Part 3

“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.

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.

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.