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figures is two, and the answer is read from the same index as the multiplicand, so that the quotient has one figure.

64 × 24

15.36. The sum of figures is two, but the answer is read at a different index to the multiplicand, and therefore the quotient has two figures.

12 × 0014201704. The sum of the figure is 0, but the answer is read at the same index with the multiplicand, and therefore the quotient has one less, or minus one.

64

0024

1536. The sum of the figures is 0, and the answer is read at a different index to the multiplicand, and therefore the quotient has 0 figures.

48 42 x 64·34 = 3115·3. The sum of the figures is 4, and the answer is read at a different index to the multiplicand, and therefore the quotient has four figures.

DIVISION.

Rule.-Place divisor to fixed index, and the upper or lower movable index to the dividend, according as the first figure in the divisor is greater or less than the first figure in the dividend. Then move the cylinder so that the fixed index is at 100, and read the quotient at one of the movable indices.

The number of figures in the quotient is the algebraic difference between the number of figures in the dividend and divisor, if it is not read upon

the same index as the dividend.

It is one more

than that difference if read upon the same index.

Examples.-146863

=

23.3, as the difference is 2, and the quotient is not read upon the same index as the dividend.

1468125 = 11·7, as the difference is 1, and the quotient is read upon the same index as the dividend, and therefore has two figures.

· 146863 = ·00233, as the difference is 2, and the quotient is not read upon the same index as the dividend, and therefore has 2 figures.

146800125 = 1174000, as the difference is 4-(-2) = 4+2 = 6, and the quotient is read upon the same index as the dividend, and therefore has seven figures.

MULTIPLICATION AND DIVISION.

Rule.-Move the cylinder so as to place the denominator to the fixed index. Then place movable index to one of the numerators. Then move the cylinder so that the fixed index points to the other numerator, and read the quotient at one of movable indices.

The number of figures in the quotient is the algebraic difference between the sum of the number of figures in the numerator and in the denominator, if it is read upon the same index as a factor of the numerator. It is one more than that difference if read upon the other index.

Example.

4854 × 32.6

536

= 295.22, as the differ

ence is 4+2 - 3 = 3, and the quotient is read at the same index as either 4854 or 32.6 is placed.

⚫0764 × ⚫032
14.63

= 000167, as the difference is

(− 1) + ( − 1) −24, and the quotient is not read upon the index, that either 0764 or ⚫032 is placed, and therefore the number of figures is 3.

To multiply three numbers together when one of them is a constant in frequent use.

Rule. Find the reciprocal of the constant by division, and use it as the divisor in the preceding rule.

RATIO.

When either of the movable indices is at one number and the fixed index at another, and the cylinder is turned into any other position, though the numbers at the indices will be different, their ratio will remain constant.

Example.-To convert francs and centimes into sterling money, supposing exchange 25f. 45c. for 11. The ratio between centimes and pence is 2545 to 240. Place the cylinder so that the fixed index is at 2545, and make one of the movable indices point to 240. Then on moving the cylinder to read off different numbers of centimes at the fixed index, the corresponding value in pence will be read at the movable index.

Wages Table.-To find the wages for different times at 358. per week of 57 hours. Place the

the

cylinder so that the fixed index is at 57, and make one of the movable indices point to 420, number of pence in 358. Then on moving the cylinder to read off different numbers of hours at the fixed index, the corresponding wages in pence will be read at the movable index.

PROPORTION.

To find a third proportion to two numbersabb:c

bx b

c =

a

Proceed according to rule for multi

plication and division.

To find a fourth proportion to three numbersa b c d

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To obtain the square, cube, and fourth power of a number. The quickest way with this rule is by direct multiplication.

For higher powers and roots. Place the upper movable index (c) to the number, and read the scales (n and m). These together give the mantissa of the logarithm of the number. To this the index has to be added. The index of the logarithm of a number greater than unity is one less than the number of figures in the integral part of that number. Thus the index of 5432 is 3, of 543.2 is 2, of 54 32 is 1, and of 5·432 is 0.

Multiply or divide the resulting number by the power or root, as shown above. Then place the cylinder so that it reads on the scales (n and m) the decimal part of the quotient. The power or root is then at the index (c). In the result the number of figures before the decimal point is one more than the number in the integral part of the above quotient.

The scale (n) is read from the lowest line of the top spiral and (m) from the vertical edge of the scale (n).

Examples.-513, on placing (c) to 500, scale (n) reads 68 and scale (m) 01897, which gives the logarithm of 569897, the index being 0. Then 69897 x 13 = 9.08661. Now placing the cylinder so that it reads 08661 on scales (n and m) the index (c) reads 12207, and the required power is 1220700000, having 10 figures, as the integral part of the above quotient is 9.

741 on placing (c) to 741, scale (n) reads ⚫86 and scale (m) 00982 which gives the logarithm of 741-2.86982, the index being 2. Then 2.86982557396. Now placing the cylinder so that it reads 57396 on scales (n and m) thẹ index (c) reads 37495, and the required root is 3.7495, having one figure before the decimal point, as the integral part of the above quotient is 0.

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POWERS OF DECIMAL FRACTIONS.

To avoid the use of negative indices, which often lead to erroneous results unless they are

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