Part 6

For the division of a constant dividend by a variable divisor, set the cursor to the dividend on D and bring the divisor on C successively to the cursor, reading the corresponding quotients on D under the index of C. Another method which avoids moving the slide is explained in the section on “Multiplication and Division with the Slide Inverted.”

CONTINUED DIVISION, if we can so call such an expression as

(3·14)/(785 × 0·00021 × 4·3 × 64·4) = 0·0688,

may be worked by repeating as follows:—Set 7·85 on C to 3·14 on D, bring cursor to index of C, 2·1 on C to cursor, cursor to index, 4·3 to cursor, cursor to index, 6·44 to cursor, and under index of C read 688 on D as the significant figures of the answer.

For the number of figures in the result, we deduct the sum of the number of digits in the several factors and add 1 for each time the slide projects to the right, which in this case occurs once. There are 3 + (−3) + 1 + 2 = 3 denominator digits, 1 numerator digit, and 1 is to be added to the difference. Therefore there are 1 − 3 + 1 = −1 digits in the answer, which is therefore 0·0688. The foregoing method of working may confuse the beginner, who is apt to fall into the process of continued multiplication. For this reason, until familiarity with combined methods has been acquired, the product of the several denominators should be first found by the continued multiplication process, and the figures in this product determined. Then divide the numerator by this product to obtain the result.

As the denominator product will be read on D, we may avoid resetting the slide by bringing the numerator on C to this product and reading the result on C over the index of D. The slide and rule have here changed places; hence if rules are followed for the number of figures in the result, 1 must be added to the difference of digits, when the rule projects to the right of the slide.

The author’s method of recording the number of times division is performed with the slide to the right is by vertical memorandum marks, thus |. The full significance of these memo-marks will appear in the following section.

For a rough calculation to fix the decimal point, in this example we move the decimal points in the factors, obtaining

(3)/(0·8 × 2 × 4 × 6) = (3)/(40) = 0·075.

THE USE OF THE UPPER SCALES FOR MULTIPLICATION AND DIVISION.

Many prefer to use the upper scales A and B, in preference to C and D. The disadvantage is that as the scales are only one-half the length of C or D, the graduation does not permit of the same degree of accuracy being obtained as when working with the lower scales. But the result can always be read directly from the rule without ever having to change the position of the slide after it has been initially set. Hence, it obviates the uncertainty as to the direction in which the slide is to be moved in making a setting.

When the A and B scales are employed, it is understood that the left-hand pair of scales are to be used in the same manner as C and D, and so far the rules relating to the latter are entirely applicable. But in this case the slide is always moved to the right, so that in multiplication the product is found either upon the left or right scales of A. If it is found on the left A scale, the rule for the number of digits in the product is found as for the C and D scales, and is equal to the sum of the digits in the two factors, minus 1; but if found on the right-hand A scale, the number of digits in the product is equal to the sum of the digits in the two factors.

In division, similar modifications are necessary. If when moving the slide to the right the division can be completely effected by using the L.H. scale of A, the quotient (read on A above the L.H. of index B) has a number of digits equal to the number in the dividend, less the number in the divisor, plus 1. But if the division necessitates the use of both the A scales, the number of digits in the quotient equals the number in the dividend, less the number in the divisor.

RECIPROCALS.

A special case of division to be considered is the determination of the reciprocal of a number n, or (1)/(n). Following the ordinary rule for division, it is evident that setting n on C to 1 on D, gives (1)/(n) on D under 1 on C. It is more important to observe that by inverting the operation—setting 1 (or 10) on C to n on D—we can read (1)/(n) on C over 1 (or 10) on D. Hence whenever a result is read on D under an index of C, we can also read its reciprocal on C over whichever index of D is available.

The Number of Digits in a Reciprocal is obvious when n = 10, 100, or any power (p) of 10. Thus (1)/(10) = 0·1; (1)/(100) = 0·01; (1)/(10^{p}) = 1 preceded by p − 1 cyphers. For all other cases we have the rule:—Subtract from 1 the number of digits in the number.

EX.—(1)/(339) = 0·00295.

There are 3 digits in the number; hence, there are 1 − 3 = −2 digits in the answer.

EX.—(1)/(0·0000156) = 64,100.

There are −4 digits in the number; hence, there are 1 − (−4) = 5 digits in the result.

CONTINUED MULTIPLICATION AND DIVISION.

By combining the rules for multiplication and division, we can readily evaluate expressions of the form (a)/(b) × (c)/(d) × (e)/(f) × (g)/(h) = x. The simplest case, (a × c)/(b) can be solved by one setting of the slide. Take as an example, (14·45 × 60)/(8·5) = 102. Setting 8·5 on C to 14·45 on D, we can, if desired, read 1·7 on D under 1 on C, as the quotient. However, we are not concerned with this, but require its multiplication by 60, and the slide being already set for this operation, we at once read under 60 on C the result, 102, on D. The figures in the answer are obvious.

When there are more factors to take into account, we place the cursor over 102 on D, bring the next divisor on C to the cursor, move the cursor to the next multiplier on C, bring the next divisor on C to the cursor, and so on, until all the factors have been dealt with. Note that only the first factor and the result are read on D; also that the cursor is moved for multiplying and the slide for dividing.

Number of Digits in Result in Combined Multiplication and Division.—For those who use rules the author’s method of determining the decimal point in combined multiplication and division may be used. Each time multiplication is performed with the slide projecting to the right, make a − mark; each time division is effected with the slide to the right, make a | mark; but allow the | marks to cancel the − marks as far as they will. Subtract the sum of the digits in the denominator from the sum of digits in the numerator, and to this difference add any uncancelled memo-marks, if of | character, or subtract them if of − character.

EX.—(43·5 × 29·4 × 51 × 32)/(27 × 3·83 × 10·5 × 1·31) = 1468.

Set 27 on C to 43·5 on D, and as with this division the slide is to the right, make the first ⵏ mark. Bring cursor to 29·4 on C, and as in this multiplication the slide is to the right, make the first − mark, cancelling as shown. Setting 3·83 on C to the cursor, requires the second ⵏ mark, which, however, is cancelled in turn by the multiplication by 51. The division by 10·5 requires the third ⵏ mark, and after multiplying by 32 (requiring no mark) the final division by 1·31 requires the fourth ⵏ mark. Then, as there are 8 numerator digits, 6 denominator, and 2 uncancelled memo-marks (which, being 1, are additive) we have

Number of digits in result = 8 − 6 + 2 = 4.

Had the uncancelled marks been − in character, the number of digits would have been 8 − 6 − 2 = 0.