The slide rule is extensively used in engineering, science and commerce for rapidly performing calculations involving multiplication and division which have to be accurate to not more than three or four decimal places. It can also be used for such operations such as involution (raising to a power) and evolution (extraction of a root) and for calculations with trigonometric functions (sine, cosine, tangent, cotangent).

In addition to those for general use there are many different types of special-purpose slide rules. What they all have in common is logarithmic scales. The slide rule, also known as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for scientific functions such as roots, logarithms and trigonometry, but does not generally perform addition or subtraction.

To understand how a slide rule works it is essential to know about logarithms. In general, the logarithm of a value a to base b is the exponent n denoting the power to which b must be raised in order to obtain a; i.e., if a=bn , then log ba =n, where a itself is called the antilogarithm. Hence logarithms are exponents. In particular, so-called common, or Briggsian, logarithms are exponents of the base 10, this being the most convenient base for purposes of general computation. (In theory any other base could be adopted for a system of logarithms. Of special important besides common logarithms, are the natural, or naperian, logarithms, which have the value e=2.718 … for their base and are important in higher mathematical analysis).

Since 1=100, 10 = 101, 100=102, 1000=103 etc., the common logarithms of 1,10,100,1000 etc. are equal to 0,1,2,3, etc. This is set forth in Table 1. Any number between 1 and 10 has a logarithm which lies between 0 and 1; numbers between 10 and 100 have logarithms between 1 and 2 etc. Whereas the logarithms of whole powers of 10 are whole numbers, the logarithms of intermediate values are decimal fractions, as exemplified by the logarithms of numbers between 0 and 10 listed in table 2.

When two or more values which are powers of the same base are multiplied, their product is obtained by adding the exponents. For example: 102x105 = 100x1000 =100,000 = 105. Similarly, division is done by subtraction of exponents.

For e.g. 105 /103= 1,00,000 /1000= 100 =102 In general 10m x 10n = 10m+n and 10m/10n =10m-n. Thus multiplication and division can be reduced to the simpler operations of addition and subtraction if, instead of the actual numbers, the logarithms of those numbers are employed. For this purpose the logarithms of the numbers to be multiplied or divided are looked up in suitable tables and are added or subtracted. The result thus obtained is the logarithm of the required answer, which can be looked up (as the antilogarithm) in the same logarithm tables. Example: to calculate 2 x 3 using logarithms: log 2x3 = log 2 + log 3 = 0.301 +0.477 = 0.778; the antilogarithm of 0.778 is 6.

An ordinary slide rule consists of the actual rule, the slide, and the transparent cursor with a hairline. Various logarithmic scales are engraved on the rule and the slide (Fig.2). When the rule is closed the pairs of scales A and B, and C and D, respectively coincide. The divisions on A and B extend from 1 to 100; those on C and D extend from 1 to 10.

To determine the square of a value, the hairline is moved to that value on Scale C and D, and the square is indicated by the hairline on scale B or A (Fig.3). Square roots are obtained by the reverse procedure (Fig.4). The scale K (above A in Fig.2) gives the cubes of the values on scale D (Fig.3). Conversely, scale D gives the cube roots of the values on K (Fig.4).

For multiplication, the scales C and D are preferably used (Fig.5). It may occur that the slide has to be moved too far to the right to give a reading of the answer. In that case the slide should be moved back to the left until the figure 10 instead of the figure 1 on scale C is opposite the first factor of the multiplication. The result is then obtained in the usual way on scale D, opposite the second factor on scale C. Alternatively, the scales A and B may be used, though with reduced accuracy of the readings.

Division is performed as indicated in Figs 7 and 8. Operations involving both multiplication and division may be carried out in accordance with Fig.9. On a standard slide rule there is, between the scales C and D, another scale, usually marked CI. It gives the reciprocals of the values on scale C (Fig.10). For the values on scale B the scale CI gives the reciprocals of the square roots; for example, a value a on scale B corresponds to 1/va on scale CI. Reciprocals can be used advantageously for multiplying (dividing by the reciprocal value) and also for complex multiplications and divisions involving several factors (Fig.11).