Part 27

(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, 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^{10n} 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 nth power or the nth 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^{100n}.

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.

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).