Fig. 101. The slide-rule consists of four lines, viz., A, B, C, D; A being on the upper edge of the rule, B being on the upper edge of the slide, C being on the lower edge of the slide, and D being on the lower edge of the rule. The lines A and B work together, and the lines C and D work together. The divisions and numbers on A and B are exactly alike; and, when closed, they stand thus: -

But, if 1 on the slide B is set to 2 on the rule A, then the numbers will stand thus: -

It will be seen that the proportion of 2 to 1 runs throughout, each number on A being the product of the number immediately underneath, on B, multiplied by 2; or, inversely, each number on B being the result of dividing the number immediately above, on A, by 2.

If 1 on the slide B is set to 3 on the rule A, the numbers will stand thus: -

It will be seen that the proportion 3/1 runs throughout, each number on A being the product of the number immediately underneath, on B, multiplied by 3; or, inversely, each number on B being the result of dividing the mini-ber immediately above, on A, by 3.

The C and D lines are relatively different, each number on the slide C being the square or self multiple of the number immediately underneath, on the rule D; or, inversely, each number on D being the square root of each number immediately above it, on C*

The numbers and divisions are to be read decimally; for the spaces are, or are supposed to be, divided and subdivided into tens and tenths. The ordinary reading of the divisions on the lines A, B, and C, is, beginning at the left, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, which is marked 1; 11, which is not numbered; 12; then the intermediate numbers, 13, 14, 15, etc., which are not numbered, up to 20, which is marked 2; then the intermediate numbers, 21, 22, 23, etc., up to 30, which is marked 3; then continuing on to 40, which is marked 4; 50, which is marked 5; 60, marked 6; 70, marked 7; 80, marked 8; 90, marked 9 ; and 100, which is marked 10. Between 1 and 2 are the first division beyond 1 is l 2/100, the second is 1 4/100, the fifth is 1 10/100 or 1 1/10; the division next to 2 is 1 98/100, etc. Between 2 and 3 the divisions are tenths and half-tenths, a half-tenth being 5/100. From 3 to 10 the divisions are nil tenths; from 10 to 20 each subdivision represents 2/10; the first division beyond 20 represents 20 5/10; the second division represents 21; the third division represents 21 5/10; the fourth represents 22, and so on up to 30; from 30 up to 100 each division represents 1.

10 principal divisions, which indicate 1 plus any number of tenths. The first principal division beyond 1 indicates 1 1/100, the second division indicates 1 2/10, and so on up to 2. Each of these principal divisions between 1 and 2 are subdivided into 5 parts, each part representing 2/100: so

* The square of any number is the result obtained by multiplying that number by itself: thus the square of 2 is 2 x2=4; the square of 3 is 3 x 3 = 9. The square root of any number is that number which, when multiplied oy itself, will produce the given number: thus the square root of 4 is 2, since 2x2=-4; the square root of 9 is 3, since 3x3 = 9.

These numbers, marked 1, 2, 3, etc., are arbitrary, and have no fixed values; for, beginning at the left, 1 might represent 10; 2 would represent 20; each of the principal divisions between 1 and 2, which in the ordinary reading represented tenths, would represent 1; each of the subdivisions, which in the ordinary reading represented 2/100, would represent 2/10; 3 would represent 30; the number which formerly represented 10 would represent 100; the number which formerly represented 12 would represent 120; the number formerly representing 20 would represent 200, and so on, the value of the whole line being increased tenfold; or, 1 at the left might represent 100, 2 would represent 200, and so on, the value of the whole line being increased one hundred-fold. On different lines, 1 may bear different values in working out a problem. For example: Multiply 40 by 5. We set 1, which is on the line B, to 5, on the line A: above 4, which we will call 40, on the line B, we find 20 on the line A; but, since we have increased the value of one of the divisions tenfold its ordinary value, we must increase the result the same, which gives us 200 as the answer. The line D is divided the same as A, B, and C are, from 1 to 10, only on a larger scale; and 1 on this line may represent 1, 10, or 100, the same as the other lines. It will require considerable practice to readily and correctly read the numbers and tenths or hundredths on the slide-rule, with the different values which 1 may bear; and, in practising, it would be well for the beginner to compare the answers he obtains with some printed tables that are correct. If his answers do not agree with the tables, he has made an error somewhere, which must be rectified. By considerable and careful practice he will become expert in the use of the slide-rule.

Multiplication by the Slide-rule. - Rule. Set 1 on the line B to the number on A, which is used as the multiplier: then above the number on B, which is used as a multiplicand, find the answer on the line A.

Examples. - To multiply 4 by 5, we set 1 on the line B to 4 on the line A: then above 5 on the line B we find the answer 20 on the line A.

To multiply 3 1/2 by 2 1/2, we set 1 on the line B to 2 1/2 on the line A: then above 3 1/2 on the line B we find the answer 8 3/4 on the line A.

To multiply 30 by 4, we set 1 on the line B to 4 on the line A: then above 3, which we will call 30, on the line B, we find 12 on the line A. Now, as we have increased the value of three tenfold over its ordinary value, we must increase the result tenfold to get the answer: 10 times 12 equal 120, the required answer.

To multiply 35 by 25, we set 1 on the line B to 2 1/2 (2.5, or 2 5/10), which we will call 25, on the line A: then above 3 1/2 (3.5, or 3 5/10), which we will call 35, on the line B, we find 8.75 (8 71/10 = 8 75/100) on the line A. Now, as we have increased the value of 2 1/2 tenfold, and also have increased the value of 3 1/2 to tenfold its ordinary value, we must increase the result ten times tenfold, which is one hundred-fold: one hundred times 8.75 (8 75/100) is 875, the required answer.

Division by the Slide-rule. - Rule. Set the number indicating the divisor on the line B under the number indicating the dividend on the line A: then above 1 on the line B find the answer on the line A.

Examples. - To divide 24 by 6, we set 6 on the line B under 24 on the line A: then above 1 on the line B we find the answer 4 on the line A.

To divide 260 by 13, we set 13 on the line B under 26, which we will call 260, on the line A: then over 1 on the line B we find 2 on the line A. But, since we have increased the value of 26 tenfold its ordinary value, we must increase the result tenfold: ten times 2 equal 20, the required answer.

To divide 3,500 by 50, we set 5, which we will call 50, on the line B under 35, which we will call 3,500, on the line A: then above 1 on the line B we find 7 on the line A. Now, to find how many fold to increase this result, we divide the number of times we increased the value of 35, which we increased one hundred-fold, by the number of times we increased the value of 5, which was tenfold; 100 divided by 10 equals 10, so we must increase the result tenfold; ten times 7 equal 70, the required answer.

Proportion by the Slide-rule. - Example 1. As 3 is to 12, so is 5 to the answer. We set 3 on the line B under 12 on the line A: then above 5 on the line B we find the answer, 20, on the line A.

Example 2. - As 2 1/2 is to 5 1/4, so is 3 to the answer. We set 2 1/2 on the line B under 5 1/4 on the line A: then above 3 on the line B we find the answer, 6 3/10 on the line A.

Example 3. - If my wages are $2.75 per day (working 10 hours), how much will be due me when I have worked 6 days and 4 hours? We reduce the days and parts of a day to hours: 6 days and 4 hours equal 64 hours: so we state our proportion as follows. As 10 is to 64, so is $2.75 to the answer. We set 1, which we will call 10, on the line B, under 6 4/10 (6.4), which we will call 64, on the line A: then above 2 7.5 /10 (2.75, or 2 3/4) on the line B we find the answer, 17 3/5, on the line A. $17 3/5 equal $17. 60.

Example 4. - If I pay a man $15 per week, what should I pay him for 8 1/2 days' work? We may state our proportion as follows. As 6 is to 8 1/2, so is 15 to the answer. We set 6 on the line B under 8 1/2 on the line A: then above 15 on the line B we find the answer, 21 1/4, on the line A. $21 1/4 equal $21.25.