1. Wishing to know the distance from A to the top of a tower, I measured the height of the tower and found it to be 40 feet; also the distance from A to B, and found it 30 feet what was the distance from A to C ?

:

RULE.-Square the base and square the perpendicular, add the results, and then extract the square root of their sum.

311. To find one side when we know the hypothenuse and the other side.

1. The length of a ladder which will reach from the middle of a street 80 feet wide to the eaves of a house, is 50 feet: what is the height of the house?

ANALYSIS. Since the square of the length of the ladder is equal to the sum of the squares of half the street and the height of the house, the square of the length of the ladder diminished by the square of half the street will be equal to the square of the height of the house hence,

RULE.-Square the hypothenuse and the known side, and take the difference; the square root of the difference will be the other

side.

EXAMPLES.

1. A general having an army of 117649 men, wished to form them into a square : how many should he place on each front?

2. In a square piece of pavement there are 48841 stones, of equal size, one foot square: what is the length of one side of the pavement?

3. In the centre of a square garden, there is an artificial circular pond covering an area of 810 square feet, which is

311. When you know the hypothenuse and one side, how do the other side?

of the whole garden: how many rods of fence will enclose the garden?

4. Let it be required to lay out 67A. 2R. of land in the form of a rectangle, the longer side of which is to be three times as great as the less: what is its length and width ?

5. A farmer wishes to set out an orchard of 3200 dwarf pear trees. He has a field which is twice as long as it is wide which he appropriates to this purpose, setting the trees 12 feet apart each way: how many trees will there be in a row, each way, and how much land will they occupy?

6. There is a wall 45 feet high built upon the bank of a stream 60 feet wide: how long must a ladder be that will reach from the outside of the stream to the top of the wall?

7. A boy having lodged his kite in the top of a tree, finds that by letting out the whole length of his line, which he knows to be 225 feet, it will reach the ground 180 feet from the foot of the tree what is the height of the tree?

8. There are two buildings standing on opposite sides of the street, one 39 feet, and the other 49 feet from the ground to the eaves. The foot of a ladder 65 feet long rests upon the ground at a point between them, from which it will touch the eaves of either building: what is the width of the street?

9. A tree 120 feet high was broken off in a storm, the top striking 40 feet from the roots, and the broken end resting upon the stump allowing the ground to be a horizontal plane, what was the height of the part standing?

10. What will be the distance from corner to corner, through the centre of a cube, whose dimensions are 5 feet on a side?

11. Two vessels start from the same point, one sails due north at the rate of 10 miles an hour, the other due west at the rate of 14 miles an hour: how far apart will they be at the end of 2 days, supposing the surface of the earth to be a plane?

12. How much more will it cost to fence 10 acres of land, in the form of a rectangle, the length of which is four times its breadth, than if it were in the form of a square, the cost of the fence being $2,50 a rod?

13. What is the diameter of a cylindrical reservoir containing 9 times as much water as one 25 feet in diameter, the heights being the same ?*

14. If a cylindrical cistern 8 feet in diameter will hold 120 barrels, what must be the diameter of a cistern of the same depth to hold 1500 barrels ?

15. If a pipe 3 inches in diameter will discharge 400 gallons in 3 minutes, what must be the diameter of a pipe that will discharge 1600 gallons in the same time?

16. What length of rope must be attached to a halter 4 feet long that a horse may feed over 2 acres of ground?

17. Three men bought a grindstone, which was four feet in diameter: how much must each grind off to use up his share of the stone?

CUBE ROOT.

312. The CUBE ROOT of a number is one of three equal factors of the number.

To extract the cube root of a number is to find a factor which multiplied into itself twice, will produce the given number. Thus, 2 is the cube root of 8; for, 2 × 2 × 2 = 8: and 3 is the cube root of 27; for, 3 × 3 × 3 = 27.

9,

1, 2, 3, 4, 5, 6, 7, 8, 1 8 27 64 125 216 343 512 729 The numbers in the first line are the cube roots of the corresponding numbers of the second. The numbers of the second line are called perfect cubes. A number is a perfect cube when

312. What is the cube root of a number? When is a number a perfect cube? How many perfect cubes are there between 1 and 1000?

* NOTE-If two volumes have the same altitude, their contents will be to each other in the same proportion as their bases; and if the bases are similar figures (that is, of like form,) they will be to each other as the squares of their diameters, or other like dimensions.

it has three exact factors. By examining the numbers of the

two lines we see,

1st. That the cube of units cannot give a higher order than hundreds.

2d. That since the cube of one ten (10) is 1000 and the cube of 9 tens (90), 729000, the cube of tens will not give a lower denomination than thousands, nor a higher denomination than hundreds of thousands.

Hence, if a number contains more than three figures, its cube root will contain more than one; if it contains more than six, its root will contain more than two, and so on; every additional three figures giving one additional figure in the root, and the figures which remain at the left hand, although less than three, will also give a figure in the root. This law explains the reason for pointing off into periods of three figures each.

313. Let us now see how the cube of any number, as 16, is formed. Sixteen is composed of 1 ten and 6 units, and may be written, 10+ 6. To find the cube of 16 multiply the number by itself twice.

To do this we place the number thus,

product by the units,

10+6, we must

16

=

10+

6

1. By examining the parts of this number, it is seen that the first part 1000 is the cube of the tens; that is,

10 x 10 x 10 = 1000.

313. Of how many parts is the cube of a number composed? What are they?

2. The second part 1800 is three times the square of the tens multiplied by the units; that is,

3 × (10)2 × 6 = 3 × 100 × 6 = 1800.

3. The third part 1080 is three times the square of the units multiplied by the tens; that is,

3 x 62 x 10 = 3 × 36 × 10 1080.

4. The fourth part is the cube of the units; that is, 62 = 6 × 6 × 6 = 216.

1. What is the cube root of the number 4096?

ANALYSIS. Since the number contains more than three figures, we

know that the root will contain at

least units and tens.

Separating the three right-hand figures from the 4, we know that the cube of the tens will be found in the 4; and 1 is the greatest cube in 4.

OPERATION.

4 096(16

1

12 x 3 =3)30 (9-8-7-6 16 4 096.

Hence, we place the root 1 on the right, and this is the tens of the required root. We then cube 1 and subtract the result from 4, and to the remainder we bring down the first figure 0 of the next period.

We have seen that the second part of the cube of 16, viz., 1800, is three times the square of the tens multiplied by the units; and hence, it can have no significant figure of a less denomination than hundreds. It must, therefore, make up a part of the 30 hundreds above. But this 30 hundreds also contains all the hundreds which come from the 3d and 4th parts of the cube of 16. If it were not so, the 30 hundreds, divided by three times the square of the tens, would give the unit figure exactly.

Forming a divisor of three times the square of the tens, we find the quotient to be ten; but this we know to be too large. Placing 9 in the root and cubing 19, we find the result to be 6859. Then trying 8 we find the cube of 18 still too large; but when we take 6 we find the exact number. Hence, the cube root of 4096 is 16.

314. Hence, to find the cube root of a number:

RULE.-I. Separate the given number into periods of three figures each, by placing a dot over the place of units, a second