many independent equations as there are unknown terms used in the statements.

When there is more than one unknown quantity in the statements, we so combine the equations as to eliminate the unknown quantities by addition and subtraction.

Ratio is the relation of one quantity to another of the same kind. Ratio is expressed by the quotient of the first quantity divided by the second.

Ratio may be written by use of the colon, by use of the sign of division, or in fractional form.

Proportion is an equality of ratios. In a proportion the product of the extremes is equal to the product of the means.

The corresponding sides of similar triangles are proportional. By the use of this principle we may calculate heights or distances not easily measured in the ordinary way.

A simple measuring instrument may easily be made by which we may apply the principle of triangular measurement in finding the proportion of dimensions of distant objects, and in calculating heights and distances not easily measured in the direct manner.

The principles of leverage are of very common The law for calculating the effect of the use of a lever is:

use.

Power

tance.

X

power distance = weight x weight dis

The wheel and axle, the windlass and capstan, are applications of the lever. The law for calculating their effect is:

Power x length of crank=load × radius of axle.

The pulley may be used either to change the direction of the pull or to increase its force. The law governing its effect is:

Power x number of separate parts of the rope that sustains the weight = the weight.

The inclined plane is used to raise or lower heavy loads by passing them up or down an incline. The law for calculating the gain by its use is :

Power: load height raised: length of incline.

=

CHAPTER XV

SQUARE ROOT

375. If a square field area is 16 square units. square of 4, and 4 the square root of 16.

has a side of 4 units, its 16 is therefore called the

376. Find the squares of the following:

1. 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18.

2. .2, .03, .004, .0005, .0125, .025, .875, .75, .125.

2 3 4
5'

6 7

3. 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 8.

4. 2345, 8076, 785, 5032, 45.25, 320, 2.003, 4.05, 99, 77.

377. From a study of the results in the last article write the square roots of the following:

1. 16. Ans. 4. Proof: 4 x 4 = 16. (Prove every answer.)

2. 64, 81, .25, .0049, .0016, 625, 8100.

4 16 64 36 49

3.

16409

18

9' 25' 81' 49' 64' 100' 144'

4. 121, 169, 196, 225, 289, 361, 256, 324, 625, 676.

TERMS USED IN SQUARES AND SQUARE ROOT

378. 1. The square of a quantity is indicated by writing a small figure 2 at the right-hand upper corner of the symbol for the quantity, thus 22, x2, .7854, etc. This figure, thus used to indicate the number of times the quantity is used as a factor, is called an exponent.

2. To indicate that the root of a quantity is to be taken, the radical sign is used as follows:

√16, √x, √2500, √16+9, etc.

To find the Square Root of Numbers, when it cannot be readily Seen

Note. Full explanation, step by step, of the "rule" for finding square root is often given. It does not seem necessary to do this, as the universal application of the plan is so easily tested, and every result so easily proved, that it seems wise to accept it for the results which it gives.

379. The following plan may be used to find the square root:

1. Point off the number into periods of two figures each, beginning at units' place.

2. Find the greatest root of the left-hand period and write it as the first figure of the answer.

3. Subtract the square of this first root figure from the left-hand period, and to the remainder annex the next period.

4. Double this root figure, and see how many

times the dividend, exclusive of its right-hand figure, will contain the result.

5. Write the number of times as the next answer figure, and also at the right of the trial divisor.

6. Proceed as in division and repeat the processes until the required root is found.

To prove the result, multiply the answer by itself, to see if it produces the original number.

Applications of Square Root

380. 1. Find the side of a square field which has an area of 2809 square feet.

CONVENIENT ARRANGEMENT OF WRITTEN WORK

28'09(53 53 ft. Ans.

25

103)3 09

3.09

Proof: 53 x 53=2809.

2. Find the side of a square field which has an area of 6889 square yards.

3. What is the distance around a square field which contains 6241 square yards?

4. A box which is square on the bottom and 5 feet in height contains 80 cubic feet. What is the length of the sides?

5. The area of a circle is equal to the square of its diameter multiplied by .7854. Find the diameter of a circle which has an area of 19.6350 square feet.