking the readings by mental arithmetic.
+To the same end, find squares, square roots, etc., comparing the results
+with the actual values as given in tables. Practise setting both slide
+and cursor to values taken at random. Aim at accuracy; speed will come
+with practice.
+
+When in doubt as to any method of working, verify by making a simple
+calculation of the same form.
+
+Follow the orthodox methods of working until entirely confident in the
+use of the instrument, and even then do not readily make a change. If
+any altered procedure is adopted, first work a simple case and guard
+carefully against unconsciously lapsing into the usual method during the
+operation.
+
+Unless the calculation is of a straightforward character, time taken in
+considering how best to attack it (rearranging the expression if
+desirable) is generally time well spent.
+
+In setting two values together, set the cursor to one of them on the
+rule, and bring the other, on the slide, to the cursor line.
+
+In multiplying factors, as 57 × 0·1256, take the fractional value first.
+It is easier to set 1 on C to 1256 on D and read under 57 on C, than to
+reverse the procedure. When both values are eye-estimated, set the
+cursor to the second factor on C and read the result on D, under the
+cursor line.
+
+In continuous operations avoid moving the slide further than necessary,
+by taking the factors in that order which will keep the scale readings
+as close together as possible.
+
+
+                       SQUARES AND SQUARE ROOTS.
+
+We have seen that the relation which the upper scales bear to the lower
+set is such that over any number on D is its square on A, and,
+conversely, under any number on A is its square root on D, the same
+remarks applying to the C and B scales on the slide. Taking the values
+engraved on the rule, we have on D, numbers lying between 1 and 10, and
+on A the corresponding squares extending from 1 to 100. Hence the
+squares of numbers between 1 and 10, or the roots of numbers between 1
+and 100, can be read off on the rule by the aid of the cursor. All other
+cases are brought within these ranges of values by factorising with
+powers of 10, as before explained.
+
+The more practical rule is the following:—
+
+_To Find the Square of a Number_, set the cursor to the number on D and
+read the required square on A under the cursor. The rule for
+
+_The Number of Digits in a Square_ is easily deducible from the rule for
+multiplication. If the square is read on the _left_ scale of A, it will
+contain _twice_ the number of digits in the original number _less_ 1; if
+it is read on the _right_ scale of A, it will contain _twice_ the number
+of digits in the original number.
+
+
+  EX.—Find the square of 114.
+
+  Placing the cursor to 114 on D, it is seen that the coinciding number
+  on A is 13. As the result is read off on the _left_ scale of A, the
+  number of digits will be (3 × 2) − 1 = 5, and the answer is read as
+  13,000. The true result is 12,996.
+
+  EX.—Find the square of 0·0093.
+
+  The cursor being placed to 93 on D, the number on A is found to be
+  865. The result is read on the _right_ scale of A, so the number of
+  digits = −2 × 2 = −4, and the answer is read as 0·0000865
+  [0·00008649].
+
+
+_Square Root._—The foregoing rules suggest the method of procedure in
+the inverse operation of extracting the square root of a given number,
+which will be found on the D scale opposite the number on the A scale.
+It is necessary to observe, however, that if the number consists of an
+_odd_ number of digits, it is to be taken on the _left-hand_ portion of
+the A scale, and the number of digits in the root = (N + 1)/(2), N being
+the number of digits in the original number. When there is an even
+number of digits in the number, it is to be taken on the _right-hand_
+portion of the A scale, and the root contains _one-half_ the number of
+digits in the original number.
+
+
+  EX.—Find the square root of 36,500.
+
+  As there is an _odd_ number of digits, placing the cursor to 365 on
+  the L.H. A scale gives 191 on D. By the rule there are (N + 1)/(2) =
+  (5 + 1)/(2) = 3 digits in the required root, which is therefore read
+  as 191 [191·05].
+
+  EX.—Find √(0·0098.)
+
+  Placing the cursor to 98 on the right-hand scale of A (since −2 is an
+  _even_ number of digits), it is seen that the coinciding number on D
+  is 99. As the number of digits in the number is −2, the number of
+  digits in the root will be (−2)/(2) = −1. It will therefore be read as
+  0·099 [0·09899+].
+
+  EX.—Find √(0·098).
+
+  The number of digits is −1, so under 98 on the left scale of A, we
+  find 313 on D. By the rule the number in the root will be (−1 +1)/(2)
+  = 0, and the root is therefore read as 0·313 [0·313049+].
+
+  EX.—Find √(0·149.)
+
+  As the number of digits (0) is _even_, the cursor is set to 149 on the
+  right-hand scale of A, giving 386 on D. By the rule, the number of
+  digits in the root will be (0)/(2) = 0, and the root will be read as
+  0·386 [0·38605+].
+
+
+Another method of extracting the square root, by which more accurate
+readings may generally be obtained, is by using the C and D scales only,
+with the slide inverted. If there is an _odd_ number of digits in the
+number, the _right_ index, or if an even number of digits the _left_
+index, of the inverted scale Ɔ is placed so as to coincide with the
+number on D of which the root is sought. Then with the cursor, the
+number is found on D which coincides with the same number on Ɔ, which
+number is the root sought.
+
+
+  EX.—Find √(22·2.)
+
+  Placing the left index of Ɔ to 222 on D, the two equal coinciding
+  numbers on Ɔ and D are found to be 4·71.
+
+
+Note that under the cursor line we have the original number, 22·2, on A,
+and from this the number of digits in the root is determined as before.
+
+The plan of finding the square of a number by ordinary multiplication is
+often very convenient. The inverse process of finding a square root by
+trial division is not to be recommended.
+
+To obtain a close value of a root or to verify one found in the usual
+way, the author has, on occasion, adopted the following plan:—Set 1 (or
+10) on B to the number on the A scale (L.H. or R.H. as the case may
+require), and bring the cursor to the number on D. If the root found is
+correct, the readings on C under the cursor and on D under the index of
+C, will be in exact agreement.
+
+If 1 on B is placed to a number _n_ on the L.H. A scale, the student
+will note that while root _n_ is read on D under 1 on C, the root of 10
+_n_ is read on D under 10 on B. Hence, if preferred, the number can be
+taken always on the first scale of A and the root read under 1 or 10 on
+B, according to whether there is an odd or even number of digits in the
+number. Obviously the second root is the first multiplied by √(10).
+
+
+                         CUBES AND CUBE ROOTS.
+
+In raising a number to the third power, a combination of the preceding
+method and ordinary multiplication is employed.
+
+TO FIND THE CUBE OF A NUMBER.—_Set the_ L.H. _or_ R.H. _index of C to
+the number on D, and opposite the number_ ON THE LEFT-HAND _scale of B
+read the cube on the_ L.H. _or_ R.H. _scale of A_.
+
+By this rule four scales are brought into requisition. Of these, the D
+scale and the L.H. B scale are _always_ employed, and are to be read as
+of equal denomination. The values assigned to the L.H. and R.H. scales
+of A will be apparent from the following considerations.
+
+Commencing with the indices of C and D coinciding, and moving the slide
+to the right, it will be seen that, working in accordance with the above
+rule, the cubes of numbers from 1 to 2·154 (= ∛(10)) will be found on
+the first or L.H. scale of A. Moving the slide still farther to the
+right, we obtain _on the_ R.H. _A scale_ cubes of numbers from 2·154 to
+4·641 (or ∛(10) to ∛(100)). Had we a _third_ repetition of the L.H. A
+scale, the L.H. index of C could be still further traversed to the
+right, and the cubes of numbers from 4·641 to 10 read off on this
+prolongation of A. But the same end can be attained by making use of the
+R.H. index of C, when, traversing the slide to the right as before, the
+cubes of numbers from 4·641 to 10 on D can be read off _on the_ L.H. _A
+scale_ over the corresponding numbers on the L.H. B scale. Hence, using
+the L.H. index of C, the readings on the L.H. A scale may be regarded
+comparatively as units, those on the R.H. A scale as tens; while for the
+hundreds we again make use of the L.H. A scale in conjunction with the
+_right-hand_ index of C.
+
+By keeping these points in view, the number of digits in the cube (N) of
+a given number (_n_) are readily deduced. Thus, if the units scale is
+used, N = 3_n_ − 2; if the tens scale, N = 3_n_ − 1; while if the
+hundreds scale be used, N = 3_n_. Placed in the form of rules:—
+
+N = 3_n_ − 2 when the product is read on the L.H. scale of A with the
+slide to the _right_ (units scale).
+
+N = 3_n_ − 1 when the product is read on the R.H. scale of A; slide to
+the _right_ (tens scale).
+
+N = 3_n_ when the product is read on the L.H. scale of A with the slide
+to the _left_ (hundreds scale).
+
+With decimals the same rule applies, but, as before, the number of
+digits must be read as −1, −2, etc., when one, two, etc., cyphers follow
+immediately after the decimal point.
+
+EX.—Find the value of 1·4^3.
+
+Placing the L.H. index of C to 1·4 on D, the reading on A opposite 1·4
+on the L.H. scale of B is found to be about 2·745 [2·744].
+
+EX.—Find the value of 26·4^3.
+
+Placing the L.H. index of C to 26·4 on D, the reading on A opposite 26·4
+on the L.H. scale of B is found to be about 18,400 [18,399·744].
+
+EX.—Find the value of 7·3^3.
+
+In this case it becomes necessary to use the R.H. index of C, which is
+set to 7·3 on D, when opposite 7·3 on the L.H. scale of B is read 389
+[389·017] on A.
+
+EX.—Find the value of 0·073^3.
+
+From the setting as before it is seen that the number of digits in the
+number must be multiplied by 3. Hence, as there is −1 digit in 0·073,
+there will be −3 in the cube, which is therefore read 0·000389.
+
+The last two examples serve to illustrate the principle of factorising
+with powers of 10. Thus
+
+ 0·073 = 7·3 × 10^{−2}; 0·073^3 = 7·3^3 × (10^{−2})^3 = 389 × 10^{−6} =
+                               0·000389.
+
+_Cube Root_ (_Direct Method_).—One method of extracting the cube root of
+a number is by an inversion of the foregoing operation. Using the same
+scales, _the slide is moved either to the right or left until under the
+given number on A is found a number on the_ L.H. _B scale, identical
+with the number simultaneously found on D under the right or left index
+of C_. This number is the required cube root.
+
+From what has already been said regarding the combined use of these
+scales in cubing, it will be evident that in extracting the cube root of
+a number, it is necessary, in order to decide which scales are to be
+used, to know the number of figures to be dealt with. We therefore (as
+in the arithmetical method of extraction) point off the given number
+into sections of three figures each, commencing at the decimal point,
+and proceeding to the left for numbers greater than unity, and to the
+right for numbers less than unity. Then if the first section of figures
+on the left consists of—
+
+1 figure, the number will evidently require to be taken on what we have
+called the “units” scale—_i.e._, on the L.H. scale of A, using the L.H.
+index of C.
+
+If of 2 figures, the number will be taken on the “tens” scale—_i.e._, on
+the R.H. scale of A, using the L.H. index of C.
+
+If of 3 figures, the number will be taken on the “hundreds”
+scale—_i.e._, on the L.H. scale of A, using the R.H. index of C.
+
+To determine the number of digits in cube roots it is only necessary to
+note that when the number is pointed off into sections as directed,
+there will be one figure in the root for every section into which the
+number is so divided, whether the _first_ section consists of 1, 2, or 3
+digits.
+
+Of numbers wholly decimal, the cube roots will be decimal, and for every
+group of _three_ 0s immediately following the decimal point, _one_ 0
+will follow the decimal point in the root. If necessary, 0s must be
+added so as to make up complete multiples of 3 figures before proceeding
+to extract the root. Thus 0·8 is to be regarded as 0·800, and 0·00008 as
+0·000080 in extracting cube roots.
+
+EX.—Find ∛(14,000.)
+
+Pointing the number off in the manner described, it is seen that there
+are _two_ figures in the first section—viz., 14. Setting the cursor to
+14 on the R.H. scale of A, the slide is moved to the right until it is
+seen that 241 on the L.H. scale of B falls under the cursor, when 241 on
+D is under the L.H. index of C. Pointing 14,000 off into sections we
+have 14 000—that is, _two_ sections. Therefore, there are two digits in
+the root, which in consequence will be read 24·1 [24·1014+].
+
+EX.—Find ∛(0·162.)
+
+As the divisional section consists of _three_ figures, we use the
+“hundreds” scale. Setting the cursor to 0·162 on the L.H. A scale, and
+using the R.H. index of C, we move the slide to the left until under the
+cursor 0·545 is found on the L.H. B scale, while the R.H. index of C
+points to 0·545 on D, which is therefore the cube root of 0·162.
+
+EX.—Find ∛(0·0002.)
+
+To make even multiples of 3 figures requires the addition of 00; we have
+then 200, the cube root of which is found to be about 5·85. Then, since
+the first divisional group consists of 0s, one 0 will follow the decimal
+point, giving ∛(0·0002) = 0·0585 [0·05848].
+
+_Cube Root (Inverted Slide Method)._—Another method of extracting the
+cube root involves the use of the inverted slide. Several methods are
+used, but the following is to be preferred:—_Set the_ L.H. _or_ R.H.
+_index of the slide to the number on A, and the number on ᗺ (i.e., B
+inverted), which coincides with the same number on D, is the required
+root._
+
+Setting the slide as directed, and using first the L.H. index of the
+slide and then the R.H. index, it is always possible to find _three_
+pairs of coincident values. To determine which of the three is the
+required result is best shown by an example.
+
+
+  EX.—Find ∛(5,) ∛(50,) and ∛(500.)
+
+  Setting the R.H. index of the slide to 5 on A, it is seen that 1·71 on
+  D coincides with 1·71 on ᗺ. Then setting the L.H. index to 5 on A,
+  further coincidences are found at 3·68 and at 7·93, the three values
+  thus found being the required roots. Note that the first root was
+  found on that portion of the D scale lying under 1 to 5 on A; the
+  second root on that portion lying under 5 to 50 on A; and the third
+  root on that portion of D lying under 50 to 100 on A. In this
+  connection, therefore, scale A may always be considered to be divided
+  into three sections—viz., 1 to _n_, _n_ to 10_n_, and 10_n_ to 100.
+  For all numbers consisting of 1, 1 + 3, 1 + 6, 1 + 9—_i.e._, of 1, 4,
+  7, 10, or −2, −5, etc., figures—the coincidence under the first
+  section is the one required. If the number has 2, 5, 8, or −1, −4, −7,
+  etc., figures, the coincidence under the second section is correct,
+  while if the number has 3, 6, 9, or 0, −3, etc., figures, the
+  coincidence under the last section is that required. The number of
+  digits in the root is determined by marking off the number into
+  sections, as already explained.
+
+
+_Cube Root (Pickworth’s Method)._—One of the principal objections to the
+two methods described is the difficulty of recollecting which scales are
+to be employed and with which index of the slide they are to be used.
+With the direct method another objection is that the readings to be
+compared are often some distance apart, the maximum distance intervening
+being _two-thirds_ of the length of the rule. To carry the eye from one
+to another is troublesome and time-taking. With the inverted scale
+method the reading of a scale reversed in direction and with the figures
+inverted is also objectionable.
+
+With the author’s method these objections are entirely obviated. The
+_same scales and index are always used_, and are read in their normal
+position. The three roots of _n_, 10_n_ and 100_n_ (_n_ being less than
+10 and not less than 1) are given with one setting and appear in their
+natural sequence, no traversing of the slide being needed. The readings
+to be compared are always close together, the maximum distance between
+them being _one-sixth_ of the length of the rule. The setting is always
+made in the earlier part of the scales where closer readings can be
+obtained, and finally, if desired, the result may be readily verified on
+the lower scales by successive multiplication.
+
+For this method two gauge points are required on C. To conveniently
+locate these, set 53 on C to 246 on D; join 1 on D to 1 on A with a
+straight-edge and with a needle point draw a short fine line on C. Set
+246 on C to 53 on D, and repeat the process at the other end of the
+rule. The gauge points thus obtained (dividing C into three equal parts)
+will be at 2·154 and 4·641, and should be marked ∛(10) and ∛(100)
+respectively.[6]
+
+
+  EX.—Find ∛(2·86,) ∛(28·6) and ∛(286).
+
+  Set cursor to 2·86 on A and drawing the slide to the right find 1·42
+  under 1 on C, when 1·42 on B is under the cursor. Then reading under
+  1, ∛(10) and ∛(100,) we have
+
+  ∛(2·86) = 1·42; ∛(28·6) = 3·06 and ∛(286) = 6·59.
+
+
+It will be seen that factorising with powers of 10, we multiply the
+initial root by ∛(10) and ∛(100). Obviously the three roots will always
+be found on D, in their natural order and at intervals of one-third the
+length of the rule. The number of digits in the roots of numbers which
+do not lie between 1 and 1000, is found as before explained.
+
+In any method of extracting cube roots in which the slide has to be
+adjusted to give equal readings on B and D, the author has found it of
+advantage to adopt the following plan:—The cursor being set to, say, 4·8
+on A, bring a near _main_ division line on B, as 1·7, to the cursor;
+then 1 on C is at 1·68 on D. The difference in the readings is two small
+divisions on D, and moving the slide forward by _one-third the space
+representing this difference_, we obtain 1·687 as the root required.
+With a little practice it is possible to obtain more accurate results by
+this method than by comparing the reading on D with that on the less
+finely-graded B scale.
+
+
+                    MISCELLANEOUS POWERS AND ROOTS.
+
+In addition to squares and cubes, certain other powers and roots may be
+readily obtained with the slide rule.
+
+_Two-thirds Power._—The value of N^⅔ is found on A over ∛̅N on D. The
+number of digits is decided by the rule for squares, working from the
+number of digits in the cube root. It will often be found preferable to
+treat N^⅔ as N ÷ ∛̅N, as in this way the magnitude of the result is much
+more readily appreciated.
+
+_Three-two Power._—N^{³⁄₂} can be obtained by cubing the square root,
+deciding the number of digits in each process. For the reason just
+given, it is preferable to regard N^{³⁄₂} as N × √̅N.
+
+_Fourth Power._—For N^4 set the index of C to N on D and over N on C
+read N^4 on A; or find the square of the square of N, deciding the
+number of digits at each step.
+
+_Fourth Root._—Similarly for ∜̅N, take the square root of the square
+root.
+
+_Four-third Power._—N^{⁴⁄₃} = N^{1·33} (useful in gas-engine diagram
+calculations) is best treated as N × ∛̅N.
+
+Other powers can be found by repeated multiplication. Thus setting 1 on
+B to N on A, we have on A, N^2 over N; N^3 over N^2; N^4 over N^3; N^5
+over N^4, etc. In the same way, setting N on B to N on D, we can read
+such values as N^¾, N^⅞, etc.
+
+
+                    POWERS AND ROOTS BY LOGARITHMS.
+
+For powers or roots other than those of the simple forms already
+discussed, it is necessary to employ the usual logarithmic process. Thus
+to find _a^n_ = _x_, we multiply the logarithm of _a_ by _n_, and find
+the number _x_ corresponding to the logarithm so obtained. Similarly, to
+find _ⁿ√̅a_ = _x_ we divide the logarithm of _a_ by _n_, and find the
+number _x_ corresponding to the resulting logarithm.
+
+_The Scale of Logarithms._—Upon the back of the slide of the Gravêt and
+similar slide rules there will be found three scales. One of
+these—usually the centre one—is divided equally throughout its entire
+length, and figured from right to left. It is sometimes marked L,
+indicating that it is a scale giving logarithms. The whole scale is
+divided primarily into ten equal parts, and each of these subdivided
+into 50 equal parts. In the recess or notch in the right-hand end of the
+rule is a reference mark, to which any of the divisions of this
+evenly-divided scale can be set.
+
+As this decimally-divided scale is equal in length to the logarithmic
+scale D, and is figured in the reverse direction, it results that when
+the slide is drawn to the right so that the L.H. index of C coincides
+with any number on D, the reading on the equally-divided scale will give
+the decimal part of the logarithm of the number taken on D. Thus if the
+L.H. index of C is placed to agree with 2 on D, the reading of the back
+scale, taken at the reference mark, will be found to be 0·301, the
+logarithm of 2. It must be distinctly borne in mind that the number so
+obtained is the _decimal part_ or _mantissa_ of the logarithm of the
+number, and that to this the characteristic must be prefixed in
+accordance with the usual rule—viz., _The integral part, or
+characteristic of a logarithm is equal to the number of digits in the
+number, minus 1. If the number is wholly decimal, the characteristic is
+equal to the number of cyphers following the decimal point, plus 1._ In
+the latter case the characteristic is negative, and is so indicated by
+having the minus sign written _over_ it.
+
+To obtain any given power or root of a number, the operation is as
+follows:—Set the L.H. index of C to the given number on D, and turning
+the rule over, read opposite the mark in the notch at the right-hand end
+of the rule, the decimal part of the logarithm of the number. Add the
+characteristic according to the above rule, and multiply by the exponent
+of the power, or divide by the exponent of the root. Place the _decimal
+part_ of the resultant reading, taken on the scale of equal parts,
+opposite the mark in the aperture of the rule, and read the answer on D
+under the L.H. index of C, pointing off the number of digits in the
+answer in accordance with the number of the characteristic of the
+resultant.
+
+
+  EX.—Evaluate 36^{1·414}.
+
+  Set 1 on C to 36 on D and read the decimal part of log. 36 on the
+  scale of logarithms on the back of the slide. This value is found to
+  be 0·556. As there are two digits in the number, the characteristic
+  will be 1; hence log. 36 = 1·556. Multiply by 1·414, using the C and D
+  scales, and obtain 2·2 as the log. of the result. Set the decimal
+  part, 0·2, on the log. scale to the mark in the notch at the end of
+  the rule and read 1585 on D under 1 on C. Since the log. of the result
+  has a characteristic 2, there will be 3 digits in the result, which is
+  therefore read as 158·5.
+
+
+This example will suffice to show the method of obtaining the nth power
+or the _n_th root of _any_ number.
+
+
+
+
+              OTHER METHODS OF OBTAINING POWERS AND ROOTS.
+
+
+A simple method of obtaining powers and roots, which may serve on
+occasion, is by scaling off proportional lengths on the D scale (or the
+A scale) of the ordinary rule. Thus, to determine the value of
+1·25^{1·67} we take the actual length 1–1·25 on D scale, and increase it
+by any convenient means in the proportion of 1 ∶ 1·67. Then with a pair
+of dividers we set off this new length from 1, and obtain 1·44 as the
+result. One convenient method of obtaining the desired ratio is by a
+pair of proportional compasses. Thus to obtain 1·52^{¹⁷⁄₁₆}, the
+compasses would be set in the ratio of 16 to 17, and the smaller end
+opened out to include 1–1·52 on the D scale; the opening in the large
+end of the compasses will then be such that setting it off from 1 we
+obtain 1·56 on D as the result sought.
+
+[Illustration: FIG. 11.]
+
+The converse procedure for obtaining the _n_th root of a number N will
+obviously resolve itself into obtaining (1)/(_n_)th of the scale length
+1-N, and need not be further considered.
+
+Simple geometrical constructions are also used for obtaining scale
+lengths in the required ratio. A series of parallel lines ruled on
+transparent celluloid or stout tracing paper may be placed in an
+inclined position on the face of the rule and adjusted so as to divide
+the scale as desired. When much work is to be done which requires values
+to be raised to some constant but comparatively low power, _n_, the
+author has found the following device of assistance:—On a piece of thin
+transparent celluloid a line OC is drawn (Fig. 11) and in this a point B
+is taken such that (OC)/(OB) is the desired ratio. It is convenient to
+make OB = 1–10 on the A scale, so that assuming we require a series of
+values of _v_^{1·35}, OB would be 12·5 cm. and OC, 16·875 cm. On these
+lines semi-circles are drawn as shown, both passing through the point O.
+
+Applying this cursor to the upper scales so that the point O is on 1 and
+the semi-circle O M B passes through _v_ on A, the larger semi-circle
+will give on A the value of _v^n_. Thus for _p_ _v^n_ = 39·5 ×
+4·9^{1·35}, set 1 on B to 39·5 on A (Fig. 12) and apply the cursor to
+the working edge of B, so that O agrees with 1 and O M B passes through
+4·9 on B. The larger semi-circle then cuts the edge of the slide on a
+point, giving 337 on A as the result required.
+
+Of course any number of semi-circles may be drawn, giving different
+ratios. If a number of evenly-spaced divisions are used as bases, the
+device affords a simple means of obtaining a succession of small powers
+or roots, while it also finds a use in determining a number of geometric
+means between two values as is required in arranging the speed gears of
+machine tools, etc. The converse operation of finding roots will be
+evident as will also many other uses for which the device is of service.
+
+[Illustration: FIG. 12.]
+
+The lines should be drawn in Indian ink with a very sharp pen and on the
+_under_ side of the celluloid so that the lines lie in close contact
+with the face of the rule.
+
+_The Radial Cursor_, another device for the same purpose, is always used
+in conjunction with the upper scales. As will be seen from Fig. 13, the
+body of the cursor P carries a graduated bar S which can be removed in a
+direction transverse to the rule, and adjusted to any desired position.
+Pivoted to the lower end of S is a radial arm R of transparent celluloid
+on which a centre line is engraved.
+
+A reference to the illustration will show that the principle involved is
+that of similar triangles, the width of the slide being used as one of
+the elements. Thus, to take a simple case, if 2 on S is set to the index
+on P, and 1 on B is brought to N on A, then by swinging the radial arm
+until its centre line agrees with 1 on C, we can read N^2 on A.
+Evidently, since in the two similar triangles A O N^2 and N _t_ N^2 the
+length of A O is twice that of N _t_, it results that A N^2 = 2 A N. In
+general, then, to find the _n_th power of a number, we set the cursor to
+1 or 10 on A, bring _n_ on the cross bar S to the index on the cursor,
+and 1 on B to N on A. Then to 1 on C we set the line on the radial arm,
+and under the latter read N^{_n_} on A. The inverse proceeding for
+finding the _n_th root will be obvious.
+
+[Illustration: FIG. 13.]
+
+An advantage offered by this and analogous methods of obtaining powers
+and roots is that the result is obtained on the ordinary scale of the
+rule, and hence it can be taken directly into any further calculation
+which may be necessary.
+
+
+                          COMBINED OPERATIONS.
+
+Thus far the various operations have been separately considered, and we
+now pass on to a consideration of the methods of working for solving the
+various formulæ met with in technical calculations. We propose to
+explain the methods of dealing with a few of the more generally used
+expressions, as this will suffice to suggest the procedure in dealing
+with other and more intricate calculations. In solving the following
+problems, both the upper and lower scales are used, and the relative
+value of the several scales must be observed throughout. Thus, in
+solving such an expression as √((74·5)/(15·8)) = 6·86, the division is
+first effected by setting 15·8 on B to 745 on A. From the relation of
+the two parts of the upper scales (page 37) we know that such values as
+7·45, 745, etc., will be taken on the _left-hand_ A and B scales, while
+values as 15·8, 1580, etc., will be taken on the _right-hand_ A and B
+scales. Hence, 15·8 on the R.H. B scale is set to 745 on the L.H. A
+scale, and the result read on D under the index of C. Had both values
+been taken on the L.H. A and B scales, or both on the R.H. A and B
+scales, the results would have corresponded to _x_ = √((7·45)/(1·58)) =
+2·17, or to _x_ =√((74·5)/(15·8)) = 2·17, _i.e_., to (6·86)/(√(10)).
+Hence if a wrong choice of scales has been made, we can correct the
+result by multiplying or dividing by √(10) as the case may require. If
+the result is read on D, set to it the centre index (10) of B and read
+the corrected result under the index of C.
+
+To solve _a_ × _b_^2 = _x_. Set the index of C to _b_ on D, and over _a_
+on B read _x_ on A.
+
+To solve (_a_^2)/(_b_) = _x_. Set _b_ on B to _a_ on D by using the
+cursor, and over index of B read _x_ on A.
+
+To solve (_b_)/(_a_^2) = _x_. Set _a_ on C to _b_ on A, and over 1 on B
+read _x_ on A.
+
+To solve (_a_ × _b_^2)/(_c_) = _x_. Set _c_ on B to _b_ on D, and over
+_a_ on B read _x_ on A.
+
+To solve (_a_ × _b_)^2 = _x_. Set 1 on C to _a_ on D, and over _b_ on C
+read _x_ on A.
+
+To solve ((_a_)/(_b_))^2 = _x_. Set _b_ on C to _a_ on D, and over 1 on
+C read _x_ on A.
+
+To solve √(_a_ × _b_) = _x_. Set 1 on B to _a_ on A, and under _b_ on B
+read _x_ on D.
+
+To solve √((_a_)/(_b_)) = _x_. Set _b_ on B to _a_ on A, and under 1 on
+C read _x_ on D.
+
+To solve _a_ (_b_)/(_c_^2) = _x_. Set _b_ on C to _c_ on D and over _a_
+on B read _x_ on A.
+
+To solve _c_√((_a_)/(_b_)) = _x_. Set _b_ on B to _a_ on A, and under
+_c_ on C read _x_ on D.
+
+To solve (√_̅a_)/(_b_) = _x_. Set _b_ on C to _a_ on A, and under 1 on C
+read _x_ on D.
+
+To solve (_a_)/(√_̅b_) = _x_. Set _b_ on B to _a_ on D, and under 1 on C
+read _x_ on D.
+
+To solve _b_√_̅a_ = _x_. Set 1 on C to _b_ on D, and under _a_ on B read
+_x_ on D.
+
+To solve √(_a_^3) = _x_. Treat as _a_√_̅a_.
+
+To solve _a_√(_b_^3) = _x_. Treat as _a_√_̅b_ × _b_.
+
+To solve (√_̅a_^3)/(_b_) = _x_. Treat as (√_̅a_ × _a_)/(_b_).
+
+To solve √((_a_^3)/(_b_)) = _x_. Treat as (√_̅a_ × _a_)/(√_̅b_) =
+√((_a_)/(_b_)) × _a_.
+
+To solve √((_a_ × _b_)/(_c_)) = _x_. Set _c_ on B to _a_ on A, and under
+_b_ on B read _x_ on D.
+
+To solve (_a_ × _b_)/(√_̅c_) = _x_. Set _c_ on B to _b_ an D, and under
+_a_ on C read _x_ on D.
+
+To solve √((_a_^2 × _b_)/(_c_)) = _x_. Set _c_ on B to _a_ on D, and
+under _b_ on B read _x_ on D.
+
+To solve (_a_^2 × _b_^2)/(_c_) = _x_. Set _c_ on B to _a_ on D, and over
+_b_ on C read _x_ on A.
+
+To solve (_a_√_̅b_)/(_c_) = _x_. Set _c_ on C to _b_ on A, and under _a_
+on C read _x_ on D.
+
+To solve ((_a_ × √_̅b_)/(_c_))^2 = _x_. Set _c_ on C to _a_ on D,
+and over _b_ on B read _x_ on A.
+
+
+                    HINTS ON EVALUATING EXPRESSIONS.
+
+As a general rule, the use of cubes and higher powers should be avoided
+whenever possible. Thus, in the foregoing section, we recommend treating
+an expression of the form _a_√(_b_^3) as _a_ × _b_ × √_̅b_; the
+magnitudes of the values thus met with are more easily appreciated by
+the beginner, and mistakes in estimating the large numbers involved in
+cubing are avoided.
+
+
+  EX.—7·3 × √(57^3) = 3140.
+
+  Set 1 on C to 57 on D; bring cursor to 57 on B (R.H., since 57 has an
+  _even_ number of digits); bring 1 on C to cursor, and under 7·3 on C
+  read 3140 on D. As a rough estimate we have √(57), about 8; 8 × 57,
+  about 400; 400 × 7, gives 2800, showing the result consists of 4
+  figures.
+
+
+An expression of the form _a_∛(_b_^2), or _a_ _b_^⅔, is better dealt
+with by rearranging as _a_ × (_b_)/(∛_b_).
+
+
+  EX.—3·64∛(4·32^2) = 9·65.
+
+  Set cursor to 4·32 on A, and move the slide until 1·63 is found
+  simultaneously under the cursor on B and on D under 1 on C; bring
+  cursor to 1 on C; 4·32 on C to cursor, and _over_ 3·64 on D read 9·65
+  on C. (Note that in this case it is convenient to read the answer on
+  the _slide_; see page 22). From the slide rule we know ∛(4·32) = about
+  1·6; this into 4·32 is roughly 3; 3·64 × 3 is about 10, showing the
+  answer to be 9·65.
+
+
+Similarly products of the form _a_ × _b_^{⁴⁄₃} are best dealt with as
+_a_ × _b_ × ∛_b_.
+
+Factorising expressions sometimes simplifies matters, as, for instance,
+in _x_^4 − _y_^4 = (_x_^2 + _y_^2)(_x_^2 − _y_^2). Here, working with
+the fourth powers involves large numbers and the troublesome
+determination of the number of digits in each factor; but squares are
+read on the rule at once, the number of digits is obvious, and, in
+general, the method should give a more accurate result. Take the
+expression, D_{1} = ∛((D^4 − _d_^4)/(D)) giving the diameter D_{1} of a
+solid shaft equal in torsional strength to a hollow shaft whose external
+and internal diameters are D and _d_ respectively. Rearranging as D_{1}
+= ∛(((D^2 + _d_^2)(D^2 − _d_^2))/(D)) and taking, as an example, D = 15
+in. and _d_ = 7 in., we have D^2 + _d_^2 = 274 and D^2 − _d_^2 = 176;
+hence D_1 = ∛((274 × 176)/(15)) = ∛(3210) = 14·75 in.
+
+_Reversed Scale Notation._—With expressions of the form 1 − _x_, or 100
+− _x_, it is often convenient to regard the scales as having their
+notation reversed, _i.e._, to read the scale backwards. When this is
+done the D scale is read as shown on the lower line—
+
+            Direct Notation   1  2  3  4  5  6  7  8  9  10
+                 D Scale
+            Reversed Notation 9  8  7  6  5  4  3  2  1  0
+
+The new reading can be found by subtracting the ordinary reading from 1,
+10, 100, etc., according to the value assigned to the R.H. index, but
+actually it is unnecessary to make this calculation, as with a little
+practice it is quite an easy matter to read both the main and
+subdivisions in the reversed order. Applications are found in plotting
+curves, trigonometrical formulæ, etc.
+
+
+  EX.—Find the per cent. of slip of a screw propeller from
+
+  100 − S = (10133V)/(PR)
+
+  taking the speed, V, as 15 knots, the pitch of the propeller, P, as 27
+  ft. 6 in., and the revolutions per minute, R, as 60.
+
+  Set 27·5 on B to 10133 on A (N.B.—Take the setting near the _centre_
+  index of A); bring the cursor to 15 on B and 60 on B to cursor.
+  Reading the L.H. A scale backwards, the slip, S, = 8 per cent. is
+  found on A over 10 on B.
+
+
+_Percentage Calculations._—To increase a quantity by _x_ per cent. we
+multiply by 100 + _x_; to diminish a quantity by _x_ per cent. we
+multiply by 100 − _x_. Hence, to add _x_ per cent., set 100 + _x_ on C
+to 1 on D and read new values on D under original values on C. To deduct
+_x_ per cent. read the D scale backwards from 10 and set R.H. index of C
+to _x_ per cent. so read. Then read as before.
+
+
+                             GAUGE POINTS.
+
+Special graduations, marking the position of constant factors which
+frequently enter into engineering calculations, are found on most slide
+rules. Usually the values of π = 3·1416 and (π)/(4) = 0·7854—the “gauge
+points” for calculating the circumference and area of a circle—are
+marked on the upper scales. The first should be given on the lower
+scales also. Marks _c_ and _c_^1 are sometimes found on the lower scales
+at 1·128 = √((4)/(π)) and at 3·568 = √((40)/(π)). These are useful in
+calculating the contents of cylinders and are thus derived:—Cubic
+contents of cylinder of diameter _d_ and length _l_ = (π)/(4)_d_^2_l_;
+substituting for (π)/(4) its reciprocal (4)/(π), the formula becomes
+(_d_^2)/(1·273 × _l_), and by taking the square root of the fractional
+part we have (_d_)/(1·128)^2 × _l_. This is now in a very convenient
+form, since by setting the gauge point _c_ on C to _d_ on D, we can read
+over _l_ on B the cubic contents on A. This example indicates the
+principle to be followed in arranging gauge points. Successive
+multiplication is avoided by substituting the reciprocal of the
+constant, thus bringing the expression into the form (_a_ × _b_)/(_c_),
+which, as we know, can be resolved by one setting of the slide. The
+advantage of dividing _d_ before squaring is also evident. The mark
+_c_^1 = _c_ × √(10) is used if it is necessary to draw the slide more
+than one-half its length to the right.
+
+A gauge point, M, at 31·83 = (100)/(π) is found on the upper scales of
+some rules. Setting this point on B to the diameter of a cylinder on A,
+the circumference is read over 1 or 100 on B or the area of the curved
+surface over the length on B.
+
+As another example of establishing a gauge point, we will take the
+formula for the theoretical delivery of pumps. If _d_ is the diameter of
+the plunger in inches, _l_ the length of stroke in feet, and Q the
+delivery in gallons, we have
+
+Q = _d_^2 × (π)/(4) × _l_ × (12)/(277). (N.B.—277 cubic inches = 1
+gallon.)
+
+Multiplying out the constant quantities and taking its reciprocal, we
+readily transform the statement into Q = (_d_^2_l_)/(29·4) or
+((_d_)/(5·42))^2 × _l_. Hence set gauge point 5·42 on C to _d_ on
+D and over length of stroke in feet on B, read delivery in gallons per
+stroke on A; or over piston speed in feet per minute on B, read
+theoretical delivery in gallons per minute on A.
+
+Several examples of gauge points will be found in the section on
+calculating the weights of metal (see pages 59 and 60). In most cases
+their derivation will be evident from what has been said above. In the
+case of the weight of spheres, we have Vol. = 0·5236_d_^3, and this
+multiplied by the weight of 1 cubic inch of the material will give the
+weight W in lb. Hence for cast-iron, W = 0·5236 × _d_^3 × 0·26, which is
+conveniently transformed into W = (_d_ × _d_^2)/(7·35) as in the example
+on page 60.
+
+With these examples no difficulty should be experienced in establishing
+gauge points for any calculation in which constant factors recur.
+
+_Marking Gauge Points._—The practice of marking gauge points by lines
+extending to the working edge of the scale is not to be recommended, as
+it confuses the ordinary reading of the scales. Generally speaking,
+gauge points are only required occasionally, and if they are placed
+clear of the scale to which they pertain, but near enough to show the
+connection, they can be brought readily into a calculation by means of
+the cursor. Usually there is sufficient margin above the A scale and
+below the D scale for various gauge points to be marked. Another plan
+consists in cutting two nicks in the upper and lower edges of the cursor
+near the centre and about ⅛ in. apart. These centre pieces, when bent
+out, form a tongue, which are in line with the cursor line and run
+nearly in contact with the square and bevelled edges of the rule
+respectively. A fine line in the tongue can then be set to gauge points
+marked on these two edge strips, the ordinary measuring graduations
+being removed, if desired, by a piece of fine sand-paper.
+
+For gauge points marked on the face of the rule, the author prefers two
+fine lines drawn at 45°—thus, ✕—and crossing in the exact point which it
+is required to indicate. With the “cross” gauge point the meeting lines
+facilitate the placing of the cursor, and an exact setting is readily
+made.[7] All lines should be drawn in Indian ink with a very sharp
+drawing pen. For a more permanent marking the Indian ink may be rubbed
+up in glacial acetic acid or the special ink for celluloid may be used.
+If any difficulty is found in writing the distinguishing signs against
+the gauge point, the inscription may be formed by a succession of small
+dots made with a sharp pricker.
+
+
+                  EXAMPLES IN TECHNICAL CALCULATIONS.
+
+In order to illustrate the practical value of the slide rule, we now
+give a number of examples which will doubtless be sufficient to suggest
+the methods of working with other formulæ. A few of the rules give
+results which are approximate only, but in all cases the degree of
+accuracy obtained is well within the possible reading of the scales. In
+many cases the rules given may be modified, if desired, by varying the
+constants. In most of the examples the particular formula employed will
+be evident from the solution, but in a few of the more complicated cases
+a separate statement has been given.
+
+
+                           MENSURATION, ETC.
+
+Given the chord _c_ of a circular arc, and the vertical height _h_, to
+find the diameter _d_ of the circle.
+
+Set the height _h_ on B to half the chord on D, and over 1 on B read _x_
+on A. Then _x_ + _h_ = _d_.
+
+
+  EX.—_c_ = 6; _h_ = 2; find _d_. Set 2 on B to 3 on D, and over 1 on B
+  read 4·5 on A. Then 4·5 + 2 = 6·5 = _d_.
+
+
+Given the radius of a circle _r_, and the number of degrees _n_ in an
+arc, to find the length _l_ of the arc.
+
+Set _r_ on C to 57·3 on D, and over any number of degrees _n_ on D read
+the (approximate) length of the arc on C.
+
+
+  EX.—_r_ = 24; _n_ = 30; find _l_.
+
+  Set 24 on C to 57·3 on D, and over 30 on D read 12·56 = _l_ on C.
+
+
+Given the diameter _d_ of a circle in _inches_, to find the
+circumference _c_ in _feet_.
+
+Set 191 on C to 50 on D, and under any diameter in inches on C read
+circumference _c_, in feet on D.
+
+
+  EX.—Find the circumference in feet of a pulley 17 in. in diameter. Set
+  191 on C to 50 on D, and under 17 on C read 4·45 ft. on D.
+
+
+Given the diameter of a circle, to find its area.
+
+Set 0·7854 on B to 10 (centre index) on A and over any diameter on D
+read area on B.
+
+When the rule has a special graduation line = 0·7854, on the right-hand
+scale of B, set this line to the R.H. index of A and read off as above.
+If only π is marked, set this special graduation on B to 4 on A.
+
+On the C and D scales of some rules a gauge point marked _c_ will be
+found indicating √((4)/(π)) = 1·1286. In this case, therefore, set 1 on
+C to gauge point _c_ on D, and read area on A as above. If the gauge
+point _c_′ is used, divide the result by 10. Or set _c_ on C, to
+diameter on D, and over index of B read area on A. Cursors are supplied,
+having _two_ lines ruled on the glass, the interval between them being
+equal to (4)/(π) = 1·273 on the A scale. In this case, if the right hand
+of the two cursor lines be set to the diameter on D, the _area_ will be
+read on A under the _left_-hand cursor line. For diameters less than
+1·11 it is necessary to set the middle index of B to the L.H. index of
+A, reading the areas on the L.H. B scale. The confusion which in general
+work is sometimes caused by the use of two cursor lines might be
+obviated by making the left-hand line in two short lengths, each only
+just covering the scales.
+
+Given diameter of circle _d_ in _inches_, to find area _a_ in square
+_feet_.
+
+Set 6 on B to 11 on A, and over diameter in inches on D read area in
+square feet on B.
+
+To find the surface in square feet of boiler flues, condenser tubes,
+heating pipes, etc., having given the diameter in inches and length in
+feet.
+
+Find the circumference in feet as above and multiply by the length in
+feet.
+
+
+  EX.—Find the heating surface afforded by 160 locomotive boiler tubes
+  1¾in. in diameter and 12 ft. long.
+
+  Set 191 on C to 50 on D; bring cursor 1·75 on C, L.H. index of C to
+  cursor; cursor to 12 on C; 1 on C to cursor; and under 160 on C read
+  880 sq. ft. of heating surface on D.
+
+
+If the dimensions are in the same denomination and the rule has a gauge
+point M at 31·83 (= (100)/(π)), set this mark on B to diameter of
+cylinder on A, and read cylindrical surface on A over length on B.
+
+To find the side _s_ of a square, equal in area to a given rectangle of
+length _l_ and breadth _b_.
+
+Set R.H. or L.H. index of B to _l_ on A, and under _b_ on B read _s_ on
+D.
+
+
+  EX.—Find the side of a square equal in area to a rectangle in which
+  _l_ = 31 ft. and _b_ = 5 ft.
+
+  Set the (R.H.) index of B to 31 on A, and under 5 on B read 12·45 ft.
+  on D.
+
+
+To find various lengths _l_ and breadths _b_ of a rectangle, to give a
+constant area _a_.
+
+Invert the slide and set the index of Ɔ to the given area on D. Then
+opposite any length _l_ on Ɔ find the corresponding breadth _b_ on D.
+
+
+  EX.—Find the corresponding breadths of rectangular sheets, 16, 18, 24,
+  36, and 60 ft. long, to give a constant area of 72 sq. ft.
+
+  Set the R.H. index of Ɔ to 72 on D, and opposite 16, 18, 24, 36, and
+  60 on Ɔ read 4·5, 4, 3, 2, and 1·2 ft., the corresponding breadths on
+  D.
+
+
+To find the contents in cubic feet of a cylinder of diameter _d_ in
+inches and length _l_ in feet.
+
+Find area in feet as before, and multiply by the length.
+
+If dimensions are all in inches or feet, set the mark _c_ (= 1·128) on C
+to diameter on D and over length on B, read cubic contents on A.
+
+To find the area of an ellipse.
+
+Set 205 on C to 161 on D; bring cursor to length of major axis on C, 1
+on C to cursor, and under length of minor axis on C read area on D.
+
+
+  EX.—Find the area of an ellipse the major and minor axes of which are
+  16 in. and 12 in. in length respectively.
+
+  Set 205 on C to 161 on D; bring cursor to 16 on C, 1 on C to cursor,
+  and under 12 on C read 150·8 in. on D.
+
+
+To find the surface of spheres.
+
+Set 3·1416 on B to R.H. or L.H. index of A, and over diameter on D read
+by the aid of the cursor, the convex surface on B.
+
+To find the cubic contents of spheres.
+
+Set 1·91 on B to diameter on A, and over diameter on C read cubic
+contents on A.
+
+
+                           WEIGHTS OF METALS.
+
+To find the weight in lb. per lineal foot of square bars of metal.
+
+Set index of B to weight of 12 cubic inches of the metal (_i.e._, one
+lineal foot, 1 square inch in section) on A, and over the side of the
+square in inches on C read weight in lb. on A.
+
+
+  EX.—Find the weight per foot length of 4½in. square wrought-iron bars.
+
+  Set middle index of B to 3·33 on A, and over 4½ on C read 67·5 lb. on
+  A.
+
+
+(N.B.—For other metals use the corresponding constant in column (2),
+below).
+
+To find the weight in lb. per lineal foot of round bars.
+
+Set R.H. or L.H. index of B to weight of 12 cylindrical inches of the
+metal on A (column (4), below), and opposite the diameter of the bar in
+inches on C, read weight in lb. per lineal foot on A.
+
+
+  EX.—Find the weight of 1 lineal foot of 2 in. round cast steel.
+
+  Set L.H. index of B to 2·68 on A, and over 2 on C read 10·7 lb. on A.
+
+
+To find the weight of flat bars in lb. per lineal foot.
+
+Set the breadth in inches on C to (1)/(weight of 12 cub. in.) of the
+metal (column (3), below) on D, and above the thickness on D read weight
+in lb. per lineal foot on C.
+
+
+  EX.—Find the weight per lineal foot of bar steel, 4½in. wide and ⅝in.
+  thick.
+
+  Set 4·5 on C to 0·294 on D, and over 0·625 on D read 9·56 lb. per
+  lineal foot on C.
+
+
+To find the weight per square foot of sheet metal, set the weight per
+cubic foot of the metal (col. 1) on C to 12 on D, and
+
+       ────────────┬──────────┬──────────┬──────────┬───────────
+                   │   (1)    │(2)       │   (3)    │    (4)
+                   │Weight in │Weight of │ (1)/(Wt. │ Weight of
+         Metals.   │ lb. per  │12 cubic  │of 12 cub.│    12
+                   │cubic ft. │in.       │   in.)   │cylindrical
+                   │          │          │          │    in.
+       ────────────┼──────────┼──────────┼──────────┼───────────
+       Wrought iron│480       │3·33      │0·300     │2·62
+       Cast iron   │450       │3·125     │0·320     │2·45
+       Cast steel  │490       │3·40      │0·294     │2·68
+       Copper      │550       │3·82      │0·262     │3·00
+       Aluminium   │168       │1·166     │0·085     │0·915
+       Brass       │520       │3·61      │0·277     │2·83
+       Lead        │710       │4·93      │0·203     │3·87
+       Tin         │462       │3·21      │0·312     │2·52
+       Zinc (cast) │430       │2·98      │0·335     │2·34
+       „  (sheet)  │450       │3·125     │0·320     │2·45
+       ────────────┴──────────┴──────────┴──────────┴───────────
+
+above the thickness of the plate in inches on D read weight in lb. per
+square foot on C.
+
+
+  EX.—Find the weight in lb. per square foot of aluminium sheet ⅜in.
+  thick.
+
+  Set 168 on C to 12 on D, and over 0·375 on D read 5·25 lb. on C.
+
+
+To find the weight of pipes in lb. per lineal foot.
+
+Set mean diameter of the pipe in inches (_i.e._, internal diameter
+_plus_ the thickness, or external diameter _minus_ the thickness) on C
+to the constant given below on D, and over the thickness on D read
+weight in lb. per lineal foot on C.
+
+       ┌────────────┬─────────────────────┬─────────────────────┐
+       │  Metals.   │ Constant for Pipes. │Constant for Spheres.│
+       ├────────────┼─────────────────────┼─────────────────────┤
+       │Wrought iron│       0·0955        │        6·87         │
+       │Cast iron   │       0·1020        │        7·35         │
+       │Steel       │       0·0936        │        6·73         │
+       │Brass       │       0·0882        │        6·35         │
+       │Copper      │       0·0834        │        6·00         │
+       │Lead        │       0·0646        │        4·65         │
+       └────────────┴─────────────────────┴─────────────────────┘
+
+
+  EX.—Find the weight per foot of cast-iron piping 4 in. internal
+  diameter and ½in. thick.
+
+  Set 4·5 on C to 0·102 on D, and over 0·5 on D read 22·1 lb. on C, the
+  required weight.
+
+
+To find the weight in lb. of spheres or balls, given the diameter in
+inches. (W = 0·5236_d_^3 × wt. of 1 cub. in. of material).
+
+Set the constant for spheres (given above) on B to diameter in inches on
+A, and over diameter on C read weight in lb. on A.
+
+
+  EX.—Find the weight of a cast-iron ball 7½in. in diameter.
+
+  Set 7·35 on B to 7·5 on A, and over 7·5 on C read 57·7 lb. on A.
+
+
+To find diameter in inches of a sphere of given weight.
+
+Set the cursor to the given weight in lb. on A, and move the slide until
+the same number is found on C under the cursor that is simultaneously
+found on A over the constant for the sphere on B.
+
+
+  EX.—Find diameter in inches of a sphere of cast-iron to weigh 7½lb.
+
+  Setting the cursor to 7·5 on A, and moving the slide, it is found that
+  when 3·8 on C falls under the cursor, 3·8 on A is simultaneously found
+  over 7·35 on B. The required diameter is therefore 3·8 in.
+
+
+The rules for cubes and cube roots (page 40) should be kept in view in
+solving the last two examples.
+
+
+                            FALLING BODIES.
+
+To find velocity in feet per second of a falling body, given the time of
+fall in seconds.
+
+Set index on C to time of fall on D, and under 32·2 on C read velocity
+in feet per second on D.
+
+To find velocity in feet per second, given distance fallen through in
+feet.
+
+Set 1 on C to distance fallen through on A, and under 64·4 on B read
+velocity in feet per second on D.
+
+
+  EX.—Find velocity acquired by falling through 14 ft.
+
+  Set (R.H.) index of C to 14 on A, and under 64·4 on B read 30 ft. per
+  second on D.
+
+
+To find distance fallen through in feet in a given time.
+
+Set index of C to time in seconds on D, and over 16·1 on B read distance
+fallen through in feet on A.
+
+
+                           CENTRIFUGAL FORCE.
+
+To find the centrifugal force of a revolving mass in lb.
+
+Set 2940 on B to revolutions per minute on D; bring cursor to weight in
+lb. on B; index of B to cursor, and over radius in feet on B read
+centrifugal force in lb. on A.
+
+To find the centrifugal stress in lb. per square inch, in rims of
+revolving wheels of cast iron.
+
+Set 61·3 on C to the mean diameter of the wheel in feet on D, and over
+revolutions per minute on C read stress per square inch on A.
+
+
+  EX.—Find the stress per square inch in a cast-iron fly-wheel rim 8 ft.
+  in diameter and running at 120 revolutions per minute.
+
+  Set 61·3 on C to 8 on D, and over 120 on C read 245 lb. per square
+  inch on A.
+
+
+                           THE STEAM ENGINE.
+
+Given the stroke and number of revolutions per minute, to find the
+piston speed.
+
+Set stroke in inches on C to 6 on D, and over number of revolutions on D
+read piston speed in feet per minute on C.
+
+To find cubic feet of steam in a cylinder at cut-off, given diameter of
+cylinder and period of admission in inches.
+
+Set 2200 on B to cylinder diameter on D, and over period of admission on
+B read cubic feet of steam on A.
+
+
+  EX.—Cylinder diameter 26 in., stroke 40 in., cut-off at ⅝ of stroke.
+  Find cubic feet of steam used (theoretically) per stroke.
+
+  Set 2200 on B to 26 on D, and over 40 × ⅝ or 25 in. on B, read 7·68
+  cub. ft. on A, as the number of cubic feet of steam used per stroke.
+
+
+Given the diameter of a cylinder in inches, and the pressure in lb. per
+square inch, to find the load on the piston in tons.
+
+Set pressure in lb. per square inch on B to 2852 on A, and over cylinder
+diameter in inches on D read load on piston in tons on B.
+
+
+  EX.—Steam pressure 180 lb. per square inch; cylinder diameter, 42 in.
+  Find load in tons on piston.
+
+  Set 180 on B to 2852 on A, and over 42 on D read 111 tons, the gross
+  load, on B.
+
+
+Given admission period and absolute initial pressure of steam in a
+cylinder, to find the pressure at various points in the expansion period
+(isothermal expansion).
+
+Invert the slide and set the admission period, in inches, on Ɔ to the
+initial pressure on D; then under any point in the expansion stroke on Ɔ
+find the corresponding pressure on D.
+
+
+  EX.—Admission period 12 in., stroke 42 in., initial pressure 80 lb.
+  per square inch. Find pressure at successive fifths of the expansion
+  period.
+
+  Set 12 on Ɔ to 80 on D, and opposite 18, 24, 30, 36 and 42 in. of the
+  whole stroke on Ɔ find the corresponding pressures on D:—53·3, 40, 32,
+  26·6 and 22·8 lb. per square inch.
+
+
+To find the mean pressure constant for isothermally expanding steam,
+given the cut-off as a fraction of the stroke.
+
+Find the logarithm of the ratio of the expansion _r_, by the method
+previously explained (page 46). Prefix the characteristic and to the
+number thus obtained, on D, set 1 on C. Then under 2·302 on C read _x_
+on D. To _x_ + 1 on D set _r_ on C, and under index of C read mean
+pressure constant on D. The latter, multiplied by the initial pressure,
+gives the mean forward pressure throughout the stroke. (N.B.—Common log.
+× 2·302 = hyperbolic log.)
+
+
+  EX.—Find the mean pressure constant for a cut-off of ¼th, or a ratio
+  of expansion of 4.
+
+  Set (L.H.) index of C to 4 on D, and on the reverse side of the slide
+  read 0·602 on the logarithmic scale. The characteristic = 0; hence to
+  0·602 on D set (R.H.) index of C, and under 2·302 on C read 1·384 on
+  D. Add 1, and to 2·384 thus obtained on D set _r_ (= 4) on C, and
+  under 1 on C read 0·596, the mean pressure constant required.
+
+
+Mean pressure constants for the most usual degrees of cut-off are given
+below:—
+
+         Cut-off in fractions of stroke Mean pressure constant
+                       ¾                        0·968
+                      ⁷⁄₁₀                      0·952
+                       ⅔                        0·934
+                       ⅝                        0·919
+                       ⅗                        0·913
+                       ½                        0·846
+                       ⅖                        0·766
+                       ⅜                        0·750
+                       ⅓                        0·699
+                      ³⁄₁₀                      0·664
+                       ¼                        0·596
+                       ⅕                        0·522
+                       ⅙                        0·465
+                       ⅐                        0·421
+                       ⅛                        0·385
+                       ⅑                        0·355
+                       ⅒                        0·330
+                      ¹⁄₁₁                      0·309
+                      ¹⁄₁₂                      0·290
+                      ¹⁄₁₃                      0·274
+                      ¹⁄₁₄                      0·260
+                      ¹⁄₁₅                      0·247
+                      ¹⁄₁₆                      0·236
+
+To find mean pressure:—Set 1 on C to constant on D, and under initial
+pressure on C read mean pressure on D.
+
+Given the absolute initial pressure, length of stroke, and admission
+period, to find the absolute pressure at any point in the expansion
+period, it being assumed that the steam expands adiabatically. (P_{2} =
+(P_{1})/(R^{¹⁰⁄₉}) in which P_{1} = initial pressure and P_{2} the
+pressure corresponding to a ratio of expansion R.)
+
+Set L.H. index of C to ratio of expansion on D, and read on the back of
+the slide the decimal of the logarithm. Add the characteristic, and to
+the number thus obtained on D set 9 on C, and read off the value found
+on D under the index of C. Set this number on the logarithmic scale to
+the index mark, in the opening on the back of the rule, and under L.H.
+index of C read the value of R^{¹⁰⁄₉} on D. The initial pressure divided
+by this value gives the corresponding pressure due to the expansion.
+
+
+  EX.—Absolute initial pressure 120 lb. per square inch; stroke, 4 ft.;
+  cut-off ¼. Find the respective pressures when ½ and ¾ths of the stroke
+  have been completed.
+
+  In the first case R = 2. Therefore setting the L.H. index of C to 2 on
+  D, we find the decimal of the logarithm on the back of the slide to be
+  0·301. The characteristic is 0, so placing 9 on C to 0·301 on D, we
+  read 0·334 as the value under the R.H. index of C. (N.B.—In locating
+  the decimal point it is to be observed that the log. of R has been
+  multiplied by 10, in accordance with the terms of the above
+  expression.) Setting this number on the logarithmic scale to the back
+  index, the value of R^{¹⁰⁄₉} is found on D, under the L.H. index of C,
+  to be 2·16. Setting 120 on C to this value, it is found that the
+  pressure at ½ stroke, read on C over the R.H. index of D, is 55·5 lb.
+  per square inch. In a similar manner, the pressure when ¾ths of the
+  stroke is completed is found to be 35·4 lb. per square inch.
+
+
+For other conditions of expanding steam, or for gas or air, the method
+of procedure is similar to the above.
+
+To find the horse-power of an engine, having given the mean _effective_
+pressure, the cylinder diameter, stroke, and number of revolutions per
+minute.
+
+To cylinder diameter on D set 145 on C; bring cursor to stroke in feet
+on B, 1 on B to cursor, cursor to number of revolutions on B, 1 on B to
+cursor, and over mean effective pressure on B find horse-power on A.
+
+(N.B.—If stroke is in inches, use 502 in place of 145 given above.)
+
+
+  EX.—Find the indicated horse-power, given cylinder diameter 27 in.,
+  mean effective pressure 38 lb. per square inch, stroke 32 in.,
+  revolutions 57 per minute.
+
+  Set 502 on C to 27 on D, bring cursor to 32 on B, 1 on B to cursor,
+  cursor to 57 on B, 1 on B to cursor, and over 38 on B read 200 I.H.P.
+  on A.
+
+
+To determine the horse-power of a compound engine, invert the slide and
+set the diameter of the _high_-pressure cylinder on Ɔ to the cut-off in
+that cylinder on A. Use the number then found on A over the diameter of
+the _low_-pressure cylinder on Ɔ as the cut-off in that cylinder,
+working with the same pressure and piston speed, and calculate the
+horse-power as for a single cylinder.
+
+To find the cylinder ratio in compound engines, invert the slide and set
+index of Ɔ to diameter of the low-pressure cylinder on D. Then over the
+diameter of the high-pressure cylinder on C, read cylinder ratio on A.
+
+
+  EX.—Diameter of high-pressure cylinder 7¾in., low-pressure 15 in. Find
+  cylinder ratio.
+
+  Set index on Ɔ to 15 on D, and over 7·75 on Ɔ read 3·75, the required
+  ratio, on A.
+
+
+The cylinder ratios of triple or quadruple-expansion engines may be
+similarly determined.
+
+
+  EX.—In a quadruple-expansion engine, the cylinders are 18, 26, 37, and
+  54 inches in diameter. Find the respective ratios of the high, first
+  intermediate, and second intermediate cylinders to the low-pressure.
+
+  Set (R.H.) index of Ɔ to 54 on D, and over 18, 26, and 37 on Ɔ read 9,
+  4·31, and 2·13, the required ratios, on A.
+
+
+Given the mean effective pressures in lb. per square inch in each of the
+three cylinders of a triple-expansion engine, the I.H.P. to be developed
+in each cylinder, and the piston speed, to find the respective cylinder
+diameters.
+
+Set 42,000 on B to piston speed on A; bring cursor to mean effective
+pressure in low-pressure cylinder on B, index of B to cursor, and under
+I.H.P. on A read low-pressure cylinder diameter on C. To find the
+diameters of the high-pressure and intermediate-pressure cylinders,
+invert the slide and place the mean pressure in the low-pressure
+cylinder on ᗺ to the diameter of that cylinder on D. Then under the
+respective mean pressures on ᗺ read corresponding cylinder diameters on
+D.
+
+
+  EX.—The mean effective pressures in the cylinders of a
+  triple-expansion engine are:—L.P., 10·32; I.M.P., 27·5; and H.P., 77·5
+  lb. per square inch. The piston speed is 650 ft. per minute, and the
+  I.H.P. developed in each cylinder, 750. Find the cylinder diameters.
+
+  Set 42,000 on B to 650 on A, and bring cursor to 10·32 on B. Bring
+  index of B to cursor, and under 750 on A read 68·5 in. on C, the L.P.
+  cylinder diameter. Invert the slide, and placing 10·32 on ᗺ to 68·5 on
+  D, read, under 27·5 on ᗺ, the I.M.P. cylinder diameter = 42 in., on D;
+  also under 77·5 on ᗺ read the H.P. cylinder diameter = 25 in., on D.
+
+
+To compute brake or dynamometrical horse-power.
+
+Set 525 on C to the total weight in lb. acting at the end of the lever
+(or pull of spring balance in lb.) on D; set cursor to length of lever
+in feet on C, bring 1 on C to cursor, and under number of revolutions
+per minute on C find brake horse-power on D.
+
+Given cylinder diameter and piston speed in feet per minute, to find
+diameter of steam pipe, assuming the maximum velocity of the steam to be
+6000 ft. per minute.
+
+Set 6000 on B to cylinder diameter on D, and under piston speed on B
+read steam pipe diameter on D.
+
+Given the number of revolutions per minute of a Watt governor, to find
+the vertical height in inches, from the plane of revolution of the balls
+to the point of suspension.
+
+Set revolutions per minute on C to 35,200 on A, and over index of B read
+height on A.
+
+Given the weight in lb. of the rim of a cast-iron fly-wheel, to find the
+sectional area of the rim in square inches.
+
+Set the mean diameter of the wheel in feet on C to 0·102 on D, and under
+weight of rim on C find area on D.
+
+Given the consumption of coal in tons per week of 56 hours, and the
+I.H.P., to find the coal consumed per I.H.P. per hour.
+
+Set I.H.P. on C to 40 on D, and under weekly consumption on C read lb.
+of coal per I.H.P., per hour on D.
+
+
+  EX.—Find coal used per I.H.P. per hour, when 24 tons is the weekly
+  consumption for 300 I.H.P.
+
+  Set 300 on C to 40 on D, and under 24 on C read 3·2 lb. per I.H.P. per
+  hour on D.
+
+
+(N.B.—For any other number of working hours per week divide 2240 by the
+number of working hours, and use the quotient in place of 40 as above.)
+
+To find the tractive force of a locomotive.
+
+Set diameter of driving wheel in inches on B to diameter of cylinder in
+inches on D, and over the stroke in inches on B read on A, tractive
+force in lb. for each lb. of effective pressure on the piston.
+
+
+                             STEAM BOILERS.
+
+To find the bursting pressure of a cylindrical boiler shell, having
+given the diameter of shell and the thickness and ultimate strength of
+the material.
+
+Set the diameter of the shell in inches on C to twice the thickness of
+the plate on D, and under strength of material per square inch on C read
+bursting pressure in lb. per square inch on D.
+
+
+  EX.—Find the bursting pressure of a cylindrical boiler shell 7 ft. 6
+  in. in diameter, with plates ½in. thick, assuming an ultimate strength
+  of 50,000 lb. per square inch.
+
+  Set 90 on C to 1·0 on D, and under 50,000 on C find 555 lb. on D.
+
+
+To find working pressure for Fox’s corrugated furnaces by Board of Trade
+rule.
+
+Set the least outside diameter in inches on C to 14,000 on D, and under
+thickness in inches on C read working pressure on D in lb. per square
+inch.
+
+To find diameter _d_ in inches, of round steel for safety valve springs
+by Board of Trade rule.
+
+Set 8000 on C to load on spring in lb. on D, and under the mean diameter
+of the spring in inches on C read _d_^3 on D. Then extract the cube root
+as per rule.
+
+
+                     SPEED RATIOS OF PULLEYS, ETC.
+
+Given the diameter of a pulley and its number of revolutions per minute,
+to find the circumferential velocity of the pulley or the speed of
+ropes, belts, etc., driven thereby.
+
+Set diameter of pulley in inches on C to 3·82 on D, and over revolutions
+per minute on D read speed in feet per minute on C.
+
+
+  EX.—Find the speed of a belt driven by a pulley 53 in. in diameter and
+  running at 180 revolutions per minute.
+
+  Set 53 on C to 3·82 on D, and over 180 on D read 2500 ft. per minute
+  on C.
+
+  EX.—Find the speed of the pitch line of a spur wheel 3 ft. 6 in. in
+  diameter running at 60 revolutions per minute.
+
+  Set 42 in. on C to 3·82 on D, and over 60 on D read 660 ft. per minute
+  on C.
+
+
+Given diameter and number of revolutions per minute of a driving pulley,
+and the diameter of the driven pulley, to find the number of revolutions
+of the latter.
+
+Invert the slide and set diameter of driving pulley on Ɔ to given number
+of its revolutions on D; then opposite diameter of any driven pulley on
+Ɔ read its number of revolutions on D.
+
+
+  EX.—Diameter of driving pulley 10 ft.; revolutions per minute 55;
+  diameter of driven pulley 2 ft. 9 in. Find number of revolutions per
+  minute of latter.
+
+  Set 10 on Ɔ to 55 on D, and opposite 2·75 on Ɔ read 200 revolutions on
+  D.
+
+
+                            BELTS AND ROPES.
+
+To find the ratio of tensions in the two sides of a belt, given the
+coefficient of friction between belt and pulley μ and the number of
+degrees θ in the arc of contact (log. R = (μθ)/(132)).
+
+Set 132 on C to the coefficient of friction on D, and read off the value
+found on D under the number of degrees in the arc of contact on C. Place
+this value on the scale of equal parts on the back of the slide, to the
+index mark in the aperture, and read the required ratio on D under the
+L.H. index of C.
+
+
+  EX.—Find the tension ratio in a belt, assuming a coefficient of
+  friction of 0·3 and an arc of contact of 120 degrees.
+
+  Set 132 on C to 0·3 on D, and under 120 on C read 0·273. Place this on
+  the scale to the index on the back of the rule, and under the L.H.
+  index C read 1·875 on D, the required ratio.
+
+
+Given belt velocity and horse-power to be transmitted, to find the
+requisite width of belt, taking the effective tension at 50 lb. per inch
+of width.
+
+Set 660 on C to velocity in feet per minute on D, and opposite
+horse-power on D find width of belt in inches on C.
+
+Given velocity and width of belt, to find horse-power transmitted.
+
+Set 660 on C to velocity on D, and under width on C find horse-power
+transmitted on D.
+
+(N.B.—For any other effective tension, instead of 660 use as a gauge
+point:—33,000 ÷ tension.)
+
+Given speed and diameter of a cotton driving rope, to find power
+transmitted, disregarding centrifugal action, and assuming an effective
+working tension of 200 lb. per square inch of rope.
+
+Set 210 on B to 1·75 on D, and over speed in feet per minute on B read
+horse-power on A.
+
+
+  EX.—Find the power transmitted by a 1¾in. rope running at 4000 ft. per
+  minute.
+
+    Set 210 on B to 1·75 on D, and over 4000 on B read 58·3 horse-power
+    on A.
+
+
+Find the “centrifugal tension” in the previous example, taking the
+weight per foot of the rope as = 0·27_d_^2.
+
+Set 655 on C to the diameter, 1·75 in., on D, and over the speed, 4000
+ft. on C, read centrifugal tension = 114 lb. on A.
+
+
+                              SPUR WHEELS.
+
+Given diameter and pitch of a spur wheel, to find number of teeth.
+
+Set pitch on C to π (3·1416) on D, and under any diameter on C read
+number of teeth on D.
+
+Given diameter and number of teeth in a spur wheel, to find the pitch.
+
+Set diameter on C to number of teeth on D, and read pitch on C opposite
+3·1416 on D.
+
+Given the distance between the centres of a pair of spur wheels and the
+number of revolutions of each, to determine their diameters.
+
+To twice the distance between the centres on D, set the sum of the
+number of revolutions on C, and under the revolutions of each wheel on C
+find the respective wheel diameters on D.
+
+
+  EX.—The distance between the centres of two spur wheels is 37·5 in.,
+  and they are required to make 21 and 24 revolutions in the same time.
+  Find their respective diameters.
+
+    Set 21 + 24 = 45 on C to 75 (or 37·5 × 2) on D, and under 21 and 24
+    on C find 35 and 40 in. on D as the respective diameters.
+
+
+To find the power transmitted by toothed wheels, given the pitch
+diameter _d_ in inches, the number of revolutions per minute _n_, and
+the pitch _p_ in inches, by the rule, H.P. = (_n_ _d_ _p_^2)/(400).
+
+Set 400 on B to pitch in inches on D; set cursor to d on B, 1 on B to
+cursor, and over any number of revolutions n on B read power transmitted
+on A.
+
+
+  EX.—Find the horse-power capable of being transmitted by a spur wheel
+  7 ft. in diameter, 3 in. pitch, and running at 90 revolutions per
+  minute.
+
+  Set 400 on B to 3 on D; bring cursor to 84 in. on B, 1 on B to cursor,
+  and over 90 revolutions on B read 170, the horse-power transmitted, on
+  A.
+
+
+                             SCREW-CUTTING.
+
+Given the number of threads per inch in the guide screw, to find the
+wheels to cut a screw of given pitch.
+
+Set threads per inch in guide screw on C, to the number of threads per
+inch to be cut on D. Then opposite any number of teeth in the wheel on
+the mandrel on C, is the number of teeth in the wheel to be placed on
+the guide screw on D.
+
+
+                         STRENGTH OF SHAFTING.
+
+Given the diameter _d_ of a steel shaft, and the number of revolutions
+per minute _n_, to find the horse-power from:—
+
+H.P. = _d_^3 × _n_ × 0·02.
+
+Set 1 on C to _d_ on D, and bring cursor to _d_ on B. Bring 50 on B to
+cursor, and over number of revolutions on B read H.P. on A.
+
+
+  EX.—Find horse-power transmitted by a 3 in. steel shaft at 110
+  revolutions per minute.
+
+  Set 1 on C to 3 on D, and bring cursor to 3 on B. Bring 50 on B to
+  cursor, and over 110 on B read 59·4 horse-power on A.
+
+
+Given the horse-power to be transmitted and the number of revolutions of
+a steel shaft, to find the diameter.
+
+Set revolutions on B to horse-power on A, and bring cursor to 50 on B.
+Then move the slide until the same number is found on B under the cursor
+that is simultaneously found on D under the index of C. This number is
+the diameter required.
+
+To find the deflection _k_ in inches, of a round steel shaft of diameter
+_d_, under a uniformly distributed load in lb. _w_, and supported by
+bearings, the centres of which are _l_ feet apart (_k_ = (_w_
+_l_^3)/(78,000_d_^4)).
+
+Modifying the form of this expression slightly, we proceed as
+follows:—Set _d_ on C to _l_ on D, and bring the cursor to the same
+number on B that is found on D under the index of C. Bring _d_ on B to
+cursor, cursor to _w_ on B, 78,800 on B to cursor, and read deflection
+on A over index of B.
+
+
+  EX.—Find the deflection in inches of a round steel shaft 3½in.
+  diameter, carrying a uniformly distributed load of 3200 lb., the
+  distance apart of the centres of support being 9 ft.
+
+    Set 3·5 on C to 9 on D, and read 2·57 on D, under the L.H. index of
+    C. Set cursor to 2·57 on B, and bring 3·5 on B to cursor, cursor to
+    3200 on B, 78,000 on B to cursor, and over L.H. index of B read
+    0·199 in., the required deflection on A.
+
+
+To find the diameter of a shaft subject to twisting only, given the
+twisting moment in inch-lb. and the allowable stress in lb. per square
+inch.
+
+Set the stress in lb. per square inch on B to the twisting moment in
+inch-lb. on A, and bring cursor to 5·1 on B. Then move the slide until
+the same number is found on B under the cursor that is simultaneously
+found on D under the index of C.
+
+
+  EX.—A steel shaft is subjected to a twisting moment of 2,700,000
+  inch-lb. Determine the diameter if the allowable stress is taken at
+  9000 lb. per square inch.
+
+    Set 9000 on B to 2,700,000 on A, and bring the cursor to 5·1 on B.
+    Moving the slide to the left, it is found that when 11·51 on the
+    R.H. scale of B is under the cursor, the L.H. index of C is opposite
+    11·51 on D. This, then, is the required diameter of the shaft.
+
+
+(N.B.—The rules for the scales to be used in finding the cube root (page
+42) must be carefully observed in working these examples.)
+
+
+                          MOMENTS OF INERTIA.
+
+To find the moment of inertia of a square section about an axis formed
+by one of its diagonals (I = (_s_^4)/(12)).
+
+Set index of C to the length of the side of square _s_ on D; bring
+cursor to _s_ on C, 12 on B to cursor, and over index of B read moment
+of inertia on A.
+
+To find the moment of inertia of a rectangular section about an axis
+parallel to one side and perpendicular to the plane of bending.
+
+Set index of C to the height or depth _h_ of the section, and bring
+cursor to _h_ on B. Set 12 on B to cursor, and over breadth _b_ of the
+section on B read moment of inertia on A.
+
+
+  EX.—Find the moment of inertia of a rectangular section of which _h_ =
+  14 in. and _b_ = 7 in.
+
+    Set index of C to 14 on D, and cursor to 14 on B. Bring 12 on B to
+    cursor, and over 7 on B read 1600 on A.
+
+
+                   DISCHARGE FROM PUMPS, PIPES, ETC.
+
+To find the theoretical delivery of pumps, in gallons per stroke.
+
+Set 29·4 on B to the diameter of the plunger in inches on D, and over
+length of stroke in feet on B read theoretical delivery in gallons per
+stroke on A.
+
+(N.B.—A deduction of from 20 to 40 per cent. should be made to allow for
+slip.)
+
+To find loss of head of water in feet due to friction in pipes (Prony’s
+rule).
+
+Set diameter of pipe in feet on B to velocity of water in feet per
+second on D and bring cursor to 2·25 on B; bring 1 on B to cursor, and
+over length of pipe in miles on B, read loss of head of water in feet,
+on A.
+
+To find velocity in feet per second, of water in pipes (Blackwell’s
+rule).
+
+Set 2·3 on B to diameter of pipe in feet on A, and under inclination of
+pipe in feet per mile on B read velocity in feet per second on D.
+
+To find the discharge over weirs in cubic feet per minute and per foot
+of width. (Discharge = 214√(_h_^3))
+
+Set 0·00467 on C to the head in feet _h_ on D, and under _h_ on B read
+discharge on D.
+
+To find the theoretical velocity of water flowing under a given head in
+feet.
+
+Set index of B to head in feet on A, and under 64·4 on B read
+theoretical velocity in feet per second on D.
+
+
+                      HORSE-POWER OF WATER WHEELS.
+
+To find the effective horse-power of a Poncelet water wheel.
+
+Set 880 on C to cubic feet of flow of water per minute on D, and under
+height of fall in feet on C, read effective horse-power on D.
+
+For breast water wheels use 960, and for overshot wheels 775, in place
+of 880 as above.
+
+
+                        ELECTRICAL ENGINEERING.
+
+To find the resistance per mile, in ohms, of copper wire of high
+conductivity, at 60° F. the diameter being given in mils. (1 mil. =
+0·001 in.).
+
+Set diameter of wire in mils. on C to 54,900 on A, and over R.H. or L.H.
+index of B read resistance in ohms on A.
+
+
+  EX.—Find the resistance per mile of a copper wire 64 mils. in
+  diameter.
+
+    Set 64 on C to 54,900 on A, and over R.H. index of B read 13·4 ohms
+    on A.
+
+
+To find the weight of copper wire in lb. per mile.
+
+Set 7·91 on C to diameter of wire in mils. on D, and over index of B
+read weight per mile on A.
+
+Given electromotive force and current, to find electrical horse-power.
+
+Set 746 on C to electromotive force in volts on D, and under current in
+ampères on C read electrical horse-power on D.
+
+Given the resistance of a circuit in ohms and current in ampères, to
+find the energy absorbed in horse-power.
+
+Set 746 on B to current on D, and over resistance on B read energy
+absorbed in H.P. on A.
+
+
+  EX.—Find the H.P. expended in sending a current of 15 ampères through
+  a circuit of 220 ohms resistance.
+
+  Set 746 on B to 15 on D, and over 220 on B read 66·3 H.P. on A.
+
+
+                              COMMERCIAL.
+
+To add on percentages.
+
+Set 100 on C to 100 + given percentage on D, and under original number
+on C read result on D.
+
+To deduct percentages.
+
+Set R.H. index of C to 100 − the given percentage on D, and under
+original number on C read result on D.
+
+
+  EX.—From £16 deduct 7½ per cent.
+
+    Set 10 on C to 92·5 on D and under 16 on C, read 14·8 = £14, 16s. on
+    D.
+
+
+To calculate simple interest.
+
+Set 1 on C to rate per cent. on D; bring cursor to period on C and 1 on
+C to cursor. Then opposite any sum on C find simple interest on D.
+
+For interest per annum.
+
+Set R.H. index on C to rate on D, and opposite principal on C read
+interest on D.
+
+
+  EX.—Find the amount with simple interest of £250 at 8 per cent., and
+  for a period of 1 year and 9 months.
+
+    Set 1 on C to 8 on D; bring cursor to 1·75 on C, and 1 on C to
+    cursor; then opposite 250 on C read £35, the interest, on D. Then
+    250 + 35 = £285 = the amount.
+
+
+To calculate compound interest.
+
+Set the L.H. index of C to the amount of £1 at the given rate of
+interest on D, and find the logarithm of this by reading on the reverse
+side of the rule, as explained on page 46. Multiply the logarithm, so
+found, by the period, and set the result, on the scale of equal parts,
+to the index on the under-side of the rule; then opposite any sum on C
+read the amount (including compound interest) on D.
+
+
+  EX.—Find the amount of £500 at 5 per cent. for 6 years, with compound
+  interest.
+
+    Set L.H. index of C to £1·05 on D, and read at the index on the
+    scale of equal parts on the under-side of rule, 0·0212. Multiply by
+    6, we obtain 0·1272, which, on the scale of equal parts, is placed
+    to the index in the notch at the end of the rule. Then opposite 500
+    on C read £670 on D, the amount required, including compound
+    interest.
+
+
+                      MISCELLANEOUS CALCULATIONS.
+
+To calculate percentages of compositions.
+
+Set weight (or volume) of sample on C, to weight (or volume) of
+substance considered, on D; then under index of C read required
+percentage on D.
+
+
+  EX.—A sample of coal weighing 1·25 grms. contains 0·04425 grm. of ash.
+  Find the percentage of ash.
+
+    Set 1·25 on C to 0·04425 on D, and under index on C read 3·54, the
+    required percentage of ash on D.
+
+
+Given the steam pressure P and the diameter _d_ in millimetres, of the
+throat of an injector, to find the weight W, of water delivered in lb.
+per hour from W = (_d_^2√̅P)/(0·505).
+
+Set 0·505 on C to P on A; bring cursor to _d_ on C and index of C to
+cursor. Then under _d_ on C read delivery of water on D.
+
+To find the pressure of wind per square foot, due to a given velocity in
+miles per hour.
+
+Set 1 on B to 2 on A, and over the velocity in miles per hour on D read
+pressure in lb. per square foot on B.
+
+To find the kinetic energy of a moving body.
+
+Set 64·4 on B to velocity in feet per second on D, and over weight of
+body in lb. on B read kinetic energy or accumulated work in foot-lb. on
+A.
+
+
+
+
+                      TRIGONOMETRICAL APPLICATIONS
+
+
+_Scales._—Not the least important feature of the modern slide rule is
+the provision of the special scales on the under-side of the slide, and
+by the use of which, in conjunction with the ordinary scales on the
+rule, a large variety of trigonometrical computations may be readily
+performed.
+
+Three scales will be found on the reverse or under-side of the slide of
+the ordinary Gravêt or Mannheim rule. One of these is the evenly-divided
+scale or scale of equal parts referred to in previous sections, and by
+which, as explained, the decimal parts or mantissæ of logarithms of
+numbers may be obtained. Usually this scale is the centre one of the
+three, but in some rules it will be found occupying the lowest position,
+in which case some little modification of the following instructions
+will be necessary. The requisite transpositions will, however, be
+evident when the purposes of the scales are understood. The upper of the
+three scales, usually distinguished by the letter S, is a scale giving
+the logarithms of the sines of angles, and is used to determine the
+natural sines of angles of from 35 minutes to 90 degrees. The notation
+of this scale will be evident on inspection. The main divisions 1, 2, 3,
+etc., represent the degrees of angles; but the values of the
+subdivisions differ according to their position on the scale. Thus, if
+any primary space is subdivided into 12 parts, each of the latter will
+be read as 5 minutes (5′), since 1° = 60′.
+
+_Sines of Angles._—To find the sine of an angle the slide is placed in
+the groove, with the under-side uppermost, and the end division lines or
+indices on the slide, coinciding with the right and left indices of the
+A scale. Then over the given angle on S is read the value of the sine of
+the angle on A. If the result is found on the left scale of A (1 to 10),
+the logarithmic characteristic is −2; if it is found on the right-hand
+side (10 to 100), it is −1. In other words, results on the right-hand
+scale are prefixed by the decimal point only, while those on the
+left-hand scale are to be preceded by a cypher also. Thus:—
+
+              Sine 2° 40′ = 0·0465; sine 15° 40′ = 0·270.
+
+Multiplication and division of the sines of angles are performed in the
+same manner as ordinary calculations, excepting that the slide has its
+under-face placed uppermost, as just explained. Thus to multiply sine
+15° 40′ by 15, the R.H. index of S is brought to 15 on A, and opposite
+15° 40′ on S is found 4·05 on A. Again, to divide 142 by sine 16° 30′,
+we place 16° 30′ on S to 142 on A, and over R.H. index of S read 500 on
+A.
+
+The rules for the number of integers in the results are thus determined:
+Let N be the number of integers in the multiplier M or in the dividend
+D. Then the number of integers P, in the product or Q, in the quotient
+are as follows:—
+
+ When the result is found to the right of M or D,   │P = N − 2│Q = N
+   and in the same scale                            │         │
+ When the result is found to the right of M or D,   │P = N − 1│Q = N + 1
+   and in the other scale                           │         │
+ When the result is found to the left of M or D, and│P = N − 1│Q = N + 1
+   in the other scale                               │         │
+ When the result is found to the left of M or D, and│P = N    │Q = N + 2
+   in the same scale                                │         │
+
+If the division is of the form (20° 30′)/(50), the result cannot be read
+off directly on the face of the rule. Thus, if in the above example 20°
+30′ on S, is placed to agree with 50 on the right-hand scale of A, the
+result found on S under the R.H. index of A is 44° 30′. The required
+numerical value can then be found: (1) By placing the slide with all
+indices coincident when opposite 44° 30′ on S will be found 0·007 on A;
+or (2) In the ordinary form of rule, by reading off on the scale B
+opposite the index mark in the opening on the under-side of the rule.
+The above rules for the number of integers in the quotient do not apply
+in this case.
+
+If it is required to find the sine of an angle simply, this may be done
+with the slide in its ordinary position, with scale B under A. The given
+angle on scale S is then set to the index on the under-side of the rule,
+and the value of the sine is read off on B under the right index of A.
+
+Owing to the rapidly diminishing differences of the values of the sines
+as the upper end of the scale is approached, the sines of angles between
+60° and 90° cannot be accurately determined in the foregoing manner. It
+is therefore advisable to calculate the value of the sine by means of
+the formula:
+
+Sine θ = 1 − 2 sin^2 (90 − θ)/(2).
+
+To determine the value of sin^2 (90 − θ)/(2). With the slide in the
+normal position, set the value of (90 − θ)/(2). on S to the index on the
+under-side of the rule, and read off the value _x_ on B under the R.H.
+index of A. Without moving the slide find _x_ on A, and read under it on
+B the value required.
+
+
+  EX.—Find value of sine 79° 40′.
+
+  Sine 79° 40′ = 1 − 2sin^2 5° 10′.
+
+  But sine 5° 10′ = 0·0900, and under this value on A is 0·0081 on B.
+  Therefore sine 79° 40′ = 1 − 0·0162 = 0·9838.
+
+
+The sines of very small angles, being very nearly proportional to the
+angles themselves, are found by direct reading. To facilitate this, some
+rules are provided with two marks, one of which, a single accent (′),
+corresponds to the logarithm of (1)/(sine 1′) and is found at the number
+3438. The other mark—a double accent (″)—corresponds to the logarithm of
+(1)/(sine 1″) and is found at the number 206,265. In some rules these
+marks are found on either the A or the B scales; sometimes they are on
+both. In either case the angle on the one scale is placed so as to
+coincide with the significant mark on the other, and the result read off
+on the first-named scale opposite the index of the second.
+
+In sines of angles under 3″, the number of integers in the result is −5;
+while it is −4 for angles from 3″ to 21″; −3 from 21″ to 3′ 27″; and −2
+from 3′ 27″ to 34′ 23″.
+
+
+  EX.—Find sine 6′.
+
+  Placing the significant mark for minutes coincident with 6, the value
+  opposite the index is found to be 175, and by the rule above this is
+  to be read 0·00175. For angles in seconds the other significant mark
+  is used; while angles expressed in minutes and seconds are to be first
+  reduced to seconds. Thus, 3′ 10″ = 190″.
+
+
+_Tangents of Angles._—There remains to be considered the third scale
+found on the back of the slide, and usually distinguished from the
+others by being lettered T. In most of the more recent forms of rule
+this scale is placed near the lower edge of the slide, but in some
+arrangements it is found to be the centre scale of the three. Again, in
+some rules this scale is figured in the same direction as the scale of
+sines—viz., from left to right,—while in others the T scale is reversed.
+In both cases there is now usually an aperture formed in the back of the
+left extremity of the rule, with an index mark similar to that already
+referred to in connection with the scale of sines. Considering what has
+been referred to as the more general arrangement, the method of
+determining the tangents of angles may be thus explained:—
+
+The tangent scale will be found to commence, in some rules, at about
+34′, or, precisely, at the angle whose tangent is 0·01. More usually,
+however, the scale will be found to commence at about 5° 43′, or at the
+angle whose tangent is 0·1. The other extremity of the scale corresponds
+in all cases to 45°, or the angle whose tangent is 1. This explanation
+will suggest the method of using the scale, however it may be arranged.
+If the graduations commence with 34′, the T scale is to be used in
+conjunction with the right and left scales of A; while if they commence
+with 5° 43′ it is to be used in conjunction with the D scale.
+
+In the former case the slide is to be placed in the rule so that the T
+scale is adjacent to the A scales, and, with the right and left indices
+coinciding, when opposite any angle on T will be found its tangent on A.
+From what has been said above, it follows that the tangents read on the
+L.H. scale of A have values extending from 0·01 to 0·1; while those read
+on the R.H. scale of A have values from 0·1 to 1·0. Otherwise expressed,
+to the values of any tangent read on the L.H. scale of A a cypher is to
+be prefixed; while if found on the R.H. scale, it is read directly as a
+decimal.
+
+
+  EX.—Find tan. 3° 50′.
+
+  Placing the slide as directed, the reading on A opposite 3° 50′ on T
+  is found to be 67. As this is found on the L.H. scale of A, it is to
+  be read as 0·067.
+
+
+  EX.—Find tan. 17° 45′.
+
+  Here the reading on A opposite 17° 45′ on T is 32, and as it is found
+  on the R.H. scale of A it is read as 0·32.
+
+
+As in the case of the scale of sines, the tangents may be found without
+reversing the slide, when a fixed index is provided in the back of the
+rule for the T scale.
+
+We revert now to a consideration of those rules in which a single
+tangent scale is provided. It will be understood that in this case the
+slide is placed so that the scale T is adjacent to the D scale, and that
+when the indices of both are placed in agreement, the value of the
+tangent of any angle on T (from 5° 43′ to 45°) may be read off on D, the
+result so found being read as wholly decimal. Thus tan. 13° 20′ is read
+0·237.
+
+If a back index is provided, the slide is used in its normal position,
+when, setting the angle on the tangent scale to this index, the result
+can be read on C over the L.H. index of D.
+
+The tangents of angles above 45° are obtained by the formula: Tan. θ =
+(1)/(tan. (90 − θ)). For all angles from 45° to (90° − 5° 43′) we
+proceed as follows:—Place (90 − θ) on T to the R.H. index of D, and read
+tan. θ on D under the L.H. index of T. The first figure in the value
+thus obtained is to be read as an integer. Thus, to find tan. 71° 20′ we
+place 90° − 71° 20′ = 18° 40′ on T, to the R.H. index of D, and under
+the L.H. index of T read 2·96, the required tangent.
+
+The tangents of angles less than 40′ are sensibly proportional to the
+angles themselves, and as they may therefore be considered as sines,
+their value is determined by the aid of the single and double accent
+marks on the sine scale, as previously explained. The rules for the
+number of integers are the same as for the sines.
+
+Multiplication and division of tangents may be quite readily effected.
+
+
+  EX.—Tan. 21° 50′ × 15 = 6.
+
+  Set L.H. index of T to 15 on D, and under 21° 50′ on T read 6 on D.
+
+
+  EX.—Tan. 72° 40′ × 117 = 375.
+
+  Set (90° − 72° 40′) = 17° 20′ on T to 117 on D, and under R.H. index
+  of T read 375 on D.
+
+
+_Cosines of Angles._—The cosines of angles may be determined by placing
+the scale S with its indices coinciding with those of A, and when
+opposite (90 − θ) on S is read cos. θ on A. If the result is read on the
+L.H. scale of A, a cypher is to be prefixed to the value read; while if
+it is read on the R.H. scale of A, the value is read directly as a
+decimal. Thus, to determine cos. 86° 30′ we find opposite (90° − 86°
+30′) = 3° 30′ on S, 61° on A, and as this is on the L.H. scale the
+result is read 0·061. Again, to find cos. 59° 20′ we read opposite (90°
+− 59° 20′) or 30° 40′ on S, 51 on A, and as this is found on the R.H.
+scale of A, it is read 0·51.
+
+In finding the cosines of small angles it will be seen that direct
+reading on the rule becomes impossible for angles of less than 20°. It
+is advisable in such cases to adopt the method described for determining
+the _sines_ of the _large_ angles of which the complements are sought.
+
+_Cotangents of Angles._—From the methods of finding the tangents of
+angles previously described, it will be apparent that the cotangents of
+angles may also be obtained with equal facility. For angles between 5°
+45′ and 45°, the procedure is the same as that for finding tangents of
+angles greater than 45°. Thus, the angle on scale T is brought to the
+R.H. index of D, and the cotangent read off on D under the L.H. index of
+T. The first figure of the result so found is to be read as an integer.
+
+If the angle (θ) lies between 45° and 84° 15′, the slide is placed so
+that the indices of T coincide with those of D, and the result is then
+read off on D opposite (90 − θ) on T. In this case the value is wholly
+decimal.
+
+_Secants of Angles._—The secants of angles are readily found by bringing
+(90 − θ) on S to the R.H. index of A and reading the result on A over
+the L.H. index of S. If the value is found on the L.H. scale of A, the
+first figure is to be read as an integer; while if the result is read on
+the R.H. scale of A, the first _two_ figures are to be regarded as
+integers.
+
+_Cosecants of Angles._—The cosecants of angles are found by placing the
+angle on S to the R.H. index of A, and reading the value found on A over
+the L.H. index of S. If the result is read on the L.H. scale of A, the
+first figure is to be read as an integer; while if the result is found
+on the R.H. scale of A, the first _two_ figures are to be read as
+integers.
+
+It will be noted that some of the rules here given for determining the
+several trigonometrical functions of angles apply only to those forms of
+rules in which a single scale of tangents T is used, reading from left
+to right. For the other arrangements of the scale, previously referred
+to, some slight modification of the method of procedure in finding the
+tangents and cotangents of angles will be necessary; but as in each case
+the nature and extent of this modification is evident, no further
+directions are required.
+
+
+
+
+                THE SOLUTION OF RIGHT-ANGLED TRIANGLES.
+
+
+From the foregoing explanation of the manner of determining the
+trigonometrical functions of angles, the methods of solving right-angled
+triangles will be readily perceived, and only a few examples need
+therefore be given.
+
+Let _a_ and _b_ represent the sides and _c_ the hypothenuse of a
+right-angled triangle, and _a_° and _b_° the angles opposite to the
+sides. Then of the possible cases we will take
+
+(1.) Given _c_ and _a_°, to find _a_, _b_, and _b_°.
+
+The angle _b_° = 90 − _a_°, while _a_ = _c_ sin _a_° and _b_ = _c_ sin
+_b_°. To find _a_, therefore, the index of S is set to _c_ on A, and the
+value of _a_ read on A opposite _a_° on S. In the same manner the value
+of _b_ is obtained.
+
+
+  EX.—Given in a right-angled triangle _c_ = 9 ft. and _a_° = 30°. Find
+  _a_, _b_, and _b_°.
+
+  The angle _b_° = 90 − 30 = 60°. To find _a_, set R.H. index of S to 9
+  on A, and over 30° on S read _a_ = 4·5 ft. on A. Also, with the slide
+  in the same position, read _b_ = 7·8 ft. [7·794] on A over 60° on S.
+
+
+(2.) Given _a_ and _c_, to determine _a_°, _b_°, and _b_.
+
+In this case advantage is taken of the fact that in every triangle the
+sides are proportional to the sines of the opposite angles. Therefore,
+as in this case the hypothenuse c subtends a right angle, of which the
+sine = 1, the R.H. index (or 90°) on S is set to the length of _c_ on A,
+when under _a_ on A is found _a_° on S. Hence _b_° and _b_ may be
+determined.
+
+(3.) Given _a_ and _a_°, to find _b_, _c_, and _b_°.
+
+Here _b_° = (90 − _a_°), and the solution is similar to the foregoing.
+
+(4.) Given _a_ and _b_, to find _a_°, _b_°, and _c_.
+
+To find _a_°, we have tan. _a_° = _a_/_b_, which in the above example
+will be (4·5)/(7·8) = 0·577. Therefore, placing the slide so that the
+indices of T coincide with those of D, we read opposite 0·577 on D the
+value of _a_° = 30°. The hypothenuse _c_ is readily obtained from _c_ =
+_a_/(sin _a_°).
+
+
+
+
+               THE SOLUTION OF OBLIQUE-ANGLED TRIANGLES.
+
+
+Using the same letters as before to designate the three sides and the
+subtending angles of oblique-angled triangles, we have the following
+cases:—
+
+(1.) Given one side and two angles, as _a_, _a_°, and _b_°, to find _b_,
+_c_, and _c_°.
+
+In the first place, _c_° = 180° − (_a_° + _b_°); also we note that, as
+the sides are proportional to the sines of the opposite angles, _b_ =
+(_a_ sine _b_°)/(sine _a_°) and _c_ = (_a_ sine _c_°)/(sine _a_°).
+
+Taking as an example, _a_ = 45, _a_° = 57°, and _b_° = 63°, we have _c_°
+= 180 − (57 + 63) = 60°. To find _b_ and _c_, set _a_° on S to _a_ on A,
+and read off on A above 63° and 60° the values of _b_ (= 47·8) and _c_
+(= 46·4) respectively.
+
+(2.) Given _a_, _b_, and _a_°, to find _b_°, _c_°, and _c_.
+
+In this case the angle _a_° on S is placed under the length of side _a_
+on A and under _b_ on A is found the angle _b_° on S. The angle _c_° =
+180 − (_a_° + _b_°), whence the length _c_ can be read off on A over
+_c_° on S.
+
+(3.) Given the sides and the included angle, to find the other side and
+the remaining angles.
+
+If, for example, there are given _a_ = 65, _b_ = 42, and the included
+angle _c_° = 55°, we have (_a_ + _b_) ∶ (_a_ − _b_) = tan. (_a_° +
+_b_°)/(2) ∶ tan. (_a_° − _b_°)/(2). Then, since _a_° + _b_° = 180° − 55°
+= 125°, it follows that (_a_° + _b_°)/(2) = (125°)/(2) = 62° 30′.
+
+By the rule for tangents of angles greater than 45°, we find tan. 62°
+30′ = 1·92. Inserting in the above proportion the values thus found, we
+have 107 ∶ 23 = 1·92 ∶ tan. (_a_° − _b_°)/(2). From this it is found
+that the value of the tangent is 0·412, and placing the slide with all
+indices coinciding, it is seen that this value on D corresponds to an
+angle of 22° 25′. Therefore, since (_a_° + _b_°)/(2) = 62° 30′, and
+(_a_° − _b_°)/(2) = 22° 25′, it follows that _a_° = 84° 55′, and _b_° =
+40° 5′. Finally, to determine the side _c_, we have _c_ = (_a_ sin
+_c_°)/(sin _a_°) as before.
+
+
+                PRACTICAL TRIGONOMETRICAL APPLICATIONS.
+
+A few examples illustrative of the application of the methods of
+determining the functions of angles, etc., described in the preceding
+section, will now be given.
+
+To find the chord of an arc, having given the included angle and the
+radius.
+
+With the slide placed in the rule with the C and D scales outward, bring
+one-half of the given angle on S to the index mark in the back of the
+rule, and read the chord on B under twice the radius on A.
+
+
+  EX.—Required the chord of an arc of 15°, the radius being 23 in.
+
+  Set 7° 30′ on S to the index mark in the back of the rule, and under
+  46 on A read 6 in., the required length of chord on B.
+
+
+To find the area of a triangle, given two sides and the included angle.
+
+Set the angle on S to the index mark on the back of the rule, and bring
+cursor to 2 on B. Then bring the length of one side on B to cursor,
+cursor to 1 on B, the length of the other side on B to cursor, and read
+area on B under index of A.
+
+
+  EX.—The sides of a triangle are 5 and 6 ft. in length respectively,
+  and they include an angle of 20°. Find the area.
+
+  Set 20 on S to index mark, bring cursor to 2 on B, 5 on B to cursor,
+  cursor to 1 on B, 6 on B to cursor, and under 1 on A read the area =
+  5·13 sq. ft. on B.
+
+
+To find the number of degrees in a gradient, given the rise per cent.
+
+Place the slide with the indices of T coincident with those of D, and
+over the rate per cent. on D read number of degrees in the slope on T.
+
+As the arrangement of rule we have chiefly considered has only a single
+T scale, it will be seen that only solutions of the above problem
+involving slopes between 10 and 100 per cent. can be directly read off.
+For smaller angles, one of the formulæ for the determination of the
+tangents of submultiple angles must be used.
+
+In rules having a double T scale (which is used with the A scale) the
+value in degrees of any slope from 1 to 100 per cent. can be directly
+read off on A.
+
+To find the number of degrees, when the gradient is expressed as 1 in
+_x_.
+
+Place the index of T to _x_ on D, and over index of D read the required
+angle in degrees on T.
+
+
+  EX.—Find the number of degrees in a gradient of 1 in 3·8.
+
+  Set 1 on T to 3·8 on D, and over R.H. index of D read 14° 45′ on T.
+
+
+Given the lap, the lead and the travel of an engine slide valve, to find
+the angle of advance.
+
+Set (lap + lead) on B to half the travel of the valve on A, and read the
+angle of advance on S at the index mark on the back of the rule.
+
+
+  EX.—Valve travel 4½in., lap 1 in., lead ⁵⁄₁₆in. Find angle of advance.
+
+  Set 1⁵⁄₁₆ = 1·312 on B to 2·25 on A, and read 35° 40′ on S opposite
+  the index on the back of the rule.
+
+
+Given the angular advance θ, the lap and the travel of a slide valve, to
+find the cut-off in percentage of the stroke.
+
+Place the lap on B to half the travel of valve on A, and read on S the
+angle (the supplement of the _angle of the eccentric_) found opposite
+the index in the back of the rule. To this angle, add the angle of
+advance and deduct the sum from 180°, thus obtaining the _angle of the
+crank_ at the point of cut-off. To the cosine of the supplement of this
+angle, add 1 and multiply the result by 50, obtaining the percentage of
+stroke completed when cut-off occurs.
+
+
+  EX.—Given the angular advance = 35° 40′, the valve travel = 4½in., and
+  the lap = 1 in., find the angle of the crank at cut-off and the
+  admission period expressed as a percentage of the stroke.
+
+  Set 1 on B to 2·25 on A, and read off on S opposite the index, the
+  supplement of the angle of the eccentric = 26° 20′. Then 180° − (35°
+  40′ + 26° 20′) = 118° = the crank angle at the point of cut-off.
+  Further, cos. 118° = cos. 62° = sin (90° − 62°) = sin 28°, and placing
+  28° on S to the back index, the cosine, read on B under R.H. index of
+  A, is found to be 0·469. Adding 1 and placing the L.H. index of C to
+  the result, 1·469, on D, we read off under 50 on C, the required
+  period of admission = 73·4 per cent. on D.
+
+
+The trigonometrical scales are useful for evaluating certain formulæ.
+Thus in the following expressions, if we find the angle _a_ such that
+sin. _a_ = _k_, we can write:—
+
+  (_k_)/(√1 − _k^2_) = tan. _a_; (√1 − _k^2_)/(_k_) = cot. _a_; √(1 −
+                        _k^2_) = cos. _a_; etc.
+
+In the first expression, take _k_ = 0·298. Place the slide with the sine
+scale outward and with its indices agreeing with the indices of the
+rule. Set the cursor to 0·298 on the (R.H.) scale of A, and read 17° 20′
+on the sine scale as the angle required. Then under 17° 20′ on the
+tangent scale, read 0·312 on D as the result.
+
+
+                   SLIDE RULES WITH LOG.-LOG. SCALES.
+
+For occasional requirements, the method described on page 45 of
+determining powers and roots other than the square and cube, is quite
+satisfactory. When, however, a number of such calculations are to be
+made, the process may be simplified considerably by the use of what are
+known as _log.-log._, _logo-log._, or _logometric_ scales, in
+conjunction with the ordinary scales of the rule. The principle involved
+will be understood from a consideration of those rules for logarithmic
+computation (page 8) which refer to powers and roots. From these it is
+seen that while for the multiplication and division of numbers we _add_
+their logarithms, for involution and evolution we require to _multiply_
+or _divide_ the logarithms of the numbers by the exponent of the power
+or root as the case may be. Thus to find 3^{2.3}, we have (log. 3) × 2·3
+= log. _x_, and by the ordinary method described on page 45 we should
+determine log. 3 by the aid of the scale L on the back of the slide,
+multiply this by 2·3 by using the C and D scales in the usual manner,
+transfer the result to scale L, and read the value of _x_ on D under 1
+on C. By the simpler method, first proposed by Dr. P. M. Roget,[8] the
+multiplication of log. 3 by 2·3 is effected in the same way as with any
+two ordinary factors—_i.e._, by adding their logarithms and finding the
+number corresponding to the resulting logarithm. In this case we have
+log. (log. 3) + log. 2·3 = log. (log. _x_). The first of the three terms
+is obviously the _logarithm of the logarithm_ of 3, the second is the
+simple logarithm of 2·3, and the third the _logarithm of the logarithm
+of_ the answer. Hence, if we have a scale so graduated that the
+distances from the point of origin represent the logarithms of the
+logarithms (the log.-logs.) of the numbers engraved upon it, then by
+using this in conjunction with the ordinary scale of logarithms, we can
+effect the required multiplication in a manner which is both expeditious
+and convenient. Slightly varying arrangements of the log.-log. scale,
+sometimes referred to as the “P line,” have been introduced from time to
+time, but latterly the increasing use of exponential formulæ in
+thermodynamic, electrical, and physical calculations has led to a
+revival of interest in Dr. Roget’s invention, and various arrangements
+of rules with log.-log. scales are now available.
+
+_The Davis Log.-Log. Rule._—In the rule introduced by Messrs. John Davis
+& Son Limited, Derby, the log.-log. scales are placed upon a separate
+slide—a plan which has the advantage of leaving the rule intact for all
+ordinary purposes, while providing a length of 40 in. for the log.-log.
+scales.
+
+In the 10 in. Davis rule one face of the slide, marked E, has two
+log.-log. scales for numbers greater than unity, the lower extending
+from 1·07 to 2, and the upper continuing the graduations from 2 to 1000.
+On the reverse face of the slide, marked -E, are two log.-log. scales
+for numbers less than unity, the upper extending from 0·001 to 0·5, and
+the lower continuing the graduations from 0·5 to 0·933. Both sets of
+scales are used in conjunction _with the lower or D scale of the rule_,
+which is to be primarily regarded as running from 1 to 10, and
+constitutes a scale of exponents. In the 20 in. rule the log.-log.
+scales are more extensive, and are used in conjunction with the upper or
+A scale of the rule (1 to 100); in what follows, however, the 10 in.
+rule is more particularly referred to.
+
+It has been explained that on the log.-log. scale the distance of any
+numbered graduation from the point of origin represents the log.-log. of
+the number. The point of origin will obviously be that graduation whose
+log.-log. = 0. This is seen to be 10, since log. (log. 10) = log. 1 = 0.
+Hence, confining attention to the E scale, to locate the graduation 20,
+we have log. (log. 20) = log. 1·301 = 0·11397, so that if the scale D is
+25 cm. long, the distance between 10 and 20 on the corresponding
+log.-log. scale would be 113·97 ÷ 4 = 28·49 mm. For numbers less than 10
+the resulting log.-logs. will be negative, and the distances will be
+spaced off from the point of origin in a negative direction—_i.e._, from
+right to left. Thus, to locate the graduation 5, we have
+
+   log. (log. 5) = log. 0·699 = ̅1·844; _i.e._, −1 + 0·844 or −0·156;
+
+so that the graduation marked 5 would be placed 156 ÷ 4 = 39 mm. distant
+from 10 in a _negative_ direction, and proceeding in a similar manner,
+the scale may be extended in either direction. In the -E scale, the
+notation runs in the reverse direction to that of the E scale, but in
+all other respects it is precisely analogous, the distance from the
+point of origin (0·1 in this case) to any graduation _x_ representing
+log. [-log. _x_.]. It follows that of the similarly situated graduations
+on the two scales, those on the -E scale are the _reciprocals_ of those
+on the E scale. This may be readily verified by setting, say, 10 on E to
+(R.H.) 1 on D, when turning to the back of the rule we find 0·1 on -E
+agreeing with the index mark in the aperture at the right-hand extremity
+of the rule.
+
+In using the log.-log. scales it is important to observe (1) that the
+values engraved on the scale are definite and unalterable (_e.g._, 1·2
+can only be read as 1·2 and not as 120, 0·0012, etc., as with the
+ordinary scales); (2) that the upper portion of each scale should be
+regarded as forming a prolongation to the right of the lower portion;
+and (3) that immediately above any value on the lower portion of the
+scale is found the 10th power of that value on the upper portion of the
+scale. Keeping these points in view, if we set 1·1 on E to 1 on D we
+find over 2 on D the value of 1·1^2 = 1·21 on E. Similarly, over 3 we
+find 1·1^3 = 1·331, and so on. Then, reading across the slide, we have,
+over 2, the value of 1·1^{2 × 10} = 1·1^{20} = 6·73, and over 3 we have
+1·1^{3 × 10} = 1·1^{30} = 17·4. Hence the rule:—_To find the value of
+x^n, set x on E to 1 on D, and over n on D read x^n on E._
+
+With the slide set as above, the 8th, 9th, etc., powers of 1·1 cannot be
+read off; but it is seen that, according to (2) in the foregoing, the
+missing portion of the E scale is that part of the upper scale (2 to
+about 2·6) which is outside the rule to the left. Hence placing 1·1 to
+10 on D, the 8th, 9th, etc., powers of 1·1 will be read off _on the
+upper part_ of the E scale. In general, then,
+
+If _x_ on the _lower_ line is set to 1 on D, then _x^n_ is read directly
+on that line and _x_^{10_n_} on the upper line.
+
+If _x_ on the _upper_ line is set to 1 on D, then _x^n_ is read directly
+on that line and _x_^{_ⁿ⁄₁₀_} on the lower line.
+
+If _x_ on the _lower_ line is set to 10 on D, then _x_^{_ⁿ⁄₁₀_} is read
+directly on that line and _x^n_ on the upper line.
+
+If _x_ on the _upper_ line is set to 10 on D, then _x_^{_ⁿ⁄₁₀_} is read
+directly on that line and _x_^{_ⁿ⁄₁₀₀_} on the lower line.
+
+These rules are conveniently exhibited in the accompanying diagram (Fig.
+14). They are equally applicable to both the E and -E scales of the 10
+in. rule, and include practically all the instruction required for
+determining the _n_th power or the _n_th root of a number. They do not
+apply directly to the 20 in. rule, however, for here the relation of the
+lower and upper scales will be _x^n_ and _x_^{100_n_}.
+
+
+  EX.—Find 1·167^{2·56}.
+
+  Set 1·167 on E to 1 on D, and over 2·56 on D read 1·485 on E.
+
+
+  EX.—Find 4·6^{1·61}.
+
+  Set 4·6 on upper E scale to 1 on D, and over 1·61 on D read 11·7
+  (11·67) on E.
+
+
+  EX.—Find 1·4^{0·27} and 1·4^{2·7}.
+
+  Set 1·4 on E to 10 on D, and over 2·7 on D read 1·095 = 1·4^{0·27} on
+  lower E scale and 2·48 = 1·4^{2·7} on upper E scale.
+
+
+[Illustration: FIG. 14.]
+
+
+  EX.—Find 46^{0·0184} and 46^{0·184}.
+
+  Set 46 on upper E scale to 10 on D, and over 1·84 on D read 1·073 on
+  lower E scale and 2·022 (2·0228) on upper E scale.
+
+
+  EX.—Find 0·074^{1·15}.
+
+  Using the -E scale, set 0·074 to 1 on D, and over 1·15 on D read 0·05
+  on -E.
+
+
+The method of determining the root of a number will be obvious from the
+preceding examples.
+
+
+  EX.—Find ^{1.4}√(17) and ^{14}√(17).
+
+  Set 17 on E to 1·4 on D, and over 1 on D read 7·56 on upper E scale
+  and 1·224 on lower E scale.
+
+  EX.—Find ^{0·031}√(0·914).
+
+  Set 0·914 on -E to 3·1 on D, and over 10 on D read 0·055 on upper -E
+  scale.
+
+
+When the exponent _n_ is fractional, it is often possible to obtain the
+result directly with one setting of the slide. Thus to determine
+1·135^{¹⁷⁄₁₆} by the first method we find ¹⁷⁄₁₆ = 1·0625, and placing
+1·135 on E to 1 on D, read 1·144 on E over 1·0625 on D. By the direct
+method we place 1·135 on the E scale on 1·6 on D, and over 1·7 on D read
+1·144 on E. It will be seen that since the scale D is assumed to run
+from 1 to 10 we are unable to read 16 and 17 on this scale; but it is
+obvious that the _ratios_ (1·7)/(1·6) and (17)/(16) are identical, and
+it is with the ratio only that we are, in effect, concerned.
+
+Since an expression of the form _x_^{-_n_} = (1)/(_x^n_) or
+((1)/(_x_))^{_n_}, the required value may be obtained by first
+determining the reciprocal of _x_ and proceeding as before. By using
+both the direct and reciprocal log.-log. scales (E and -E) in
+conjunction however, the required value can be read directly from the
+rule, and the preliminary calculation entirely avoided. In the Davis
+form of rule, the result can be read on the -E scale, used in
+conjunction with the D scale of the rule, _x_ on E being set to the
+index mark in the aperture in the back of the rule.
+
+
+  EX.—Find the value of 1·195^{−1·65}.
+
+  Set 1·195 on E to the index in the left aperture in the back of the
+  rule, and over 1·65 on D read 0·745 on the -E scale.
+
+
+It may be noted in passing that the log.-log. scale affords a simple
+means for determining the logarithm or anti-logarithm of a number to any
+base. For this purpose it is necessary to set the base of the given
+system on E to 1 on D, when _under_ any number on E will be found its
+logarithm on D. Thus, for common logs., we set the base 10 on E to 1 on
+D, and under 100 we find 2, the required log. Similarly we read log. 20
+= 1·301; log. 55 = 1·74; log. 550 = 2·74, etc. Reading reversely, over
+1·38 on D we find its antilog. 24 on E; also antilog. 1·58 = 38;
+antilog. 1·19 = 15·5, etc.
+
+For logs. of numbers under 10 we set the base 10 to 10 on D; hence the
+readings on D will be read as one-tenth their apparent value. Thus log.
+3 = 0·477; log. 5·25 = 0·72; antilog. 0·415 = 2·6; antilog. 0·525 =
+3.·35, etc.
+
+The logs. of the numbers on the lower half of the E scale will also be
+found on the D scale; but a consideration of Fig. 14 will show that this
+will be read as _one-tenth_ its face value if the base is set to 1 on D,
+and as _one-hundredth_ if the base is set to 10.
+
+For natural, hyperbolic, or Napierian logarithms, the base is 2·718. A
+special line marked ε or _e_ serves to locate the exact position of this
+value on the E scale, and placing this to 1 on D we read log._{_e_} 4·35
+= 1·47; log._{_e_} 7·4 = 2·0; antilog._{_e_} × 2·89 = 18, etc. The other
+parts of the scale are read as already described for common logs.
+Calculations involving powers of _e_ are frequently met with, and these
+are facilitated by using the special graduation line referred to, as
+will be readily understood.
+
+If it is required to determine the power or root of a number which does
+not appear on either of the log.-log. scales, we may break up the number
+into factors. Usually it is convenient to make one of the factors a
+power of 10.
+
+
+  EX.—3950^{1·97} = 3·95^{1·97} × 10^{3 × 1·97} = 3·95^{1·97} ×
+  10^{5·91}.
+
+  Then 3·95^{1·97} = 15, and 10^{5·91} (or antilog.) 5·91 = 812,000.
+  Hence, 15 × 812,000 = 12,180,000 is the result sought.
+
+
+Numbers which are to be found in the higher part of the log.-log. scale
+may often be factorised in this way, and greater accuracy obtained than
+by direct reading.
+
+The form of log.-log. rule which has been mainly dealt with in the
+foregoing gives a scale of comparatively long range, and the only
+objection to the arrangement adopted is the use of a separate slide.
+
+_The Jackson-Davis Double Slide Rule._—In this instrument a pair of
+aluminium clips enable the log.-log. slide to be temporarily attached to
+the lower edge of the ordinary rule, and used, by means of a special
+cursor, in conjunction with the C scale of the ordinary slide. In this
+way both the log.-log. and ordinary scales are available without the
+trouble of replacing one slide by the other. Since the scale of
+exponents is now on the slide, the value of _x^n_ will be obtained by
+setting 1 on C to _x_ on E and reading the result on E under _n_ on C.
+
+By using a pair of log.-log. slides, one in the rule and one clamped to
+the edge by the clips, we have an arrangement which is very useful in
+deducing empirical formulæ of the type _y_ = _x^n_.
+
+_The Yokota Slide Rule._—In this instrument the log.-log. scales are
+placed on the face of the rule, each set comprising three lines. These,
+for numbers greater than 1, are found above the A scale while the three
+reciprocal log.-log. lines are below the D scale. Both sets are used in
+conjunction with the C scale on the slide. Other features of this rule
+are:—The ordinary scales are 10 in. long instead of 25 cm. as hitherto
+usual; hence the logarithms of numbers can be read on the ordinary scale
+of inches on the edge of the rule. There is a scale of cubes in the
+centre of the slide and on the back of the slide there is a scale of
+secants in addition to the sine and tangent scales.
+
+[Illustration: FIG. 15.]
+
+_The Faber Log.-log. Rule._—In this instrument shown in Fig. 15, the two
+log.-log. scales are placed on the face of the rule. One section,
+extending from 1·1 to 2·9, is placed above the A scale, and the other
+section, extending from 2·9 to 100,000, is placed below the D scale.
+These scales are used in conjunction with the C scale of the slide in
+the manner previously described. The width of the rule is increased
+slightly, but the arrangement is more convenient than that formerly
+employed, wherein the log.-log. scales were placed on the bevelled edge
+of the rule and read by a tongue projecting from the cursor.
+
+[Illustration: FIG. 16.]
+
+Another novel feature of this rule is the provision of two special
+scales at the bottom of the groove, to which a bevelled metal index or
+marker on the left end of the slide can be set. The upper of these
+scales is for determining the efficiency of dynamos and electric motors;
+the lower for determining the loss of potential in an electric circuit.
+
+_The Perry Log.-log. Rule._—In this rule, introduced by Messrs. A. G.
+Thornton, Limited, Manchester, the log.-log. scales are arranged as in
+Fig. 16, the E scale, running from 1·1 to 10,000, being placed above the
+A scale of the rule, and the -E or E^{−1} scale running from 0·93 to
+0·0001, below the D scale of the rule. These scales are read in
+conjunction with the B scales on the slide by the aid of the cursor.
+
+The following tabular statement embodies all the instructions required
+for using this form of log.-log. slide rule:—
+
+                       When _x_ is greater than 1.
+
+ _x^n_        Set 1 on B to _x_ on E; over _n_ on B read _x^n_ on E
+ _x_^{-_n_}   Set 1 on B to _x_ on E; under _n_ on B read _x_^{-_n_} on
+                E^{−1}
+ _x_^{_ⁱ⁄ₙ_}  Set _n_ on B to _x_ on E; over 1 on B read _x_^{_ⁱ⁄ₙ_} on
+                E
+ _x_^{_⁻ⁱ⁄ₙ_} Set _n_ on B to _x_ on E; under 1 on B read _x_^{_⁻ⁱ⁄ₙ_}
+                on E^{−1}
+
+                        When _x_ is less than 1.
+
+ _x^n_        Set 1 on B to _x_ on E^{−1}; under _n_ on B read _x^n_ on
+                E^{−1}
+ _x_^{-_n_}   Set 1 on B to _x_ on E^{−1}; over _n_ on B read _x_^{-_n_}
+                on E
+ _x_^{_ⁱ⁄ₙ_}  Set _n_ on B to _x_ on E^{−1}; under 1 on B read
+                _x_^{_ⁱ⁄ₙ_} on E^{−1}
+ _x_^{_⁻ⁱ⁄ₙ_} Set _n_ on B to _x_ on E^{−1}; over 1 on B read
+                _x_^{_⁻ⁱ⁄ₙ_} on E
+
+If 10 on B is used in place of 1 on B, read _x_^{_ⁿ⁄₁₀_} in place of
+_x^n_ on E, and _x_^{-_ⁿ⁄₁₀_} in place of _x_^{-_n_} on E^{−1}. If 100
+on B is used, these readings are to be taken as _x_^{_ⁿ⁄₁₀₀_} and
+_x_^{-_ⁿ⁄₁₀₀_} respectively.
+
+In rules with no -E scale the value of _x_^{-_n_} is obtained by the
+usual rules for reciprocals. We may either determine _x^n_ and find its
+reciprocal or, first find the reciprocal of _x_ and raise it to the
+_n_th power. The first method should be followed when the number _x_ is
+found on the E scale.
+
+
+  EX.—3·45^{−1·82} = 0·105.
+
+  Set 1 on C to 3·45 on E, and under 1·82 on C read 9·51 on C. Then set
+  1 on B to 9·5 on A, and under index of A read 0·105 on B.
+
+
+When _x_ is less than 1 the second method is more suitable.
+
+
+  EX.—0·23^{−1·77} = ((1)/(0·23))^{1·77} = 4·35^{1·77} = 13·5
+
+  Set 1 on B to 0·23 on A, and under index of A read (1)/(0·23) = 4·35
+  on B.
+
+  Set 1 on C to 4·35 on E, and under 1·77 on C read 13·5 on E.
+
+
+As with the Davis rule, the exponent scale C will be read as ⅒th its
+face value if its R.H. index (10) is used in place of 1.
+
+
+                     SPECIAL TYPES OF SLIDE RULES.
+
+In addition, to the new forms of log.-log. slide rules previously
+described, several other arrangements have been recently introduced,
+notably a series by Mr. A. Nestler, of Lahr (London: A. Fastlinger, Snow
+Hill). These comprise the “Rietz,” the “Precision,” the “Universal,” and
+the “Fix” slide rules.
+
+THE RIETZ RULE.—In this rule the usual scales A, B, C, and D, are
+provided, while at the upper edge is a scale, which, being three times
+the range of the D scale, enables cubes and cube roots to be directly
+evaluated and also _n_^{³⁄₂} and _n_^⅔.
+
+A scale at the lower edge of the rule gives the mantissa of the
+logarithms of the numbers on D.
+
+THE PRECISION SLIDE RULE.—In this rule the scales are so arranged that
+the accuracy of a 20 in. rule is obtainable in a length of 10 in. This
+is effected by dividing a 20 in. (50 cm.) scale length into two parts
+and placing these on the working edges of the rule and slide. On the
+upper and lower margins of the face of the rule are the two parts of
+what corresponds to the A scale in the ordinary rule; while in the
+centre of the slide is the scale of logarithms which, used in
+conjunction with the 50 cm. scales on the slide, is virtually twice the
+length of that ordinarily obtainable in a 10 in. rule. The same remark
+applies to the trigonometrical scales on the under face of the slide.
+Both the sine and tangent scales are in two adjacent lengths, while on
+the edge of the stock of the rule, below the cursor groove, is a scale
+of sines of small angles from 1° 49′ to 5° 44′. This is referred to the
+50 cm. scales by an index projection on the cursor.
+
+If C and C′ are the two parts of the scale on the slide and D and D′ the
+corresponding scales on the rule, it is clear that in multiplying two
+factors 1 on C can only be set directly to the upper scale D; while 10
+on C′ can only be set directly to the lower scale D′. Hence if the first
+factor is greater than about 3·2, the cursor must be used to bring 1 on
+C to the first factor on D′. Similarly, in division, numerators and
+denominators which occur on C and D′ or on C′ and D cannot be placed in
+direct coincidence but must be set by the aid of the cursor.
+
+Any uncertainty in reading the result can be avoided by observing the
+following rule: _If in setting the index_ (1 _or_ 10) _in
+multiplication, or in setting the numerator to the denominator in
+division, it is necessary to cross the slide, then it will also be
+necessary to cross the slide to read the product or quotient._
+
+THE UNIVERSAL SLIDE RULE.—In this instrument the stock carries two
+similar scales running from 1 to 10, to which the slide can be set.
+Above the upper one is the logarithm scale and under the lower one the
+scale of squares 1 to 100. On the edge of the stock of the rule, under
+the cursor groove, is a scale running from 1 to 1000. An index
+projecting from the cursor enables this scale to be used with the scales
+on the face of the rule, giving cubes, cube roots, etc.
+
+On the slide, the lower scale is an ordinary scale, 1 to 10. The centre
+scale is the first part of a scale giving the values of sin _n_ cos _n_,
+this scale being continued along the upper edge of the slide (marked
+“sin-cos”) up to the graduation 50. On the remainder of this line is a
+scale running from right to left (0 to 50) and giving the value of
+cos^2_n_. In surveying, these scales greatly facilitate the calculations
+for the horizontal distance between the observer’s station and any
+point, and the difference in height of these two points.
+
+On the back of the slide are scales for the sines and tangents of
+angles. The values of the sines and tangents of angles from 34′ to 5°
+44′ differ little from one another, and the one centre scale suffices
+for both functions of these small angles.
+
+THE FIX SLIDE RULE.—This is a standard rule in all respects, except that
+the A scale is displaced by a distance (π)/(4) so that over 1 on D is
+found 0·7854 on A. This enables calculations relating to the area and
+cubic contents of cylinders to be determined very readily.
+
+THE BEGHIN SLIDE RULE.—We have seen that a disadvantage attending the
+use of the ordinary C and D scales, is that it is occasionally necessary
+to traverse the slide through its own length in order to change the
+indices or to bring other parts of the slide into a readable position
+with regard to the stock. To obviate this disadvantage, Tserepachinsky
+devised an ingenious arrangement which has since been used in various
+rules, notably in the Beghin slide rule made by Messrs. Tavernier-Gravêt
+of Paris. In this rule the C and D scales are used as in the standard
+rule, but in place of the A and B scales, we have another pair of C and
+D scales, displaced by one-half the length of the rule. The lower pair
+of scales may therefore be regarded as running from 10^{_n_} to 10^{_n_
++ 1}, and the upper pair as running from √(10) × 10^{_n_} to √(10) ×
+10^{_n_ + 1}. With this arrangement, _without moving the slide more than
+half its length_, to the left or right, it is always possible to compare
+_all values between_ 1 _and_ 10 _on the two scales_. This is a great
+advantage especially in continuous working.
+
+Another commendable feature of the Beghin rule is the presence of a
+reversed C scale in the centre of the slide, thus enabling such
+calculations as _a_ × _b_ × _c_ to be made with one setting of the
+slide. On the back of the slide are three scales, the lowest of which,
+used with the D scale, is a scale of squares (corresponding to the
+ordinary B scale), while on the upper edge is a scale of sines from 5°
+44′ to 90°, and in the centre, a scale of tangents from 5° 43′ to 45°.
+On the square edge of the stock, under the cursor groove, is the
+logarithm scale, while on the same edge, above the cursor groove, are a
+series of gauge points. All these values are referred to the face scales
+by index marks on the cursor.
+
+THE ANDERSON SLIDE RULE.—The principle of dividing a long scale into
+sections as in the Precision rule, has been extended in the Anderson
+slide rule made by Messrs. Casella & Co., London, and shown in Fig. 17.
+In this the slide carries a scale in four sections, used in conjunction
+with an exactly similar set of scale-lines in the upper part of the
+stock. On the lower part of the stock is a scale in eight sections
+giving the square roots of the upper values. In order to set the index
+of the slide to values in the stock, two indices of transparent
+celluloid are fixed to the slide extending over the face of the rule as
+shown in the illustration. As each scale section is 30 cm. in length,
+the upper lines correspond to a single scale of nearly 4 ft., and the
+lower set to one of nearly 8 ft. in length, giving a correspondingly
+large increase in the number of subdivisions of these scales, and
+consequently much greater accuracy.
+
+In order to decide upon which line a result is to be found, sets of
+“line numbers” are marked at each end of the rule and slide and also on
+the metal frame of the cursor. In multiplication, the line number of the
+product is the sum of the line numbers of the factors if the left index
+is used, or 1 more than this sum if the right index is used. The
+illustration shows the multiplication of 2 by 4. The left index is set
+to 2 (line number, 1), and the cursor set to 4 on the slide (line
+number, 2); hence, as the left index is used, the result is found on
+line No. 3. Similar rules are readily established for division. The
+column of line numbers headed 0 is used for units, that headed 4 for
+tens, and so on; one column is given for tenths, headed −4. The square
+root scale bears similar line numbers, so that the square root of any
+value on the upper scales is found on the correspondingly figured line
+below.
+
+[Illustration: FIG. 17.]
+
+THE MULTIPLEX SLIDE RULE differs from the ordinary form of rule in the
+arrangement of the B scale. The right-hand section of this scale runs
+from left to right as ordinarily arranged, but the left-hand section
+runs in the reverse direction, and so furnishes a reciprocal scale. At
+the bottom of the groove, under the slide, there is a scale running from
+1 to 1000, which is used in conjunction with the D scale, readings being
+referred thereto by a metal index on the end of the slide. By this means
+cubes, cube roots, etc., can be read off directly. Messrs. Eugene
+Dietzgen & Co., New York, are the makers.
+
+THE “LONG” SLIDE RULE has one scale in two sections along the upper and
+lower parts of the stock, as in the “Precision” rule. The scale on the
+slide is similarly divided, but the graduations run in the reverse
+direction, corresponding to an inverted slide. Hence the rules for
+multiplication and division are the reverse of those usually followed
+(page 30). On the back of the slide is a single scale 1–10, and a scale
+1–1000, giving cubes of this single scale. By using the first in
+conjunction with the scales on the stock, squares may be read, while in
+conjunction with the cube scale, various expressions involving squares,
+cubes and their roots may be evaluated.
+
+HALL’S NAUTICAL SLIDE RULE consists of two slides fitting in grooves in
+the stock, and provided with eight scales, two on each slide, and one on
+each edge of each groove. While fulfilling the purposes of an ordinary
+slide rule, it is of especial service to the practical navigator in
+connection with such problems as the “reduction of an ex-meridian sight”
+and the “correction of chronometer sights for error in latitude.” The
+rule, which has many other applications of a similar character, is made
+by Mr. J. H. Steward, Strand, London.
+
+
+                         LONG-SCALE SLIDE RULES
+
+It has been shown that the degree of accuracy attainable in slide-rule
+calculations depends upon the length of scale employed. Considerations
+of general convenience, however, render simple straight-scale rules of
+more than 20 in. in length inadmissible, so that inventors of long-scale
+slide rules, in order to obtain a high degree of precision, combined
+with convenience in operation, have been compelled to modify the
+arrangement of scales usually employed. The principal methods adopted
+may be classed under three varieties: (1) The use of a long scale in
+sectional lengths, as in Hannyngton’s Extended Slide Rule and Thacher’s
+Calculating Instrument; (2) the employment of a long scale laid in
+spiral form upon a disc, as in Fearnley’s Universal Calculator and
+Schuerman’s Calculating Instrument; and (3) the adoption of a long scale
+wound helically upon a cylinder, of which Fuller’s and the “R.H.S.”
+Calculating Rules are examples.
+
+FULLER’S CALCULATING RULE.—This instrument, which is shown in Fig. 18,
+consists of a cylinder _d_ capable of being moved up and down and around
+the cylindrical stock _f_, which is held by the handle. The logarithmic
+scale-line is arranged in the form of a helix upon the surface of the
+cylinder _d_, and as it is equivalent to a straight scale of 500 inches,
+or 41 ft. 8 in., it is possible to obtain four, and frequently five,
+figures in a result.
+
+Upon reference to the figure it will be seen that three indices are
+employed. Of these, that lettered _b_ is fixed to the handle; while two
+others, _c_ and _a_ (whose distance apart is equal to the axial length
+of the complete helix), are fixed to the innermost cylinder _g_. This
+latter cylinder slides telescopically in the stock _f_, enabling the
+indices to be placed in any required position relatively to _d_. Two
+other scales are provided, one (_m_) at the upper end of the cylinder
+_d_, and the other (_n_) on the movable index.
+
+[Illustration: FIG. 18.]
+
+In using the instrument a given number on _d_ is set to the fixed index
+_b_, and either _a_ or _c_ is brought to another number on the scale.
+This establishes a ratio, and if the cylinder is now moved so as to
+bring any number to _b_, the fourth term of the proportion will be found
+under _a_ or _c_. Of course, in multiplication, one factor is brought to
+_b_, and _a_ or _c_ brought to 100. The other factor is then brought to
+_a_ or _c_, and the result read off under _b_. Problems involving
+continuous multiplication, or combined multiplication and division, are
+very readily dealt with. Thus, calling the fixed index F, the upper
+movable index A, and the lower movable index B, we have for _a_ × _b_ ×
+_c_:—Bring _a_ to F; A to 100; _b_ to A or B; A to 100; _c_ to A or B
+and read the product at F.
+
+The maximum number of figures in a product is the sum of the number of
+figures in the factors and this results when all the factors except the
+first have to be brought to B. Each time a factor is brought to A, 1 is
+to be deducted from that sum.
+
+For division, as _a_/(_m_ × _n_), bring _a_ to F; A or B to _m_; 100 to
+A; A or B to _a_; 100 to A and read the quotient at F.
+
+[Illustration: FIG. 19.]
+
+The maximum number of figures in the quotient is the difference between
+the sum of the number of figures in the numerator factors and those of
+the denominator factors, _plus_ 1 for each factor of the denominator and
+this results when A has to be set to all the factors of the denominator
+and all the factors of the numerator except the first brought to B. Each
+time B is set to a denominator factor or a numerator factor is brought
+to A, 1 is to be deducted.
+
+Logarithms of numbers are obtained by using the scales _m_ and _n_ and
+hence powers and roots of any magnitude may be obtained by the procedure
+already fully explained. The instrument illustrated is made by Messrs.
+W. F. Stanley & Co., Limited, London.
+
+THE “R.H.S.” CALCULATOR.—In this calculator, designed by Prof. R. H.
+Smith, the scale-line, which is 50 in. long, is also arranged in a
+spiral form (Fig. 19), but in this case it is wrapped around the central
+portion of a tube which is about ¾in. in diameter and 9½in. long. A
+slotted holder, capable of sliding upon the plain portions of this tube,
+is provided with four horns, these being formed at the ends of the two
+wide openings through which the scale is read. An outer ring carrying
+two horns completes the arrangement.
+
+One of the horns of the holder being placed in agreement with the first
+factor, and one of the horns of the ring with the second factor, the
+holder is moved until the third factor falls under the same horn of the
+ring, when the resulting fourth term will be found under the same (right
+or left) horn of the holder, at either end of the slot. In
+multiplication, 100 or 1000 is taken for the second factor in the above
+proportion, as already explained in connection with Fuller’s rule;
+indeed, generally, the mode of operation is essentially similar to that
+followed with the former instrument.
+
+The scale shown on one edge of the opening in the holder, together with
+the circular scale at the top of the spiral, enables the mantissæ of
+logarithms of numbers to be obtained, and thus problems involving powers
+and roots may be dealt with quite readily. This instrument is supplied
+by Mr. J. H. Steward, London.
+
+THACHER’S CALCULATING INSTRUMENT, shown in Fig. 20, consists of a
+cylinder 4 in. in diameter and 18 in. long, which can be given both a
+rotary and a longitudinal movement within an open framework composed of
+twenty triangular bars. These bars are connected to rings at their ends,
+which can be rotated in standards fixed to the baseboard. The scale on
+the cylinder consists of forty sectional lengths, but of each scale-line
+that part which appears on the right-hand half of the cylinder is
+repeated on the left-hand half, one line in advance. Hence each half of
+the cylinder virtually contains two complete scales following round in
+regular order. On the lower lines of the triangular bars are scales
+exactly corresponding to those on the cylinder, while upon the upper
+lines of the bars and not in contact with the slide is a scale of square
+roots.
+
+[Illustration: FIG. 20.]
+
+By rotating the slide any line on it may be brought opposite any line in
+frame and by a longitudinal movement any graduation on these lines may
+be brought into agreement. The whole can be rotated in the supporting
+standards in order to bring any reading into view. As shown in the
+illustration, a magnifier is provided, this being conveniently mounted
+on a bar, along which it can be moved as required.
+
+SECTIONAL LENGTH OR GRIDIRON SLIDE RULES.—The idea of breaking up a long
+scale into sectional lengths is due to Dr. J. D. Everett, who described
+such a gridiron type of slide rule in 1866. Hannyngton’s Extended Slide
+Rule is on the same principle. Both instruments have the lower scale
+repeated. H. Cherry (1880) appears to have been the first to show that
+such duplication could be avoided by providing two fixed index points in
+addition to the natural indices of the scale. These additional indices
+are shown at 10′ and 100′ in Fig. 21, which represents the lower sheet
+of Cherry’s Calculator on a reduced scale. The upper member of the
+calculator consists of a transparent sheet ruled with parallel lines,
+which coincide with the lines of the lower scale when the indices of
+both are placed in agreement. To multiply one number by another, one of
+the indices on the upper sheet is placed to one of the factors, and the
+position of whichever index falls under the transparent sheet is noted
+on the latter. Bringing the latter point to the other factor, the result
+is found under whichever index lies on the card. In other arrangements
+the inventor used transparent scales, the graduations running in a
+reverse direction to those of the lower scale. In this case, a factor on
+the upper scale is set to the other factor on the lower, and the result
+read at the available index.
+
+[Illustration: FIG. 21.]
+
+PROELL’S POCKET CALCULATOR is an application of the last-named
+principle. It comprises a lower card arranged as Fig. 21, with an upper
+sheet of transparent celluloid on which is a similar scale running in
+the reverse direction. For continued multiplication and division, a
+needle (supplied with the instrument) is used as a substitute for a
+cursor, to fix the position of the intermediate results. A series of
+index points on the lower card enable square and cube roots to be
+extracted very readily. This calculator is supplied by Messrs. John J.
+Griffin & Sons, Ltd., London.
+
+
+
+
+                         CIRCULAR CALCULATORS.
+
+
+Although the 10 in. slide rule is probably the most serviceable form of
+calculating instrument for general purposes, many prefer the more
+portable circular calculator, of which many varieties have been
+introduced during recent years. The advantages of this type are: It is
+more compact and conveniently carried in the waistcoat pocket. The
+scales are continuous, so that no traversing of the slide from 1 to 10
+is required. The dial can be set quickly to any value; there is no
+trouble with tight or ill-fitting slides. The disadvantages of most
+forms are: Many problems involve more operations than a straight rule.
+The results being read under fingers or pointers, an error due to
+parallax is introduced, so that the results generally are not so
+accurate as with a straight rule. The inner scales are short, and
+therefore are read with less accuracy. Special scale circles are needed
+for cubes and cube roots. The slide cannot be reversed or inverted.
+
+[Illustration: FIG. 22.]
+
+[Illustration: FIG. 23.]
+
+THE BOUCHER CALCULATOR.—This circular calculator resembles a
+stem-winding watch, being about 2 in. in diameter and ⁹⁄₁₆in. in
+thickness. The instrument has two dials, the back one being fixed, while
+the front one, Fig. 22 (showing the form made by Messrs. W. F. Stanley,
+London), turns upon the large centre arbor shown. This movement is
+effected by turning the milled head of the stem-winder. The small centre
+axis, which is turned by rotating the milled head at the side of the
+case, carries two fine needle pointers, one moving over each dial, and
+so fixed on the axis that one pointer always lies evenly over the other.
+A fine index or pointer fixed to the case in line with the axis of the
+winding stem, extends over the four scales of the movable dial as shown.
+Of these scales, the second from the outer is the ordinary logarithmic
+scale, which in this instrument corresponds to a straight scale of about
+4¾in. in length. The two inner circles give the square roots of the
+numbers on the primary logarithmic scale, the smaller circle containing
+the square roots of values between 1 and 3·162 (= √(10)), while the
+other section corresponds to values between 3·162 and 10. The outer
+circle is a scale of logarithms of sines of angles, the corresponding
+sines of which can be read off on the ordinary scale.
+
+On the fixed or back dial there are also four scales, these being
+arranged as in Fig. 23. The outer of these is a scale of equal parts,
+while the three inner scales are separate sections of a scale giving the
+cube roots of the numbers taken on the ordinary logarithmic scale and
+referred thereto by means of the pointers. In dividing this cube-root
+scale into sections, the same method is adopted as in the case of the
+square-root scale. Thus, the smallest circle contains the cube roots of
+numbers between 1 and 10, and is therefore graduated from 1 to 2·154;
+the second circle contains the cube roots of numbers between 10 and 100,
+being graduated from 2·154 to 4·657; while the third section, in which
+are found the cube roots of numbers between 100 and 1000, carries the
+graduations from 4·657 to 10.
+
+What has been said in an earlier section regarding the notation of the
+slide rule may in general be taken to apply to the scales of the Boucher
+calculator. The manner of using the instrument is, however, not quite so
+evident, although from what follows it will be seen that the operative
+principle—that of variously combining lengths of a logarithmic scale—is
+essentially similar. In this case, however, it is seen that in place of
+the straight scale-lengths shown in Fig. 4, we require to add or
+subtract arc-lengths of the circular scales, while, further, it is
+evident that in the absence of a fixed scale (corresponding to the stock
+of the slide rule) these operations cannot be directly performed as in
+the ordinary form of instrument. However, by the aid of the fixed index
+and the movable pointer, we can effect the desired combination of the
+scale-lengths in the following manner. Assuming it is desired to
+multiply 2 by 3, the dial is turned in a backward direction until 2 on
+the ordinary scale lies under the fixed index, after which the movable
+pointer is set to 1 on the scale. As now set, it is clear that the
+arc-length 1–2 is spaced off between the fixed index and the movable
+pointer, and it now only remains to add to this definite arc-length a
+further length of 1–3. To do this we turn the dial still further
+backward until the arc 1–3 has passed under the movable pointer, when
+the result, 6, is read under the fixed index. A little consideration
+will show that any other scale length may be added to that included
+between the fixed and movable pointers, or, in other words, any number
+on the scale may be multiplied by 2 by bringing the number to the
+movable pointer and reading the result under the fixed index. The rule
+for multiplication is now evident.
+
+_Rule for Multiplication._—_Set one factor to the fixed index and bring
+the pointer to 1 on the scale; set the other factor to the pointer and
+read the result under the fixed index._
+
+With the explanation just given, the process of division needs little
+explanation. It is clear that to divide 6 by 3, an arc-length 1–3 is to
+be taken from a length 1–6. To this end we set 6 to the index
+(corresponding in effect to passing a length 1–6 to the left of that
+reference point) and set the pointer to the divisor 3. As now set, the
+arc 1–6 is included between 1 on the scale and the index, while the arc
+1–3 is included between 1 on the scale and the pointer. Obviously if the
+dial is now turned forward until 1 on the scale agrees with the pointer,
+an arc 1–3 will have been deducted from the larger arc 1–6, and the
+remainder, representing the result of this operation, will be read under
+the index as 2.
+
+_Rule for Division._—_Set the dividend to the fixed index, and the
+pointer to the divisor; turn the dial until 1 on the scale agrees with
+the pointer, and read the result under the fixed index._
+
+The foregoing method being an inversion of the rule for multiplication,
+is easily remembered and is generally advised. Another plan is, however,
+preferable when a series of divisions are to be effected with a constant
+divisor—_i.e._, when _b_ in (_a_)/(_b_) = _x_ is constant. In this case
+1 on the scale is set to the index and the pointer set to _b_; then if
+any value of a is brought to the pointer, the quotient _x_ will be found
+under the index.
+
+_Combined Multiplication and Division_, as (_a_ × _b_ × _c_)/(_m_ × _n_)
+= _x_, can be readily performed, while cases of continued multiplication
+evidently come under the same category, since _a_ × _b_ × _c_ = (_a_ ×
+_b_ × _c_)/(1 × 1) = _x_. Such cases as _a_/(_m_ × _n_ × _r_) = _x_ are
+regarded as (_a_ × 1 × 1 × 1)/(_m_ × _n_ × _r_) = _x_; while (_a_ × _b_
+× _c_)/(_m_) = _x_ is similarly modified, taking the form (_a_ × _b_ ×
+_c_)/(_m_ × 1) = _x_. In all cases the expression must be arranged so
+that there is _one more factor in the numerator_ than _in the
+denominator_, _1’s being introduced as often as required_. The simple
+operations of multiplication and division involve a similar disposition
+of factors, since from the rules given it is evident that _m_ × _n_ is
+actually regarded as (_m_ × _n_)/(1), while (_m_)/(_n_) becomes in
+effect (_m_ × 1)/(_n_). It is important to note the general
+applicability of this arrangement-rule, as it will be found of great
+assistance in solving more complicated expressions.
+
+As with the ordinary form of slide rule, the factors in such an
+expression as (_a_ × _b_ × _c_)/(_m_ × _n_) = _x_ are taken in the
+order:—1st factor of numerator; 1st factor of denominator; 2nd factor of
+numerator; 2nd factor of denominator, and so on; the 1st factor as _a_
+being set to the index, and the result _x_ being finally read at the
+same point of reference.
+
+
+  EX.—(39 × 14·2 × 6·3)/(1·37 × 19) = 134.
+
+  Commence by setting 39 to the index, and the pointer to 1·37; bring
+  14·2 to the pointer; pointer to 19; 6·3 to the pointer, and read the
+  result 134 at the index.
+
+
+It should be noted that after the first factor is set to the fixed
+index, the _pointer_ is set to each of the _dividing_ factors as they
+enter into the calculation, while the _dial_ is moved for each of the
+_multiplying_ factors. Thus the dial is first moved (setting the first
+factor to the index), then the pointer, then the dial, and so on.
+
+_Number of Digits in the Result._—If rules are preferred to the plan of
+roughly estimating the result, the general rules given on pages 21 and
+25 should be employed for simple cases of multiplication and division.
+For combined multiplication and division, modify the expression, if
+necessary, by introducing 1’s, as already explained, and subtract the
+sum of the denominator digits from the sum of numerator digits. Then
+proceed by the author’s rule, as follows:—
+
+_Always turn dial to the_ LEFT; _i.e._, _against the hands of a watch_.
+
+_Note dial movements only; ignore those of the pointer._
+
+_Each time 1 on dial agrees with or passes fixed index_, ADD _1 to the
+above difference of digits_.
+
+_Each time 1 on dial agrees with or passes pointer_, DEDUCT _1 from the
+above difference of digits_.
+
+Treat continued multiplication in the same way, counting the 1’s used as
+denominator digits as one less than the number of multiplied factors.
+
+
+  EX.—(8·6 × 0·73 × 1·02)/(3·5 × 0·23) = 7·95 [7·95473+].
+
+  Set 8·6 to index and pointer to 3·5. Bring 0·73 to pointer (noting
+  that 1 on the scale passes the index) and set pointer to 0·23. Set
+  1·02 to pointer (noting that 1 on the scale passes the pointer) and
+  read under index 7·95. There are 1 + 0 + 1 = 2 numerator digits and 1
+  + 0 = 1 denominator digit; while 1 is to be added and 1 deducted as
+  per rule. But as the latter cancel, the digits in the result will be 2
+  − 1 = 1.
+
+
+When moving the dial to the left will cause 1 on the dial to pass _both_
+index and pointer (thus cancelling), the dial may be turned back to make
+the setting.
+
+It will be understood that when 1 is the _first_ numerator, and 1 on the
+dial is therefore set to the index, no digit addition will be made for
+this, as the actual operation of calculating has not been commenced.
+
+In the Stanley-Boucher calculator (Fig. 23) a small centre scale is
+added, on which a finger indicates automatically the number of digits to
+be added or deducted; the method of calculating, however, differs from
+the foregoing. To avoid turning back to 0 at the commencement of each
+calculation, a circle is ground on the glass face, so that a pencil mark
+can be made thereon to show the position of the finger when commencing a
+calculation.
+
+_To Find the Square of a Number._— Set the number, on one or other of
+the square root scales, to the index, and read the required square on
+the ordinary scale.
+
+_To Find the Square Root of a Number._—Set the number to the index, and
+if there is an _odd_ number of digits in the number, read the root on
+the inner circle; if an even number, on the second circle.
+
+_To Find the Cube of a Number._—Set 1 on the ordinary scale to the
+index, and the pointer (on the back dial) to the number on one of the
+three cube-root scales. Then under the pointer read the cube on the
+ordinary scale.
+
+_To Find the Cube Root of a Number._—Set 1 to index, and pointer to
+number. Then read the cube root under the pointer on one of the three
+inner circles on the back dial. If the number has
+
+      1, 4, 7, 10 or −2, −5, etc., digits, use the inner circle.
+      2, 5, 8, 11 or −1, −4, etc.,    „      „     second circle.
+      3, 6, 9, 12 or −0, −3, etc.,    „      „     third circle.
+
+_For Powers or Roots of Higher Denomination._—Set 1 to index, the
+pointer to the number on the ordinary scale, and read on the outer
+circle on the back dial the mantissa of the logarithm. Add the
+characteristic (see p. 46), multiply by the power or divide by the root,
+and set the pointer to the mantissa of the result on this outer circle.
+Under the pointer on the ordinary scale read the number, obtaining the
+number of figures from the characteristic.
+
+_To Find the Sines of Angles._—Set 1 to index, pointer to the angle on
+the outer circle, and read under the pointer the _natural sine_ on the
+ordinary scale; also under the pointer on the outer circle of the back
+dial read the _logarithmic sine_.
+
+THE HALDEN CALCULEX.—After the introduction of the Boucher calculator in
+1876, circular instruments, such as the Charpentier calculator, were
+introduced, in which a disc turned within a fixed ring, so that scales
+on the faces of both could be set together and ratios established as on
+the slide rule. Cultriss’s Calculating Disc is another instrument on the
+same principle. The Halden Calculex, of which half-size illustrations
+are given in Figs. 24 and 25, represents a considerable improvement upon
+these early instruments. It consists of an outer metal ring carrying a
+fixed-scale ring, within which is a dial. On each side of this dial are
+flat milled heads, so that by holding these between the thumb and
+forefinger the dial can be set quickly and conveniently. The protecting
+glass discs, which are not fixed in the metal ring but are arranged to
+turn therein, carry fine cursor lines, and as these are on the side next
+to the scales a very close setting can be made quite free from the
+effects of parallax. This construction not only avoids the use of
+mechanism, with its risk of derangement, but reduces the bulk of the
+instrument very considerably, the thickness being about ¼in.
+
+On the front face, Fig. 24, the fixed ring carries an outer
+evenly-divided scale, giving logarithms, and an ordinary scale, 1–10,
+which works in conjunction with a similar scale on the edge of the dial.
+The two inner circles give the square roots of values on the main scales
+as in the Boucher calculator. On the back face, Fig. 25, the ring bears
+an outer scale, giving sines of angles from 6° to 90° and an ordinary
+scale, 1–10, as on the front face. The scales on the dial are all
+reversed in direction (running from right to left), the outer one
+consisting of an ordinary (but inverse) scale, 1–10, while the three
+inner circles give the cube roots of values on this inverse scale. As
+the fine cursor lines extend over all the scales, a variety of
+calculations can be effected very readily and accurately.
+
+[Illustration: FIG. 24.]
+
+[Illustration: FIG. 25.]
+
+SPERRY’S POCKET CALCULATOR, made by the Keuffel and Esser Company, New
+York (Fig. 26), has two rotating dials, each with its own pointer and
+fixed index. The S dial has an outer scale of equal parts, an ordinary
+logarithmic scale, and a square-root scale. The L dial has a single
+logarithmic scale arranged spirally, in three sections, giving a scale
+length of 12½in. The pointers are turned by the small milled head, which
+is concentric with the milled thumb-nut by which the two dials are
+rotated. The gearing is such that both the L dial and its pointer rotate
+three times as fast as the S dial and pointer. All the usual
+calculations can be made with the spiral scale, as with the Boucher
+calculator, and the result read off on one or other of the three
+scale-sections. Frequently the point at which to read the result is
+obvious, but otherwise a reference to the single scale on the S dial
+will show on which of the three spirals the result is to be found.
+
+[Illustration: FIG. 26.]
+
+_The K and E Calculator_, also made by the Keuffel and Esser Company, is
+shown in Figs. 27 and 28. It has two dials, of which only one revolves.
+This, as shown in Fig. 27, has an ordinary logarithmic scale and a scale
+of squares. There is an index line engraved on the glass of the
+instrument. The fixed dial has a scale of tangents, a scale of equal
+parts and a scale of sines, the latter being on a two-turn spiral. The
+pointers, which move together, are turned by a milled nut and the
+movable dial by a thumb-nut, as in Sperry’s Calculator, Fig. 26.
+
+[Illustration: FIG. 27.]
+
+[Illustration: FIG. 28.]
+
+
+                 SLIDE RULES FOR SPECIAL CALCULATIONS.
+
+ENGINE POWER COMPUTER.—A typical example of special slide rules is shown
+in Fig. 29, which represents, on a scale of about half full size, the
+author’s Power Computer for Steam, Gas, and Oil Engines. This, as will
+be seen, consists of a stock, on the lower portion of which is a scale
+of cylinder diameters, while the upper portion carries a scale of
+horse-powers. In the groove between these scales are two slides, also
+carrying scales, and capable of sliding in edge contact with the stock
+and with each other.
+
+This instrument gives directly the brake horse-power of any steam, gas,
+or oil engine; the indicated horse-power, the dimensions of an engine to
+develop a given power, and the mechanical efficiency of an engine. The
+calculation of piston speed, velocity ratios of pulleys and gear wheels,
+the circumferential speed of pulleys, and the velocity of belts and
+ropes driven thereby, are among the other principal purposes for which
+the computer may be employed.
+
+[Illustration: FIG. 29.]
+
+THE SMITH-DAVIS PIECEWORK BALANCE CALCULATOR has two scales, 11 feet
+long, having a range from 1d. to £20, and marked so that they can be
+used either for money or time calculations. The scales are placed on the
+rims of two similar wheels and so arranged that the divided edges come
+together. The wheels are mounted on a spindle carried at each end in the
+bearings of a supporting stand. The wheels are pressed together by a
+spring, and move as one.
+
+To set the scales one to the other, a treadle gear is arranged to take
+the pressure of the spring so that when the fixed wheel is held by the
+left hand the free wheel can be rotated by the right hand in either
+direction. When the amount of the balance has been set to the combined
+weekly wage the treadle is released locking the two wheels together,
+when the whole can be turned and the amounts respectively due to each
+man read off opposite his weekly wage. The Smith-Davis Premium
+Calculator is on the same principle but the scales are about 4 feet 6
+inches long and the wheels spring-controlled. Both instruments are
+supplied by Messrs. John Davis & Son, Ltd., Derby.
+
+THE BAINES SLIDE RULE.—In this rule, invented by Mr. H. M. Baines,
+Lahore, four slides carrying scales are arranged to move, each in edge
+contact with the next. The slides are kept in contact and given the
+desired relative movement one to the other, by being attached (at the
+back), to a jointed parallelogram. On this principle which is of general
+application, the inventor has made a rule for the solution of problems
+covered by Flamant’s formula for the flow of water in cast-iron pipes:—V
+= 76·28_d_^{⁵⁄₇}_s_^{⁴⁄₇}, in which _s_ is the sine of the inclination
+or loss of head; _d_ the diameter of the pipe in inches and V the
+velocity in feet per second. The formula Q = AV is also included in the
+scope of the rule, Q being the discharge in cubic feet per second and A
+the cross sectional area of the pipe in square inches.
+
+FARMAR’S PROFIT-CALCULATING RULE.—The application of the slide rule to
+commercial calculations has been often attempted, but the degree of
+accuracy required necessitates the use of a long scale, and generally
+this results in a cumbersome instrument. In Farmar’s Profit-calculating
+Rule the money scale is arranged in ten sections, these being mounted in
+parallel form on a roller which takes the place of the upper scale of an
+ordinary rule. The roller, which is ¾in. in diameter, is carried in
+brackets secured to each end of the stock, so that by rotating the
+roller any section of the money scale can be brought into reading with
+the scale on the upper edge of the slide and with which the roller is in
+contact. This scale gives percentages, and enables calculations to be
+made showing profit on turnover, profit on cost, and discount. The lower
+scale on the slide, and that on the stock adjacent to it, are similar to
+the A and B scales of an ordinary rule. The instrument is supplied by
+Messrs. J. Casartelli & Son, Manchester.
+
+
+              CONSTRUCTIONAL IMPROVEMENTS IN SLIDE RULES.
+
+The attention of instrument makers is now being given to the devising of
+means for ensuring the smooth and even working of the slide in the stock
+of the rule. In some cases very good results are obtained by slitting
+the back of the stock to give more elasticity.
+
+In the rules made by Messrs. John Davis & Son, a metal strip, slightly
+curved in cross section as shown at A (Fig. 30), runs for the full
+length of the stock to which it is fastened at intervals. Near each end
+of the rule, openings about 1 in. long are made in the metal backing
+through which the scales on the back of the slide can be read. To
+prevent warping under varying climatic conditions both the stock of the
+rule and the slide are of composite construction. The base of the stock
+is of mahogany, while the grooved sides, firmly secured to the base, are
+of boxwood. Similarly the centre portion of the slide is of mahogany and
+the tongued sides of boxwood. Celluloid also enters into the
+construction, a strip of this material being laid along the bottom of
+the groove in the stock. A fine groove runs along the centre of this
+strip in order to give elasticity and to allow the sides of the stock to
+be pressed together slightly to adjust the fitting of the slide. As a
+further means of adjustment the makers fit metal clips at each end of
+the rule, so that by tightening two small screws the stock can be closed
+on the slide when necessary.
+
+[Illustration: FIG. 30.]
+
+[Illustration: FIG. 32.]
+
+[Illustration: FIG. 31.]
+
+In the rule made by the Keuffel and Esser Company of New York, one strip
+is made adjustable (Fig. 32).
+
+
+                  THE ACCURACY OF SLIDE RULE RESULTS.
+
+The degree of accuracy obtainable with the slide rule depends primarily
+upon the length of the scale employed, but the accuracy of the
+graduations, the eyesight of the operator, and, in particular, his
+ability to estimate interpolated values, are all factors which affect
+the result. Using the lower scales and working carefully the error
+should not greatly exceed 0·15 per cent. with short calculations. With
+successive settings, the discrepancy need not necessarily be greater, as
+the errors may be neutralised; but with rapid working the percentage
+error may be doubled. However, much depends upon the graduation of the
+scales. Rules in which one or more of the indices have been thickened to
+conceal some slight inaccuracy should be avoided. The line on the cursor
+should be sharp and fine and both slide and cursor should move smoothly
+or good work cannot be done. Occasionally a little vaseline or clean
+tallow should be applied to the edges of the slide and cursor.
+
+That the percentage error is constant throughout the scale is seen by
+setting 1 on C to 1·01 on D, when under 2 is 2·02; under 3, 3·03; under
+5, 5·05, etc., the several readings showing a uniform error of 1 per
+cent.
+
+A method of obtaining a closer reading of a first setting or of a result
+on D has been suggested to the author by Mr. M. Ainslie, B.Sc. If any
+graduation, as 4 on C, is set to 3 on D, it is seen that 4 main
+divisions on C (40–44) are equal in scale length to 3 main divisions on
+D (30–33). Hence, very approximately, 1 division on C is equal to 0·75
+of a division on D, this ratio being shown, of course, on D under 10 on
+C. Suppose √(4·3) to be required. Setting the cursor to 4·3 on A, it is
+seen that the root is something more than 2·06. Move the slide until a
+main division is found on C, which exactly corresponds to the interval
+between 2 and the cursor line, on D. The division 27–28 just fits,
+giving a reading under 10 on C, of 74. Hence the root is read as 2·074.
+For the higher parts of the scale, the subdivisions, 1–1·1, etc., are
+used in place of main divisions. The method is probably more interesting
+than useful, since in most operations the inaccuracies introduced in
+making settings will impose a limit on the reliable figures of the
+result.
+
+For the majority of engineering calculations, the slide rule will give
+an accuracy consistent with the accuracy of the data usually available.
+For some purposes, however, _logarithmic section paper_ (the use of
+which the author has advocated for the last twenty years) will be found
+especially useful, more particularly in calculations involving
+exponential formulæ.
+
+
+
+
+                               APPENDIX.
+
+
+      NEW SLIDE RULES—FIFTH ROOTS, ETC.—THE SOLUTION OF ALGEBRAIC
+ EQUATIONS—GAUGE POINTS AND SIGNS ON SLIDE RULES—TABLES AND DATA—SLIDE
+                            RULE DATA SLIPS.
+
+THE PICKWORTH SLIDE RULE.—In this rule, made by Mr. A. W. Faber, the
+novel feature is the provision of a scale of cubes (F) in the stock or
+body of the rule. From Fig. 33 it will be seen that the scale is fixed
+on the bevelled side of a slotted recess in the back of the rule. The
+slide carries an index mark, which is seen through the slot and can be
+set to any graduation of the scale; in its normal position it agrees
+with 1 on the scale. The C scale on the face of the rule is divided into
+three equal parts by two special division lines, marked II. and III.,
+which, together with the initial graduation 1 of the scale, serve for
+setting or reading off values on the D scale. Similar division lines are
+marked on the D scale.
+
+[Illustration: FIG. 33.]
+
+In using the rule for cubes or cube roots the slide is drawn to the
+right, this movement never exceeding one-third of the length of the D
+scale. With this limited movement, and with a single setting of the
+slide, the values of ∛_̅a_, ∛(_a_ × 10), and ∛(_a_ × 100)) (_a_ being
+less than 10 and not less than 1) are given simultaneously and without
+any uncertainty as to the scales to use or the values to be read off.
+
+_To Find the Cube of a Number._—The marks II. and III. on D divide that
+scale into three equal sections. If the number to be cubed is in the
+first section, I. on C is set to it; if in the second section, II. on C
+is set to it; if in the third section, III. on C is set to it. Then,
+under the index mark on the back of the slide will be found the
+significant figures of the cube on the scale F. If I. on C was used for
+the setting, the cube contains 1 digit; if II. was used, 2 digits; if
+III. was used, 3 digits. If the first figure of the number to be cubed
+is not in the units place, the decimal point is moved through _n_ places
+so as to bring the first significant figure into the units place, the
+cube found as above, and the decimal point moved in the _reverse
+direction_ through 3_n_ places.
+
+_To Find the Cube Root of a Number._—The index mark is set to the
+significant figures of the number on scale F, and the cube root is read
+on D under I., II. or III. on C, according as the number has 1, 2 or 3
+digits preceding the decimal point. Numbers which have 1, 2 or 3 figures
+preceding the decimal point are dealt with directly. Numbers of any
+other form are brought to one of the above forms by moving the decimal
+point 3 places (or such multiple of 3 places as may be required), the
+root found and its decimal point moved 1 place for each 3-place
+movement, but in the _reverse direction_.
+
+THE “ELECTRO” SLIDE RULE.—In this special rule for electrical
+calculations, made by Mr. A. Nestler, the upper scales run from 0·1 to
+1000, and are marked “Amp.” and “sq. mm.” respectively. The lower scale
+on the slide running from 1 to 10,000 is marked M (metres), while the
+lower scale on the rule (0·1 to 100) is marked “Volt.” The latter scale
+is so displaced that 10 on M agrees with 0·173 on the Volt scale. The
+four factors involved are the current strength (in Amp.); the area of a
+conductor (in sq. mm.); the length of the conductor (in metres); and the
+permissible loss of potential (in volts). Having given any three of
+these, the fourth can be found very readily. On the back of the slide
+are a scale of squares, a scale of cubes and a single scale
+corresponding to the D scale of an ordinary rule. Hence, by reversing
+the slide, it is possible to obtain the 2nd, 3rd and 4th powers and
+roots of numbers. In another form of the rule, the scale of metres is
+replaced by one of yards, while instead of the area of the conductor in
+sq. mm., the corresponding “gauge” sizes of wires are given.
+
+THE “POLYPHASE” SLIDE RULE.—This instrument, made by the Keuffel & Esser
+Company, New York, has, in addition to the usual scales, a scale of
+cubes on the vertical edge of the stock of the rule, while in the centre
+of the slide there is a reversed C scale; _i.e._, a scale exactly
+similar to an ordinary C scale but with the graduations running from
+right to left. The rule is specially useful for the solution of problems
+containing combinations of three factors and problems involving squares,
+square roots, cubes, cube roots and many of the higher powers and roots.
+It is specially adapted for electrical and hydraulic work.
+
+THE LOG-LOG DUPLEX SLIDE RULE.—The same makers have introduced a log-log
+duplex slide rule, in which the log-log scale is in three sections,
+placed one above the other, these occupying the position usually taken
+up by the A scale. These scales are used in the manner already described
+(page 86), but some advantage is obtained by the manner in which the
+complete log-log scale is divided, the limits being _e_^{¹⁄₁₀₀} to _e_^⅒
+(on Scale L.L. 1); _e_^⅒ to _e_ (on Scale L.L. 2); and _e_ to _e_^{10}
+(on Scale L.L. 3), _e_ being the base of natural or hyperbolic
+logarithms (2·71828). In this way a total log-log range of from 1·01 to
+22,000 is provided, meeting all practical requirements. These log-log
+scales are read in conjunction with a C scale placed at the upper edge
+of the slide. A similar C scale, but reversed in direction, is placed at
+the lower edge of the slide, this having red figures to distinguish it
+readily. The adjacent scale on the body of the rule is an ordinary D
+scale, and under this is an equally-divided scale giving the common
+logarithms of values on D. In the centre of the slide is a scale of
+tangents.
+
+It will be understood that a “duplex” rule consists of two side strips
+securely clamped together at the two ends, forming the body of the rule,
+the slide moving between them; hence both front and back faces of the
+rule and slide are available, graduations on the one side being referred
+to those on the other by the cursor which extends around the whole. In
+this instrument, the scales on the back face are the ordinary scales of
+the standard rule with the addition of a scale of sines which is placed
+in the centre of the slide. It will be evident that this instrument is
+capable of dealing with a very wide range of problems involving
+exponential and trigonometrical formulæ.
+
+SMALL SLIDE RULES WITH MAGNIFYING CURSORS.—Several makers now supply 5
+in. rules having the full graduations of a 10 in. rule, and fitted with
+a magnifying cursor (Fig. 34). This forms a compact instrument for the
+pocket, but owing to the closeness of the graduations it is not usually
+possible to make a setting of the slide without using the cursor. This,
+of course, involves more movements than with the ordinary instrument. It
+is also very necessary to use the magnifying cursor in a _direct_ light,
+if accurate readings are to be obtained. If these slight inconveniences
+are to be tolerated, the principle could be extended, a 10 in. rule
+being marked as fully as a 20 in., and fitted with a magnifying cursor.
+The author has endeavoured, but without success, to induce makers to
+introduce such a rule.
+
+The magnifying cursor, supplied by Messrs. A. G. Thornton, Limited, has
+a lens which fills the entire cursor. It has a powerful magnifying
+effect, and the change from the natural to the magnified reading is less
+abrupt than with the semicircular lens.
+
+[Illustration: FIG. 34.]
+
+THE CHEMIST’S SLIDE RULE.—A slide rule, specially adapted for chemical
+calculations, has been introduced recently by Mr. A. Nestler. In this
+instrument the C and D scales are as usually arranged; but, in place of
+the A and B scales, there are a number of gauge points or marks denoting
+the atomic and molecular weights of the most important elements and
+combinations. The scales on the back of the slide are similarly
+arranged, so that by reversing the slide the operations can be extended
+very considerably. The rule finds its chief use in the calculation of
+analyses. Thus, to find the percentage of chlorine if _s_ grammes of a
+substance have been used and the precipitate of Ag.Cl. weighs _a_
+grammes, we have the equation, _x_ = (Cl.)/(Ag.Cl.) × (_a_)/(_s_).
+Hence, the mark Ag.Cl. on the upper scale of the slide is set to the
+mark Cl. on the upper scale of the rule, when under _a_ on the C scale
+is found the quantity of chlorine on D. By setting the cursor to this
+value and bringing _s_ on C to the cursor, the percentage required can
+be read on C over 10 on D.
+
+The rule is also adapted to the solution of various other chemical and
+electro-chemical calculations.
+
+THE STELFOX SLIDE RULE.—This rule, shown in Fig. 35, has a stock 5 in.
+long, fitted with a 10 in. slide jointed in the middle of its length by
+means of long dowels. By separating the parts the compactness of a 5 in.
+rule is obtained. The upper scales on the rule and slide resemble the
+usual A and B scales. The D scale on the lower part of the stock is in
+two sections, the second portion being placed below the first, as shown
+in the illustration. The centre scale on the slide corresponds to the
+usual C scale, while on the lower edge of the slide is a similar scale,
+but with the index (1) in the middle of its length. The arrangement
+avoids the necessity of resetting the slide, as is sometimes necessary
+with the ordinary rule, and in general it combines the accuracy of a 10
+in. rule with the compactness of a 5 in. rule; but a more frequent use
+of the cursor is necessary. This rule is made by Messrs. John Davis &
+Son, Limited, Derby.
+
+[Illustration: FIG. 35.]
+
+ELECTRICAL SLIDE RULE.—Another rule by the same makers, specially useful
+for electrical engineers, has the usual scales on the working edges of
+the rule and slide, while in the middle of the slide is placed a scale
+of cubes. A log-log scale in two sections is provided; the power
+portion, running from 1·07 to 2, is found on the lower part of the
+stock, and the upper portion, running from 2 to 10^3, on the upper part
+of the stock. The uppermost scale on the stock is in two parts, of which
+that to the left, running from 20 to 100 and marked “Dynamo,” gives the
+efficiencies of dynamos; that on the right, running from 20 to 100 and
+marked “Motor,” gives the efficiencies of electric motors. The lowest
+scale on the stock, marked “Volt,” gives the loss of potential in copper
+conductors. The ordinary upper scale on the stock is marked L (length of
+lead) at the left, and KW (kilowatts) at the right; the ordinary upper
+scale on the slide is marked A (ampères) and mm^2 (sectional area) at
+the left, and HP (horse-power) at the right. Additional lines on the
+cursor enable the electrical calculations to be made either in British
+or metric units.
+
+THE PICOLET CIRCULAR SLIDE RULE.—A simple form of circular calculator,
+made by Mr. L. E. Picolet of Philadelphia, is shown in Fig. 36. It
+consists of a base disc of stout celluloid on which turns a smaller disc
+of thin celluloid. A cursor formed of transparent celluloid is folded
+over the discs, and is attached so that the friction between the cursor
+and the inner disc enables the latter to be turned by moving the former.
+By holding both discs the cursor can be adjusted as required. The
+adjacent scales run in opposite directions, so that multiplication and
+division are performed as with the inverted slide in an ordinary rule.
+The outer scale, which is two-thirds the length of the main scale,
+enables cube roots to be found. Square roots are readily determined and
+continuous multiplication and division conveniently effected. Modified
+forms of this neatly made little instrument are also available.
+
+[Illustration: FIG. 36.]
+
+OTHER RECENT SLIDE RULES.—Among other special types of slide rule,
+mention should be made of the _Jakin_ 10 in. rule for surveyors, made by
+Messrs. John Davis & Son, Limited, Derby. By the provision of a series
+of short subsidiary scales, the multiplication of a sine or tangent of
+an angle by a number can be obtained to an accuracy of 1 in 10,000. The
+_Davis-Lee-Bottomley_ slide rule, by the same makers, has special scales
+provided for circle spacing. The division of a circle into a number of
+equal parts, often required in spacing rivets, bolts, etc., and in
+setting out the teeth of gearwheels, is readily effected by the aid of
+this instrument. The _Cuntz_ slide rule is a very comprehensive
+instrument, having a stock about 2¼ in. wide, with the slide near the
+lower edge. Above the slide are eleven scales, referable to the main
+scales by the cursor. These scales enable squares and square roots,
+cubes and cube roots, and areas and circumferences of circles to be
+obtained by direct reading. A much more compact instrument could be
+obtained by removing one-half the scales to the back of the rule and
+using a double cursor.
+
+[Illustration: FIG. 37.]
+
+In one form of 10 in. rule, supplied by Mr. W. H. Harling, London, the
+body of the rule is made of well-seasoned cane, with the usual celluloid
+facings. The rule has a metal back, enabling the fit of the slide to be
+regulated. This backing extends the full length of the rule, openings
+about 1 in. long being provided at each end, enabling the scales on the
+back of the slide to be set with greater facility than is possible with
+the notched recesses usually adopted. The author has long endeavoured,
+but without success, to induce makers to fit windows of glass or
+celluloid in place of the notched recesses. This would allow the
+graduation of the S and T scales to be set more accurately, and enable
+both to be used at each end of the rule—an advantage in certain
+trigonometrical calculations. It would have the further advantage of
+permitting each alternate graduation of the evenly-divided or logarithm
+scale to be placed at opposite sides of one central line, enabling the
+reading to be made more accurately and conveniently.
+
+Many special slide rules have lately been devised for determining the
+time necessary to perform various machine-tool operations and for
+analogous purposes, while attention has again been given to rules for
+calculating the weights of iron and steel bars, plates, etc.
+
+THE DAVIS-STOKES FIELD GUNNERY SLIDE RULE.—This rule, which is adapted
+for calculations involved in “encounter” and “entrenched” field gunnery,
+is designed for the 18 pr. quick-firing gun. The upper and lower
+portions of the boxwood stock are united by a flexible centre of
+celluloid, thus providing grooves front and rear to receive boxwood
+slides. Each of the nineteen scales is marked with its name, and
+corresponding scales are coloured red or black. The front edge is
+bevelled and carries a scale of 1 in 20,000. The rule solves
+displacement problems, map angles of sight, changes of corrector and
+range corrections for changes in temperature, wind and barometer, etc. A
+special feature for displacement calculations is the provision of a 50
+yd. sub-base angle scale, by which the apex angle is read at one
+setting.
+
+THE DAVIS-MARTIN WIRELESS SLIDE RULE.—In wireless telegraphy it is
+frequently necessary to determine wave-length, capacity or
+self-induction when one or other of the factors of the equation, λ =
+59·6√(LC) is unknown. The Davis-Martin wireless rule is designed to
+simplify such calculations. The upper scale in the stock (inductance)
+runs from 10,000 to 1,000,000; the adjacent scale on the slide
+(capacity) runs from 0·0001 to 0·01 but in the reverse direction. The
+lower scale on the stock (wave-length) runs from 100 to 1000, giving
+square roots of the upper scale; while on the lower edge of the scale
+are several arrows to suit the various denominations in which the
+wave-length and capacity may be expressed.
+
+IMPROVED CURSORS.—In some slide-rule operations, notably in those
+involved in solving quadratic and cubic equations, it not infrequently
+happens that readings are obscured by the frame of the cursor. Frameless
+cursors have been introduced to obviate this defect. A piece of thick
+transparent celluloid is sometimes employed, but this is liable to
+become scratched in use. Fig. 37 shows a recent form of frameless glass
+cursor made by the Keuffel & Esser Company, Hoboken, N.J., which is
+satisfactory in every way.
+
+Cursors having three hair lines are now fitted to some rules, the
+distance apart of the lines being equal to the interval 0·7854–1 on the
+A scale.
+
+THE DAVIS-PLETTS SLIDE RULE.—In this rule a single log.-log. scale and
+its reciprocal scale are arranged opposite the ordinary upper log.
+scale. Thus, common logarithms can be read directly, while by taking
+advantage of the properties of characteristics and mantissas of common
+logarithms, the scale can be extended indefinitely. As 10 is the highest
+number on the log.-log. scale, it is carried down to within 0·025 of
+unity. The reading of log.-log. values above 10 is effected in a very
+simple manner. There is also a scale in the centre of the slide which,
+used in conjunction with the upper log. scale enables the natural
+logarithm of any number between 0·0001 and 10,000 to be read direct,
+while any number on the upper log. scale can be multiplied or divided by
+_e^x_ if the latter is between these limits. On the back of the slide
+are scales for all circular and hyperbolic functions, these being used
+in conjunction with the upper log. scales.
+
+THE CROMPTON-GALLAGHER BOILER EFFICIENCY CALCULATOR has a stock in the
+thickness of which is a slot admitting a chart which can be moved at
+right angles to the two separate slides. On the bevelled edge of one
+slide, the graduations are continued so as to read against curves on the
+chart, through an opening in the stock.
+
+THE DAVIS-GRINSTED COMPLEX CALCULATOR.—This slide rule is of
+considerable service in connection with calculations involving the
+conversion of complex quantities from the form _a_ + _j_ _b_ to the form
+R∠θ, and _vice versa_. The usual process of conversion necessitates
+repeated reference to trigonometrical tables, and is both tedious and
+time-taking. The Complex Calculator enables the conversions to be
+effected without reference to tables and with the minimum expenditure of
+time and labour.
+
+The rule, which is about 16 in. long, has five scales. The upper one (A)
+is an ordinary logarithmic scale thrice-repeated. The adjacent scales on
+the slide comprise (1) a logarithmic scale of tangents (B) ranging from
+0·1° to 45°, and (2) a logarithmic scale of secants (C) from 0° to 45°.
+The lower scales D and E are identical with the A scale, and are
+provided to enable multiplication, etc., to be performed without the
+need for a separate slide rule. Readings can be transferred from A to
+the lower scales by means of the cursor.
+
+In using the rule to convert _a_ + _j_ _b_ to R∠θ, the index (45°) of
+the B scale is set to the larger component and the cursor to the smaller
+component, on scale A. Then θ (or its complement if _b_ is greater than
+_a_) is read on B under the cursor. The cursor is then set to θ on the C
+scale, and R is read on A under the cursor. The rule is made by Messrs.
+John Davis & Son, Limited, Derby.
+
+
+                  THE SOLUTION OF ALGEBRAIC EQUATIONS.
+
+The slide rule finds an interesting application in the solution of
+equations of the second and third degree; and although the process is
+essentially one of trial and error, it may often serve as an efficient
+substitute for the more laborious algebraic methods, particularly when
+the conditions of the problem or the operator’s knowledge of the theory
+of equations enables some idea to be obtained as to the character of the
+result sought. The principle may be thus briefly explained:—If 1 on C is
+set to _x_ on D (Fig. 38), we find _x_(_x_) = _x_^2 on D under _x_ on C.
+If, however, with the slide set as before, instead of reading under _x_,
+we read under _x_ + _m_ on C, the result on D will now be _x_(_x_ + _m_)
+= _x_^2 + _mx_ = _q_. Hence to solve the equation _x_^2 + _mx_ − _q_ =
+0, we reverse the above process, and setting the cursor to _q_ on D, we
+move the slide until the number on C under the cursor, and that on D
+under 1 on C, _differ by m_. It is obvious from the setting that the
+_product_ of these numbers = _q_, and as their difference = _m_, they
+are seen to be the roots of the equation as required. For the equation
+_x_^2 − _mx_ + _q_ = 0, we require _m_ to equal the _sum_ of the roots.
+Hence, setting the cursor as before to _q_ on D, we move the slide until
+the number on C under the cursor, and that on D under 1 on C, are
+_together equal to_ _m_, these numbers being the roots sought. The
+alternative equations _x_^2 − _mx_ − _q_ = 0, and _x_^2 + _mx_ + _q_ = 0
+are deducible from the others by changing the signs of the roots, and
+need not be further considered.
+
+[Illustration: FIG. 38.]
+
+
+  EX.—Find the roots of _x_^2 − 8_x_ + 9 = 0.
+
+  Set the cursor to 9 on D, and move the slide to the right until when
+  6·64 is found under the cursor, 1·355 on D is under 1 on C. These
+  numbers are the roots required.
+
+
+The upper scales can of course be used; indeed, in general they are to
+be preferred.
+
+
+  EX.—Find the roots of _x_^2 + 12·8_x_ + 39·4 = 0.
+
+  Set the cursor to 39·4 on A, and move the slide to the right until we
+  read 7·65 on B under the cursor, and 5·15 on A over 1 on B. The roots
+  are therefore −7·65 and −5.15.
+
+
+With a little consideration of the relative value of the upper and lower
+scales, the student interested will readily perceive how equations of
+the third degree may be similarly resolved. The subject is not of
+sufficient general importance to warrant a detailed examination being
+made of the several expressions which can be dealt with in the manner
+suggested; but the author gives the following example as affording some
+indication of the adaptability of the method to practical calculations.
+
+
+  EX.—A hollow copper ball, 7·5 in. in diameter and 2 lb. in weight,
+  floats in water. To what depth will it sink?
+
+  The water displaced = 27·7 × 2 = 55·4 cub. in. The cubic contents of
+  the immersed segment will be (π)/(3)(3_r_ _x_^2 − _x_^3), _r_ being
+  the radius and _x_ the depth of immersion. Hence (π)/(3)(3_r_ _x_^2 −
+  _x_^3) = 55·4, and 11·25_x_^2 − _x_^3 = 52·9.
+
+  To solve this equation we place the cursor to 52·9 on A, and move the
+  slide until the reading on D under 1 and that on B under the cursor
+  together amount to 11·25. In this way find 2·45 on D under 1, with 8·8
+  on B under the cursor _c_, _c_, as a pair of values of which the sum
+  is 11·25. Hence we conclude that _x_ = 2·45 in. is the result sought.
+
+  With the rule thus set (Fig. 39) the student will note that the slide
+  is displaced to the right by an amount which represents _x_ on D, and
+  therefore _x_^2 on A; while the length on B from 1 to the cursor line
+  represents 11·25 − _x_. Hence the upper scale setting gives
+  _x_^2(11·25 − _x_) = 11·25_x_^2 − _x_^3 = 52·9 as required.
+
+
+[Illustration: FIG. 39.]
+
+When in doubt as to the method to be pursued in any given case, the
+student should work synthetically, building up a simple example of an
+analogous character to that under consideration, and so deducing the
+plan to be followed in the reverse process.
+
+
+                    SCREW-CUTTING GEAR CALCULATIONS.
+
+The slide rule has long found a useful application in connection with
+the gear calculations necessary in screw-cutting, helical gear-cutting,
+and spiral gear work.
+
+SINGLE GEARS.—For simple cases of screw-cutting in the lathe it is only
+necessary to set the threads per inch to be cut to the threads per inch
+in the guide screw (or the pitch in inches in each case, if more
+convenient). Then any pair of coinciding values on the two scales will
+give possible pairs of wheels.
+
+
+  EX.—Find wheels to cut a screw of 1⅝ threads per inch with a guide
+  screw of 2 threads per inch.
+
+  Setting 1·625 on C to 2 on D, it is seen that 80 (driver) and 65
+  (driven) are possible wheels.
+
+
+COMPOUND GEARS.—When wheels so found are of inconvenient size, a
+compound train is used, consisting (usually) of two drivers and two
+driven wheels, the product of the two former and the product of the two
+latter being in the same ratio as the simple wheels. Thus with 60 and 40
+as drivers, and 65 and 30 as driven, we have, (60 × 40)/(65 × 30) =
+(2400)/(1950) = (2)/(1·625) as before.
+
+With the slide set as above, values convenient for splitting up into
+suitable wheels are readily obtainable. Thus, (1600)/(1300);
+(2400)/(1950); (4000)/(3250); (4800)/(3900) are a few suggestive values
+which may be readily factorised.
+
+SLIDE RULES FOR SCREW-CUTTING CALCULATIONS.—Special circular and
+straight slide rules for screw-cutting gear calculations have long been
+employed. For compound gears these usually entail the use of six scales,
+two on each of the two slides and two on the stock. The upper scale on
+the stock may be a scale of threads per inch to be cut, the adjacent
+scale (on the upper slide) a scale of threads per inch in the guide
+screw. Setting the guide screw-graduation to the threads to be cut, the
+lower slide is adjusted until a convenient pair of drivers is found in
+coincidence on the central pair of scales, while a pair of driven wheels
+are in coincidence on the two lower scales.
+
+Some years ago, a slide rule was introduced by which compound gears
+could be obtained with a single slide. Assuming the set of wheels
+usually provided—20 to 120 teeth advancing by 5 teeth—the products of 20
+× 25, 20 × 30, etc., up to 115 × 120 were calculated. These products
+were laid out along each of the two lower scales. The upper scales were
+a scale of threads per inch to be cut and a scale of the threads per
+inch of various guide screws. Setting the guide screw-graduation to the
+threads to be cut, any coinciding graduations on the lower scales gave
+the required pairs of drivers and driven wheels.
+
+FRACTIONAL PITCH CALCULATIONS.—The author has long advocated the use of
+the slide rule for determining the wheels necessary for cutting
+fractional pitch threads, and it is gratifying to find its value in this
+connection is now being appreciated. For the best results a good 20 in.
+rule is desirable, but with care very close approximations can be found
+with an accurate 10 in. rule. In any case a magnifying cursor or a hand
+reading-glass is of great assistance.
+
+
+  EX.—Find wheels to cut a thread of 0·70909 in. pitch; guide screw, 2
+  threads per inch.
+
+  To 0·70909 on D, set 0·5 (guide screw pitch in inches) on C. To make
+  this setting as accurately as possible, the method described on page
+  112 may be used. Set 10 on C to about 91 on D, and note that the
+  interval 77–78 on C represents 0·91 of the interval 70–71 on D. Set
+  the cursor to 78 on C and bring 5 to the cursor. The slide is then set
+  so that 5 on C agrees with 7·091 on D.
+
+
+Inspection of the two scales shows various coinciding factors in the
+ratio required. The most accurate is seen to be (55 on C)/(78 on D).
+These values may be split up into (55 × 50)/(65 × 60) to form a suitable
+compound train of gears.
+
+
+                 GAUGE POINTS AND SIGNS ON SLIDE RULES.
+
+Many slide rules have the sign (Prod.)/(−1) at the right-hand end of the
+D scale, while on the left is (Quot.)/(+1.) It is somewhat unfortunate
+that these signs refer to rules for determining the number of digits in
+products and quotients, which are used to a considerable extent on the
+Continent, and conflict with those used in this country. By the
+Continental method the number of digits in a product is equal to the sum
+of the digits in the two factors, if the result is obtained on the LEFT
+_of the first factor_; but if the result is found on the RIGHT of the
+first factor, it is equal to this sum − 1. The sign (Prod.)/(−1) the
+_right_-hand end of the D scale provides a visible reminder of this
+rule.
+
+Similarly for division:—The number of digits in a quotient is equal to
+the number of the digits in the dividend, minus those in the divisor, if
+the quotient appears on the RIGHT _of the dividend_, and to this
+difference + 1, if the quotient appears on the LEFT of the dividend. The
+sign (Quot.)/(+1) at the _left_-hand end of the D scale provides a
+visible reminder of this rule.
+
+The sign
+
+                                  +ⵏ–
+                                  ⟵ⵏ⟶
+                                  –ⵏ+
+
+found at both ends of the A scale is of general application but of
+questionable utility. It is assumed to represent a fraction, the
+vertical line indicating the position of the decimal point. If the
+number 455 is to be dealt with in a multiplication on the lower scales,
+we may suppose the decimal point moved two places to the left, giving
+4·55, a value which can be actually found on the scale. If we use this
+value, then to the number of digits in this result, as many must be
+added as the number of places (two in this case) by which the decimal
+point was moved. If the point is moved to the right, the number of
+places must be subtracted. Similarly, in division, if the decimal point
+in the divisor is moved _n_ places to the left, then _n_ places must be
+subtracted at the end of the operation; while if the point is moved
+through _n_ places to the right, then _n_ places must be added. The sign
+referred to, which, of course, applies to all scales, completely
+indicates these processes and is submitted as a reminder of the
+procedure to be followed by those using the method described.
+
+The signs π, _c_, _c′_, and M are explained in the Section on “Gauge
+Points,” p. 53.
+
+On some rules additional signs are found on the D scale. One, locating
+the value (180 × 60)/(π) = 3437·74 and hence giving the number of
+minutes in a radian, is marked ρ′. Another, representing the value (180
+× 60 × 60)/(π) = 206265, and hence giving the number of seconds in a
+radian is marked ρ″. A third point, marked ρ_{˶}, placed at the value
+(200 × 100 × 100)/(π) = 636620, is used when the newer graduation of the
+circle is employed.
+
+These gauge points are useful when converting angles into circular
+measure, or _vice versa_, and also for determining the functions of
+small angles.
+
+A gauge point is sometimes marked at 1146 on the A and B scales. This is
+known as the “Gunner’s Mark,” and is used in artillery calculations
+involving angles of less than 20°, when, for the purpose in view, the
+tangent and circular measure of the angle may be regarded as equal. For
+this constant, the angle is taken in minutes, the auxiliary base in
+feet, and the base in yards. The auxiliary base in feet on B is set to
+the angle in minutes on A when over 1146 on B is the base in yards on A.
+The value (1)/(1146) = (π × 3)/(180 × 60).
+
+
+                            TABLES AND DATA.
+
+
+                         MENSURATION FORMULAE.
+
+
+  Area of a parallelogram = base × height.
+
+  Area of rhombus = ½ product of the diagonals.
+
+  Area of a triangle = ½ base × perpendicular height.
+
+  Area of equilateral triangle = square of side × 0·433.
+
+  Area of trapezium = ½ sum of two parallel sides × height.
+
+  Area of any right-lined figure of four or more unequal sides is found
+  by dividing it into triangles, finding area of each and adding
+  together.
+
+  Area of regular polygon = (1) length of one side × number of sides ×
+  radius of inscribed circle; or (2) the sum of the triangular areas
+  into which the figures may be divided.
+
+  Circumference of a circle = diameter × 3·1416.
+
+  Circumference of circle circumscribing a square = side × 4·443.
+
+  Circumference of circle = side of equal square × 3·545.
+
+  Length of arc of circle = radius × degrees in arc × 0·01745.
+
+  Area of a circle = square of diameter × 0·7854.
+
+  Area of sector of a circle = length of arc × ½ radius.
+
+  Area of segment of a circle = area of sector − area of triangle.
+
+  Side of square of area equal to a circle = diameter × 0·8862.
+
+  Diameter of circle equal in area to square = side of square × 1·1284.
+
+  Side of square inscribed in circle = diameter of circle × 0·707.
+
+  Diameter of circle circumscribing a square = side of square × 1·414.
+
+  Area of square = area of inscribed circle × 1·2732.
+
+  Area of circle circumscribing square = square of side × 1·5708.
+
+  Area of square = area of circumscribing circle × 0·6366.
+
+  Area of a parabola = base x ⅔ height.
+
+  Area of an ellipse = major axis × minor axis × 0·7854.
+
+  Surface of prism or cylinder = (area of two ends) + (length ×
+  perimeter).
+
+  Volume of prism or cylinder = area of base × height.
+
+  Surface of pyramid or cone = ½(slant height × perimeter of base) +
+  area of base.
+
+  Volume of pyramid or cone = (⅓)(area of base × perpendicular height).
+
+  Surface of sphere = square of diameter × 3·1416.
+
+  Volume of sphere = cube of diameter × 0·5236.
+
+  Volume of hexagonal prism = square of side × 2·598 × height.
+
+  Volume of paraboloid = ½ volume of circumscribing cylinder.
+
+  Volume of ring (circular section) = mean diameter of ring × 2·47 ×
+  square of diameter of section.
+
+
+               SPECIFIC GRAVITY AND WEIGHT OF MATERIALS.
+
+                                METALS.
+ ─────────────────────┬───────────────┬───────────────┬───────────────
+        METAL.        │   Specific    │  Weight of 1  │  Weight of 1
+                      │   Gravity.    │Cub. Ft. (Lb.).│Cub. In. (Lb.).
+ ─────────────────────┼───────────────┼───────────────┼───────────────
+ Aluminium, Cast      │           2·56│            160│         0·0927
+ Aluminium, Bronze    │           7·68│            475│          0·275
+ Antimony             │           6·71│            418│          0·242
+ Bismuth              │           9·90│            617│          0·357
+ Brass, Cast          │           8·10│            505│          0·293
+ „    Wire            │          8·548│            533│          0·309
+ Copper, Sheet        │          8·805│            549│          0·318
+ „    Wire            │          8·880│            554│          0·321
+ Gold                 │         19·245│           1200│          0·695
+ Gun metal            │           8·56│            534│          0·310
+ Iron, Wrought (mean) │          7·698│            480│          0·278
+ „   Cast (mean)      │          7·217│            450│          0·261
+ Lead, Milled Sheet   │         11·418│            712│          0·412
+ Manganese            │          8·012│            499│          0·289
+ Mercury              │         13·596│            849│          0·491
+ Nickel, Cast         │           8·28│            516│          0·300
+ Phosphor Bronze, Cast│           8·60│          536·8│          0·310
+ Platinum             │         21·522│           1342│          0·778
+ Silver               │         10·505│            655│          0·380
+ Steel (mean)         │          7·852│          489·6│          0·283
+ Tin                  │          7·409│            462│          0·268
+ Zinc, Sheet          │           7·20│            449│          0·260
+ „ Cast               │           6·86│            428│          0·248
+ ─────────────────────┴───────────────┴───────────────┴───────────────
+
+                       MISCELLANEOUS SUBSTANCES.
+                   ────────────┬──────────┬──────────
+                    SUBSTANCE. │ Specific │Weight of
+                               │ Gravity. │1 Cub. In.
+                               │          │  (Lb.).
+                   ────────────┼──────────┼──────────
+                   Asbestos    │ 2·1–2·80 │·076-·101
+                   Brick       │   1·90   │   ·069
+                   Cement      │2·72–3·05 │·0984-·109
+                   Clay        │   2·0    │   ·072
+                   Coal        │   1·37   │  ·0495
+                   Coke        │   0·5    │  ·0181
+                   Concrete    │   2·0    │   ·072
+                   Fire-brick  │   2·30   │   ·083
+                   Granite     │ 2·5–2·75 │·051-·100
+                   Graphite    │ 1·8–2·35 │·065-·085
+                   Sand-stone  │   2·3    │   ·083
+                   Slate       │   2·8    │   ·102
+                   Wood—       │          │
+                     Beech     │   0·75   │  ·0271
+                     Cork      │   0·24   │  ·0087
+                     Elm       │   0·58   │   ·021
+                     Fir       │   0·56   │  ·0203
+                     Oak       │ ·62-·85  │·025-·031
+                     Pine      │   0·47   │   ·017
+                     Teak      │   0·80   │   ·029
+                   ────────────┴──────────┴──────────
+
+                    ULTIMATE STRENGTH OE MATERIALS.
+ ──────────────────┬────────────┬────────────┬────────────┬────────────
+     MATERIAL.     │ Tension in │Compression │Shearing in │ Modulus of
+                   │lb. per sq. │ in lb. per │lb. per sq. │ Elasticity
+                   │    in.     │  sq. in.   │    in.     │ in lb. per
+                   │            │            │            │  sq. in.
+ ──────────────────┼────────────┼────────────┼────────────┼────────────
+ Cast Iron         │   11,000 to│   50,000 to│            │  14,000,000
+                   │      30,000│     130,000│            │          to
+                   │            │            │            │  23,000,000
+     „        aver.│      16,000│      95,000│      11,000│
+ Wrought Iron      │   40,000 to│            │            │  26,000,000
+                   │      70,000│            │            │          to
+                   │            │            │            │  31,000,000
+      „       aver.│      50,000│      50,000│      40,000│
+ Soft Steel        │   60,000 to│            │            │  30,000,000
+                   │     100,000│            │            │          to
+                   │            │            │            │  36,000,000
+ Soft Steel   aver.│      80,000│      70,000│      55,000│
+ Cast Steel   aver.│     120,000│            │            │  15,000,000
+                   │            │            │            │          to
+                   │            │            │            │  17,000,000
+ Copper, Cast      │      19,000│      58,000│            │
+ „     Wrought     │      34,000│            │            │  16,000,000
+ Brass, Cast       │      18,000│      10,500│            │   9,170,000
+ Gun Metal         │      34,000│            │            │  11,500,000
+ Phosphor Bronze   │      58,000│            │      43,000│  13,500,000
+ Wood, Ash         │      17,000│       9,300│       1,400│
+ „    Beech        │      16,000│       8,500│            │
+ „    Pine         │      11,000│       6,000│         650│   1,400,000
+ „    Oak          │      15,000│      10,000│       2,300│   1,500,000
+ Leather           │       4,200│            │            │      25,000
+ ──────────────────┴────────────┴────────────┴────────────┴────────────
+
+                        POWERS, ROOTS, ETC., OF USEFUL FACTORS.
+       _n_       │(1)/(_n_)│ _n_^2 │  _n_^3   │ √_̅n_ │ (1)/(√_̅n_) │ ∛_̅n_ │ (1)/(∛_̅n_)
+ ────────────────┼─────────┼───────┼──────────┼──────┬┴───────────┬─┴────┬──┴─────────
+    π = 3·142    │    0·318│  9·870│    31·006│ 1·772│       0·564│ 1·465│       0·683
+    2π= 6·283    │    0·159│ 39·478│   248·050│ 2·507│       0·399│ 1·845│       0·542
+ (π)/(2) = 1·571 │    0·637│  2·467│     3·878│ 1·253│       0·798│ 1·162│       0·860
+ (π)/(3) = 1·047 │    0·955│  1·097│     1·148│ 1·023│       0·977│ 1·016│       0·985
+ (4)/(3)π = 4·189│    0·239│ 17·546│    73·496│ 2·047│       0·489│ 1·612│       0·622
+ (π)/(4) = 0·785 │    1·274│  0·617│     0·484│ 0·886│       1·128│ 0·923│       1·084
+ (π)/(6) = 0·524 │    1·910│  0·274│     0·144│ 0·724│       1·382│ 0·806│       1·241
+   π^2 = 9·870   │    0·101│ 97·409│   961·390│ 3·142│       0·318│ 2·145│       0·466
+   π^3 = 31·006  │    0·032│961·390│29,809·910│ 5·568│       1·796│ 3·142│       0·318
+ (π)/(32) = 0·098│   10·186│ 0·0095│     0·001│ 0·313│       3·192│ 0·461│       2·168
+   _g_  = 32·2   │    0·031│1036·84│ 33,386·24│ 5·674│       0·176│ 3·181│       0·314
+   2_g_ = 64·4   │    0·015│4147·36│   267,090│ 8·025│       0·125│ 4·007│       0·249
+ ────────────────┴─────────┴───────┴──────────┴──────┴────────────┴──────┴────────────
+
+
+                         HYDRAULIC EQUIVALENTS.
+
+  1 foot head = 0·434 lb. per square inch.
+  1 lb. per square inch = 2·31 ft. head.
+  1 imperial gallon = 277·274 cubic inches.
+  1 imperial gallon = 0·16045 cubic foot.
+  1 imperial gallon = 10 lb.
+  1 cubic foot of water = 62·32 lb. = 6·232 imperial gallons.
+  1 cubic foot of sea water = 64·00 lb.
+  1 cubic inch of water = 0·03616 lb.
+  1 cubic inch of sea water = 0·037037 lb.
+  1 cylindrical foot of water = 48·96 lb.
+  1 cylindrical inch of water = 0·0284 lb.
+  A column of water 12 in. long 1 in. square = 0·434 lb.
+  A column of water 12 in. long 1 in. diameter = 0·340 lb.
+  Capacity of a 12 in. cube = 6·232 gallons.
+  Capacity of a 1 in. square 1 ft. long = 0·0434 gallon.
+  Capacity of a 1 ft. diameter 1 ft. long = 4·896 gallons.
+  Capacity of a cylinder 1 in. diameter 1 ft. long = 0·034 gallon.
+  Capacity of a cylindrical inch = 0·002832 gallon.
+  Capacity of a cubic inch = 0·003606 gallon.
+  Capacity of a sphere 12 in. diameter = 3·263 gallons.
+  Capacity of a sphere 1 in. diameter = 0·00188 gallon.
+  1 imperial gallon = 1·2 United States gallon.
+  1 imperial gallon = 4·543 litres of water.
+  1 United States gallon = 231·0 cubic inches.
+  1 United States gallon = 0·83 imperial gallon.
+  1 United States gallon = 3·8 litres of water.
+  1 cubic foot of water = 7·476 United States gallons.
+  1 cubic foot of water = 28·375 litres of water.
+  1 litre of water = 0·22 imperial gallon.
+  1 litre of water = 0·264 United States gallon.
+  1 litre of water = 61·0 cubic inches.
+  1 litre of water = 0·0353 cubic foot.
+
+ ─────────────────────────────────────────────────────────────────────────
+                    EQUIVALENTS OF POUNDS AVOIRDUPOIS.
+ ─┬───────┬────────────┬────────────────┬────────────────┬────────────────
+  │  10   │    100     │      1000      │     10,000     │    100,000
+ ─┼───────┼────────────┼────────────────┼────────────────┼────────────────
+  │qr. lb.│cwt. qr. lb.│ton cwt. qr. lb.│ton cwt. qr. lb.│ton cwt. qr. lb.
+ 1│  0  10│   0   3  16│  0    8   3  20│  4    9   1   4│ 44   12   3  12
+ 2│  0  20│   1   3   4│  0   17   3  12│  8   18   2   8│ 89    5   2  24
+ 3│  1   2│   2   2  20│  1    6   3   4│ 13    7   3  12│133   18   2   8
+ 4│  1  12│   3   2   8│  1   15   2  24│ 17   17   0  16│178   11   1  20
+ 5│  1  22│   4   1  24│  2    4   2  16│ 22    6   1  20│223    4   1   4
+ 6│  2   4│   5   1  12│  2   13   2   8│ 26   15   2  24│267   17   0  16
+ 7│  2  14│   6   1   0│  3    2   2   0│ 31    5   0   0│312   10   0   0
+ 8│  2  24│   7   0  16│  3   11   1  20│ 35   14   1   4│357    2   3  12
+ 9│  3   6│   8   0   4│  4    0   1  12│ 40    3   2   8│401   15   2  24
+ ─┴───────┴────────────┴────────────────┴────────────────┴────────────────
+
+
+                       TRIGONOMETRICAL FUNCTIONS.
+
+
+                        RIGHT-ANGLED TRIANGLES.
+
+[Illustration: [Right-angled Triangle]]
+
+Sin. A = (_a_)/(_b_) Sec. A = (_b_)/(_c_) Tan. A = (_a_)/(_c_)
+
+Cos. A = (_c_)/(_b_) Cosec. A = (_b_)/(_a_) Cotan. A = (_c_)/(_a_)
+
+Versin. A = (_b_ − _c_)/(_b_). Coversin. A = (_b_ − _a_)/(_b_).
+
+ ───────┬─────────┬─────────────────────────────────────────────────────
+ Given. │Required.│                      Formulæ.
+ ───────┼─────────┼─────────────────────────────────────────────────────
+ _a_,_b_│ A,C,_c_ │Sin. A = (_a_)/(_b_)    Cos. C = (_a_)/(_b_)    _c_ =
+        │         │                √((_b + a_)(_b − a_))
+        │         │
+ _a_,_c_│ A,C,_b_ │Tan. A = (_a_)/(_c_)  Cotan. B = (_a_)/(_c_)    _b_ =
+        │         │                  √(_a_^2 + _c_^2)
+        │         │
+  A,_a_ │C,_c_,_b_│     C = 90° − A    _c_ = _a_ × Cotan. A  _b_ =
+        │         │                   (_a_)/(Sin. A)
+        │         │
+  A,_b_ │C,_a_,_c_│C = 90° − A    _a_ = _b_ × Sin. A    _c_ = _b_ × Cos.
+        │         │                          A
+        │         │
+  A,_c_ │C,_a_,_b_│     C = 90° − A    _a_ = _c_ × Tan. A    _b_ =
+        │         │                   (_c_)/(Cos. A)
+        │         │
+ ───────┴─────────┴─────────────────────────────────────────────────────
+
+
+                       OBLIQUE-ANGLED TRIANGLES.
+
+_s_ = ½(_a + b + c_)
+
+[Illustration: [Oblique-angled Triangle]]
+
+ ───────────┬─────────┬─────────────────────────────────────────────────
+   Given.   │         │                    Formulæ.
+ ───────────┼─────────┼─────────────────────────────────────────────────
+  A,B,C,_a_ │  Area=  │(_a_^2 × Sin. B × Sin. C) ÷ 2 Sin. A
+  A,_b_,_c_ │    „    │½(_c_ × _b_ × Sin. A)
+ _a_,_b_,_c_│    „    │√(_s_(_s_ − _a_)(_s_ − _b_)(_s_ − _c_))
+ ───────────┼─────────┼─────────────────────────────────────────────────
+   Given.   │Required.│                    Formulæ.
+ ───────────┼─────────┼─────────────────────────────────────────────────
+   A,C,_a_  │   _c_   │           _c_ = _a_(Sin. C)/(Sin. A)
+ ───────────┼─────────┼─────────────────────────────────────────────────
+  A,_a_,_c_ │    C    │           Sin. C = (_c_ Sin. A)/(_a_)
+ ───────────┼─────────┼─────────────────────────────────────────────────
+  _a_,_c_,B │    A    │    Tan. A = (_a_ Sin. B)/(_c_ − _a_ Cos. B)
+ ───────────┼─────────┼─────────────────────────────────────────────────
+ _a_,_b_,_c_│    A    │Sin. ½A = √(((_s_ − _b_)(_s_ − _c_))/(_b_ × _c_))
+      „     │    „    │   Cos. ½A = √((_s_(_s_ − _a_))/(_b_ × _c_));
+      „     │    „    │ Tan. ½A = √(((_s_ − _b_)(_s_ − _c_))/(_s_(_s_ −
+            │         │                     _a_)))
+ ───────────┴─────────┴─────────────────────────────────────────────────
+
+
+                            COMPOUND ANGLES.
+
+  Sin. (A + B) = Sin. A Cos. B + Cos. A Sin. B.
+  Sin. (A − B) = Sin. A Cos. B − Cos. A Sin. B.
+  Cos. (A + B) = Cos. A Cos. B − Sin. A Sin. B.
+  Cos. (A − B) = Cos. A Cos. B + Sin. A Sin. B.
+
+Tan. (A + B) = (Tan. A + Tan. B)/(1 − Tan. A Tan. B).
+
+Tan. (A − B) = (Tan. A − Tan. B)/(1 + Tan. A Tan. B).
+
+
+       SLIDE RULE DATA SLIPS, COMPILED BY C. N. PICKWORTH, WH.SC.
+
+ (_It is suggested that this page be removed by cutting through the above
+ line, and selected portions of the Sectional Data Slips attached to the
+                        back of the Slide Rule._)
+
+                             ¹⁄₃₂  │0·03125
+                             ¹⁄₁₆  │0·0625
+                             ³⁄₃₂  │0·09375
+                               ⅛   │0·125
+                             ⁵⁄₃₂  │0·15625
+                             ³⁄₁₆  │0·1875
+                             ⁷⁄₃₂  │0·21875
+                               ¼   │0·25
+                             ⁹⁄₃₂  │0·28125
+                             ⁵⁄₁₆  │0·3125
+                             ¹¹⁄₃₂ │0·34375
+                               ⅜   │0·375
+                             ¹³⁄₃₂ │0·40625
+                             ⁷⁄₁₆  │0·4375
+                             ¹⁵⁄₃₂ │0·46875
+                             ¹⁷⁄₃₂ │0·53125
+                             ⁹⁄₁₆  │0·5625
+                             ¹⁹⁄₃₂ │0·59375
+                               ⅝   │0·625
+                             ²¹⁄₃₂ │0·65625
+                             ¹¹⁄₁₆ │0·6875
+                             ²³⁄₃₂ │0·71875
+                               ¾   │0·75
+                             ²⁵⁄₃₂ │0·78125
+                             ¹³⁄₁₆ │0·8125
+                             ²⁷⁄₃₂ │0·84375
+                               ⅞   │0·875
+                             ²⁹⁄₃₂ │0·90625
+                             ¹⁵⁄₁₆ │0·9375
+                             ³¹⁄₃₂ │0·96875
+
+Circ. of circle = 3·1416 _d_.
+
+Area „ „ = 0·7854 _d_^2.
+
+Sq. eq. area to cir., _s_ = 0·886 _d_.
+
+Circle eq. to sq., _d_ = 1·128 _s_.
+
+Sq. inscbd. in circ., _s_ = 0·707 _d_.
+
+Circsb. circ. of sq., _d_ = 1·414 _s_.
+
+Area of ellipse = 0.7854 _a_ × _b_.
+
+Surface of sphere = 3·1416 _d_^2.
+
+Volume „ „ = 0·5236 _d_^3.
+
+  „ „ cone = 0·2618 _d_^2 _h_.
+
+Radian = (180°)/(π) = 57·29 deg.
+
+Base of nat. or hyp. log. = e = 2·7183.
+
+Nat. or hyp. log. = com. log. × 2·3026.
+
+g (at London) 32·18 ft. per sec., per sec.
+
+Abs. temp. = deg. F. + 461° = deg. C. + 274°.
+
+C.° = (5)/(9)(F.° − 32°); F.° = (9)/(5)C.° + 32°.
+
+Cal. pr.—Ther. units per lb.: Coal, 14,300;
+
+  petrol’m, 20,000; coal gas per cu. ft., 700.
+
+Sp. heat:—Wt. iron, 0·1138; C.I., 0·1298;
+
+  copper, brass, 0·095; lead, 0·0314.
+
+Inch = 25·4 mil’metres; mil’metre = 0·03937 in.
+
+Foot = 0·3048 metres; metre = 3·2809 feet.
+
+Yard = 0·91438 metre; metre = 1·0936 yards.
+
+Mile = 1·6093 kilomtrs.; kilomtr. = 0·6213 mile.
+
+Sq. in. = 6·4513 sq. cm.; sq. cm. = 0·155 sq. in.
+
+Sq. ft. = 9·29 sq. decmtr.; sq. decmtr. = 0·1076 sq. ft.
+
+Sq. yd. = 0·836 sq. metre; sq. metre = 1·196 sq. yds.
+
+Sq. ml. = 258·9 hectares; hectare = 0·00386 sq. ml.
+
+Cu. in. = 16·386 c. cm.; c. cm. = 0·06102 cu. in.
+
+Cu. ft. = 0·0283 c. metre; c. metre = 35·316 cu. ft.
+
+Grain = 0·0648 gramme; gram. = 15·43 grs.
+
+Ounce = 28·35 grams.; „ = 0·03527 oz.
+
+Pound = 0·4536 kilogm.; kilogm. = 2·204 lb.
+
+Ton = 1·016 tonnes; tonne = 0·9842 ton.
+
+Mile per hr. = 1·466 ft., or 44·7 cm., per sec.
+
+Lb. per cu. in. = 0·0276 kilogram per cu. cm.
+
+Kilogram per cu. cm. = 36·125 lb. per cu. in.
+
+Lb. per cu. ft. = 16·019 kilogm. per cu. mtre.
+
+Grain per gall. = 0·01426 gramme per litre.
+
+Gramme per litre = 70·116 grains per gall.
+
+                   Ultimate Strength│Lb. per Sq. in.
+                           „        │Tens’n.│Comp’n.
+                   ─────────────────┼───────┼───────
+                   Wt. iron         │ 50,000│ 50,000
+                   Cast „           │ 16,000│ 95,000
+                   Steel            │ 80,000│ 70,000
+                   Copper           │ 21,000│ 50,000
+                   Brass            │ 18,000│ 10,500
+                   Lead             │  2,500│  7,000
+                   Pine             │ 11,000│  6,000
+                   Oak              │ 15,000│ 10,000
+
+           Weight of Metals.│ Cub. In. │ Cub. Ft. │12 Cu. In.
+           ─────────────────┼──────────┼──────────┼──────────
+           Wt. iron         │     0·277│       480│      3·33
+           Cast „           │     0·260│       450│      3·12
+           Steel            │     0·283│       490│      3·40
+           Copper           │     0·318│       550│      3·82
+           Brass            │     0·300│       520│      3·61
+           Zinc             │     0·248│       430│      2·98
+           Alumin’m         │     0.096│       168│      1·16
+           Lead             │     0.411│       710│      4·93
+
+ Lb. per sq. in. = 2·31 ft. water = 2·04 in. mercury = 0·0703 kilo. per
+    sq. cm.
+ Atmosphere = 14·7 lb. per sq. in. = 33·94 ft. water = 1·0335 „ „
+ Ft. hd. water = 0·433 lb. per sq. in. = 62·35 lb. per sq. ft. = 0·0304 „
+    „
+ Cub. ft. of water = 62·35 lb. = 0·0278 ton = 28·315 litres = 7·48 U.S.
+    galls.
+ Gall. (Imp.) = 277·27 cu. in. = 0·1604 cu. ft. = 10 lb. water = 4·544
+    litres.
+ Litre = 1·76 pints = 0·22 gall. = 61 cu. in. = 0·0353 cu. ft. = 0·264
+    U.S. gall.
+ Horse-power = 33,000 ft.-lb. per min. = 0·746 kilowatt = 42·4 heat units
+    per min.
+ Heat unit = 778 ft.-lb. = 1055 watt-sec. = 107·5 kilogrammetres = 0·252
+    calorie.
+ Foot-pound = 0·00129 heat unit = 1·36 joules = 0·1383 kilogrammetres.
+ Kilowatt = 1·34 H.P. = 44,240 ft.-lb. per min. = 3412 heat units per
+    hour.
+
+-----
+
+Footnote 1:
+
+  It will be recognised that n is the characteristic of the logarithm of
+  the original number.
+
+Footnote 2:
+
+  The special case in which the numerator is 1, 10, or any power of 10
+  must be treated by the rule for reciprocals (page 27).
+
+Footnote 3:
+
+  The possible need for traversing the slide, to change the indices,
+  when using the C and D scales, is not considered as a setting.
+
+Footnote 4:
+
+  The reader may be reminded that cross-multiplication of the factors in
+  any such slide rule setting will give a constant product, _e.g._, 20 ×
+  94·5 = 27 × 70.
+
+Footnote 5:
+
+  In this case cross _dividing_ gives a constant quotient, _e.g._, 8 ÷ 3
+  = 4 ÷ 1·5. Since the upper scale is now a scale of reciprocals, the
+  ratio is really
+
+                               O   ⅛   ¼
+                              ───────────
+                               D  1·5  3
+
+Footnote 6:
+
+  These lines should not be brought to the working edge of the scale but
+  should terminate in the horizontal line which forms the border of the
+  finer graduations, their value being read into the calculation by
+  means of the cursor (see page 55).
+
+Footnote 7:
+
+  The same principle may be applied to the cursor.
+
+Footnote 8:
+
+  Philosophical Transactions of the Royal Society, 1815.
+
+------------------------------------------------------------------------
+
+
+
+
+                         _BY THE SAME AUTHOR._
+
+
+                       LOGARITHMS FOR BEGINNERS.
+
+“An extremely useful and much-needed little work, giving a complete
+explanation of the theory and use of logarithms, by a teacher of great
+clearness and good style.”—_The Mining Journal._
+
+                           1s. 8d. Post Free.
+
+
+                        THE INDICATOR HANDBOOK.
+
+Comprising “The Indicator: Its Construction and Application” and “The
+Indicator Diagram: Its Analysis and Calculation.” Complete in One
+Volume.
+
+                          7s. 10d. Post Free.
+
+“Mr. Pickworth’s judgment is always sound, and is evidently derived from
+a personal acquaintance with indicator work.”—_The Engineer._
+
+
+          POWER COMPUTER FOR STEAM, GAS AND OIL ENGINES, Etc.
+
+“Accurate, expeditious and thoroughly practical.... We can confidently
+recommend it, and engineers will find it a great boon in undertaking
+tests, etc.”—_The Electrician._
+
+                           7s. 6d. Post Free.
+
+------------------------------------------------------------------------
+
+
+
+
+                            ADVERTISEMENTS.
+
+
+                        LOGARITHMS FOR BEGINNERS
+
+For a full and intelligent appreciation of the Slide Rule and its
+various applications an elementary knowledge of logarithms is necessary.
+All that is required will be found in this little work, which gives a
+simple, detailed and practical explanation of logarithms and their uses,
+particular care having been taken to elucidate all difficult points by
+the aid of a number of worked examples.
+
+                  Seventh Edition, 1s. 8d. Post Free.
+
+
+                             POWER COMPUTER
+                                  for
+                   STEAM, GAS, AND OIL ENGINES, Etc.
+
+Gives The Indicated Horse-Power of Steam, Gas, and Oil Engines—The Brake
+Horse-Power of Steam, Gas, and Oil Engines—The Size of Engine Necessary
+to Develop any Given Power—The Mechanical Efficiency of an Engine—The
+Ratio of Compound Engine Cylinders—The Piston Speed of an Engine—The
+Delivery of Pumps with any Efficiency—The Horse-Power of Belting—The Rim
+Speeds of Wheels, Speeds of Ropes, Belts, etc.—Speed Ratios of Pulleys,
+Gearing, etc.
+
+       Pocket size, in neat case, with instructions and examples.
+
+                        Post Free, 7s. 6d. net.
+
+
+                C. N. PICKWORTH, Withington, Manchester
+
+ ┌─────────────────────────────────────────────────────────────────────┐
+ │         W. P. THOMPSON,                     G. C. DYMOND,           │
+ │ F.C.S., M.I.Mech.E., F.I.C.P.A.        M.I.Mech.E., F.I.C.P.A.      │
+ │                                                                     │
+ │                        W. P. Thompson & Co.,                        │
+ │                    12 CHURCH STREET, LIVERPOOL,                     │
+ │                      CHARTERED PATENT AGENTS.                       │
+ │                                                                     │
+ │           H. E. POTTS,                     J. V. ARMSTRONG,         │
+ │  M.Sc., Hon. Chem., F.I.C.P.A.          M.Text.I., F.I.C.P.A.       │
+ │                                                                     │
+ │                        W. H. BEESTON, R.P.A.                        │
+ └─────────────────────────────────────────────────────────────────────┘
+
+------------------------------------------------------------------------
+
+
+
+
+                                BRITISH
+                              SLIDE RULES
+
+                                                           for all
+                                                             ARTS and
+                                                             INDUSTRIES
+
+                                                               including
+
+[Illustration: [Slide Rule]]
+
+  _LOG-O-LOG
+  DR. YOKOTA’S
+  SURVEYORS’
+  WIRELESS
+  GUNNERY
+  ELECTRICAL RULES, Etc._
+
+                            SEND FOR LIST 55
+
+MADE BY—
+
+                     JOHN DAVIS & SON (Derby), Ltd.
+                        ALL SAINTS’ WORKS, DERBY
+
+------------------------------------------------------------------------
+
+
+
+
+                           K & E Slide Rules
+
+are constantly growing in popularity, and they can now be obtained from
+the leading houses in our line throughout the United Kingdom.
+
+[Illustration: [Slide Rule]]
+
+  We manufacture a complete line of ENGINE-DIVIDED SLIDE RULES, and call
+special attention to our Patent Adjustment, ensuring smooth working of
+the Slide; also to our new “Frameless” Indicator, which hides no figures
+on the Rule.
+
+[Illustration: [Thacher’s Calculating Instrument]]
+
+THACHER’S CALCULATING INSTRUMENT, for solving problems in
+multiplication, division, or combinations of the two; has upwards of
+33,000 divisions. Results can be obtained to the fourth and usually to
+the fifth place of figures with a surprising degree of accuracy.
+
+                              We also make
+
+ ALL METAL, CIRCULAR, STADIA, CHEMISTS’, ELECTRICAL, and OTHER SPECIAL
+                              SLIDE RULES
+                   _DESCRIPTIVE CIRCULARS ON REQUEST_
+
+                          KEUFFEL & ESSER CO.
+  127 Fulton St., NEW YORK      General Office and Factories, HOBOKEN,
+                                  N.J.
+             CHICAGO − ST. LOUIS − SAN FRANCISCO − MONTREAL
+
+                          _DRAWING MATERIALS_
+                _MATHEMATICAL and SURVEYING INSTRUMENTS_
+                           _MEASURING TAPES_
+
+------------------------------------------------------------------------
+
+
+
+
+[Illustration: 6 in. Standard with magnifying Cursor complete in pocket
+case, 5/-]
+
+                                 NORTON
+                                   &
+                                GREGORY
+                                  LTD.
+
+
+                              Head Office
+
+          CASTLE LANE, WESTMINSTER,            LONDON, S.W. 1
+
+                                Branches
+
+                       71 QUEEN STREET, GLASGOW.
+   PHOENIX HOUSE, QUEEN STREET and       SANDHILL, NEWCASTLE-ON-TYNE.
+
+
+                SLIDE RULES in Stock, from 17/6 to 27/6
+
+             Special Quotations to the Trade for Quantities
+
+
+  For particulars of Surveying, Measuring and Mathematical Instruments,
+      Appliances and Material of all kinds for the Drawing Office, write
+      to the Head Office.
+
+[Illustration: NORTON & GREGORY LTD]
+
+------------------------------------------------------------------------
+
+
+
+
+                        NORTON & GREGORY, LTD.,
+                                London.
+
+
+                               “DIAMOND”
+                          DRAWING INSTRUMENTS
+                   Manufactured at our London Works.
+
+                   CENTRE SCREW SPRING BOW HALF SET.
+
+[Illustration: [Centre Screw Spring Bow Half Set]]
+
+4 inch Spring Bow Half Set centre screw adjustment, with interchangeable
+needle, pen, and pencil points Price 17/6
+
+The Centre Screw Spring Bow Half Set of Compasses, as illustrated,
+possesses the advantage of COMBINING IN ONE INSTRUMENT THE SET OF THREE
+SEPARATE SPRING BOWS hitherto in use, while the centre screw makes for
+ease and accuracy of manipulation, at the same time providing a radius
+of over 2 inches, or double that of the old pattern.
+
+This instrument is less expensive than the set of 3 bows, while
+considerably stronger in construction.
+
+The fixed needle point is shouldered.
+
+This illustration is given as an indication of the various Drawing
+Instruments manufactured by us.
+
+Illustrated Booklet giving full particulars and prices of other
+Instruments and Cases of Instruments sent on application.
+
+
+   Specially arranged Sets of Instruments made for Colleges, Schools,
+                          Technical Institutes
+
+                  Estimates submitted on Application.
+
+
+                      _Write to our Head Office_:
+               CASTLE LANE, WESTMINSTER, LONDON, S.W. 1.
+
+------------------------------------------------------------------------
+
+
+                   DRAWING AND SURVEYING INSTRUMENTS
+
+                           A. G. THORNTON Ltd.
+     SLIDE RULES FOR          Paragon Works        ACCURATE SECTIONAL
+        ENGINEERS            2 King St. West        PAPERS AND CLOTHS
+                               MANCHESTER
+
+  D 1916 Illustrated Catalogue, just published, in two editions; Drawing
+     Office (448 pages); Draughtsman’s (160 pages): the most complete
+                         Catalogues in the trade.
+
+              _CONTRACTORS TO H.M. WAR OFFICE AND ADMIRALTY_
+ _Manufacturers also of Drawing Materials and Drawing Office Stationary._
+
+          (ALSO AT MINERVA WORKS AND ALBERT MILLS, MANCHESTER.)
+
+
+                        MATHEMATICAL INSTRUMENTS
+                         SURVEYING INSTRUMENTS
+                              SLIDE RULES
+                       For Students and Engineers
+
+     MANNHEIM, POLYPHASE, DUPLEX, ELECTRICAL, LOG-LOG, AND CALCULEX
+
+                           J. H. STEWARD LTD.
+                      Scientific Instrument Makers
+
+                    406 STRAND, and 457 WEST STRAND
+                             LONDON, W.C. 2
+
+------------------------------------------------------------------------
+
+
+
+
+                    _A True Friend and Trusty Guide_
+
+
+                                  THE
+                           ‘HALDEN CALCULEX’
+
+[Illustration: [Halden Calculex]]
+
+                     ACTUAL SIZE| | | BRITISH MADE
+
+           The handiest and most perfect form of Slide Rule.
+            Does all that can be done with a straight rule.
+              Complete in Case, with book of instructions,
+                            27/6 post free.
+
+           J. HALDEN & CO., LTD., 8 ALBERT SQUARE MANCHESTER
+
+   _Depots_—London, Newcastle-on-Tyne, Birmingham, Glasgow, and Leeds
+
+------------------------------------------------------------------------
+
+
+[Illustration: [Rope]]
+
+                              ENGINEERING,
+                               SURVEYING
+                                  AND
+                              MATHEMATICAL
+                              INSTRUMENTS,
+                                  ETC.
+
+
+                              SLIDE RULES.
+
+                        JOSEPH CASARTELLI & SON,
+                     43 MARKET STREET, MANCHESTER.
+               Tel. No. 2958 City.| | | Established 1790.
+
+ROPE DRIVING
+
+  Is the most EFFICIENT and most ECONOMICAL METHOD of Power
+  Transmission.
+
+The LAMBETH Cotton Driving Rope
+
+  Is the most EFFICIENT and most ECONOMICAL ROPE for Power Transmission.
+
+[Illustration: Made 4 Strand or 3 Strand.]
+
+SPECIAL FEATURES:
+
+  LESS STRETCH THAN ANY OTHER ROPE. MORE PLIABLE THAN ANY OTHER ROPE.
+  GREATER DRIVING POWER THAN ANY OTHER ROPE.
+
+              THOMAS HART LTD., Lambeth Works, BLACKBURN.
+
+------------------------------------------------------------------------
+
+
+
+
+                          TRANSCRIBER’S NOTES
+
+
+ Page           Changed from                      Changed to
+
+   24 the right, so the number of      the right, so the number of
+      digits in the answer = 3 − 2 × 1 digits in the answer = 3 − 2 + 1
+      = 2                              = 2
+
+  116 grammes, we have the equation,   grammes, we have the equation,
+      _x_ × (Cl.)/(Ag.Cl.) ×           _x_ = (Cl.)/(Ag.Cl.) ×
+      (_a_)/(_s_). Hence, the mark     (_a_)/(_s_). Hence, the mark
+
+ ● Typos fixed; non-standard spelling and dialect retained.
+ ● Used numbers for footnotes, placing them all at the end of the last
+     chapter.
+ ● Enclosed italics font in _underscores_.
+ ● Enclosed bold font in =equals=.
+ ● The caret (^) serves as a superscript indicator, applicable to
+     individual characters (like 2^d) and even entire phrases (like
+     1^{st}).
+ ● Subscripts are shown using an underscore (_) with curly braces { },
+     as in H_{2}O.
+
+
+
+*** END OF THE PROJECT GUTENBERG EBOOK 75904 ***
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+  <body>   
+<div style='text-align:center'>*** START OF THE PROJECT GUTENBERG EBOOK 75904 ***</div>
+
+<div class='tnotes covernote'>
+
+<p class='c000'><strong>Transcriber’s Note:</strong></p>
+
+<p class='c000'>New original cover art included with this eBook is granted to the public domain.</p>
+
+</div>
+
+<div class='titlepage'>
+
+<div>
+  <h1 class='c001'><span class='large'>THE</span><br> SLIDE RULE:<br> <span class='large'>A PRACTICAL MANUAL</span></h1>
+</div>
+
+<div class='nf-center-c0'>
+<div class='nf-center c002'>
+    <div><span class='small'>BY</span></div>
+    <div class='c003'><span class='xlarge'>CHARLES N. PICKWORTH</span></div>
+    <div class='c003'><span class='small'>WHITWORTH SCHOLAR; EDITOR OF “THE MECHANICAL WORLD”; AUTHOR OF “LOGARITHMS FOR BEGINNERS”; “THE INDICATOR: ITS CONSTRUCTION AND APPLICATION”; “THE INDICATOR DIAGRAM: ITS ANALYSIS AND CALCULATION,” ETC.</span></div>
+    <div class='c003'><em>SEVENTEENTH EDITION</em></div>
+    <div class='c003'><span class='sc'>Manchester:</span></div>
+    <div><span class='sc'>Emmott and Co., Limited,</span></div>
+    <div><span class='sc'>65 King Street;</span></div>
+    <div class='c003'><span class='sc'>New York:</span></div>
+    <div><span class='sc'>D. Van Nostrand Co.,</span></div>
+    <div><span class='sc'>8 Warren Street.</span></div>
+    <div class='c003'><span class='sc'>London:</span></div>
+    <div><span class='sc'>Emmott and Co., Limited,</span></div>
+    <div><span class='sc'>20 Bedford Street, W.C.</span></div>
+    <div class='c003'><span class='sc'>and</span></div>
+    <div><span class='sc'>Pitman and Sons, Limited,</span></div>
+    <div><span class='sc'>Parker St., Kingsway, W.C. 2.</span></div>
+    <div class='c003'>[<em>Three Shillings and Sixpence net</em>]</div>
+  </div>
+</div>
+
+</div>
+
+<div class='nf-center-c0'>
+<div class='nf-center c004'>
+    <div><em>All rights reserved.</em></div>
+  </div>
+</div>
+
+<div class='chapter'>
+  <h2 class='c005'>PREFACE TO THE FIFTEENTH EDITION.</h2>
+</div>
+
+<p class='drop-capa0_0_6 c006'>Several new slide rules for special calculations are described in
+this edition, and the contents further extended to include a
+section dealing with screw-cutting gear calculations by the slide rule—an
+application of the instrument to which attention has been given
+recently.</p>
+
+<p class='c007'>Mention should be made of the fact that some of the special slide
+rules described in previous editions are no longer obtainable. As,
+however, the descriptive notes may be of service to those possessing
+the instruments, and are, in some measure, of general interest, they
+have been allowed to remain in the present issue.</p>
+
+<p class='c007'>The author tenders his thanks to the many who have evinced their
+appreciation of his efforts to popularise the subject; also for the many
+kind hints and suggestions which he has received from time to time,
+and with a continuance of which he trusts to be favoured in the
+future.</p>
+
+<div class='lg-container-r'>
+  <div class='linegroup'>
+    <div class='group'>
+      <div class='line'>C. N. P.</div>
+    </div>
+  </div>
+</div>
+
+<p class='c007'><span class='sc'>Withington, Manchester</span>, <em>November 1917</em>.</p>
+
+<h3 class='c008'>PREFACE TO THE SEVENTEENTH EDITION.</h3>
+
+<p class='c009'>The sustained demand for this very successful work having
+resulted in the early call for a new edition, the opportunity
+has been taken to introduce descriptions of new slide rules and to
+effect some slight revisions.</p>
+
+<div class='lg-container-r'>
+  <div class='linegroup'>
+    <div class='group'>
+      <div class='line'>C. N. P.</div>
+    </div>
+  </div>
+</div>
+
+<p class='c007'><span class='sc'>Withington, Manchester</span>, <em>December 1920</em>.</p>
+
+<div class='chapter'>
+  <h2 class='c005'>CONTENTS.</h2>
+</div>
+
+<table class='table0'>
+  <tr>
+    <th class='c010' colspan='2'>&#160;</th>
+    <th class='c011'>PAGE</th>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Introductory</td>
+    <td class='c011'><a href='#Page_5'>5</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>The Mathematical Principle of the Slide Rule</td>
+    <td class='c011'><a href='#Page_6'>6</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Notation by Powers of 10</td>
+    <td class='c011'><a href='#Page_8'>8</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>The Mechanical Principle of the Slide Rule</td>
+    <td class='c011'><a href='#Page_9'>9</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>The Primitive Slide Rule</td>
+    <td class='c011'><a href='#Page_10'>10</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>The Modern Slide Rule</td>
+    <td class='c011'><a href='#Page_12'>12</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>The Notation of the Slide Rule</td>
+    <td class='c011'><a href='#Page_14'>14</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>The Cursor or Runner</td>
+    <td class='c011'><a href='#Page_17'>17</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Multiplication</td>
+    <td class='c011'><a href='#Page_19'>19</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Division</td>
+    <td class='c011'><a href='#Page_24'>24</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>The Use of the Upper Scales for Multiplication and Division</td>
+    <td class='c011'><a href='#Page_26'>26</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Reciprocals</td>
+    <td class='c011'><a href='#Page_27'>27</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Continued Multiplication and Division</td>
+    <td class='c011'><a href='#Page_28'>28</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Multiplication and Division with the Slide Inverted</td>
+    <td class='c011'><a href='#Page_30'>30</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Proportion</td>
+    <td class='c011'><a href='#Page_31'>31</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>General Hints on the Elementary Uses of the Slide Rule</td>
+    <td class='c011'><a href='#Page_36'>36</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Squares and Square Roots</td>
+    <td class='c011'><a href='#Page_37'>37</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Cubes and Cube Roots</td>
+    <td class='c011'><a href='#Page_40'>40</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Miscellaneous Powers and Roots</td>
+    <td class='c011'><a href='#Page_45'>45</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Power and Roots by Logarithms</td>
+    <td class='c011'><a href='#Page_45'>45</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Other Methods of Obtaining Powers and Roots</td>
+    <td class='c011'><a href='#Page_47'>47</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Combined Operations</td>
+    <td class='c011'><a href='#Page_49'>49</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Hints on Evaluating Expressions</td>
+    <td class='c011'><a href='#Page_52'>52</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Gauge Points</td>
+    <td class='c011'><a href='#Page_53'>53</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Examples in Technical Calculations</td>
+    <td class='c011'><a href='#Page_56'>56</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Trigonometrical Application</td>
+    <td class='c011'><a href='#Page_74'>74</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Slide Rules with Log-log Scales</td>
+    <td class='c011'><a href='#Page_84'>84</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Special Types of Slide Rules</td>
+    <td class='c011'><a href='#Page_92'>92</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Long-Scale Slide Rules</td>
+    <td class='c011'><a href='#Page_96'>96</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Circular Calculators</td>
+    <td class='c011'><a href='#Page_101'>101</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Slide Rules for Special Calculations</td>
+    <td class='c011'><a href='#Page_109'>109</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Construction Improvements in Slide Rules</td>
+    <td class='c011'><a href='#Page_110'>110</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>The Accuracy of Slide Rule Results</td>
+    <td class='c011'><a href='#Page_111'>111</a></td>
+  </tr>
+  <tr>
+    <td class='c010' colspan='2'>Appendix:—</td>
+    <td class='c011'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c010'>&#160;</td>
+    <td class='c010'>New Slide Rules</td>
+    <td class='c011'><a href='#Page_113'>113</a></td>
+  </tr>
+  <tr>
+    <td class='c010'>&#160;</td>
+    <td class='c010'>The Solution of Algebraic Equations</td>
+    <td class='c011'><a href='#Page_122'>122</a></td>
+  </tr>
+  <tr>
+    <td class='c010'>&#160;</td>
+    <td class='c010'>Screw-Cutting Gear Calculations</td>
+    <td class='c011'><a href='#Page_124'>124</a></td>
+  </tr>
+  <tr>
+    <td class='c010'>&#160;</td>
+    <td class='c010'>Gauge Points and Signs on Slide Rules</td>
+    <td class='c011'><a href='#Page_126'>126</a></td>
+  </tr>
+  <tr>
+    <td class='c010'>&#160;</td>
+    <td class='c010'>Tables and Data</td>
+    <td class='c011'><a href='#Page_128'>128</a></td>
+  </tr>
+  <tr>
+    <td class='c010'>&#160;</td>
+    <td class='c010'>Slide Rule Data Slips</td>
+    <td class='c011'><a href='#Page_133'>133</a></td>
+  </tr>
+</table>
+
+<div class='chapter ph1'>
+
+<div class='nf-center-c0'>
+<div class='nf-center c004'>
+    <div>THE SLIDE RULE.</div>
+  </div>
+</div>
+
+</div>
+
+<div>
+  <span class='pageno' id='Page_5'>5</span>
+  <h2 class='c005'>INTRODUCTORY.</h2>
+</div>
+
+<p class='drop-capa0_0_6 c006'>The slide rule may be defined as an instrument for mechanically
+effecting calculations by logarithms. Those familiar
+with logarithms and their use will recognise that the slide rule
+provides what is in effect a concisely arranged table of logarithms,
+together with a simple and convenient means for adding and
+subtracting any selected values. Those, however, who have no
+acquaintance with logarithms will find that only an elementary
+knowledge of the subject is necessary to enable them to make full
+use of the slide rule. It is true that for simple slide-rule operations,
+as multiplication and division, a knowledge of logarithms is
+unnecessary; indeed, many who have no conscious understanding
+of logarithms make good use of the instrument. But this involves
+a blind reliance upon rules without an appreciation of their origin
+or limitations, and this, in turn, engenders a want of confidence in
+the results of any but the simplest operations, and prevents the
+fullest use being made of the instrument. For this reason a brief,
+but probably sufficient <em>résumé</em> of the principles of logarithmic
+calculation will be given. Those desiring a more detailed explanation
+are referred to the writer’s “Logarithms for Beginners.”</p>
+
+<p class='c007'>The slide rule enables various arithmetical, algebraical and
+trigonometrical processes to be performed with ease and rapidity,
+and with sufficient accuracy for most practical purposes. A grasp
+of the simple fundamental principles which underlie its operation,
+together with a little patient practice, are all that are necessary to
+acquire facility in using the instrument, and few who have become
+proficient in this system of calculating would willingly revert to
+the laborious arithmetical processes.</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_6'>6</span>
+  <h2 class='c005'>THE MATHEMATICAL PRINCIPLE OF THE SLIDE RULE.</h2>
+</div>
+
+<p class='c012'>Logarithms may be defined as a series of numbers in <em>arithmetical</em>
+progression, as 0, 1, 2, 3, 4, etc., which bear a definite relationship
+to another series of numbers in <em>geometrical</em> progression, as 1, 2, 4,
+8, 16, etc. A more precise definition is:—The logarithm of a
+number to any base, is the <em>index of the power</em> to which the base
+must be raised to equal the given number. In the logarithms in
+general use, known as <em>common logarithms</em>, and with which we are
+alone concerned, 10 is the base selected. The general definition
+may therefore be stated in the following modified form:—<em>The
+common logarithm of a number is the index of the power to which
+10 must be raised to equal the given number.</em> Applying this rule
+to a simple case, as 100 = 10<sup>2</sup>, we see that the base 10 must be
+squared (<em>i.e.</em>, raised to the 2nd power) in order to equal 100, the
+number selected. Therefore, as 2 is the index of the power to
+which 10 must be raised to equal 100, it follows from our definition
+that 2 is the common logarithm of 100. Similarly the common
+logarithm of 1000 will be 3, while proceeding in the opposite
+direction the common log. of 10 must equal 1. Tabulating these
+results and extending, we have:—</p>
+
+<table class='table1'>
+  <tr>
+    <td class='c010'>Numbers</td>
+    <td class='c013'>10,000</td>
+    <td class='c013'>1000</td>
+    <td class='c013'>100</td>
+    <td class='c013'>10</td>
+    <td class='c014'>1</td>
+  </tr>
+  <tr>
+    <td class='c010'>Logarithms</td>
+    <td class='c013'>4</td>
+    <td class='c013'>3</td>
+    <td class='c013'>2</td>
+    <td class='c013'>1</td>
+    <td class='c014'>0</td>
+  </tr>
+</table>
+
+<p class='c007'>It will now be evident that for numbers</p>
+
+<table class='table1'>
+  <tr>
+    <td class='c013'>between</td>
+    <td class='c015'>1</td>
+    <td class='c013'>and</td>
+    <td class='c015'>10</td>
+    <td class='c013'>the logs. will be between</td>
+    <td class='c013'>0</td>
+    <td class='c013'>and</td>
+    <td class='c014'>1</td>
+  </tr>
+  <tr>
+    <td class='c013'>„</td>
+    <td class='c015'>10</td>
+    <td class='c013'>„</td>
+    <td class='c015'>100</td>
+    <td class='c013'>„&#8196; &#8196; &#8196; &#8196; &#8196; „</td>
+    <td class='c013'>1</td>
+    <td class='c013'>„</td>
+    <td class='c014'>2</td>
+  </tr>
+  <tr>
+    <td class='c013'>„</td>
+    <td class='c015'>100</td>
+    <td class='c013'>„</td>
+    <td class='c015'>1000</td>
+    <td class='c013'>„&#8196; &#8196; &#8196; &#8196; &#8196; „</td>
+    <td class='c013'>2</td>
+    <td class='c013'>„</td>
+    <td class='c014'>3</td>
+  </tr>
+  <tr>
+    <td class='c013'>„</td>
+    <td class='c015'>1000</td>
+    <td class='c013'>„</td>
+    <td class='c015'>10,000</td>
+    <td class='c013'>„&#8196; &#8196; &#8196; &#8196; &#8196; „</td>
+    <td class='c013'>3</td>
+    <td class='c013'>„</td>
+    <td class='c014'>4</td>
+  </tr>
+</table>
+
+<p class='c007'>In other words, the logarithms of numbers between 1 and 10 will
+be wholly fractional (<em>i.e.</em>, decimal); the logs. of numbers between
+10 and 100 will be 1 <em>followed by a decimal quantity</em>; the logs. of
+numbers between 100 and 1000 will be 2 followed by a decimal
+quantity, and so on. These decimal quantities for numbers from
+1 to 10 (which are the logarithms of this particular series) are
+as follows:—</p>
+
+<table class='table1'>
+  <tr>
+    <td class='c010'>Numbers</td>
+    <td class='c013'>1</td>
+    <td class='c013'>2</td>
+    <td class='c013'>3</td>
+    <td class='c013'>4</td>
+    <td class='c013'>5</td>
+    <td class='c013'>6</td>
+    <td class='c013'>7</td>
+    <td class='c013'>8</td>
+    <td class='c013'>9</td>
+    <td class='c014'>10</td>
+  </tr>
+  <tr>
+    <td class='c010'>Logarithms</td>
+    <td class='c013'>0</td>
+    <td class='c013'>0·301</td>
+    <td class='c013'>0·477</td>
+    <td class='c013'>0·602</td>
+    <td class='c013'>0·699</td>
+    <td class='c013'>0·778</td>
+    <td class='c013'>0·845</td>
+    <td class='c013'>0·903</td>
+    <td class='c013'>0·954</td>
+    <td class='c014'>1·000</td>
+  </tr>
+</table>
+
+<p class='c016'><span class='pageno' id='Page_7'>7</span>Combining the two tables, we can complete the logarithms. Thus
+for 3 multiplied successively by 10, we have:—</p>
+
+<table class='table1'>
+  <tr>
+    <td class='c010'>Numbers</td>
+    <td class='c017'>3</td>
+    <td class='c017'>30</td>
+    <td class='c017'>300</td>
+    <td class='c017'>3000</td>
+    <td class='c017'>30,000</td>
+    <td class='c018' rowspan='2'>etc.</td>
+  </tr>
+  <tr>
+    <td class='c010'>Logarithms</td>
+    <td class='c017'>0·477</td>
+    <td class='c017'>1·477</td>
+    <td class='c017'>2·477</td>
+    <td class='c017'>3·477</td>
+    <td class='c017'>4·477</td>
+    
+  </tr>
+</table>
+
+<p class='c007'>We see from this that for numbers having the <em>same significant
+figure</em> (or figures), 3 in this case, the decimal part or <em>mantissa</em> of
+the logarithm is the same, but that the integral part or <em>characteristic</em>
+is always <em>one less than the number of figures before the decimal
+point</em>.</p>
+
+<p class='c007'>For numbers less than 1 the same plan is followed. Thus
+extending our first table downwards, we have:—</p>
+
+<table class='table1'>
+  <tr>
+    <td class='c010'>Numbers</td>
+    <td class='c017'>1</td>
+    <td class='c017'>0·1</td>
+    <td class='c017'>0·01</td>
+    <td class='c017'>0·001</td>
+    <td class='c017'>0·0001</td>
+    <td class='c018' rowspan='2'>etc.</td>
+  </tr>
+  <tr>
+    <td class='c010'>Logarithms</td>
+    <td class='c017'>0</td>
+    <td class='c017'>−1</td>
+    <td class='c017'>−2</td>
+    <td class='c017'>−3</td>
+    <td class='c017'>−4</td>
+    
+  </tr>
+</table>
+<p class='c016'>so that for 3 divided successively by 10, we have:—</p>
+
+<table class='table1'>
+  <tr>
+    <td class='c010'>Numbers</td>
+    <td class='c017'>3</td>
+    <td class='c017'>0·3</td>
+    <td class='c017'>0·03</td>
+    <td class='c017'>0·003</td>
+    <td class='c017'>0·0003</td>
+    <td class='c018' rowspan='2'>etc.</td>
+  </tr>
+  <tr>
+    <td class='c010'>Logarithms</td>
+    <td class='c017'>0·477</td>
+    <td class='c017'> &#x0305;1·477</td>
+    <td class='c017'> &#x0305;2·477</td>
+    <td class='c017'> &#x0305;3·477</td>
+    <td class='c017'> &#x0305;4·477</td>
+    
+  </tr>
+</table>
+
+<p class='c007'>Here again we see that with the same significant figures in the
+numbers, the mantissa of the logarithm has always the same
+(<em>positive</em>) value, but the characteristic is <em>one more</em> than the <em>number
+of 0’s immediately following the decimal point</em>, and is <em>negative</em>, as
+indicated by the minus sign written over it. Only the decimal
+parts of the logarithms of numbers between 1 and 10 are given
+in the usual tables, for, as shown above, the logarithms of all tenfold
+multiples or submultiples of a number can be obtained at
+once by modifying the characteristic in accordance with the rules
+given.</p>
+
+<p class='c007'>An examination of the two rows of figures giving the logarithms
+of numbers from 1 to 10 will reveal some striking peculiarities,
+and at the same time serve to illustrate the principle of logarithmic
+calculation. First, it will be noticed that the addition of any two
+of the logarithms gives the logarithm of the <em>product</em> of these two
+numbers. Thus, the addition of log. 2 and log. 4 = 0·301 + 0·602
+= 0·903, and this is seen to be the logarithm of 8, that is, of 2 × 4.
+Conversely, the difference of the logarithms of two numbers gives
+the logarithm of the <em>quotient</em> resulting from the division of these
+two numbers. Thus, log. 8 − log. 2 = 0·903 − 0·301 = 0·602, which is
+the log. of 4, or of 8 ÷ 2.</p>
+
+<p class='c007'><span class='pageno' id='Page_8'>8</span>One other important point is to be noted. If the logarithm of
+any number is <em>multiplied</em> by 2, 3, or any other quantity, whole or
+fractional, the result is the logarithm of the original number,
+raised to the 2nd, 3rd, or other power respectively. Thus, multiplying
+the log. of 3 by 2, we obtain 0·477 × 2 = 0·954, and this is
+seen to be the log. of 9, that is, of 3 raised to the 2nd power, or 3
+<em>squared</em>. Again, log. 2 multiplied by 3 = 0·903—that is, the log. of
+8, or of 2 raised to the 3rd power, or 2 <em>cubed</em>. Conversely, dividing
+the logarithm of any original number by any number <em>n</em>, we obtain
+the logarithm of the <em>n</em>th root of the original number. Thus, log.
+8 ÷ 3 = 0·903 ÷ 3 = 0·301, and is therefore equal to log. 2 or to the
+log. of the <em>cube root</em> of 8.</p>
+
+<p class='c007'>Only simple logs. have been taken in these examples, but the
+student will understand that the same reasoning applies, whatever
+the number. Thus for 20<sup>3</sup> we prefix the characteristic (1 in this
+case) to log. 2, giving 1·301. Multiplying by 3, we have 3·903 as
+the resulting logarithm, and as its characteristic is 3, we know
+that it corresponds to the number 8000. Hence 20<sup>3</sup> = 8000.</p>
+
+<p class='c007'>In this brief explanation is included all that need now be said
+with regard to the properties of logarithms. The main facts to
+be borne clearly in mind are:—(1.) That to find the <em>product</em> of
+two numbers, the logarithms of the numbers are to be <em>added</em>
+together, the result being the logarithm of the product required,
+the value of which can then be determined. (2.) That in finding
+the <em>quotient</em> resulting from the division of one number by another,
+<em>the difference</em> of the logarithms of the numbers gives the logarithm
+of the quotient, from which the value of the latter can be ascertained.
+(3.) That to find the result of <em>raising a number to the nth
+power</em>, we <em>multiply</em> the logarithm of the number by <em>n</em>, thus obtaining
+the logarithm, and hence the value, of the desired result. And
+(4.) That to find the n<em>th root of a number</em>, we <em>divide</em> the logarithm
+of the number by <em>n</em>, this giving the logarithm of the result, from
+which its value may be determined.</p>
+
+<div class='chapter'>
+  <h2 class='c005'>NOTATION BY POWERS OF 10.</h2>
+</div>
+
+<p class='c012'>A convenient method of representing an arithmetical quantity is
+to split it up into two factors, of which the first is the original
+number, with the decimal point moved so as to immediately follow
+the first significant figure, and the second, 10<sup><em>n</em></sup> where <em>n</em> is the
+<span class='pageno' id='Page_9'>9</span>number of places the decimal point has been moved, this index
+being <em>positive</em> for numbers greater than 1, and <em>negative</em> for numbers
+less than 1.<a id='r1'></a><a href='#f1' class='c019'><sup>[1]</sup></a> In this system, therefore, we regard 3,610,000 as
+3·61 × 1,000,000, and write it as 3·61 × 10<sup>6</sup>. Similarly 361 = 3·61 x 10<sup>2</sup>;
+0·0361 (=
+<span class='fraction'><span class='under'>3·61</span><br>100</span>)
+= 3·61 × 10<sup>−2</sup>; 0·0000361 = 3·61 × 10<sup>−5</sup>, etc. To restore
+a number to its original form, we have only to move the
+decimal point through the number of places indicated by the
+index, moving to the right if the index is positive and to the
+left (prefixing 0’s) if negative. This method, which should be
+cultivated for ordinary arithmetical work, is substantially that
+followed in calculating by the slide rule. Thus with the slide rule
+the multiplication of 63,200 by 0·0035 virtually resolves itself into
+6·32 × 10<sup>4</sup> × 3·5 × 10<sup>−3</sup> or 6·32 × 3·5 × 10<sup>4–3</sup> = 22·12 x 10<sup>1</sup> = 221·2. It will
+be seen later, however, that the result can be arrived at by a more
+direct, if less systematic, method of working.</p>
+
+<div class='chapter'>
+  <h2 class='c005'>THE MECHANICAL PRINCIPLE OF THE SLIDE RULE.</h2>
+</div>
+
+<div id='f_001'  class='figcenter id001'>
+<img src='images/f_001.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 1.</span></p>
+</div>
+</div>
+
+<p class='c012'>The mechanical principle involved in the slide rule is of a very
+simple character. In Fig. <a href='#f_001'>1</a>, A and B represent two rules divided
+into 10 equal parts, the division lines being numbered consecutively
+as shown. If the rule B is moved to the right until 0 on B is
+opposite 3 on A, it is seen that any number on A is equal to the
+coinciding number on B, plus 3. Thus opposite 4 on B is 7 on A.
+The reason is obvious. By moving B to the right, we add to a
+length 0·3, another length 0·4, the result read off on A being 7.
+Evidently, the same result would have been obtained if a length
+0·4 had been added, by means of a pair of dividers, to the length
+0·3 on the scale A. By means of the slide B, however, the addition
+is more readily effected, and, what is of much greater importance,
+<span class='pageno' id='Page_10'>10</span>the result of adding 3 to <em>any one of the numbers</em> within range, on
+the lower scale, is <em>immediately</em> seen by reading the adjacent
+number on A.</p>
+
+<p class='c007'>Of course, subtraction can be quite as readily performed.
+Thus, to subtract 4 from 7, we require to deduct from 0·7 on the
+A scale, a length 0·4 on B. We do this by placing 4 on B under
+7 on A, when over 0 on B we find 3, on A. It is here evident that
+the <em>difference</em> of any pair of coinciding numbers on the scales is
+constantly equal to 3.</p>
+
+<div id='f_002'  class='figcenter id001'>
+<img src='images/f_002.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 2.</span></p>
+</div>
+</div>
+
+<p class='c007'>An important modification results if the slide-scale B is inverted
+as in Fig. <a href='#f_002'>2</a>. In this case, to find the sum of 4 and 3 we
+require to place the 4 of the A scale to 3 on the B scale, and the
+result is read on A over 0 on B. Here it will be noted, the <em>sum</em>
+of any pair of coinciding numbers on the scales is constant and
+equal to 7. This case, therefore, resembles that of the immediately
+preceding one, except that the <em>sum</em>, instead of the <em>difference</em>, of any
+pair of coinciding numbers is constant.</p>
+
+<p class='c007'>To find the difference of two factors, the converse operation is
+necessary. Thus, to subtract 4 from 7, 0 on B is placed opposite 7
+on A, and over 4 on B is found 3 on A.</p>
+
+<p class='c007'>From these examples it will be seen that with the slide <em>inverted</em>
+the methods of operation are the reverse of those used when the
+slide is in its normal position.</p>
+
+<p class='c007'>It will be understood that although we have only considered
+the primary divisions of the scales, the remarks apply equally to
+any subdivisions into which the primary spaces of the scales might
+be divided. Further, we note that the length of scale taken to
+represent a unit is quite arbitrary.</p>
+
+<div class='chapter'>
+  <h2 class='c005'>THE PRIMITIVE SLIDE RULE.</h2>
+</div>
+
+<p class='c012'>The application of the foregoing principles to the slide rule can
+be shown most conveniently by describing the construction of a
+simple form of slide rule:—Take a strip of card about 11 in. long
+and 2 in. wide; draw a line down the centre of its width, and
+<span class='pageno' id='Page_11'>11</span>mark off two points, 10 in. apart. Draw cross lines at these points
+and figure them 1 and 10 on each side, as in Fig. <a href='#f_003'>3</a>. Next mark off
+lengths of 3·01, 4·77, 6·02, 6·99, 7·78, 8·45, 9·03 and 9·54 inches,
+from the line marked 1. Draw cross lines as before, and figure
+these lines, 2, 3, 4, 5, 6, 7, 8 and 9. To fill in the intermediate
+divisions of the scale, take the logs, of 1·1, 1·2, 1·3, etc. (from a
+table), multiply each by 10, and thus obtain the distances from 1,
+at which the several subdivisions are to be placed. Mark these
+1·2, 1·3, 1·4, etc., and complete the scale, making the interpolated
+division marks shorter to facilitate reading, as with an ordinary
+measuring rule. Cutting the card cleanly down the centre line,
+we have the essentials of the slide rule.</p>
+
+<div id='f_003'  class='figcenter id001'>
+<img src='images/f_003.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 3.</span></p>
+</div>
+</div>
+
+<p class='c007'>The fundamental principle of the slide rule is now evident:—Each
+scale is graduated in such a manner that the <em>distance of any
+number from 1 is proportional to the logarithm of that number</em>.</p>
+
+<div id='f_004'  class='figcenter id001'>
+<img src='images/f_004.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 4.</span></p>
+</div>
+</div>
+
+<p class='c007'>“We know that to find the product of 2 × 3 by logarithms, we
+add 0·301, or log. 2, to 0·477, the log. of 3, obtaining 0·778, or log.
+6. With our primitive slide rule we place 1 on the lower scale to
+3·01 in. (which we have marked 2) on the upper scale (Fig. <a href='#f_004'>4</a>). Then
+over 4·77 in. on the lower scale (which we marked 3), we have 7·78 in.
+(which we marked 6) on the upper scale. Conversely, to divide
+6 by 3, we place 3 on the lower scale in agreement with 6 on the
+upper, and over 1 on the lower scale read 2 on the upper scale.
+This method of adding and subtracting scale lengths will be seen
+to be identical with that used in the simple case shown in Fig. <a href='#f_001'>1</a>.</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_12'>12</span>
+  <h2 class='c005'>THE MODERN SLIDE RULE.</h2>
+</div>
+
+<p class='c012'>The modern form of slide rule, variously
+styled the Gravêt, the Tavernier-Gravêt,
+and the Mannheim rule, is frequently made
+of boxwood, but all the leading instrument
+makers now supply rules made of boxwood
+or mahogany, and faced with celluloid, the
+white surface of which brings out the
+graduations much more distinctly than lines
+engraved on a boxwood surface. The celluloid
+facings should not be polished, as a dull
+surface is much less fatiguing to the eyes.
+The most generally used, and on the whole
+the most convenient size of rule, is about
+10½in. long, 1¼in. wide, and about ⅜in. thick;
+but 5 in., 8 in., 15 in., 20 in., 24 in. and 40 in.
+rules are also made. In the centre of the stock
+of the rule a movable slip is fitted, which
+constitutes the slide, and corresponds to the
+lower of the two rules of our rudimentary
+examples.</p>
+
+<div id='f_005'  class='figcenter id001'>
+<img src='images/f_005.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 5.</span></p>
+</div>
+</div>
+
+<p class='c007'>From Fig. <a href='#f_005'>5</a>, which is a representation
+of the face of a Gravêt or Mannheim slide
+rule, it will be seen that four series of logarithmic
+graduations or scale-lines are employed,
+the upper and lower being engraved
+on the stock or body of the rule, while the
+other two are engraved upon the slide. The
+two upper sets of graduations are exactly
+alike in every particular, and the lower
+sets are also similar. It is usual to identify
+the two upper scale-lines by the letters A
+and B, and the two lower by the letters C
+and D, as indicated in the figure at the
+left-hand extremities of the scales.</p>
+
+<p class='c007'>Referring to the scales C and D, these
+will each be seen to be a development
+of the elementary scales of Fig. <a href='#f_003'>3</a>, but
+<span class='pageno' id='Page_13'>13</span>in this case each principal space is subdivided, more or
+less minutely. The principle, however, is exactly the same,
+so that by moving the slide (carrying scale C), multiplication
+and division can be mechanically performed in the manner
+described.</p>
+
+<p class='c007'>The upper scale-line A consists of two exactly similar scales,
+placed end to end, the first lying between <span class='sc'>Il</span> and <span class='sc'>Ic</span>, and the
+second between <span class='sc'>Ic</span> and <span class='sc'>Ir</span>. The first of these scales will be designated
+the <em>left-hand A scale</em>, and the second the <em>right-hand A scale</em>.
+Similarly the coinciding scales on the slide are the <em>left-hand B
+scale</em> and the <em>right-hand B scale</em>. Each of these four scales is
+divided (as finely as convenient) as in the case of the C and D
+scales, but, of course, they are exactly one half the length of the
+latter.</p>
+
+<p class='c007'>The two end graduations of both the C and D scales are known
+as the <em>left-</em> and <em>right-hand indices</em> of these scales. Sometimes they
+are figured 1 and 10 respectively; sometimes both are marked 1.
+Similarly <span class='sc'>Il</span> and <span class='sc'>Ir</span> are the left- and right-hand indices of the A
+and B lines, while <span class='sc'>Ic</span>  is the centre index of these scales. Other
+division lines usually found on the face of the rule are one on the
+left-hand A and B scales, indicating the ratio of the circumference
+of a circle to its diameter, π = 3·1416; and a line on the right-hand
+B scale marking the position of <span class='fraction'>π<br><span class='vincula'>4</span></span> = 0·7854, used in calculating
+the areas of circles. Reference will be made hereafter to the
+scales on the under-side of the slide, and we need now only add
+that one of the edges of the rule, usually bevelled, is generally
+graduated in millimetres, while the other edge has engraved on it
+a scale of inches divided into eighths or tenths. On the bottom
+face inside the groove of the rule either one or the other of these
+scales is continued in such a manner that by drawing the slide out
+to the right and using the scale inside the rule, in conjunction
+with the corresponding scale on the edge, it is possible to measure
+20 inches in the one case, or nearly 500 millimetres in the other.
+On the back of the rule there is usually a collection of data, for
+which the slips given at the end of this work may often be substituted
+with advantage.</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_14'>14</span>
+  <h2 class='c005'>THE NOTATION OF THE SLIDE RULE.</h2>
+</div>
+
+<p class='c012'>Hitherto our attention has been confined to a consideration of
+the primary divisions of the scales. The same principle of graduation
+is, however, used throughout; and after what has been said,
+this part of the subject need not be further enlarged upon. Some
+explanation of the method of reading the scales is necessary, as
+facility in using the instrument depends in a very great measure
+upon the dexterity of the operator in assigning the correct value
+to each division on the rule. By reference to Fig. <a href='#f_005'>5</a>, it will be
+seen that each of the primary spacings in the several scales is
+invariably subdivided into ten; but since the lengths of the
+successive primary divisions rapidly diminish, it is impossible to
+subdivide each main space into the same number of parts that
+the space 1–2 can be subdivided. This variable spacing of the
+scales is at first confusing to the student, but with a little practice
+the difficulty is soon overcome.</p>
+
+<p class='c007'>With the C or D scale, it will be noticed that the length of the
+interval 1–2 is sufficient to allow each of the 10 subdivisions to be
+again divided into 10 parts, so that the whole interval 1–2 is
+divided into 100. The shorter main space 2–3, and the still shorter
+one 3–4, only allow of the 10 subdivisions of each being divided
+into five parts. Each of these main spaces is therefore divided
+into 50 parts. For the remainder of the scale each of the 10
+subdivisions of each main space is divided into two parts only; so
+that from the main division 4 to the end of the scale the primary
+spaces are divided into 20 parts only.</p>
+
+<p class='c007'>In the upper scales A or B, it will be found that—as the space
+1–2 is of only half the length of the corresponding space on C or D—the
+10 subdivisions of this interval are divided into five parts
+only. Similarly each of the 10 subdivisions of the intervals 2–3,
+3–4, and 4–5 are further divided into two parts only, while
+for the remainder of the scale only the 10 subdivisions are
+possible, owing to the rapidly diminishing lengths of the primary
+spacings.</p>
+
+<p class='c007'>The values actually given on the rule run from 1 to 10 on
+the lower scales and from 1 to 100 on the upper scales, and, as
+explained on page <a href='#Page_9'>9</a>, all factors are brought within these ranges
+of values by multiplying or dividing them by powers of 10. By
+following this plan, we virtually regard each factor as merely a
+<span class='pageno' id='Page_15'>15</span>series of significant figures, and make the necessary modification
+due to the “powers of 10” when fixing the position of the decimal
+point in the answer.</p>
+
+<p class='c007'>Many, however, find it convenient in practice to regard the
+values on the rule as multiplied or divided by such powers of 10
+as may be necessary to suit the factors entering into the calculation.
+If this plan is adopted, the values given to each graduation
+of the scales will depend on that given to the left index figure (1)
+of the lower scales, this being any multiple or submultiple of 10.
+Thus <span class='sc'>Il</span> on the D scale may be regarded as 1, 10, 100, 1000, etc., or
+as 0·1, 0·01, 0·001, 0·0001, etc.; but once the initial value is assigned
+to the index, the ratio of value must be maintained throughout the
+whole scale. For example, if 1 on C is taken to represent 10, the
+main divisions 2, 3, 4, etc., will be read as 20, 30, 40, etc. On
+the other hand, if the fourth main division is read as 0·004, then
+the left index figure of the scale will be read as 0·001. The figured
+subdivisions of the main space 1–2 are to be read as 11, 12, 13, 14,
+15, 16, 17, 18 and 19—if the index represents 10,—and as corresponding
+multiples for any other value of the index.</p>
+
+<p class='c007'>Independently considered, these remarks apply equally to the
+A or B scale, but in this case the notation is continued through
+the second half of the scale, the figures of which are to be read as
+tenfold values of the corresponding figures in the first half of the
+scale.</p>
+
+<p class='c007'>The reading of the intermediate divisions will, of course, be
+determined by the values assigned to the main divisions. Thus, if
+<span class='sc'>Il</span> on D is read as 1, then each of the smallest subdivisions of the
+space 1–2 will be read as 0·01, and each of the smallest subdivisions
+of the spaces 2–3 or 3–4 as 0·02, while for the remainder of the
+scale the smallest subdivisions are read as 0·05. In the A or B
+scale the subdivisions of the space 1–2 of the first half of the scale
+are (if <span class='sc'>Il</span> = 1) read as 0·02, 0·04, etc.; for the divisions 2–3, 3–4, and
+4–5, the smallest intervals are read as 0·05 of the primary spaces,
+and from 5 to the centre index of the scale the divisions represent
+0·1 of each main interval. Passing the centre index, which is, now
+read as 10, the smallest subdivisions immediately following are
+read 10·2, 10·4, etc., until 20·0 is reached; then we read 20·5, 21·0,
+21·5 22·0, etc., until the figured main division 5 is reached. The
+remainder of the scale is read 51, 52, 53, etc., up to 100, the right-hand
+index.</p>
+
+<p class='c007'><span class='pageno' id='Page_16'>16</span>Further subdivision of any of the spaces of the rule can be
+effected by the eye, and after a little practice the operator will
+become quite expert in estimating any intermediate value. It
+affords good practice to set 1 on C to 1·04, 1·09, etc. on D, and to
+read the values on D, under 4, 6, 8, etc. on C. As the exact
+results are easily calculated mentally, the student, by this means,
+will receive better instruction in estimating intermediate results
+than can be given by any diagram.</p>
+
+<p class='c007'>Some rules will be found figured as shown in Fig. <a href='#f_005'>5</a>; in others,
+the right-hand upper scales are marked 10, 20, 30, etc. Again,
+others are marked decimally, the lower scales and the left-hand
+upper scales being figured 1, 1·1, 1·2, 1·3&#160;... 2·5, etc. The
+latter form has advantages from the point of view of the
+beginner.</p>
+
+<p class='c007'>The method of reading the A and B scales, just given, applies
+only when these scales are regarded as altogether independent of
+the lower pair of scales C and D. Some operators prefer to use
+the A and B scales, and some the C and D scales, for the ordinary
+operations of proportion, multiplication, and division. Each
+method has its advantages, as will be shown, but in the more
+complex calculations, as involution and evolution, etc., the relation
+of the upper scales to the lower scales becomes a very important
+factor.</p>
+
+<p class='c007'>The distance 1–10 on the upper scales is one-half of the distance
+1–10 on the lower scales. Hence any distance from 1, taken on
+the upper scales, represents <em>twice the logarithm</em> which the same
+distance represents on the lower scales. In other words, the length
+which represents log. N on D, would represent 2 log. N on A;
+and, conversely, the length which represents log. N on A, would
+represent <span class='fraction'><span class='under'>log. N</span><br>2</span> on D.</p>
+
+<p class='c007'>Now we have seen (page <a href='#Page_8'>8</a>) that multiplying the log. of a
+number by 2 gives the log. of the square of the number. Hence,
+above any number on D we find its <em>square</em> on A, or, conversely,
+below any number on A, we find its <em>square root</em> on D. Thus,
+above 2 we find 4; under 49, we find 7 and so on. Obviously the
+same relation exists between the B and C scales.</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_17'>17</span>
+  <h2 class='c005'>THE CURSOR OR RUNNER.</h2>
+</div>
+
+<p class='c012'>All modern slide rules are now fitted with a <em>cursor</em> or <em>runner</em>,
+which usually consists of a light metal frame moving under spring
+control in grooves in the edges of the stock of the rule. This
+frame carries a piece of glass, mica or transparent celluloid, about
+1 in. square, across the centre of which a fine reference line is
+drawn exactly at right angles to the line of scales. To “set the
+cursor” to any value on the scales of the rule, the frame is taken
+between the thumb and forefinger and adjusted in position until
+the line falls exactly upon the graduation, or upon an estimated
+value, between a pair of graduations, as the case may be. Having
+fixed one number in this way, another value on either of the scales
+on the slide may be similarly adjusted in reference to the cursor
+line. The cursor will be found very convenient in making such
+settings, especially when either or both of the numbers are located
+by eye estimation. It also finds a very important use in referring
+the readings of the upper scale to those of the lower, or <em>vice versa</em>,
+while as an aid in continued multiplication and division and complex
+calculations generally, its value is inestimable.</p>
+
+<p class='c007'><em>Multiple Line Cursors.</em>—Cursors can be obtained with <em>two</em> lines,
+the distance between them being that between 7·854 and 10 on
+the A scale. The use of this cursor is explained on page <a href='#Page_57'>57</a>.
+Another multiple line cursor has short lines engraved on it, corresponding
+to the main graduations from 95 to 105 on the respective
+scales. This is useful for adding or deducting small percentages.</p>
+
+<p class='c007'><em>The Broken Line Cursor.</em>—To facilitate setting, broken line
+cursors are made, in which the hair-line is not continued across
+the scales, but has two gaps, as shown in Fig. <a href='#f_006'>6</a>.</p>
+
+<p class='c007'><em>The Pointed Cursor</em> has an index or pointer, extending over the
+bevelled edge of the rule, on which is a scale of inches. It is
+useful for summing the lengths of the ordinates of indicator
+diagrams, and also for plotting lengths representing the logarithms
+of numbers, sometimes required in graphic calculations.</p>
+
+<p class='c007'><em>The Goulding Cursor.</em>—It has been pointed out that in order to
+obtain the third or fourth figure of a reading on the 10 in. slide
+rule, it is frequently necessary to depend upon the operator’s
+ability to mentally subdivide the space within which the reading
+falls. This subdivision can be mechanically effected by the aid of
+the Goulding Cursor (Fig. <a href='#f_007'>7</a>), which consists of a frame fitting
+<span class='pageno' id='Page_18'>18</span>into the usual grooves in the rule, and carrying a metal plate faced
+with celluloid, upon which is engraved a triangular scale A B C.
+The portion carrying the chisel edges E is not fixed to the cursor
+proper, but slides on the latter, so that the index marks on the
+projecting prongs can be moved slightly along the scales of the
+rule, this movement being effected by the short end of the bent
+lever F working in the slot as shown. D is a pointer which can
+be moved along F under spring control. As illustrating the
+method of use, we will assume that 1 on C is placed to 155 on D,
+and that we require to read the value on D under 27 on C. This
+is seen to lie between 4150 and 4200, so setting the pointer D to
+the line B C—always the first operation—we move the whole
+along the rule until the index line on the lower prong agrees with
+4200. We then move F across the scale until the index line agrees
+with 4100, set the pointer D to the line A C, and move the lever
+back until the index line agrees with 27 on the slide. It will then
+be found that the pointer D gives 85 on A B as the value of the
+supplementary figures, and hence the complete reading is 4185.</p>
+
+<div id='f_006'  class='figleft id002'>
+<img src='images/f_006.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 6.</span></p>
+</div>
+</div>
+
+<div id='f_008'  class='figright id002'>
+<img src='images/f_008.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 8.</span></p>
+</div>
+</div>
+
+<div id='f_007'  class='figleft id003'>
+<img src='images/f_007.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 7.</span></p>
+</div>
+</div>
+
+<div id='f_009'  class='figright id004'>
+<img src='images/f_009.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 9.</span></p>
+</div>
+</div>
+
+<div class='section'>
+
+</div>
+<p class='c007'><em>Magnifying Cursors</em> are of assistance in reading the scales, and
+in a good and direct light are very helpful. In one form an
+ordinary lens is carried by two light arms hinged to the upper
+and lower edges of the cursor, so that it can be folded down to the
+face of the rule when not in use. A more compact form, shown in
+<span class='pageno' id='Page_19'>19</span>Fig. <a href='#f_008'>8</a>, consists of a strip of plano-convex glass, on the under-side
+of which is the hair-line. In a cursor made by Nestler of Lahr,
+the plano-convex strip is fixed on the ordinary cursor. The
+magnifying power is about 2, so that a 5 in. rule, having the same
+number of graduations as a 10 in. rule, can be read with equal
+facility, by the aid of this cursor.</p>
+
+<p class='c007'>The Digit-registering Cursor, supplied by Mr. A. W. Faber,
+London, and shown in Fig. <a href='#f_009'>9</a>, has a semicircular scale running
+from 0 at the centre upward to −6 and downward to +6. A
+small finger enables the operator to register the number of digits
+to be added or subtracted at the end of a lengthy operation, as
+explained at page <a href='#Page_28'>28</a>.</p>
+
+<div class='chapter'>
+  <h2 class='c005'>MULTIPLICATION.</h2>
+</div>
+
+<p class='c012'>In the preliminary notes it was shown that by mechanically
+adding two lengths representing the logarithms of two numbers, we
+can obtain the <em>product</em> of these numbers; while by subtracting one
+log. length from another, the number represented by the latter is
+divided by the number represented by the former. Hence, using
+the C and D scales, we have the</p>
+
+<p class='c007'><span class='sc'>Rule for Multiplication.</span>—<em>Set the index of the C scale to one of the
+factors on D, and under the other factor on C, find the product on D.</em></p>
+
+<div id='f_010'  class='figcenter id001'>
+<img src='images/f_010.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 10.</span></p>
+</div>
+</div>
+
+<p class='c007'>Thus, to find the product of 2 × 4, the slide is moved to the right
+until the left index (1) of C is brought over 2 on D, when under the
+other factor (4) on C, is found the required product (8) on D.
+Following along the slide, to the right, we find that beyond 5 on C
+(giving 10 on D), we have no scale below the projecting slide (Fig.
+<a href='#f_010'>10</a>). If we imagine the D scale prolonged to the right, we should
+have a repetition of the earlier portion, but, as with the two parts
+<span class='pageno' id='Page_20'>20</span>of the A scales, the repeated portion would be of tenfold value, and
+10 on C would agree with 20 on the prolonged D scale. We turn
+this fact to account by moving the slide to the left until 10 on C
+agrees with 2 on D, and we can then read off such results as
+2 × 6 = 12; 2 × 8 = 16, etc., remembering that as the scale is now of
+tenfold value, there will be two figures in the result. Hence, for
+those who prefer rules, we have the</p>
+
+<p class='c007'><span class='sc'>Rule for the Number of Digits in a Product.</span>—<em>If the product
+is read with the slide projecting to the</em> <span class='fss'>LEFT, ADD THE NUMBER
+OF THE DIGITS IN THE TWO FACTORS</span>; <em>if read with the slide to the</em>
+<span class='fss'>RIGHT</span>, <em>deduct 1 from this sum</em>.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—25 × 70 = 1750.</p>
+
+<p class='c021'>The product is found with the slide projecting to the <em>left</em>, so
+the number of digits in the product = 2 + 2 = 4.</p>
+
+<p class='c021'><span class='sc'>Ex.</span>—3·6 × 25 = 90.</p>
+
+<p class='c021'>The slide projects to the <em>right</em>, and the number of digits in the
+product is therefore 1 + 2 − 1 = 2.</p>
+
+<p class='c021'><span class='sc'>Ex.</span>—0·025 × 0·7 = 0·0175.</p>
+
+<p class='c021'>The product is obtained with the slide projecting to the <em>left</em>,
+and the number of digits is therefore −1 + 0 = −1.</p>
+
+<p class='c021'><span class='sc'>Ex.</span>—0·000184 × 0·005 = 0·00000092.</p>
+
+<p class='c021'>The sum of the number of digits in the two factors = −3 + (−2)
+= −5, but as the slide projects to the <em>right</em>, the number of digits
+will be −5 − 1 = −6.</p>
+
+<p class='c012'>From the last two examples it will be seen that when the first
+significant figure of a decimal factor does not immediately follow
+the decimal point, the minus sign is to be prefixed to the number
+of digits, counting as many digits <em>minus</em> as there are 0’s following
+the decimal point. Thus, 0·03 has −1 digit, 0·0035 has −2 digits,
+and so on. Some little care is necessary to ensure these minus
+values being correctly taken into account in determining the
+number of digits in the answer. For this reason many prefer to
+treat decimal factors as whole numbers, and to locate the decimal
+point according to the usual rules for the multiplication of decimals.
+Thus, in the last example we take 184 × 5 = 920, but as by the usual
+rule the product must contain 6 + 3 = 9 decimal places, we prefix
+six cyphers, obtaining 0·00000092. When both factors consist of
+integers as well as decimals, the number of digits in the product,
+and therefore the position of the decimal point, will be determined
+by the usual rule for whole numbers.</p>
+
+<p class='c007'><span class='pageno' id='Page_21'>21</span>Another method of determining the number of digits in a
+product deserves mention, which, not being dependent upon the
+position of the slide, is applicable to all calculating instruments.</p>
+
+<p class='c007'><span class='sc'>General Rule for Number of Digits in a Product.</span>—<em>When
+the first significant figure in the product is smaller than in</em> <span class='fss'>EITHER</span> <em>of
+the factors, the number of digits in the product is equal to the</em> <span class='fss'>SUM</span> <em>of
+the digits in the two factors. When the contrary is the case, the
+number of digits is 1</em> <span class='fss'>LESS</span> <em>than the sum of the digits in the two
+factors. When the first figures are the same, those following must be
+compared.</em></p>
+
+<p class='c007'><em>Estimation of the Figures in a Product.</em>—We have given rules for
+those who prefer to decide the number of figures by this means,
+but experience will show that to make the best use of the instrument,
+the result, as read on the rule, should be regarded merely
+as the <em>significant figures of the answer</em>, the position of the decimal
+point, if not obvious, being decided by a very rough mental calculation.
+In very many instances, the magnitude of the result will
+be evident from the conditions of the problem—<em>e.g.</em>, whether the
+answer should be 0·3 in., 3 in., or 30 in.; or 10 tons, 0·1 ton, 100
+tons, etc. In those cases where the magnitude of the answer cannot
+be estimated, and the factors contain many figures, or have a
+number of 0’s following the decimal point, the use of notation by
+powers of 10 (page <a href='#Page_8'>8</a>) is of considerable assistance; but more
+usually it will be found, that a very rough calculation will settle the
+point with comparatively little trouble. Considerable practice is
+needed to work rapidly and with certainty, when using rules.
+Moreover, the experience thus acquired is confined to slide-rule
+work. The same time spent in practising the “rough approximation”
+method will enable reliable results to be obtained
+rapidly, with the advantage that the method is applicable to
+calculations generally. However, the choice of methods is a
+matter of personal preference. Both methods will be given, but
+whichever plan is followed, the student is strongly advised to
+cultivate the habit of forming an idea of the magnitude of the
+result.</p>
+
+<p class='c022'><span class='sc'>Ex.</span>—33·6 × 236 = 7930.</p>
+
+<p class='c023'>Setting 1 on C to 33·6 on D, we read under 236 on D and find 793
+on D, as the significant figures of the answer. A rough calculation,
+as 30 × 200 = 6000, indicates that the result will consist of 4 figures,
+and is therefore to be read as 7930.</p>
+
+<p class='c024'><span class='pageno' id='Page_22'>22</span><span class='sc'>Ex.</span>—17,300 × 3780 = 65,400,000.</p>
+
+<p class='c024'>By factorising with powers of 10</p>
+
+<div class='nf-center-c0'>
+<div class='nf-center c025'>
+    <div>1·73 × 10<sup>4</sup> × 3·78 × 10<sup>3</sup> = 1·73 × 3·78 × 10<sup>7</sup>.</div>
+  </div>
+</div>
+
+<p class='c023'>Setting 1 on C to 1·73 on D, we read, under 3·78 on C, the result
+of the simple multiplication, as 6·54. Multiplying by 10<sup>7</sup> moves the
+decimal point 7 places to the right, and the answer is 65,400,000.</p>
+
+<p class='c012'>If it is required to find a series of products of which one of the
+factors is <em>constant</em>, set 1 on C to the constant factor on D and read
+the several products on D, under the respective variable factors.</p>
+
+<p class='c007'>If the factors are required which will give a constant <em>product</em>
+(really a case of division), set the cursor to the constant product on
+D. Then obviously, as the slide is moved along, any pair of factors
+found simultaneously under the cursor line on C, and on D under
+index of C, will give the product. A better method of working
+will be explained when we deal with the inversion of the slide.</p>
+
+<p class='c007'>It is sometimes useful to remember that although we usually
+set the slide to the rule, we can obtain the result equally well by
+setting the rule to the slide. Thus, bringing 1 (or 10) on D to 2
+on C, we find on C, <em>over</em> any other factor, <em>n</em> on D, the product of
+2 × <em>n</em>. But note that the slide and rule have now changed places,
+and if we use rules for the number of digits in the result, we
+must now deduct 1 from the sum of the digits in the factors,
+when the <em>rule projects</em> to the <em>right of the slide</em>.</p>
+
+<p class='c007'>With the ordinary 10 in. rule it will be found in general that
+the extent to which the C and D scales are subdivided is such as
+to enable not more than three figures in either factor being dealt
+with. For the same reason it is impossible to directly read more
+than the first three figures of any product, although it is often
+possible—by mentally dividing the smallest space involved in the
+reading—to correctly determine the fourth figure of a product.
+Necessarily this method is only reliable when used in the earlier
+parts of the C and D scales. However, the last numeral of a
+three-figure, and in some cases the last of a four-figure, product
+can be readily ascertained by an inspection of the factors.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—19 × 27 = 513. Placing the <span class='fss'>L.H.</span> index of C to 19 on D,
+we find opposite 27 on C, the product, which lies between 510 and
+515. A glance at the factors, however, is sufficient to decide that
+the third figure must be 3, since the product of 9 and 7 is 63, and
+the last figure of this product must be the last figure in the answer.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—79 × 91 = 7189.</p>
+
+<p class='c007'><span class='pageno' id='Page_23'>23</span>In this case the division line 91 on C indicates on D that the
+answer lies between 7180 and 7190. As the last figure must be 9,
+it is at once inferred that the last two figures are 89.</p>
+
+<p class='c007'>When there are more than three figures in either or both of the
+factors, the fourth and following figures to the right must be
+neglected. It is well to note, however, that if the first neglected
+figure is 5, or greater than 5, it will generally be advisable to
+increase by 1 the third figure of the factor employed. Generally
+it will suffice to make this increase in one of the two factors only,
+but it is obvious that in some cases greater accuracy will be
+obtained by increasing both factors in this way.</p>
+
+<p class='c007'><span class='sc'>Continued Multiplication.</span>—To find the product of more
+than two factors, we make use of the cursor to mark the position
+of successive products (the value of which does not concern us) as
+the several factors are taken into the calculation. Setting the index
+of C to the 1st factor on D, we bring the line of the cursor to the
+2nd factor on C, then the index of C to the cursor, the cursor to
+the 3rd factor, index of C to cursor, and so on, reading the final product
+on D under the last factor on C. (Note that the 1st factor and
+the result are read on D; all intermediate readings are taken on C.)</p>
+
+<p class='c007'>If the rule for the number of digits in a product is used, it is
+necessary to note the number of times multiplication is effected
+with the slide projecting to the right. This number, deducted
+from the sum of the digits of the several factors, gives the number
+of digits in the product. Ingenious devices have been adopted to
+record the number of times the slide projects to the right, but
+some of these are very inconvenient. The author’s method is to
+record each time the slide so projects, by a minus mark, thus −.
+These can be noted down in any convenient manner, and the sum
+of the marks so obtained deducted from the sum of the digits in
+the several factors, gives the number of digits in the product as
+before explained.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—42 × 71 × 1·5 × 0·32 × 121 = 173,200.</p>
+
+<p class='c007'>The product given, which is that read on the rule, is obtained
+as follows:—Set <span class='fss'>R.H.</span> index of C to 42 on D, and bring the cursor
+to 71 on C. Next bring the <span class='fss'>L.H.</span> index of C to the cursor, and the
+latter to 1·5 on C. This multiplication is effected with the slide
+to the right, and a memorandum of this fact is kept by making a
+mark −. Bring the <span class='fss'>R.H.</span> index of C to the cursor and the latter to
+0·32 on C. Then set the <span class='fss'>L.H.</span> index of C to the cursor and read
+<span class='pageno' id='Page_24'>24</span>the result, 1732, on D under 121 on C, while as a slide again
+projects to the right, a second − memo-mark is recorded. There
+are 2 + 2 + 1 + 0 + 3 = 8 digits in the factors, and as there were 2 −
+marks recorded during the operation, there will be 8 − 2 = 6 digits
+in the product, which will therefore read 173,200 (173,194·56).</p>
+
+<p class='c007'>For a very rough evaluation of the result, we note that 1·5 × 0·3
+is about 0·5; hence, as a clue to the number of figures we have</p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>40 × 70 × 60 = 3000 × 60 = 180,000.</div>
+  </div>
+</div>
+
+<div class='chapter'>
+  <h2 class='c005'>DIVISION.</h2>
+</div>
+
+<p class='c012'>The instructions for multiplication having been given in some detail,
+a full discussion of the inverse process of division will be unnecessary.</p>
+
+<p class='c007'><span class='sc'>Rule for Division.</span>—<em>Place the divisor on C, opposite the dividend
+on D, and read the quotient on D under the index of C.</em></p>
+
+<p class='c007'><span class='sc'>Ex.</span>—225 ÷ 18 = 12·5.</p>
+
+<p class='c007'>Bringing 18 on C to 225 on D, we find 12·5 under the <span class='fss'>L.H.</span>
+index of C.</p>
+
+<p class='c007'>As in multiplication, the factors are treated as whole numbers,
+and the position of the decimal point afterwards decided according
+to the following rule, which, as will be seen, is the reverse of that
+for multiplication:—</p>
+
+<p class='c007'><span class='sc'>Rule for the Number of Digits in a Quotient.</span>—<em>If the
+quotient is read with the slide projecting to the</em> <span class='fss'>LEFT</span>, <em>subtract the
+number of digits in the divisor from those in the dividend; but if
+read with the slide to the</em> <span class='fss'>RIGHT, ADD</span> <em>1 to this difference</em>.<a id='r2'></a><a href='#f2' class='c019'><sup>[2]</sup></a></p>
+
+<p class='c007'>In the above example the quotient is read off with the slide to
+the right, so the number of digits in the answer = 3 − 2 <a id='t24'></a>+ 1 = 2.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—0·000221 ÷ 0·017 = 0·013.</p>
+
+<p class='c007'>Here the number of digits in the dividend is −3, and in the divisor
+−1. The difference is −2; but as the result is obtained with the
+slide to the right, this result must be increased by 1, so that the number
+of digits in the quotient is −2 + 1 = −1, giving the answer as 0·013.</p>
+
+<p class='c007'>If preferred, the result can be obtained in the manner referred
+to when considering the multiplication of decimals. Thus, treating
+the above as whole numbers, we find that the result of dividing
+221 by 17 = 13, since the difference in the number of digits in the
+factors, which is 1, is, owing to the position of the slide, increased
+by 1, giving 2 as the number of digits in the answer. Then by the
+<span class='pageno' id='Page_25'>25</span>rules for the division of decimals we know that the number of
+decimal places in the quotient is equal to 6 − 3 = 3, showing that a
+cypher is to be prefixed to the result read on the rule.</p>
+
+<p class='c007'>As in multiplication, so in division, we have a</p>
+
+<p class='c007'><span class='sc'>General Rule for Number of Digits in a Quotient.</span>—<em>When
+the first significant figure in the</em> <span class='fss'>DIVISOR</span> <em>is greater than that in the</em>
+<span class='fss'>DIVIDEND</span><em>, the number of digits in the quotient is found by subtracting
+the digits in the divisor from those in the dividend. When the contrary
+is the case, 1</em> <span class='fss'>IS TO BE ADDED</span> <em>to this difference. When the first figures
+are the same, those following must be compared.</em></p>
+
+<p class='c007'><span class='sc'>Estimation of the Figures in a Quotient.</span>—The method of
+roughly estimating the number of figures in a quotient needs little
+explanation.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—3·95 ÷ 5340 = 0·00074.</p>
+
+<p class='c020'>Setting 534 on C to 3·95 on D we read under the (<span class='fss'>R.H.</span>) index of
+C, the significant figures on D, which are 74. Then 3·9 ÷ 5 is about 0·8
+and 0·8 ÷ 1000 gives 0·0008 as a rough estimate.</p>
+
+<p class='c012'><span class='sc'>Ex.</span>—0·00000285 ÷ 0·000197 = 0·01446.</p>
+
+<p class='c020'>Regarding this as 2·85 × 10<sup>−6</sup> ÷ 1·97 × 10<sup>−4</sup> we divide 2·85 by 1·97
+and obtain 1·446. Dividing the powers of 10 we have 10<sup>−6</sup> ÷ 10<sup>−4</sup> =
+10<sup>−2</sup>, so the decimal point is to be moved two places to the left and the
+answer is read as 0·01446.</p>
+
+<p class='c012'>Another method of dividing deserves mention as of special
+service when dividing a number of quantities by a <em>constant divisor</em>:—Set
+the index of C to the divisor on D and over any dividend on D,
+read the quotient on C.</p>
+
+<p class='c007'>For the division of a <em>constant dividend</em> by a variable divisor,
+set the cursor to the dividend on D and bring the divisor on C
+successively to the cursor, reading the corresponding quotients on
+D under the index of C. Another method which avoids moving
+the slide is explained in the section on “Multiplication and Division
+with the Slide Inverted.”</p>
+
+<p class='c007'><span class='sc'>Continued Division</span>, if we can so call such an expression as
+<span class='fraction'>3·14<br><span class='vincula'>785 × 0·00021 × 4·3 × 64·4</span></span> = 0·0688,
+may be worked by repeating as follows:—Set 7·85 on C to 3·14 on
+D, bring cursor to index of C, 2·1 on C to cursor, cursor to index,
+4·3 to cursor, cursor to index, 6·44 to cursor, and under index of C
+read 688 on D as the significant figures of the answer.</p>
+
+<p class='c007'>For the number of figures in the result, we deduct the sum of
+the number of digits in the several factors and add 1 for each
+<span class='pageno' id='Page_26'>26</span>time the slide projects to the right, which in this case occurs once.
+There are 3 + (−3) + 1 + 2 = 3 denominator digits, 1 numerator digit,
+and 1 is to be added to the difference. Therefore there are
+1 − 3 + 1 = −1 digits in the answer, which is therefore 0·0688. The
+foregoing method of working may confuse the beginner, who is apt
+to fall into the process of continued multiplication. For this
+reason, until familiarity with combined methods has been acquired,
+the product of the several denominators should be first found by
+the continued multiplication process, and the figures in this product
+determined. Then divide the numerator by this product to obtain
+the result.</p>
+
+<p class='c007'>As the denominator product will be read on D, we may avoid
+resetting the slide by bringing the numerator on C to this product
+and reading the result on C <em>over</em> the index of D. The slide and
+rule have here changed places; hence if rules are followed for the
+number of figures in the result, 1 must be added to the difference
+of digits, when the <em>rule projects</em> to the <em>right of the slide</em>.</p>
+
+<p class='c007'>The author’s method of recording the number of times division
+is performed with the slide to the right is by vertical memorandum
+marks, thus |. The full significance of these memo-marks will
+appear in the following section.</p>
+
+<p class='c007'>For a rough calculation to fix the decimal point, in this example
+we move the decimal points in the factors, obtaining
+<span class='fraction'>3<br><span class='vincula'>0·8 × 2 × 4 × 6</span></span> = <span class='fraction'>3<br><span class='vincula'>40</span></span> = 0·075.</p>
+
+<div class='chapter'>
+  <h2 class='c005'>THE USE OF THE UPPER SCALES FOR MULTIPLICATION AND DIVISION.</h2>
+</div>
+
+<p class='c012'>Many prefer to use the upper scales A and B, in preference to C
+and D. The disadvantage is that as the scales are only one-half
+the length of C or D, the graduation does not permit of the same
+degree of accuracy being obtained as when working with the lower
+scales. But the result can always be read directly from the rule
+without ever having to change the position of the slide after it has
+been initially set. Hence, it obviates the uncertainty as to the
+direction in which the slide is to be moved in making a setting.</p>
+
+<p class='c007'>When the A and B scales are employed, it is understood that
+the left-hand pair of scales are to be used in the same manner as
+C and D, and so far the rules relating to the latter are entirely
+applicable. But in this case the slide is always moved to the
+<span class='pageno' id='Page_27'>27</span>right, so that in multiplication the product is found either upon
+the left or right scales of A. If it is found on the left A scale,
+the rule for the number of digits in the product is found as for
+the C and D scales, and is equal to the <em>sum of the digits in the two
+factors, minus 1</em>; but if found on the right-hand A scale, the
+number of digits in the product is equal to the sum of the digits
+in the two factors.</p>
+
+<p class='c007'>In division, similar modifications are necessary. If when
+moving the slide to the right the division can be completely
+effected by using the <span class='fss'>L.H.</span> scale of A, the quotient (read on A above
+the <span class='fss'>L.H.</span> of index B) has a number of digits equal to the number in
+the dividend, less the number in the divisor, <em>plus 1</em>. But if the
+division necessitates the use of both the A scales, the number of
+digits in the quotient equals the number in the dividend, less the
+number in the divisor.</p>
+
+<div class='chapter'>
+  <h2 class='c005'>RECIPROCALS.</h2>
+</div>
+
+<p class='c012'>A special case of division to be considered is the determination of
+the <em>reciprocal</em> of a number <em>n</em>, or <span class='fraction'>1<br><span class='vincula'><em>n</em></span></span>. Following the ordinary rule for
+division, it is evident that setting <em>n</em> on C to 1 on D, gives <span class='fraction'>1<br><span class='vincula'><em>n</em></span></span> on D
+under 1 on C. It is more important to observe that by inverting
+the operation—setting 1 (or 10) on C to <em>n</em> on D—we can read <span class='fraction'>1<br><span class='vincula'><em>n</em></span></span> on
+C over 1 (or 10) on D. Hence whenever a result is read on D
+under an index of C, we can also read its reciprocal on C over
+whichever index of D is available.</p>
+
+<p class='c007'><em>The Number of Digits in a Reciprocal</em> is obvious when <em>n</em> = 10,
+100, or any power (<em>p</em>) of 10. Thus <span class='fraction'>1<br><span class='vincula'>10</span></span> = 0·1; <span class='fraction'>1<br><span class='vincula'>100</span></span> = 0·01; <span class='fraction'>1<br><span class='vincula'>10<sup><em>p</em></sup></span></span> = 1
+preceded by <em>p</em> − 1 cyphers. For all other cases we have the rule:—<em>Subtract
+from 1 the number of digits in the number.</em></p>
+
+<p class='c007'><span class='sc'>Ex.</span>—<span class='fraction'>1<br><span class='vincula'>339</span></span> = 0·00295.</p>
+
+<p class='c007'>There are 3 digits in the number; hence, there are 1 − 3 = −2
+digits in the answer.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—<span class='fraction'>1<br><span class='vincula'>0·0000156</span></span> = 64,100.</p>
+
+<p class='c007'>There are −4 digits in the number; hence, there are 1 − (−4) = 5
+digits in the result.</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_28'>28</span>
+  <h2 class='c005'>CONTINUED MULTIPLICATION AND DIVISION.</h2>
+</div>
+
+<p class='c012'>By combining the rules for multiplication and division, we can
+readily evaluate expressions of the form <span class='fraction'><span class='under'><em>a</em></span><br><em>b</em></span> × <span class='fraction'><span class='under'><em>c</em></span><br><em>d</em></span> × <span class='fraction'><span class='under'><em>e</em></span><br><em>f</em></span> × <span class='fraction'><span class='under'><em>g</em></span><br><em>h</em></span> = <em>x</em>. The
+simplest case, <span class='fraction'><span class='under'><em>a</em> × <em>c</em></span><br><em>b</em></span> can be solved by one setting of the slide.<a id='r3'></a><a href='#f3' class='c019'><sup>[3]</sup></a>
+Take as an example, <span class='fraction'><span class='under'>14·45 × 60</span><br>8·5</span> = 102. Setting 8·5 on C to 14·45 on
+D, we can, if desired, read 1·7 on D under 1 on C, as the quotient.
+However, we are not concerned with this, but require its multiplication
+by 60, and the slide being already set for this operation, we
+at once read under 60 on C the result, 102, on D. The figures in
+the answer are obvious.</p>
+
+<p class='c007'>When there are more factors to take into account, we place the
+cursor over 102 on D, bring the next divisor on C to the cursor,
+move the cursor to the next multiplier on C, bring the next divisor
+on C to the cursor, and so on, until all the factors have been dealt
+with. Note that only the first factor and the result are read on
+D; also <em>that the cursor is moved for multiplying and the slide for
+dividing</em>.</p>
+
+<p class='c007'><em>Number of Digits in Result in Combined Multiplication and
+Division.</em>—For those who use rules the author’s method of determining
+the decimal point in combined multiplication and division
+may be used. Each time <em>multiplication</em> is performed with the slide
+projecting to the <em>right</em>, make a − mark; each time <em>division</em> is
+effected with the slide to the right, make a | mark; <em>but allow
+the</em> | <em>marks to cancel the</em> − <em>marks as far as they will</em>. Subtract
+the sum of the digits in the denominator from the sum of digits in
+the numerator, and to this difference <em>add</em> any uncancelled memo-marks,
+if of | character, or <em>subtract</em> them if of − character.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—<span class='fraction'>43·5 × 29·4 × 51 × 32<br><span class='vincula'>27 × 3·83 × 10·5 × 1·31</span></span> = 1468.</p>
+
+<div class='sidenote'>ⵜ<br>ⵜ<br>ⵏ<br>ⵏ</div>
+
+<p class='c007'>Set 27 on C to 43·5 on D, and as with this <em>division</em> the
+slide is to the right, make the first ⵏ mark. Bring cursor
+to 29·4 on C, and as in this <em>multiplication</em> the slide is to
+the right, make the first − mark, cancelling as shown.
+<span class='pageno' id='Page_29'>29</span>Setting 3·83 on C to the cursor, requires the
+second ⵏ mark, which, however, is cancelled in
+turn by the multiplication by 51. The division
+by 10·5 requires the third ⵏ mark, and after multiplying
+by 32 (requiring no mark) the final division
+by 1·31 requires the fourth ⵏ mark. Then, as there
+are 8 numerator digits, 6 denominator, and 2 uncancelled
+memo-marks (which, being 1, are additive)
+we have</p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>Number of digits in result = 8 − 6 + 2 = 4.</div>
+  </div>
+</div>
+
+<p class='c007'>Had the uncancelled marks been − in character, the number of
+digits would have been 8 − 6 − 2 = 0.</p>
+
+<p class='c007'>For quantities less than 0·1 the digit place numbers will be
+<em>negative</em>. The troublesome addition of these may be avoided by
+transferring them to the opposite side and treating them as
+positive.</p>
+
+<table class='table1'>
+  <tr>
+    <td class='c026'>&#160;</td>
+    <td class='c017'>&#160;</td>
+    <td class='c017'>&#160;</td>
+    <td class='c017'><em>2</em></td>
+    <td class='c017'>&#160;</td>
+    <td class='c017'><em>4</em></td>
+    <td class='c027'>&#160;</td>
+  </tr>
+  <tr>
+ <td class='c026' rowspan='2'>Thus:—</td>
+ <td class='bbt c017'>0·00356</td>
+ <td class='bbt c017'>×</td>
+ <td class='bbt c017'>27·1</td>
+ <td class='bbt c017'>×</td>
+ <td class='bbt c017'>0·08375</td>
+ <td class='c027' rowspan='2'>= 288</td>
+  </tr>
+  <tr>
+    
+    <td class='c017'>0·1426</td>
+    <td class='c017'>×</td>
+    <td class='c017'>9·85</td>
+    <td class='c017'>×</td>
+    <td class='c017'>0·00002</td>
+    
+  </tr>
+  <tr>
+    <td class='c026'>&#160;</td>
+    <td class='c017'><em>2</em></td>
+    <td class='c017'>&#160;</td>
+    <td class='c017'><em>1</em></td>
+    <td class='c017'>&#160;</td>
+    <td class='c017'><em>1</em></td>
+    <td class='c027'>&#160;</td>
+  </tr>
+</table>
+
+<p class='c007'>The first numerator, 0·00356, has −2 digits. Note this by
+placing 2 <em>below the lower line</em> as shown. 27·1 has 2 digits; place 2
+over it. 0·08375 has −1 digit; hence place 1 <em>below the lower line</em>.
+The first denominator has no digits; the second, 9·85, has 1 digit;
+hence place 1 under it. 0·00002 has −4 digits; place 4 <em>above the
+upper line</em>. The sum of the top series is 2 + 4 = 6; of the bottom
+series 2 + 1 + 1 = 4. Subtracting the bottom from the top, we have
+6 − 4 = 2 digits, to which 1 has to be added for an uncancelled
+memo-mark, and the result is read as 288.</p>
+
+<p class='c007'>Moving the decimal point often facilitates matters. Thus,
+<span class='fraction'>32·4 × 0·98 × 432 × 0·0217<br><span class='vincula'>4·71 × 0·175 × 0·00000621 × 412000</span></span> is much more conveniently dealt
+with when re-arranged as <span class='fraction'>32·4 × 9·8 × 432 × 2·17<br><span class='vincula'>4·71 × 17·5 × 6·21 × 4·12</span></span> = 141.</p>
+
+<p class='c007'>To determine the number of figures in the result by rough
+cancelling and mental calculation, we note that 4·71 enters 432
+about 100 times; 9·8 enters 17·5 about 2; 6·21 into 32·4 about 5;
+and 2·17 into 4·12 about 2. This gives <span class='fraction'><span class='under'>500</span><br>4</span> = 125, showing that
+the result contains 3 digits. From the slide rule we read 141,
+which is therefore the result sought.</p>
+
+<p class='c007'><span class='pageno' id='Page_30'>30</span>The occasional traversing of the slide through the rule, to
+interchange the indices—a contingency which the use of the C and
+D scales always involves—may often be avoided by a very simple
+expedient. Such an example as <span class='fraction'>6·19 × 31·2 × 422<br><span class='vincula'>1120 × 8·86 × 2.09</span></span> = 3·93 is sometimes
+cited as a particularly difficult case. Working through the
+expression as given, two traversings of the slide are necessary;
+but by taking the factors in the slightly different order,
+<span class='fraction'>6·19 × 31·2 × 422<br><span class='vincula'>8·86 × 2·09 × 1120</span></span>, <em>so that the significant figures of each pair are more
+nearly alike</em>, we not only avoid any traversing the slide, but we
+also reduce the extent to which the slide is moved to effect the
+several divisions.</p>
+
+<p class='c007'>Such cases as <span class='fraction'><em>a</em> × <em>b</em><br><span class='vincula'><em>c</em> × <em>d</em> × <em>e</em> × <em>f</em> × <em>g</em></span></span> or <span class='fraction'><span class='under'><em>a</em> × <em>b</em> × <em>c</em> × <em>d</em> × <em>e</em></span><br><em>f</em> × <em>g</em></span> really resolve
+themselves into <span class='fraction'><em>a</em> × <em>b</em> × 1 × 1 × 1<br><span class='vincula'><em>c</em> × <em>d</em> × <em>e</em> × <em>f</em> × <em>g</em></span></span> and <span class='fraction'><span class='under'><em>a</em> × <em>b</em> × <em>c</em> × <em>d</em> × <em>e</em></span><br><em>f</em> × <em>g</em> × 1 × 1 × 1</span>, but, of course,
+if rules are used to locate the decimal point, the 1’s so (mentally) introduced
+are not to be counted as additional figures in the factors.</p>
+
+<div class='chapter'>
+  <h2 class='c005'>MULTIPLICATION AND DIVISION WITH THE SLIDE INVERTED.</h2>
+</div>
+
+<p class='c012'>If the slide be inverted in the rule but with the same face
+uppermost, so that the Ɔ  scale lies adjacent to the A scale, and the
+right and left indices of the slide and rule are placed in coincidence,
+we find the product of any number on D by the coincident number
+on Ɔ (readily referred to each other by the cursor) is always 10.
+Hence, by reading the numbers on Ɔ as decimals, we have over
+any unit number on D, its <em>reciprocal</em> on Ɔ. Thus 2 on D is found
+opposite 0·5 on Ɔ; 3 on D opposite to 0·333; while opposite 8 on
+Ɔ is 0·125 on D, etc. The reason of this is that the sum of the
+lengths of the slide and rule corresponding to the factors, is
+always equal to the length corresponding to the product—in this
+case, 10.</p>
+
+<p class='c007'>It will be seen that if we attempt to apply the ordinary rule
+for multiplication, with the slide inverted, we shall actually be
+multiplying the one factor taken on D by the <em>reciprocal</em> of the
+other taken on Ɔ. But multiplying by the <em>reciprocal of a number</em>
+is equivalent to <em>dividing</em> by that number, and <em>dividing</em> a factor by
+the <em>reciprocal</em> of a number is equivalent to <em>multiplying</em> by that
+<span class='pageno' id='Page_31'>31</span>number. It follows that with the slide inverted the operations of
+multiplication and division are reversed, as are also the rules for
+the number of digits in the product and the position of the decimal
+point. Hence, in multiplying with the slide inverted, we place
+(by the aid of the cursor) one factor on Ɔ opposite the other factor
+on D, and read the result on D under either index of Ɔ. It follows
+that with the slide thus set, any pair of coinciding factors on Ɔ
+and D will give the same constant product found on D under the
+index of Ɔ. One useful application of this fact is found in selecting
+the scantlings of rectangular sections of given areas or in
+deciding upon the dimensions of rectangular sheets, plates, cisterns,
+etc. Thus by placing the index of Ɔ to 72 on D, it is readily seen
+that a plate having an area of 72 sq. ft. may have sides 8 by 9 ft.,
+6 by 12, 5 by 14·4, 4 by 18, 3 by 24, 2 by 36, with innumerable
+intermediate values. Many other useful applications of a similar
+character will suggest themselves.</p>
+
+<div class='chapter'>
+  <h2 class='c005'>PROPORTION.</h2>
+</div>
+
+<p class='c012'>With the slide in the ordinary position and with the indices of
+the C and D scales in exact agreement, the <em>ratio</em> of the corresponding
+divisions of these scales is 1. If the slide is moved so that
+1 on C agrees with 2 on D, we know that under any number <em>n</em> on
+C is <em>n</em> × 2 on D, so that if we read numerators on C and denominators
+on D we have</p>
+
+<table class='table1'>
+  <tr>
+ <td class='bbt c013'>C</td>
+ <td class='bbt c013'>1</td>
+ <td class='bbt c013'>1·5</td>
+ <td class='bbt c013'>2</td>
+ <td class='bbt c013'>3</td>
+ <td class='bbt c014'>4</td>
+  </tr>
+  <tr>
+    <td class='c013'>D1</td>
+    <td class='c013'>2</td>
+    <td class='c013'>3</td>
+    <td class='c013'>4</td>
+    <td class='c013'>6</td>
+    <td class='c014'>8.</td>
+  </tr>
+</table>
+
+<p class='c016'>In other words, the numbers on D bear to the coinciding numbers
+on C a ratio of 2 to 1. Obviously the same condition will obtain
+no matter in what position the slide may be placed. The rule for
+proportion, which is apparent from the foregoing, may be expressed
+as follows:—</p>
+
+<p class='c007'><span class='sc'>Rule for Proportion.</span>—<em>Set the first term of a proportion on the
+C scale to the second term on the D scale, and opposite the third term
+on the C scale read the fourth term on the D scale.</em></p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the 4th term in the proportion of 20&#8201;∶&#8201;27&#8201;∷&#8201;70&#8201;∶&#8201;<em>x</em>.
+Set 20 on C to 27 on D, and opposite 70 on C read 94·5 on D. Thus</p>
+
+<table class='table1'>
+  <tr>
+ <td class='bbt c013'>C</td>
+ <td class='bbt c013'>20</td>
+ <td class='bbt c014'>70</td>
+  </tr>
+  <tr>
+    <td class='c013'>D</td>
+    <td class='c013'>27</td>
+    <td class='c014'>94·5.</td>
+  </tr>
+</table>
+
+<p class='c012'>It will be evident that this is merely a case of combined
+multiplication and division of the form, <span class='fraction'><span class='under'>20 × 70</span><br>27</span> = 94·5. Hence,
+<span class='pageno' id='Page_32'>32</span>given any three terms of a proportion, we set the 1st to the 2nd,
+or the 3rd to the 4th, as the case may be, and opposite the other
+given term read the term required.<a id='r4'></a><a href='#f4' class='c019'><sup>[4]</sup></a></p>
+
+<p class='c007'>Thus, in reducing vulgar fractions to decimals, the decimal
+equivalent of <span class='fraction'>3<br><span class='vincula'>16</span></span> is determined by placing 3 on C to 16 on D, when
+over the index or 1 of D we read 0·1875 on C. In this case the
+terms are 3&#8201;∶&#8201;16&#8201;∷&#8201;<em>x</em>&#8201;∶&#8201;1. For the inverse operation—to find a
+vulgar fraction equivalent to a given decimal—the given decimal
+fraction on C is set to the index of D, and then opposite any
+denominator on D is the corresponding numerator of the fraction
+on C.</p>
+
+<p class='c007'>If the index of C be placed to agree with 3·1416 on D, it will
+be clear from what has been said that this ratio exists throughout
+between the numbers of the two scales. Therefore, against any
+<em>diameter</em> of a circle on C will be found the corresponding <em>circumference</em>
+on D. In the same way, by setting 1 on C to the appropriate
+conversion factor on D, we can convert a series of values in one
+denomination to their equivalents in another denomination. In
+this connection the following table of conversion factors will be
+found of service. If the A and B scales are used instead of the
+C and D scales, a complete set of conversions will be at once
+obtained. In this case, however, the left-hand A and B scales
+should be used for the initial setting, any values read on the right-hand
+A or B scales being read as of tenfold value. With the C and
+D scales a portion of the one scale will project beyond the other.
+To read this portion of the scale, the cursor or runner is brought
+to whichever index of the C scale falls within the rule, and the
+slide moved until the other index of the C scale coincides with
+the cursor, when the remainder of the equivalent values can then
+be read off. It must be remembered that if the slide is moved in
+the direction of notation (to the <em>right</em>), the values read thereon
+have a tenfold <em>greater</em> value; if the slide is moved to the <em>left</em>, the
+readings thereon are <em>decreased</em> in a tenfold degree. Although
+preferred by many, in the form given, the case is obviously one of
+multiplication, and is so treated in the Data Slips at the end of
+the book.</p>
+
+<table class='table2'>
+  <tr><td class='c028' colspan='3'><span class='pageno' id='Page_33'>33</span></td></tr>
+  <tr><th class='c028' colspan='3'>TABLE OF CONVERSION FACTORS.</th></tr>
+  <tr><th class='c028' colspan='3'><span class='sc'>Geometrical Equivalents.</span></th></tr>
+  <tr>
+ <th class='btt bbt c029'><span class='sc'>Scale C.</span></th>
+ <th class='btt bbt blt c029'><span class='sc'>Scale D.</span></th>
+ <th class='btt bbt blm c030'>If C = 1,<br>D =</th>
+  </tr>
+  <tr>
+    <td class='c031'>Diameter of circle</td>
+ <td class='blt c032'>Circumference of circle</td>
+ <td class='blm c033'>3·1416</td>
+  </tr>
+  <tr>
+    <td class='c031'>„         „</td>
+ <td class='blt c032'>Side of inscribed square</td>
+ <td class='blm c033'>0·707</td>
+  </tr>
+  <tr>
+    <td class='c031'>„         „</td>
+ <td class='blt c032'>„ equal square</td>
+ <td class='blm c033'>0·886</td>
+  </tr>
+  <tr>
+    <td class='c031'>„         „</td>
+ <td class='blt c032'>„ „ equilateral triangle</td>
+ <td class='blm c033'>1·346</td>
+  </tr>
+  <tr>
+    <td class='c031'>Circum. of circle</td>
+ <td class='blt c032'>„ inscribed square</td>
+ <td class='blm c033'>0·225</td>
+  </tr>
+  <tr>
+    <td class='c031'>„         „</td>
+ <td class='blt c032'>„ equal square</td>
+ <td class='blm c033'>0·282</td>
+  </tr>
+  <tr>
+    <td class='c031'>Side of square</td>
+ <td class='blt c032'>Diagonal of square</td>
+ <td class='blm c033'>1·414</td>
+  </tr>
+  <tr>
+    <td class='c031'>Square inch</td>
+ <td class='blt c032'>Circular inch</td>
+ <td class='blm c033'>1·273</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>Area of circle</td>
+ <td class='bbt blt c032'>Area of inscribed square</td>
+ <td class='bbt blm c033'>0·636</td>
+  </tr>
+  <tr><th class='c028' colspan='3'><span class='sc'>Measures of Length.</span></th></tr>
+  <tr>
+ <td class='btt c031'>Inches</td>
+ <td class='btt blt c032'>Millimetres</td>
+ <td class='btt blm c033'>25·40</td>
+  </tr>
+  <tr>
+    <td class='c031'>„</td>
+ <td class='blt c032'>Centimetres</td>
+ <td class='blm c033'>2·54</td>
+  </tr>
+  <tr>
+    <td class='c031'>8ths of an inch</td>
+ <td class='blt c032'>Millimetres</td>
+ <td class='blm c033'>3·175</td>
+  </tr>
+  <tr>
+    <td class='c031'>16ths „      „</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>1·587</td>
+  </tr>
+  <tr>
+    <td class='c031'>32nds „      „</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>0·794</td>
+  </tr>
+  <tr>
+    <td class='c031'>64ths „      „</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>0·397</td>
+  </tr>
+  <tr>
+    <td class='c031'>Feet</td>
+ <td class='blt c032'>Metres</td>
+ <td class='blm c033'>0·3048</td>
+  </tr>
+  <tr>
+    <td class='c031'>Yards</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>0·9144</td>
+  </tr>
+  <tr>
+    <td class='c031'>Chains</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>20·116</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>Miles</td>
+ <td class='bbt blt c032'>Kilometres</td>
+ <td class='bbt blm c033'>1·609</td>
+  </tr>
+  <tr><th class='c028' colspan='3'><span class='sc'>Measures of Area.</span></th></tr>
+  <tr>
+ <td class='btt c031'>Square inches</td>
+ <td class='btt blt c032'>Square centimetres</td>
+ <td class='btt blm c033'>6·46</td>
+  </tr>
+  <tr>
+    <td class='c031'>Circular  „</td>
+ <td class='blt c032'>„ „</td>
+ <td class='blm c033'>5·067</td>
+  </tr>
+  <tr>
+    <td class='c031'>Square feet</td>
+ <td class='blt c032'>„ metres</td>
+ <td class='blm c033'>0·0929</td>
+  </tr>
+  <tr>
+    <td class='c031'>„   yards</td>
+ <td class='blt c032'>„ „</td>
+ <td class='blm c033'>0·836</td>
+  </tr>
+  <tr>
+    <td class='c031'>„   miles</td>
+ <td class='blt c032'>„ kilometres</td>
+ <td class='blm c033'>2·59</td>
+  </tr>
+  <tr>
+    <td class='c031'>„     „</td>
+ <td class='blt c032'>Hectares</td>
+ <td class='blm c033'>259·00</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>Acres</td>
+ <td class='bbt blt c032'>„</td>
+ <td class='bbt blm c033'>0·4046</td>
+  </tr>
+  <tr><th class='c028' colspan='3'><span class='sc'>Measures of Capacity.</span></th></tr>
+  <tr>
+ <td class='btt c031'>Cubic inches</td>
+ <td class='btt blt c032'>Cubic centimetres</td>
+ <td class='btt blm c033'>16·38</td>
+  </tr>
+  <tr>
+    <td class='c031'>„     „</td>
+ <td class='blt c032'>Imperial gallons</td>
+ <td class='blm c033'>0·00360</td>
+  </tr>
+  <tr>
+    <td class='c031'>„     „</td>
+ <td class='blt c032'>U.S. gallons</td>
+ <td class='blm c033'>0·00432</td>
+  </tr>
+  <tr>
+    <td class='c031'>„     „</td>
+ <td class='blt c032'>Litres</td>
+ <td class='blm c033'>0·01638</td>
+  </tr>
+  <tr>
+    <td class='c031'>Cubic feet</td>
+ <td class='blt c032'>Cubic metres</td>
+ <td class='blm c033'>0·0283</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    „</td>
+ <td class='blt c032'>Imperial gallons</td>
+ <td class='blm c033'>6·23</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    „</td>
+ <td class='blt c032'>U.S. gallons</td>
+ <td class='blm c033'>7·48</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    „</td>
+ <td class='blt c032'>Litres</td>
+ <td class='blm c033'>28·37</td>
+  </tr>
+  <tr>
+    <td class='c031'>„   yards</td>
+ <td class='blt c032'>Cubic metres</td>
+ <td class='blm c033'>0·764</td>
+  </tr>
+  <tr>
+    <td class='c031'>Imperial gallons</td>
+ <td class='blt c032'>Litres</td>
+ <td class='blm c033'>4·54</td>
+  </tr>
+  <tr>
+    <td class='c031'>„        „</td>
+ <td class='blt c032'>U.S. gallons</td>
+ <td class='blm c033'>1·200</td>
+  </tr>
+  <tr>
+    <td class='c031'>Bushels</td>
+ <td class='blt c032'>Cubic metres</td>
+ <td class='blm c033'>0·0363</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>„</td>
+ <td class='bbt blt c032'>„ feet</td>
+ <td class='bbt blm c033'>1·283</td>
+  </tr>
+  <tr><th class='c028' colspan='3'><span class='sc'>Measures of Weight.</span></th></tr>
+  <tr>
+ <td class='btt c031'><span class='pageno' id='Page_34'>34</span>Grains</td>
+ <td class='btt blt c032'>Grammes</td>
+ <td class='btt blm c033'>0·0648</td>
+  </tr>
+  <tr>
+    <td class='c031'>Ounces (Troy)</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>31·103</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    (Avoird.)</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>28·35</td>
+  </tr>
+  <tr>
+    <td class='c031'>„       „</td>
+ <td class='blt c032'>Kilogrammes</td>
+ <td class='blm c033'>0·02835</td>
+  </tr>
+  <tr>
+    <td class='c031'>Pounds (Troy)</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>0·3732</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    (Avoird.)</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>0·4536</td>
+  </tr>
+  <tr>
+    <td class='c031'>Hundredweights</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>50·802</td>
+  </tr>
+  <tr>
+    <td class='c031'>Tons</td>
+ <td class='blt c032'>„</td>
+ <td class='blm c033'>1016·4</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>„</td>
+ <td class='bbt blt c032'>Metric tonnes</td>
+ <td class='bbt blm c033'>1·016</td>
+  </tr>
+  <tr><th class='c028' colspan='3'><span class='sc'>Compound Factors—Velocities.</span></th></tr>
+  <tr>
+ <td class='btt c031'>Feet per second</td>
+ <td class='btt blt c032'>Metres per second</td>
+ <td class='btt blm c033'>0·3048</td>
+  </tr>
+  <tr>
+    <td class='c031'>„       „</td>
+ <td class='blt c032'>„ minute</td>
+ <td class='blm c033'>18·288</td>
+  </tr>
+  <tr>
+    <td class='c031'>„       „</td>
+ <td class='blt c032'>Miles per hour</td>
+ <td class='blm c033'>0.682</td>
+  </tr>
+  <tr>
+    <td class='c031'>„      minute</td>
+ <td class='blt c032'>Meters per second</td>
+ <td class='blm c033'>0·00508</td>
+  </tr>
+  <tr>
+    <td class='c031'>„       „</td>
+ <td class='blt c032'>„ minute</td>
+ <td class='blm c033'>0·3048</td>
+  </tr>
+  <tr>
+    <td class='c031'>„       „</td>
+ <td class='blt c032'>Miles per hour</td>
+ <td class='blm c033'>0·01136</td>
+  </tr>
+  <tr>
+    <td class='c031'>Yards per „</td>
+ <td class='blt c032'>„ „</td>
+ <td class='blm c033'>0·0341</td>
+  </tr>
+  <tr>
+    <td class='c031'>Miles per hour</td>
+ <td class='blt c032'>Metres per minute</td>
+ <td class='blm c033'>26·82</td>
+  </tr>
+  <tr>
+    <td class='c031'>Knots</td>
+ <td class='blt c032'>„ „</td>
+ <td class='blm c033'>30·88</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>„</td>
+ <td class='bbt blt c032'>Miles per hour</td>
+ <td class='bbt blm c033'>1·151</td>
+  </tr>
+  <tr><th class='c028' colspan='3'><span class='sc'>Compound Factors—Pressures.</span></th></tr>
+  <tr>
+ <td class='btt c031'>Pounds per sq. inch</td>
+ <td class='btt blt c032'>Grammes per sq. mm.</td>
+ <td class='btt blm c033'>0·7031</td>
+  </tr>
+  <tr>
+    <td class='c031'>„         „</td>
+ <td class='blt c032'>Kilos. per sq. centimetre</td>
+ <td class='blm c033'>0·0703</td>
+  </tr>
+  <tr>
+    <td class='c031'>„         „</td>
+ <td class='blt c032'>Atmospheres</td>
+ <td class='blm c033'>0·068</td>
+  </tr>
+  <tr>
+    <td class='c031'>„         „</td>
+ <td class='blt c032'>Head of water in inches</td>
+ <td class='blm c033'>27·71</td>
+  </tr>
+  <tr>
+    <td class='c031'>„         „</td>
+ <td class='blt c032'>„ „ feet</td>
+ <td class='blm c033'>2·309</td>
+  </tr>
+  <tr>
+    <td class='c031'>„         „</td>
+ <td class='blt c032'>„ „ metres</td>
+ <td class='blm c033'>0·757</td>
+  </tr>
+  <tr>
+    <td class='c031'>„         „</td>
+ <td class='blt c032'>Inches of Mercury</td>
+ <td class='blm c033'>2·04</td>
+  </tr>
+  <tr>
+    <td class='c031'>Inches of water</td>
+ <td class='blt c032'>Pounds per square inch</td>
+ <td class='blm c033'>0·0361</td>
+  </tr>
+  <tr>
+    <td class='c031'>„       „</td>
+ <td class='blt c032'>Inches of mercury</td>
+ <td class='blm c033'>0·0714</td>
+  </tr>
+  <tr>
+    <td class='c031'>„       „</td>
+ <td class='blt c032'>Pounds per square foot</td>
+ <td class='blm c033'>5·20</td>
+  </tr>
+  <tr>
+    <td class='c031'>Inches of mercury</td>
+ <td class='blt c032'>Atmospheres</td>
+ <td class='blm c033'>0·0333</td>
+  </tr>
+  <tr>
+    <td class='c031'>Atmospheres</td>
+ <td class='blt c032'>Metres of water</td>
+ <td class='blm c033'>10·34</td>
+  </tr>
+  <tr>
+    <td class='c031'>„</td>
+ <td class='blt c032'>Kilos. per sq. cm.</td>
+ <td class='blm c033'>1·033</td>
+  </tr>
+  <tr>
+    <td class='c031'>Feet of water</td>
+ <td class='blt c032'>Pounds per square foot</td>
+ <td class='blm c033'>62·35</td>
+  </tr>
+  <tr>
+    <td class='c031'>„     „</td>
+ <td class='blt c032'>Atmospheres</td>
+ <td class='blm c033'>0·0294</td>
+  </tr>
+  <tr>
+    <td class='c031'>„     „</td>
+ <td class='blt c032'>Inches of mercury</td>
+ <td class='blm c033'>0·883</td>
+  </tr>
+  <tr>
+    <td class='c031'>Pounds per sq. foot</td>
+ <td class='blt c032'>„ „</td>
+ <td class='blm c033'>0·01417</td>
+  </tr>
+  <tr>
+    <td class='c031'>„        „</td>
+ <td class='blt c032'>Kilos. per square metre</td>
+ <td class='blm c033'>4·883</td>
+  </tr>
+  <tr>
+    <td class='c031'>„        „</td>
+ <td class='blt c032'>Atmospheres</td>
+ <td class='blm c033'>0·000472</td>
+  </tr>
+  <tr>
+    <td class='c031'>Pounds per sq. yard</td>
+ <td class='blt c032'>Kilos. per square metre</td>
+ <td class='blm c033'>0·5425</td>
+  </tr>
+  <tr>
+    <td class='c031'>Tons per sq. inch</td>
+ <td class='blt c032'>„ square mm.</td>
+ <td class='blm c033'>1·575</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>„ sq. foot</td>
+ <td class='bbt blt c032'>Tonnes per square metre</td>
+ <td class='bbt blm c033'>10·936</td>
+  </tr>
+  <tr><th class='c028' colspan='3'><span class='sc'>Compound Factors—Weights, Capacities, etc.</span></th></tr>
+  <tr>
+ <td class='btt c031'><span class='pageno' id='Page_35'>35</span>Pounds per lineal ft.</td>
+ <td class='btt blt c032'>Kilos. per lineal metre</td>
+ <td class='btt blm c033'>1·488</td>
+  </tr>
+  <tr>
+    <td class='c031'>„ per lineal yd.</td>
+ <td class='blt c032'>„ „ „</td>
+ <td class='blm c033'>0·496</td>
+  </tr>
+  <tr>
+    <td class='c031'>„ per lineal mile</td>
+ <td class='blt c032'>Kilos. per kilometre</td>
+ <td class='blm c033'>0·2818</td>
+  </tr>
+  <tr>
+    <td class='c031'>Tons     „       „</td>
+ <td class='blt c032'>Tonnes „</td>
+ <td class='blm c033'>0·6313</td>
+  </tr>
+  <tr>
+    <td class='c031'>Feet     „       „</td>
+ <td class='blt c032'>Metres „</td>
+ <td class='blm c033'>1·894</td>
+  </tr>
+  <tr>
+    <td class='c031'>Pounds per cubic in.</td>
+ <td class='blt c032'>Grammes per cubic cm.</td>
+ <td class='blm c033'>27·68</td>
+  </tr>
+  <tr>
+    <td class='c031'>„  per cubic ft.</td>
+ <td class='blt c032'>Kilos. per cubic metre</td>
+ <td class='blm c033'>16·02</td>
+  </tr>
+  <tr>
+    <td class='c031'>„  per cubic yd.</td>
+ <td class='blt c032'>„ „ „</td>
+ <td class='blm c033'>0·593</td>
+  </tr>
+  <tr>
+    <td class='c031'>Tons per cubic yard</td>
+ <td class='blt c032'>Tonnes „ „</td>
+ <td class='blm c033'>1·329</td>
+  </tr>
+  <tr>
+    <td class='c031'>Cubic yds. per pound</td>
+ <td class='blt c032'>Cubic metres per kilo.</td>
+ <td class='blm c033'>1·685</td>
+  </tr>
+  <tr>
+    <td class='c031'>„      per ton</td>
+ <td class='blt c032'>„ „ per tonne</td>
+ <td class='blm c033'>0·7525</td>
+  </tr>
+  <tr>
+    <td class='c031'>Cubic inch of water</td>
+ <td class='blt c032'>Weight in pounds</td>
+ <td class='blm c033'>0·03608</td>
+  </tr>
+  <tr>
+    <td class='c031'>Cubic feet of water</td>
+ <td class='blt c032'>„ „</td>
+ <td class='blm c033'>62·35</td>
+  </tr>
+  <tr>
+    <td class='c031'>„          „</td>
+ <td class='blt c032'>„ kilos</td>
+ <td class='blm c033'>28·23</td>
+  </tr>
+  <tr>
+    <td class='c031'>„          „</td>
+ <td class='blt c032'>Imperial gallons</td>
+ <td class='blm c033'>6·235</td>
+  </tr>
+  <tr>
+    <td class='c031'>„          „</td>
+ <td class='blt c032'>U.S. gallons</td>
+ <td class='blm c033'>7·48</td>
+  </tr>
+  <tr>
+    <td class='c031'>Litre of water</td>
+ <td class='blt c032'>Cubic inches</td>
+ <td class='blm c033'>61·025</td>
+  </tr>
+  <tr>
+    <td class='c031'>Gallons of water</td>
+ <td class='blt c032'>Weight in kilos</td>
+ <td class='blm c033'>4·54</td>
+  </tr>
+  <tr>
+    <td class='c031'>Pounds of fresh water</td>
+ <td class='blt c032'>Pounds of sea water</td>
+ <td class='blm c033'>1·026</td>
+  </tr>
+  <tr>
+    <td class='c031'>Grains per gallon</td>
+ <td class='blt c032'>Grammes per litre</td>
+ <td class='blm c033'>0·01426</td>
+  </tr>
+  <tr>
+    <td class='c031'>Pounds per gallon</td>
+ <td class='blt c032'>Kilos. per litre</td>
+ <td class='blm c033'>0·0998</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>„ per U.S. gal.</td>
+ <td class='bbt blt c032'>„ „</td>
+ <td class='bbt blm c033'>0·115</td>
+  </tr>
+  <tr><th class='c028' colspan='3'><span class='sc'>Compound Factors—Power Units, etc.</span></th></tr>
+  <tr>
+ <td class='btt c031'>British Ther. Units.</td>
+ <td class='btt blt c032'>Kilogrammetres.</td>
+ <td class='btt blm c033'>108</td>
+  </tr>
+  <tr>
+    <td class='c031'>„          „</td>
+ <td class='blt c032'>Joules</td>
+ <td class='blm c033'>1058</td>
+  </tr>
+  <tr>
+    <td class='c031'>„          „</td>
+ <td class='blt c032'>Calories (Fr. Ther. units)</td>
+ <td class='blm c033'>0·252</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    „   per sq. ft.</td>
+ <td class='blt c032'>„ per square metre</td>
+ <td class='blm c033'>2·713</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    „   per pound</td>
+ <td class='blt c032'>„ per kilogramme</td>
+ <td class='blm c033'>0·555</td>
+  </tr>
+  <tr>
+    <td class='c031'>Pounds per sq. ft.</td>
+ <td class='blt c032'>Dynes, per sq. cm.</td>
+ <td class='blm c033'>479</td>
+  </tr>
+  <tr>
+    <td class='c031'>Foot-pounds</td>
+ <td class='blt c032'>Kilogrammetres</td>
+ <td class='blm c033'>0·1382</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    „</td>
+ <td class='blt c032'>Joules</td>
+ <td class='blm c033'>1·356</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    „</td>
+ <td class='blt c032'>Thermal Units</td>
+ <td class='blm c033'>0·00129</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    „</td>
+ <td class='blt c032'>Calorie</td>
+ <td class='blm c033'>0·000324</td>
+  </tr>
+  <tr>
+    <td class='c031'>Foot-tons</td>
+ <td class='blt c032'>Tonne-metres</td>
+ <td class='blm c033'>0·333</td>
+  </tr>
+  <tr>
+    <td class='c031'>Horse-power</td>
+ <td class='blt c032'>Force decheval (Fr.H.P.)</td>
+ <td class='blm c033'>1·014</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    „</td>
+ <td class='blt c032'>Kilowatts</td>
+ <td class='blm c033'>0·746</td>
+  </tr>
+  <tr>
+    <td class='c031'>Pounds per H.P.</td>
+ <td class='blt c032'>Kilos. per cheval</td>
+ <td class='blm c033'>0·447</td>
+  </tr>
+  <tr>
+    <td class='c031'>Square feet per H. P.</td>
+ <td class='blt c032'>Square metres per cheval</td>
+ <td class='blm c033'>0·0196</td>
+  </tr>
+  <tr>
+    <td class='c031'>Cubic     „      „</td>
+ <td class='blt c032'>Cubic „ „</td>
+ <td class='blm c033'>0·0279</td>
+  </tr>
+  <tr>
+    <td class='c031'>Watts</td>
+ <td class='blt c032'>Ther. Units per hour</td>
+ <td class='blm c033'>3·44</td>
+  </tr>
+  <tr>
+    <td class='c031'>„</td>
+ <td class='blt c032'>Foot-pounds per second</td>
+ <td class='blm c033'>0·73</td>
+  </tr>
+  <tr>
+    <td class='c031'>„</td>
+ <td class='blt c032'>„ per minute</td>
+ <td class='blm c033'>44·24</td>
+  </tr>
+  <tr>
+    <td class='c031'>Watt-hours</td>
+ <td class='blt c032'>Kilogrammetres</td>
+ <td class='blm c033'>367</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    „</td>
+ <td class='blt c032'>Joules</td>
+ <td class='blm c033'>3600</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>Kilogrammetres</td>
+ <td class='bbt blt c032'>„</td>
+ <td class='bbt blm c033'>9·806</td>
+  </tr>
+</table>
+
+<p class='c007'><span class='pageno' id='Page_36'>36</span><em>Inverse Proportion.</em>—If “more” requires “less,” or “less”
+requires “more,” the case is one of <em>inverse</em> proportion, and
+although it will be seen that this form of proportion is quite
+readily dealt with by the preceding method, the working is
+simplified to some extent by inverting the slide so that the C
+scale is adjacent to the A scale. By the aid of the cursor, the
+values on the inverted C (or  Ɔ) scale, and on the D scale, can be
+then read off. These will now constitute a series of inverse ratios.
+For example, in the proportion</p>
+
+<table class='table1'>
+  <tr>
+ <td class='btt bbt c013'>Ɔ</td>
+ <td class='btt bbt c013'>8</td>
+ <td class='btt bbt c014'>4</td>
+  </tr>
+  <tr>
+    <td class='c013'>D</td>
+    <td class='c013'>1·5</td>
+    <td class='c014'>3</td>
+  </tr>
+</table>
+
+<p class='c016'>the 4 on the Ɔ scale is brought opposite 3 on D, when under 8 on
+Ɔ is found 1·5 on D.<a id='r5'></a><a href='#f5' class='c019'><sup>[5]</sup></a></p>
+
+<h3 class='c008'>GENERAL HINTS ON THE ELEMENTARY USES OF THE SLIDE RULE.</h3>
+
+<p class='c009'>Before the more complex operations of involution, evolution, etc.,
+are considered, a few general hints on the use of the slide rule
+for elementary operations may be of service, especially as these
+will serve to enforce some of the more important points brought
+out in the preceding sections.</p>
+
+<p class='c007'>Always use the slide rule in as <em>direct</em> a light as possible.</p>
+
+<p class='c007'>Study the manner in which the scales are divided. Follow the
+graduations of the C and D scales from 1 to 10, noting the values
+given by each successive graduation and how these values change
+as we follow along to the right. Do the same with the two halves
+of the A and B scales and note the difference in the value of the
+subdivisions, due to the shorter scale-lengths.</p>
+
+<p class='c007'>Practise reading values by setting 1 on C to some value on D
+and reading under 2, 3, 4, etc., on C, checking the readings by
+mental arithmetic. To the same end, find squares, square roots,
+etc., comparing the results with the actual values as given in tables.
+Practise setting both slide and cursor to values taken at random.
+Aim at accuracy; speed will come with practice.</p>
+
+<p class='c007'><span class='pageno' id='Page_37'>37</span>When in doubt as to any method of working, verify by making
+a simple calculation of the same form.</p>
+
+<p class='c007'>Follow the orthodox methods of working until entirely confident
+in the use of the instrument, and even then do not readily make a
+change. If any altered procedure is adopted, first work a simple
+case and guard carefully against unconsciously lapsing into the
+usual method during the operation.</p>
+
+<p class='c007'>Unless the calculation is of a straightforward character, time
+taken in considering how best to attack it (rearranging the
+expression if desirable) is generally time well spent.</p>
+
+<p class='c007'>In setting two values together, set the cursor to one of them
+on the rule, and bring the other, on the slide, to the cursor line.</p>
+
+<p class='c007'>In multiplying factors, as 57 × 0·1256, take the fractional value
+first. It is easier to set 1 on C to 1256 on D and read under 57 on
+C, than to reverse the procedure. When both values are eye-estimated,
+set the cursor to the second factor on C and read the
+result on D, under the cursor line.</p>
+
+<p class='c007'>In continuous operations avoid moving the slide further than
+necessary, by taking the factors in that order which will keep the
+scale readings as close together as possible.</p>
+
+<h3 class='c008'>SQUARES AND SQUARE ROOTS.</h3>
+
+<p class='c009'>We have seen that the relation which the upper scales bear to
+the lower set is such that over any number on D is its square on
+A, and, conversely, under any number on A is its square root on
+D, the same remarks applying to the C and B scales on the slide.
+Taking the values engraved on the rule, we have on D, numbers
+lying between 1 and 10, and on A the corresponding squares extending
+from 1 to 100. Hence the squares of numbers between 1
+and 10, or the roots of numbers between 1 and 100, can be read off
+on the rule by the aid of the cursor. All other cases are brought
+within these ranges of values by factorising with powers of 10, as
+before explained.</p>
+
+<p class='c007'>The more practical rule is the following:—</p>
+
+<p class='c007'><em>To Find the Square of a Number</em>, set the cursor to the number
+on D and read the required square on A under the cursor. The
+rule for</p>
+
+<p class='c007'><em>The Number of Digits in a Square</em> is easily deducible from the rule
+for multiplication. If the square is read on the <em>left</em> scale of A, it
+<span class='pageno' id='Page_38'>38</span>will contain <em>twice</em> the number of digits in the original number <em>less</em>
+1; if it is read on the <em>right</em> scale of A, it will contain <em>twice</em> the
+number of digits in the original number.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the square of 114.</p>
+
+<p class='c021'>Placing the cursor to 114 on D, it is seen that the coinciding
+number on A is 13. As the result is read off on the <em>left</em> scale of A,
+the number of digits will be (3 × 2) − 1 = 5, and the answer is read
+as 13,000. The true result is 12,996.</p>
+
+<p class='c021'><span class='sc'>Ex.</span>—Find the square of 0·0093.</p>
+
+<p class='c021'>The cursor being placed to 93 on D, the number on A is found
+to be 865. The result is read on the <em>right</em> scale of A, so the
+number of digits = −2 × 2 = −4, and the answer is read as 0·0000865
+[0·00008649].</p>
+
+<p class='c012'><em>Square Root.</em>—The foregoing rules suggest the method of procedure
+in the inverse operation of extracting the square root of a
+given number, which will be found on the D scale opposite the
+number on the A scale. It is necessary to observe, however, that
+if the number consists of an <em>odd</em> number of digits, it is to be taken
+on the <em>left-hand</em> portion of the A scale, and the number of digits
+in the root = <span class='fraction'><span class='under'>N + 1</span><br>2</span>, N being the number of digits in the original
+number. When there is an even number of digits in the number,
+it is to be taken on the <em>right-hand</em> portion of the A scale, and
+the root contains <em>one-half</em> the number of digits in the original
+number.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the square root of 36,500.</p>
+
+<p class='c021'>As there is an <em>odd</em> number of digits, placing the cursor to
+365 on the <span class='fss'>L.H.</span> A scale gives 191 on D. By the rule there are
+<span class='fraction'><span class='under'>N + 1</span><br>2</span> = <span class='fraction'><span class='under'>5 + 1</span><br>2</span> = 3 digits in the required root, which is therefore
+read as 191 [191·05].</p>
+
+<p class='c021'><span class='sc'>Ex.</span>—Find √<span class='vincula'>0·0098.</span></p>
+
+<p class='c021'>Placing the cursor to 98 on the right-hand scale of A (since −2
+is an <em>even</em> number of digits), it is seen that the coinciding number
+on D is 99. As the number of digits in the number is −2, the
+number of digits in the root will be <span class='fraction'><span class='under'>−2</span><br>2</span> = −1. It will therefore
+be read as 0·099 [0·09899+].</p>
+
+<p class='c021'><span class='sc'>Ex.</span>—Find √<span class='vincula'>0·098</span>.</p>
+
+<p class='c021'>The number of digits is −1, so under 98 on the left scale of A,
+<span class='pageno' id='Page_39'>39</span>we find 313 on D. By the rule the number in the root will
+be <span class='fraction'><span class='under'>−1 +1</span><br>2</span> = 0, and the root is therefore read as 0·313 [0·313049+].</p>
+
+<p class='c021'><span class='sc'>Ex.</span>—Find √<span class='vincula'>0·149.</span></p>
+
+<p class='c021'>As the number of digits (0) is <em>even</em>, the cursor is set to 149 on
+the right-hand scale of A, giving 386 on D. By the rule, the
+number of digits in the root will be <span class='fraction'>0<br><span class='vincula'>2</span></span> = 0, and the root will be
+read as 0·386 [0·38605+].</p>
+
+<p class='c012'>Another method of extracting the square root, by which more
+accurate readings may generally be obtained, is by using the C and
+D scales only, with the slide inverted. If there is an <em>odd</em> number
+of digits in the number, the <em>right</em> index, or if an even number of
+digits the <em>left</em> index, of the inverted scale Ɔ is placed so as to
+coincide with the number on D of which the root is sought. Then
+with the cursor, the number is found on D which coincides with
+the same number on Ɔ, which number is the root sought.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find √<span class='vincula'>22·2.</span></p>
+
+<p class='c021'>Placing the left index of Ɔ to 222 on D, the two equal coinciding
+numbers on Ɔ and D are found to be 4·71.</p>
+
+<p class='c012'>Note that under the cursor line we have the original number,
+22·2, on A, and from this the number of digits in the root is
+determined as before.</p>
+
+<p class='c007'>The plan of finding the square of a number by ordinary multiplication
+is often very convenient. The inverse process of finding
+a square root by trial division is not to be recommended.</p>
+
+<p class='c007'>To obtain a close value of a root or to verify one found in the
+usual way, the author has, on occasion, adopted the following
+plan:—Set 1 (or 10) on B to the number on the A scale (<span class='fss'>L.H.</span> or
+<span class='fss'>R.H.</span> as the case may require), and bring the cursor to the number
+on D. If the root found is correct, the readings on C under
+the cursor and on D under the index of C, will be in exact agreement.</p>
+
+<p class='c007'>If 1 on B is placed to a number <em>n</em> on the <span class='fss'>L.H.</span> A scale, the student
+will note that while root <em>n</em> is read on D under 1 on C, the root of
+10 <em>n</em> is read on D under 10 on B. Hence, if preferred, the number
+can be taken always on the first scale of A and the root read under
+1 or 10 on B, according to whether there is an odd or even number
+of digits in the number. Obviously the second root is the first
+multiplied by √<span class='vincula'>10</span>.</p>
+
+<div>
+  <span class='pageno' id='Page_40'>40</span>
+  <h3 class='c008'>CUBES AND CUBE ROOTS.</h3>
+</div>
+
+<p class='c009'>In raising a number to the third power, a combination of the
+preceding method and ordinary multiplication is employed.</p>
+
+<p class='c007'><span class='sc'>To Find the Cube of a Number.</span>—<em>Set the</em> <span class='fss'>L.H.</span> <em>or</em> <span class='fss'>R.H.</span> <em>index of
+C to the number on D, and opposite the number</em> <span class='fss'>ON THE LEFT-HAND</span>
+<em>scale of B read the cube on the</em> <span class='fss'>L.H.</span> <em>or</em> <span class='fss'>R.H.</span> <em>scale of A</em>.</p>
+
+<p class='c007'>By this rule four scales are brought into requisition. Of these,
+the D scale and the <span class='fss'>L.H.</span> B scale are <em>always</em> employed, and are to
+be read as of equal denomination. The values assigned to the <span class='fss'>L.H.</span>
+and <span class='fss'>R.H.</span> scales of A will be apparent from the following considerations.</p>
+
+<p class='c007'>Commencing with the indices of C and D coinciding, and
+moving the slide to the right, it will be seen that, working in
+accordance with the above rule, the cubes of numbers from 1 to
+2·154 (= ∛<span class='vincula'>10</span>) will be found on the first or <span class='fss'>L.H.</span> scale of A. Moving
+the slide still farther to the right, we obtain <em>on the</em> <span class='fss'>R.H.</span> <em>A scale</em>
+cubes of numbers from 2·154 to 4·641 (or ∛<span class='vincula'>10</span> to ∛<span class='vincula'>100</span>). Had we
+a <em>third</em> repetition of the <span class='fss'>L.H.</span> A scale, the <span class='fss'>L.H.</span> index of C could be
+still further traversed to the right, and the cubes of numbers from
+4·641 to 10 read off on this prolongation of A. But the same end
+can be attained by making use of the <span class='fss'>R.H.</span> index of C, when,
+traversing the slide to the right as before, the cubes of numbers
+from 4·641 to 10 on D can be read off <em>on the</em> <span class='fss'>L.H.</span> <em>A scale</em> over the
+corresponding numbers on the <span class='fss'>L.H.</span> B scale. Hence, using the
+<span class='fss'>L.H.</span> index of C, the readings on the <span class='fss'>L.H.</span> A scale may be regarded
+comparatively as units, those on the <span class='fss'>R.H.</span> A scale as tens; while
+for the hundreds we again make use of the <span class='fss'>L.H.</span> A scale in conjunction
+with the <em>right-hand</em> index of C.</p>
+
+<p class='c007'>By keeping these points in view, the number of digits in the
+cube (N) of a given number (<em>n</em>) are readily deduced. Thus, if the
+units scale is used, N = 3<em>n</em> − 2; if the tens scale, N = 3<em>n</em> − 1; while if
+the hundreds scale be used, N = 3<em>n</em>. Placed in the form of rules:—</p>
+
+<p class='c007'>N = 3<em>n</em> − 2 when the product is read on the <span class='fss'>L.H.</span> scale of A with
+the slide to the <em>right</em> (units scale).</p>
+
+<p class='c007'>N = 3<em>n</em> − 1 when the product is read on the <span class='fss'>R.H.</span> scale of A;
+slide to the <em>right</em> (tens scale).</p>
+
+<p class='c007'>N = 3<em>n</em> when the product is read on the <span class='fss'>L.H.</span> scale of A with the
+slide to the <em>left</em> (hundreds scale).</p>
+
+<p class='c007'><span class='pageno' id='Page_41'>41</span>With decimals the same rule applies, but, as before, the number
+of digits must be read as −1, −2, etc., when one, two, etc., cyphers
+follow immediately after the decimal point.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—Find the value of 1·4<sup>3</sup>.</p>
+
+<p class='c007'>Placing the <span class='fss'>L.H.</span> index of C to 1·4 on D, the reading on A
+opposite 1·4 on the <span class='fss'>L.H.</span> scale of B is found to be about 2·745
+[2·744].</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—Find the value of 26·4<sup>3</sup>.</p>
+
+<p class='c007'>Placing the <span class='fss'>L.H.</span> index of C to 26·4 on D, the reading on A
+opposite 26·4 on the <span class='fss'>L.H.</span> scale of B is found to be about 18,400
+[18,399·744].</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—Find the value of 7·3<sup>3</sup>.</p>
+
+<p class='c007'>In this case it becomes necessary to use the <span class='fss'>R.H.</span> index of C,
+which is set to 7·3 on D, when opposite 7·3 on the <span class='fss'>L.H.</span> scale of B
+is read 389 [389·017] on A.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—Find the value of 0·073<sup>3</sup>.</p>
+
+<p class='c007'>From the setting as before it is seen that the number of digits
+in the number must be multiplied by 3. Hence, as there is −1 digit
+in 0·073, there will be −3 in the cube, which is therefore read
+0·000389.</p>
+
+<p class='c007'>The last two examples serve to illustrate the principle of
+factorising with powers of 10. Thus</p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>0·073 = 7·3 × 10<sup>−2</sup>; 0·073<sup>3</sup> = 7·3<sup>3</sup> × (10<sup>−2</sup>)<sup>3</sup> = 389 × 10<sup>−6</sup> = 0·000389.</div>
+  </div>
+</div>
+
+<p class='c007'><em>Cube Root</em> (<em>Direct Method</em>).—One method of extracting the cube
+root of a number is by an inversion of the foregoing operation.
+Using the same scales, <em>the slide is moved either to the right or left
+until under the given number on A is found a number on the</em> <span class='fss'>L.H.</span>
+<em>B scale, identical with the number simultaneously found on D under
+the right or left index of C</em>. This number is the required cube
+root.</p>
+
+<p class='c007'>From what has already been said regarding the combined use
+of these scales in cubing, it will be evident that in extracting the
+cube root of a number, it is necessary, in order to decide which
+scales are to be used, to know the number of figures to be dealt
+with. We therefore (as in the arithmetical method of extraction)
+point off the given number into sections of three figures each,
+commencing at the decimal point, and proceeding to the left for
+numbers greater than unity, and to the right for numbers less
+than unity. Then if the first section of figures on the left consists
+of—</p>
+
+<p class='c007'><span class='pageno' id='Page_42'>42</span>1 figure, the number will evidently require to be taken on
+what we have called the “units” scale—<em>i.e.</em>, on the <span class='fss'>L.H.</span> scale of A,
+using the <span class='fss'>L.H.</span> index of C.</p>
+
+<p class='c007'>If of 2 figures, the number will be taken on the “tens”
+scale—<em>i.e.</em>, on the <span class='fss'>R.H.</span> scale of A, using the <span class='fss'>L.H.</span> index of C.</p>
+
+<p class='c007'>If of 3 figures, the number will be taken on the “hundreds”
+scale—<em>i.e.</em>, on the <span class='fss'>L.H.</span> scale of A, using the <span class='fss'>R.H.</span> index of C.</p>
+
+<p class='c007'>To determine the number of digits in cube roots it is only
+necessary to note that when the number is pointed off into sections
+as directed, there will be one figure in the root for every section
+into which the number is so divided, whether the <em>first</em> section consists
+of 1, 2, or 3 digits.</p>
+
+<p class='c007'>Of numbers wholly decimal, the cube roots will be decimal, and
+for every group of <em>three</em> 0s immediately following the decimal
+point, <em>one</em> 0 will follow the decimal point in the root. If necessary,
+0s must be added so as to make up complete multiples of 3
+figures before proceeding to extract the root. Thus 0·8 is to
+be regarded as 0·800, and 0·00008 as 0·000080 in extracting cube
+roots.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—Find ∛<span class='vincula'>14,000.</span></p>
+
+<p class='c007'>Pointing the number off in the manner described, it is seen
+that there are <em>two</em> figures in the first section—viz., 14. Setting the
+cursor to 14 on the <span class='fss'>R.H.</span> scale of A, the slide is moved to the right
+until it is seen that 241 on the <span class='fss'>L.H.</span> scale of B falls under the
+cursor, when 241 on D is under the <span class='fss'>L.H.</span> index of C. Pointing
+14,000 off into sections we have 14 000—that is, <em>two</em> sections.
+Therefore, there are two digits in the root, which in consequence
+will be read 24·1 [24·1014+].</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—Find ∛<span class='vincula'>0·162.</span></p>
+
+<p class='c007'>As the divisional section consists of <em>three</em> figures, we use the
+“hundreds” scale. Setting the cursor to 0·162 on the <span class='fss'>L.H.</span> A
+scale, and using the <span class='fss'>R.H.</span> index of C, we move the slide to the left
+until under the cursor 0·545 is found on the <span class='fss'>L.H.</span> B scale, while the
+<span class='fss'>R.H.</span> index of C points to 0·545 on D, which is therefore the cube
+root of 0·162.</p>
+
+<p class='c007'><span class='sc'>Ex.</span>—Find ∛<span class='vincula'>0·0002.</span></p>
+
+<p class='c007'>To make even multiples of 3 figures requires the addition of
+00; we have then 200, the cube root of which is found to be about
+5·85. Then, since the first divisional group consists of 0s, one 0
+will follow the decimal point, giving ∛<span class='vincula'>0·0002</span> = 0·0585 [0·05848].</p>
+
+<p class='c007'><span class='pageno' id='Page_43'>43</span><em>Cube Root (Inverted Slide Method).</em>—Another method of extracting
+the cube root involves the use of the inverted slide. Several
+methods are used, but the following is to be preferred:—<em>Set the</em>
+<span class='fss'>L.H.</span> <em>or</em> <span class='fss'>R.H.</span> <em>index of the slide to the number on A, and the number
+on ᗺ (i.e., B inverted), which coincides with the same number on D,
+is the required root.</em></p>
+
+<p class='c007'>Setting the slide as directed, and using first the <span class='fss'>L.H.</span> index of
+the slide and then the <span class='fss'>R.H.</span> index, it is always possible to find <em>three</em>
+pairs of coincident values. To determine which of the three is the
+required result is best shown by an example.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find ∛<span class='vincula'>5,</span> ∛<span class='vincula'>50,</span> and ∛<span class='vincula'>500.</span></p>
+
+<p class='c021'>Setting the <span class='fss'>R.H.</span> index of the slide to 5 on A, it is seen that
+1·71 on D coincides with 1·71 on ᗺ. Then setting the <span class='fss'>L.H.</span> index
+to 5 on A, further coincidences are found at 3·68 and at 7·93,
+the three values thus found being the required roots. Note that
+the first root was found on that portion of the D scale lying under
+1 to 5 on A; the second root on that portion lying under 5 to 50
+on A; and the third root on that portion of D lying under 50 to
+100 on A. In this connection, therefore, scale A may always be
+considered to be divided into three sections—viz., 1 to <em>n</em>, <em>n</em> to 10<em>n</em>,
+and 10<em>n</em> to 100. For all numbers consisting of 1, 1 + 3, 1 + 6, 1 + 9—<em>i.e.</em>,
+of 1, 4, 7, 10, or −2, −5, etc., figures—the coincidence under
+the first section is the one required. If the number has 2, 5, 8, or
+−1, −4, −7, etc., figures, the coincidence under the second section
+is correct, while if the number has 3, 6, 9, or 0, −3, etc., figures,
+the coincidence under the last section is that required. The
+number of digits in the root is determined by marking off the
+number into sections, as already explained.</p>
+
+<p class='c012'><em>Cube Root (Pickworth’s Method).</em>—One of the principal objections
+to the two methods described is the difficulty of recollecting which
+scales are to be employed and with which index of the slide they
+are to be used. With the direct method another objection is that
+the readings to be compared are often some distance apart, the
+maximum distance intervening being <em>two-thirds</em> of the length of
+the rule. To carry the eye from one to another is troublesome
+and time-taking. With the inverted scale method the reading of
+a scale reversed in direction and with the figures inverted is also
+objectionable.</p>
+
+<p class='c007'>With the author’s method these objections are entirely obviated.
+The <em>same scales and index are always used</em>, and are read in their
+<span class='pageno' id='Page_44'>44</span>normal position. The three roots of <em>n</em>, 10<em>n</em> and 100<em>n</em> (<em>n</em> being less
+than 10 and not less than 1) are given with one setting and appear
+in their natural sequence, no traversing of the slide being needed.
+The readings to be compared are always close together, the maximum
+distance between them being <em>one-sixth</em> of the length of the rule.
+The setting is always made in the earlier part of the scales where
+closer readings can be obtained, and finally, if desired, the result may
+be readily verified on the lower scales by successive multiplication.</p>
+
+<p class='c007'>For this method two gauge points are required on C. To conveniently
+locate these, set 53 on C to 246 on D; join 1 on D to
+1 on A with a straight-edge and with a needle point draw a short
+fine line on C. Set 246 on C to 53 on D, and repeat the process
+at the other end of the rule. The gauge points thus obtained
+(dividing C into three equal parts) will be at 2·154 and 4·641, and
+should be marked ∛<span class='vincula'>10</span> and ∛<span class='vincula'>100</span> respectively.<a id='r6'></a><a href='#f6' class='c019'><sup>[6]</sup></a></p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find ∛<span class='vincula'>2·86,</span> ∛<span class='vincula'>28·6</span> and ∛<span class='vincula'>286</span>.</p>
+
+<p class='c021'>Set cursor to 2·86 on A and drawing the slide to the right
+find 1·42 under 1 on C, when 1·42 on B is under the cursor. Then
+reading under 1, ∛<span class='vincula'>10</span> and ∛<span class='vincula'>100,</span> we have</p>
+
+<p class='c021'>∛<span class='vincula'>2·86</span> = 1·42; ∛<span class='vincula'>28·6</span> = 3·06 and ∛<span class='vincula'>286</span> = 6·59.</p>
+
+<p class='c012'>It will be seen that factorising with powers of 10, we multiply
+the initial root by ∛<span class='vincula'>10</span> and ∛<span class='vincula'>100</span>. Obviously the three roots will
+always be found on D, in their natural order and at intervals of
+one-third the length of the rule. The number of digits in the
+roots of numbers which do not lie between 1 and 1000, is found as
+before explained.</p>
+
+<p class='c007'>In any method of extracting cube roots in which the slide has
+to be adjusted to give equal readings on B and D, the author has
+found it of advantage to adopt the following plan:—The cursor
+being set to, say, 4·8 on A, bring a near <em>main</em> division line on B,
+as 1·7, to the cursor; then 1 on C is at 1·68 on D. The difference
+in the readings is two small divisions on D, and moving the slide
+forward by <em>one-third the space representing this difference</em>, we obtain
+1·687 as the root required. With a little practice it is possible to
+obtain more accurate results by this method than by comparing
+the reading on D with that on the less finely-graded B scale.</p>
+
+<div>
+  <span class='pageno' id='Page_45'>45</span>
+  <h3 class='c008'>MISCELLANEOUS POWERS AND ROOTS.</h3>
+</div>
+
+<p class='c009'>In addition to squares and cubes, certain other powers and
+roots may be readily obtained with the slide rule.</p>
+
+<p class='c007'><em>Two-thirds Power.</em>—The value of N<sup>⅔</sup> is found on A over ∛̅N on
+D. The number of digits is decided by the rule for squares,
+working from the number of digits in the cube root. It will often
+be found preferable to treat N<sup>⅔</sup> as N ÷ ∛̅N, as in this way the
+magnitude of the result is much more readily appreciated.</p>
+
+<p class='c007'><em>Three-two Power.</em>—N<sup>³⁄₂</sup> can be obtained by cubing the square
+root, deciding the number of digits in each process. For the
+reason just given, it is preferable to regard N<sup>³⁄₂</sup> as N × √̅N.</p>
+
+<p class='c007'><em>Fourth Power.</em>—For N<sup>4</sup> set the index of C to N on D and over
+N on C read N<sup>4</sup> on A; or find the square of the square of N,
+deciding the number of digits at each step.</p>
+
+<p class='c007'><em>Fourth Root.</em>—Similarly for ∜̅N, take the square root of the
+square root.</p>
+
+<p class='c007'><em>Four-third Power.</em>—N<sup>⁴⁄₃</sup> = N<sup>1·33</sup> (useful in gas-engine diagram
+calculations) is best treated as N × ∛̅N.</p>
+
+<p class='c007'>Other powers can be found by repeated multiplication. Thus
+setting 1 on B to N on A, we have on A, N<sup>2</sup> over N; N<sup>3</sup> over N<sup>2</sup>;
+N<sup>4</sup> over N<sup>3</sup>; N<sup>5</sup> over N<sup>4</sup>, etc. In the same way, setting N on B
+to N on D, we can read such values as N<sup>¾</sup>, N<sup>⅞</sup>, etc.</p>
+
+<h3 class='c008'>POWERS AND ROOTS BY LOGARITHMS.</h3>
+
+<p class='c009'>For powers or roots other than those of the simple forms
+already discussed, it is necessary to employ the usual logarithmic
+process. Thus to find <em>a<sup>n</sup></em> = <em>x</em>, we multiply the logarithm of <em>a</em> by <em>n</em>,
+and find the number <em>x</em> corresponding to the logarithm so obtained.
+Similarly, to find <em>ⁿ√̅a</em> = <em>x</em> we divide the logarithm of <em>a</em> by <em>n</em>, and
+find the number <em>x</em> corresponding to the resulting logarithm.</p>
+
+<p class='c007'><em>The Scale of Logarithms.</em>—Upon the back of the slide of the
+Gravêt and similar slide rules there will be found three scales.
+One of these—usually the centre one—is divided equally throughout
+its entire length, and figured from right to left. It is sometimes
+marked L, indicating that it is a scale giving logarithms.
+The whole scale is divided primarily into ten equal parts, and each
+of these subdivided into 50 equal parts. In the recess or notch in
+the right-hand end of the rule is a reference mark, to which any
+of the divisions of this evenly-divided scale can be set.</p>
+
+<p class='c007'><span class='pageno' id='Page_46'>46</span>As this decimally-divided scale is equal in length to the
+logarithmic scale D, and is figured in the reverse direction, it
+results that when the slide is drawn to the right so that the <span class='fss'>L.H.</span>
+index of C coincides with any number on D, the reading on the
+equally-divided scale will give the decimal part of the logarithm
+of the number taken on D. Thus if the <span class='fss'>L.H.</span> index of C is placed
+to agree with 2 on D, the reading of the back scale, taken at the
+reference mark, will be found to be 0·301, the logarithm of 2. It
+must be distinctly borne in mind that the number so obtained is
+the <em>decimal part</em> or <em>mantissa</em> of the logarithm of the number, and
+that to this the characteristic must be prefixed in accordance with
+the usual rule—viz., <em>The integral part, or characteristic of a logarithm
+is equal to the number of digits in the number, minus 1. If the
+number is wholly decimal, the characteristic is equal to the number of
+cyphers following the decimal point, plus 1.</em> In the latter case the
+characteristic is negative, and is so indicated by having the minus
+sign written <em>over</em> it.</p>
+
+<p class='c007'>To obtain any given power or root of a number, the operation
+is as follows:—Set the <span class='fss'>L.H.</span> index of C to the given number on
+D, and turning the rule over, read opposite the mark in the notch
+at the right-hand end of the rule, the decimal part of the logarithm
+of the number. Add the characteristic according to the above
+rule, and multiply by the exponent of the power, or divide by
+the exponent of the root. Place the <em>decimal part</em> of the resultant
+reading, taken on the scale of equal parts, opposite the mark in
+the aperture of the rule, and read the answer on D under the <span class='fss'>L.H.</span>
+index of C, pointing off the number of digits in the answer in
+accordance with the number of the characteristic of the resultant.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Evaluate 36<sup>1·414</sup>.</p>
+
+<p class='c021'>Set 1 on C to 36 on D and read the decimal part of log. 36 on
+the scale of logarithms on the back of the slide. This value is
+found to be 0·556. As there are two digits in the number, the
+characteristic will be 1; hence log. 36 = 1·556. Multiply by 1·414,
+using the C and D scales, and obtain 2·2 as the log. of the result.
+Set the decimal part, 0·2, on the log. scale to the mark in the notch
+at the end of the rule and read 1585 on D under 1 on C. Since
+the log. of the result has a characteristic 2, there will be 3 digits
+in the result, which is therefore read as 158·5.</p>
+
+<p class='c012'>This example will suffice to show the method of obtaining the
+nth power or the <em>n</em>th root of <em>any</em> number.</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_47'>47</span>
+  <h2 class='c005'>OTHER METHODS OF OBTAINING POWERS AND ROOTS.</h2>
+</div>
+
+<p class='c012'>A simple method of obtaining powers and roots, which may serve
+on occasion, is by scaling off proportional lengths on the D scale
+(or the A scale) of the ordinary rule. Thus, to determine the value
+of 1·25<sup>1·67</sup> we take the actual length 1–1·25 on D scale, and increase
+it by any convenient means in the proportion of 1&#8201;∶&#8201;1·67. Then
+with a pair of dividers we set off this new length from 1, and
+obtain 1·44 as the result. One convenient method of obtaining the
+desired ratio is by a pair of proportional compasses. Thus to
+obtain 1·52<sup>¹⁷⁄₁₆</sup>, the compasses would be set in the ratio of 16 to 17,
+and the smaller end opened out to include 1–1·52 on the D scale;
+the opening in the large end of the compasses will then be such
+that setting it off from 1 we
+obtain 1·56 on D as the result
+sought.</p>
+
+<div id='f_011'  class='figleft id005'>
+<img src='images/f_011.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 11.</span></p>
+</div>
+</div>
+
+<p class='c007'>The converse procedure for
+obtaining the <em>n</em>th root of a
+number N will obviously resolve
+itself into obtaining <span class='fraction'>1<br><span class='vincula'><em>n</em></span></span>th
+of the scale length 1-N, and
+need not be further considered.</p>
+
+<p class='c007'>Simple geometrical constructions are also used for obtaining
+scale lengths in the required ratio. A series of parallel lines ruled
+on transparent celluloid or stout tracing paper may be placed in an
+inclined position on the face of the rule and adjusted so as to divide
+the scale as desired. When much work is to be done which requires
+values to be raised to some constant but comparatively low power,
+<em>n</em>, the author has found the following device of assistance:—On a
+piece of thin transparent celluloid a line OC is drawn (Fig. <a href='#f_011'>11</a>) and
+in this a point B is taken such that <span class='fraction'>OC<br><span class='vincula'>OB</span></span> is the desired ratio. It is
+convenient to make OB = 1–10 on the A scale, so that assuming
+we require a series of values of <em>v</em><sup>1·35</sup>, OB would be 12·5 cm. and OC,
+16·875 cm. On these lines semi-circles are drawn as shown, both
+passing through the point O.</p>
+
+<p class='c007'><span class='pageno' id='Page_48'>48</span>Applying this cursor to the upper scales so that the point O is
+on 1 and the semi-circle O M B passes through <em>v</em> on A, the larger
+semi-circle will give on A the value of <em>v<sup>n</sup></em>. Thus for <em>p</em> <em>v<sup>n</sup></em> = 39·5 × 4·9<sup>1·35</sup>,
+set 1 on B to 39·5 on A (Fig. <a href='#f_012'>12</a>) and apply the cursor to the
+working edge of B, so that O agrees with 1 and O M B passes
+through 4·9 on B. The larger semi-circle then cuts the edge of the
+slide on a point, giving 337 on A as the result required.</p>
+
+<p class='c007'>Of course any number of semi-circles may be drawn, giving
+different ratios. If a number of evenly-spaced divisions are used
+as bases, the device affords a simple means of obtaining a succession
+of small powers or roots, while it also finds a use in determining a
+number of geometric means between two values as is required in
+arranging the speed gears of machine tools, etc. The converse
+operation of finding roots will be evident as will also many other
+uses for which the device is of service.</p>
+
+<div id='f_012'  class='figcenter id001'>
+<img src='images/f_012.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 12.</span></p>
+</div>
+</div>
+
+<p class='c007'>The lines should be drawn in Indian ink with a very sharp pen
+and on the <em>under</em> side of the celluloid so that the lines lie in close
+contact with the face of the rule.</p>
+
+<p class='c007'><em>The Radial Cursor</em>, another device for the same purpose, is
+always used in conjunction with the upper scales. As will be seen
+from Fig. <a href='#f_013'>13</a>, the body of the cursor P carries a graduated bar S
+which can be removed in a direction transverse to the rule, and
+adjusted to any desired position. Pivoted to the lower end of S
+is a radial arm R of transparent celluloid on which a centre line is
+engraved.</p>
+
+<p class='c007'>A reference to the illustration will show that the principle
+involved is that of similar triangles, the width of the slide being
+<span class='pageno' id='Page_49'>49</span>used as one of the elements. Thus, to take a simple case, if 2 on S
+is set to the index on P, and 1 on B is brought to N on A, then by
+swinging the radial arm until its centre line agrees with 1 on C,
+we can read N<sup>2</sup> on A. Evidently, since in the two similar triangles
+A&#8201;O&#8201;N<sup>2</sup> and N&#8201;<em>t</em>&#8201;N<sup>2</sup> the length of A&#8201;O is twice that of N&#8201;<em>t</em>, it results
+that A&#8201;N<sup>2</sup> = 2&#8201;A&#8201;N. In general, then, to find the <em>n</em>th power of a
+number, we set the cursor to 1 or 10 on A, bring <em>n</em> on the cross
+bar S to the index on the cursor, and 1 on B to N on A. Then to
+1 on C we set the line on the radial arm, and under the latter read
+N<sup><em>n</em></sup> on A. The inverse proceeding for finding the <em>n</em>th root will be
+obvious.</p>
+
+<div id='f_013'  class='figcenter id001'>
+<img src='images/f_013.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 13.</span></p>
+</div>
+</div>
+
+<p class='c007'>An advantage offered by this and analogous methods of obtaining
+powers and roots is that the result is obtained on the ordinary
+scale of the rule, and hence it can be taken directly into any
+further calculation which may be necessary.</p>
+
+<h3 class='c008'>COMBINED OPERATIONS.</h3>
+
+<p class='c009'>Thus far the various operations have been separately considered,
+and we now pass on to a consideration of the methods of working for
+solving the various formulæ met with in technical calculations. We
+propose to explain the methods of dealing with a few of the more
+generally used expressions, as this will suffice to suggest the procedure
+in dealing with other and more intricate calculations. In solving
+<span class='pageno' id='Page_50'>50</span>the following problems, both the upper and lower scales are used,
+and the relative value of the several scales must be observed
+throughout. Thus, in solving such an expression as <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>74·5</span></span><br>15·8</span> = 6·86,
+the division is first effected by setting 15·8 on B to 745 on A.
+From the relation of the two parts of the upper scales (page <a href='#Page_37'>37</a>)
+we know that such values as 7·45, 745, etc., will be taken on the
+<em>left-hand</em> A and B scales, while values as 15·8, 1580, etc., will be
+taken on the <em>right-hand</em> A and B scales. Hence, 15·8 on the <span class='fss'>R.H.</span>
+B scale is set to 745 on the <span class='fss'>L.H.</span> A scale, and the result read on D
+under the index of C. Had both values been taken on the <span class='fss'>L.H.</span> A and
+B scales, or both on the <span class='fss'>R.H.</span> A and B scales, the results would have
+corresponded to <em>x</em> = <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>7·45</span></span><br>1·58</span> = 2·17, or to <em>x</em> =<sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>74·5</span></span><br>15·8</span> = 2·17, <em>i.e</em>., to
+<span class='fraction'><span class='under'>6·86</span><br>√<span class='vincula'>10</span></span>. Hence if a wrong choice of scales has been made, we can
+correct the result by multiplying or dividing by √<span class='vincula'>10</span> as the case
+may require. If the result is read on D, set to it the centre
+index (10) of B and read the corrected result under the index of C.</p>
+
+<p class='c007'>To solve <em>a</em> × <em>b</em><sup>2</sup> = <em>x</em>.  Set the index of C to <em>b</em> on D, and over <em>a</em> on B read <em>x</em> on A.</p>
+
+<p class='c007'>To solve <span class='fraction'><span class='under'><em>a</em><sup>2</sup></span><br><em>b</em></span> = <em>x</em>.  Set <em>b</em> on B to <em>a</em> on D by using the cursor, and over index of B read <em>x</em> on A.</p>
+
+<p class='c007'>To solve <span class='fraction'><em>b</em><br><span class='vincula'><em>a</em><sup>2</sup></span></span> = <em>x</em>.  Set <em>a</em> on C to <em>b</em> on A, and over 1 on B read <em>x</em> on A.</p>
+
+<p class='c007'>To solve <span class='fraction'><span class='under'><em>a</em> × <em>b</em><sup>2</sup></span><br><em>c</em></span> = <em>x</em>.  Set <em>c</em> on B to <em>b</em> on D, and over <em>a</em> on B read <em>x</em> on A.</p>
+
+<p class='c007'>To solve (<em>a</em> × <em>b</em>)<sup>2</sup> = <em>x</em>.  Set 1 on C to <em>a</em> on D, and over <em>b</em> on C read <em>x</em> on A.</p>
+
+<p class='c007'>To solve (<span class='fraction'><em>a</em><br><span class='vincula'><em>b</em></span></span>)<sup>2</sup> = <em>x</em>.  Set <em>b</em> on C to <em>a</em> on D, and over 1 on C read <em>x</em> on A.</p>
+
+<p class='c007'>To solve √<span class='vincula'><em>a</em> × <em>b</em></span> = <em>x</em>.  Set 1 on B to <em>a</em> on A, and under <em>b</em> on B read <em>x</em> on D.</p>
+
+<p class='c007'>To solve <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'><em>a</em></span></span><br><em>b</em></span> = <em>x</em>.  Set <em>b</em> on B to <em>a</em> on A, and under 1 on C read <em>x</em> on D.</p>
+
+<p class='c007'><span class='pageno' id='Page_51'>51</span>To solve <em>a</em> <span class='fraction'><em>b</em><br><span class='vincula'><em>c</em><sup>2</sup></span></span> = <em>x</em>.  Set <em>b</em> on C to <em>c</em> on D and over <em>a</em> on B read <em>x</em> on A.</p>
+
+<p class='c007'>To solve <em>c</em><sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'><em>a</em></span></span><br><em>b</em></span> = <em>x</em>.  Set <em>b</em> on B to <em>a</em> on A, and under <em>c</em> on C read <em>x</em> on D.</p>
+
+<p class='c007'>To solve <span class='fraction'><span class='under'>√<em> &#x0305;a</em></span><br><em>b</em></span> = <em>x</em>.  Set <em>b</em> on C to <em>a</em> on A, and under 1 on C read <em>x</em> on D.</p>
+
+<p class='c007'>To solve <span class='fraction'><em>a</em><br><span class='vincula'>√<em> &#x0305;b</em></span></span> = <em>x</em>.  Set <em>b</em> on B to <em>a</em> on D, and under 1 on C read <em>x</em> on D.</p>
+
+<p class='c007'>To solve <em>b</em>√<em> &#x0305;a</em> = <em>x</em>.  Set 1 on C to <em>b</em> on D, and under <em>a</em> on B read <em>x</em> on D.</p>
+
+<p class='c007'>To solve √<span class='vincula'><em>a</em><sup>3</sup></span> = <em>x</em>.  Treat as <em>a</em>√<em> &#x0305;a</em>.</p>
+
+<p class='c007'>To solve <em>a</em>√<span class='vincula'><em>b</em><sup>3</sup></span> = <em>x</em>.  Treat as <em>a</em>√<em> &#x0305;b</em> × <em>b</em>.</p>
+
+<p class='c007'>To solve <span class='fraction'><span class='under'>√<em> &#x0305;a</em><sup>3</sup></span><br><em>b</em></span> = <em>x</em>.  Treat as <span class='fraction'><span class='under'>√<em> &#x0305;a</em> × <em>a</em></span><br><em>b</em></span>.</p>
+
+<p class='c007'>To solve <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'><em>a</em><sup>3</sup></span></span><br><em>b</em></span> = <em>x</em>.  Treat as <span class='fraction'><span class='under'>√<em> &#x0305;a</em> × <em>a</em></span><br>√<em> &#x0305;b</em></span> = <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'><em>a</em></span></span><br><em>b</em></span> × <em>a</em>.</p>
+
+<p class='c007'>To solve <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'><em>a</em> × <em>b</em></span></span><br><em>c</em></span> = <em>x</em>.  Set <em>c</em> on B to <em>a</em> on A, and under <em>b</em> on B read <em>x</em> on D.</p>
+
+<p class='c007'>To solve <span class='fraction'><span class='under'><em>a</em> × <em>b</em></span><br>√<em> &#x0305;c</em></span> = <em>x</em>.  Set <em>c</em> on B to <em>b</em> an D, and under <em>a</em> on C read <em>x</em> on D.</p>
+
+<p class='c007'>To solve <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'><em>a</em><sup>2</sup> × <em>b</em></span></span><br><em>c</em></span> = <em>x</em>.  Set <em>c</em> on B to <em>a</em> on D, and under <em>b</em> on B read <em>x</em> on D.</p>
+
+<p class='c007'>To solve <span class='fraction'><span class='under'><em>a</em><sup>2</sup> × <em>b</em><sup>2</sup></span><br><em>c</em></span> = <em>x</em>.  Set <em>c</em> on B to <em>a</em> on D, and over <em>b</em> on C read <em>x</em> on A.</p>
+
+<p class='c007'>To solve <span class='fraction'><span class='under'><em>a</em>√<em> &#x0305;b</em></span><br><em>c</em></span> = <em>x</em>.  Set <em>c</em> on C to <em>b</em> on A, and under <em>a</em> on C read <em>x</em> on D.</p>
+
+<p class='c007'>To solve <span class='c035'><sub><span class='c034'>(</span></sub></span><span class='fraction'><span class='under'><em>a</em> × √<em> &#x0305;b</em></span><br><em>c</em></span><span class='c035'><sub><span class='c034'>)</span></sub></span><sup>2</sup> = <em>x</em>.  Set <em>c</em> on C to <em>a</em> on D, and over <em>b</em> on B read <em>x</em> on A.</p>
+
+<div>
+  <span class='pageno' id='Page_52'>52</span>
+  <h3 class='c008'>HINTS ON EVALUATING EXPRESSIONS.</h3>
+</div>
+
+<p class='c009'>As a general rule, the use of cubes and higher powers should be
+avoided whenever possible. Thus, in the foregoing section, we
+recommend treating an expression of the form <em>a</em>√<span class='vincula'><em>b</em><sup>3</sup></span> as <em>a</em> × <em>b</em> × √<em> &#x0305;b</em>;
+the magnitudes of the values thus met with are more easily appreciated
+by the beginner, and mistakes in estimating the large numbers
+involved in cubing are avoided.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—7·3 × √<span class='vincula'>57<sup>3</sup></span> = 3140.</p>
+
+<p class='c021'>Set 1 on C to 57 on D; bring cursor to 57 on B (<span class='fss'>R.H.</span>, since 57
+has an <em>even</em> number of digits); bring 1 on C to cursor, and under
+7·3 on C read 3140 on D. As a rough estimate we have √<span class='vincula'>57</span>,
+about 8; 8 × 57, about 400; 400 × 7, gives 2800, showing the result
+consists of 4 figures.</p>
+
+<p class='c012'>An expression of the form <em>a</em>∛<span class='vincula'><em>b</em><sup>2</sup></span>, or <em>a</em> <em>b</em><sup>⅔</sup>, is better dealt with by
+rearranging as <em>a</em> × <span class='fraction'><em>b</em><br><span class='vincula'>∛<em>b</em></span></span>.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—3·64∛<span class='vincula'>4·32<sup>2</sup></span> = 9·65.</p>
+
+<p class='c021'>Set cursor to 4·32 on A, and move the slide until 1·63 is found
+simultaneously under the cursor on B and on D under 1 on C;
+bring cursor to 1 on C; 4·32 on C to cursor, and <em>over</em> 3·64 on D
+read 9·65 on C. (Note that in this case it is convenient to read the
+answer on the <em>slide</em>; see page <a href='#Page_22'>22</a>). From the slide rule we know
+∛<span class='vincula'>4·32</span> = about 1·6; this into 4·32 is roughly 3; 3·64 × 3 is about 10,
+showing the answer to be 9·65.</p>
+
+<p class='c012'>Similarly products of the form <em>a</em> × <em>b</em><sup>⁴⁄₃</sup> are best dealt with as
+<em>a</em> × <em>b</em> × ∛<em>b</em>.</p>
+
+<p class='c007'>Factorising expressions sometimes simplifies matters, as, for instance,
+in <em>x</em><sup>4</sup> − <em>y</em><sup>4</sup> = (<em>x</em><sup>2</sup> + <em>y</em><sup>2</sup>)(<em>x</em><sup>2</sup> − <em>y</em><sup>2</sup>). Here, working with the fourth
+powers involves large numbers and the troublesome determination
+of the number of digits in each factor; but squares are read on
+the rule at once, the number of digits is obvious, and, in general,
+the method should give a more accurate result. Take the expression,
+D<sub>1</sub> = <sub><span class='c034'>∛</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>D<sup>4</sup> − <em>d</em><sup>4</sup></span></span><br>D</span> giving the diameter D<sub>1</sub> of a solid shaft
+equal in torsional strength to a hollow shaft whose external and
+internal diameters are D and <em>d</em> respectively. Rearranging as
+D<sub>1</sub> = <sub><span class='c034'>∛</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>(D<sup>2</sup> + <em>d</em><sup>2</sup>)(D<sup>2</sup> − <em>d</em><sup>2</sup>)</span></span><br>D</span> and taking, as an example, D = 15 in.
+<span class='pageno' id='Page_53'>53</span>and <em>d</em> = 7 in., we have D<sup>2</sup> + <em>d</em><sup>2</sup> = 274 and D<sup>2</sup> − <em>d</em><sup>2</sup> = 176; hence
+D_1 = <sub><span class='c034'>∛</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>274 × 176</span></span><br>15</span> = ∛<span class='vincula'>3210</span> = 14·75 in.</p>
+
+<p class='c007'><em>Reversed Scale Notation.</em>—With expressions of the form 1 − <em>x</em>,
+or 100 − <em>x</em>, it is often convenient to regard the scales as having
+their notation reversed, <em>i.e.</em>, to read the scale backwards. When
+this is done the D scale is read as shown on the lower line—</p>
+
+<table class='table1'>
+  <tr>
+    <td class='c010'>Direct Notation</td>
+    <td class='c013'>1</td>
+    <td class='c013'>2</td>
+    <td class='c013'>3</td>
+    <td class='c013'>4</td>
+    <td class='c013'>5</td>
+    <td class='c013'>6</td>
+    <td class='c013'>7</td>
+    <td class='c013'>8</td>
+    <td class='c013'>9</td>
+    <td class='c014'>10</td>
+  </tr>
+  <tr>
+    <td class='c013'>D Scale</td>
+    <td class='c013'>&#160;</td>
+    <td class='c013'>&#160;</td>
+    <td class='c013'>&#160;</td>
+    <td class='c013'>&#160;</td>
+    <td class='c013'>&#160;</td>
+    <td class='c013'>&#160;</td>
+    <td class='c013'>&#160;</td>
+    <td class='c013'>&#160;</td>
+    <td class='c013'>&#160;</td>
+    <td class='c014'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c010'>Reversed Notation</td>
+    <td class='c013'>9</td>
+    <td class='c013'>8</td>
+    <td class='c013'>7</td>
+    <td class='c013'>6</td>
+    <td class='c013'>5</td>
+    <td class='c013'>4</td>
+    <td class='c013'>3</td>
+    <td class='c013'>2</td>
+    <td class='c013'>1</td>
+    <td class='c014'>0</td>
+  </tr>
+</table>
+
+<p class='c016'>The new reading can be found by subtracting the ordinary reading
+from 1, 10, 100, etc., according to the value assigned to the <span class='fss'>R.H.</span>
+index, but actually it is unnecessary to make this calculation, as
+with a little practice it is quite an easy matter to read both the
+main and subdivisions in the reversed order. Applications are
+found in plotting curves, trigonometrical formulæ, etc.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the per cent. of slip of a screw propeller from</p>
+
+<p class='c021'>100 − S = <span class='fraction'><span class='under'>10133V</span><br>PR</span></p>
+
+<p class='c021'>taking the speed, V, as 15 knots, the pitch of the propeller, P, as
+27 ft. 6 in., and the revolutions per minute, R, as 60.</p>
+
+<p class='c021'>Set 27·5 on B to 10133 on A (N.B.—Take the setting near the
+<em>centre</em> index of A); bring the cursor to 15 on B and 60 on B to
+cursor. Reading the <span class='fss'>L.H.</span> A scale backwards, the slip, S, = 8 per
+cent. is found on A over 10 on B.</p>
+
+<p class='c012'><em>Percentage Calculations.</em>—To increase a quantity by <em>x</em> per cent.
+we multiply by 100 + <em>x</em>; to diminish a quantity by <em>x</em> per cent. we
+multiply by 100 − <em>x</em>. Hence, to add <em>x</em> per cent., set 100 + <em>x</em> on C to
+1 on D and read new values on D under original values on C. To
+deduct <em>x</em> per cent. read the D scale backwards from 10 and set <span class='fss'>R.H.</span>
+index of C to <em>x</em> per cent. so read. Then read as before.</p>
+
+<h3 class='c008'>GAUGE POINTS.</h3>
+
+<p class='c009'>Special graduations, marking the position of constant factors which
+frequently enter into engineering calculations, are found on most
+slide rules. Usually the values of π = 3·1416 and <span class='fraction'>π<br><span class='vincula'>4</span></span> = 0·7854—the
+“gauge points” for calculating the circumference and area of a
+circle—are marked on the upper scales. The first should be given
+<span class='pageno' id='Page_54'>54</span>on the lower scales also. Marks <em>c</em> and <em>c</em><sup>1</sup> are sometimes found on
+the lower scales at 1·128 = <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>4</span></span><br>π</span> and at 3·568 = <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>40</span></span><br>π</span>. These are
+useful in calculating the contents of cylinders and are thus
+derived:—Cubic contents of cylinder of diameter <em>d</em> and length <em>l</em> =
+<span class='fraction'>π<br><span class='vincula'>4</span></span><em>d</em><sup>2</sup><em>l</em>; substituting for <span class='fraction'>π<br><span class='vincula'>4</span></span> its reciprocal <span class='fraction'>4<br><span class='vincula'>π</span></span>, the formula becomes
+<span class='fraction'><em>d</em><sup>2</sup><br><span class='vincula'>1·273 × <em>l</em></span></span>, and by taking the square root of the fractional part we
+have <span class='fraction'><em>d</em><br><span class='vincula'>1·128</span></span><sup>2</sup> × <em>l</em>. This is now in a very convenient form, since
+by setting the gauge point <em>c</em> on C to <em>d</em> on D, we can read over <em>l</em> on
+B the cubic contents on A. This example indicates the principle
+to be followed in arranging gauge points. Successive multiplication
+is avoided by substituting the reciprocal of the constant, thus
+bringing the expression into the form <span class='fraction'><span class='under'><em>a</em> × <em>b</em></span><br><em>c</em></span>, which, as we know,
+can be resolved by one setting of the slide. The advantage of
+dividing <em>d</em> before squaring is also evident. The mark <em>c</em><sup>1</sup> = <em>c</em> × √<span class='vincula'>10</span>
+is used if it is necessary to draw the slide more than one-half its
+length to the right.</p>
+
+<p class='c007'>A gauge point, M, at 31·83 = <span class='fraction'><span class='under'>100</span><br>π</span> is found on the upper scales of
+some rules. Setting this point on B to the diameter of a cylinder
+on A, the circumference is read over 1 or 100 on B or the area of
+the curved surface over the length on B.</p>
+
+<p class='c007'>As another example of establishing a gauge point, we will take
+the formula for the theoretical delivery of pumps. If <em>d</em> is the
+diameter of the plunger in inches, <em>l</em> the length of stroke in feet,
+and Q the delivery in gallons, we have</p>
+
+<p class='c007'>Q = <em>d</em><sup>2</sup> × <span class='fraction'>π<br><span class='vincula'>4</span></span> × <em>l</em> × <span class='fraction'>12<br><span class='vincula'>277</span></span>. (N.B.—277 cubic inches = 1 gallon.)</p>
+
+<p class='c007'>Multiplying out the constant quantities and taking its reciprocal,
+we readily transform the statement into Q = <span class='fraction'><span class='under'><em>d</em><sup>2</sup><em>l</em></span><br>29·4</span> or <sub><span class='c034'>(</span></sub><span class='fraction'><em>d</em><br><span class='vincula'>5·42</span></span><sub><span class='c034'>)</span></sub><sup>2</sup> × <em>l</em>.
+Hence set gauge point 5·42 on C to <em>d</em> on D and over length of
+stroke in feet on B, read delivery in gallons per stroke on A; or
+over piston speed in feet per minute on B, read theoretical delivery
+in gallons per minute on A.</p>
+
+<p class='c007'>Several examples of gauge points will be found in the section
+<span class='pageno' id='Page_55'>55</span>on calculating the weights of metal (see pages <a href='#Page_59'>59</a> and <a href='#Page_60'>60</a>). In
+most cases their derivation will be evident from what has been
+said above. In the case of the weight of spheres, we have Vol. =
+0·5236<em>d</em><sup>3</sup>, and this multiplied by the weight of 1 cubic inch of the
+material will give the weight W in lb. Hence for cast-iron, W =
+0·5236 × <em>d</em><sup>3</sup> × 0·26, which is conveniently transformed into W = <span class='fraction'><span class='under'><em>d</em> × <em>d</em><sup>2</sup></span><br>7·35</span>
+as in the example on page <a href='#Page_60'>60</a>.</p>
+
+<p class='c007'>With these examples no difficulty should be experienced in
+establishing gauge points for any calculation in which constant
+factors recur.</p>
+
+<p class='c007'><em>Marking Gauge Points.</em>—The practice of marking gauge points
+by lines extending to the working edge of the scale is not to be
+recommended, as it confuses the ordinary reading of the scales.
+Generally speaking, gauge points are only required occasionally,
+and if they are placed clear of the scale to which they pertain,
+but near enough to show the connection, they can be brought
+readily into a calculation by means of the cursor. Usually there
+is sufficient margin above the A scale and below the D scale for
+various gauge points to be marked. Another plan consists in
+cutting two nicks in the upper and lower edges of the cursor near
+the centre and about ⅛ in. apart. These centre pieces, when bent
+out, form a tongue, which are in line with the cursor line and run
+nearly in contact with the square and bevelled edges of the rule respectively.
+A fine line in the tongue can then be set to gauge points
+marked on these two edge strips, the ordinary measuring graduations
+being removed, if desired, by a piece of fine sand-paper.</p>
+
+<p class='c007'>For gauge points marked on the face of the rule, the author
+prefers two fine lines drawn at 45°—thus, ✕—and crossing in the
+exact point which it is required to indicate. With the “cross”
+gauge point the meeting lines facilitate the placing of the cursor, and
+an exact setting is readily made.<a id='r7'></a><a href='#f7' class='c019'><sup>[7]</sup></a> All lines should be drawn in
+Indian ink with a very sharp drawing pen. For a more permanent
+marking the Indian ink may be rubbed up in glacial acetic acid or
+the special ink for celluloid may be used. If any difficulty is found
+in writing the distinguishing signs against the gauge point, the
+inscription may be formed by a succession of small dots made with
+a sharp pricker.</p>
+
+<div>
+  <span class='pageno' id='Page_56'>56</span>
+  <h3 class='c008'>EXAMPLES IN TECHNICAL CALCULATIONS.</h3>
+</div>
+
+<p class='c009'>In order to illustrate the practical value of the slide rule,
+we now give a number of examples which will doubtless be
+sufficient to suggest the methods of working with other formulæ.
+A few of the rules give results which are approximate only,
+but in all cases the degree of accuracy obtained is well
+within the possible reading of the scales. In many cases the
+rules given may be modified, if desired, by varying the constants.
+In most of the examples the particular formula employed will be
+evident from the solution, but in a few of the more complicated
+cases a separate statement has been given.</p>
+
+<h4 class='c036'><span class='sc'>Mensuration, Etc.</span></h4>
+
+<p class='c009'>Given the chord <em>c</em> of a circular arc, and the vertical height <em>h</em>,
+to find the diameter <em>d</em> of the circle.</p>
+
+<p class='c007'>Set the height <em>h</em> on B to half the chord on D, and over 1 on B
+read <em>x</em> on A. Then <em>x</em> + <em>h</em> = <em>d</em>.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—<em>c</em> = 6; <em>h</em> = 2; find <em>d</em>.
+Set 2 on B to 3 on D, and over 1 on B read 4·5 on A. Then 4·5 + 2
+= 6·5 = <em>d</em>.</p>
+
+<p class='c012'>Given the radius of a circle <em>r</em>, and the number of degrees <em>n</em> in
+an arc, to find the length <em>l</em> of the arc.</p>
+
+<p class='c007'>Set <em>r</em> on C to 57·3 on D, and over any number of degrees <em>n</em> on
+D read the (approximate) length of the arc on C.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—<em>r</em> = 24; <em>n</em> = 30; find <em>l</em>.</p>
+
+<p class='c021'>Set 24 on C to 57·3 on D, and over 30 on D read 12·56 = <em>l</em> on C.</p>
+
+<p class='c012'>Given the diameter <em>d</em> of a circle in <em>inches</em>, to find the circumference
+<em>c</em> in <em>feet</em>.</p>
+
+<p class='c007'>Set 191 on C to 50 on D, and under any diameter in inches on
+C read circumference <em>c</em>, in feet on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the circumference in feet of a pulley 17 in. in diameter.
+Set 191 on C to 50 on D, and under 17 on C read 4·45 ft. on D.</p>
+
+<p class='c012'>Given the diameter of a circle, to find its area.</p>
+
+<p class='c007'>Set 0·7854 on B to 10 (centre index) on A and over any diameter
+on D read area on B.</p>
+
+<p class='c007'>When the rule has a special graduation line = 0·7854, on the
+right-hand scale of B, set this line to the <span class='fss'>R.H.</span> index of A and read
+off as above. If only π is marked, set this special graduation on
+B to 4 on A.</p>
+
+<p class='c007'><span class='pageno' id='Page_57'>57</span>On the C and D scales of some rules a gauge point
+marked <em>c</em> will be found indicating <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>4</span></span><br>π</span> = 1·1286. In this case,
+therefore, set 1 on C to gauge point <em>c</em> on D, and read area on A
+as above. If the gauge point <em>c</em>′ is used, divide the result by 10.
+Or set <em>c</em> on C, to diameter on D, and over index of B read area on A.
+Cursors are supplied, having <em>two</em> lines ruled on the glass, the
+interval between them being equal to <span class='fraction'>4<br><span class='vincula'>π</span></span> = 1·273 on the A scale. In
+this case, if the right hand of the two cursor lines be set to the
+diameter on D, the <em>area</em> will be read on A under the <em>left</em>-hand
+cursor line. For diameters less than 1·11 it is necessary to set the
+middle index of B to the <span class='fss'>L.H.</span> index of A, reading the areas on the
+<span class='fss'>L.H.</span> B scale. The confusion which in general work is sometimes
+caused by the use of two cursor lines might be obviated by making
+the left-hand line in two short lengths, each only just covering the
+scales.</p>
+
+<p class='c007'>Given diameter of circle <em>d</em> in <em>inches</em>, to find area <em>a</em> in square
+<em>feet</em>.</p>
+
+<p class='c007'>Set 6 on B to 11 on A, and over diameter in inches on D read
+area in square feet on B.</p>
+
+<p class='c007'>To find the surface in square feet of boiler flues, condenser
+tubes, heating pipes, etc., having given the diameter in inches and
+length in feet.</p>
+
+<p class='c007'>Find the circumference in feet as above and multiply by the
+length in feet.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the heating surface afforded by 160 locomotive boiler tubes
+1¾in. in diameter and 12 ft. long.</p>
+
+<p class='c021'>Set 191 on C to 50 on D; bring cursor 1·75 on C, <span class='fss'>L.H.</span> index of C to
+cursor; cursor to 12 on C; 1 on C to cursor; and under 160 on C read
+880 sq. ft. of heating surface on D.</p>
+
+<p class='c012'>If the dimensions are in the same denomination and the rule has
+a gauge point M at 31·83 <sub><span class='c034'>(</span></sub>= <span class='fraction'><span class='under'>100</span><br>π</span><sub><span class='c034'>)</span></sub>, set this mark on B to diameter of
+cylinder on A, and read cylindrical surface on A over length on B.</p>
+
+<p class='c007'>To find the side <em>s</em> of a square, equal in area to a given rectangle
+of length <em>l</em> and breadth <em>b</em>.</p>
+
+<p class='c007'>Set <span class='fss'>R.H.</span> or <span class='fss'>L.H.</span> index of B to <em>l</em> on A, and under <em>b</em> on B read <em>s</em>
+on D.</p>
+
+<p class='c020'><span class='pageno' id='Page_58'>58</span><span class='sc'>Ex.</span>—Find the side of a square equal in area to a rectangle in which
+<em>l</em> = 31 ft. and <em>b</em> = 5 ft.</p>
+
+<p class='c021'>Set the (<span class='fss'>R.H.</span>) index of B to 31 on A, and under 5 on B read
+12·45 ft. on D.</p>
+
+<p class='c012'>To find various lengths <em>l</em> and breadths <em>b</em> of a rectangle, to give
+a constant area <em>a</em>.</p>
+
+<p class='c007'>Invert the slide and set the index of Ɔ to the given area on D.
+Then opposite any length <em>l</em> on Ɔ find the corresponding breadth
+<em>b</em> on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the corresponding breadths of rectangular sheets, 16, 18,
+24, 36, and 60 ft. long, to give a constant area of 72 sq. ft.</p>
+
+<p class='c021'>Set the <span class='fss'>R.H.</span> index of Ɔ to 72 on D, and opposite 16, 18, 24, 36,
+and 60 on Ɔ read 4·5, 4, 3, 2, and 1·2 ft., the corresponding breadths
+on D.</p>
+
+<p class='c012'>To find the contents in cubic feet of a cylinder of diameter <em>d</em> in
+inches and length <em>l</em> in feet.</p>
+
+<p class='c007'>Find area in feet as before, and multiply by the length.</p>
+
+<p class='c007'>If dimensions are all in inches or feet, set the mark <em>c</em> (= 1·128)
+on C to diameter on D and over length on B, read cubic contents
+on A.</p>
+
+<p class='c007'>To find the area of an ellipse.</p>
+
+<p class='c007'>Set 205 on C to 161 on D; bring cursor to length of major
+axis on C, 1 on C to cursor, and under length of minor axis on C
+read area on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the area of an ellipse the major and minor axes of which
+are 16 in. and 12 in. in length respectively.</p>
+
+<p class='c021'>Set 205 on C to 161 on D; bring cursor to 16 on C, 1 on C to
+cursor, and under 12 on C read 150·8 in. on D.</p>
+
+<p class='c012'>To find the surface of spheres.</p>
+
+<p class='c007'>Set 3·1416 on B to <span class='fss'>R.H.</span> or <span class='fss'>L.H.</span> index of A, and over diameter
+on D read by the aid of the cursor, the convex surface on B.</p>
+
+<p class='c007'>To find the cubic contents of spheres.</p>
+
+<p class='c007'>Set 1·91 on B to diameter on A, and over diameter on C read
+cubic contents on A.</p>
+
+<h4 class='c036'><span class='sc'>Weights of Metals.</span></h4>
+
+<p class='c009'>To find the weight in lb. per lineal foot of square bars of metal.</p>
+
+<p class='c007'>Set index of B to weight of 12 cubic inches of the metal (<em>i.e.</em>,
+one lineal foot, 1 square inch in section) on A, and over the side of
+the square in inches on C read weight in lb. on A.</p>
+
+<p class='c020'><span class='pageno' id='Page_59'>59</span><span class='sc'>Ex.</span>—Find the weight per foot length of 4½in. square wrought-iron
+bars.</p>
+
+<p class='c021'>Set middle index of B to 3·33 on A, and over 4½ on C read 67·5
+lb. on A.</p>
+
+<p class='c012'>(N.B.—For other metals use the corresponding constant in
+column (2), below).</p>
+
+<p class='c007'>To find the weight in lb. per lineal foot of round bars.</p>
+
+<p class='c007'>Set <span class='fss'>R.H.</span> or <span class='fss'>L.H.</span> index of B to weight of 12 cylindrical inches of
+the metal on A (column (4), below), and opposite the diameter of
+the bar in inches on C, read weight in lb. per lineal foot on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the weight of 1 lineal foot of 2 in. round cast steel.</p>
+
+<p class='c021'>Set <span class='fss'>L.H.</span> index of B to 2·68 on A, and over 2 on C read 10·7 lb. on A.</p>
+
+<p class='c012'>To find the weight of flat bars in lb. per lineal foot.</p>
+
+<p class='c007'>Set the breadth in inches on C to <span class='fraction'>1<br><span class='vincula'>weight of 12 cub. in.</span></span> of the
+metal (column (3), below) on D, and above the thickness on D
+read weight in lb. per lineal foot on C.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the weight per lineal foot of bar steel, 4½in. wide and
+⅝in. thick.</p>
+
+<p class='c021'>Set 4·5 on C to 0·294 on D, and over 0·625 on D read 9·56 lb. per
+lineal foot on C.</p>
+
+<p class='c012'>To find the weight per square foot of sheet metal, set the
+weight per cubic foot of the metal (col. 1) on C to 12 on D, and</p>
+
+<table class='table2'>
+  <tr>
+ <th class='btt bbt c037'>Metals.</th>
+ <th class='btt bbt blt c037'>(1)<br>Weight in lb. per cubic ft.</th>
+ <th class='btt bbt blt c038'>(2)<br>Weight of 12 cubic in.</th>
+ <th class='btt bbt blt c037'>(3)<br><span class='fraction'>1<br><span class='vincula'>Wt. of 12 cub. in.</span></span></th>
+ <th class='btt bbt blt c037'>(4)<br>Weight of 12 cylindrical in.</th>
+  </tr>
+  <tr>
+    <td class='c039'>Wrought iron</td>
+ <td class='blt c038'>480</td>
+ <td class='blt c038'>3·33</td>
+ <td class='blt c038'>0·300</td>
+ <td class='blt c038'>2·62</td>
+  </tr>
+  <tr>
+    <td class='c039'>Cast iron</td>
+ <td class='blt c038'>450</td>
+ <td class='blt c038'>3·125</td>
+ <td class='blt c038'>0·320</td>
+ <td class='blt c038'>2·45</td>
+  </tr>
+  <tr>
+    <td class='c039'>Cast steel</td>
+ <td class='blt c038'>490</td>
+ <td class='blt c038'>3·40</td>
+ <td class='blt c038'>0·294</td>
+ <td class='blt c038'>2·68</td>
+  </tr>
+  <tr>
+    <td class='c039'>Copper</td>
+ <td class='blt c038'>550</td>
+ <td class='blt c038'>3·82</td>
+ <td class='blt c038'>0·262</td>
+ <td class='blt c038'>3·00</td>
+  </tr>
+  <tr>
+    <td class='c039'>Aluminium</td>
+ <td class='blt c038'>168</td>
+ <td class='blt c038'>1·166</td>
+ <td class='blt c038'>0·085</td>
+ <td class='blt c038'>0·915</td>
+  </tr>
+  <tr>
+    <td class='c039'>Brass</td>
+ <td class='blt c038'>520</td>
+ <td class='blt c038'>3·61</td>
+ <td class='blt c038'>0·277</td>
+ <td class='blt c038'>2·83</td>
+  </tr>
+  <tr>
+    <td class='c039'>Lead</td>
+ <td class='blt c038'>710</td>
+ <td class='blt c038'>4·93</td>
+ <td class='blt c038'>0·203</td>
+ <td class='blt c038'>3·87</td>
+  </tr>
+  <tr>
+    <td class='c039'>Tin</td>
+ <td class='blt c038'>462</td>
+ <td class='blt c038'>3·21</td>
+ <td class='blt c038'>0·312</td>
+ <td class='blt c038'>2·52</td>
+  </tr>
+  <tr>
+    <td class='c039'>Zinc (cast)</td>
+ <td class='blt c038'>430</td>
+ <td class='blt c038'>2·98</td>
+ <td class='blt c038'>0·335</td>
+ <td class='blt c038'>2·34</td>
+  </tr>
+  <tr>
+ <td class='bbt c039'>„ (sheet)</td>
+ <td class='bbt blt c038'>450</td>
+ <td class='bbt blt c038'>3·125</td>
+ <td class='bbt blt c038'>0·320</td>
+ <td class='bbt blt c038'>2·45</td>
+  </tr>
+</table>
+
+<p class='c016'>above the thickness of the plate in inches on D read weight in lb.
+per square foot on C.</p>
+
+<p class='c020'><span class='pageno' id='Page_60'>60</span><span class='sc'>Ex.</span>—Find the weight in lb. per square foot of aluminium sheet ⅜in.
+thick.</p>
+
+<p class='c021'>Set 168 on C to 12 on D, and over 0·375 on D read 5·25 lb. on C.</p>
+
+<p class='c012'>To find the weight of pipes in lb. per lineal foot.</p>
+
+<p class='c007'>Set mean diameter of the pipe in inches (<em>i.e.</em>, internal diameter
+<em>plus</em> the thickness, or external diameter <em>minus</em> the thickness) on C
+to the constant given below on D, and over the thickness on D
+read weight in lb. per lineal foot on C.</p>
+
+<table class='table2'>
+  <tr>
+ <th class='btt bbt blt brt c029'>Metals.</th>
+ <th class='btt bbt brt c029'>Constant for Pipes.</th>
+ <th class='btt bbt brt c029'>Constant for Spheres.</th>
+  </tr>
+  <tr>
+ <td class='blt brt c032'>Wrought iron</td>
+ <td class='brt c029'>0·0955</td>
+ <td class='brt c029'>6·87</td>
+  </tr>
+  <tr>
+ <td class='blt brt c032'>Cast iron</td>
+ <td class='brt c029'>0·1020</td>
+ <td class='brt c029'>7·35</td>
+  </tr>
+  <tr>
+ <td class='blt brt c032'>Steel</td>
+ <td class='brt c029'>0·0936</td>
+ <td class='brt c029'>6·73</td>
+  </tr>
+  <tr>
+ <td class='blt brt c032'>Brass</td>
+ <td class='brt c029'>0·0882</td>
+ <td class='brt c029'>6·35</td>
+  </tr>
+  <tr>
+ <td class='blt brt c032'>Copper</td>
+ <td class='brt c029'>0·0834</td>
+ <td class='brt c029'>6·00</td>
+  </tr>
+  <tr>
+ <td class='bbt blt brt c032'>Lead</td>
+ <td class='bbt brt c029'>0·0646</td>
+ <td class='bbt brt c029'>4·65</td>
+  </tr>
+</table>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the weight per foot of cast-iron piping 4 in. internal
+diameter and ½in. thick.</p>
+
+<p class='c021'>Set 4·5 on C to 0·102 on D, and over 0·5 on D read 22·1 lb. on C,
+the required weight.</p>
+
+<p class='c012'>To find the weight in lb. of spheres or balls, given the diameter
+in inches. (W = 0·5236<em>d</em><sup>3</sup> × wt. of 1 cub. in. of material).</p>
+
+<p class='c007'>Set the constant for spheres (given above) on B to diameter in
+inches on A, and over diameter on C read weight in lb. on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the weight of a cast-iron ball 7½in. in diameter.</p>
+
+<p class='c021'>Set 7·35 on B to 7·5 on A, and over 7·5 on C read 57·7 lb. on A.</p>
+
+<p class='c012'>To find diameter in inches of a sphere of given weight.</p>
+
+<p class='c007'>Set the cursor to the given weight in lb. on A, and move
+the slide until the same number is found on C under the cursor
+that is simultaneously found on A over the constant for the
+sphere on B.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find diameter in inches of a sphere of cast-iron to weigh 7½lb.</p>
+
+<p class='c021'>Setting the cursor to 7·5 on A, and moving the slide, it is found
+that when 3·8 on C falls under the cursor, 3·8 on A is simultaneously
+found over 7·35 on B. The required diameter is therefore 3·8 in.</p>
+
+<p class='c012'>The rules for cubes and cube roots (page <a href='#Page_40'>40</a>) should be kept in
+view in solving the last two examples.</p>
+
+<div>
+  <span class='pageno' id='Page_61'>61</span>
+  <h4 class='c036'><span class='sc'>Falling Bodies.</span></h4>
+</div>
+
+<p class='c009'>To find velocity in feet per second of a falling body, given the
+time of fall in seconds.</p>
+
+<p class='c007'>Set index on C to time of fall on D, and under 32·2 on C read
+velocity in feet per second on D.</p>
+
+<p class='c007'>To find velocity in feet per second, given distance fallen
+through in feet.</p>
+
+<p class='c007'>Set 1 on C to distance fallen through on A, and under 64·4 on
+B read velocity in feet per second on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find velocity acquired by falling through 14 ft.</p>
+
+<p class='c021'>Set (<span class='fss'>R.H.</span>) index of C to 14 on A, and under 64·4 on B read 30 ft.
+per second on D.</p>
+
+<p class='c012'>To find distance fallen through in feet in a given time.</p>
+
+<p class='c007'>Set index of C to time in seconds on D, and over 16·1 on B
+read distance fallen through in feet on A.</p>
+
+<h4 class='c036'><span class='sc'>Centrifugal Force.</span></h4>
+
+<p class='c009'>To find the centrifugal force of a revolving mass in lb.</p>
+
+<p class='c007'>Set 2940 on B to revolutions per minute on D; bring cursor
+to weight in lb. on B; index of B to cursor, and over radius in
+feet on B read centrifugal force in lb. on A.</p>
+
+<p class='c007'>To find the centrifugal stress in lb. per square inch, in rims of
+revolving wheels of cast iron.</p>
+
+<p class='c007'>Set 61·3 on C to the mean diameter of the wheel in feet on D,
+and over revolutions per minute on C read stress per square inch
+on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the stress per square inch in a cast-iron fly-wheel rim 8 ft.
+in diameter and running at 120 revolutions per minute.</p>
+
+<p class='c021'>Set 61·3 on C to 8 on D, and over 120 on C read 245 lb. per square
+inch on A.</p>
+
+<h4 class='c036'><span class='sc'>The Steam Engine.</span></h4>
+
+<p class='c009'>Given the stroke and number of revolutions per minute, to find
+the piston speed.</p>
+
+<p class='c007'>Set stroke in inches on C to 6 on D, and over number of
+revolutions on D read piston speed in feet per minute on C.</p>
+
+<p class='c007'>To find cubic feet of steam in a cylinder at cut-off, given
+diameter of cylinder and period of admission in inches.</p>
+
+<p class='c007'>Set 2200 on B to cylinder diameter on D, and over period of
+admission on B read cubic feet of steam on A.</p>
+
+<p class='c020'><span class='pageno' id='Page_62'>62</span><span class='sc'>Ex.</span>—Cylinder diameter 26 in., stroke 40 in., cut-off at ⅝ of stroke.
+Find cubic feet of steam used (theoretically) per stroke.</p>
+
+<p class='c021'>Set 2200 on B to 26 on D, and over 40 × ⅝ or 25 in. on B, read
+7·68 cub. ft. on A, as the number of cubic feet of steam used per
+stroke.</p>
+
+<p class='c012'>Given the diameter of a cylinder in inches, and the pressure in
+lb. per square inch, to find the load on the piston in tons.</p>
+
+<p class='c007'>Set pressure in lb. per square inch on B to 2852 on A, and over
+cylinder diameter in inches on D read load on piston in tons on B.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Steam pressure 180 lb. per square inch; cylinder diameter, 42 in.
+Find load in tons on piston.</p>
+
+<p class='c021'>Set 180 on B to 2852 on A, and over 42 on D read 111 tons, the
+gross load, on B.</p>
+
+<p class='c012'>Given admission period and absolute initial pressure of steam
+in a cylinder, to find the pressure at various points in the expansion
+period (isothermal expansion).</p>
+
+<p class='c007'>Invert the slide and set the admission period, in inches, on Ɔ
+to the initial pressure on D; then under any point in the
+expansion stroke on Ɔ find the corresponding pressure on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Admission period 12 in., stroke 42 in., initial pressure 80 lb. per
+square inch. Find pressure at successive fifths of the expansion period.</p>
+
+<p class='c021'>Set 12 on Ɔ to 80 on D, and opposite 18, 24, 30, 36 and 42 in. of
+the whole stroke on Ɔ find the corresponding pressures on D:—53·3,
+40, 32, 26·6 and 22·8 lb. per square inch.</p>
+
+<p class='c012'>To find the mean pressure constant for isothermally expanding
+steam, given the cut-off as a fraction of the stroke.</p>
+
+<p class='c007'>Find the logarithm of the ratio of the expansion <em>r</em>, by the
+method previously explained (page <a href='#Page_46'>46</a>). Prefix the characteristic
+and to the number thus obtained, on D, set 1 on C. Then under
+2·302 on C read <em>x</em> on D. To <em>x</em> + 1 on D set <em>r</em> on C, and under
+index of C read mean pressure constant on D. The latter,
+multiplied by the initial pressure, gives the mean forward pressure
+throughout the stroke. (N.B.—Common log. × 2·302 = hyperbolic
+log.)</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the mean pressure constant for a cut-off of ¼th, or a ratio
+of expansion of 4.</p>
+
+<p class='c021'>Set (<span class='fss'>L.H.</span>) index of C to 4 on D, and on the reverse side of the slide
+read 0·602 on the logarithmic scale. The characteristic = 0; hence to
+0·602 on D set (<span class='fss'>R.H.</span>) index of C, and under 2·302 on C read 1·384 on
+D. Add 1, and to 2·384 thus obtained on D set <em>r</em> (= 4) on C, and
+under 1 on C read 0·596, the mean pressure constant required.</p>
+
+<p class='c012'><span class='pageno' id='Page_63'>63</span>Mean pressure constants for the most usual degrees of cut-off
+are given below:—</p>
+
+<table class='table1'>
+  <tr>
+    <th class='c013'>Cut-off in fractions of stroke</th>
+    <th class='c014'>Mean pressure constant</th>
+  </tr>
+  <tr>
+    <td class='c013'>¾</td>
+    <td class='c014'>0·968</td>
+  </tr>
+  <tr>
+    <td class='c013'>⁷⁄₁₀</td>
+    <td class='c014'>0·952</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅔</td>
+    <td class='c014'>0·934</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅝</td>
+    <td class='c014'>0·919</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅗</td>
+    <td class='c014'>0·913</td>
+  </tr>
+  <tr>
+    <td class='c013'>½</td>
+    <td class='c014'>0·846</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅖</td>
+    <td class='c014'>0·766</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅜</td>
+    <td class='c014'>0·750</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅓</td>
+    <td class='c014'>0·699</td>
+  </tr>
+  <tr>
+    <td class='c013'>³⁄₁₀</td>
+    <td class='c014'>0·664</td>
+  </tr>
+  <tr>
+    <td class='c013'>¼</td>
+    <td class='c014'>0·596</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅕</td>
+    <td class='c014'>0·522</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅙</td>
+    <td class='c014'>0·465</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅐</td>
+    <td class='c014'>0·421</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅛</td>
+    <td class='c014'>0·385</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅑</td>
+    <td class='c014'>0·355</td>
+  </tr>
+  <tr>
+    <td class='c013'>⅒</td>
+    <td class='c014'>0·330</td>
+  </tr>
+  <tr>
+    <td class='c013'>¹⁄₁₁</td>
+    <td class='c014'>0·309</td>
+  </tr>
+  <tr>
+    <td class='c013'>¹⁄₁₂</td>
+    <td class='c014'>0·290</td>
+  </tr>
+  <tr>
+    <td class='c013'>¹⁄₁₃</td>
+    <td class='c014'>0·274</td>
+  </tr>
+  <tr>
+    <td class='c013'>¹⁄₁₄</td>
+    <td class='c014'>0·260</td>
+  </tr>
+  <tr>
+    <td class='c013'>¹⁄₁₅</td>
+    <td class='c014'>0·247</td>
+  </tr>
+  <tr>
+    <td class='c013'>¹⁄₁₆</td>
+    <td class='c014'>0·236</td>
+  </tr>
+</table>
+
+<p class='c007'>To find mean pressure:—Set 1 on C to constant on D, and
+under initial pressure on C read mean pressure on D.</p>
+
+<p class='c007'>Given the absolute initial pressure, length of stroke, and
+admission period, to find the absolute pressure at any point in
+the expansion period, it being assumed that the steam expands
+adiabatically. (P<sub>2</sub> = <span class='fraction'>P<sub>1</sub><br><span class='vincula'>R<sup>¹⁰⁄₉</sup></span></span> in which P<sub>1</sub> = initial pressure and P<sub>2</sub>
+the pressure corresponding to a ratio of expansion R.)</p>
+
+<p class='c007'>Set <span class='fss'>L.H.</span> index of C to ratio of expansion on D, and read on the
+back of the slide the decimal of the logarithm. Add the characteristic,
+and to the number thus obtained on D set 9 on C, and
+read off the value found on D under the index of C. Set this
+number on the logarithmic scale to the index mark, in the opening
+on the back of the rule, and under <span class='fss'>L.H.</span> index of C read the value
+of R<sup>¹⁰⁄₉</sup> on D. The initial pressure divided by this value gives the
+corresponding pressure due to the expansion.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Absolute initial pressure 120 lb. per square inch; stroke, 4 ft.;
+cut-off ¼. Find the respective pressures when ½ and ¾ths of the stroke
+have been completed.</p>
+
+<p class='c021'>In the first case R = 2. Therefore setting the <span class='fss'>L.H.</span> index of C to 2
+on D, we find the decimal of the logarithm on the back of the slide to
+be 0·301. The characteristic is 0, so placing 9 on C to 0·301 on D, we
+read 0·334 as the value under the <span class='fss'>R.H.</span> index of C. (N.B.—In
+locating the decimal point it is to be observed that the log. of R has
+been multiplied by 10, in accordance with the terms of the above
+expression.) Setting this number on the logarithmic scale to the back
+index, the value of R<sup>¹⁰⁄₉</sup> is found on D, under the <span class='fss'>L.H.</span> index of C, to
+be 2·16. Setting 120 on C to this value, it is found that the pressure
+at ½ stroke, read on C over the <span class='fss'>R.H.</span> index of D, is 55·5 lb. per square
+inch. In a similar manner, the pressure when ¾ths of the stroke is
+completed is found to be 35·4 lb. per square inch.</p>
+
+<p class='c012'>For other conditions of expanding steam, or for gas or air, the
+method of procedure is similar to the above.</p>
+
+<p class='c007'><span class='pageno' id='Page_64'>64</span>To find the horse-power of an engine, having given the mean
+<em>effective</em> pressure, the cylinder diameter, stroke, and number of
+revolutions per minute.</p>
+
+<p class='c007'>To cylinder diameter on D set 145 on C; bring cursor to stroke
+in feet on B, 1 on B to cursor, cursor to number of revolutions on
+B, 1 on B to cursor, and over mean effective pressure on B find
+horse-power on A.</p>
+
+<p class='c007'>(N.B.—If stroke is in inches, use 502 in place of 145 given
+above.)</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the indicated horse-power, given cylinder diameter 27 in.,
+mean effective pressure 38 lb. per square inch, stroke 32 in., revolutions 57
+per minute.</p>
+
+<p class='c021'>Set 502 on C to 27 on D, bring cursor to 32 on B, 1 on B to cursor,
+cursor to 57 on B, 1 on B to cursor, and over 38 on B read 200 I.H.P.
+on A.</p>
+
+<p class='c012'>To determine the horse-power of a compound engine, invert the
+slide and set the diameter of the <em>high</em>-pressure cylinder on Ɔ to
+the cut-off in that cylinder on A. Use the number then found on
+A over the diameter of the <em>low</em>-pressure cylinder on Ɔ as the cut-off
+in that cylinder, working with the same pressure and piston
+speed, and calculate the horse-power as for a single cylinder.</p>
+
+<p class='c007'>To find the cylinder ratio in compound engines, invert the
+slide and set index of Ɔ to diameter of the low-pressure cylinder
+on D. Then over the diameter of the high-pressure cylinder on
+C, read cylinder ratio on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Diameter of high-pressure cylinder 7¾in., low-pressure 15 in.
+Find cylinder ratio.</p>
+
+<p class='c021'>Set index on Ɔ to 15 on D, and over 7·75 on Ɔ read 3·75, the
+required ratio, on A.</p>
+
+<p class='c012'>The cylinder ratios of triple or quadruple-expansion engines
+may be similarly determined.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—In a quadruple-expansion engine, the cylinders are 18, 26, 37,
+and 54 inches in diameter. Find the respective ratios of the high, first
+intermediate, and second intermediate cylinders to the low-pressure.</p>
+
+<p class='c021'>Set (<span class='fss'>R.H.</span>) index of Ɔ to 54 on D, and over 18, 26, and 37 on Ɔ read
+9, 4·31, and 2·13, the required ratios, on A.</p>
+
+<p class='c012'>Given the mean effective pressures in lb. per square inch in
+each of the three cylinders of a triple-expansion engine, the I.H.P.
+to be developed in each cylinder, and the piston speed, to find the
+respective cylinder diameters.</p>
+
+<p class='c007'><span class='pageno' id='Page_65'>65</span>Set 42,000 on B to piston speed on A; bring cursor to mean
+effective pressure in low-pressure cylinder on B, index of B to
+cursor, and under I.H.P. on A read low-pressure cylinder diameter
+on C. To find the diameters of the high-pressure and intermediate-pressure
+cylinders, invert the slide and place the mean
+pressure in the low-pressure cylinder on ᗺ to the diameter of that
+cylinder on D. Then under the respective mean pressures on ᗺ
+read corresponding cylinder diameters on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—The mean effective pressures in the cylinders of a triple-expansion
+engine are:—L.P., 10·32; I.M.P., 27·5; and H.P., 77·5 lb. per square
+inch. The piston speed is 650 ft. per minute, and the I.H.P. developed
+in each cylinder, 750. Find the cylinder diameters.</p>
+
+<p class='c021'>Set 42,000 on B to 650 on A, and bring cursor to 10·32 on B.
+Bring index of B to cursor, and under 750 on A read 68·5 in. on C, the
+L.P. cylinder diameter. Invert the slide, and placing 10·32 on ᗺ to
+68·5 on D, read, under 27·5 on ᗺ, the I.M.P. cylinder diameter
+= 42 in., on D; also under 77·5 on ᗺ read the H.P. cylinder diameter
+= 25 in., on D.</p>
+
+<p class='c012'>To compute brake or dynamometrical horse-power.</p>
+
+<p class='c007'>Set 525 on C to the total weight in lb. acting at the end of the
+lever (or pull of spring balance in lb.) on D; set cursor to length
+of lever in feet on C, bring 1 on C to cursor, and under number of
+revolutions per minute on C find brake horse-power on D.</p>
+
+<p class='c007'>Given cylinder diameter and piston speed in feet per minute,
+to find diameter of steam pipe, assuming the maximum velocity of
+the steam to be 6000 ft. per minute.</p>
+
+<p class='c007'>Set 6000 on B to cylinder diameter on D, and under piston
+speed on B read steam pipe diameter on D.</p>
+
+<p class='c007'>Given the number of revolutions per minute of a Watt
+governor, to find the vertical height in inches, from the plane of
+revolution of the balls to the point of suspension.</p>
+
+<p class='c007'>Set revolutions per minute on C to 35,200 on A, and over index
+of B read height on A.</p>
+
+<p class='c007'>Given the weight in lb. of the rim of a cast-iron fly-wheel, to
+find the sectional area of the rim in square inches.</p>
+
+<p class='c007'>Set the mean diameter of the wheel in feet on C to 0·102 on D,
+and under weight of rim on C find area on D.</p>
+
+<p class='c007'>Given the consumption of coal in tons per week of 56 hours,
+and the I.H.P., to find the coal consumed per I.H.P. per hour.</p>
+
+<p class='c007'>Set I.H.P. on C to 40 on D, and under weekly consumption on
+C read lb. of coal per I.H.P., per hour on D.</p>
+
+<p class='c020'><span class='pageno' id='Page_66'>66</span><span class='sc'>Ex.</span>—Find coal used per I.H.P. per hour, when 24 tons is the weekly
+consumption for 300 I.H.P.</p>
+
+<p class='c021'>Set 300 on C to 40 on D, and under 24 on C read 3·2 lb. per I.H.P.
+per hour on D.</p>
+
+<p class='c012'>(N.B.—For any other number of working hours per week
+divide 2240 by the number of working hours, and use the quotient
+in place of 40 as above.)</p>
+
+<p class='c007'>To find the tractive force of a locomotive.</p>
+
+<p class='c007'>Set diameter of driving wheel in inches on B to diameter of
+cylinder in inches on D, and over the stroke in inches on B read
+on A, tractive force in lb. for each lb. of effective pressure on the
+piston.</p>
+
+<h4 class='c036'><span class='sc'>Steam Boilers.</span></h4>
+
+<p class='c009'>To find the bursting pressure of a cylindrical boiler shell,
+having given the diameter of shell and the thickness and ultimate
+strength of the material.</p>
+
+<p class='c007'>Set the diameter of the shell in inches on C to twice the thickness
+of the plate on D, and under strength of material per square
+inch on C read bursting pressure in lb. per square inch on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the bursting pressure of a cylindrical boiler shell 7 ft. 6 in.
+in diameter, with plates ½in. thick, assuming an ultimate strength of
+50,000 lb. per square inch.</p>
+
+<p class='c021'>Set 90 on C to 1·0 on D, and under 50,000 on C find 555 lb. on D.</p>
+
+<p class='c012'>To find working pressure for Fox’s corrugated furnaces by
+Board of Trade rule.</p>
+
+<p class='c007'>Set the least outside diameter in inches on C to 14,000 on D,
+and under thickness in inches on C read working pressure on D in
+lb. per square inch.</p>
+
+<p class='c007'>To find diameter <em>d</em> in inches, of round steel for safety valve
+springs by Board of Trade rule.</p>
+
+<p class='c007'>Set 8000 on C to load on spring in lb. on D, and under the
+mean diameter of the spring in inches on C read <em>d</em><sup>3</sup> on D. Then
+extract the cube root as per rule.</p>
+
+<h4 class='c036'><span class='sc'>Speed Ratios of Pulleys, Etc.</span></h4>
+
+<p class='c009'>Given the diameter of a pulley and its number of revolutions
+per minute, to find the circumferential velocity of the pulley or
+the speed of ropes, belts, etc., driven thereby.</p>
+
+<p class='c007'>Set diameter of pulley in inches on C to 3·82 on D, and over
+revolutions per minute on D read speed in feet per minute on C.</p>
+
+<p class='c020'><span class='pageno' id='Page_67'>67</span><span class='sc'>Ex.</span>—Find the speed of a belt driven by a pulley 53 in. in diameter
+and running at 180 revolutions per minute.</p>
+
+<p class='c021'>Set 53 on C to 3·82 on D, and over 180 on D read 2500 ft. per
+minute on C.</p>
+
+<p class='c021'><span class='sc'>Ex.</span>—Find the speed of the pitch line of a spur wheel 3 ft. 6 in. in
+diameter running at 60 revolutions per minute.</p>
+
+<p class='c021'>Set 42 in. on C to 3·82 on D, and over 60 on D read 660 ft. per
+minute on C.</p>
+
+<p class='c012'>Given diameter and number of revolutions per minute of a
+driving pulley, and the diameter of the driven pulley, to find the
+number of revolutions of the latter.</p>
+
+<p class='c007'>Invert the slide and set diameter of driving pulley on Ɔ to
+given number of its revolutions on D; then opposite diameter of
+any driven pulley on Ɔ read its number of revolutions on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Diameter of driving pulley 10 ft.; revolutions per minute 55;
+diameter of driven pulley 2 ft. 9 in. Find number of revolutions per minute
+of latter.</p>
+
+<p class='c021'>Set 10 on Ɔ to 55 on D, and opposite 2·75 on Ɔ read 200 revolutions
+on D.</p>
+
+<h4 class='c036'><span class='sc'>Belts and Ropes.</span></h4>
+
+<p class='c009'>To find the ratio of tensions in the two sides of a belt, given
+the coefficient of friction between belt and pulley μ and the
+number of degrees θ in the arc of contact <sub><span class='c034'>(</span></sub>log. R = <span class='fraction'>μθ<br><span class='vincula'>132</span></span><sub><span class='c034'>)</span></sub>.</p>
+
+<p class='c007'>Set 132 on C to the coefficient of friction on D, and read off the
+value found on D under the number of degrees in the arc of contact
+on C. Place this value on the scale of equal parts on the
+back of the slide, to the index mark in the aperture, and read the
+required ratio on D under the <span class='fss'>L.H.</span> index of C.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the tension ratio in a belt, assuming a coefficient of friction
+of 0·3 and an arc of contact of 120 degrees.</p>
+
+<p class='c021'>Set 132 on C to 0·3 on D, and under 120 on C read 0·273. Place
+this on the scale to the index on the back of the rule, and under the
+<span class='fss'>L.H.</span> index C read 1·875 on D, the required ratio.</p>
+
+<p class='c012'>Given belt velocity and horse-power to be transmitted, to find
+the requisite width of belt, taking the effective tension at 50 lb.
+per inch of width.</p>
+
+<p class='c007'>Set 660 on C to velocity in feet per minute on D, and opposite
+horse-power on D find width of belt in inches on C.</p>
+
+<p class='c007'>Given velocity and width of belt, to find horse-power transmitted.</p>
+
+<p class='c007'>Set 660 on C to velocity on D, and under width on C find
+horse-power transmitted on D.</p>
+
+<p class='c007'><span class='pageno' id='Page_68'>68</span>(N.B.—For any other effective tension, instead of 660 use as a
+gauge point:—33,000 ÷ tension.)</p>
+
+<p class='c007'>Given speed and diameter of a cotton driving rope, to find
+power transmitted, disregarding centrifugal action, and assuming
+an effective working tension of 200 lb. per square inch of rope.</p>
+
+<p class='c007'>Set 210 on B to 1·75 on D, and over speed in feet per minute
+on B read horse-power on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the power transmitted by a 1¾in. rope running at 4000 ft.
+per minute.</p>
+
+<p class='c023'>Set 210 on B to 1·75 on D, and over 4000 on B read 58·3 horse-power
+on A.</p>
+
+<p class='c012'>Find the “centrifugal tension” in the previous example, taking
+the weight per foot of the rope as = 0·27<em>d</em><sup>2</sup>.</p>
+
+<p class='c007'>Set 655 on C to the diameter, 1·75 in., on D, and over the speed,
+4000 ft. on C, read centrifugal tension = 114 lb. on A.</p>
+
+<h4 class='c036'><span class='sc'>Spur Wheels.</span></h4>
+
+<p class='c009'>Given diameter and pitch of a spur wheel, to find number of
+teeth.</p>
+
+<p class='c007'>Set pitch on C to π (3·1416) on D, and under any diameter on
+C read number of teeth on D.</p>
+
+<p class='c007'>Given diameter and number of teeth in a spur wheel, to find
+the pitch.</p>
+
+<p class='c007'>Set diameter on C to number of teeth on D, and read pitch on
+C opposite 3·1416 on D.</p>
+
+<p class='c007'>Given the distance between the centres of a pair of spur wheels
+and the number of revolutions of each, to determine their
+diameters.</p>
+
+<p class='c007'>To twice the distance between the centres on D, set the sum of
+the number of revolutions on C, and under the revolutions of each
+wheel on C find the respective wheel diameters on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—The distance between the centres of two spur wheels is 37·5 in.,
+and they are required to make 21 and 24 revolutions in the same time.
+Find their respective diameters.</p>
+
+<p class='c023'>Set 21 + 24 = 45 on C to 75 (or 37·5 × 2) on D, and under 21 and 24
+on C find 35 and 40 in. on D as the respective diameters.</p>
+
+<p class='c012'>To find the power transmitted by toothed wheels, given
+the pitch diameter <em>d</em> in inches, the number of revolutions
+per minute <em>n</em>, and the pitch <em>p</em> in inches, by the rule, H.P.
+= <span class='fraction'><span class='under'><em>n</em> <em>d</em> <em>p</em><sup>2</sup></span><br>400</span>.</p>
+
+<p class='c007'><span class='pageno' id='Page_69'>69</span>Set 400 on B to pitch in inches on D; set cursor to d on
+B, 1 on B to cursor, and over any number of revolutions n on B
+read power transmitted on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the horse-power capable of being transmitted by a spur
+wheel 7 ft. in diameter, 3 in. pitch, and running at 90 revolutions per
+minute.</p>
+
+<p class='c021'>Set 400 on B to 3 on D; bring cursor to 84 in. on B, 1 on B to
+cursor, and over 90 revolutions on B read 170, the horse-power transmitted,
+on A.</p>
+<h4 class='c036'><span class='sc'>Screw-Cutting.</span></h4>
+
+<p class='c009'>Given the number of threads per inch in the guide screw, to
+find the wheels to cut a screw of given pitch.</p>
+
+<p class='c007'>Set threads per inch in guide screw on C, to the number of
+threads per inch to be cut on D. Then opposite any number
+of teeth in the wheel on the mandrel on C, is the number of
+teeth in the wheel to be placed on the guide screw on D.</p>
+
+<h4 class='c036'><span class='sc'>Strength of Shafting.</span></h4>
+
+<p class='c009'>Given the diameter <em>d</em> of a steel shaft, and the number of
+revolutions per minute <em>n</em>, to find the horse-power from:—</p>
+
+<p class='c007'>H.P. = <em>d</em><sup>3</sup> × <em>n</em> × 0·02.</p>
+
+<p class='c007'>Set 1 on C to <em>d</em> on D, and bring cursor to <em>d</em> on B. Bring 50
+on B to cursor, and over number of revolutions on B read H.P.
+on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find horse-power transmitted by a 3 in. steel shaft at 110
+revolutions per minute.</p>
+
+<p class='c021'>Set 1 on C to 3 on D, and bring cursor to 3 on B. Bring 50 on B
+to cursor, and over 110 on B read 59·4 horse-power on A.</p>
+
+<p class='c012'>Given the horse-power to be transmitted and the number of
+revolutions of a steel shaft, to find the diameter.</p>
+
+<p class='c007'>Set revolutions on B to horse-power on A, and bring cursor to
+50 on B. Then move the slide until the same number is found on
+B under the cursor that is simultaneously found on D under the
+index of C. This number is the diameter required.</p>
+
+<p class='c007'>To find the deflection <em>k</em> in inches, of a round steel shaft of
+diameter <em>d</em>, under a uniformly distributed load in lb. <em>w</em>, and
+supported by bearings, the centres of which are <em>l</em> feet apart
+(<em>k</em> = <span class='fraction'><em>w</em> <em>l</em><sup>3</sup><br><span class='vincula'>78,000<em>d</em><sup>4</sup></span></span>).</p>
+
+<p class='c007'>Modifying the form of this expression slightly, we proceed as
+follows:—Set <em>d</em> on C to <em>l</em> on D, and bring the cursor to the same
+<span class='pageno' id='Page_70'>70</span>number on B that is found on D under the index of C. Bring <em>d</em>
+on B to cursor, cursor to <em>w</em> on B, 78,800 on B to cursor, and read
+deflection on A over index of B.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the deflection in inches of a round steel shaft 3½in.
+diameter, carrying a uniformly distributed load of 3200 lb., the distance
+apart of the centres of support being 9 ft.</p>
+
+<p class='c023'>Set 3·5 on C to 9 on D, and read 2·57 on D, under the <span class='fss'>L.H.</span>
+index of C. Set cursor to 2·57 on B, and bring 3·5 on B to cursor,
+cursor to 3200 on B, 78,000 on B to cursor, and over <span class='fss'>L.H.</span> index of B
+read 0·199 in., the required deflection on A.</p>
+
+<p class='c012'>To find the diameter of a shaft subject to twisting only, given
+the twisting moment in inch-lb. and the allowable stress in lb. per
+square inch.</p>
+
+<p class='c007'>Set the stress in lb. per square inch on B to the twisting
+moment in inch-lb. on A, and bring cursor to 5·1 on B. Then
+move the slide until the same number is found on B under the
+cursor that is simultaneously found on D under the index of C.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—A steel shaft is subjected to a twisting moment of 2,700,000
+inch-lb. Determine the diameter if the allowable stress is taken at 9000
+lb. per square inch.</p>
+
+<p class='c023'>Set 9000 on B to 2,700,000 on A, and bring the cursor to 5·1 on
+B. Moving the slide to the left, it is found that when 11·51 on the
+<span class='fss'>R.H.</span> scale of B is under the cursor, the <span class='fss'>L.H.</span> index of C is opposite
+11·51 on D. This, then, is the required diameter of the shaft.</p>
+
+<p class='c012'>(N.B.—The rules for the scales to be used in finding the cube
+root (page <a href='#Page_42'>42</a>) must be carefully observed in working these
+examples.)</p>
+
+<h4 class='c036'><span class='sc'>Moments of Inertia.</span></h4>
+
+<p class='c009'>To find the moment of inertia of a square section about an axis
+formed by one of its diagonals <sub><span class='c034'>(</span></sub>I = <span class='fraction'><span class='under'><em>s</em><sup>4</sup></span><br>12</span><sub><span class='c034'>)</span></sub>.</p>
+
+<p class='c007'>Set index of C to the length of the side of square <em>s</em> on D; bring
+cursor to <em>s</em> on C, 12 on B to cursor, and over index of B read
+moment of inertia on A.</p>
+
+<p class='c007'>To find the moment of inertia of a rectangular section about an
+axis parallel to one side and perpendicular to the plane of bending.</p>
+
+<p class='c007'>Set index of C to the height or depth <em>h</em> of the section, and
+bring cursor to <em>h</em> on B. Set 12 on B to cursor, and over breadth
+<em>b</em> of the section on B read moment of inertia on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the moment of inertia of a rectangular section of which
+<em>h</em> = 14 in. and <em>b</em> = 7 in.</p>
+
+<p class='c023'>Set index of C to 14 on D, and cursor to 14 on B. Bring 12 on B
+to cursor, and over 7 on B read 1600 on A.</p>
+
+<div>
+  <span class='pageno' id='Page_71'>71</span>
+  <h4 class='c036'><span class='sc'>Discharge from Pumps, Pipes, Etc.</span></h4>
+</div>
+
+<p class='c009'>To find the theoretical delivery of pumps, in gallons per
+stroke.</p>
+
+<p class='c007'>Set 29·4 on B to the diameter of the plunger in inches on D,
+and over length of stroke in feet on B read theoretical delivery in
+gallons per stroke on A.</p>
+
+<p class='c007'>(N.B.—A deduction of from 20 to 40 per cent. should be made
+to allow for slip.)</p>
+
+<p class='c007'>To find loss of head of water in feet due to friction in pipes
+(Prony’s rule).</p>
+
+<p class='c007'>Set diameter of pipe in feet on B to velocity of water in feet
+per second on D and bring cursor to 2·25 on B; bring 1 on B to
+cursor, and over length of pipe in miles on B, read loss of head of
+water in feet, on A.</p>
+
+<p class='c007'>To find velocity in feet per second, of water in pipes
+(Blackwell’s rule).</p>
+
+<p class='c007'>Set 2·3 on B to diameter of pipe in feet on A, and under
+inclination of pipe in feet per mile on B read velocity in feet per
+second on D.</p>
+
+<p class='c007'>To find the discharge over weirs in cubic feet per minute and
+per foot of width. (Discharge = 214√<span class='vincula'><em>h</em><sup>3</sup></span>)</p>
+
+<p class='c007'>Set 0·00467 on C to the head in feet <em>h</em> on D, and under <em>h</em> on B
+read discharge on D.</p>
+
+<p class='c007'>To find the theoretical velocity of water flowing under a given
+head in feet.</p>
+
+<p class='c007'>Set index of B to head in feet on A, and under 64·4 on B read
+theoretical velocity in feet per second on D.</p>
+
+<h4 class='c036'><span class='sc'>Horse-Power of Water Wheels.</span></h4>
+
+<p class='c009'>To find the effective horse-power of a Poncelet water wheel.</p>
+
+<p class='c007'>Set 880 on C to cubic feet of flow of water per minute on D,
+and under height of fall in feet on C, read effective horse-power
+on D.</p>
+
+<p class='c007'>For breast water wheels use 960, and for overshot wheels 775,
+in place of 880 as above.</p>
+
+<h4 class='c036'><span class='sc'>Electrical Engineering.</span></h4>
+
+<p class='c009'>To find the resistance per mile, in ohms, of copper wire of
+high conductivity, at 60° F. the diameter being given in mils.
+(1 mil. = 0·001 in.).</p>
+
+<p class='c007'><span class='pageno' id='Page_72'>72</span>Set diameter of wire in mils. on C to 54,900 on A, and over
+<span class='fss'>R.H.</span> or <span class='fss'>L.H.</span> index of B read resistance in ohms on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the resistance per mile of a copper wire 64 mils. in diameter.</p>
+
+<p class='c023'>Set 64 on C to 54,900 on A, and over <span class='fss'>R.H.</span> index of B read 13·4
+ohms on A.</p>
+
+<p class='c012'>To find the weight of copper wire in lb. per mile.</p>
+
+<p class='c007'>Set 7·91 on C to diameter of wire in mils. on D, and over index
+of B read weight per mile on A.</p>
+
+<p class='c007'>Given electromotive force and current, to find electrical horse-power.</p>
+
+<p class='c007'>Set 746 on C to electromotive force in volts on D, and under
+current in ampères on C read electrical horse-power on D.</p>
+
+<p class='c007'>Given the resistance of a circuit in ohms and current in ampères,
+to find the energy absorbed in horse-power.</p>
+
+<p class='c007'>Set 746 on B to current on D, and over resistance on B read
+energy absorbed in H.P. on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the H.P. expended in sending a current of 15 ampères
+through a circuit of 220 ohms resistance.</p>
+
+<p class='c021'>Set 746 on B to 15 on D, and over 220 on B read 66·3 H.P. on A.</p>
+
+<h4 class='c036'><span class='sc'>Commercial.</span></h4>
+
+<p class='c009'>To add on percentages.</p>
+
+<p class='c007'>Set 100 on C to 100 + given percentage on D, and under original
+number on C read result on D.</p>
+
+<p class='c007'>To deduct percentages.</p>
+
+<p class='c007'>Set <span class='fss'>R.H.</span> index of C to 100 − the given percentage on D, and
+under original number on C read result on D.</p>
+
+<p class='c020'><span class='sc'>Ex</span>.—From £16 deduct 7½ per cent.</p>
+
+<p class='c023'>Set 10 on C to 92·5 on D and under 16 on C, read 14·8 = £14, 16s.
+on D.</p>
+
+<p class='c012'>To calculate simple interest.</p>
+
+<p class='c007'>Set 1 on C to rate per cent. on D; bring cursor to period
+on C and 1 on C to cursor. Then opposite any sum on C find
+simple interest on D.</p>
+
+<p class='c007'>For interest per annum.</p>
+
+<p class='c007'>Set <span class='fss'>R.H.</span> index on C to rate on D, and opposite principal on C
+read interest on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the amount with simple interest of £250 at 8 per cent.,
+and for a period of 1 year and 9 months.</p>
+
+<p class='c023'>Set 1 on C to 8 on D; bring cursor to 1·75 on C, and 1 on C to
+cursor; then opposite 250 on C read £35, the interest, on D. Then
+250 + 35 = £285 = the amount.</p>
+
+<p class='c012'><span class='pageno' id='Page_73'>73</span>To calculate compound interest.</p>
+
+<p class='c007'>Set the <span class='fss'>L.H.</span> index of C to the amount of £1 at the given rate
+of interest on D, and find the logarithm of this by reading on the
+reverse side of the rule, as explained on page <a href='#Page_46'>46</a>. Multiply the
+logarithm, so found, by the period, and set the result, on the scale
+of equal parts, to the index on the under-side of the rule; then
+opposite any sum on C read the amount (including compound
+interest) on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the amount of £500 at 5 per cent. for 6 years, with compound
+interest.</p>
+
+<p class='c023'>Set <span class='fss'>L.H.</span> index of C to £1·05 on D, and read at the index on the
+scale of equal parts on the under-side of rule, 0·0212. Multiply by 6,
+we obtain 0·1272, which, on the scale of equal parts, is placed to the
+index in the notch at the end of the rule. Then opposite 500 on C
+read £670 on D, the amount required, including compound interest.</p>
+
+<h4 class='c036'><span class='sc'>Miscellaneous Calculations.</span></h4>
+
+<p class='c009'>To calculate percentages of compositions.</p>
+
+<p class='c007'>Set weight (or volume) of sample on C, to weight (or volume)
+of substance considered, on D; then under index of C read
+required percentage on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—A sample of coal weighing 1·25 grms. contains 0·04425 grm. of
+ash. Find the percentage of ash.</p>
+
+<p class='c023'>Set 1·25 on C to 0·04425 on D, and under index on C read 3·54,
+the required percentage of ash on D.</p>
+
+<p class='c012'>Given the steam pressure P and the diameter <em>d</em> in millimetres,
+of the throat of an injector, to find the weight W, of water
+delivered in lb. per hour from W = <span class='fraction'><span class='under'><em>d</em><sup>2</sup>√̅P</span><br>0·505</span>.</p>
+
+<p class='c007'>Set 0·505 on C to P on A; bring cursor to <em>d</em> on C and index of
+C to cursor. Then under <em>d</em> on C read delivery of water on D.</p>
+
+<p class='c007'>To find the pressure of wind per square foot, due to a given
+velocity in miles per hour.</p>
+
+<p class='c007'>Set 1 on B to 2 on A, and over the velocity in miles per hour
+on D read pressure in lb. per square foot on B.</p>
+
+<p class='c007'>To find the kinetic energy of a moving body.</p>
+
+<p class='c007'>Set 64·4 on B to velocity in feet per second on D, and over
+weight of body in lb. on B read kinetic energy or accumulated
+work in foot-lb. on A.</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_74'>74</span>
+  <h2 class='c005'>TRIGONOMETRICAL APPLICATIONS</h2>
+</div>
+
+<p class='c012'><em>Scales.</em>—Not the least important feature of the modern slide
+rule is the provision of the special scales on the under-side of the
+slide, and by the use of which, in conjunction with the ordinary
+scales on the rule, a large variety of trigonometrical computations
+may be readily performed.</p>
+
+<p class='c007'>Three scales will be found on the reverse or under-side of the
+slide of the ordinary Gravêt or Mannheim rule. One of these is
+the evenly-divided scale or scale of equal parts referred to in
+previous sections, and by which, as explained, the decimal parts or
+mantissæ of logarithms of numbers may be obtained. Usually
+this scale is the centre one of the three, but in some rules it will
+be found occupying the lowest position, in which case some little
+modification of the following instructions will be necessary. The
+requisite transpositions will, however, be evident when the
+purposes of the scales are understood. The upper of the three
+scales, usually distinguished by the letter S, is a scale giving the
+logarithms of the sines of angles, and is used to determine the
+natural sines of angles of from 35 minutes to 90 degrees. The
+notation of this scale will be evident on inspection. The main
+divisions 1, 2, 3, etc., represent the degrees of angles; but the
+values of the subdivisions differ according to their position on the
+scale. Thus, if any primary space is subdivided into 12 parts, each
+of the latter will be read as 5 minutes (5′), since 1° = 60′.</p>
+
+<p class='c007'><em>Sines of Angles.</em>—To find the sine of an angle the slide is placed
+in the groove, with the under-side uppermost, and the end division
+lines or indices on the slide, coinciding with the right and left
+indices of the A scale. Then over the given angle on S is
+read the value of the sine of the angle on A. If the result is found
+on the left scale of A (1 to 10), the logarithmic characteristic is
+−2; if it is found on the right-hand side (10 to 100), it is −1. In
+other words, results on the right-hand scale are prefixed by the
+decimal point only, while those on the left-hand scale are to be
+preceded by a cypher also. Thus:—</p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>Sine 2° 40′ = 0·0465; sine 15° 40′ = 0·270.</div>
+  </div>
+</div>
+
+<p class='c007'>Multiplication and division of the sines of angles are performed
+in the same manner as ordinary calculations, excepting
+<span class='pageno' id='Page_75'>75</span>that the slide has its under-face placed uppermost, as just explained.
+Thus to multiply sine 15° 40′ by 15, the <span class='fss'>R.H.</span> index of S is brought
+to 15 on A, and opposite 15° 40′ on S is found 4·05 on A. Again,
+to divide 142 by sine 16° 30′, we place 16° 30′ on S to 142 on A,
+and over <span class='fss'>R.H.</span> index of S read 500 on A.</p>
+
+<p class='c007'>The rules for the number of integers in the results are thus
+determined: Let N be the number of integers in the multiplier M
+or in the dividend D. Then the number of integers P, in the
+product or Q, in the quotient are as follows:—</p>
+
+<table class='table2'>
+  <tr>
+    <td class='c031'>When the result is found to the right of M or D, and in the same scale</td>
+ <td class='blt c040'>P = N − 2</td>
+ <td class='blt c040'>Q = N</td>
+  </tr>
+  <tr>
+    <td class='c031'>When the result is found to the right of M or D, and in the other scale</td>
+ <td class='blt c040'>P = N − 1</td>
+ <td class='blt c040'>Q = N + 1</td>
+  </tr>
+  <tr>
+    <td class='c031'>When the result is found to the left of M or D, and in the other scale</td>
+ <td class='blt c040'>P = N − 1</td>
+ <td class='blt c040'>Q = N + 1</td>
+  </tr>
+  <tr>
+    <td class='c031'>When the result is found to the left of M or D, and in the same scale</td>
+ <td class='blt c040'>P = N</td>
+ <td class='blt c040'>Q = N + 2</td>
+  </tr>
+</table>
+
+<p class='c007'>If the division is of the form <span class='fraction'><span class='under'>20° 30′</span><br>50</span>, the result cannot be
+read off directly on the face of the rule. Thus, if in the above example
+20° 30′ on S, is placed to agree with 50 on the right-hand scale
+of A, the result found on S under the <span class='fss'>R.H.</span> index of A is 44° 30′.
+The required numerical value can then be found: (1) By placing
+the slide with all indices coincident when opposite 44° 30′ on S
+will be found 0·007 on A; or (2) In the ordinary form of rule, by
+reading off on the scale B opposite the index mark in the opening
+on the under-side of the rule. The above rules for the number of
+integers in the quotient do not apply in this case.</p>
+
+<p class='c007'>If it is required to find the sine of an angle simply, this may
+be done with the slide in its ordinary position, with scale B under
+A. The given angle on scale S is then set to the index on the
+under-side of the rule, and the value of the sine is read off on B
+under the right index of A.</p>
+
+<p class='c007'>Owing to the rapidly diminishing differences of the values of
+the sines as the upper end of the scale is approached, the sines of
+angles between 60° and 90° cannot be accurately determined in the
+foregoing manner. It is therefore advisable to calculate the value
+of the sine by means of the formula:</p>
+
+<p class='c007'>Sine θ = 1 − 2 sin<sup>2</sup> <span class='fraction'><span class='under'>90 − θ</span><br>2</span>.</p>
+
+<p class='c007'><span class='pageno' id='Page_76'>76</span>To determine the value of sin<sup>2</sup> <span class='fraction'><span class='under'>90 − θ</span><br>2</span>. With the slide in the
+normal position, set the value of <span class='fraction'><span class='under'>90 − θ</span><br>2</span>. on S to the index on the
+under-side of the rule, and read off the value <em>x</em> on B under the
+<span class='fss'>R.H.</span> index of A. Without moving the slide find <em>x</em> on A, and read
+under it on B the value required.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find value of sine 79° 40′.</p>
+
+<p class='c021'>Sine 79° 40′ = 1 − 2sin<sup>2</sup> 5° 10′.</p>
+
+<p class='c021'>But sine 5° 10′ = 0·0900, and under this value on A is 0·0081 on
+B. Therefore sine 79° 40′ = 1 − 0·0162 = 0·9838.</p>
+
+<p class='c012'>The sines of very small angles, being very nearly proportional
+to the angles themselves, are found by direct reading. To
+facilitate this, some rules are provided with two marks, one of
+which, a single accent (′), corresponds to the logarithm of <span class='fraction'>1<br><span class='vincula'>sine 1′</span></span>
+and is found at the number 3438. The other mark—a double
+accent (″)—corresponds to the logarithm of <span class='fraction'>1<br><span class='vincula'>sine 1″</span></span> and is found at
+the number 206,265. In some rules these marks are found on
+either the A or the B scales; sometimes they are on both.
+In either case the angle on the one scale is placed so as to
+coincide with the significant mark on the other, and the result
+read off on the first-named scale opposite the index of the
+second.</p>
+
+<p class='c007'>In sines of angles under 3″, the number of integers in the
+result is −5; while it is −4 for angles from 3″ to 21″; −3 from
+21″ to 3′ 27″; and −2 from 3′ 27″ to 34′ 23″.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find sine 6′.</p>
+
+<p class='c021'>Placing the significant mark for minutes coincident with 6,
+the value opposite the index is found to be 175, and by the rule
+above this is to be read 0·00175. For angles in seconds the other
+significant mark is used; while angles expressed in minutes and
+seconds are to be first reduced to seconds. Thus, 3′ 10″ = 190″.</p>
+
+<p class='c012'><em>Tangents of Angles.</em>—There remains to be considered the third
+scale found on the back of the slide, and usually distinguished
+from the others by being lettered T. In most of the more recent
+forms of rule this scale is placed near the lower edge of the slide,
+but in some arrangements it is found to be the centre scale of the
+three. Again, in some rules this scale is figured in the same
+<span class='pageno' id='Page_77'>77</span>direction as the scale of sines—viz., from left to right,—while in
+others the T scale is reversed. In both cases there is now usually
+an aperture formed in the back of the left extremity of the rule,
+with an index mark similar to that already referred to in
+connection with the scale of sines. Considering what has been
+referred to as the more general arrangement, the method of
+determining the tangents of angles may be thus explained:—</p>
+
+<p class='c007'>The tangent scale will be found to commence, in some rules, at
+about 34′, or, precisely, at the angle whose tangent is 0·01. More
+usually, however, the scale will be found to commence at about
+5° 43′, or at the angle whose tangent is 0·1. The other extremity
+of the scale corresponds in all cases to 45°, or the angle
+whose tangent is 1. This explanation will suggest the method
+of using the scale, however it may be arranged. If the
+graduations commence with 34′, the T scale is to be used in conjunction
+with the right and left scales of A; while if they
+commence with 5° 43′ it is to be used in conjunction with the
+D scale.</p>
+
+<p class='c007'>In the former case the slide is to be placed in the rule so that
+the T scale is adjacent to the A scales, and, with the right and
+left indices coinciding, when opposite any angle on T will be found
+its tangent on A. From what has been said above, it follows that
+the tangents read on the <span class='fss'>L.H.</span> scale of A have values extending
+from 0·01 to 0·1; while those read on the <span class='fss'>R.H.</span> scale of A have
+values from 0·1 to 1·0. Otherwise expressed, to the values of
+any tangent read on the <span class='fss'>L.H.</span> scale of A a cypher is to be prefixed;
+while if found on the <span class='fss'>R.H.</span> scale, it is read directly as a
+decimal.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find tan. 3° 50′.</p>
+
+<p class='c021'>Placing the slide as directed, the reading on A opposite 3° 50′
+on T is found to be 67. As this is found on the <span class='fss'>L.H.</span> scale of A,
+it is to be read as 0·067.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find tan. 17° 45′.</p>
+
+<p class='c021'>Here the reading on A opposite 17° 45′ on T is 32, and as it is
+found on the <span class='fss'>R.H.</span> scale of A it is read as 0·32.</p>
+
+<p class='c012'>As in the case of the scale of sines, the tangents may be found
+without reversing the slide, when a fixed index is provided in the
+back of the rule for the T scale.</p>
+
+<p class='c007'>We revert now to a consideration of those rules in which a
+single tangent scale is provided. It will be understood that in this
+<span class='pageno' id='Page_78'>78</span>case the slide is placed so that the scale T is adjacent to the D
+scale, and that when the indices of both are placed in agreement,
+the value of the tangent of any angle on T (from 5° 43′ to 45°)
+may be read off on D, the result so found being read as wholly
+decimal. Thus tan. 13° 20′ is read 0·237.</p>
+
+<p class='c007'>If a back index is provided, the slide is used in its normal
+position, when, setting the angle on the tangent scale to this
+index, the result can be read on C over the <span class='fss'>L.H.</span> index
+of D.</p>
+
+<p class='c007'>The tangents of angles above 45° are obtained by the formula:
+Tan. θ = <span class='fraction'>1<br><span class='vincula'>tan. (90 − θ)</span></span>. For all angles from 45° to (90° − 5° 43′)
+we proceed as follows:—Place (90 − θ) on T to the <span class='fss'>R.H.</span> index of
+D, and read tan. θ on D under the <span class='fss'>L.H.</span> index of T. The first
+figure in the value thus obtained is to be read as an integer. Thus,
+to find tan. 71° 20′ we place 90° − 71° 20′ = 18° 40′ on T, to the <span class='fss'>R.H.</span>
+index of D, and under the <span class='fss'>L.H.</span> index of T read 2·96, the required
+tangent.</p>
+
+<p class='c007'>The tangents of angles less than 40′ are sensibly proportional
+to the angles themselves, and as they may therefore be considered
+as sines, their value is determined by the aid of the single and
+double accent marks on the sine scale, as previously explained.
+The rules for the number of integers are the same as for the
+sines.</p>
+
+<p class='c007'>Multiplication and division of tangents may be quite readily
+effected.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Tan. 21° 50′ × 15 = 6.</p>
+
+<p class='c021'>Set <span class='fss'>L.H.</span> index of T to 15 on D, and under 21° 50′ on T read 6
+on D.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Tan. 72° 40′ × 117 = 375.</p>
+
+<p class='c021'>Set (90° − 72° 40′) = 17° 20′ on T to 117 on D, and under <span class='fss'>R.H.</span>
+index of T read 375 on D.</p>
+
+<p class='c012'><em>Cosines of Angles.</em>—The cosines of angles may be determined by
+placing the scale S with its indices coinciding with those of A, and
+when opposite (90 − θ) on S is read cos. θ on A. If the result is
+read on the <span class='fss'>L.H.</span> scale of A, a cypher is to be prefixed to the value
+read; while if it is read on the <span class='fss'>R.H.</span> scale of A, the value is read
+directly as a decimal. Thus, to determine cos. 86° 30′ we find
+opposite (90° − 86° 30′) = 3° 30′ on S, 61° on A, and as this is on the
+<span class='fss'>L.H.</span> scale the result is read 0·061. Again, to find cos. 59° 20′ we
+<span class='pageno' id='Page_79'>79</span>read opposite (90° − 59° 20′) or 30° 40′ on S, 51 on A, and as this is
+found on the <span class='fss'>R.H.</span> scale of A, it is read 0·51.</p>
+
+<p class='c007'>In finding the cosines of small angles it will be seen that direct
+reading on the rule becomes impossible for angles of less than 20°.
+It is advisable in such cases to adopt the method described for determining
+the <em>sines</em> of the <em>large</em> angles of which the complements
+are sought.</p>
+
+<p class='c007'><em>Cotangents of Angles.</em>—From the methods of finding the tangents
+of angles previously described, it will be apparent that the cotangents
+of angles may also be obtained with equal facility. For
+angles between 5° 45′ and 45°, the procedure is the same as that for
+finding tangents of angles greater than 45°. Thus, the angle on
+scale T is brought to the <span class='fss'>R.H.</span> index of D, and the cotangent read
+off on D under the <span class='fss'>L.H.</span> index of T. The first figure of the result so
+found is to be read as an integer.</p>
+
+<p class='c007'>If the angle (θ) lies between 45° and 84° 15′, the slide is placed
+so that the indices of T coincide with those of D, and the result is
+then read off on D opposite (90 − θ) on T. In this case the value is
+wholly decimal.</p>
+
+<p class='c007'><em>Secants of Angles.</em>—The secants of angles are readily found by
+bringing (90 − θ) on S to the <span class='fss'>R.H.</span> index of A and reading the result
+on A over the <span class='fss'>L.H.</span> index of S. If the value is found on the <span class='fss'>L.H.</span>
+scale of A, the first figure is to be read as an integer; while if the
+result is read on the <span class='fss'>R.H.</span> scale of A, the first <em>two</em> figures are to be
+regarded as integers.</p>
+
+<p class='c007'><em>Cosecants of Angles.</em>—The cosecants of angles are found by
+placing the angle on S to the <span class='fss'>R.H.</span> index of A, and reading the
+value found on A over the <span class='fss'>L.H.</span> index of S. If the result is read
+on the <span class='fss'>L.H.</span> scale of A, the first figure is to be read as an integer;
+while if the result is found on the <span class='fss'>R.H.</span> scale of A, the first <em>two</em>
+figures are to be read as integers.</p>
+
+<p class='c007'>It will be noted that some of the rules here given for determining
+the several trigonometrical functions of angles apply only to
+those forms of rules in which a single scale of tangents T is used,
+reading from left to right. For the other arrangements of the
+scale, previously referred to, some slight modification of the method
+of procedure in finding the tangents and cotangents of angles will
+be necessary; but as in each case the nature and extent of this
+modification is evident, no further directions are required.</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_80'>80</span>
+  <h2 class='c005'>THE SOLUTION OF RIGHT-ANGLED TRIANGLES.</h2>
+</div>
+
+<p class='c012'>From the foregoing explanation of the manner of determining the
+trigonometrical functions of angles, the methods of solving right-angled
+triangles will be readily perceived, and only a few examples
+need therefore be given.</p>
+
+<p class='c007'>Let <em>a</em> and <em>b</em> represent the sides and <em>c</em> the hypothenuse of a
+right-angled triangle, and <em>a</em>° and <em>b</em>° the angles opposite to the sides.
+Then of the possible cases we will take</p>
+
+<p class='c007'>(1.) Given <em>c</em> and <em>a</em>°, to find <em>a</em>, <em>b</em>, and <em>b</em>°.</p>
+
+<p class='c007'>The angle <em>b</em>° = 90 − <em>a</em>°, while <em>a</em> = <em>c</em> sin <em>a</em>° and <em>b</em> = <em>c</em> sin <em>b</em>°. To find
+<em>a</em>, therefore, the index of S is set to <em>c</em> on A, and the value of <em>a</em> read
+on A opposite <em>a</em>° on S. In the same manner the value of <em>b</em> is
+obtained.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Given in a right-angled triangle <em>c</em> = 9 ft. and <em>a</em>° = 30°.
+Find <em>a</em>, <em>b</em>, and <em>b</em>°.</p>
+
+<p class='c021'>The angle <em>b</em>° = 90 − 30 = 60°. To find <em>a</em>, set <span class='fss'>R.H.</span> index of S to 9
+on A, and over 30° on S read <em>a</em> = 4·5 ft. on A. Also, with the slide
+in the same position, read <em>b</em> = 7·8 ft. [7·794] on A over 60° on S.</p>
+
+<p class='c012'>(2.) Given <em>a</em> and <em>c</em>, to determine <em>a</em>°, <em>b</em>°, and <em>b</em>.</p>
+
+<p class='c007'>In this case advantage is taken of the fact that in every triangle
+the sides are proportional to the sines of the opposite angles.
+Therefore, as in this case the hypothenuse c subtends a right angle,
+of which the sine = 1, the <span class='fss'>R.H.</span> index (or 90°) on S is set to the
+length of <em>c</em> on A, when under <em>a</em> on A is found <em>a</em>° on S. Hence <em>b</em>°
+and <em>b</em> may be determined.</p>
+
+<p class='c007'>(3.) Given <em>a</em> and <em>a</em>°, to find <em>b</em>, <em>c</em>, and <em>b</em>°.</p>
+
+<p class='c007'>Here <em>b</em>° = (90 − <em>a</em>°), and the solution is similar to the foregoing.</p>
+
+<p class='c007'>(4.) Given <em>a</em> and <em>b</em>, to find <em>a</em>°, <em>b</em>°, and <em>c</em>.</p>
+
+<p class='c007'>To find <em>a</em>°, we have tan. <em>a</em>° = <em>a</em>/<em>b</em>, which in the above example
+will be <span class='fraction'>4·5<br><span class='vincula'>7·8</span></span> = 0·577. Therefore, placing the slide so that the indices
+of T coincide with those of D, we read opposite 0·577 on D the
+value of <em>a</em>° = 30°. The hypothenuse <em>c</em> is readily obtained from
+<em>c</em> = <em>a</em>/(sin <em>a</em>°).</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_81'>81</span>
+  <h2 class='c005'>THE SOLUTION OF OBLIQUE-ANGLED TRIANGLES.</h2>
+</div>
+
+<p class='c012'>Using the same letters as before to designate the three sides
+and the subtending angles of oblique-angled triangles, we have
+the following cases:—</p>
+
+<p class='c007'>(1.) Given one side and two angles, as <em>a</em>, <em>a</em>°, and <em>b</em>°, to find <em>b</em>, <em>c</em>,
+and <em>c</em>°.</p>
+
+<p class='c007'>In the first place, <em>c</em>° = 180° − (<em>a</em>° + <em>b</em>°); also we note that, as
+the sides are proportional to the sines of the opposite angles,
+<em>b</em> = <span class='fraction'><span class='under'><em>a</em> sine <em>b</em>°</span><br>sine <em>a</em>°</span> and <em>c</em> = <span class='fraction'><span class='under'><em>a</em> sine <em>c</em>°</span><br>sine <em>a</em>°</span>.</p>
+
+<p class='c007'>Taking as an example, <em>a</em> = 45, <em>a</em>° = 57°, and <em>b</em>° = 63°, we have <em>c</em>° =
+180 − (57 + 63) = 60°. To find <em>b</em> and <em>c</em>, set <em>a</em>° on S to <em>a</em> on A, and
+read off on A above 63° and 60° the values of <em>b</em> (= 47·8) and <em>c</em>
+(= 46·4) respectively.</p>
+
+<p class='c007'>(2.) Given <em>a</em>, <em>b</em>, and <em>a</em>°, to find <em>b</em>°, <em>c</em>°, and <em>c</em>.</p>
+
+<p class='c007'>In this case the angle <em>a</em>° on S is placed under the length of side
+<em>a</em> on A and under <em>b</em> on A is found the angle <em>b</em>° on S. The angle
+<em>c</em>° = 180 − (<em>a</em>° + <em>b</em>°), whence the length <em>c</em> can be read off on A over
+<em>c</em>° on S.</p>
+
+<p class='c007'>(3.) Given the sides and the included angle, to find the other
+side and the remaining angles.</p>
+
+<p class='c007'>If, for example, there are given <em>a</em> = 65, <em>b</em> = 42, and the included
+angle <em>c</em>° = 55°, we have (<em>a</em> + <em>b</em>)&#8201;∶&#8201;(<em>a</em> − <em>b</em>) = tan. <span class='fraction'><span class='under'><em>a</em>° + <em>b</em>°</span><br>2</span>&#8201;∶&#8201;tan. <span class='fraction'><span class='under'><em>a</em>° − <em>b</em>°</span><br>2</span>.
+Then, since <em>a</em>° + <em>b</em>° = 180° − 55° = 125°, it follows that <span class='fraction'><span class='under'><em>a</em>° + <em>b</em>°</span><br>2</span> = <span class='fraction'><span class='under'>125°</span><br>2</span> =
+62° 30′.</p>
+
+<p class='c007'>By the rule for tangents of angles greater than 45°, we find
+tan. 62° 30′ = 1·92. Inserting in the above proportion the values
+thus found, we have 107&#8201;∶&#8201;23 = 1·92&#8201;∶&#8201;tan. <span class='fraction'><span class='under'><em>a</em>° − <em>b</em>°</span><br>2</span>. From this it
+is found that the value of the tangent is 0·412, and placing the
+slide with all indices coinciding, it is seen that this value on D
+corresponds to an angle of 22° 25′. Therefore, since <span class='fraction'><span class='under'><em>a</em>° + <em>b</em>°</span><br>2</span> = 62°
+30′, and <span class='fraction'><span class='under'><em>a</em>° − <em>b</em>°</span><br>2</span> = 22° 25′, it follows that <em>a</em>° = 84° 55′, and <em>b</em>° = 40° 5′.
+Finally, to determine the side <em>c</em>, we have <em>c</em> = <span class='fraction'><span class='under'><em>a</em> sin <em>c</em>°</span><br>sin <em>a</em>°</span> as before.</p>
+
+<div>
+  <span class='pageno' id='Page_82'>82</span>
+  <h3 class='c008'>PRACTICAL TRIGONOMETRICAL APPLICATIONS.</h3>
+</div>
+
+<p class='c009'>A few examples illustrative of the application of the methods
+of determining the functions of angles, etc., described in the
+preceding section, will now be given.</p>
+
+<p class='c007'>To find the chord of an arc, having given the included angle
+and the radius.</p>
+
+<p class='c007'>With the slide placed in the rule with the C and D scales outward,
+bring one-half of the given angle on S to the index mark in
+the back of the rule, and read the chord on B under twice the
+radius on A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Required the chord of an arc of 15°, the radius being 23 in.</p>
+
+<p class='c021'>Set 7° 30′ on S to the index mark in the back of the rule, and under
+46 on A read 6 in., the required length of chord on B.</p>
+
+<p class='c012'>To find the area of a triangle, given two sides and the included
+angle.</p>
+
+<p class='c007'>Set the angle on S to the index mark on the back of the rule,
+and bring cursor to 2 on B. Then bring the length of one side on
+B to cursor, cursor to 1 on B, the length of the other side on B to
+cursor, and read area on B under index of A.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—The sides of a triangle are 5 and 6 ft. in length respectively, and
+they include an angle of 20°. Find the area.</p>
+
+<p class='c021'>Set 20 on S to index mark, bring cursor to 2 on B, 5 on B to cursor,
+cursor to 1 on B, 6 on B to cursor, and under 1 on A read the area =
+5·13 sq. ft. on B.</p>
+
+<p class='c012'>To find the number of degrees in a gradient, given the rise per
+cent.</p>
+
+<p class='c007'>Place the slide with the indices of T coincident with those of
+D, and over the rate per cent. on D read number of degrees in the
+slope on T.</p>
+
+<p class='c007'>As the arrangement of rule we have chiefly considered has only
+a single T scale, it will be seen that only solutions of the above
+problem involving slopes between 10 and 100 per cent. can be
+directly read off. For smaller angles, one of the formulæ for the
+determination of the tangents of submultiple angles must be used.</p>
+
+<p class='c007'>In rules having a double T scale (which is used with the A
+scale) the value in degrees of any slope from 1 to 100 per cent. can
+be directly read off on A.</p>
+
+<p class='c007'>To find the number of degrees, when the gradient is expressed
+as 1 in <em>x</em>.</p>
+
+<p class='c007'><span class='pageno' id='Page_83'>83</span>Place the index of T to <em>x</em> on D, and over index of D read the
+required angle in degrees on T.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the number of degrees in a gradient of 1 in 3·8.</p>
+
+<p class='c021'>Set 1 on T to 3·8 on D, and over <span class='fss'>R.H.</span> index of D read 14° 45′
+on T.</p>
+
+<p class='c012'>Given the lap, the lead and the travel of an engine slide valve,
+to find the angle of advance.</p>
+
+<p class='c007'>Set (lap + lead) on B to half the travel of the valve on A, and
+read the angle of advance on S at the index mark on the back of
+the rule.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Valve travel 4½in., lap 1 in., lead ⁵⁄₁₆in. Find angle of advance.</p>
+
+<p class='c021'>Set 1⁵⁄₁₆ = 1·312 on B to 2·25 on A, and read 35° 40′ on S opposite
+the index on the back of the rule.</p>
+
+<p class='c012'>Given the angular advance θ, the lap and the travel of a slide
+valve, to find the cut-off in percentage of the stroke.</p>
+
+<p class='c007'>Place the lap on B to half the travel of valve on A, and read
+on S the angle (the supplement of the <em>angle of the eccentric</em>) found
+opposite the index in the back of the rule. To this angle, add the
+angle of advance and deduct the sum from 180°, thus obtaining
+the <em>angle of the crank</em> at the point of cut-off. To the cosine of the
+supplement of this angle, add 1 and multiply the result by 50,
+obtaining the percentage of stroke completed when cut-off occurs.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Given the angular advance = 35° 40′, the valve travel = 4½in.,
+and the lap = 1 in., find the angle of the crank at cut-off and the admission
+period expressed as a percentage of the stroke.</p>
+
+<p class='c021'>Set 1 on B to 2·25 on A, and read off on S opposite the
+index, the supplement of the angle of the eccentric = 26° 20′. Then
+180° − (35° 40′ + 26° 20′) = 118° = the crank angle at the point of cut-off.
+Further, cos. 118° = cos. 62° = sin (90° − 62°) = sin 28°, and placing 28°
+on S to the back index, the cosine, read on B under <span class='fss'>R.H.</span> index of
+A, is found to be 0·469. Adding 1 and placing the <span class='fss'>L.H.</span> index of C to
+the result, 1·469, on D, we read off under 50 on C, the required period
+of admission = 73·4 per cent. on D.</p>
+
+<p class='c012'>The trigonometrical scales are useful for evaluating certain
+formulæ. Thus in the following expressions, if we find the angle
+<em>a</em> such that sin. <em>a</em> = <em>k</em>, we can write:—</p>
+
+<p class='c007'><span class='fraction'><em>k</em><br><span class='vincula'>√<span class='vincula'>1 <em>k<sup>2</sup></em></span></span></span> = tan. <em>a</em>; <span class='fraction'><span class='under'>√<span class='vincula'>1 − <em>k<sup>2</sup></em></span></span><br><em>k</em></span> = cot. <em>a</em>; √<span class='vincula'>1 − <em>k<sup>2</sup></em></span> = cos. <em>a</em>; etc.</p>
+
+<p class='c007'>In the first expression, take <em>k</em> = 0·298. Place the slide with the
+sine scale outward and with its indices agreeing with the indices
+of the rule. Set the cursor to 0·298 on the (<span class='fss'>R.H.</span>) scale of A, and
+read 17° 20′ on the sine scale as the angle required. Then under
+17° 20′ on the tangent scale, read 0·312 on D as the result.</p>
+
+<div>
+  <span class='pageno' id='Page_84'>84</span>
+  <h3 class='c008'>SLIDE RULES WITH LOG.-LOG. SCALES.</h3>
+</div>
+
+<p class='c009'>For occasional requirements, the method described on page <a href='#Page_45'>45</a>
+of determining powers and roots other than the square and cube,
+is quite satisfactory. When, however, a number of such calculations
+are to be made, the process may be simplified considerably
+by the use of what are known as <em>log.-log.</em>, <em>logo-log.</em>, or <em>logometric</em>
+scales, in conjunction with the ordinary scales of the rule. The
+principle involved will be understood from a consideration of
+those rules for logarithmic computation (page <a href='#Page_8'>8</a>) which refer to
+powers and roots. From these it is seen that while for the multiplication
+and division of numbers we <em>add</em> their logarithms, for
+involution and evolution we require to <em>multiply</em> or <em>divide</em> the
+logarithms of the numbers by the exponent of the power or root
+as the case may be. Thus to find 3<sup>2.3</sup>, we have (log. 3) × 2·3 = log.
+<em>x</em>, and by the ordinary method described on page <a href='#Page_45'>45</a> we should
+determine log. 3 by the aid of the scale L on the back of the slide,
+multiply this by 2·3 by using the C and D scales in the usual
+manner, transfer the result to scale L, and read the value of <em>x</em> on
+D under 1 on C. By the simpler method, first proposed by Dr.
+P. M. Roget,<a id='r8'></a><a href='#f8' class='c019'><sup>[8]</sup></a> the multiplication of log. 3 by 2·3 is effected in the
+same way as with any two ordinary factors—<em>i.e.</em>, by adding their
+logarithms and finding the number corresponding to the resulting
+logarithm. In this case we have log. (log. 3) + log. 2·3 = log. (log.
+<em>x</em>). The first of the three terms is obviously the <em>logarithm of the
+logarithm</em> of 3, the second is the simple logarithm of 2·3, and the
+third the <em>logarithm of the logarithm of</em> the answer. Hence, if we
+have a scale so graduated that the distances from the point of
+origin represent the logarithms of the logarithms (the log.-logs.)
+of the numbers engraved upon it, then by using this in conjunction
+with the ordinary scale of logarithms, we can effect the
+required multiplication in a manner which is both expeditious
+and convenient. Slightly varying arrangements of the log.-log.
+scale, sometimes referred to as the “P line,” have been introduced
+from time to time, but latterly the increasing use of exponential
+formulæ in thermodynamic, electrical, and physical calculations
+has led to a revival of interest in Dr. Roget’s invention, and various
+arrangements of rules with log.-log. scales are now available.</p>
+
+<p class='c007'><span class='pageno' id='Page_85'>85</span><em>The Davis Log.-Log. Rule.</em>—In the rule introduced by Messrs.
+John Davis &#38; Son Limited, Derby, the log.-log. scales are placed
+upon a separate slide—a plan which has the advantage of leaving
+the rule intact for all ordinary purposes, while providing a length
+of 40 in. for the log.-log. scales.</p>
+
+<p class='c007'>In the 10 in. Davis rule one face of the slide, marked E, has
+two log.-log. scales for numbers greater than unity, the lower
+extending from 1·07 to 2, and the upper continuing the graduations
+from 2 to 1000. On the reverse face of the slide, marked
+-E, are two log.-log. scales for numbers less than unity, the upper
+extending from 0·001 to 0·5, and the lower continuing the graduations
+from 0·5 to 0·933. Both sets of scales are used in conjunction
+<em>with the lower or D scale of the rule</em>, which is to be primarily
+regarded as running from 1 to 10, and constitutes a scale of
+exponents. In the 20 in. rule the log.-log. scales are more extensive,
+and are used in conjunction with the upper or A scale of the
+rule (1 to 100); in what follows, however, the 10 in. rule is more
+particularly referred to.</p>
+
+<p class='c007'>It has been explained that on the log.-log. scale the distance of
+any numbered graduation from the point of origin represents
+the log.-log. of the number. The point of origin will obviously
+be that graduation whose log.-log. = 0. This is seen to be 10, since
+log. (log. 10) = log. 1 = 0. Hence, confining attention to the E scale,
+to locate the graduation 20, we have log. (log. 20) = log. 1·301 =
+0·11397, so that if the scale D is 25 cm. long, the distance between
+10 and 20 on the corresponding log.-log. scale would be 113·97 ÷ 4
+= 28·49 mm. For numbers less than 10 the resulting log.-logs. will
+be negative, and the distances will be spaced off from the point
+of origin in a negative direction—<em>i.e.</em>, from right to left. Thus, to
+locate the graduation 5, we have</p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>log. (log. 5) = log. 0·699 = ̅1·844; <em>i.e.</em>, −1 + 0·844 or −0·156;</div>
+  </div>
+</div>
+
+<p class='c016'>so that the graduation marked 5 would be placed 156 ÷ 4 = 39 mm.
+distant from 10 in a <em>negative</em> direction, and proceeding in a similar
+manner, the scale may be extended in either direction. In the
+-E scale, the notation runs in the reverse direction to that of the E
+scale, but in all other respects it is precisely analogous, the distance
+from the point of origin (0·1 in this case) to any graduation <em>x</em>
+representing log. [-log. <em>x</em>.]. It follows that of the similarly
+situated graduations on the two scales, those on the -E scale are
+the <em>reciprocals</em> of those on the E scale. This may be readily verified
+<span class='pageno' id='Page_86'>86</span>by setting, say, 10 on E to (<span class='fss'>R.H.</span>) 1 on D, when turning to the back
+of the rule we find 0·1 on -E agreeing with the index mark in the
+aperture at the right-hand extremity of the rule.</p>
+
+<p class='c007'>In using the log.-log. scales it is important to observe (1) that
+the values engraved on the scale are definite and unalterable (<em>e.g.</em>,
+1·2 can only be read as 1·2 and not as 120, 0·0012, etc., as with the
+ordinary scales); (2) that the upper portion of each scale should
+be regarded as forming a prolongation to the right of the lower
+portion; and (3) that immediately above any value on the lower
+portion of the scale is found the 10th power of that value on the
+upper portion of the scale. Keeping these points in view, if we
+set 1·1 on E to 1 on D we find over 2 on D the value of 1·1<sup>2</sup> = 1·21
+on E. Similarly, over 3 we find 1·1<sup>3</sup> = 1·331, and so on. Then,
+reading across the slide, we have, over 2, the value of 1·1<sup>2 × 10</sup> = 1·1<sup>20</sup>
+= 6·73, and over 3 we have 1·1<sup>3 × 10</sup> = 1·1<sup>30</sup> = 17·4. Hence the rule:—<em>To
+find the value of x<sup>n</sup>, set x on E to 1 on D, and over n on D read
+x<sup>n</sup> on E.</em></p>
+
+<p class='c007'>With the slide set as above, the 8th, 9th, etc., powers of 1·1
+cannot be read off; but it is seen that, according to (2) in the
+foregoing, the missing portion of the E scale is that part of the
+upper scale (2 to about 2·6) which is outside the rule to the left.
+Hence placing 1·1 to 10 on D, the 8th, 9th, etc., powers of 1·1 will
+be read off <em>on the upper part</em> of the E scale. In general, then,</p>
+
+<p class='c007'>If <em>x</em> on the <em>lower</em> line is set to 1 on D, then <em>x<sup>n</sup></em> is read directly
+on that line and <em>x</em><sup>10<em>n</em></sup> on the upper line.</p>
+
+<p class='c007'>If <em>x</em> on the <em>upper</em> line is set to 1 on D, then <em>x<sup>n</sup></em> is read directly
+on that line and <em>x</em><sup><em>ⁿ⁄₁₀</em></sup> on the lower line.</p>
+
+<p class='c007'>If <em>x</em> on the <em>lower</em> line is set to 10 on D, then <em>x</em><sup><em>ⁿ⁄₁₀</em></sup> is read
+directly on that line and <em>x<sup>n</sup></em> on the upper line.</p>
+
+<p class='c007'>If <em>x</em> on the <em>upper</em> line is set to 10 on D, then <em>x</em><sup><em>ⁿ⁄₁₀</em></sup> is read
+directly on that line and <em>x</em><sup><em>ⁿ⁄₁₀₀</em></sup> on the lower line.</p>
+
+<p class='c007'>These rules are conveniently exhibited in the accompanying
+diagram (Fig. <a href='#f_014'>14</a>). They are equally applicable to both the E and
+-E scales of the 10 in. rule, and include practically all the
+instruction required for determining the <em>n</em>th power or the <em>n</em>th
+root of a number. They do not apply directly to the 20 in. rule,
+however, for here the relation of the lower and upper scales will
+be <em>x<sup>n</sup></em> and <em>x</em><sup>100<em>n</em></sup>.</p>
+
+<p class='c020'><span class='pageno' id='Page_87'>87</span><span class='sc'>Ex.</span>—Find 1·167<sup>2·56</sup>.</p>
+
+<p class='c021'>Set 1·167 on E to 1 on D, and over 2·56 on D read 1·485 on E.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find 4·6<sup>1·61</sup>.</p>
+
+<p class='c021'>Set 4·6 on upper E scale to 1 on D, and over 1·61 on D read 11·7
+(11·67) on E.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find 1·4<sup>0·27</sup> and 1·4<sup>2·7</sup>.</p>
+
+<p class='c021'>Set 1·4 on E to 10 on D, and over 2·7 on D read 1·095 = 1·4<sup>0·27</sup> on
+lower E scale and 2·48 = 1·4<sup>2·7</sup> on upper E scale.</p>
+
+<div id='f_014'  class='figcenter id006'>
+<img src='images/f_014.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 14.</span></p>
+</div>
+</div>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find 46<sup>0·0184</sup> and 46<sup>0·184</sup>.</p>
+
+<p class='c021'>Set 46 on upper E scale to 10 on D, and over 1·84 on D read 1·073
+on lower E scale and 2·022 (2·0228) on upper E scale.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find 0·074<sup>1·15</sup>.</p>
+
+<p class='c021'>Using the -E scale, set 0·074 to 1 on D, and over 1·15 on D read
+0·05 on -E.</p>
+
+<p class='c012'>The method of determining the root of a number will be
+obvious from the preceding examples.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find <sup>1.4</sup>√<span class='vincula'>17</span> and <sup>14</sup>√<span class='vincula'>17</span>.</p>
+
+<p class='c021'>Set 17 on E to 1·4 on D, and over 1 on D read 7·56 on upper E
+scale and 1·224 on lower E scale.</p>
+
+<p class='c021'><span class='pageno' id='Page_88'>88</span><span class='sc'>Ex.</span>—Find <sup>0·031</sup>√<span class='vincula'>0·914</span>.</p>
+
+<p class='c021'>Set 0·914 on -E to 3·1 on D, and over 10 on D read 0·055 on
+upper -E scale.</p>
+
+<p class='c012'>When the exponent <em>n</em> is fractional, it is often possible to
+obtain the result directly with one setting of the slide. Thus to
+determine 1·135<sup>¹⁷⁄₁₆</sup> by the first method we find ¹⁷⁄₁₆ = 1·0625, and
+placing 1·135 on E to 1 on D, read 1·144 on E over 1·0625 on D.
+By the direct method we place 1·135 on the E scale on 1·6 on D,
+and over 1·7 on D read 1·144 on E. It will be seen that since the
+scale D is assumed to run from 1 to 10 we are unable to read 16
+and 17 on this scale; but it is obvious that the <em>ratios</em> <span class='fraction'>1·7<br><span class='vincula'>1·6</span></span> and <span class='fraction'>17<br><span class='vincula'>16</span></span> are
+identical, and it is with the ratio only that we are, in effect,
+concerned.</p>
+
+<p class='c007'>Since an expression of the form <em>x</em><sup>-<em>n</em></sup> = <span class='fraction'>1<br><span class='vincula'><em>x<sup>n</sup></em></span></span> or (<span class='fraction'>1<br><span class='vincula'><em>x</em></span></span>)<sup><em>n</em></sup>, the required
+value may be obtained by first determining the reciprocal of <em>x</em>
+and proceeding as before. By using both the direct and reciprocal
+log.-log. scales (E and -E) in conjunction however, the required
+value can be read directly from the rule, and the preliminary
+calculation entirely avoided. In the Davis form of rule, the result
+can be read on the -E scale, used in conjunction with the D scale
+of the rule, <em>x</em> on E being set to the index mark in the aperture in
+the back of the rule.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the value of 1·195<sup>−1·65</sup>.</p>
+
+<p class='c021'>Set 1·195 on E to the index in the left aperture in the back of the
+rule, and over 1·65 on D read 0·745 on the -E scale.</p>
+
+<p class='c012'>It may be noted in passing that the log.-log. scale affords a
+simple means for determining the logarithm or anti-logarithm of
+a number to any base. For this purpose it is necessary to set the
+base of the given system on E to 1 on D, when <em>under</em> any number
+on E will be found its logarithm on D. Thus, for common logs.,
+we set the base 10 on E to 1 on D, and under 100 we find 2, the
+required log. Similarly we read log. 20 = 1·301; log. 55 = 1·74;
+log. 550 = 2·74, etc. Reading reversely, over 1·38 on D we find its
+antilog. 24 on E; also antilog. 1·58 = 38; antilog. 1·19 = 15·5, etc.</p>
+
+<p class='c007'>For logs. of numbers under 10 we set the base 10 to 10 on D;
+hence the readings on D will be read as one-tenth their apparent
+value. Thus log. 3 = 0·477; log. 5·25 = 0·72; antilog. 0·415 = 2·6;
+antilog. 0·525 = 3.·35, etc.</p>
+
+<p class='c007'><span class='pageno' id='Page_89'>89</span>The logs. of the numbers on the lower half of the E scale
+will also be found on the D scale; but a consideration of
+Fig. <a href='#f_014'>14</a> will show that this will be read as <em>one-tenth</em> its face
+value if the base is set to 1 on D, and as <em>one-hundredth</em> if the
+base is set to 10.</p>
+
+<p class='c007'>For natural, hyperbolic, or Napierian logarithms, the base is
+2·718. A special line marked ε or <em>e</em> serves to locate the exact
+position of this value on the E scale, and placing this to 1 on D
+we read log.<sub><em>e</em></sub> 4·35 = 1·47; log.<sub><em>e</em></sub> 7·4 = 2·0; antilog.<sub><em>e</em></sub> × 2·89 = 18, etc.
+The other parts of the scale are read as already described for
+common logs. Calculations involving powers of <em>e</em> are frequently
+met with, and these are facilitated by using the special graduation
+line referred to, as will be readily understood.</p>
+
+<p class='c007'>If it is required to determine the power or root of a number
+which does not appear on either of the log.-log. scales, we may
+break up the number into factors. Usually it is convenient to
+make one of the factors a power of 10.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—3950<sup>1·97</sup> = 3·95<sup>1·97</sup> × 10<sup>3 × 1·97</sup> = 3·95<sup>1·97</sup> × 10<sup>5·91</sup>.</p>
+
+<p class='c021'>Then 3·95<sup>1·97</sup> = 15, and 10<sup>5·91</sup> (or antilog.) 5·91 = 812,000. Hence,
+15 × 812,000 = 12,180,000 is the result sought.</p>
+
+<p class='c012'>Numbers which are to be found in the higher part of the log.-log.
+scale may often be factorised in this way, and greater accuracy
+obtained than by direct reading.</p>
+
+<p class='c007'>The form of log.-log. rule which has been mainly dealt with in
+the foregoing gives a scale of comparatively long range, and the
+only objection to the arrangement adopted is the use of a separate
+slide.</p>
+
+<p class='c007'><em>The Jackson-Davis Double Slide Rule.</em>—In this instrument a
+pair of aluminium clips enable the log.-log. slide to be temporarily
+attached to the lower edge of the ordinary rule, and used, by
+means of a special cursor, in conjunction with the C scale of the
+ordinary slide. In this way both the log.-log. and ordinary scales
+are available without the trouble of replacing one slide by the
+other. Since the scale of exponents is now on the slide, the value
+of <em>x<sup>n</sup></em> will be obtained by setting 1 on C to <em>x</em> on E and reading the
+result on E under <em>n</em> on C.</p>
+
+<p class='c007'>By using a pair of log.-log. slides, one in the rule and one
+clamped to the edge by the clips, we have an arrangement which
+is very useful in deducing empirical formulæ of the type <em>y</em> = <em>x<sup>n</sup></em>.</p>
+
+<p class='c007'><span class='pageno' id='Page_90'>90</span><em>The Yokota Slide Rule.</em>—In this instrument the log.-log. scales
+are placed on the face of the rule, each set comprising three lines.
+These, for numbers greater than 1, are found above the A scale
+while the three reciprocal log.-log. lines are below the D scale.
+Both sets are used in conjunction with the C scale on the slide.
+Other features of this rule are:—The ordinary scales are 10 in. long
+instead of 25 cm. as hitherto usual; hence the logarithms of numbers
+can be read on the ordinary scale of inches on the edge of
+the rule. There is a scale of cubes in the centre of the slide and
+on the back of the slide there is a scale of secants in addition to
+the sine and tangent scales.</p>
+
+<div id='f_015'  class='figcenter id001'>
+<img src='images/f_015.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 15.</span></p>
+</div>
+</div>
+
+<p class='c007'><em>The Faber Log.-log. Rule.</em>—In this instrument shown in Fig. <a href='#f_015'>15</a>,
+the two log.-log. scales are placed on the face of the rule. One
+section, extending from 1·1 to 2·9, is placed above the A scale, and
+the other section, extending from 2·9 to 100,000, is placed below
+the D scale. These scales are used in conjunction with the C
+scale of the slide in the manner previously described. The width
+of the rule is increased slightly, but the arrangement is more
+convenient than that formerly employed, wherein the log.-log.
+scales were placed on the bevelled edge of the rule and read by
+a tongue projecting from the cursor.</p>
+
+<div id='f_016'  class='figcenter id001'>
+<img src='images/f_016.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 16.</span></p>
+</div>
+</div>
+
+<p class='c007'><span class='pageno' id='Page_91'>91</span>Another novel feature of this rule is the provision of two
+special scales at the bottom of the groove, to which a bevelled
+metal index or marker on the left end of the slide can be set.
+The upper of these scales is for determining the efficiency of
+dynamos and electric motors; the lower for determining the loss
+of potential in an electric circuit.</p>
+
+<p class='c007'><em>The Perry Log.-log. Rule.</em>—In this rule, introduced by Messrs.
+A. G. Thornton, Limited, Manchester, the log.-log. scales are arranged
+as in Fig. <a href='#f_016'>16</a>, the E scale, running from 1·1 to 10,000, being placed
+above the A scale of the rule, and the -E or E<sup>−1</sup> scale running from
+0·93 to 0·0001, below the D scale of the rule. These scales are read
+in conjunction with the B scales on the slide by the aid of the cursor.</p>
+
+<p class='c007'>The following tabular statement embodies all the instructions
+required for using this form of log.-log. slide rule:—</p>
+
+<table class='table1'>
+  <tr><th class='c028' colspan='2'>When <em>x</em> is greater than 1.</th></tr>
+  <tr>
+    <td class='c041'>&#160;</td>
+    <td class='c042'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c041'><em>x<sup>n</sup></em></td>
+    <td class='c042'>Set 1 on B to <em>x</em> on E; over <em>n</em> on B read <em>x<sup>n</sup></em> on E</td>
+  </tr>
+  <tr>
+    <td class='c041'><em>x</em><sup>-<em>n</em></sup></td>
+    <td class='c042'>Set 1 on B to <em>x</em> on E; under <em>n</em> on B read <em>x</em><sup>-<em>n</em></sup> on E<sup>−1</sup></td>
+  </tr>
+  <tr>
+    <td class='c041'><em>x</em><sup><em>ⁱ⁄ₙ</em></sup></td>
+    <td class='c042'>Set <em>n</em> on B to <em>x</em> on E; over 1 on B read <em>x</em><sup><em>ⁱ⁄ₙ</em></sup> on E</td>
+  </tr>
+  <tr>
+    <td class='c041'><em>x</em><sup><em>⁻ⁱ⁄ₙ</em></sup></td>
+    <td class='c042'>Set <em>n</em> on B to <em>x</em> on E; under 1 on B read <em>x</em><sup><em>⁻ⁱ⁄ₙ</em></sup> on E<sup>−1</sup></td>
+  </tr>
+  <tr>
+    <td class='c041'>&#160;</td>
+    <td class='c042'>&#160;</td>
+  </tr>
+  <tr><th class='c028' colspan='2'>When <em>x</em> is less than 1.</th></tr>
+  <tr>
+    <td class='c041'>&#160;</td>
+    <td class='c042'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c041'><em>x<sup>n</sup></em></td>
+    <td class='c042'>Set 1 on B to <em>x</em> on E<sup>−1</sup>; under <em>n</em> on B read <em>x<sup>n</sup></em> on E<sup>−1</sup></td>
+  </tr>
+  <tr>
+    <td class='c041'><em>x</em><sup>-<em>n</em></sup></td>
+    <td class='c042'>Set 1 on B to <em>x</em> on E<sup>−1</sup>; over <em>n</em> on B read <em>x</em><sup>-<em>n</em></sup> on E</td>
+  </tr>
+  <tr>
+    <td class='c041'><em>x</em><sup><em>ⁱ⁄ₙ</em></sup></td>
+    <td class='c042'>Set <em>n</em> on B to <em>x</em> on E<sup>−1</sup>; under 1 on B read <em>x</em><sup><em>ⁱ⁄ₙ</em></sup> on E<sup>−1</sup></td>
+  </tr>
+  <tr>
+    <td class='c041'><em>x</em><sup><em>⁻ⁱ⁄ₙ</em></sup></td>
+    <td class='c042'>Set <em>n</em> on B to <em>x</em> on E<sup>−1</sup>; over 1 on B read <em>x</em><sup><em>⁻ⁱ⁄ₙ</em></sup> on E</td>
+  </tr>
+</table>
+
+<p class='c007'>If 10 on B is used in place of 1 on B, read <em>x</em><sup><em>ⁿ⁄₁₀</em></sup> in place of
+<em>x<sup>n</sup></em> on E, and <em>x</em><sup>-<em>ⁿ⁄₁₀</em></sup> in place of <em>x</em><sup>-<em>n</em></sup> on E<sup>−1</sup>. If 100 on B is used, these
+readings are to be taken as <em>x</em><sup><em>ⁿ⁄₁₀₀</em></sup> and <em>x</em><sup>-<em>ⁿ⁄₁₀₀</em></sup> respectively.</p>
+
+<p class='c007'>In rules with no -E scale the value of <em>x</em><sup>-<em>n</em></sup> is obtained by the
+usual rules for reciprocals. We may either determine <em>x<sup>n</sup></em> and find
+its reciprocal or, first find the reciprocal of <em>x</em> and raise it to the
+<em>n</em>th power. The first method should be followed when the
+number <em>x</em> is found on the E scale.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—3·45<sup>−1·82</sup> = 0·105.</p>
+
+<p class='c021'>Set 1 on C to 3·45 on E, and under 1·82 on C read 9·51 on C.
+Then set 1 on B to 9·5 on A, and under index of A read 0·105 on B.</p>
+
+<p class='c012'><span class='pageno' id='Page_92'>92</span>When <em>x</em> is less than 1 the second method is more suitable.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—0·23<sup>−1·77</sup> = (<span class='fraction'>1<br><span class='vincula'>0·23</span></span>)<sup>1·77</sup> = 4·35<sup>1·77</sup> = 13·5</p>
+
+<p class='c021'>Set 1 on B to 0·23 on A, and under index of A read <span class='fraction'>1<br><span class='vincula'>0·23</span></span> = 4·35 on B.</p>
+
+<p class='c021'>Set 1 on C to 4·35 on E, and under 1·77 on C read 13·5 on E.</p>
+
+<p class='c012'>As with the Davis rule, the exponent scale C will be read as
+⅒th its face value if its <span class='fss'>R.H.</span> index (10) is used in place of 1.</p>
+
+<h3 class='c008'>SPECIAL TYPES OF SLIDE RULES.</h3>
+
+<p class='c009'>In addition, to the new forms of log.-log. slide rules previously
+described, several other arrangements have been recently
+introduced, notably a series by Mr. A. Nestler, of Lahr
+(London: A. Fastlinger, Snow Hill). These comprise the
+“Rietz,” the “Precision,” the “Universal,” and the “Fix”
+slide rules.</p>
+
+<p class='c007'><span class='sc'>The Rietz Rule.</span>—In this rule the usual scales A, B, C, and D,
+are provided, while at the upper edge is a scale, which, being three
+times the range of the D scale, enables cubes and cube roots to be
+directly evaluated and also <em>n</em><sup>³⁄₂</sup> and <em>n</em><sup>⅔</sup>.</p>
+
+<p class='c007'>A scale at the lower edge of the rule gives the mantissa of the
+logarithms of the numbers on D.</p>
+
+<p class='c007'><span class='sc'>The Precision Slide Rule.</span>—In this rule the scales are so
+arranged that the accuracy of a 20 in. rule is obtainable in a length
+of 10 in. This is effected by dividing a 20 in. (50 cm.) scale length
+into two parts and placing these on the working edges of the rule
+and slide. On the upper and lower margins of the face of the
+rule are the two parts of what corresponds to the A scale in the
+ordinary rule; while in the centre of the slide is the scale of
+logarithms which, used in conjunction with the 50 cm. scales on
+the slide, is virtually twice the length of that ordinarily obtainable
+in a 10 in. rule. The same remark applies to the trigonometrical
+scales on the under face of the slide. Both the sine and tangent
+scales are in two adjacent lengths, while on the edge of the stock
+of the rule, below the cursor groove, is a scale of sines of small
+angles from 1° 49′ to 5° 44′. This is referred to the 50 cm. scales
+by an index projection on the cursor.</p>
+
+<p class='c007'>If C and C′ are the two parts of the scale on the slide and
+D and D′ the corresponding scales on the rule, it is clear that in
+<span class='pageno' id='Page_93'>93</span>multiplying two factors 1 on C can only be set directly to the
+upper scale D; while 10 on C′ can only be set directly to the lower
+scale D′. Hence if the first factor is greater than about 3·2, the
+cursor must be used to bring 1 on C to the first factor on D′.
+Similarly, in division, numerators and denominators which occur
+on C and D′ or on C′ and D cannot be placed in direct coincidence
+but must be set by the aid of the cursor.</p>
+
+<p class='c007'>Any uncertainty in reading the result can be avoided by
+observing the following rule: <em>If in setting the index</em> (1 <em>or</em> 10) <em>in
+multiplication, or in setting the numerator to the denominator in
+division, it is necessary to cross the slide, then it will also be necessary
+to cross the slide to read the product or quotient.</em></p>
+
+<p class='c007'><span class='sc'>The Universal Slide Rule.</span>—In this instrument the stock
+carries two similar scales running from 1 to 10, to which the slide
+can be set. Above the upper one is the logarithm scale and
+under the lower one the scale of squares 1 to 100. On the edge
+of the stock of the rule, under the cursor groove, is a scale
+running from 1 to 1000. An index projecting from the cursor
+enables this scale to be used with the scales on the face of the
+rule, giving cubes, cube roots, etc.</p>
+
+<p class='c007'>On the slide, the lower scale is an ordinary scale, 1 to 10. The
+centre scale is the first part of a scale giving the values of sin <em>n</em>
+cos <em>n</em>, this scale being continued along the upper edge of the slide
+(marked “sin-cos”) up to the graduation 50. On the remainder of
+this line is a scale running from right to left (0 to 50) and giving
+the value of cos<sup>2</sup><em>n</em>. In surveying, these scales greatly facilitate the
+calculations for the horizontal distance between the observer’s
+station and any point, and the difference in height of these two
+points.</p>
+
+<p class='c007'>On the back of the slide are scales for the sines and tangents
+of angles. The values of the sines and tangents of angles from
+34′ to 5° 44′ differ little from one another, and the one centre scale
+suffices for both functions of these small angles.</p>
+
+<p class='c007'><span class='sc'>The Fix Slide Rule.</span>—This is a standard rule in all respects,
+except that the A scale is displaced by a distance <span class='fraction'>π<br><span class='vincula'>4</span></span> so that over
+1 on D is found 0·7854 on A. This enables calculations relating
+to the area and cubic contents of cylinders to be determined very
+readily.</p>
+
+<p class='c007'><span class='pageno' id='Page_94'>94</span><span class='sc'>The Beghin Slide Rule.</span>—We have seen that a disadvantage
+attending the use of the ordinary C and D scales, is that it is
+occasionally necessary to traverse the slide through its own
+length in order to change the indices or to bring other parts of
+the slide into a readable position with regard to the stock. To
+obviate this disadvantage, Tserepachinsky devised an ingenious
+arrangement which has since been used in various rules, notably
+in the Beghin slide rule made by Messrs. Tavernier-Gravêt of
+Paris. In this rule the C and D scales are used as in the standard
+rule, but in place of the A and B scales, we have another pair of
+C and D scales, displaced by one-half the length of the rule.
+The lower pair of scales may therefore be regarded as running
+from 10<sup><em>n</em></sup> to 10<sup><em>n</em> + 1</sup>, and the upper pair as running from √<span class='vincula'>10</span> × 10<sup><em>n</em></sup> to
+√<span class='vincula'>10</span> × 10<sup><em>n</em> + 1</sup>. With this arrangement, <em>without moving the slide
+more than half its length</em>, to the left or right, it is always possible
+to compare <em>all values between</em> 1 <em>and</em> 10 <em>on the two scales</em>. This is
+a great advantage especially in continuous working.</p>
+
+<p class='c007'>Another commendable feature of the Beghin rule is the
+presence of a reversed C scale in the centre of the slide,
+thus enabling such calculations as <em>a</em> × <em>b</em> × <em>c</em> to be made with
+one setting of the slide. On the back of the slide are three
+scales, the lowest of which, used with the D scale, is a scale of
+squares (corresponding to the ordinary B scale), while on the
+upper edge is a scale of sines from 5° 44′ to 90°, and in the
+centre, a scale of tangents from 5° 43′ to 45°. On the square edge
+of the stock, under the cursor groove, is the logarithm scale,
+while on the same edge, above the cursor groove, are a series
+of gauge points. All these values are referred to the face
+scales by index marks on the cursor.</p>
+
+<p class='c007'><span class='sc'>The Anderson Slide Rule.</span>—The principle of dividing a long
+scale into sections as in the Precision rule, has been extended in
+the Anderson slide rule made by Messrs. Casella &#38; Co., London,
+and shown in Fig. <a href='#f_017'>17</a>. In this the slide carries a scale in four
+sections, used in conjunction with an exactly similar set of scale-lines
+in the upper part of the stock. On the lower part of the
+stock is a scale in eight sections giving the square roots of the
+upper values. In order to set the index of the slide to values in
+the stock, two indices of transparent celluloid are fixed to the
+slide extending over the face of the rule as shown in the illustration.
+As each scale section is 30 cm. in length, the upper lines
+<span class='pageno' id='Page_95'>95</span>correspond to a single scale of nearly 4 ft., and
+the lower set to one of nearly 8 ft. in length,
+giving a correspondingly large increase in the
+number of subdivisions of these scales, and
+consequently much greater accuracy.</p>
+
+<p class='c007'>In order to decide upon which line a result is
+to be found, sets of “line numbers” are marked
+at each end of the rule and slide and also on the
+metal frame of the cursor. In multiplication, the
+line number of the product is the sum of the line
+numbers of the factors if the left index is used,
+or 1 more than this sum if the right index is
+used. The illustration shows the multiplication
+of 2 by 4. The left index is set to 2 (line
+number, 1), and the cursor set to 4 on the slide
+(line number, 2); hence, as the left index is
+used, the result is found on line No. 3. Similar
+rules are readily established for division. The
+column of line numbers headed 0 is used for
+units, that headed 4 for tens, and so on; one
+column is given for tenths, headed −4. The
+square root scale bears similar line numbers, so
+that the square root of any value on the upper
+scales is found on the correspondingly figured
+line below.</p>
+
+<div id='f_017'  class='figcenter id001'>
+<img src='images/f_017.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 17.</span></p>
+</div>
+</div>
+
+<p class='c007'><span class='sc'>The Multiplex Slide Rule</span> differs from the
+ordinary form of rule in the arrangement of the
+B scale. The right-hand section of this scale
+runs from left to right as ordinarily arranged,
+but the left-hand section runs in the reverse
+direction, and so furnishes a reciprocal scale.
+At the bottom of the groove, under the slide,
+there is a scale running from 1 to 1000, which
+is used in conjunction with the D scale, readings
+being referred thereto by a metal index on the
+end of the slide. By this means cubes, cube
+roots, etc., can be read off directly. Messrs.
+Eugene Dietzgen &#38; Co., New York, are the
+makers.</p>
+
+<p class='c007'><span class='pageno' id='Page_96'>96</span><span class='sc'>The “Long” Slide Rule</span> has one scale in two sections along
+the upper and lower parts of the stock, as in the “Precision” rule.
+The scale on the slide is similarly divided, but the graduations run
+in the reverse direction, corresponding to an inverted slide. Hence
+the rules for multiplication and division are the reverse of those
+usually followed (page <a href='#Page_30'>30</a>). On the back of the slide is a single
+scale 1–10, and a scale 1–1000, giving cubes of this single scale.
+By using the first in conjunction with the scales on the stock,
+squares may be read, while in conjunction with the cube scale,
+various expressions involving squares, cubes and their roots may
+be evaluated.</p>
+
+<p class='c007'><span class='sc'>Hall’s Nautical Slide Rule</span> consists of two slides fitting in
+grooves in the stock, and provided with eight scales, two on each
+slide, and one on each edge of each groove. While fulfilling the
+purposes of an ordinary slide rule, it is of especial service to the
+practical navigator in connection with such problems as the
+“reduction of an ex-meridian sight” and the “correction of
+chronometer sights for error in latitude.” The rule, which has
+many other applications of a similar character, is made by Mr. J.
+H. Steward, Strand, London.</p>
+
+<h3 class='c008'>LONG-SCALE SLIDE RULES</h3>
+
+<p class='c009'>It has been shown that the degree of accuracy attainable in
+slide-rule calculations depends upon the length of scale employed.
+Considerations of general convenience, however, render simple
+straight-scale rules of more than 20 in. in length inadmissible, so
+that inventors of long-scale slide rules, in order to obtain a high
+degree of precision, combined with convenience in operation, have
+been compelled to modify the arrangement of scales usually
+employed. The principal methods adopted may be classed under
+three varieties: (1) The use of a long scale in sectional lengths, as
+in Hannyngton’s Extended Slide Rule and Thacher’s Calculating
+Instrument; (2) the employment of a long scale laid in spiral form
+upon a disc, as in Fearnley’s Universal Calculator and Schuerman’s
+Calculating Instrument; and (3) the adoption of a long scale
+wound helically upon a cylinder, of which Fuller’s and the
+“R.H.S.” Calculating Rules are examples.</p>
+
+<p class='c007'><span class='sc'>Fuller’s Calculating Rule.</span>—This instrument, which is
+shown in Fig. <a href='#f_018'>18</a>, consists of a cylinder <em>d</em> capable of being moved
+<span class='pageno' id='Page_97'>97</span>up and down and around the cylindrical stock <em>f</em>, which is
+held by the handle. The logarithmic scale-line is arranged
+in the form of a helix upon the surface of the cylinder <em>d</em>, and
+as it is equivalent to a straight scale of 500 inches, or 41 ft.
+8 in., it is possible to obtain four, and frequently five, figures
+in a result.</p>
+
+<p class='c007'>Upon reference to the figure it will be
+seen that three indices are employed. Of these,
+that lettered <em>b</em> is fixed to the handle; while
+two others, <em>c</em> and <em>a</em> (whose distance apart is
+equal to the axial length of the complete helix),
+are fixed to the innermost cylinder <em>g</em>. This
+latter cylinder slides telescopically in the stock
+<em>f</em>, enabling the indices to be placed in any
+required position relatively to <em>d</em>. Two other
+scales are provided, one (<em>m</em>) at the upper end
+of the cylinder <em>d</em>, and the other (<em>n</em>) on the
+movable index.</p>
+
+<div id='f_018'  class='figright id005'>
+<img src='images/f_018.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 18.</span></p>
+</div>
+</div>
+
+<p class='c007'>In using the instrument a given number on <em>d</em>
+is set to the fixed index <em>b</em>, and either <em>a</em> or <em>c</em> is
+brought to another number on the scale. This
+establishes a ratio, and if the cylinder is now
+moved so as to bring any number to <em>b</em>, the fourth
+term of the proportion will be found under <em>a</em>
+or <em>c</em>. Of course, in multiplication, one factor
+is brought to <em>b</em>, and <em>a</em> or <em>c</em> brought to 100. The
+other factor is then brought to <em>a</em> or <em>c</em>, and
+the result read off under <em>b</em>. Problems involving
+continuous multiplication, or combined multiplication
+and division, are very readily dealt
+with. Thus, calling the fixed index F, the
+upper movable index A, and the lower movable
+index B, we have for <em>a</em> × <em>b</em> × <em>c</em>:—Bring <em>a</em> to F;
+A to 100; <em>b</em> to A or B; A to 100; <em>c</em> to A or B and read the
+product at F.</p>
+
+<p class='c007'>The maximum number of figures in a product is the sum
+of the number of figures in the factors and this results
+when all the factors except the first have to be brought to B.
+Each time a factor is brought to A, 1 is to be deducted from
+that sum.</p>
+
+<p class='c007'><span class='pageno' id='Page_98'>98</span>For division, as <em>a</em>/(<em>m</em> × <em>n</em>), bring <em>a</em> to F; A or B to <em>m</em>; 100 to A;
+A or B to <em>a</em>; 100 to A and read the quotient at F.</p>
+
+<div id='f_019'  class='figleft id005'>
+<img src='images/f_019.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 19.</span></p>
+</div>
+</div>
+
+<p class='c007'>The maximum number of figures in the quotient is the difference
+between the sum of the number of figures in
+the numerator factors and those of the denominator
+factors, <em>plus</em> 1 for each factor of the denominator and
+this results when A has to be set to all the factors
+of the denominator and all the factors of the
+numerator except the first brought to B. Each time
+B is set to a denominator factor or a numerator
+factor is brought to A, 1 is to be deducted.</p>
+
+<p class='c007'>Logarithms of numbers are obtained by using
+the scales <em>m</em> and <em>n</em> and hence powers and roots of
+any magnitude may be obtained by the procedure
+already fully explained. The instrument illustrated
+is made by Messrs. W. F. Stanley &#38; Co., Limited,
+London.</p>
+
+<p class='c007'><span class='sc'>The “R.H.S.” Calculator.</span>—In this calculator,
+designed by Prof. R. H. Smith, the scale-line, which
+is 50 in. long, is also arranged in a spiral form (Fig.
+<a href='#f_019'>19</a>), but in this case it is wrapped around the central
+portion of a tube which is about ¾in. in diameter
+and 9½in. long. A slotted holder, capable of sliding
+upon the plain portions of this tube, is provided
+with four horns, these being formed at the ends of
+the two wide openings through which the scale is
+read. An outer ring carrying two horns completes
+the arrangement.</p>
+
+<p class='c007'>One of the horns of the holder being placed in
+agreement with the first factor, and one of the horns
+of the ring with the second factor, the holder is
+moved until the third factor falls under the same
+horn of the ring, when the resulting fourth term will
+be found under the same (right or left) horn of
+the holder, at either end of the slot. In multiplication, 100 or
+1000 is taken for the second factor in the above proportion,
+as already explained in connection with Fuller’s rule; indeed,
+generally, the mode of operation is essentially similar to that
+followed with the former instrument.</p>
+
+<p class='c007'><span class='pageno' id='Page_99'>99</span>The scale shown on one edge of the opening in the holder,
+together with the circular scale at the top of the spiral, enables the
+mantissæ of logarithms of numbers to be obtained, and thus
+problems involving powers and roots may be dealt with quite
+readily. This instrument is supplied by Mr. J. H. Steward,
+London.</p>
+
+<p class='c007'><span class='sc'>Thacher’s Calculating Instrument</span>, shown in Fig. <a href='#f_020'>20</a>, consists
+of a cylinder 4 in. in diameter and 18 in. long, which can be given
+both a rotary and a longitudinal movement within an open framework
+composed of twenty triangular bars. These bars are connected
+to rings at their ends, which can be rotated in standards
+fixed to the baseboard. The scale on the cylinder consists of forty
+sectional lengths, but of each scale-line that part which appears on
+the right-hand half of the cylinder is repeated on the left-hand
+half, one line in advance. Hence each half of the cylinder virtually
+contains two complete scales following round in regular order. On
+the lower lines of the triangular bars are scales exactly corresponding
+to those on the cylinder, while upon the upper lines of the bars
+and not in contact with the slide is a scale of square roots.</p>
+
+<div id='f_020'  class='figcenter id001'>
+<img src='images/f_020.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 20.</span></p>
+</div>
+</div>
+
+<p class='c007'>By rotating the slide any line on it may be brought opposite
+any line in frame and by a longitudinal movement any graduation
+on these lines may be brought into agreement. The whole can be
+rotated in the supporting standards in order to bring any reading
+into view. As shown in the illustration, a magnifier is provided,
+this being conveniently mounted on a bar, along which it can be
+moved as required.</p>
+
+<p class='c007'><span class='sc'>Sectional Length or Gridiron Slide Rules.</span>—The idea of
+breaking up a long scale into sectional lengths is due to Dr. J. D.
+Everett, who described such a gridiron type of slide rule in 1866.
+<span class='pageno' id='Page_100'>100</span>Hannyngton’s Extended Slide Rule is on the same principle. Both
+instruments have the lower scale repeated. H. Cherry (1880)
+appears to have been the first to show that such duplication could
+be avoided by providing two fixed index points in addition to the
+natural indices of the scale. These additional indices are shown at
+10′ and 100′ in Fig. <a href='#f_021'>21</a>, which represents the lower sheet of Cherry’s
+Calculator on a reduced scale. The upper member of the calculator
+consists of a transparent sheet ruled with parallel lines, which
+coincide with the lines of the lower scale when the indices of both
+are placed in agreement. To multiply one number by another, one
+of the indices on the upper sheet is placed to one of the factors,
+and the position of whichever index falls under the transparent
+sheet is noted on the latter. Bringing the latter point to the other
+factor, the result is found under whichever index lies on the card.
+In other arrangements the inventor used transparent scales, the
+graduations running in a reverse direction to those of the lower
+scale. In this case, a factor on the upper scale is set to the
+other factor on the lower, and the result read at the available
+index.</p>
+
+<div id='f_021'  class='figcenter id001'>
+<img src='images/f_021.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 21.</span></p>
+</div>
+</div>
+
+<p class='c007'><span class='sc'>Proell’s Pocket Calculator</span> is an application of the last-named
+principle. It comprises a lower card arranged as Fig. <a href='#f_021'>21</a>,
+with an upper sheet of transparent celluloid on which is a similar
+scale running in the reverse direction. For continued multiplication
+and division, a needle (supplied with the instrument) is used as a
+substitute for a cursor, to fix the position of the intermediate
+results. A series of index points on the lower card enable square
+and cube roots to be extracted very readily. This calculator is
+supplied by Messrs. John J. Griffin &#38; Sons, Ltd., London.</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_101'>101</span>
+  <h2 class='c005'>CIRCULAR CALCULATORS.</h2>
+</div>
+
+<p class='c012'>Although the 10 in. slide rule is probably the most serviceable
+form of calculating instrument for general purposes, many prefer
+the more portable circular calculator, of which many varieties have
+been introduced during recent years. The advantages of this type
+are: It is more compact and conveniently carried in the waistcoat
+pocket. The scales are continuous, so that no traversing of
+the slide from 1 to 10 is required. The dial can be set quickly to
+any value; there is no trouble with tight or ill-fitting slides. The
+disadvantages of most forms are: Many problems involve more
+operations than a straight rule. The results being read under
+fingers or pointers, an error due to parallax is introduced, so that
+the results generally are not so accurate as with a straight rule.
+The inner scales are short, and therefore are read with less
+accuracy. Special scale circles are needed for cubes and cube
+roots. The slide cannot be reversed or inverted.</p>
+
+<div id='f_022'  class='figleft id005'>
+<img src='images/f_022.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 22.</span></p>
+</div>
+</div>
+
+<div id='f_023'  class='figright id005'>
+<img src='images/f_023.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 23.</span></p>
+</div>
+</div>
+
+<p class='c007'><span class='sc'>The Boucher Calculator.</span>—This circular calculator resembles
+a stem-winding watch, being about 2 in. in diameter and ⁹⁄₁₆in. in
+thickness. The instrument has two dials, the back one being fixed,
+while the front one, Fig. <a href='#f_022'>22</a> (showing the form made by Messrs.
+W. F. Stanley, London), turns upon the large centre arbor shown.
+This movement is effected by turning the milled head of the stem-winder.
+The small centre axis, which is turned by rotating the
+milled head at the side of the case, carries two fine needle pointers,
+<span class='pageno' id='Page_102'>102</span>one moving over each dial, and so fixed on the axis that one pointer
+always lies evenly over the other. A fine index or pointer fixed
+to the case in line with the axis of the winding stem, extends over
+the four scales of the movable dial as shown. Of these scales, the
+second from the outer is the ordinary logarithmic scale, which in
+this instrument corresponds to a straight scale of about 4¾in. in
+length. The two inner circles give the square roots of the numbers
+on the primary logarithmic scale, the smaller circle containing the
+square roots of values between 1 and 3·162 (= √<span class='vincula'>10</span>), while the
+other section corresponds to values between 3·162 and 10. The
+outer circle is a scale of logarithms of sines of angles, the
+corresponding sines of which can be read off on the ordinary
+scale.</p>
+
+<p class='c007'>On the fixed or back dial there are also four scales, these
+being arranged as in Fig. <a href='#f_023'>23</a>. The outer of these is a scale of equal
+parts, while the three inner scales are separate sections of a scale
+giving the cube roots of the numbers taken on the ordinary
+logarithmic scale and referred thereto by means of the pointers.
+In dividing this cube-root scale into sections, the same method is
+adopted as in the case of the square-root scale. Thus, the smallest
+circle contains the cube roots of numbers between 1 and 10, and is
+therefore graduated from 1 to 2·154; the second circle contains
+the cube roots of numbers between 10 and 100, being graduated
+from 2·154 to 4·657; while the third section, in which are found
+the cube roots of numbers between 100 and 1000, carries the
+graduations from 4·657 to 10.</p>
+
+<p class='c007'>What has been said in an earlier section regarding the notation
+of the slide rule may in general be taken to apply to the scales of
+the Boucher calculator. The manner of using the instrument is,
+however, not quite so evident, although from what follows it will
+be seen that the operative principle—that of variously combining
+lengths of a logarithmic scale—is essentially similar. In this case,
+however, it is seen that in place of the straight scale-lengths shown
+in Fig. <a href='#f_004'>4</a>, we require to add or subtract arc-lengths of the circular
+scales, while, further, it is evident that in the absence of a fixed
+scale (corresponding to the stock of the slide rule) these operations
+cannot be directly performed as in the ordinary form of instrument.
+However, by the aid of the fixed index and the movable pointer,
+we can effect the desired combination of the scale-lengths in the
+following manner. Assuming it is desired to multiply 2 by 3, the
+<span class='pageno' id='Page_103'>103</span>dial is turned in a backward direction until 2 on the ordinary scale
+lies under the fixed index, after which the movable pointer is set
+to 1 on the scale. As now set, it is clear that the arc-length 1–2
+is spaced off between the fixed index and the movable pointer, and
+it now only remains to add to this definite arc-length a further
+length of 1–3. To do this we turn the dial still further backward
+until the arc 1–3 has passed under the movable pointer, when the
+result, 6, is read under the fixed index. A little consideration
+will show that any other scale length may be added to that
+included between the fixed and movable pointers, or, in other
+words, any number on the scale may be multiplied by 2 by bringing
+the number to the movable pointer and reading the result
+under the fixed index. The rule for multiplication is now
+evident.</p>
+
+<p class='c007'><em>Rule for Multiplication.</em>—<em>Set one factor to the fixed index
+and bring the pointer to 1 on the scale; set the other factor to the
+pointer and read the result under the fixed index.</em></p>
+
+<p class='c007'>With the explanation just given, the process of division needs
+little explanation. It is clear that to divide 6 by 3, an arc-length
+1–3 is to be taken from a length 1–6. To this end we set 6 to
+the index (corresponding in effect to passing a length 1–6 to the
+left of that reference point) and set the pointer to the divisor 3.
+As now set, the arc 1–6 is included between 1 on the scale and the
+index, while the arc 1–3 is included between 1 on the scale and
+the pointer. Obviously if the dial is now turned forward until 1
+on the scale agrees with the pointer, an arc 1–3 will have been
+deducted from the larger arc 1–6, and the remainder, representing
+the result of this operation, will be read under the index
+as 2.</p>
+
+<p class='c007'><em>Rule for Division.</em>—<em>Set the dividend to the fixed index, and the
+pointer to the divisor; turn the dial until 1 on the scale agrees with
+the pointer, and read the result under the fixed index.</em></p>
+
+<p class='c007'>The foregoing method being an inversion of the rule for multiplication,
+is easily remembered and is generally advised. Another
+plan is, however, preferable when a series of divisions are to be
+effected with a constant divisor—<em>i.e.</em>, when <em>b</em> in <span class='fraction'><span class='under'><em>a</em></span><br><em>b</em></span> = <em>x</em> is constant.
+In this case 1 on the scale is set to the index and the pointer set
+to <em>b</em>; then if any value of a is brought to the pointer, the quotient
+<em>x</em> will be found under the index.</p>
+
+<p class='c007'><span class='pageno' id='Page_104'>104</span><em>Combined Multiplication and Division</em>, as <span class='fraction'><span class='under'><em>a</em> × <em>b</em> × <em>c</em></span><br><em>m</em> × <em>n</em></span> = <em>x</em>, can be
+readily performed, while cases of continued multiplication evidently
+come under the same category, since <em>a</em> × <em>b</em> × <em>c</em> = <span class='fraction'><span class='under'><em>a</em> × <em>b</em> × <em>c</em></span><br>1 × 1</span> = <em>x</em>.
+Such cases as <em>a</em>/(<em>m</em> × <em>n</em> × <em>r</em>) = <em>x</em> are regarded as <span class='fraction'><span class='under'><em>a</em> × 1 × 1 × 1</span><br><em>m</em> × <em>n</em> × <em>r</em></span> = <em>x</em>; while
+<span class='fraction'><span class='under'><em>a</em> × <em>b</em> × <em>c</em></span><br><em>m</em></span> = <em>x</em> is similarly modified, taking the form <span class='fraction'><span class='under'><em>a</em> × <em>b</em> × <em>c</em></span><br><em>m</em> × 1</span> = <em>x</em>. In
+all cases the expression must be arranged so that there is <em>one
+more factor in the numerator</em> than <em>in the denominator</em>, <em>1’s being
+introduced as often as required</em>. The simple operations of multiplication
+and division involve a similar disposition of factors, since
+from the rules given it is evident that <em>m</em> × <em>n</em> is actually regarded as
+<span class='fraction'><span class='under'><em>m</em> × <em>n</em></span><br>1</span>, while <span class='fraction'><em>m</em><br><span class='vincula'><em>n</em></span></span> becomes in effect <span class='fraction'><span class='under'><em>m</em> × 1</span><br><em>n</em></span>. It is important to note the
+general applicability of this arrangement-rule, as it will be found
+of great assistance in solving more complicated expressions.</p>
+
+<p class='c007'>As with the ordinary form of slide rule, the factors in such an
+expression as <span class='fraction'><span class='under'><em>a</em> × <em>b</em> × <em>c</em></span><br><em>m</em> × <em>n</em></span> = <em>x</em> are taken in the order:—1st factor of
+numerator; 1st factor of denominator; 2nd factor of numerator;
+2nd factor of denominator, and so on; the 1st factor as <em>a</em> being set
+to the index, and the result <em>x</em> being finally read at the same point
+of reference.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—<span class='fraction'><span class='under'>39 × 14·2 × 6·3</span><br>1·37 × 19</span> = 134.</p>
+
+<p class='c021'>Commence by setting 39 to the index, and the pointer to 1·37;
+bring 14·2 to the pointer; pointer to 19; 6·3 to the pointer, and
+read the result 134 at the index.</p>
+
+<p class='c012'>It should be noted that after the first factor is set to the fixed
+index, the <em>pointer</em> is set to each of the <em>dividing</em> factors as they
+enter into the calculation, while the <em>dial</em> is moved for each of the
+<em>multiplying</em> factors. Thus the dial is first moved (setting the
+first factor to the index), then the pointer, then the dial, and
+so on.</p>
+
+<p class='c007'><em>Number of Digits in the Result.</em>—If rules are preferred to the plan
+of roughly estimating the result, the general rules given on pages
+<a href='#Page_21'>21</a> and <a href='#Page_25'>25</a> should be employed for simple cases of multiplication
+and division. For combined multiplication and division, modify
+<span class='pageno' id='Page_105'>105</span>the expression, if necessary, by introducing 1’s, as already explained,
+and subtract the sum of the denominator digits from the sum
+of numerator digits. Then proceed by the author’s rule, as
+follows:—</p>
+
+<p class='c007'><em>Always turn dial to the</em> <span class='fss'>LEFT</span>; <em>i.e.</em>, <em>against the hands of a watch</em>.</p>
+
+<p class='c007'><em>Note dial movements only; ignore those of the pointer.</em></p>
+
+<p class='c007'><em>Each time 1 on dial agrees with or passes fixed index</em>, <span class='fss'>ADD</span> <em>1 to the
+above difference of digits</em>.</p>
+
+<p class='c007'><em>Each time 1 on dial agrees with or passes pointer</em>, <span class='fss'>DEDUCT</span> <em>1 from
+the above difference of digits</em>.</p>
+
+<p class='c007'>Treat continued multiplication in the same way, counting the
+1’s used as denominator digits as one less than the number of
+multiplied factors.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—<span class='fraction'><span class='under'>8·6 × 0·73 × 1·02</span><br>3·5 × 0·23</span> = 7·95 [7·95473+].</p>
+
+<p class='c021'>Set 8·6 to index and pointer to 3·5. Bring 0·73 to pointer
+(noting that 1 on the scale passes the index) and set pointer to
+0·23. Set 1·02 to pointer (noting that 1 on the scale passes the
+pointer) and read under index 7·95. There are 1 + 0 + 1 = 2
+numerator digits and 1 + 0 = 1 denominator digit; while 1 is to be
+added and 1 deducted as per rule. But as the latter cancel, the
+digits in the result will be 2 − 1 = 1.</p>
+
+<p class='c012'>When moving the dial to the left will cause 1 on the dial to
+pass <em>both</em> index and pointer (thus cancelling), the dial may be
+turned back to make the setting.</p>
+
+<p class='c007'>It will be understood that when 1 is the <em>first</em> numerator, and 1
+on the dial is therefore set to the index, no digit addition will be
+made for this, as the actual operation of calculating has not been
+commenced.</p>
+
+<p class='c007'>In the Stanley-Boucher calculator (Fig. <a href='#f_023'>23</a>) a small centre scale
+is added, on which a finger indicates automatically the number of
+digits to be added or deducted; the method of calculating,
+however, differs from the foregoing. To avoid turning back
+to 0 at the commencement of each calculation, a circle is
+ground on the glass face, so that a pencil mark can be made
+thereon to show the position of the finger when commencing a
+calculation.</p>
+
+<p class='c007'><em>To Find the Square of a Number.</em>— Set the number, on one or
+other of the square root scales, to the index, and read the required
+square on the ordinary scale.</p>
+
+<p class='c007'><span class='pageno' id='Page_106'>106</span><em>To Find the Square Root of a Number.</em>—Set the number to the
+index, and if there is an <em>odd</em> number of digits in the number, read
+the root on the inner circle; if an even number, on the second
+circle.</p>
+
+<p class='c007'><em>To Find the Cube of a Number.</em>—Set 1 on the ordinary scale to
+the index, and the pointer (on the back dial) to the number on
+one of the three cube-root scales. Then under the pointer read
+the cube on the ordinary scale.</p>
+
+<p class='c007'><em>To Find the Cube Root of a Number.</em>—Set 1 to index, and pointer
+to number. Then read the cube root under the pointer on one of
+the three inner circles on the back dial. If the number has</p>
+
+<table class='table1'>
+  <tr>
+    <td class='c041'>1, 4, 7, 10 or −2, −5, etc.,</td>
+    <td class='c013'>digits, use the</td>
+    <td class='c043'>inner circle.</td>
+  </tr>
+  <tr>
+    <td class='c041'>2, 5, 8, 11 or −1, −4, etc.,</td>
+    <td class='c013'>„      „</td>
+    <td class='c043'>second circle.</td>
+  </tr>
+  <tr>
+    <td class='c041'>3, 6, 9, 12 or −0, −3, etc.,</td>
+    <td class='c013'>„      „</td>
+    <td class='c043'>third circle.</td>
+  </tr>
+</table>
+
+<p class='c007'><em>For Powers or Roots of Higher Denomination.</em>—Set 1 to index,
+the pointer to the number on the ordinary scale, and read on the
+outer circle on the back dial the mantissa of the logarithm. Add
+the characteristic (see p. <a href='#Page_46'>46</a>), multiply by the power or divide
+by the root, and set the pointer to the mantissa of the result on
+this outer circle. Under the pointer on the ordinary scale read
+the number, obtaining the number of figures from the characteristic.</p>
+
+<p class='c007'><em>To Find the Sines of Angles.</em>—Set 1 to index, pointer to the
+angle on the outer circle, and read under the pointer the <em>natural
+sine</em> on the ordinary scale; also under the pointer on the outer
+circle of the back dial read the <em>logarithmic sine</em>.</p>
+
+<p class='c007'><span class='sc'>The Halden Calculex.</span>—After the introduction of the Boucher
+calculator in 1876, circular instruments, such as the Charpentier
+calculator, were introduced, in which a disc turned within a fixed
+ring, so that scales on the faces of both could be set together and
+ratios established as on the slide rule. Cultriss’s Calculating Disc
+is another instrument on the same principle. The Halden Calculex,
+of which half-size illustrations are given in Figs. <a href='#f_024'>24</a> and <a href='#f_025'>25</a>,
+represents a considerable improvement upon these early instruments.
+It consists of an outer metal ring carrying a fixed-scale
+ring, within which is a dial. On each side of this dial are flat
+milled heads, so that by holding these between the thumb and
+forefinger the dial can be set quickly and conveniently. The
+protecting glass discs, which are not fixed in the metal ring but
+<span class='pageno' id='Page_107'>107</span>are arranged to turn therein, carry fine cursor lines, and as these
+are on the side next to the scales a very close setting can be made
+quite free from the effects of parallax. This construction not only
+avoids the use of mechanism, with its risk of derangement, but
+reduces the bulk of the instrument very considerably, the thickness
+being about ¼in.</p>
+
+<p class='c007'>On the front face, Fig. <a href='#f_024'>24</a>, the fixed ring carries an outer evenly-divided
+scale, giving logarithms, and an ordinary scale, 1–10, which
+works in conjunction with a similar scale on the edge of the dial.
+The two inner circles give the square roots of values on the main
+scales as in the Boucher calculator. On the back face, Fig. <a href='#f_025'>25</a>, the
+ring bears an outer scale, giving sines of angles from 6° to 90° and
+an ordinary scale, 1–10, as on the front face. The scales on the
+dial are all reversed in direction (running from right to left), the
+outer one consisting of an ordinary (but inverse) scale, 1–10, while
+the three inner circles give the cube roots of values on this inverse
+scale. As the fine cursor lines extend over all the scales, a variety
+of calculations can be effected very readily and accurately.</p>
+
+<div id='f_024'  class='figleft id005'>
+<img src='images/f_024.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 24.</span></p>
+</div>
+</div>
+
+<div id='f_025'  class='figright id005'>
+<img src='images/f_025.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 25.</span></p>
+</div>
+</div>
+
+<p class='c007'><span class='sc'>Sperry’s Pocket Calculator</span>, made by the Keuffel and Esser
+Company, New York (Fig. <a href='#f_026'>26</a>), has two rotating dials, each with
+its own pointer and fixed index. The S dial has an outer scale of
+equal parts, an ordinary logarithmic scale, and a square-root
+scale. The L dial has a single logarithmic scale arranged spirally,
+in three sections, giving a scale length of 12½in. The pointers are
+turned by the small milled head, which is concentric with the
+milled thumb-nut by which the two dials are rotated. The gearing
+<span class='pageno' id='Page_108'>108</span>is such that both the L dial and its pointer rotate three times as
+fast as the S dial and pointer. All the usual calculations can be
+made with the spiral scale, as with the Boucher calculator, and
+the result read off on one or other of the three scale-sections.
+Frequently the point at which to read the result is obvious, but
+otherwise a reference to the single scale on the S dial will show on
+which of the three spirals the result is to be found.</p>
+
+<div id='f_026'  class='figcenter id001'>
+<img src='images/f_026.jpg' alt='S. Dial. L. Dial.' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 26.</span></p>
+</div>
+</div>
+
+<p class='c007'><em>The K and E Calculator</em>, also made by the Keuffel and Esser
+Company, is shown in Figs. <a href='#f_027'>27</a> and <a href='#f_028'>28</a>. It has two dials, of which
+only one revolves. This, as shown in Fig. <a href='#f_027'>27</a>, has an ordinary
+logarithmic scale and a scale of squares. There is an index line
+engraved on the glass of the instrument. The fixed dial has a
+scale of tangents, a scale of equal parts and a scale of sines, the
+latter being on a two-turn spiral. The pointers, which move
+together, are turned by a milled nut and the movable dial by a
+thumb-nut, as in Sperry’s Calculator, Fig. <a href='#f_026'>26</a>.</p>
+
+<div id='f_027'  class='figleft id005'>
+<img src='images/f_027.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 27.</span></p>
+</div>
+</div>
+
+<div id='f_028'  class='figright id005'>
+<img src='images/f_028.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 28.</span></p>
+</div>
+</div>
+
+<div>
+  <span class='pageno' id='Page_109'>109</span>
+  <h3 class='c008'>SLIDE RULES FOR SPECIAL CALCULATIONS.</h3>
+</div>
+
+<p class='c009'><span class='sc'>Engine Power Computer.</span>—A typical example of special slide
+rules is shown in Fig. <a href='#f_029'>29</a>, which represents, on a scale of about
+half full size, the author’s Power Computer for Steam, Gas, and
+Oil Engines. This, as will be seen, consists of a stock, on the
+lower portion of which is a scale of cylinder diameters, while the
+upper portion carries a scale of horse-powers. In the groove
+between these scales are two slides, also carrying scales, and capable
+of sliding in edge contact with the stock and with each other.</p>
+
+<p class='c007'>This instrument gives directly the brake horse-power of any
+steam, gas, or oil engine; the indicated horse-power, the dimensions
+of an engine to develop a given power, and the mechanical efficiency
+of an engine. The calculation of piston speed, velocity ratios of
+pulleys and gear wheels, the circumferential speed of pulleys, and
+the velocity of belts and ropes driven thereby, are among the
+other principal purposes for which the computer may be employed.</p>
+
+<div id='f_029'  class='figcenter id001'>
+<img src='images/f_029.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 29.</span></p>
+</div>
+</div>
+
+<p class='c007'><span class='sc'>The Smith-Davis Piecework Balance Calculator</span> has two
+scales, 11 feet long, having a range from 1d. to £20, and marked
+so that they can be used either for money or time calculations.
+The scales are placed on the rims of two similar wheels and so
+arranged that the divided edges come together. The wheels are
+mounted on a spindle carried at each end in the bearings of a
+supporting stand. The wheels are pressed together by a spring,
+and move as one.</p>
+
+<p class='c007'>To set the scales one to the other, a treadle gear is arranged to
+take the pressure of the spring so that when the fixed wheel is
+held by the left hand the free wheel can be rotated by the right
+hand in either direction. When the amount of the balance has
+been set to the combined weekly wage the treadle is released
+<span class='pageno' id='Page_110'>110</span>locking the two wheels together, when the whole can be turned
+and the amounts respectively due to each man read off opposite
+his weekly wage. The Smith-Davis Premium Calculator is on the
+same principle but the scales are about 4 feet 6 inches long and
+the wheels spring-controlled. Both instruments are supplied by
+Messrs. John Davis &#38; Son, Ltd., Derby.</p>
+
+<p class='c007'><span class='sc'>The Baines Slide Rule.</span>—In this rule, invented by Mr.
+H. M. Baines, Lahore, four slides carrying scales are arranged to
+move, each in edge contact with the next. The slides are kept in
+contact and given the desired relative movement one to the other,
+by being attached (at the back), to a jointed parallelogram. On
+this principle which is of general application, the inventor has
+made a rule for the solution of problems covered by Flamant’s
+formula for the flow of water in cast-iron pipes:—V = 76·28<em>d</em><sup>⁵⁄₇</sup><em>s</em><sup>⁴⁄₇</sup>,
+in which <em>s</em> is the sine of the inclination or loss of head; <em>d</em> the
+diameter of the pipe in inches and V the velocity in feet per
+second. The formula Q = AV is also included in the scope of the
+rule, Q being the discharge in cubic feet per second and A the
+cross sectional area of the pipe in square inches.</p>
+
+<p class='c007'><span class='sc'>Farmar’s Profit-calculating Rule.</span>—The application of the
+slide rule to commercial calculations has been often attempted, but
+the degree of accuracy required necessitates the use of a long scale,
+and generally this results in a cumbersome instrument. In Farmar’s
+Profit-calculating Rule the money scale is arranged in ten sections,
+these being mounted in parallel form on a roller which takes the
+place of the upper scale of an ordinary rule. The roller, which is
+¾in. in diameter, is carried in brackets secured to each end of the
+stock, so that by rotating the roller any section of the money scale
+can be brought into reading with the scale on the upper edge of the
+slide and with which the roller is in contact. This scale gives percentages,
+and enables calculations to be made showing profit on
+turnover, profit on cost, and discount. The lower scale on the slide,
+and that on the stock adjacent to it, are similar to the A and B
+scales of an ordinary rule. The instrument is supplied by Messrs.
+J. Casartelli &#38; Son, Manchester.</p>
+
+<h3 class='c008'>CONSTRUCTIONAL IMPROVEMENTS IN SLIDE RULES.</h3>
+
+<p class='c009'>The attention of instrument makers is now being given to the
+devising of means for ensuring the smooth and even working of
+the slide in the stock of the rule. In some cases very good results
+<span class='pageno' id='Page_111'>111</span>are obtained by slitting the back of the stock to give more
+elasticity.</p>
+
+<p class='c007'>In the rules made by Messrs. John Davis &#38; Son, a metal strip,
+slightly curved in cross section as shown at A (Fig. <a href='#f_030'>30</a>), runs for the
+full length of the stock to which it is fastened at intervals. Near
+each end of the rule, openings about 1 in. long are made in the metal
+backing through which the scales on the back of the slide can be
+read. To prevent warping under varying climatic conditions both
+the stock of the rule and the slide are of composite construction.
+The base of the stock is of mahogany, while the grooved sides,
+firmly secured to the base, are of boxwood. Similarly the centre
+portion of the slide is of mahogany and the tongued sides of boxwood.
+Celluloid also enters into the construction, a strip of this
+material being laid along the bottom of the groove in the stock.
+A fine groove runs along the centre of this strip in order to give
+elasticity and to allow the sides of the stock to be pressed together
+slightly to adjust the fitting of the slide. As a further means of
+adjustment the makers fit metal clips at each end of the rule, so
+that by tightening two small screws the stock can be closed on the
+slide when necessary.</p>
+
+<div id='f_030'  class='figleft id007'>
+<img src='images/f_030.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 30.</span></p>
+</div>
+</div>
+
+<div id='f_032'  class='figright id007'>
+<img src='images/f_032.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 32.</span></p>
+</div>
+</div>
+
+<div id='f_031'  class='figcenter id008'>
+<img src='images/f_031.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 31.</span></p>
+</div>
+</div>
+
+<p class='c007'>In the rule made by the Keuffel and Esser Company of New
+York, one strip is made adjustable (Fig. <a href='#f_032'>32</a>).</p>
+
+<h3 class='c008'>THE ACCURACY OF SLIDE RULE RESULTS.</h3>
+
+<p class='c009'>The degree of accuracy obtainable with the slide rule depends
+primarily upon the length of the scale employed, but the accuracy
+of the graduations, the eyesight of the operator, and, in particular,
+his ability to estimate interpolated values, are all factors which
+<span class='pageno' id='Page_112'>112</span>affect the result. Using the lower scales and working carefully
+the error should not greatly exceed 0·15 per cent. with short calculations.
+With successive settings, the discrepancy need not
+necessarily be greater, as the errors may be neutralised; but with
+rapid working the percentage error may be doubled. However,
+much depends upon the graduation of the scales. Rules in which
+one or more of the indices have been thickened to conceal some slight
+inaccuracy should be avoided. The line on the cursor should be
+sharp and fine and both slide and cursor should move smoothly
+or good work cannot be done. Occasionally a little vaseline or clean
+tallow should be applied to the edges of the slide and cursor.</p>
+
+<p class='c007'>That the percentage error is constant throughout the scale is
+seen by setting 1 on C to 1·01 on D, when under 2 is 2·02; under
+3, 3·03; under 5, 5·05, etc., the several readings showing a uniform
+error of 1 per cent.</p>
+
+<p class='c007'>A method of obtaining a closer reading of a first setting or of a
+result on D has been suggested to the author by Mr. M. Ainslie,
+B.Sc. If any graduation, as 4 on C, is set to 3 on D, it is seen
+that 4 main divisions on C (40–44) are equal in scale length to 3
+main divisions on D (30–33). Hence, very approximately, 1 division
+on C is equal to 0·75 of a division on D, this ratio being shown, of
+course, on D under 10 on C. Suppose √<span class='vincula'>4·3</span> to be required. Setting
+the cursor to 4·3 on A, it is seen that the root is something
+more than 2·06. Move the slide until a main division is found on
+C, which exactly corresponds to the interval between 2 and the
+cursor line, on D. The division 27–28 just fits, giving a reading
+under 10 on C, of 74. Hence the root is read as 2·074. For the
+higher parts of the scale, the subdivisions, 1–1·1, etc., are used in
+place of main divisions. The method is probably more interesting
+than useful, since in most operations the inaccuracies introduced
+in making settings will impose a limit on the reliable figures of the
+result.</p>
+
+<p class='c007'>For the majority of engineering calculations, the slide rule will
+give an accuracy consistent with the accuracy of the data usually
+available. For some purposes, however, <em>logarithmic section paper</em>
+(the use of which the author has advocated for the last twenty
+years) will be found especially useful, more particularly in calculations
+involving exponential formulæ.</p>
+
+<div class='chapter'>
+  <span class='pageno' id='Page_113'>113</span>
+  <h2 class='c005'>APPENDIX.</h2>
+</div>
+<h3 class='c036'>NEW SLIDE RULES—FIFTH ROOTS, ETC.—THE SOLUTION OF ALGEBRAIC EQUATIONS—GAUGE POINTS AND SIGNS ON SLIDE RULES—TABLES AND DATA—SLIDE RULE DATA SLIPS.</h3>
+
+<p class='c009'><span class='sc'>The Pickworth Slide Rule.</span>—In this rule, made by Mr. A. W.
+Faber, the novel feature is the provision of a scale of cubes (F) in
+the stock or body of the rule. From Fig. <a href='#f_033'>33</a> it will be seen that
+the scale is fixed on the bevelled side of a slotted recess in the back
+of the rule. The slide carries an index mark, which is seen through
+the slot and can be set to any graduation of the scale; in its normal
+position it agrees with 1 on the scale. The C scale on the face of
+the rule is divided into three equal parts by two special division
+lines, marked II. and III., which, together with the initial
+graduation 1 of the scale, serve for setting or reading off values
+on the D scale. Similar division lines are marked on the D
+scale.</p>
+
+<div id='f_033'  class='figcenter id001'>
+<img src='images/f_033.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 33.</span></p>
+</div>
+</div>
+
+<p class='c007'>In using the rule for cubes or cube roots the slide is drawn to
+the right, this movement never exceeding one-third of the length
+of the D scale. With this limited movement, and with a single
+setting of the slide, the values of ∛<em> &#x0305;a</em>, ∛<span class='vincula'><em>a</em> × 10</span>, and ∛<span class='vincula'><em>a</em> × 100)</span>
+(<em>a</em> being less than 10 and not less than 1) are given simultaneously
+and without any uncertainty as to the scales to use or the values
+to be read off.</p>
+
+<p class='c007'><em>To Find the Cube of a Number.</em>—The marks II. and III. on D
+divide that scale into three equal sections. If the number to be
+<span class='pageno' id='Page_114'>114</span>cubed is in the first section, I. on C is set to it; if in the second
+section, II. on C is set to it; if in the third section, III. on C is
+set to it. Then, under the index mark on the back of the slide
+will be found the significant figures of the cube on the scale F.
+If I. on C was used for the setting, the cube contains 1 digit; if
+II. was used, 2 digits; if III. was used, 3 digits. If the first
+figure of the number to be cubed is not in the units place, the
+decimal point is moved through <em>n</em> places so as to bring the first
+significant figure into the units place, the cube found as above, and
+the decimal point moved in the <em>reverse direction</em> through 3<em>n</em> places.</p>
+
+<p class='c007'><em>To Find the Cube Root of a Number.</em>—The index mark is set to
+the significant figures of the number on scale F, and the cube root
+is read on D under I., II. or III. on C, according as the number
+has 1, 2 or 3 digits preceding the decimal point. Numbers which
+have 1, 2 or 3 figures preceding the decimal point are dealt with
+directly. Numbers of any other form are brought to one of the
+above forms by moving the decimal point 3 places (or such
+multiple of 3 places as may be required), the root found and its
+decimal point moved 1 place for each 3-place movement, but in the
+<em>reverse direction</em>.</p>
+
+<p class='c007'><span class='sc'>The “Electro” Slide Rule.</span>—In this special rule for electrical
+calculations, made by Mr. A. Nestler, the upper scales run from 0·1
+to 1000, and are marked “Amp.” and “sq. mm.” respectively. The
+lower scale on the slide running from 1 to 10,000 is marked M
+(metres), while the lower scale on the rule (0·1 to 100) is marked
+“Volt.” The latter scale is so displaced that 10 on M agrees with
+0·173 on the Volt scale. The four factors involved are the current
+strength (in Amp.); the area of a conductor (in sq. mm.); the length
+of the conductor (in metres); and the permissible loss of potential
+(in volts). Having given any three of these, the fourth can be
+found very readily. On the back of the slide are a scale of squares,
+a scale of cubes and a single scale corresponding to the D scale of
+an ordinary rule. Hence, by reversing the slide, it is possible to
+obtain the 2nd, 3rd and 4th powers and roots of numbers. In
+another form of the rule, the scale of metres is replaced by one of
+yards, while instead of the area of the conductor in sq. mm., the
+corresponding “gauge” sizes of wires are given.</p>
+
+<p class='c007'><span class='sc'>The “Polyphase” Slide Rule.</span>—This instrument, made by the
+Keuffel &#38; Esser Company, New York, has, in addition to the
+usual scales, a scale of cubes on the vertical edge of the stock of
+<span class='pageno' id='Page_115'>115</span>the rule, while in the centre of the slide there is a reversed C
+scale; <em>i.e.</em>, a scale exactly similar to an ordinary C scale but with
+the graduations running from right to left. The rule is specially
+useful for the solution of problems containing combinations of three
+factors and problems involving squares, square roots, cubes, cube
+roots and many of the higher powers and roots. It is specially
+adapted for electrical and hydraulic work.</p>
+
+<p class='c007'><span class='sc'>The Log-log Duplex Slide Rule.</span>—The same makers have
+introduced a log-log duplex slide rule, in which the log-log scale is
+in three sections, placed one above the other, these occupying the
+position usually taken up by the A scale. These scales are used in
+the manner already described (page <a href='#Page_86'>86</a>), but some advantage is obtained
+by the manner in which the complete log-log scale is divided,
+the limits being <em>e</em><sup>¹⁄₁₀₀</sup> to <em>e</em><sup>⅒</sup> (on Scale L.L. 1); <em>e</em><sup>⅒</sup> to <em>e</em> (on Scale L.L. 2);
+and <em>e</em> to <em>e</em><sup>10</sup> (on Scale L.L. 3), <em>e</em> being the base of natural or hyperbolic
+logarithms (2·71828). In this way a total log-log range of from
+1·01 to 22,000 is provided, meeting all practical requirements. These
+log-log scales are read in conjunction with a C scale placed at the
+upper edge of the slide. A similar C scale, but reversed in direction,
+is placed at the lower edge of the slide, this having red figures to
+distinguish it readily. The adjacent scale on the body of the rule
+is an ordinary D scale, and under this is an equally-divided scale
+giving the common logarithms of values on D. In the centre of
+the slide is a scale of tangents.</p>
+
+<p class='c007'>It will be understood that a “duplex” rule consists of two side
+strips securely clamped together at the two ends, forming the body
+of the rule, the slide moving between them; hence both front and
+back faces of the rule and slide are available, graduations on the
+one side being referred to those on the other by the cursor which
+extends around the whole. In this instrument, the scales on the
+back face are the ordinary scales of the standard rule with the
+addition of a scale of sines which is placed in the centre of the slide.
+It will be evident that this instrument is capable of dealing with a
+very wide range of problems involving exponential and trigonometrical
+formulæ.</p>
+
+<p class='c007'><span class='sc'>Small Slide Rules with Magnifying Cursors.</span>—Several makers
+now supply 5 in. rules having the full graduations of a 10 in. rule,
+and fitted with a magnifying cursor (Fig. <a href='#f_034'>34</a>). This forms a compact
+instrument for the pocket, but owing to the closeness of the graduations
+it is not usually possible to make a setting of the slide
+<span class='pageno' id='Page_116'>116</span>without using the cursor. This, of course, involves more movements
+than with the ordinary instrument. It is also very necessary
+to use the magnifying cursor in a <em>direct</em> light, if accurate readings
+are to be obtained. If these slight inconveniences are to be
+tolerated, the principle could be extended, a 10 in. rule being
+marked as fully as a 20 in., and fitted with a magnifying cursor.
+The author has endeavoured, but without success, to induce makers
+to introduce such a rule.</p>
+
+<p class='c007'>The magnifying cursor, supplied by Messrs. A. G. Thornton,
+Limited, has a lens which fills the entire cursor. It has a powerful
+magnifying effect, and the change from the natural to the magnified
+reading is less abrupt than with the semicircular lens.</p>
+
+<div id='f_034'  class='figcenter id001'>
+<img src='images/f_034.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 34.</span></p>
+</div>
+</div>
+
+<p class='c007'><span class='sc'>The Chemist’s Slide Rule.</span>—A slide rule, specially adapted
+for chemical calculations, has been introduced recently by Mr. A.
+Nestler. In this instrument the C and D scales are as usually
+arranged; but, in place of the A and B scales, there are a number
+of gauge points or marks denoting the atomic and molecular
+weights of the most important elements and combinations. The
+scales on the back of the slide are similarly arranged, so that by
+reversing the slide the operations can be extended very considerably.
+The rule finds its chief use in the calculation of
+analyses. Thus, to find the percentage of chlorine if <em>s</em> grammes of
+a substance have been used and the precipitate of Ag.Cl. weighs <em>a</em>
+grammes, we have the equation, <em>x</em> <a id='t116'></a>= <span class='fraction'>Cl.<br><span class='vincula'>Ag.Cl.</span></span> × <span class='fraction'><em>a</em><br><span class='vincula'><em>s</em></span></span>. Hence, the mark
+Ag.Cl. on the upper scale of the slide is set to the mark Cl. on the
+upper scale of the rule, when under <em>a</em> on the C scale is found the
+quantity of chlorine on D. By setting the cursor to this value and
+bringing <em>s</em> on C to the cursor, the percentage required can be
+read on C over 10 on D.</p>
+
+<p class='c007'>The rule is also adapted to the solution of various other
+chemical and electro-chemical calculations.</p>
+
+<p class='c007'><span class='pageno' id='Page_117'>117</span><span class='sc'>The Stelfox Slide Rule.</span>—This rule, shown in Fig. <a href='#f_035'>35</a>, has a
+stock 5 in. long, fitted with a 10 in. slide jointed in the middle of
+its length by means of long dowels. By separating the parts the
+compactness of a 5 in. rule is obtained. The upper scales on the rule
+and slide resemble the usual A and B scales. The D scale on the
+lower part of the stock is in two sections, the second portion being
+placed below the first, as shown in the illustration. The centre scale
+on the slide corresponds to the usual C scale, while on the lower edge
+of the slide is a similar scale, but with the index (1) in the middle of
+its length. The arrangement avoids the necessity of resetting the
+slide, as is sometimes necessary with the ordinary rule, and in general
+it combines the accuracy of a 10 in. rule with the compactness of a
+5 in. rule; but a more frequent use of the cursor is necessary. This
+rule is made by Messrs. John Davis &#38; Son, Limited, Derby.</p>
+
+<div id='f_035'  class='figcenter id001'>
+<img src='images/f_035.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 35.</span></p>
+</div>
+</div>
+
+<p class='c007'><span class='sc'>Electrical Slide Rule.</span>—Another rule by the same makers,
+specially useful for electrical engineers, has the usual scales on the
+working edges of the rule and slide, while in the middle of the slide
+is placed a scale of cubes. A log-log scale in two sections is provided;
+the power portion, running from 1·07 to 2, is found on the lower part
+of the stock, and the upper portion, running from 2 to 10<sup>3</sup>, on the
+upper part of the stock. The uppermost scale on the stock is in two
+parts, of which that to the left, running from 20 to 100 and marked
+“Dynamo,” gives the efficiencies of dynamos; that on the right,
+running from 20 to 100 and marked “Motor,” gives the efficiencies
+of electric motors. The lowest scale on the stock, marked “Volt,”
+gives the loss of potential in copper conductors. The ordinary upper
+scale on the stock is marked L (length of lead) at the left, and KW
+(kilowatts) at the right; the ordinary upper scale on the slide is
+marked A (ampères) and mm<sup>2</sup> (sectional area) at the left, and HP
+(horse-power) at the right. Additional lines on the cursor enable
+the electrical calculations to be made either in British or metric
+units.</p>
+
+<p class='c007'><span class='pageno' id='Page_118'>118</span><span class='sc'>The Picolet Circular Slide Rule.</span>—A simple form of circular
+calculator, made by Mr. L. E. Picolet of Philadelphia, is shown in Fig.
+<a href='#f_036'>36</a>. It consists of a base disc of stout celluloid on which turns a
+smaller disc of thin celluloid. A cursor formed of transparent
+celluloid is folded over the discs, and is attached so that the friction
+between the cursor and the inner disc enables the latter to be turned
+by moving the former. By holding both discs the cursor can be
+adjusted as required. The adjacent scales run in opposite directions,
+so that multiplication and division are performed as with the inverted
+slide in an ordinary rule. The outer scale, which is two-thirds the
+length of the main scale, enables cube roots to be found. Square
+roots are readily determined and continuous multiplication and
+division conveniently effected. Modified forms of this neatly made
+little instrument are also available.</p>
+
+<div id='f_036'  class='figcenter id006'>
+<img src='images/f_036.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 36.</span></p>
+</div>
+</div>
+
+<p class='c007'><span class='sc'>Other Recent Slide Rules.</span>—Among other special types of
+slide rule, mention should be made of the <em>Jakin</em> 10 in. rule for
+surveyors, made by Messrs. John Davis &#38; Son, Limited, Derby.
+By the provision of a series of short subsidiary scales, the multiplication
+of a sine or tangent of an angle by a number can be obtained
+to an accuracy of 1 in 10,000. The <em>Davis-Lee-Bottomley</em> slide rule,
+by the same makers, has special scales provided for circle spacing.
+The division of a circle into a number of equal parts, often required
+<span class='pageno' id='Page_119'>119</span>in spacing rivets, bolts, etc., and in setting out the teeth of gearwheels,
+is readily effected by the aid of this instrument. The <em>Cuntz</em>
+slide rule is a very comprehensive instrument, having a stock about
+2¼ in. wide, with the slide near the lower edge. Above the slide are
+eleven scales, referable to the main scales by the cursor. These scales
+enable squares and square roots, cubes and cube roots, and areas
+and circumferences of circles to be obtained by direct reading. A
+much more compact instrument could be obtained by removing one-half
+the scales to the back of the rule and using a double cursor.</p>
+
+<div id='f_037'  class='figcenter id006'>
+<img src='images/f_037.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 37.</span></p>
+</div>
+</div>
+
+<p class='c007'>In one form of 10 in. rule, supplied by Mr. W. H. Harling, London,
+the body of the rule is made of well-seasoned cane, with the usual
+celluloid facings. The rule has a metal back, enabling the fit of the
+slide to be regulated. This backing extends the full length of the
+rule, openings about 1 in. long being provided at each end, enabling
+the scales on the back of the slide to be set with greater facility than
+is possible with the notched recesses usually adopted. The author
+has long endeavoured, but without success, to induce makers to fit
+windows of glass or celluloid in place of the notched recesses. This
+would allow the graduation of the S and T scales to be set more
+accurately, and enable both to be used at each end of the rule—an
+advantage in certain trigonometrical calculations. It would have
+the further advantage of permitting each alternate graduation of the
+evenly-divided or logarithm scale to be placed at opposite sides of
+one central line, enabling the reading to be made more accurately
+and conveniently.</p>
+
+<p class='c007'>Many special slide rules have lately been devised for determining
+the time necessary to perform various machine-tool operations
+and for analogous purposes, while attention has again been
+<span class='pageno' id='Page_120'>120</span>given to rules for calculating the weights of iron and steel bars,
+plates, etc.</p>
+
+<p class='c007'><span class='sc'>The Davis-Stokes Field Gunnery Slide Rule.</span>—This rule,
+which is adapted for calculations involved in “encounter” and
+“entrenched” field gunnery, is designed for the 18 pr. quick-firing
+gun. The upper and lower portions of the boxwood stock are
+united by a flexible centre of celluloid, thus providing grooves
+front and rear to receive boxwood slides. Each of the nineteen
+scales is marked with its name, and corresponding scales are coloured
+red or black. The front edge is bevelled and carries a scale of 1 in
+20,000. The rule solves displacement problems, map angles of sight,
+changes of corrector and range corrections for changes in temperature,
+wind and barometer, etc. A special feature for displacement
+calculations is the provision of a 50 yd. sub-base angle scale, by
+which the apex angle is read at one setting.</p>
+
+<p class='c007'><span class='sc'>The Davis-Martin Wireless Slide Rule.</span>—In wireless telegraphy
+it is frequently necessary to determine wave-length, capacity
+or self-induction when one or other of the factors of the equation,
+λ = 59·6√<span class='vincula'>LC</span> is unknown. The Davis-Martin wireless rule is
+designed to simplify such calculations. The upper scale in the
+stock (inductance) runs from 10,000 to 1,000,000; the adjacent
+scale on the slide (capacity) runs from 0·0001 to 0·01 but in the
+reverse direction. The lower scale on the stock (wave-length) runs
+from 100 to 1000, giving square roots of the upper scale; while on
+the lower edge of the scale are several arrows to suit the various
+denominations in which the wave-length and capacity may be
+expressed.</p>
+
+<p class='c007'><span class='sc'>Improved Cursors.</span>—In some slide-rule operations, notably in
+those involved in solving quadratic and cubic equations, it not infrequently
+happens that readings are obscured by the frame of the
+cursor. Frameless cursors have been introduced to obviate this
+defect. A piece of thick transparent celluloid is sometimes employed,
+but this is liable to become scratched in use. Fig. <a href='#f_037'>37</a> shows a recent
+form of frameless glass cursor made by the Keuffel &#38; Esser Company,
+Hoboken, N.J., which is satisfactory in every way.</p>
+
+<p class='c007'>Cursors having three hair lines are now fitted to some rules, the
+distance apart of the lines being equal to the interval 0·7854–1 on
+the A scale.</p>
+
+<p class='c007'><span class='sc'>The Davis-Pletts Slide Rule.</span>—In this rule a single log.-log.
+scale and its reciprocal scale are arranged opposite the ordinary
+<span class='pageno' id='Page_121'>121</span>upper log. scale. Thus, common logarithms can be read directly,
+while by taking advantage of the properties of characteristics and
+mantissas of common logarithms, the scale can be extended indefinitely.
+As 10 is the highest number on the log.-log. scale, it is
+carried down to within 0·025 of unity. The reading of log.-log.
+values above 10 is effected in a very simple manner. There is also
+a scale in the centre of the slide which, used in conjunction with
+the upper log. scale enables the natural logarithm of any number
+between 0·0001 and 10,000 to be read direct, while any number
+on the upper log. scale can be multiplied or divided by <em>e<sup>x</sup></em> if the
+latter is between these limits. On the back of the slide are scales
+for all circular and hyperbolic functions, these being used in
+conjunction with the upper log. scales.</p>
+
+<p class='c007'><span class='sc'>The Crompton-Gallagher Boiler Efficiency Calculator</span>
+has a stock in the thickness of which is a slot admitting a chart
+which can be moved at right angles to the two separate slides.
+On the bevelled edge of one slide, the graduations are continued so
+as to read against curves on the chart, through an opening in the
+stock.</p>
+
+<p class='c007'><span class='sc'>The Davis-Grinsted Complex Calculator.</span>—This slide rule is
+of considerable service in connection with calculations involving
+the conversion of complex quantities from the form <em>a</em> + <em>j</em> <em>b</em> to the
+form R∠θ, and <em>vice versa</em>. The usual process of conversion necessitates
+repeated reference to trigonometrical tables, and is both
+tedious and time-taking. The Complex Calculator enables the
+conversions to be effected without reference to tables and with the
+minimum expenditure of time and labour.</p>
+
+<p class='c007'>The rule, which is about 16 in. long, has five scales. The upper
+one (A) is an ordinary logarithmic scale thrice-repeated. The
+adjacent scales on the slide comprise (1) a logarithmic scale of
+tangents (B) ranging from 0·1° to 45°, and (2) a logarithmic scale
+of secants (C) from 0° to 45°. The lower scales D and E are identical
+with the A scale, and are provided to enable multiplication, etc.,
+to be performed without the need for a separate slide rule. Readings
+can be transferred from A to the lower scales by means of the cursor.</p>
+
+<p class='c007'>In using the rule to convert <em>a</em> + <em>j</em> <em>b</em> to R∠θ, the index (45°) of the
+B scale is set to the larger component and the cursor to the smaller
+component, on scale A. Then θ (or its complement if <em>b</em> is greater
+than <em>a</em>) is read on B under the cursor. The cursor is then set to θ
+on the C scale, and R is read on A under the cursor. The rule
+is made by Messrs. John Davis &#38; Son, Limited, Derby.</p>
+
+<div>
+  <span class='pageno' id='Page_122'>122</span>
+  <h3 class='c008'>THE SOLUTION OF ALGEBRAIC EQUATIONS.</h3>
+</div>
+
+<p class='c009'>The slide rule finds an interesting application in the solution of
+equations of the second and third degree; and although the process
+is essentially one of trial and error, it may often serve as an
+efficient substitute for the more laborious algebraic methods,
+particularly when the conditions of the problem or the operator’s
+knowledge of the theory of equations enables some idea to be
+obtained as to the character of the result sought. The principle
+may be thus briefly explained:—If 1 on C is set to <em>x</em> on D (Fig.
+<a href='#f_038'>38</a>), we find <em>x</em>(<em>x</em>) = <em>x</em><sup>2</sup> on D under <em>x</em> on C. If, however, with the
+slide set as before, instead of reading under <em>x</em>, we read under <em>x</em> + <em>m</em>
+on C, the result on D will now be <em>x</em>(<em>x</em> + <em>m</em>) = <em>x</em><sup>2</sup> + <em>mx</em> = <em>q</em>.
+Hence to solve the equation <em>x</em><sup>2</sup> + <em>mx</em> − <em>q</em> = 0, we reverse the
+above process, and setting the cursor to <em>q</em> on D, we move the slide
+until the number on C under the cursor, and that on D under 1 on
+C, <em>differ by m</em>. It is obvious from the setting that the <em>product</em> of
+these numbers = <em>q</em>, and as their difference = <em>m</em>, they are seen to be
+the roots of the equation as required. For the equation <em>x</em><sup>2</sup> − <em>mx</em>
++ <em>q</em> = 0, we require <em>m</em> to equal the <em>sum</em> of the roots. Hence,
+setting the cursor as before to <em>q</em> on D, we move the slide until the
+number on C under the cursor, and that on D under 1 on C, are
+<em>together equal to</em> <em>m</em>, these numbers being the roots sought. The
+alternative equations <em>x</em><sup>2</sup> − <em>mx</em> − <em>q</em> = 0, and <em>x</em><sup>2</sup> + <em>mx</em> + <em>q</em> = 0 are
+deducible from the others by changing the signs of the roots, and
+need not be further considered.</p>
+
+<div id='f_038'  class='figcenter id001'>
+<img src='images/f_038.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 38.</span></p>
+</div>
+</div>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the roots of <em>x</em><sup>2</sup> − 8<em>x</em> + 9 = 0.</p>
+
+<p class='c021'>Set the cursor to 9 on D, and move the slide to the right until
+when 6·64 is found under the cursor, 1·355 on D is under 1 on C.
+These numbers are the roots required.</p>
+
+<p class='c012'>The upper scales can of course be used; indeed, in general
+they are to be preferred.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find the roots of <em>x</em><sup>2</sup> + 12·8<em>x</em> + 39·4 = 0.</p>
+
+<p class='c021'>Set the cursor to 39·4 on A, and move the slide to the right
+until we read 7·65 on B under the cursor, and 5·15 on A over 1 on
+B. The roots are therefore −7·65 and −5.15.</p>
+
+<p class='c012'><span class='pageno' id='Page_123'>123</span>With a little consideration of the relative value of the upper
+and lower scales, the student interested will readily perceive how
+equations of the third degree may be similarly resolved. The subject
+is not of sufficient general importance to warrant a detailed
+examination being made of the several expressions which can be
+dealt with in the manner suggested; but the author gives the
+following example as affording some indication of the adaptability
+of the method to practical calculations.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—A hollow copper ball, 7·5 in. in diameter and 2 lb. in
+weight, floats in water. To what depth will it sink?</p>
+
+<p class='c021'>The water displaced = 27·7 × 2 = 55·4 cub. in. The cubic
+contents of the immersed segment will be <span class='fraction'>π<br><span class='vincula'>3</span></span>(3<em>r</em> <em>x</em><sup>2</sup> − <em>x</em><sup>3</sup>), <em>r</em> being the
+radius and <em>x</em> the depth of immersion. Hence <span class='fraction'>π<br><span class='vincula'>3</span></span>(3<em>r</em> <em>x</em><sup>2</sup> − <em>x</em><sup>3</sup>) = 55·4,
+and 11·25<em>x</em><sup>2</sup> − <em>x</em><sup>3</sup> = 52·9.</p>
+
+<p class='c021'>To solve this equation we place the cursor to 52·9 on A, and
+move the slide until the reading on D under 1 and that on B under
+the cursor together amount to 11·25. In this way find 2·45 on D
+under 1, with 8·8 on B under the cursor <em>c</em>, <em>c</em>, as a pair of values of
+which the sum is 11·25. Hence we conclude that <em>x</em> = 2·45 in. is
+the result sought.</p>
+
+<p class='c021'>With the rule thus set (Fig. <a href='#f_039'>39</a>) the student will note that the
+slide is displaced to the right by an amount which represents <em>x</em> on
+D, and therefore <em>x</em><sup>2</sup> on A; while the length on B from 1 to the
+cursor line represents 11·25 − <em>x</em>. Hence the upper scale setting
+gives <em>x</em><sup>2</sup>(11·25 − <em>x</em>) = 11·25<em>x</em><sup>2</sup> − <em>x</em><sup>3</sup> = 52·9 as required.</p>
+
+<div id='f_039'  class='figcenter id001'>
+<img src='images/f_039.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p><span class='sc'>Fig. 39.</span></p>
+</div>
+</div>
+
+<p class='c012'>When in doubt as to the method to be pursued in any given
+case, the student should work synthetically, building up a simple
+example of an analogous character to that under consideration, and
+so deducing the plan to be followed in the reverse process.</p>
+
+<div>
+  <span class='pageno' id='Page_124'>124</span>
+  <h3 class='c008'>SCREW-CUTTING GEAR CALCULATIONS.</h3>
+</div>
+
+<p class='c009'>The slide rule has long found a useful application in connection with
+the gear calculations necessary in screw-cutting, helical gear-cutting,
+and spiral gear work.</p>
+
+<p class='c007'><span class='sc'>Single Gears.</span>—For simple cases of screw-cutting in the lathe it is
+only necessary to set the threads per inch to be cut to the threads per
+inch in the guide screw (or the pitch in inches in each case, if more
+convenient). Then any pair of coinciding values on the two scales
+will give possible pairs of wheels.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find wheels to cut a screw of 1⅝ threads per inch with a
+guide screw of 2 threads per inch.</p>
+
+<p class='c021'>Setting 1·625 on C to 2 on D, it is seen that 80 (driver) and 65
+(driven) are possible wheels.</p>
+
+<p class='c012'><span class='sc'>Compound Gears.</span>—When wheels so found are of inconvenient
+size, a compound train is used, consisting (usually) of two drivers
+and two driven wheels, the product of the two former and the
+product of the two latter being in the same ratio as the simple
+wheels. Thus with 60 and 40 as drivers, and 65 and 30 as driven,
+we have, <span class='fraction'>60 × 40<br><span class='vincula'>65 × 30</span></span> = <span class='fraction'>2400<br><span class='vincula'>1950</span></span> = <span class='fraction'>2<br><span class='vincula'>1·625</span></span> as before.</p>
+
+<p class='c007'>With the slide set as above, values convenient for splitting up
+into suitable wheels are readily obtainable. Thus, <span class='fraction'>1600<br><span class='vincula'>1300</span></span>; <span class='fraction'>2400<br><span class='vincula'>1950</span></span>;
+<span class='fraction'>4000<br><span class='vincula'>3250</span></span>; <span class='fraction'>4800<br><span class='vincula'>3900</span></span> are a few suggestive values which may be readily
+factorised.</p>
+
+<p class='c007'><span class='sc'>Slide Rules for Screw-cutting Calculations.</span>—Special circular
+and straight slide rules for screw-cutting gear calculations
+have long been employed. For compound gears these usually
+entail the use of six scales, two on each of the two slides and two
+on the stock. The upper scale on the stock may be a scale of
+threads per inch to be cut, the adjacent scale (on the upper slide) a
+scale of threads per inch in the guide screw. Setting the guide
+screw-graduation to the threads to be cut, the lower slide is adjusted
+until a convenient pair of drivers is found in coincidence on the
+central pair of scales, while a pair of driven wheels are in coincidence
+on the two lower scales.</p>
+
+<p class='c007'><span class='pageno' id='Page_125'>125</span>Some years ago, a slide rule was introduced by which compound
+gears could be obtained with a single slide. Assuming the set of
+wheels usually provided—20 to 120 teeth advancing by 5 teeth—the
+products of 20 × 25, 20 × 30, etc., up to 115 × 120 were calculated.
+These products were laid out along each of the two lower scales.
+The upper scales were a scale of threads per inch to be cut and a
+scale of the threads per inch of various guide screws. Setting the
+guide screw-graduation to the threads to be cut, any coinciding
+graduations on the lower scales gave the required pairs of drivers
+and driven wheels.</p>
+
+<p class='c007'><span class='sc'>Fractional Pitch Calculations.</span>—The author has long advocated
+the use of the slide rule for determining the wheels necessary
+for cutting fractional pitch threads, and it is gratifying to find its
+value in this connection is now being appreciated. For the best
+results a good 20 in. rule is desirable, but with care very close approximations
+can be found with an accurate 10 in. rule. In any case
+a magnifying cursor or a hand reading-glass is of great assistance.</p>
+
+<p class='c020'><span class='sc'>Ex.</span>—Find wheels to cut a thread of 0·70909 in. pitch; guide
+screw, 2 threads per inch.</p>
+
+<p class='c021'>To 0·70909 on D, set 0·5 (guide screw pitch in inches) on C. To
+make this setting as accurately as possible, the method described on
+page <a href='#Page_112'>112</a> may be used. Set 10 on C to about 91 on D, and note
+that the interval 77–78 on C represents 0·91 of the interval 70–71
+on D. Set the cursor to 78 on C and bring 5 to the cursor. The
+slide is then set so that 5 on C agrees with 7·091 on D.</p>
+
+<p class='c012'>Inspection of the two scales shows various coinciding factors
+in the ratio required. The most accurate is seen to be <span class='fraction'>55 on C<br><span class='vincula'>78 on D</span></span>.
+These values may be split up into <span class='fraction'>55 × 50<br><span class='vincula'>65 × 60</span></span> to form a suitable compound
+train of gears.</p>
+
+<div>
+  <span class='pageno' id='Page_126'>126</span>
+  <h3 class='c008'>GAUGE POINTS AND SIGNS ON SLIDE RULES.</h3>
+</div>
+
+<p class='c009'>Many slide rules have the sign <span class='fraction'><span class='under'>Prod.</span><br>−1</span> at the right-hand end of the
+D scale, while on the left is <span class='fraction'><span class='under'>Quot.</span><br>+1.</span> It is somewhat unfortunate
+that these signs refer to rules for determining the number of digits
+in products and quotients, which are used to a considerable extent
+on the Continent, and conflict with those used in this country. By
+the Continental method the number of digits in a product is equal
+to the sum of the digits in the two factors, if the result is obtained on
+the <span class='fss'>LEFT</span> <em>of the first factor</em>; but if the result is found on the <span class='fss'>RIGHT</span> of
+the first factor, it is equal to this sum − 1. The sign <span class='fraction'><span class='under'>Prod.</span><br>−1</span> the
+<em>right</em>-hand end of the D scale provides a visible reminder of this
+rule.</p>
+
+<p class='c007'>Similarly for division:—The number of digits in a quotient is
+equal to the number of the digits in the dividend, minus those in
+the divisor, if the quotient appears on the <span class='fss'>RIGHT</span> <em>of the dividend</em>, and
+to this difference + 1, if the quotient appears on the <span class='fss'>LEFT</span> of the
+dividend. The sign <span class='fraction'><span class='under'>Quot.</span><br>+1</span> at the <em>left</em>-hand end of the D scale provides
+a visible reminder of this rule.</p>
+
+<p class='c007'>The sign <img src='images/i_126.jpg' class='height2' alt=''> found at both ends of the A scale is of general
+application but of questionable utility. It is assumed to represent
+a fraction, the vertical line indicating the position of the decimal
+point. If the number 455 is to be dealt with in a multiplication
+on the lower scales, we may suppose the decimal point moved
+two places to the left, giving 4·55, a value which can be actually
+found on the scale. If we use this value, then to the number of
+digits in this result, as many must be added as the number of
+places (two in this case) by which the decimal point was moved.
+If the point is moved to the right, the number of places must
+be subtracted. Similarly, in division, if the decimal point in the
+divisor is moved <em>n</em> places to the left, then <em>n</em> places must be
+subtracted at the end of the operation; while if the point is moved
+through <em>n</em> places to the right, then <em>n</em> places must be added. The
+<span class='pageno' id='Page_127'>127</span>sign referred to, which, of course, applies to all scales, completely
+indicates these processes and is submitted as a reminder of the
+procedure to be followed by those using the method described.</p>
+
+<p class='c007'>The signs π, <em>c</em>, <em>c′</em>, and M are explained in the Section on
+“Gauge Points,” p. <a href='#Page_53'>53</a>.</p>
+
+<p class='c007'>On some rules additional signs are found on the D scale. One,
+locating the value <span class='fraction'><span class='under'>180 × 60</span><br>π</span> = 3437·74 and hence giving the number
+of minutes in a radian, is marked ρ′. Another, representing the
+value <span class='fraction'><span class='under'>180 × 60 × 60</span><br>π</span> = 206265, and hence giving the number of
+seconds in a radian is marked ρ″. A third point, marked ρ<sub>˶</sub>,
+placed at the value <span class='fraction'><span class='under'>200 × 100 × 100</span><br>π</span> = 636620, is used when the
+newer graduation of the circle is employed.</p>
+
+<p class='c007'>These gauge points are useful when converting angles into
+circular measure, or <em>vice versa</em>, and also for determining the
+functions of small angles.</p>
+
+<p class='c007'>A gauge point is sometimes marked at 1146 on the A and B
+scales. This is known as the “Gunner’s Mark,” and is used in
+artillery calculations involving angles of less than 20°, when, for
+the purpose in view, the tangent and circular measure of the angle
+may be regarded as equal. For this constant, the angle is taken
+in minutes, the auxiliary base in feet, and the base in yards. The
+auxiliary base in feet on B is set to the angle in minutes on A
+when over 1146 on B is the base in yards on A. The value
+<span class='fraction'>1<br><span class='vincula'>1146</span></span> = <span class='fraction'>π × 3<br><span class='vincula'>180 × 60</span></span>.</p>
+
+<div>
+  <span class='pageno' id='Page_128'>128</span>
+  <h3 class='c008'>TABLES AND DATA.</h3>
+</div>
+<h4 class='c036'>MENSURATION FORMULAE.</h4>
+
+<p class='c020'>Area of a parallelogram = base × height.</p>
+
+<p class='c021'>Area of rhombus = ½ product of the diagonals.</p>
+
+<p class='c021'>Area of a triangle = ½ base × perpendicular height.</p>
+
+<p class='c021'>Area of equilateral triangle = square of side × 0·433.</p>
+
+<p class='c021'>Area of trapezium = ½ sum of two parallel sides × height.</p>
+
+<p class='c021'>Area of any right-lined figure of four or more unequal sides is found
+by dividing it into triangles, finding area of each and adding together.</p>
+
+<p class='c021'>Area of regular polygon = (1) length of one side × number of sides
+× radius of inscribed circle; or (2) the sum of the triangular
+areas into which the figures may be divided.</p>
+
+<p class='c021'>Circumference of a circle = diameter × 3·1416.</p>
+
+<p class='c021'>Circumference of circle circumscribing a square = side × 4·443.</p>
+
+<p class='c021'>Circumference of circle = side of equal square × 3·545.</p>
+
+<p class='c021'>Length of arc of circle = radius × degrees in arc × 0·01745.</p>
+
+<p class='c021'>Area of a circle = square of diameter × 0·7854.</p>
+
+<p class='c021'>Area of sector of a circle = length of arc × ½ radius.</p>
+
+<p class='c021'>Area of segment of a circle = area of sector − area of triangle.</p>
+
+<p class='c021'>Side of square of area equal to a circle = diameter × 0·8862.</p>
+
+<p class='c021'>Diameter of circle equal in area to square = side of square × 1·1284.</p>
+
+<p class='c021'>Side of square inscribed in circle = diameter of circle × 0·707.</p>
+
+<p class='c021'>Diameter of circle circumscribing a square = side of square × 1·414.</p>
+
+<p class='c021'>Area of square = area of inscribed circle × 1·2732.</p>
+
+<p class='c021'>Area of circle circumscribing square = square of side × 1·5708.</p>
+
+<p class='c021'>Area of square = area of circumscribing circle × 0·6366.</p>
+
+<p class='c021'>Area of a parabola = base x ⅔ height.</p>
+
+<p class='c021'>Area of an ellipse = major axis × minor axis × 0·7854.</p>
+
+<p class='c021'>Surface of prism or cylinder = (area of two ends) + (length × perimeter).</p>
+
+<p class='c021'>Volume of prism or cylinder = area of base × height.</p>
+
+<p class='c021'>Surface of pyramid or cone = ½(slant height × perimeter of base)
++ area of base.</p>
+
+<p class='c021'>Volume of pyramid or cone = (⅓)(area of base × perpendicular height).</p>
+
+<p class='c021'>Surface of sphere = square of diameter × 3·1416.</p>
+
+<p class='c021'>Volume of sphere = cube of diameter × 0·5236.</p>
+
+<p class='c021'>Volume of hexagonal prism = square of side × 2·598 × height.</p>
+
+<p class='c021'>Volume of paraboloid = ½ volume of circumscribing cylinder.</p>
+
+<p class='c021'>Volume of ring (circular section) = mean diameter of ring × 2·47
+× square of diameter of section.</p>
+
+<div>
+  <span class='pageno' id='Page_129'>129</span>
+  <h4 class='c036'>SPECIFIC GRAVITY AND WEIGHT OF MATERIALS.</h4>
+</div>
+
+<table class='table3'>
+  <tr><th class='c028' colspan='4'><span class='sc'>Metals.</span></th></tr>
+  <tr>
+ <th class='btt bbt c029'><span class='sc'>Metal.</span></th>
+ <th class='btt bbt blt c029'>Specific Gravity.</th>
+ <th class='btt bbt blt c029'>Weight of 1 Cub. Ft. (Lb.).</th>
+ <th class='btt bbt blt c029'>Weight of 1 Cub. In. (Lb.).</th>
+  </tr>
+  <tr>
+    <td class='c031'>Aluminium, Cast</td>
+ <td class='blt c044'>2·56</td>
+ <td class='blt c044'>160</td>
+ <td class='blt c044'>0·0927</td>
+  </tr>
+  <tr>
+    <td class='c031'>Aluminium, Bronze</td>
+ <td class='blt c044'>7·68</td>
+ <td class='blt c044'>475</td>
+ <td class='blt c044'>0·275</td>
+  </tr>
+  <tr>
+    <td class='c031'>Antimony</td>
+ <td class='blt c044'>6·71</td>
+ <td class='blt c044'>418</td>
+ <td class='blt c044'>0·242</td>
+  </tr>
+  <tr>
+    <td class='c031'>Bismuth</td>
+ <td class='blt c044'>9·90</td>
+ <td class='blt c044'>617</td>
+ <td class='blt c044'>0·357</td>
+  </tr>
+  <tr>
+    <td class='c031'>Brass, Cast</td>
+ <td class='blt c044'>8·10</td>
+ <td class='blt c044'>505</td>
+ <td class='blt c044'>0·293</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    Wire</td>
+ <td class='blt c044'>8·548</td>
+ <td class='blt c044'>533</td>
+ <td class='blt c044'>0·309</td>
+  </tr>
+  <tr>
+    <td class='c031'>Copper, Sheet</td>
+ <td class='blt c044'>8·805</td>
+ <td class='blt c044'>549</td>
+ <td class='blt c044'>0·318</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    Wire</td>
+ <td class='blt c044'>8·880</td>
+ <td class='blt c044'>554</td>
+ <td class='blt c044'>0·321</td>
+  </tr>
+  <tr>
+    <td class='c031'>Gold</td>
+ <td class='blt c044'>19·245</td>
+ <td class='blt c044'>1200</td>
+ <td class='blt c044'>0·695</td>
+  </tr>
+  <tr>
+    <td class='c031'>Gun metal</td>
+ <td class='blt c044'>8·56</td>
+ <td class='blt c044'>534</td>
+ <td class='blt c044'>0·310</td>
+  </tr>
+  <tr>
+    <td class='c031'>Iron, Wrought (mean)</td>
+ <td class='blt c044'>7·698</td>
+ <td class='blt c044'>480</td>
+ <td class='blt c044'>0·278</td>
+  </tr>
+  <tr>
+    <td class='c031'>„   Cast (mean)</td>
+ <td class='blt c044'>7·217</td>
+ <td class='blt c044'>450</td>
+ <td class='blt c044'>0·261</td>
+  </tr>
+  <tr>
+    <td class='c031'>Lead, Milled Sheet</td>
+ <td class='blt c044'>11·418</td>
+ <td class='blt c044'>712</td>
+ <td class='blt c044'>0·412</td>
+  </tr>
+  <tr>
+    <td class='c031'>Manganese</td>
+ <td class='blt c044'>8·012</td>
+ <td class='blt c044'>499</td>
+ <td class='blt c044'>0·289</td>
+  </tr>
+  <tr>
+    <td class='c031'>Mercury</td>
+ <td class='blt c044'>13·596</td>
+ <td class='blt c044'>849</td>
+ <td class='blt c044'>0·491</td>
+  </tr>
+  <tr>
+    <td class='c031'>Nickel, Cast</td>
+ <td class='blt c044'>8·28</td>
+ <td class='blt c044'>516</td>
+ <td class='blt c044'>0·300</td>
+  </tr>
+  <tr>
+    <td class='c031'>Phosphor Bronze, Cast</td>
+ <td class='blt c044'>8·60</td>
+ <td class='blt c044'>536·8</td>
+ <td class='blt c044'>0·310</td>
+  </tr>
+  <tr>
+    <td class='c031'>Platinum</td>
+ <td class='blt c044'>21·522</td>
+ <td class='blt c044'>1342</td>
+ <td class='blt c044'>0·778</td>
+  </tr>
+  <tr>
+    <td class='c031'>Silver</td>
+ <td class='blt c044'>10·505</td>
+ <td class='blt c044'>655</td>
+ <td class='blt c044'>0·380</td>
+  </tr>
+  <tr>
+    <td class='c031'>Steel (mean)</td>
+ <td class='blt c044'>7·852</td>
+ <td class='blt c044'>489·6</td>
+ <td class='blt c044'>0·283</td>
+  </tr>
+  <tr>
+    <td class='c031'>Tin</td>
+ <td class='blt c044'>7·409</td>
+ <td class='blt c044'>462</td>
+ <td class='blt c044'>0·268</td>
+  </tr>
+  <tr>
+    <td class='c031'>Zinc, Sheet</td>
+ <td class='blt c044'>7·20</td>
+ <td class='blt c044'>449</td>
+ <td class='blt c044'>0·260</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>„ Cast</td>
+ <td class='bbt blt c044'>6·86</td>
+ <td class='bbt blt c044'>428</td>
+ <td class='bbt blt c044'>0·248</td>
+  </tr>
+</table>
+
+<table class='table2'>
+  <tr><th class='c028' colspan='4'><span class='sc'>Miscellaneous Substances.</span></th></tr>
+  <tr>
+ <th class='btt bbt c029' colspan='2'><span class='sc'>Substance.</span></th>
+ <th class='btt bbt blt c029'>Specific Gravity.</th>
+ <th class='btt bbt blt c029'>Weight of 1 Cub. In. (Lb.).</th>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Asbestos</td>
+ <td class='blt c029'>2·1–2·80</td>
+ <td class='blt c029'>·076-·101</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Brick</td>
+ <td class='blt c029'>1·90</td>
+ <td class='blt c029'>·069</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Cement</td>
+ <td class='blt c029'>2·72–3·05</td>
+ <td class='blt c029'>·0984-·109</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Clay</td>
+ <td class='blt c029'>2·0</td>
+ <td class='blt c029'>·072</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Coal</td>
+ <td class='blt c029'>1·37</td>
+ <td class='blt c029'>·0495</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Coke</td>
+ <td class='blt c029'>0·5</td>
+ <td class='blt c029'>·0181</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Concrete</td>
+ <td class='blt c029'>2·0</td>
+ <td class='blt c029'>·072</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Fire-brick</td>
+ <td class='blt c029'>2·30</td>
+ <td class='blt c029'>·083</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Granite</td>
+ <td class='blt c029'>2·5–2·75</td>
+ <td class='blt c029'>·051-·100</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Graphite</td>
+ <td class='blt c029'>1·8–2·35</td>
+ <td class='blt c029'>·065-·085</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Sand-stone</td>
+ <td class='blt c029'>2·3</td>
+ <td class='blt c029'>·083</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Slate</td>
+ <td class='blt c029'>2·8</td>
+ <td class='blt c029'>·102</td>
+  </tr>
+  <tr>
+    <td class='c031' colspan='2'>Wood—</td>
+ <td class='blt c029'>&#160;</td>
+ <td class='blt c029'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c031'>&#160;</td>
+    <td class='c031'>Beech</td>
+ <td class='blt c029'>0·75</td>
+ <td class='blt c029'>·0271</td>
+  </tr>
+  <tr>
+    <td class='c031'>&#160;</td>
+    <td class='c031'>Cork</td>
+ <td class='blt c029'>0·24</td>
+ <td class='blt c029'>·0087</td>
+  </tr>
+  <tr>
+    <td class='c031'>&#160;</td>
+    <td class='c031'>Elm</td>
+ <td class='blt c029'>0·58</td>
+ <td class='blt c029'>·021</td>
+  </tr>
+  <tr>
+    <td class='c031'>&#160;</td>
+    <td class='c031'>Fir</td>
+ <td class='blt c029'>0·56</td>
+ <td class='blt c029'>·0203</td>
+  </tr>
+  <tr>
+    <td class='c031'>&#160;</td>
+    <td class='c031'>Oak</td>
+ <td class='blt c029'>·62-·85</td>
+ <td class='blt c029'>·025-·031</td>
+  </tr>
+  <tr>
+    <td class='c031'>&#160;</td>
+    <td class='c031'>Pine</td>
+ <td class='blt c029'>0·47</td>
+ <td class='blt c029'>·017</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>&#160;</td>
+ <td class='bbt c031'>Teak</td>
+ <td class='bbt blt c029'>0·80</td>
+ <td class='bbt blt c029'>·029</td>
+  </tr>
+</table>
+
+<table class='table2'>
+  <tr><td class='c028' colspan='5'><span class='pageno' id='Page_130'>130</span></td></tr>
+  <tr><th class='c028' colspan='5'>ULTIMATE STRENGTH OE MATERIALS.</th></tr>
+  <tr>
+ <th class='btt bbt c029'><span class='sc'>Material.</span></th>
+ <th class='btt bbt blt c029'>Tension in lb. per sq. in.</th>
+ <th class='btt bbt blt c029'>Compression in lb. per sq. in.</th>
+ <th class='btt bbt blt c029'>Shearing in lb. per sq. in.</th>
+ <th class='btt bbt blt c029'>Modulus of Elasticity in lb. per sq. in.</th>
+  </tr>
+  <tr>
+    <td class='c031'>Cast Iron</td>
+ <td class='blt c044'>11,000 to 30,000</td>
+ <td class='blt c044'>50,000 to 130,000</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>14,000,000 to 23,000,000</td>
+  </tr>
+  <tr>
+    <td class='c044'>„        aver.</td>
+ <td class='blt c044'>16,000</td>
+ <td class='blt c044'>95,000</td>
+ <td class='blt c044'>11,000</td>
+ <td class='blt c044'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c031'>Wrought Iron</td>
+ <td class='blt c044'>40,000 to 70,000</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>26,000,000 to 31,000,000</td>
+  </tr>
+  <tr>
+    <td class='c044'>„       aver.</td>
+ <td class='blt c044'>50,000</td>
+ <td class='blt c044'>50,000</td>
+ <td class='blt c044'>40,000</td>
+ <td class='blt c044'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c031'>Soft Steel</td>
+ <td class='blt c044'>60,000 to 100,000</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>30,000,000 to 36,000,000</td>
+  </tr>
+  <tr>
+    <td class='c031'>Soft Steel   aver.</td>
+ <td class='blt c044'>80,000</td>
+ <td class='blt c044'>70,000</td>
+ <td class='blt c044'>55,000</td>
+ <td class='blt c044'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c031'>Cast Steel   aver.</td>
+ <td class='blt c044'>120,000</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>15,000,000 to 17,000,000</td>
+  </tr>
+  <tr>
+    <td class='c031'>Copper, Cast</td>
+ <td class='blt c044'>19,000</td>
+ <td class='blt c044'>58,000</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c031'>„     Wrought</td>
+ <td class='blt c044'>34,000</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>16,000,000</td>
+  </tr>
+  <tr>
+    <td class='c031'>Brass, Cast</td>
+ <td class='blt c044'>18,000</td>
+ <td class='blt c044'>10,500</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>9,170,000</td>
+  </tr>
+  <tr>
+    <td class='c031'>Gun Metal</td>
+ <td class='blt c044'>34,000</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>11,500,000</td>
+  </tr>
+  <tr>
+    <td class='c031'>Phosphor Bronze</td>
+ <td class='blt c044'>58,000</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>43,000</td>
+ <td class='blt c044'>13,500,000</td>
+  </tr>
+  <tr>
+    <td class='c031'>Wood, Ash</td>
+ <td class='blt c044'>17,000</td>
+ <td class='blt c044'>9,300</td>
+ <td class='blt c044'>1,400</td>
+ <td class='blt c044'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    Beech</td>
+ <td class='blt c044'>16,000</td>
+ <td class='blt c044'>8,500</td>
+ <td class='blt c044'>&#160;</td>
+ <td class='blt c044'>&#160;</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    Pine</td>
+ <td class='blt c044'>11,000</td>
+ <td class='blt c044'>6,000</td>
+ <td class='blt c044'>650</td>
+ <td class='blt c044'>1,400,000</td>
+  </tr>
+  <tr>
+    <td class='c031'>„    Oak</td>
+ <td class='blt c044'>15,000</td>
+ <td class='blt c044'>10,000</td>
+ <td class='blt c044'>2,300</td>
+ <td class='blt c044'>1,500,000</td>
+  </tr>
+  <tr>
+ <td class='bbt c031'>Leather</td>
+ <td class='bbt blt c044'>4,200</td>
+ <td class='bbt blt c044'>&#160;</td>
+ <td class='bbt blt c044'>&#160;</td>
+ <td class='bbt blt c044'>25,000</td>
+  </tr>
+</table>
+
+<table class='table2'>
+  <tr><th class='c028' colspan='8'>POWERS, ROOTS, ETC., OF USEFUL FACTORS.</th></tr>
+  <tr>
+ <th class='bbt c029'><em>n</em></th>
+ <th class='bbt blt c029'><span class='fraction'>1<br><span class='vincula'><em>n</em></span></span></th>
+ <th class='bbt blm c029'><em>n</em><sup>2</sup></th>
+ <th class='bbt blt c029'><em>n</em><sup>3</sup></th>
+ <th class='bbt blm c029'>√<em> &#x0305;n</em></th>
+ <th class='bbt blt c029'><span class='fraction'>1<br><span class='vincula'>√<em> &#x0305;n</em></span></span></th>
+ <th class='bbt blm c029'>∛<em> &#x0305;n</em></th>
+ <th class='bbt blt c029'><span class='fraction'>1<br><span class='vincula'>∛<em> &#x0305;n</em></span></span></th>
+  </tr>
+  <tr>
+    <td class='c029'>π = 3·142</td>
+ <td class='blt c044'>0·318</td>
+ <td class='blm c044'>9·870</td>
+ <td class='blt c044'>31·006</td>
+ <td class='blm c044'>1·772</td>
+ <td class='blt c044'>0·564</td>
+ <td class='blm c044'>1·465</td>
+ <td class='blt c044'>0·683</td>
+  </tr>
+  <tr>
+    <td class='c029'>2π= 6·283</td>
+ <td class='blt c044'>0·159</td>
+ <td class='blm c044'>39·478</td>
+ <td class='blt c044'>248·050</td>
+ <td class='blm c044'>2·507</td>
+ <td class='blt c044'>0·399</td>
+ <td class='blm c044'>1·845</td>
+ <td class='blt c044'>0·542</td>
+  </tr>
+  <tr>
+    <td class='c029'><span class='fraction'>π<br><span class='vincula'>2</span></span> = 1·571</td>
+ <td class='blt c044'>0·637</td>
+ <td class='blm c044'>2·467</td>
+ <td class='blt c044'>3·878</td>
+ <td class='blm c044'>1·253</td>
+ <td class='blt c044'>0·798</td>
+ <td class='blm c044'>1·162</td>
+ <td class='blt c044'>0·860</td>
+  </tr>
+  <tr>
+    <td class='c029'><span class='fraction'>π<br><span class='vincula'>3</span></span> = 1·047</td>
+ <td class='blt c044'>0·955</td>
+ <td class='blm c044'>1·097</td>
+ <td class='blt c044'>1·148</td>
+ <td class='blm c044'>1·023</td>
+ <td class='blt c044'>0·977</td>
+ <td class='blm c044'>1·016</td>
+ <td class='blt c044'>0·985</td>
+  </tr>
+  <tr>
+    <td class='c029'><span class='fraction'>4<br><span class='vincula'>3</span></span>π = 4·189</td>
+ <td class='blt c044'>0·239</td>
+ <td class='blm c044'>17·546</td>
+ <td class='blt c044'>73·496</td>
+ <td class='blm c044'>2·047</td>
+ <td class='blt c044'>0·489</td>
+ <td class='blm c044'>1·612</td>
+ <td class='blt c044'>0·622</td>
+  </tr>
+  <tr>
+    <td class='c029'><span class='fraction'>π<br><span class='vincula'>4</span></span> = 0·785</td>
+ <td class='blt c044'>1·274</td>
+ <td class='blm c044'>0·617</td>
+ <td class='blt c044'>0·484</td>
+ <td class='blm c044'>0·886</td>
+ <td class='blt c044'>1·128</td>
+ <td class='blm c044'>0·923</td>
+ <td class='blt c044'>1·084</td>
+  </tr>
+  <tr>
+    <td class='c029'><span class='fraction'>π<br><span class='vincula'>6</span></span> = 0·524</td>
+ <td class='blt c044'>1·910</td>
+ <td class='blm c044'>0·274</td>
+ <td class='blt c044'>0·144</td>
+ <td class='blm c044'>0·724</td>
+ <td class='blt c044'>1·382</td>
+ <td class='blm c044'>0·806</td>
+ <td class='blt c044'>1·241</td>
+  </tr>
+  <tr>
+    <td class='c029'>π<sup>2</sup> = 9·870</td>
+ <td class='blt c044'>0·101</td>
+ <td class='blm c044'>97·409</td>
+ <td class='blt c044'>961·390</td>
+ <td class='blm c044'>3·142</td>
+ <td class='blt c044'>0·318</td>
+ <td class='blm c044'>2·145</td>
+ <td class='blt c044'>0·466</td>
+  </tr>
+  <tr>
+    <td class='c029'>π<sup>3</sup> = 31·006</td>
+ <td class='blt c044'>0·032</td>
+ <td class='blm c044'>961·390</td>
+ <td class='blt c044'>29,809·910</td>
+ <td class='blm c044'>5·568</td>
+ <td class='blt c044'>1·796</td>
+ <td class='blm c044'>3·142</td>
+ <td class='blt c044'>0·318</td>
+  </tr>
+  <tr>
+    <td class='c029'><span class='fraction'>π<br><span class='vincula'>32</span></span> = 0·098</td>
+ <td class='blt c044'>10·186</td>
+ <td class='blm c044'>0·0095</td>
+ <td class='blt c044'>0·001</td>
+ <td class='blm c044'>0·313</td>
+ <td class='blt c044'>3·192</td>
+ <td class='blm c044'>0·461</td>
+ <td class='blt c044'>2·168</td>
+  </tr>
+  <tr>
+    <td class='c029'><em>g</em>  = 32·2</td>
+ <td class='blt c044'>0·031</td>
+ <td class='blm c044'>1036·84</td>
+ <td class='blt c044'>33,386·24</td>
+ <td class='blm c044'>5·674</td>
+ <td class='blt c044'>0·176</td>
+ <td class='blm c044'>3·181</td>
+ <td class='blt c044'>0·314</td>
+  </tr>
+  <tr>
+ <td class='bbt c029'>2<em>g</em> = 64·4</td>
+ <td class='bbt blt c044'>0·015</td>
+ <td class='bbt blm c044'>4147·36</td>
+ <td class='bbt blt c044'>267,090</td>
+ <td class='bbt blm c044'>8·025</td>
+ <td class='bbt blt c044'>0·125</td>
+ <td class='bbt blm c044'>4·007</td>
+ <td class='bbt blt c044'>0·249</td>
+  </tr>
+</table>
+
+<div>
+  <span class='pageno' id='Page_131'>131</span>
+  <h4 class='c036'>HYDRAULIC EQUIVALENTS.</h4>
+</div>
+
+<div class='lg-container-l c003'>
+  <div class='linegroup'>
+    <div class='group'>
+      <div class='line'>1 foot head = 0·434 lb. per square inch.</div>
+      <div class='line'>1 lb. per square inch = 2·31 ft. head.</div>
+      <div class='line'>1 imperial gallon = 277·274 cubic inches.</div>
+      <div class='line'>1 imperial gallon = 0·16045 cubic foot.</div>
+      <div class='line'>1 imperial gallon = 10 lb.</div>
+      <div class='line'>1 cubic foot of water = 62·32 lb. = 6·232 imperial gallons.</div>
+      <div class='line'>1 cubic foot of sea water = 64·00 lb.</div>
+      <div class='line'>1 cubic inch of water = 0·03616 lb.</div>
+      <div class='line'>1 cubic inch of sea water = 0·037037 lb.</div>
+      <div class='line'>1 cylindrical foot of water = 48·96 lb.</div>
+      <div class='line'>1 cylindrical inch of water = 0·0284 lb.</div>
+      <div class='line'>A column of water 12 in. long 1 in. square = 0·434 lb.</div>
+      <div class='line'>A column of water 12 in. long 1 in. diameter = 0·340 lb.</div>
+      <div class='line'>Capacity of a 12 in. cube = 6·232 gallons.</div>
+      <div class='line'>Capacity of a 1 in. square 1 ft. long = 0·0434 gallon.</div>
+      <div class='line'>Capacity of a 1 ft. diameter 1 ft. long = 4·896 gallons.</div>
+      <div class='line'>Capacity of a cylinder 1 in. diameter 1 ft. long = 0·034 gallon.</div>
+      <div class='line'>Capacity of a cylindrical inch = 0·002832 gallon.</div>
+      <div class='line'>Capacity of a cubic inch = 0·003606 gallon.</div>
+      <div class='line'>Capacity of a sphere 12 in. diameter = 3·263 gallons.</div>
+      <div class='line'>Capacity of a sphere 1 in. diameter = 0·00188 gallon.</div>
+      <div class='line'>1 imperial gallon = 1·2 United States gallon.</div>
+      <div class='line'>1 imperial gallon = 4·543 litres of water.</div>
+      <div class='line'>1 United States gallon = 231·0 cubic inches.</div>
+      <div class='line'>1 United States gallon = 0·83 imperial gallon.</div>
+      <div class='line'>1 United States gallon = 3·8 litres of water.</div>
+      <div class='line'>1 cubic foot of water = 7·476 United States gallons.</div>
+      <div class='line'>1 cubic foot of water = 28·375 litres of water.</div>
+      <div class='line'>1 litre of water = 0·22 imperial gallon.</div>
+      <div class='line'>1 litre of water = 0·264 United States gallon.</div>
+      <div class='line'>1 litre of water = 61·0 cubic inches.</div>
+      <div class='line'>1 litre of water = 0·0353 cubic foot.</div>
+    </div>
+  </div>
+</div>
+
+<table class='table2'>
+  <tr><th class='c028' colspan='18'>EQUIVALENTS OF POUNDS AVOIRDUPOIS.</th></tr>
+  <tr>
+ <th class='btt bbt brt c044'></th>
+ <th class='btt bbt brt c029' colspan='2'>10</th>
+ <th class='btt bbt brt c029' colspan='3'>100</th>
+ <th class='btt bbt brt c029' colspan='4'>1000</th>
+ <th class='btt bbt brt c029' colspan='4'>10,000</th>
+ <th class='btt bbt c029' colspan='4'>100,000</th>
+  </tr>
+  <tr>
+ <th class='brt c044'></th>
+    <th class='c029'>qr.</th>
+ <th class='brt c029'>lb.</th>
+    <th class='c029'>cwt.</th>
+    <th class='c029'>qr.</th>
+ <th class='brt c029'>lb.</th>
+    <th class='c029'>ton</th>
+    <th class='c029'>cwt.</th>
+    <th class='c029'>qr.</th>
+ <th class='brt c029'>lb.</th>
+    <th class='c029'>ton</th>
+    <th class='c029'>cwt.</th>
+    <th class='c029'>qr.</th>
+ <th class='brt c029'>lb.</th>
+    <th class='c029'>ton</th>
+    <th class='c029'>cwt.</th>
+    <th class='c029'>qr.</th>
+    <th class='c029'>lb.</th>
+  </tr>
+  <tr>
+ <td class='brt c044'>1</td>
+    <td class='c044'>0</td>
+ <td class='brt c044'>10</td>
+    <td class='c044'>0</td>
+    <td class='c044'>3</td>
+ <td class='brt c044'>16</td>
+    <td class='c044'>0</td>
+    <td class='c044'>8</td>
+    <td class='c044'>3</td>
+ <td class='brt c044'>20</td>
+    <td class='c044'>4</td>
+    <td class='c044'>9</td>
+    <td class='c044'>1</td>
+ <td class='brt c044'>4</td>
+    <td class='c044'>44</td>
+    <td class='c044'>12</td>
+    <td class='c044'>3</td>
+    <td class='c044'>12</td>
+  </tr>
+  <tr>
+ <td class='brt c044'>2</td>
+    <td class='c044'>0</td>
+ <td class='brt c044'>20</td>
+    <td class='c044'>1</td>
+    <td class='c044'>3</td>
+ <td class='brt c044'>4</td>
+    <td class='c044'>0</td>
+    <td class='c044'>17</td>
+    <td class='c044'>3</td>
+ <td class='brt c044'>12</td>
+    <td class='c044'>8</td>
+    <td class='c044'>18</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>8</td>
+    <td class='c044'>89</td>
+    <td class='c044'>5</td>
+    <td class='c044'>2</td>
+    <td class='c044'>24</td>
+  </tr>
+  <tr>
+ <td class='brt c044'>3</td>
+    <td class='c044'>1</td>
+ <td class='brt c044'>2</td>
+    <td class='c044'>2</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>20</td>
+    <td class='c044'>1</td>
+    <td class='c044'>6</td>
+    <td class='c044'>3</td>
+ <td class='brt c044'>4</td>
+    <td class='c044'>13</td>
+    <td class='c044'>7</td>
+    <td class='c044'>3</td>
+ <td class='brt c044'>12</td>
+    <td class='c044'>133</td>
+    <td class='c044'>18</td>
+    <td class='c044'>2</td>
+    <td class='c044'>8</td>
+  </tr>
+  <tr>
+ <td class='brt c044'>4</td>
+    <td class='c044'>1</td>
+ <td class='brt c044'>12</td>
+    <td class='c044'>3</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>8</td>
+    <td class='c044'>1</td>
+    <td class='c044'>15</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>24</td>
+    <td class='c044'>17</td>
+    <td class='c044'>17</td>
+    <td class='c044'>0</td>
+ <td class='brt c044'>16</td>
+    <td class='c044'>178</td>
+    <td class='c044'>11</td>
+    <td class='c044'>1</td>
+    <td class='c044'>20</td>
+  </tr>
+  <tr>
+ <td class='brt c044'>5</td>
+    <td class='c044'>1</td>
+ <td class='brt c044'>22</td>
+    <td class='c044'>4</td>
+    <td class='c044'>1</td>
+ <td class='brt c044'>24</td>
+    <td class='c044'>2</td>
+    <td class='c044'>4</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>16</td>
+    <td class='c044'>22</td>
+    <td class='c044'>6</td>
+    <td class='c044'>1</td>
+ <td class='brt c044'>20</td>
+    <td class='c044'>223</td>
+    <td class='c044'>4</td>
+    <td class='c044'>1</td>
+    <td class='c044'>4</td>
+  </tr>
+  <tr>
+ <td class='brt c044'>6</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>4</td>
+    <td class='c044'>5</td>
+    <td class='c044'>1</td>
+ <td class='brt c044'>12</td>
+    <td class='c044'>2</td>
+    <td class='c044'>13</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>8</td>
+    <td class='c044'>26</td>
+    <td class='c044'>15</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>24</td>
+    <td class='c044'>267</td>
+    <td class='c044'>17</td>
+    <td class='c044'>0</td>
+    <td class='c044'>16</td>
+  </tr>
+  <tr>
+ <td class='brt c044'>7</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>14</td>
+    <td class='c044'>6</td>
+    <td class='c044'>1</td>
+ <td class='brt c044'>0</td>
+    <td class='c044'>3</td>
+    <td class='c044'>2</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>0</td>
+    <td class='c044'>31</td>
+    <td class='c044'>5</td>
+    <td class='c044'>0</td>
+ <td class='brt c044'>0</td>
+    <td class='c044'>312</td>
+    <td class='c044'>10</td>
+    <td class='c044'>0</td>
+    <td class='c044'>0</td>
+  </tr>
+  <tr>
+ <td class='brt c044'>8</td>
+    <td class='c044'>2</td>
+ <td class='brt c044'>24</td>
+    <td class='c044'>7</td>
+    <td class='c044'>0</td>
+ <td class='brt c044'>16</td>
+    <td class='c044'>3</td>
+    <td class='c044'>11</td>
+    <td class='c044'>1</td>
+ <td class='brt c044'>20</td>
+    <td class='c044'>35</td>
+    <td class='c044'>14</td>
+    <td class='c044'>1</td>
+ <td class='brt c044'>4</td>
+    <td class='c044'>357</td>
+    <td class='c044'>2</td>
+    <td class='c044'>3</td>
+    <td class='c044'>12</td>
+  </tr>
+  <tr>
+ <td class='bbt brt c044'>9</td>
+ <td class='bbt c044'>3</td>
+ <td class='bbt brt c044'>6</td>
+ <td class='bbt c044'>8</td>
+ <td class='bbt c044'>0</td>
+ <td class='bbt brt c044'>4</td>
+ <td class='bbt c044'>4</td>
+ <td class='bbt c044'>0</td>
+ <td class='bbt c044'>1</td>
+ <td class='bbt brt c044'>12</td>
+ <td class='bbt c044'>40</td>
+ <td class='bbt c044'>3</td>
+ <td class='bbt c044'>2</td>
+ <td class='bbt brt c044'>8</td>
+ <td class='bbt c044'>401</td>
+ <td class='bbt c044'>15</td>
+ <td class='bbt c044'>2</td>
+ <td class='bbt c044'>24</td>
+  </tr>
+</table>
+
+<div>
+  <span class='pageno' id='Page_132'>132</span>
+  <h4 class='c036'>TRIGONOMETRICAL FUNCTIONS.</h4>
+</div>
+
+<h5 class='c036'>RIGHT-ANGLED TRIANGLES.</h5>
+
+<div  class='figleft id007'>
+<img src='images/i_132a.jpg' alt='[Right-angled Triangle]' class='ig001'>
+</div>
+
+<p class='c009'>Sin. A = <span class='fraction'><span class='under'><em>a</em></span><br><em>b</em></span>    Sec. A = <span class='fraction'><span class='under'><em>b</em></span><br><em>c</em></span>    Tan. A = <span class='fraction'><span class='under'><em>a</em></span><br><em>c</em></span></p>
+
+<p class='c007'>Cos. A = <span class='fraction'><span class='under'><em>c</em></span><br><em>b</em></span>  Cosec. A = <span class='fraction'><span class='under'><em>b</em></span><br><em>a</em></span>  Cotan. A = <span class='fraction'><span class='under'><em>c</em></span><br><em>a</em></span></p>
+
+<p class='c007'>Versin. A = <span class='fraction'><span class='under'><em>b</em> − <em>c</em></span><br><em>b</em></span>.  Coversin. A = <span class='fraction'><span class='under'><em>b</em> − <em>a</em></span><br><em>b</em></span>.</p>
+
+<table class='table2'>
+  <tr>
+ <td class='btt bbt brt c029'>Given.</td>
+ <td class='btt bbt c029'>Required.</td>
+ <td class='btt bbt blt c037'>Formulæ.</td>
+  </tr>
+  <tr>
+ <td class='brt c029'><em>a</em>,<em>b</em></td>
+    <td class='c029'>A,C,<em>c</em></td>
+ <td class='blt c037'>Sin. A = <span class='fraction'><em>a</em><br><span class='vincula'><em>b</em></span></span> Cos. C = <span class='fraction'><em>a</em><br><span class='vincula'><em>b</em></span></span> <em>c</em> = √<span class='vincula'>(<em>b + a</em>)(<em>b − a</em>)</span></td>
+  </tr>
+  <tr>
+ <td class='brt c029'>&#160;</td>
+    <td class='c029'>&#160;</td>
+ <td class='blt c037'>&#160;</td>
+  </tr>
+  <tr>
+ <td class='brt c029'><em>a</em>,<em>c</em></td>
+    <td class='c029'>A,C,<em>b</em></td>
+ <td class='blt c037'>Tan. A = <span class='fraction'><em>a</em><br><span class='vincula'><em>c</em></span></span> Cotan. B = <span class='fraction'><em>a</em><br><span class='vincula'><em>c</em></span></span> <em>b</em> = √<span class='vincula'><em>a</em><sup>2</sup> + <em>c</em><sup>2</sup></span></td>
+  </tr>
+  <tr>
+ <td class='brt c029'>&#160;</td>
+    <td class='c029'>&#160;</td>
+ <td class='blt c037'>&#160;</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>A,<em>a</em></td>
+    <td class='c029'>C,<em>c</em>,<em>b</em></td>
+ <td class='blt c037'>C = 90° − A <em>c</em> = <em>a</em> × Cotan. A <em>b</em> = <span class='fraction'><em>a</em><br><span class='vincula'>Sin. A</span></span></td>
+  </tr>
+  <tr>
+ <td class='brt c029'>&#160;</td>
+    <td class='c029'>&#160;</td>
+ <td class='blt c037'>&#160;</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>A,<em>b</em></td>
+    <td class='c029'>C,<em>a</em>,<em>c</em></td>
+ <td class='blt c037'>C = 90° − A <em>a</em> = <em>b</em> × Sin. A <em>c</em> = <em>b</em> × Cos. A</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>&#160;</td>
+    <td class='c029'>&#160;</td>
+ <td class='blt c037'>&#160;</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>A,<em>c</em></td>
+    <td class='c029'>C,<em>a</em>,<em>b</em></td>
+ <td class='blt c037'>C = 90° − A <em>a</em> = <em>c</em> × Tan. A <em>b</em> = <span class='fraction'><em>c</em><br><span class='vincula'>Cos. A</span></span></td>
+  </tr>
+  <tr>
+ <td class='bbt brt c029'>&#160;</td>
+ <td class='bbt c029'>&#160;</td>
+ <td class='bbt blt c037'>&#160;</td>
+  </tr>
+</table>
+
+<h5 class='c036'>OBLIQUE-ANGLED TRIANGLES.</h5>
+
+<p class='c009'><em>s</em> = ½(<em>a + b + c</em>)</p>
+
+<div  class='figleft id007'>
+<img src='images/i_132b.jpg' alt='[Oblique-angled Triangle]' class='ig001'>
+</div>
+
+<table class='table2'>
+  <tr>
+ <th class='btt bbt brt c037'>Given.</th>
+ <th class='btt bbt c037'>&#160;</th>
+ <th class='btt bbt blt c037'>Formulæ.</th>
+  </tr>
+  <tr>
+ <td class='brt c037'>A,B,C,<em>a</em></td>
+    <td class='c037' rowspan='3'>Area=</td>
+ <td class='blt c038'>(<em>a</em><sup>2</sup> × Sin. B × Sin. C) ÷ 2 Sin. A</td>
+  </tr>
+  <tr>
+ <td class='brt c037'>A,<em>b</em>,<em>c</em></td>
+    
+ <td class='blt c038'>½(<em>c</em> × <em>b</em> × Sin. A)</td>
+  </tr>
+  <tr>
+ <td class='brt c037'><em>a</em>,<em>b</em>,<em>c</em></td>
+    
+ <td class='blt c038'>√<span class='vincula'><em>s</em>(<em>s</em> − <em>a</em>)(<em>s</em> − <em>b</em>)(<em>s</em> − <em>c</em>)</span></td>
+  </tr>
+  <tr>
+ <td class='brt c037'><hr></td>
+    <td class='c037'><hr></td>
+ <td class='blt c037'><hr></td>
+  </tr>
+  <tr>
+ <th class='bbt brt c037'>Given.</th>
+ <th class='bbt c037'>Required.</th>
+ <th class='bbt blt c037'>Formulæ.</th>
+  </tr>
+  <tr>
+ <td class='bbt brt c037'>A,C,<em>a</em></td>
+ <td class='bbt c037'><em>c</em></td>
+ <td class='bbt blt c037'><em>c</em> = <em>a</em><span class='fraction'><span class='under'>Sin. C</span><br>Sin. A</span></td>
+  </tr>
+  <tr>
+ <td class='bbt brt c037'>A,<em>a</em>,<em>c</em></td>
+ <td class='bbt c037'>C</td>
+ <td class='bbt blt c037'>Sin. C = <span class='fraction'><em>c</em> Sin. A<br><span class='vincula'><em>a</em></span></span></td>
+  </tr>
+  <tr>
+ <td class='bbt brt c037'><em>a</em>,<em>c</em>,B</td>
+ <td class='bbt c037'>A</td>
+ <td class='bbt blt c037'>Tan. A = <span class='fraction'><em>a</em> Sin. B<br><span class='vincula'><em>c</em> − <em>a</em> Cos. B</span></span></td>
+  </tr>
+  <tr>
+ <td class='brt c037' rowspan='3'><em>a</em>,<em>b</em>,<em>c</em></td>
+    <td class='c037' rowspan='3'>A</td>
+ <td class='blt c037'>Sin. ½A = <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>(<em>s</em> − <em>b</em>)(<em>s</em> − <em>c</em>)</span></span><br><em>b</em> × <em>c</em></span></td>
+  </tr>
+  <tr>
+ 
+    
+ <td class='blt c037'>Cos. ½A = <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'><em>s</em>(<em>s</em> − <em>a</em>)</span></span><br><em>b</em> × <em>c</em></span>;</td>
+  </tr>
+  <tr>
+ 
+    
+ <td class='blt c037'>Tan. ½A = <sub><span class='c034'>√</span></sub><span class='rootfraction'><span class='vincula'><span class='under'>(<em>s</em> − <em>b</em>)(<em>s</em> − <em>c</em>)</span></span><br><em>s</em>(<em>s</em> − <em>a</em>)</span></td>
+  </tr>
+  <tr>
+ <td class='brt c037'><hr></td>
+    <td class='c037'><hr></td>
+ <td class='blt c037'><hr></td>
+  </tr>
+</table>
+
+<h5 class='c036'>COMPOUND ANGLES.</h5>
+
+<div class='lg-container-l c003'>
+  <div class='linegroup'>
+    <div class='group'>
+      <div class='line'>Sin. (A + B) = Sin. A Cos. B + Cos. A Sin. B.</div>
+      <div class='line'>Sin. (A − B) = Sin. A Cos. B − Cos. A Sin. B.</div>
+      <div class='line'>Cos. (A + B) = Cos. A Cos. B − Sin. A Sin. B.</div>
+      <div class='line'>Cos. (A − B) = Cos. A Cos. B + Sin. A Sin. B.</div>
+    </div>
+  </div>
+</div>
+
+<p class='c000'>Tan. (A + B) = <span class='fraction'>Tan. A + Tan. B<br><span class='vincula'>1 − Tan. A Tan. B</span></span>.</p>
+
+<p class='c000'>Tan. (A − B) = <span class='fraction'>Tan. A − Tan. B<br><span class='vincula'>1 + Tan. A Tan. B</span></span>.</p>
+<div>
+  <span class='pageno' id='Page_133'>133</span>
+  <h4 class='c036'>SLIDE RULE DATA SLIPS, <span class='sc'>compiled by C. N. Pickworth, Wh.Sc.</span></h4>
+</div>
+
+<div class='nf-center-c0'>
+<div class='nf-center c003'>
+    <div>(<em>It is suggested that this page be removed by cutting through the above line, and selected portions of the Sectional Data Slips attached to the back of the Slide Rule.</em>)</div>
+  </div>
+</div>
+
+<table class='table2'>
+  <tr>
+ <td class='brt c029'>¹⁄₃₂</td>
+    <td class='c040'>0·03125</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¹⁄₁₆</td>
+    <td class='c040'>0·0625</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>³⁄₃₂</td>
+    <td class='c040'>0·09375</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>⅛</td>
+    <td class='c040'>0·125</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>⁵⁄₃₂</td>
+    <td class='c040'>0·15625</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>³⁄₁₆</td>
+    <td class='c040'>0·1875</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>⁷⁄₃₂</td>
+    <td class='c040'>0·21875</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¼</td>
+    <td class='c040'>0·25</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>⁹⁄₃₂</td>
+    <td class='c040'>0·28125</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>⁵⁄₁₆</td>
+    <td class='c040'>0·3125</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¹¹⁄₃₂</td>
+    <td class='c040'>0·34375</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>⅜</td>
+    <td class='c040'>0·375</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¹³⁄₃₂</td>
+    <td class='c040'>0·40625</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>⁷⁄₁₆</td>
+    <td class='c040'>0·4375</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¹⁵⁄₃₂</td>
+    <td class='c040'>0·46875</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¹⁷⁄₃₂</td>
+    <td class='c040'>0·53125</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>⁹⁄₁₆</td>
+    <td class='c040'>0·5625</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¹⁹⁄₃₂</td>
+    <td class='c040'>0·59375</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>⅝</td>
+    <td class='c040'>0·625</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>²¹⁄₃₂</td>
+    <td class='c040'>0·65625</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¹¹⁄₁₆</td>
+    <td class='c040'>0·6875</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>²³⁄₃₂</td>
+    <td class='c040'>0·71875</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¾</td>
+    <td class='c040'>0·75</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>²⁵⁄₃₂</td>
+    <td class='c040'>0·78125</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¹³⁄₁₆</td>
+    <td class='c040'>0·8125</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>²⁷⁄₃₂</td>
+    <td class='c040'>0·84375</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>⅞</td>
+    <td class='c040'>0·875</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>²⁹⁄₃₂</td>
+    <td class='c040'>0·90625</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>¹⁵⁄₁₆</td>
+    <td class='c040'>0·9375</td>
+  </tr>
+  <tr>
+ <td class='brt c029'>³¹⁄₃₂</td>
+    <td class='c040'>0·96875</td>
+  </tr>
+</table>
+
+<p class='c000'>Circ. of circle = 3·1416 <em>d</em>.</p>
+
+<p class='c000'>Area  „    „    = 0·7854 <em>d</em><sup>2</sup>.</p>
+
+<p class='c000'>Sq. eq. area to cir., <em>s</em> = 0·886 <em>d</em>.</p>
+
+<p class='c000'>Circle eq. to sq., <em>d</em> = 1·128 <em>s</em>.</p>
+
+<p class='c000'>Sq. inscbd. in circ., <em>s</em> = 0·707 <em>d</em>.</p>
+
+<p class='c000'>Circsb. circ. of sq., <em>d</em> = 1·414 <em>s</em>.</p>
+
+<p class='c000'>Area of ellipse = 0.7854 <em>a</em> × <em>b</em>.</p>
+
+<p class='c000'>Surface of sphere = 3·1416 <em>d</em><sup>2</sup>.</p>
+
+<p class='c000'>Volume  „    „    = 0·5236 <em>d</em><sup>3</sup>.</p>
+
+<p class='c000'>  „     „   cone  = 0·2618 <em>d</em><sup>2</sup> <em>h</em>.</p>
+
+<p class='c000'>Radian = <span class='fraction'><span class='under'>180°</span><br>π</span> = 57·29 deg.</p>
+
+<p class='c000'>Base of nat. or hyp. log. = e = 2·7183.</p>
+
+<p class='c000'>Nat. or hyp. log. = com. log. × 2·3026.</p>
+
+<p class='c000'>g (at London) 32·18 ft. per sec., per sec.</p>
+
+<p class='c000'>Abs. temp. = deg. F. + 461° = deg. C. + 274°.</p>
+
+<p class='c000'>C.° = <span class='fraction'>5<br><span class='vincula'>9</span></span>(F.° − 32°); F.° = <span class='fraction'>9<br><span class='vincula'>5</span></span>C.° + 32°.</p>
+
+<p class='c000'>Cal. pr.—Ther. units per lb.: Coal, 14,300;</p>
+
+<p class='c000'>  petrol’m, 20,000; coal gas per cu. ft., 700.</p>
+
+<p class='c000'>Sp. heat:—Wt. iron, 0·1138; C.I., 0·1298;</p>
+
+<p class='c000'>  copper, brass, 0·095; lead, 0·0314.</p>
+
+<p class='c000'>Inch = 25·4 mil’metres; mil’metre = 0·03937 in.</p>
+
+<p class='c000'>Foot = 0·3048 metres; metre = 3·2809 feet.</p>
+
+<p class='c000'>Yard = 0·91438 metre; metre = 1·0936 yards.</p>
+
+<p class='c000'>Mile = 1·6093 kilomtrs.; kilomtr. = 0·6213 mile.</p>
+
+<p class='c000'>Sq. in. = 6·4513 sq. cm.; sq. cm. = 0·155 sq. in.</p>
+
+<p class='c000'>Sq. ft. = 9·29 sq. decmtr.; sq. decmtr. = 0·1076 sq. ft.</p>
+
+<p class='c000'>Sq. yd. = 0·836 sq. metre; sq. metre = 1·196 sq. yds.</p>
+
+<p class='c000'>Sq. ml. = 258·9 hectares; hectare = 0·00386 sq. ml.</p>
+
+<p class='c000'>Cu. in. = 16·386 c. cm.; c. cm. = 0·06102 cu. in.</p>
+
+<p class='c000'>Cu. ft. = 0·0283 c. metre; c. metre = 35·316 cu. ft.</p>
+
+<p class='c000'>Grain = 0·0648 gramme; gram. = 15·43 grs.</p>
+
+<p class='c000'>Ounce = 28·35 grams.;    „   = 0·03527 oz.</p>
+
+<p class='c000'>Pound = 0·4536 kilogm.; kilogm. = 2·204 lb.</p>
+
+<p class='c000'>Ton = 1·016 tonnes; tonne = 0·9842 ton.</p>
+
+<p class='c000'>Mile per hr. = 1·466 ft., or 44·7 cm., per sec.</p>
+
+<p class='c000'>Lb. per cu. in. = 0·0276 kilogram per cu. cm.</p>
+
+<p class='c000'>Kilogram per cu. cm. = 36·125 lb. per cu. in.</p>
+
+<p class='c000'>Lb. per cu. ft. = 16·019 kilogm. per cu. mtre.</p>
+
+<p class='c000'>Grain per gall. = 0·01426 gramme per litre.</p>
+
+<p class='c000'>Gramme per litre = 70·116 grains per gall.</p>
+
+<table class='table2'>
+  <tr>
+ <th class='brt c029' rowspan='2'>Ultimate Strength</th>
+    <th class='c029' colspan='2'>Lb. per Sq. in.</th>
+  </tr>
+  <tr>
+ 
+ <th class='bbt brt c029'>Tens’n.</th>
+ <th class='bbt c029'>Comp’n.</th>
+  </tr>
+  <tr>
+ <td class='brt c040'>Wt. iron</td>
+ <td class='brt c044'>50,000</td>
+    <td class='c044'>50,000</td>
+  </tr>
+  <tr>
+ <td class='brt c040'>Cast „</td>
+ <td class='brt c044'>16,000</td>
+    <td class='c044'>95,000</td>
+  </tr>
+  <tr>
+ <td class='brt c040'>Steel</td>
+ <td class='brt c044'>80,000</td>
+    <td class='c044'>70,000</td>
+  </tr>
+  <tr>
+ <td class='brt c040'>Copper</td>
+ <td class='brt c044'>21,000</td>
+    <td class='c044'>50,000</td>
+  </tr>
+  <tr>
+ <td class='brt c040'>Brass</td>
+ <td class='brt c044'>18,000</td>
+    <td class='c044'>10,500</td>
+  </tr>
+  <tr>
+ <td class='brt c040'>Lead</td>
+ <td class='brt c044'>2,500</td>
+    <td class='c044'>7,000</td>
+  </tr>
+  <tr>
+ <td class='brt c040'>Pine</td>
+ <td class='brt c044'>11,000</td>
+    <td class='c044'>6,000</td>
+  </tr>
+  <tr>
+ <td class='brt c040'>Oak</td>
+ <td class='brt c044'>15,000</td>
+    <td class='c044'>10,000</td>
+  </tr>
+</table>
+
+<table class='table2'>
+  <tr>
+ <th class='bbt brt c037'>Weight of Metals.</th>
+ <th class='bbt brt c037'>Cub. In.</th>
+ <th class='bbt brt c037'>Cub. Ft.</th>
+ <th class='bbt c037'>12 Cu. In.</th>
+  </tr>
+  <tr>
+ <td class='brt c045'>Wt. iron</td>
+ <td class='brt c046'>0·277</td>
+ <td class='brt c046'>480</td>
+    <td class='c046'>3·33</td>
+  </tr>
+  <tr>
+ <td class='brt c045'>Cast „</td>
+ <td class='brt c046'>0·260</td>
+ <td class='brt c046'>450</td>
+    <td class='c046'>3·12</td>
+  </tr>
+  <tr>
+ <td class='brt c045'>Steel</td>
+ <td class='brt c046'>0·283</td>
+ <td class='brt c046'>490</td>
+    <td class='c046'>3·40</td>
+  </tr>
+  <tr>
+ <td class='brt c045'>Copper</td>
+ <td class='brt c046'>0·318</td>
+ <td class='brt c046'>550</td>
+    <td class='c046'>3·82</td>
+  </tr>
+  <tr>
+ <td class='brt c045'>Brass</td>
+ <td class='brt c046'>0·300</td>
+ <td class='brt c046'>520</td>
+    <td class='c046'>3·61</td>
+  </tr>
+  <tr>
+ <td class='brt c045'>Zinc</td>
+ <td class='brt c046'>0·248</td>
+ <td class='brt c046'>430</td>
+    <td class='c046'>2·98</td>
+  </tr>
+  <tr>
+ <td class='brt c045'>Alumin’m</td>
+ <td class='brt c046'>0.096</td>
+ <td class='brt c046'>168</td>
+    <td class='c046'>1·16</td>
+  </tr>
+  <tr>
+ <td class='brt c045'>Lead</td>
+ <td class='brt c046'>0.411</td>
+ <td class='brt c046'>710</td>
+    <td class='c046'>4·93</td>
+  </tr>
+</table>
+
+<div class='lg-container-b'>
+  <div class='linegroup'>
+    <div class='group'>
+      <div class='line'>Lb. per sq. in. = 2·31 ft. water = 2·04 in. mercury = 0·0703 kilo. per sq. cm.</div>
+      <div class='line'>Atmosphere = 14·7 lb. per sq. in. = 33·94 ft. water = 1·0335   „        „</div>
+      <div class='line'>Ft. hd. water = 0·433 lb. per sq. in. = 62·35 lb. per sq. ft. = 0·0304  „   „</div>
+      <div class='line'>Cub. ft. of water = 62·35 lb. = 0·0278 ton = 28·315 litres = 7·48 U.S. galls.</div>
+      <div class='line'>Gall. (Imp.) = 277·27 cu. in. = 0·1604 cu. ft. = 10 lb. water = 4·544 litres.</div>
+      <div class='line'>Litre = 1·76 pints = 0·22 gall. = 61 cu. in. = 0·0353 cu. ft. = 0·264 U.S. gall.</div>
+      <div class='line'>Horse-power = 33,000 ft.-lb. per min. = 0·746 kilowatt = 42·4 heat units per min.</div>
+      <div class='line'>Heat unit = 778 ft.-lb. = 1055 watt-sec. = 107·5 kilogrammetres = 0·252 calorie.</div>
+      <div class='line'>Foot-pound = 0·00129 heat unit = 1·36 joules = 0·1383 kilogrammetres.</div>
+      <div class='line'>Kilowatt = 1·34 H.P. = 44,240 ft.-lb. per min. = 3412 heat units per hour.</div>
+    </div>
+  </div>
+</div>
+
+<hr class='c047'>
+<div class='footnote' id='f1'>
+<p class='c007'><a href='#r1'>1</a>. It will be recognised that n is the characteristic of the logarithm of
+the original number.</p>
+</div>
+<div class='footnote' id='f2'>
+<p class='c007'><a href='#r2'>2</a>. The special case in which the numerator is 1, 10, or any power of 10
+must be treated by the rule for reciprocals (page <a href='#Page_27'>27</a>).</p>
+</div>
+<div class='footnote' id='f3'>
+<p class='c007'><a href='#r3'>3</a>. The possible need for traversing the slide, to change the indices,
+when using the C and D scales, is not considered as a setting.</p>
+</div>
+<div class='footnote' id='f4'>
+<p class='c007'><a href='#r4'>4</a>. The reader may be reminded that cross-multiplication of the
+factors in any such slide rule setting will give a constant product, <em>e.g.</em>,
+20 × 94·5 = 27 × 70.</p>
+</div>
+<div class='footnote' id='f5'>
+<p class='c007'><a href='#r5'>5</a>. In this case cross <em>dividing</em> gives a constant quotient, <em>e.g.</em>,
+8 ÷ 3 = 4 ÷ 1·5. Since the upper scale is now a scale of reciprocals, the
+ratio is really</p>
+
+<table class='table1'>
+  <tr>
+ <td class='bbt c013'>O</td>
+ <td class='bbt c013'>⅛</td>
+ <td class='bbt c014'>¼</td>
+  </tr>
+  <tr>
+    <td class='c013'>D</td>
+    <td class='c013'>1·5</td>
+    <td class='c014'>3</td>
+  </tr>
+</table>
+
+</div>
+<div class='footnote' id='f6'>
+<p class='c007'><a href='#r6'>6</a>. These lines should not be brought to the working edge of the scale but
+should terminate in the horizontal line which forms the border of the
+finer graduations, their value being read into the calculation by means of
+the cursor (see page <a href='#Page_55'>55</a>).</p>
+</div>
+<div class='footnote' id='f7'>
+<p class='c007'><a href='#r7'>7</a>. The same principle may be applied to the cursor.</p>
+</div>
+<div class='footnote' id='f8'>
+<p class='c007'><a href='#r8'>8</a>. Philosophical Transactions of the Royal Society, 1815.</p>
+</div>
+<div class='pbb'>
+ <hr class='pb c003'>
+</div>
+
+<div class='border'>
+
+<div class='nf-center-c0'>
+<div class='nf-center c004'>
+    <div><em>BY THE SAME AUTHOR.</em></div>
+    <div class='c002'><span class='large'>LOGARITHMS FOR BEGINNERS.</span></div>
+  </div>
+</div>
+
+<p class='c007'>“An extremely useful and much-needed little work, giving
+a complete explanation of the theory and use of logarithms, by
+a teacher of great clearness and good style.”—<cite>The Mining
+Journal.</cite></p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>1s. 8d. Post Free.</div>
+    <div class='c002'><span class='large'>THE INDICATOR HANDBOOK.</span></div>
+  </div>
+</div>
+
+<p class='c007'>Comprising “The Indicator: Its Construction and Application”
+and “The Indicator Diagram: Its Analysis and Calculation.”
+Complete in One Volume.</p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>7s. 10d. Post Free.</div>
+  </div>
+</div>
+
+<p class='c007'>“Mr. Pickworth’s judgment is always sound, and is evidently
+derived from a personal acquaintance with indicator work.”—<cite>The
+Engineer.</cite></p>
+
+<div class='nf-center-c0'>
+<div class='nf-center c002'>
+    <div><span class='large'>POWER COMPUTER FOR STEAM, GAS AND OIL ENGINES, Etc.</span></div>
+  </div>
+</div>
+
+<p class='c007'>“Accurate, expeditious and thoroughly practical.... We can
+confidently recommend it, and engineers will find it a great
+boon in undertaking tests, etc.”—<cite>The Electrician.</cite></p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>7s. 6d. Post Free.</div>
+  </div>
+</div>
+
+</div>
+<div class='pbb'>
+ <hr class='pb c003'>
+</div>
+
+<div class='nf-center-c0'>
+<div class='nf-center c004'>
+    <div>ADVERTISEMENTS.</div>
+    <div class='c002'><span class='large'>LOGARITHMS FOR BEGINNERS</span></div>
+  </div>
+</div>
+
+<p class='c007'>For a full and intelligent appreciation of the Slide Rule and its various applications
+an elementary knowledge of logarithms is necessary. All that is
+required will be found in this little work, which gives a simple, detailed and
+practical explanation of logarithms and their uses, particular care having been
+taken to elucidate all difficult points by the aid of a number of worked examples.</p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>Seventh Edition, 1s. 8d. Post Free.</div>
+    <div class='c002'><span class='large'>POWER COMPUTER</span></div>
+    <div>for</div>
+    <div><span class='large'>STEAM, GAS, AND OIL ENGINES, Etc.</span></div>
+  </div>
+</div>
+
+<p class='c007'>Gives The Indicated Horse-Power of Steam, Gas, and Oil Engines—The Brake
+Horse-Power of Steam, Gas, and Oil Engines—The Size of Engine Necessary to
+Develop any Given Power—The Mechanical Efficiency of an Engine—The Ratio
+of Compound Engine Cylinders—The Piston Speed of an Engine—The Delivery
+of Pumps with any Efficiency—The Horse-Power of Belting—The Rim Speeds
+of Wheels, Speeds of Ropes, Belts, etc.—Speed Ratios of Pulleys, Gearing, etc.</p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>Pocket size, in neat case, with instructions and examples.</div>
+    <div class='c003'>Post Free, 7s. 6d. net.</div>
+    <div class='c002'><span class='large'>C. N. PICKWORTH, Withington, Manchester</span></div>
+  </div>
+</div>
+
+<table class='table2'>
+  <tr>
+ <td class='btt blt c029'>W. P. THOMPSON,</td>
+ <td class='btt brt c029'>G. C. DYMOND,</td>
+  </tr>
+  <tr>
+ <td class='blt c029'>F.C.S., M.I.Mech.E., F.I.C.P.A.</td>
+ <td class='brt c029'>M.I.Mech.E., F.I.C.P.A.</td>
+  </tr>
+  <tr>
+ <td class='blt c029'>&#160;</td>
+ <td class='brt c029'>&#160;</td>
+  </tr>
+  <tr>
+ <td class='blt brt c029' colspan='2'>W. P. Thompson &#38; Co.,</td>
+  </tr>
+  <tr>
+ <td class='blt brt c029' colspan='2'>12 CHURCH STREET, LIVERPOOL,</td>
+  </tr>
+  <tr>
+ <td class='blt brt c029' colspan='2'>CHARTERED PATENT AGENTS.</td>
+  </tr>
+  <tr>
+ <td class='blt c029'>&#160;</td>
+ <td class='brt c029'>&#160;</td>
+  </tr>
+  <tr>
+ <td class='blt c029'>H. E. POTTS,</td>
+ <td class='brt c029'>J. V. ARMSTRONG,</td>
+  </tr>
+  <tr>
+ <td class='blt c029'>M.Sc., Hon. Chem., F.I.C.P.A.</td>
+ <td class='brt c029'>M.Text.I., F.I.C.P.A.</td>
+  </tr>
+  <tr>
+ <td class='blt c029'>&#160;</td>
+ <td class='brt c029'>&#160;</td>
+  </tr>
+  <tr>
+ <td class='bbt blt brt c029' colspan='2'>W. H. BEESTON, R.P.A.</td>
+  </tr>
+</table>
+<div class='pbb'>
+ <hr class='pb c003'>
+</div>
+
+<div class='border'>
+
+<div class='nf-center-c0'>
+<div class='nf-center c004'>
+    <div><span class='xlarge'>BRITISH</span></div>
+    <div><span class='large'>SLIDE RULES</span></div>
+  </div>
+</div>
+
+<div class='lg-container-r'>
+  <div class='linegroup'>
+    <div class='group'>
+      <div class='line in20'>for all</div>
+      <div class='line in22'>ARTS and</div>
+      <div class='line in22'>INDUSTRIES</div>
+    </div>
+    <div class='group'>
+      <div class='line in24'>including</div>
+    </div>
+  </div>
+</div>
+
+<div  class='figcenter id006'>
+<img src='images/i_136.jpg' alt='[Slide Rule]' class='ig001'>
+</div>
+
+<div class='lg-container-l'>
+  <div class='linegroup'>
+    <div class='group'>
+      <div class='line'><span class='under'>LOG-O-LOG</span></div>
+      <div class='line'><span class='under'>DR. YOKOTA’S</span></div>
+      <div class='line'><span class='under'>SURVEYORS’</span></div>
+      <div class='line'><span class='under'>WIRELESS</span></div>
+      <div class='line'><span class='under'>GUNNERY</span></div>
+      <div class='line'><span class='under'>ELECTRICAL RULES, Etc.</span></div>
+    </div>
+  </div>
+</div>
+
+<div class='border'>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>SEND FOR LIST 55</div>
+  </div>
+</div>
+
+</div>
+
+<p class='c007'>MADE BY—</p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div><span class='large'><span class='sc'>John Davis &#38; Son</span> (Derby), Ltd.</span></div>
+    <div>ALL SAINTS’ WORKS, DERBY</div>
+  </div>
+</div>
+
+</div>
+<div class='pbb'>
+ <hr class='pb c003'>
+</div>
+<div class='border'>
+
+<div class='nf-center-c1'>
+<div class='nf-center c004'>
+    <div><span class='xlarge'>K &#38; E Slide Rules</span></div>
+  </div>
+</div>
+
+<p class='c000'>are constantly growing in popularity, and they can now be
+obtained from the leading houses in our line throughout the
+United Kingdom.</p>
+
+<div  class='figcenter id001'>
+<img src='images/i_137a.jpg' alt='[Slide Rule]' class='ig001'>
+</div>
+
+<p class='c048'>We manufacture a complete line of ENGINE-DIVIDED
+SLIDE RULES, and call special attention to our Patent
+Adjustment, ensuring smooth working of the Slide; also to
+our new “Frameless” Indicator, which hides no figures on
+the Rule.</p>
+
+<div  class='figcenter id001'>
+<img src='images/i_137b.jpg' alt='[Thacher’s Calculating Instrument]' class='ig001'>
+</div>
+
+<p class='c000'>THACHER’S CALCULATING INSTRUMENT, for
+solving problems in multiplication, division, or combinations
+of the two; has upwards of 33,000 divisions. Results can be
+obtained to the fourth and usually to the fifth place of figures
+with a surprising degree of accuracy.</p>
+
+<div class='nf-center-c1'>
+  <div class='nf-center'>
+    <div>We also make</div>
+    <div class='c003'>ALL METAL, CIRCULAR, STADIA, CHEMISTS’, ELECTRICAL, and                   OTHER SPECIAL SLIDE RULES</div>
+    <div><em>DESCRIPTIVE CIRCULARS ON REQUEST</em></div>
+    <div class='c003'>KEUFFEL &#38; ESSER CO.</div>
+    <div>127 Fulton St., NEW YORK&#8196; &#8196; &#8196; General Office and Factories, HOBOKEN, N.J.</div>
+    <div>CHICAGO − ST. LOUIS − SAN FRANCISCO − MONTREAL</div>
+    <div class='c003'><em>DRAWING MATERIALS</em></div>
+    <div><em>MATHEMATICAL and SURVEYING INSTRUMENTS</em></div>
+    <div><em>MEASURING TAPES</em></div>
+  </div>
+</div>
+
+</div>
+<div class='pbb'>
+ <hr class='pb c003'>
+</div>
+
+<div  class='figcenter id001'>
+<img src='images/i_138a.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p>6 in. Standard with magnifying Cursor complete in pocket case, 5/-</p>
+</div>
+</div>
+
+<div class='nf-center-c1'>
+<div class='nf-center c004'>
+    <div>NORTON</div>
+    <div>&#38;</div>
+    <div>GREGORY</div>
+    <div>LTD.</div>
+    <div class='c002'>Head Office</div>
+    <div class='c003'>CASTLE LANE, WESTMINSTER,            LONDON, S.W. 1</div>
+    <div class='c003'>Branches</div>
+    <div class='c003'>71 QUEEN STREET, GLASGOW.</div>
+    <div>PHOENIX HOUSE, QUEEN STREET and       SANDHILL, NEWCASTLE-ON-TYNE.</div>
+    <div class='c002'>SLIDE RULES in Stock, from 17/6 to 27/6</div>
+    <div class='c003'>Special Quotations to the Trade for Quantities</div>
+  </div>
+</div>
+
+<p class='c049'>For particulars of Surveying, Measuring and
+    Mathematical Instruments, Appliances and
+    Material of all kinds for the Drawing
+    Office, write to the Head Office.</p>
+
+<div  class='figcenter id001'>
+<img src='images/i_138b.jpg' alt='NORTON &#38; GREGORY LTD' class='ig001'>
+</div>
+<div class='pbb'>
+ <hr class='pb c003'>
+</div>
+
+<div class='border'>
+
+<div class='nf-center-c0'>
+<div class='nf-center c004'>
+    <div><span class='large'>NORTON &#38; GREGORY, LTD.,</span></div>
+    <div>London.</div>
+    <div class='c002'><span class='large'>“DIAMOND”</span></div>
+    <div><span class='xlarge'>DRAWING INSTRUMENTS</span></div>
+    <div>Manufactured at our London Works.</div>
+    <div class='c003'>CENTRE SCREW SPRING BOW HALF SET.</div>
+  </div>
+</div>
+
+<div  class='figleft id009'>
+<img src='images/i_139.jpg' alt='[Centre Screw Spring Bow Half Set]' class='ig001'>
+</div>
+
+<p class='c007'>4 inch Spring Bow Half Set centre screw
+adjustment, with interchangeable needle, pen, and
+pencil points       Price 17/6</p>
+
+<p class='c007'>The Centre Screw Spring Bow Half Set of Compasses,
+as illustrated, possesses the advantage of
+COMBINING IN ONE INSTRUMENT
+THE SET OF THREE SEPARATE SPRING
+BOWS hitherto in use, while the centre screw
+makes for ease and accuracy of manipulation, at
+the same time providing a radius of over 2 inches,
+or double that of the old pattern.</p>
+
+<p class='c007'>This instrument is less expensive than the set of 3
+bows, while considerably stronger in construction.</p>
+
+<p class='c007'>The fixed needle point is shouldered.</p>
+
+<p class='c007'>This illustration is given as an indication of the
+various Drawing Instruments manufactured by us.</p>
+
+<p class='c007'>Illustrated Booklet giving full particulars and
+prices of other Instruments and Cases of Instruments
+sent on application.</p>
+
+<div class='nf-center-c0'>
+<div class='nf-center c002'>
+    <div>Specially arranged Sets of Instruments made for Colleges, Schools, Technical Institutes</div>
+    <div class='c003'>Estimates submitted on Application.</div>
+    <div class='c002'><em>Write to our Head Office</em>:</div>
+    <div><span class='large'>CASTLE LANE, WESTMINSTER, LONDON, S.W. 1.</span></div>
+  </div>
+</div>
+
+</div>
+<div class='pbb'>
+ <hr class='pb c003'>
+</div>
+
+<div class='border'>
+
+<div class='nf-center-c0'>
+<div class='nf-center c002'>
+    <div><span class='large'>DRAWING AND SURVEYING INSTRUMENTS</span></div>
+  </div>
+</div>
+
+</div>
+
+<div  class='figcenter id010'>
+<img src='images/i_140.jpg' alt='A. G. THORNTON Ltd. Paragon Works 2 King St. West MANCHESTER' class='ig001'>
+<div class='ic010'>
+<p>SLIDE RULES FOR ENGINEERS&#8196; &#8196; &#8196; ACCURATE SECTIONAL PAPERS AND CLOTHS</p>
+</div>
+</div>
+
+<div class='border'>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>D 1916 Illustrated Catalogue, just published, in two editions; Drawing Office (448 pages); Draughtsman’s (160 pages): the most complete Catalogues in the trade.</div>
+    <div class='c003'><em>CONTRACTORS TO H.M. WAR OFFICE AND ADMIRALTY</em></div>
+    <div><em>Manufacturers also of Drawing Materials and Drawing Office Stationary.</em></div>
+    <div class='c003'>(ALSO AT MINERVA WORKS AND ALBERT MILLS, MANCHESTER.)</div>
+  </div>
+</div>
+
+</div>
+
+<div class='border'>
+
+<div class='nf-center-c0'>
+<div class='nf-center c002'>
+    <div>MATHEMATICAL INSTRUMENTS</div>
+    <div>SURVEYING INSTRUMENTS</div>
+    <div><span class='large'>SLIDE RULES</span></div>
+    <div>For Students and Engineers</div>
+    <div class='c003'>MANNHEIM, POLYPHASE, DUPLEX, ELECTRICAL, LOG-LOG, AND CALCULEX</div>
+    <div class='c003'><span class='large'>J. H. STEWARD LTD.</span></div>
+    <div>Scientific Instrument Makers</div>
+    <div class='c003'>406 STRAND, and 457 WEST STRAND</div>
+    <div>LONDON, W.C. 2</div>
+  </div>
+</div>
+
+</div>
+<div class='pbb'>
+ <hr class='pb c003'>
+</div>
+
+<div class='border'>
+
+<div class='nf-center-c0'>
+<div class='nf-center c004'>
+    <div><span class='under'>A True Friend and Trusty Guide</span></div>
+    <div class='c002'><span class='large'>THE</span></div>
+    <div><span class='xlarge'>‘HALDEN CALCULEX’</span></div>
+  </div>
+</div>
+
+<div  class='figcenter id006'>
+<img src='images/i_141.jpg' alt='[Halden Calculex]' class='ig001'>
+</div>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>ACTUAL SIZE|&#160;|&#160;|&#160;BRITISH MADE</div>
+    <div class='c003'>The handiest and most perfect form of Slide Rule.</div>
+    <div>Does all that can be done with a straight rule.</div>
+    <div>Complete in Case, with book of instructions,</div>
+    <div>27/6 post free.</div>
+    <div class='c003'>J. HALDEN &#38; CO., LTD., 8 ALBERT SQUARE MANCHESTER</div>
+    <div class='c003'><em>Depots</em>—London, Newcastle-on-Tyne, Birmingham, Glasgow, and Leeds</div>
+  </div>
+</div>
+
+</div>
+<div class='pbb'>
+ <hr class='pb c003'>
+</div>
+<div class='border'>
+
+<div  class='figleft id005'>
+<img src='images/i_142a.jpg' alt='[Rope]' class='ig001'>
+</div>
+
+<div class='nf-center-c0'>
+<div class='nf-center c002'>
+    <div>ENGINEERING,</div>
+    <div>SURVEYING</div>
+    <div>AND</div>
+    <div>MATHEMATICAL</div>
+    <div>INSTRUMENTS,</div>
+    <div>ETC.</div>
+    <div class='c002'><span class='large'>SLIDE RULES.</span></div>
+    <div class='c003'>JOSEPH CASARTELLI &#38; SON,</div>
+    <div>43 MARKET STREET, MANCHESTER.</div>
+    <div>Tel. No. 2958 City.|&#160;|&#160;|&#160;Established 1790.</div>
+  </div>
+</div>
+
+</div>
+
+<div class='border'>
+
+<p class='c007'><span class='large'>ROPE DRIVING</span></p>
+
+<p class='c050'>Is the most EFFICIENT and most ECONOMICAL METHOD
+of Power Transmission.</p>
+
+<p class='c007'><span class='large'>The LAMBETH Cotton Driving Rope</span></p>
+
+<p class='c050'>Is the most EFFICIENT and most ECONOMICAL ROPE
+for Power Transmission.</p>
+
+<div  class='figcenter id006'>
+<img src='images/i_142b.jpg' alt='' class='ig001'>
+<div class='ic001'>
+<p>Made 4 Strand or 3 Strand.</p>
+</div>
+</div>
+
+<p class='c007'>SPECIAL FEATURES:</p>
+
+<p class='c050'>LESS STRETCH THAN ANY OTHER ROPE.
+MORE PLIABLE THAN ANY OTHER ROPE.
+GREATER DRIVING POWER THAN ANY OTHER ROPE.</p>
+
+<div class='nf-center-c0'>
+  <div class='nf-center'>
+    <div>THOMAS HART LTD., Lambeth Works, BLACKBURN.</div>
+  </div>
+</div>
+
+</div>
+
+<div class='pbb'>
+ <hr class='pb c003'>
+</div>
+<div class='tnotes x-ebookmaker'>
+
+<div class='chapter ph2'>
+
+<div class='nf-center-c0'>
+<div class='nf-center c004'>
+    <div>TRANSCRIBER’S NOTES</div>
+  </div>
+</div>
+
+</div>
+
+<table class='table0'>
+  <tr>
+    <th class='c013'>Page</th>
+    <th class='c013'>Changed from</th>
+    <th class='c014'>Changed to</th>
+  </tr>
+  <tr>
+    <td class='c015'><a href='#t24'>24</a></td>
+    <td class='c041'>the right, so the number of digits in the answer = 3 − 2 × 1 = 2</td>
+    <td class='c043'>the right, so the number of digits in the answer = 3 − 2 + 1 = 2</td>
+  </tr>
+  <tr>
+    <td class='c015'><a href='#t116'>116</a></td>
+    <td class='c041'>grammes, we have the equation, <em>x</em> × <span class='fraction'>Cl.<br><span class='vincula'>Ag.Cl.</span></span> × <span class='fraction'><em>a</em><br><span class='vincula'><em>s</em></span></span>. Hence, the mark</td>
+    <td class='c043'>grammes, we have the equation, <em>x</em> = <span class='fraction'>Cl.<br><span class='vincula'>Ag.Cl.</span></span> × <span class='fraction'><em>a</em><br><span class='vincula'><em>s</em></span></span>. Hence, the mark</td>
+  </tr>
+</table>
+
+  <ul  class='ul_1'>
+    <li>Typos fixed; non-standard spelling and dialect retained.
+
+    </li>
+    <li>Used numbers for footnotes, placing them all at the end of the last chapter.
+    </li>
+  </ul>
+
+</div>
+
+<div style='text-align:center'>*** END OF THE PROJECT GUTENBERG EBOOK 75904 ***</div>
+  </body>
+  <!-- created with ppgen.py 3.57e (with regex) on 2025-04-18 15:49:07 GMT -->
+</html>
+
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diff --git a/LICENSE.txt b/LICENSE.txt
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--- /dev/null
+++ b/LICENSE.txt
@@ -0,0 +1,11 @@
+This book, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
+Procedures for determining public domain status are described in
+the "Copyright How-To" at https://www.gutenberg.org.
+
+No investigation has been made concerning possible copyrights in
+jurisdictions other than the United States. Anyone seeking to utilize
+this book outside of the United States should confirm copyright
+status under the laws that apply to them.
diff --git a/README.md b/README.md
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--- /dev/null
+++ b/README.md
@@ -0,0 +1,2 @@
+Project Gutenberg (https://www.gutenberg.org) public repository for
+book #75904 (https://www.gutenberg.org/ebooks/75904)