# Slide Rule

A Slide Rule is a mechanical device for making rapid mathematical calculations accurate to several significant figures. In particular, slide rules can be used to multiply and divide, to square a number or find a square root, to find the reciprocal of a. number, to evaluate a trigonometric function, or to find the logarithm of a number.

There are various models of slide rules, the most popular one having a long rectangular shape. The standard rectangular slide rule consists of a wooden frame, a wooden slide that can be moved left or right, and a transparent indicator with a hairline that can be moved left or right over the frame and the slide. The frame consists of two panels that are rigidly fixed together. The slide panel moves between the panels forming the frame. Scales for making various calculations appear on the three panels.

Logarithms are the basis for the construction of the scales on a slide rule. For example, the C scale, the D scale, or the A scale is constructed as follows. Some suitable length is divided from left to right into 10 parts, with the end of each successive division being at a distance proportional to the logarithms of 2, 3, 4, 5, 6, 7, 8, 9, and 10. Such a scale is said to be a decilogarithmic scale because 10 divisions are used.

## Multiplication and Division

The C and D scales are used to multiply and divide numbers. For example, consider the multiplication of 1.5 by 1.6. To perform this multiplication, the C scale left index (the "1" mark at the left end of the C scale) is slid into alignment with the D scale graduation that represents the multiplicand (1.5), and then the hairline is slid over the C scale graduation that represents the multiplier (1.6). The D scale graduation that appears under the hairline then represents the product (2.4). Note that multiplication of 15 by 16 equals 240, and so on.

Division is performed by reversing the procedure used for multiplication. In dividing 2.4 by 1.6, the hairline is slid to the D scale graduation representing the dividend (2.4), and then the slide is moved so that the C scale graduation representing the divisor (1.6) is under the hairline. The D scale graduation that appears under the C scale index then represents the quotient (1.5).

## Squares and Square Roots

The square of a number is found by moving the hairline to that number as represented on the D scale and then reading its square under the hairline on the A scale. The square root of a number is found by setting the hairline on the number on the A scale and then reading its square root under the hairline on the D scale.

## Decimal Point Location

If the numbers 155.5 and 16.05 are multiplied by slide rule, the scale reading for the result is 2495000 . . . , but the position of the decimal point is not known. One way to place the decimal point is to find the approximate answer by mental arithmetic. For instance, the factors given here can be rounded to 150 times 15. By mental arithmetic one obtains 15 x 10 x 15 = 152 x 10 = 2250. The actual result would be somewhat greater than this but certainly contains four digits to the left of the decimal point. Thus the answer must be 2495.

## Reciprocals

The CI scale is constructed so that when the hairline is over a number on the C scale, then the reciprocal of the number is under the hairline on the CI scale. For instance, if the hairline is over the 2 on the C scale, its reciprocal, 0.5 is under the hairline on the CI scale.

## Trigonometric Functions of Angles

The value of a natural sine or cosine is found by using the S and C scales. It is done simply by moving the hairline to the desired angle on the S scale and then reading the value under the hairline on the C scale. There are two scales on the S scale, one reading from left to right (6, 7, 8, 9, 10, 15, . . . , 90), and the other reading from right to left (0, 20, 30, 40, 50, . . ., 84). The former scale is used when the sine is to be evaluated, and the latter scale when the cosine is to be evaluated.

When the tangent is to be evaluated, the T and C or T and CI scales are used. If the angle is 45° or less the hairline is moved to the angle as given on the T scale from left to right, and the value of the tangent is read off the C scale. If the angle is larger than 45°, the angle is found by reading the T scale from right to left, setting the hairline at the angle as given on the T scale, and then reading the value of the tangent from the CI scale.

## Logarithm of a Number

The mantissa of a common logarithm (logarithm to the base 10) of a number is found by using the D scale and the L scale. For example, if the number is 9 or 90 or 900, the hairline is moved to the number 9 on the D scale and then the mantissa, .954, is read under the hairline on the L scale. The logarithm of 9 is 0.954, the logarithm of 90 is 1.954, and the logarithm of 900 is 2.954, with the number to the left of the decimal point called the characteristic, being the appropriate power of 10.

## Historical Background

The first slide rule, a circular one, was constructed by the English mathematician William Oughtred in the 1620's or early 1630's. He also devised a rectangular slide rule by the mid-1630's. The modern arrangement of the slide-rule scales was originated by the French mathematician Amedee Mannheim about 1850.

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