150. How long must be the boom in Fig. 34 to land the load on the 12 ft. pedestal, allowing 4 ft. clearance at the end for ropes, pulleys, etc.?

27

FIG. 34.

CHAPTER XII

MATHEMATICAL TABLES (CIRCLES, POWERS, AND ROOTS) 84. The Value of Tables.-There are certain calculations that are made thousands of times a day by different people in different parts of the world. For example, the circumferences of circles of different diameters are being calculated every day by hundreds and thousands of men. To save much of the time that is thus wasted in useless repetition, many of the common operations and their results have been "tabulated," that is, arranged in tables in the same way as are our multiplication tables in arithmetics. These tables are not learned, however, as were the multiplication tables, but are consulted each time that we have need for their assistance.

Just what tables one needs most, depends on his occupation. The machinist has use for tables of the decimal equivalents of common fractions, tables of cutting speeds, tables of change gears to use for screw cutting, etc. The draftsman would use tables of strengths and weights of different materials, safe loads for bolts, beams, etc., tables of proportions of standard machine parts of different sizes, etc. The engineer uses tables of the properties of steam, and of the horse-power of engines, boilers, etc.

There are certain mathematical tables that are of value to nearly everyone. Among these are the tables given in this chapter: Tables of Circumferences and Areas of Circles; Tables of Squares, Cubes, Square Roots, and Cube Roots of Numbers.

85. Explanation of the Tables.-The first table is to save the necessity of always multiplying the diameter by 3.1416 when we want the circumference of a circle, or of squaring the diameter and multiplying by .7854 when the area of a circle is wanted.

To find the circumference of a circle: Find, in the diameter column, the number which is the given diameter; directly across, in the next column to the right, will be found the corresponding circumference.

Examples:

Diameter 11
Diameter 27,

Diameter 90,

Circumference 3.9270
Circumference 84.823
Circumference 284.314

To find the area of a circle: Find, in the diameter column, the number which is the given diameter; directly across, in the second column to the right (the column headed "Area") will be found the area.

Examples:

Diameter 66, Area 3421.2
Diameter 17, Area 226.98
Diameter,
Area 0.3067

If the area or circumference is known and we want to get the diameter, we find the given number in the area or circumference column and read the diameter in the corresponding diameter column to the left.

The second table, that of squares, cubes, square roots, and cube roots, is especially valuable in avoiding the tedious process of extracting square or cube roots. The table is read the same as the other one. Find the given number in the first column; on a level with it, in the other columns, will be found the corresponding powers and roots, as indicated in the headings at the tops of the columns.

86. Interpolation. This is a name given to the process of finding values between those given in the tables. For example, suppose we want the circumference of a 301 in. circle. The table gives 30 and 301⁄2 and, since 30 is half way between these, its circumference will be half way between that of a 30 in. and a 30 in. circle.

Then the circumference of the 301 in. circle is just half this difference more than that of the 30-in. circle,

1
94.248+ of 1.571=95.034
2

This method enables us to increase greatly the value of tables. For most purposes the interpolation can be done quickly, and while it requires some calculating, is much shorter than the complete calculation would be. This is especially true in finding square or cube roots.

Example:

Find from the table the cube root of 736.4

3/737=9.0328

3/736=9.0287

Difference=41

.4X the difference = .4X41=16.4

9.0287
16

3/736.49.0303, Answer.

Explanation: The root of 736.4 will be between that of 736 and that of 737, and will be .4 of the difference greater than that of 736. In making this correction for the .4, we forget, for the minute, that the difference is a decimal and write it as 41 merely to save time. We then multiply it by 4, and, dropping the decimal part, add the 16 to the 90287. This gives 9.0303 as the cube root of 736.4

Hence, 736.4=9.0303

87. Roots of Numbers Greater than 1000.-For getting the cube roots of numbers greater than 1000, the easiest and most accurate way is to look in the third column headed "cubes" for a number as near as possible to our given number. Now, we know that the numbers in the first column are the cube roots of these numbers in the third column. If we can find our number in the third column, there is nothing further to do because its cube root will be directly opposite it in the first column. Examples:

Likewise, the numbers in the first column are the square roots of the numbers in the second column. But suppose we want the cube root of a number which is not found in the third column, but lies somewhere between two consecutive numbers in that column. In this case we pursue the method shown in the following example:

Example:

Find 3/621723

In the column headed "Cube" find two consecutive numbers, one larger and one smaller than 621723. These numbers are

Hence, the cube root of 621723 is more than 85 and less than 86; that is, it is 85 and a decimal, or 85+.

The decimal part is found as follows: Subtract the lesser of the two numbers found in the table from the greater and call the result the First Difference.

636056-614125=21931, First difference.

Then subtract the smaller of the two numbers in the table from the given number and call the result the Second Difference.

621723-614125=7598, Second Difference

21931

Now the first difference, 21931, is the amount that the number increases when its cube root changes from 85 to 86. Our given number is only 7598 more than the cube of 85, so its cube root will be approximately 85759 We do not want a fraction like this, so we reduce it to a decimal as follows: Divide the second difference by the first difference and annex the quotient to 85. This will give us the cube root of our number, approximately.

This is the decimal part of the root sought and the whole root is 85.346+. Hence 621723=85.346+.

This method is not exact and the third decimal place will usually be slightly off, so it is best to drop the third decimal if less than 5, or raise it to 10, if more than 5. In this case we will call the root 85.35.

88. Cube Roots of Decimals.-In getting the cube root of either a number entirely decimal, or a mixed decimal number, it is best to move the decimal point a number of periods, that is, 3, 6, 9, or 12 decimal places, sufficient to make a whole number out of the decimal. After finding the cube root, shift the decimal point in the root back to the left as many places as the number of periods that we moved the decimal point in our original number. For example, suppose that we had .621723 of which to find the cube root. Moving the decimal point two periods (of three places each) to the right gives us 621723, of which we just found the cube root to be 85.35. We moved the decimal point of our original number two periods to the right, so we must move the decimal point back two places to the left in the root; we then have .621723.8535. The following illustrations will show the principle:

Notice that there is no such similarity between the cube roots of numbers if we move the decimal point any other number of places than a multiple of three.

Care should be taken, therefore, that, if necessary to move the decimal point in finding a cube root, it should be moved an exact multiple of 3 places. If we have a decimal such as .07462, it is necessary to attach a cipher at the right, making the decimal .074620, so we can shift the decimal point 2 periods or six places. We can now find 74620 as follows: