The slide rule

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.

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

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.

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.

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.

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.

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 = 76·28d^⁵⁄₇s^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.

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.