the location of several points on scale D. The student should study these readings until he is perfectly certain that he understands them. The ordinary 10-in. logarithmic scale can be read to only three places accurately with the fourth place estimated. To multiply, locate the point on scale D corresponding to one of the numbers to be multiplied; set the unit figure, or 1, of scale C opposite that point; then locate the other number on scale C, and directly opposite on scale D will be found the answer. Lastly, determine the proper place for the decimal point in the

answer.

Examples:

1. Multiply 153 × .0425.

First, locate the point 153 on scale D and place the 1 of scale C directly opposite. (Placing the hair line of the glass indicator over the point 153 assists materially in locating the 1 of scale C properly.)

Second, find the point 425 on scale C and read the answer on scale D directly opposite this point. (Again by placing the hair line of the indicator over the point, 425, the reading of the answer is facilitated.) thus found is 65. Since .0425 is approximately or, and 2 the slide rule answer must evidently be 6.5.

100

The reading of 150 = 6,

Figure 109 illustrates the slide rule setting for this example. The dotted lines indicates the first position of the indicator. If we carry out the multiplication 153 X .0425 by "long hand," we will find that the exact answer is 6.5025, which is very close to the slide rule result. It must be emphasized, however, that the slide rule cannot be read sufficiently close to give absolutely exact answers, but the error is small and for ordinary calculations can be neglected.

2. Multiply 4.333 × 31,500.

First, locate the point 4333 on scale D and place the 1 of scale C opposite. Immediately we see that the point 315 on scale C falls beyond scale D, and apparently we need an extension on scale D to read the answer. We overcome this difficulty by placing the right-hand end, or 1, of scale C opposite the 433. This corresponds exactly to placing an extension on scale D. Second, locate the point 315 on scale C and read the answer directly opposite on scale D. The slide rule reading is about 1365. Roughly, 4.333 × 31,500 is about 4 × 30,000 120,000, which would indicate that the answer as read from the scale is 136,500.

=

To divide, locate the point on scale D corresponding to the dividend; then locate the point on scale C corresponding to the divisor, and place this point directly opposite the dividend. The quotient will be found on scale D opposite the unit figure, or 1, of scale C. This may be either end of scale C.

3. Divide .0585 by 5.41.

First, locate 585 on scale D. This is usually done with the indicator. Second, place the point 541 of scale C opposite the point 585 and read the quotient on scale D opposite the 1 of scale C. In this case it falls opposite the left-hand end of C and the reading is 108. By inspection, we

can see that the answer is about .01, which determines the quotient to be .0108.

4. Divide 375 by 52.

First, locate 375 on scale D.

Second, place the point 52 of scale C opposite 375 of scale D and read the quotient on scale D opposite the 1 of scale C. In this case it comes opposite the right-hand end of C and the reading is 721. In round numbers 52 goes into 375 about 7 times; therefore, the quotient must be 7.21.

Where a great deal of calculating is done, much time can be saved by placing the scales so as to require the minimum number of settings. This is especially true in the calculation of formulas.

5. The formula for width of belting is W

=

H X 33,000
PX V

Given,

The complete calculation can be made with two moves of the slide as follows:

Divide 40 by 70 by placing the 7 of scale C opposite the 4 of scale D. The quotient could be read opposite the 1 of scale C, but we do not read it since we immediately multiply it by 33,000. To do this we must place the 1 of scale C opposite the quotient, but it is already in this position, so all that is necessary is to locate 33 on scale C. Opposite this point on scale D is the product of 8 X 33,000. Place the hair line of the indicator over this product, without reading it, and complete the calculation by dividing by 3000. This is done by placing the 3 of scale C opposite the product of #8 × 33,000, and reading the answer on scale D opposite the 1 of scale C. The first setting is indicated in Fig. 110A, which makes the calculation #8 × 33,000, and the second setting in Fig. 110B completes the calculation. The final reading on the D scale is 628. Since 33,000 ÷ 3,000 = 11, and #8 is about, the answer is obviously 6.28.

6. The velocity in feet per minute of the rim of a flywheel is π X d X N

12

- where d is in inches. Given, d

=

8, N = 500, find V.

First multiply 3.1416 by 8. This necessitates placing the right-hand end, or 1, of scale C opposite 3.1416 on D. Set the indicator over 8 on scale C to locate the product on D without reading it. Now divide this product by 12. To do this, place the 12 of C opposite, and locate the quotient on D by setting the indicator over the 1 of scale C. Multiply by 500, by placing the right-hand end of scale C opposite the quotient last obtained, and reading the point on D which falls opposite the 5 of scale C. Figure 111 illustrates the three settings required.

This reading is 1047. To locate the decimal point make an approximate calculation of the result, as has been done in previous examples. An approximate answer is 1000, which shows that the slide rule answer is 1047.

3

C

C

146. Roots and Powers. The square of any number on the D scale can be found directly opposite on the A scale, and for square roots the opposite is true. In finding square roots, however, we must be careful to use the proper scale on A. To find the square roots of numbers having an odd number of places to the left of the decimal point, use the left-hand scale on A, and for those with an even number of places to the left of the decimal point, use the right-hand scale. Similarly for numbers smaller than 1, that is, decimals, the square roots of those having an odd number of ciphers preceding the first significant figure are opposite the left-hand scale on A, and for those having an even number of ciphers, the square roots are opposite the righthand scale.

Examples:

1. Find the squares of 24.5; 3.24; and .66.

To square 24.5, set the hair line of the indicator above 245 on D, and read the square under the hair line on the A scale. The reading on A is 6, and the answer is obviously 600, since 20 squared equals 400.

To square 3.24 and .66, set the indicator above these points on D and read their squares on A, directly opposite. The readings thus found are 105 and 436, respectively. Thus the square of 3.24 = 10.5, and the square of

.66 = .436.

To locate the decimal points in these results we see that the square of 3.24 must be a little more than 32, which is 9, and less than 42, which is 16. Therefore, the answer must be 10.5. The third number, .66, is a little less than .7. The square of .7 is .49. Therefore, the square of .66 must be less than .49. From this we can see that the square of .66 is .436.

2. Find the square root of 64; 320; and .0049.

Sixty-four (64) has an even number of places to the left of the decimal point. Therefore, we set the indicator over the 64 on the right half of A, and read the square root on D, directly opposite. The answer is 8, as we know.

In 320 there are an odd number of places on the left of the decimal point so we use the left half of A for this number. Set the indicator above 32 on A and read the square root, 17.9, beneath on D.

In .0049 there are an even number of ciphers on the left of the 4, so we use the right half of A. The reading on D, opposite 49 on A, is 7. The square root of .0049 is then .07. The rule for locating the decimal point is given in Art. 71.

Higher powers of numbers may be found by multiplying the number by itself an equivalent number of times, or by multiplying together the lower powers of the number. The more complicated slide rules frequently have what is called the log-log scale, by means of which any power of a number may be obtained directly.