Thus, the square root of 144 contains two figures, and is The square root of 1600 contains 2 figures, and

12. is 40.

The square of 30 is 900. The square of 37 is 1369, and to discover in what manner this square is formed, we will multiply 37 by 37, writing each product separately, instead of adding them as we proceed. We then see that we must add the square of the tens, twice the product of the tens by the units, and the square of the units.

37

37

49

21

21

9

1369

1369 (37

9

67) 469

469

Let us now reverse the process, and extract the square root of 1369. Pointing the periods, we find the root will consist of two figures, and the square of the tens must therefore be contained in the 13 hundreds. The greatest square in 13 is 9, and the root 3 is written as the first figure of the required root. Subtracting the square of the root already found, the remainder, 469, must contain twice the product of the tens by the units+the square of the units. To obtain the units' figure, we divide the 46 tens by 2×3-6 tens, which gives a quotient 7. Writing the 7 in the root, and also at the right of the divisor, we multiply by 7, and obtain 469, which is twice the product of the tens by the units plus the square of the units. Hence we deduce the following

RULE.

Separate the number into periods of two figures each, by placing a point over the units' figure, and another over each second figure to the left (and also to the right, if decimals are desired in the root). Write in the quotient the root of the greatest square contained in the left hand period, and subtract its square from the period.

To the remainder annex the second period, and divide the tens of the number thus formed, by twice the first quotient figure, placing the result in the root, and also at the right of the new divisor. Multiply the completed divisor by the new quotient figure, subtract the product from the dividend, and annex the third period to the remainder.

Double the root figures already found for a new trial divisor, and proceed as before, until all the periods are brought down.

When any trial divisor is not contained in the tens of

the dividend, place a zero in the root, and also at the right of the divisor, and bring down the next period.

If any figure obtained for the root proves too large, diminish it by one and repeat the work.

Approximate roots may be obtained by annexing decimal periods of two figures each.

1. Extract the square root of 529; of 961.

2. Extract the square root of 1444.

3. What is the square root of 3249? of 3969 ?

4. What is the square root of 12321?

5. What is the square root of 2256.25?

6. What is the square root of 1.907161?
7. What is the square root of 53300033424?

8. Find the square root of .0361; of .000040804.
9. Find the square root of &; ; 17; 7,290.

10. Find the approximate square root of 2, to six decimal places.

11. Find the approximate square root of 63; 1.6; .009; 27.1; 14367.

169

12. Find the square root of 318,82; 7; 7%; 43; 171.

13. How long is the side of a square field that contains 225 acres?

14. What is the side of a square which contains as much as a circle whose area is 273.5 square feet?

15. An army of 8649 men is arranged in a solid square. How many are there on each side?

16. A certain oblong field contains 40 acres, and the length is 4 times the width. Required the length and breadth.

The areas of circles are proportioned to the their diameters.

squares of

The areas of any similar figures are proportioned to the squares of their like dimensions.

The area of any circle is equal to the square of its diameter multiplied by .7854.

The circumference of a circle is equal to its diameter multiplied by 3.1416.*

The area of a triangle is equal to the base multiplied by half the height.

In any right-angled triangle, the square of the longest side is equal to the sum of the squares of the other two sides.

The distance through which bodies fall, when falling freely, are as the squares of the times. In a vacuum, a body would fall 161 ft. in 1 second. Then we have the proportion, letting n represent any number of seconds.

sec.

sec. ft.

ft.

(1)2: n2 :: 162: distance in n seconds.

Any three terms of this proportion being given, the fourth may be readily found. But it should be remarked, that in consequence of the resistance of the air, the space actually fallen through is somewhat less than that given by the formula.

17. What is the diameter of a circle that is 16 times as large as one whose diameter is 13 feet.

18. The area of a circle is 7632 feet; what is the diameter?

19. A horse is fastened to a post in the centre of a field. What is the length of a rope that will allow him to graze an acre?

20. The base of a triangle is 47 feet. What is the height, the area being 7691 square feet?

21. A ladder 75 feet long, rests against the limb of a tree that is 50 feet from the ground. How far is the foot of the ladder from the root of the tree?

22. The length of a room is 18 feet, and the width 12 feet. What is the distance between the opposite corners? What length of rope would reach from an upper corner to the opposite lower corner, the height being 10 feet?

23. The circumference of a circle is 29 rods. What is the side of a square having an equal area?

24. Two ships left the same port; one sailed 125 miles

* The more exact ratio is, 3.14159265358979323846264338328.

north, the other 100 miles east. How far were they then apart?

25. A kite accidentally lodged in the top of a tree, but the line breaking, I measure its length, which is 210 feet. What is the height of the tree, the base being 189 feet from my standing place?

26. Desiring to know the height of a precipice, I drop a stone from the summit, and observe by my watch that it strikes the ground in 3 seconds. What is the height? 27. A bag of sand is dropped from a balloon 14 miles above the surface of the earth. How long will it be in falling?

28. In what time will a stone fall to the bottom of a shaft, that is sunk 870ft. below the surface of the ground?

When one number bears the same ratio to a second as the second does to a third, the second number is called a mean proportional between the other two. Thus, in the proportion 36 :: 6 : 12, 6 is a mean proportional between 3 and 12.

The mean proportional between any two numbers is equal to the square root of their product.

29. Find a mean proportional between 7 and 252. 30. Find a mean proportional between .75 and 12. 31. Find a mean proportional between 1 and 15. 32. Find a mean proportional between and .875.

7

200

33. Find mean proportionals between 1 and 16; 5 and 6; 25 and 13; and §.

We may often discover, by simple inspection, that any given number is not a perfect square. The following are some of the principal properties of squares:

(1.) Every even square is divisible by 4. Therefore, no even number, which is not divisible by 4, can be a perfect square.

(2.) If a square number contains a prime factor, it also contains the square of that factor. Therefore, no number divisible by a prime factor, and not divisible by its square, can be a square number.

(3.) All squares terminate in 0, 1, 4, 5, 6, or 9. Therefore, no number terminated by 2, 3, 7, or 8, can be a square number.

(4.) Every square number terminated by 5, is also terminated

by 25. Therefore, no number ending in 5 can be a square, unless the 5 be preceded by a 2. We may remark moreover, that the 2 must be preceded by 0, 2, or 6.

(5.) The zeroes terminating any perfect square, must be of an even number. Therefore, no number terminating in an odd number of zeroes, can be a square number. And if the zeroes be even, unless they are preceded by a square number, the number itself is not a square. Thus 2500 is a square number, but 1500

is not.

EXTRACTION OF THE CUBE ROOT.

The cube root of a number is the number which, when raised to the third power, will produce the given number. In the following table are the numbers from 1 to 10 inclusive, and beneath them are their cubes, therefore the numbers of the second line have for their cube roots the numbers of the first.

Roots 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Cubes 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Thus we see that there are only nine integral cubes between 1 and 1000. All the other intermediate integers are imperfect cubes, and their roots can only be obtained approximately.

All perfect cubes from 1 to 1000, evidently have but one integral figure in their cube root. All numbers between 1000 or 103, and 1000000 or 1003, will have two figures in their root. And generally, if we divide a cube into periods of three figures each, by placing a point over units, and one over every third figure from units, the number of points will show the number of figures in the root.

EXAMPLES FOR THE BOARD.

In order properly to understand the principles of the cube root, the student should be provided with the following blocks:

1. A cubical block, each side measuring 3 inches, to represent the CUBE OF THE TENS.

2. Three blocks, each 3 inches square and of an inch thick, to represent THE SQUARE OF THE TENS MULTIPLIED BY THE UNits. 3. Three blocks, each of an inch square and 3 inches long, to represent THE SQUARE OF THE UNITS MULTIPLIED BY THE TENS. 4. A cubical block, each side measuring of an inch, to represent THE CUBE OF THE UNITS.