Part 10

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 10n, and 10n 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, 10n and 100n (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.

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.