DIVISION.
The instructions for multiplication having been given in some detail, a full discussion of the inverse process of division will be unnecessary.
Rule for Division.—Place the divisor on C, opposite the dividend on D, and read the quotient on D under the index of C.
Ex.—225 ÷ 18 = 12·5.
Bringing 18 on C to 225 on D, we find 12·5 under the L.H. index of C.
As in multiplication, the factors are treated as whole numbers, and the position of the decimal point afterwards decided according to the following rule, which, as will be seen, is the reverse of that for multiplication:—
Rule for the Number of Digits in a Quotient.—If the quotient is read with the slide projecting to the LEFT, subtract the number of digits in the divisor from those in the dividend; but if read with the slide to the RIGHT, ADD 1 to this difference.[2]
In the above example the quotient is read off with the slide to the right, so the number of digits in the answer = 3 − 2 + 1 = 2.
Ex.—0·000221 ÷ 0·017 = 0·013.
Here the number of digits in the dividend is −3, and in the divisor −1. The difference is −2; but as the result is obtained with the slide to the right, this result must be increased by 1, so that the number of digits in the quotient is −2 + 1 = −1, giving the answer as 0·013.
If preferred, the result can be obtained in the manner referred to when considering the multiplication of decimals. Thus, treating the above as whole numbers, we find that the result of dividing 221 by 17 = 13, since the difference in the number of digits in the factors, which is 1, is, owing to the position of the slide, increased by 1, giving 2 as the number of digits in the answer. Then by the rules for the division of decimals we know that the number of decimal places in the quotient is equal to 6 − 3 = 3, showing that a cypher is to be prefixed to the result read on the rule.
As in multiplication, so in division, we have a
General Rule for Number of Digits in a Quotient.—When the first significant figure in the DIVISOR is greater than that in the DIVIDEND, the number of digits in the quotient is found by subtracting the digits in the divisor from those in the dividend. When the contrary is the case, 1 IS TO BE ADDED to this difference. When the first figures are the same, those following must be compared.
Estimation of the Figures in a Quotient.—The method of roughly estimating the number of figures in a quotient needs little explanation.
Ex.—3·95 ÷ 5340 = 0·00074.
Setting 534 on C to 3·95 on D we read under the (R.H.) index of C, the significant figures on D, which are 74. Then 3·9 ÷ 5 is about 0·8 and 0·8 ÷ 1000 gives 0·0008 as a rough estimate.
Ex.—0·00000285 ÷ 0·000197 = 0·01446.
Regarding this as 2·85 × 10−6 ÷ 1·97 × 10−4 we divide 2·85 by 1·97 and obtain 1·446. Dividing the powers of 10 we have 10−6 ÷ 10−4 = 10−2, so the decimal point is to be moved two places to the left and the answer is read as 0·01446.
Another method of dividing deserves mention as of special service when dividing a number of quantities by a constant divisor:—Set the index of C to the divisor on D and over any dividend on D, read the quotient on C.
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.