hypothenuse is equal to the sum of the squares described on the other two sides. Thus, if ACB be a right-angled triangle, right-angled at C, then will the large square, D, described in the hypothenuse AB, be equal to the sum of the squares F and E, described on the sides AC and CB. This is called the carpenter's theorem. By counting the small squares in the large square D, you will find their number equal to that contained in the small squares F and E. In this triangle the hypothenuse AB 5, AC = 4, and CB = 3. bers having the same ratio, as 5, 4, and 3, such as = D Any num 10, 8, and 6; 20, 16, and 12, &c., will represent the sides of a right angled triangle. 385. When the base and perpendicular are known, to find the hypothenuse. ANALYSIS.-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? 0 BC = 40; AC2 AB2+BC2 AC = √2500 = 50 feet. A B Rule.-Square the base and square the perpendicular, add the results, and then extract the square root of their sum. 386. To find one side, when we know the hypothenuse and other side. 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 width of the street and the height of the house, the square of the length of the ladder diminished by the square of half the width of the street, will be equa 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 center of a square garden, there is an artificial circular pond, covering an area of 810 square feet, which is of the whole garden: how many rods of fence will inclose the garden? 4. Let it be required to lay out 67 A. 2 R. 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 twice as long as it is wide, which he appropriates to this purpose. He sets the trees 12 feet apart, and in rows that are likewise 12 feet apart: how many rows will there be, how many trees in a row, 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 one side of the stream to the top of the wall on the other? 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 center 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 height being the same? 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. 14. If a cylindrical cistern eight 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 4 feet in hiameter: how much of the radius must each grind off to use up his share of the stone? CUBE ROOT. 387. The CUBE ROOT of a number is one of its three equal factors. Thus, 2 is the cube root of 8; for, 2 x 2 x 2 = 8: and 3 is the cube root of 27; for, 3 × 3 × 3 = 27. To extract the cube root of a number, is to find one of its three equal factors. 1, 2, -3, 4, 5, 6, 7, 8, 9, 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 PERFECT CUBE is a number which has three exact equal factors. By examining the numbers in 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), 729,000, 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 ex plains the reason for pointing off into periods of three figures each. 388. Let us see how the cube of any number, as 16, is formed. Sixteen is composed of 1 ten and 6 units, and may e written, 10 + 6. To find the cube of 16 = 10 + 6, we must multiply the number by itself twice. To do this we place the number thus, 16 = 10 + 10+ 6 6 - 1000+ 1800 + 1080 + 216 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: 2. The second part 1800 is three times the square of the tens multiplied by the units; that is, 3 × (10)9 × 6 = 3 x 100 x 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, 63 = 6 X 6 × 6=216. 1. What is the cube root of the number 4096 ? ANALYSIS.--Since the num ber contains more than three figures, we know that the root will contain at least units and Separating the three right OPERATION. 4 098(16 1 1o × 3 = 3)30 (9-8-7-6 1634 096 |