tangular figure is found by multiplying the length into the breadth.

If for example, the divisions of the sides of the foregoing square, represent inches, there are 4 of them in length, and 4 in breadth, and 4×4=16: on counting the minor squares there will be found to be 16 of them.

The area of a square then, is equal to the product of one of its sides into itself; in other words, to the square of the side.

Thus, if one side of a square field be 40 rods, the area of the field=40X40=1600 square rods.

Now, as we find the area, or superficial content of a square, by squaring one of its sides, we find the side by the converse of this operation; that is, by extracting the square root of the area.

Suppose, for example, you have 2025 apple-trees, to set out in a square field, and you desire to know how many you must plant in a row, to have an orchard in a square form. The square root of 2025 is 45, the number of trees in a row.

Suppose you have 512 trees to be set out, and wish the rows to be twice the length in one direction, that they are in the other. Take half the number of trees(256), and extract the square root of it, which will dispose one half of your trees in a square, and give you the length of the shortest side. Double that number for the longest side.

Ans. 16 trees one way; 33 the other. 13. A section of land in the Western states is a square, consisting of 640 acres ; what is the length in rods of one of its sides? Ans. 320 rods.

14. A certain square pavement, contains 20736 square stones; I demand how many stones there are in one of its sides. Ans 144.

15. What must be the side of a square field, that shall contain an area equal to another field of rectangular shape, the two adjacent sides of which are 18 rods and 72 rods? Ans 36 rods. 16. Suppose 3097600 men to be drawn up in a solid square, how many men would there be on a side ? and, allow

ing each man to occupy a square yard of ground, how large a plain would contain the whole number?

S Ans. 1760 men on a side.
1 mile square.

{

DEFINITION I. A triangle is a figure of three sides, and having three angles.

II. If one of its angles be a right angle, the figure is called a right-angled triangle.

By a geometrical demonstration it is proved, that the square formed on the longest side of a right-angled triangle (that opposite the right angle,) is equal to the sum of the squares formed on the other two sides: that is, its area is equal to the areas of the other two squares. Thus the square 1, is equal to the squares 2 and 3 in the diagram above.

On this curious relation of the sides of a right-angled triangle a great many very interesting and useful calculations are based.

If for example, we know the length of the two shorter sides of such a figure, we can calculate that of the longest side (or hypotenuse,) without measuring it. We have only to each of the two sides, add their squares square together, and extract the square root of the sum.

17, Suppose for example, a surveyor has run a line due north 60 rods, and then east 80 rods, and wishes to know his distance in a right line from his first station.

60X60 3600

80x80 6400

10000

100 rods Ans.

18. If the room where you are sitting, be 28 feet long and 22 wide, what is the distance between its opposite angles. Ans. 35.6+feet.

19. If from the ground to the eaves of a house, the perpendicular height be 30 feet, and you wish to make a ladder, which being planted at the distance of 10 feet from the wall will reach the roof, of what length must it be?

Ans. 31 feet

20. A carpenter wishes to connect a post and beam by a brace, which shall enter a mortise in each, at the distance of 4 feet on the inside from the junction or angle. Of what length must the brace be to the entrance of the upper side of the mortise? Ans. 5 feet 7.8 inches.

21. A hawk perched on the top of a perpendicular tree, 77 feet high, was brought down by a sportsman standing at the distance of 14 rods, on a level with its base; what distance in yards did he shoot? Ans +yards.

As the square of the hypotenuse equals the sum of the squares of the two sides, it is evident, that the square of one of the two sides, is equal to the square of the hypotenuse less the square of the other side.

22. If a ladder 40 feet long be so planted as to reach a window 33 feet from the ground, on one side of a street, and by turning it over, without moving its foot it will reach a window 21 feet high on the other side, how wide is the street?

Here the ladder represents the hypotenuse, and the height of the window one of the sides of a right-angled triangle. The distance of the foot of the ladder from the wall, the required side.

40X40 1600
33X33-1089

511

square of the hypotenuse.

square

of the required side.

The square root of 511 will be the distance from the foot of the ladder to the wall on one side; which, added to the distance to the wall on the other side, found in the same way, will give the entire width of the street.

Ans. 56.64 feet.

23. Required the height of a May-pole, whose top bc

ing broken off by a blast of wind, struck the ground at the distance of 15 feet from its foot, and measured 39 feet. Ans. 75 feet

24. Wishing to know the distance from a station near the side of a river to a tree below, but prevented by a curve in the stream from measuring it, I run out at right angles to the line of direction 15 rods, and then 30 rods in a right line to the tree. What was the distance required? Ans. 26 rods, nearly.

25. If the distance from a point perpendicularly under a kite flying in the air, to the station of the boy holding the string, be 300 feet, and the length of string from the hand to the kite be found to measure 580 feet; how high is the kite above the surface of the ground?

Ans. 496 feet 4 inches. NOTE. In this case the string must be held to the ground when the measurement is made.

26. If the diagonal of a rectangular field measure 40 rods, and one of its sides 32 rods; what is the length of the adjacent side? Ans. 24 rods. 27. If the diagonal of a square field be 60 rods; what is the length of a side, and what the area of the field?

Length of a side 42.42 rods.
Area of the field 1800 square rods.

NOTE. The diagonal being the hypotenuse of a rightangled triangle, of which the other sides are equal, its square is double the square of each of the sides.

28. Suppose the width of a barn to be 36 feet, of what length must the rafters be, if the elevation of the ridge above the level of their foot be 14 feet? Ans. 22 feet 9 in.

NOTE. Each rafter is the hypotenuse of a triangle, and the height of the ridge one of its sides.

29. If 4200 fruit trees are to be planted, how large a square can be formed out of the number; that is, how many trees will there be in a row? Ans. 64 trees.

The square of 64-4096; consequently there will be 104 trees over, which cannot be united to the square, without destroying its figure.

There must always be a remainder, unless the given number be an exact power.

To find the side of a square that shall bear any assignable ratio to a given square:

RULE.-Multiply the given square by the ratio of the required square, and extract the square root of the product.

30. If the side of a square field be 40 rods; what will be the side of another square field that shall contain just twice the quantity?

40X40 1600×2=3200.

√3200=561+ rods Ans.

31. What would be the side of a field that would contain one half the quantity? Ans. 28.2 rods NOTE.-Multiplying by is simply taking one half the multiplicand.

To find the diameter of a circle that shall have any assignable ratio to a given circle.

RULE.-Multiply the square of the diameter, by the ratio of the required circle, and the square root of the product will be the diameter of the required circle.

32. If the bottom of a circular cistern be 5 feet in diameter; what is the diameter of another cistern, whose bottom contains just 4 times as much surface?

Ans. 10 feet.

QUESTIONS.

What is involution?

What does the exponent of a power indicate?
What relation has evolution to involution?

What is the extraction of the square root?

What is meant by the proper product of a figure?

What is the use of the points placed over the number whose root is to be extracted?

Over which places in the given number is it put?

How may you prove the correctness of your work?

Can an exact root be found to every number?

What is a right-angled triangle?

What relation is there between the square of the hypotenuse, and the squares of the two sides?

If the two sides are given, how do we find the hypotenuse?

If the hypotenuse and base are given, how is the perpendicular found?

If the hypotenuse and perpendicular are known, how do we find the base?