Part 3

The nearest perfect square to 37 is 6 * 6 = 36, so the answer should be a little more than 0.06 or .0608. All of what has been said about use of the A and D scales for squaring and extracting square root applies equally well to the B and C scales since they are identical to the A and D scales respectively.

A number of examples follow for squaring and the extraction of square root.

Example 31: square( 2 ) = 4 32: square( 15 ) = 225 33: square( 26 ) = 676 34: square( 19.65 ) = 386 35: squareroot( 64 ) = 8 36: squareroot( 6.4 ) = 2.53 37: squareroot( 498 ) = 22.5 38: squareroot( 2500 ) = 50 39: squareroot( .16 ) = .04 40: squareroot( .03 ) = .173

CUBING AND CUBE ROOT

If we take a number and multiply it by itself, and then multiply the result by the original number we get what is called the cube of the original number. This process is called cubing the number. The reverse process of finding the number which, when multiplied by itself and then by itself again, is equal to the given number, is called extracting the cube root of the given number. Thus, since 5 * 5 * 5 = 125, 125 is the cube of 5 and 5 is the cube root of 125.

To find the cube of any number on the slide rule set the indicator over the number on the D scale and read the answer on the K scale under the hair-line. To find the cube root of any number set the indicator over the number on the K scale and read the answer on the D scale under the hair-line. Just as on the A scale, where there were two places where you could set a given number, on the K scale there are three places where a number may be set. To tell which of the three to use, we must make use of the following rule.

(a) If the number is greater than one. For 1, 4, 7, 10, etc., digits to the left of the decimal point, use the left-hand third of the K scale. For 2, 5, 8, 11, etc., digits to the left of the decimal point, use the middle third of the K scale. For 3, 6, 9, 12, etc., digits to the left of the decimal point use the right-hand third of the K scale.

(b) If the number is less than one. We now tell which scale to use by counting the number of zeros to the right of the decimal point before the first digit not zero. If there are 2, 5, 8, 11, etc., zeros, use the left-hand third of the K scale. If there are 1, 4, 7, 10, etc., zeros, then use the middle third of the K scale. If there are no zeros or 3, 6, 9, 12, etc., zeros, then use the right-hand third of the K scale. For example:

Example 41: cube_root( 185 ) = 5.70

Since there are 3 digits in the given number, we set the indicator on 185 in the right-hand third of the K scale, and read the result 570 on the D scale. We can place the decimal point by thinking of the nearest perfect cube, which is 125. Therefore, the decimal point must be placed so as to give 5.70, which is nearest to 5, the cube root of 125.

Example 42: cube_root( .034 ) = .324

Since there is one zero between the decimal point and the first digit not zero, we must set the indicator over 34 on the middle third of the K scale. We read the result 324 on the D scale. The decimal point may be placed as follows:

cuberoot( .034 ) = cuberoot( 34/1000 ) = 1/10 cube_root( 34 )

The nearest perfect cube to 34 is 27, so our answer must be close to one-tenth of the cube root of 27 or nearly 0.3. Therefore, we must place the decimal point to give 0.324. A group of examples for practice in extraction of cube root follows:

Example 43: cuberoot( 64 ) = 4 44: cuberoot( 8 ) = 2 45: cuberoot( 343 ) = 7 46: cuberoot( .000715 ) = .0894 47: cuberoot( .00715 ) = .193 48: cuberoot( .0715 ) = .415 49: cuberoot( .516 ) = .803 50: cuberoot( 27.8 ) = 3.03 51: cuberoot( 5.49 ) = 1.76 52: cuberoot( 87.1 ) = 4.43

THE 1.5 AND 2/3 POWER

If the indicator is set over a given number on the A scale, the number under the hair-line on the K scale is the 1.5 power of the given number. If the indicator is set over a given number on the K scale, the number under the hair-line on the A scale is the 2/3 power of the given number.

COMBINATIONS OF PROCESSES

A slide rule is especially useful where some combination of processes is necessary, like multiplying 3 numbers together and dividing by a third. Operations of this sort may be performed in such a way that the final answer is obtained immediately without finding intermediate results.

1. Multiplying several numbers together. For example, suppose it is desired to multiply 4 * 8 * 6. Place the right-hand index of the C scale over 4 on the D scale and set the indicator over 8 on the C scale. Now, leaving the indicator where it is, move the slider till the right-hand index is under the hairline. Now, leaving the slider where it is, move the indicator until it is over 6 on the C scale, and read the result, 192, on the D scale. This may be continued indefinitely, and so as many numbers as desired may be multiplied together.

Example 53: 2.32 * 154 * .0375 * .56 = 7.54

2. Multiplication and division. Suppose we wish to do the following example:

Example 54: (4 * 15) / 2.5 = 24

First divide 4 by 2.5. Set indicator over 4 on the D scale and move the slider until 2.5 is under the hair-line. The result of this division, 1.6, appears under the left-hand index of the C scale. We do not need to write it down, however, but we can immediately move the indicator to 15 on the C scale and read the final result 24 on the D scale under the hair-line. Let us consider a more complicated problem of the same type:

Example 55: (30/7.5) * (2/4) * (4.5/5) * (1.5/3) = .9

First set indicator over 30 on the D scale and move slider until 7.5 on the C scale comes under the hairline. The intermediate result, 4, appears under the right-hand index of the C scale. We do not need to write it down but merely note it by moving the indicator until the hair-line is over the right-hand index of the C scale. Now we want to multiply this result by 2, the next factor in the numerator. Since two is out beyond the body of the rule, transfer the slider till the other (left-hand) index of the C scale is under the hair-line, and then move the indicator to 2 on the C scale. Thus, successive division and multiplication is continued until all the factors have been used. The order in which the factors are taken does not affect the result. With a little practice you will learn to take them in the order which will require the fewest settings. The following examples are for practice:

Example 56: (6/3.5) * (4/5) * (3.5/2.4) * (2.8/7) = .8

Example 57: 352 * (273/254) * (760/768) = 374