5. If at any time the trial divisor is not contained in the dividend, place a cipher in the root, annex a cipher to the trial divisor and bring down another period.

6. To locate the decimal point, remember that there will be as many figures in the root to the left of the decimal point as there were periods to the left of the decimal point in our original number.

80. The Law of Right Triangles.—One of the most useful laws of geometry is that relating to the sides of a right angled triangle. Fig. 28 shows a right angled triangle, or "right triangle," so

called because one of its angles (the one at C) is a right angle, or 90°. The longest side (c) is called the hypotenuse. "In any right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides." Written as a formula this would read

c2= b2+a2.

This can be illustrated by drawing squares on each side, as in Fig. 29, and noting that the area of the square on the hypotenuse is equal to the sum of the areas of the other two.

In using this rule, however, we do not care anything about these

areas and seldom think of them except as being the squares of numbers. It is used to find one side of such a triangle when the other two are known.

Examples:

1. If the trolley pole in Fig. 30 is 24 ft. high, and the guy wire is anchored 7 ft. from the base of the pole, what is the length of the guy wire?

The guy wire is the hypotenuse of a right triangle whose sides are 24 ft. and 7 ft.

c2= b2+a2
c2=24+72

=576+49=625

c=625-25 ft., Answer.

2. If the triangle of Fig. 31 is a right triangle having the hypotenuse c = 13 in. and the side a=5 in., what is the length of the side b? c2=b2+a2 b2=c2-a2

Hence,

b2=132-52

=169-25-144

b=144-12 in., Answer.

909

a=5

7'
FIG. 30.

C = 13

FIG. 31

This property of right triangles is also useful in laying out right angles on a large scale more accurately than it can be done with a square. This is done by using three strings, wires, or chains of such lengths that when stretched they form a right triangle. A useful set of numbers that will give this are 3, 4, and 5, since 32+42=52 (9+16=25).

Any three other numbers having the same ratios as 3, 4, and 5 can be used if desired. 6, 8, and 10; 9, 12, and 15; 12, 16, and 20; 15, 20, and 25; any of these sets of numbers can be used.

A surveyor will often use lengths of 15 ft., 20 ft., and 25 ft. on his chain to lay out a square corner; this method can also be used in aligning engines, shafting, etc.

81. Dimensions of Squares and Circles.-Square Root must be used in getting the dimensions of a square or a circle to have a given area. If the area of a square is given, the length of one side can be obtained by extracting the square root of the area. If we wish to know the diameter of a circle which shall have a certain area, we can find it by the following process:

The area is .7854× the square of the diameter or, briefly,

A = .7854XD2

If we divide the given area by .7854, we will get the area of the square constructed around the circle (see Fig. 19).

One side of this square is the same as the diameter of the circle and is equal to the square root of the area of the square.

Then, to find the diameter of a circle to have a given area: Divide the given area by .7854 and extract the square root of the quotient.

82. Dimensions of Rectangles.-Occasionally one encounters a problem in which he wants a rectangle of a certain area and knows only that the two dimensions must be in some ratio. It may be that a factory building is to cover, say 40,000 sq. ft. of ground and is to be four times as long as it is wide, or some problem of a similar nature. Suppose we take the case of this factory and see how we would proceed to find the dimensions of the building.

Example:

Wanted a factory building to cover 40,000 sq. ft. of ground. Ratio of length to breadth, 4:1. Find the dimensions.

l=4b

FIG. 32.

40000+4=10000

b2=10000
b=100

7=4X100=400.

Explanation: If we divide the total area by 4, we get 10,000 as the area of a square having the breadth b on each side. From this we find the breadth or width b to be the square root of 10,000 or 100 ft. If the length is four times as great it will be 400 ft. and the dimensions of the building will be 400 by 100.

83. Cube Root.—The Cube Root of a given number is another number which, when cubed, produces the given number. In other words, the cube root is one of the three equal factors of a number. The cube root of 8 is 2, because 23=2X2X2=8; also the cube root of 27 is 3 (since 33=27) and the cube root of 64 is 4 (since 43 = 64).

If we consider the number of which we want the cube root as representing the volume of a cubical block, then the cube root of the number will represent the length of one edge of the cube. The cube root of 1728 is 12 and a cube containing 1728 cu. in. will measure 12 in. on each edge.

There are four ways of getting cube roots: (1) by actual calculation, (2) by reference to a table of cubes or cube roots, (3) by the use of logarithms, and (4) by the use of some calculating device like the slide rule.

The use of a table is the simplest way of finding cube roots, but its value and accuracy is limited by the size of the table. Tables of cubes or cube roots are to be found in many handbooks and catalogues and should be used whenever they give the desired root with sufficient accuracy.

Logarithms give us an easy way of getting cube roots, but here also a table is necessary and the accuracy is limited by the size of the table of logarithms. The use of logarithms will be explained in a later chapter. The ordinary pocket slide rule will give the first three figures of a cube root and for many calculations this is sufficiently accurate. The method of actually calculating cube roots is very complicated and is used so seldom that one can never remember it when he needs it. Consequently, if it is necessary to hunt up a book to find how to extract the cube root, one might just as well look up a table of cube roots or a logarithm table, either of which will give the root much quicker. The next chapter contains tables of cube roots and a chapter further on explains the use of logarithms.

Note. These answers are given so that the student can see if he understands the operations of square root before proceeding further.

142. The two sides of a right triangle are 36 and 48 ft.; what is the length of the hypotenuse?

143. A square nut for a 2 in. bolt is 3 in. on each side. What is the length of the diagonal, or distance across the corners?

144. A steel stack 75 ft. high is to be supported by 4 guy wires fastened to a ring two-thirds of the way up the stack and having the other ends anchored at a distance of 50 ft. from the base and on a level with the base. How many feet of wire are necessary, allowing 20 ft. extra for fastening the ends?

145. What would be the diameter of a circular brass plate having an area of 100 sq. in.?

-17-6"

·20

FIG. 33.

146. A lineshaft and the motor which drives it are located in separate rooms as shown in Fig. 33. Calculate the exact distance between the centers of the two shafts.

147. I want to cut a rectangular sheet of drawing paper to have an area of 235 sq. in. and to be one and one-half times as long as it is wide. What I would be the dimensions of the sheet?

148. A 6 in. pipe and an 8 in. pipe both discharge into a single header. Find the diameter of the header so that it will have an area equal to that of both the pipes.

149. What would be the diameter of a 1 lb. circular cast iron weight in. thick?