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The slide rule

Chapter 22: MISCELLANEOUS POWERS AND ROOTS.
By Charles N. Pickworth
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About This Book

The manual defines the slide rule as an instrument for performing calculations by means of logarithms and provides a concise primer on common logarithms, mantissa and characteristic. It explains the mathematical principle that adding and subtracting logarithms yields products and quotients, and extends this to powers, roots, and trigonometric and algebraic operations. Practical instruction covers the layout and use of scales, slide and cursor, magnifying cursors and special rule types, with examples of routine procedures such as multiplication, division, square and root extraction, and proportional calculations. Descriptions of specialized instruments and applications include engineering tasks like screw-cutting and gear calculations, together with advice on accuracy and practice.

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 N⅔ 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 N⅔ as N ÷ ∛̅N, as in this way the magnitude of the result is much more readily appreciated.

Three-two Power.—N³⁄₂ 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 N³⁄₂ as N × √̅N.

Fourth Power.—For N4 set the index of C to N on D and over N on C read N4 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.—N⁴⁄₃ = N1·33 (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, N2 over N; N3 over N2; N4 over N3; N5 over N4, etc. In the same way, setting N on B to N on D, we can read such values as N¾, N⅞, 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 an = 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 361·414.

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 nth root of any number.