142. Calculation by the Logarithmic Scale. The same principles apply to the use of the logarithmic scale as to the actual logarithms; namely, to multiply, add the logarithms, and to divide, subtract the logarithms. The only difference is that logarithms on the scale are represented by lengths instead of numbers, and we add and subtract lengths.

To multiply two numbers, such as 2 X 3, simply lay off from the scale the length from 1 to 2 and add to it the length from 1 to 3, exactly as though you had a foot rule and laid off 2 in. and then added 3 in. to it. In Fig. 99 we have multiplied 2 X 3, step by step. In the first operation the scale is laid upon the paper and the length from 1 to 2 marked off, which distance is the logarithm of 2. In the second operation the scale is moved to the right and the length from 1 to 3, or the logarithm 3, is added to the length already measured, thus giving the sum of the two lengths, or log 2 + log 3.

From our knowledge of logarithms we know that log 2 + log 3 = log of product. We, therefore, take our logarithmic scale, in the third operation, and measuring the total length (log 2 + log 3), we find that it equals the length from 1 to 6, which is the log of 6.

Therefore,

Log 2+ Log 3 = Log 6

2 X 3 6, Answer.

=

Since in division we subtract logarithms, the process of dividing one number by another with the logarithmic scale is just the reverse of multiplying. To illustrate, let us divide 6 by 3. In this case we must subtract the log of 3 from the log of 6 to obtain the log of the quotient, which will require three operations, as shown in Fig. 100.

First operation: Mark off the length from 1 to 6 which is the log of 6.

Second operation: Move the scale to the right and subtract the length 1 to 3, or the log 3, from the log 6.

Third operation: The difference found in the second operation is the log of the quotient, so we move the scale to the left and measure the log quotient. As the scale indicates, the log quotient equals the log 2; therefore,

Thus by the aid of the logarithmic scale we have performed the multiplication and division of two numbers without having to consider the actual numerical value of the logarithms themselves.

The single logarithmic scale which we have just described and used is the most elementary form of slide rule. A similar scale was used for many years before the modern type of instrument was developed.

143. The Simple Slide Rule. If we possessed two logarithmic scales exactly alike, the three operations required for multiplication or division, Figs. 99 and 100, could be combined into a single operation, thereby effecting a considerable saving in time.

Figure 101 illustrates the method of performing the multiplication, 2 X 3, with two logarithmic scales. The log of 2 and the log of 3 are added by placing the two scales together with the 1 of scale C opposite the 2 of scale D. The total length (log 2 + log 3) can then be read from the scale D at the point which is opposite the 3 of scale C. As the figure shows, the sixth division of scale D is opposite the 3 of scale C, and it follows that

To perform a division, such as 84, the log 4 is subtracted from the log 8 by placing the 4 of scale C above the 8 of scale D, as in Fig. 102. The difference, which is the log of the quotient. is measured on the scale D.

The question now arises, how shall we multiply numbers other than those from 1 to 10, for example, 2 X 400? Apparently there is no such value as 400 on the scale.

In Fig. 103 we have extended the two logarithmic scales, C and D, so as to contain numbers from 1 to 1000, and with these scales have multiplied 2 × 400 by the same method that we used with the smaller scales.

Log 2+ Log 400 = Log 800

2 X 400 = 800, Answer.

It is evident from the figure that each of these scales could be divided into three sections, the lengths from 1 to 10, from 10 to 100, and from 100 to 1000, and also that these three sections