539. А B Subtract the square of the perpendicular from the square of the hypothenuse, and extract the square root of the remainder. Thus, if the perpendicular be 3 and the hypothenuse 5, the base will equal ✓ 52 32 = N 16 4. All triangles having the same base are to each other as their altitudes. All similar triangles, and other similar rectilineal figures, are to each other as the squares of their homologous or corresponding sides. Thus, the triangles A CE and A CD, having the same base, A C, are to each other as the altitude E C of the one is to the altitude D C of the other. Also, the triangles A C E and B C D, having their corresponding angles the same, and their sides in direct proportion, are said to be similar, and are to each other as the squares of their corresponding sides, or as (A E) is to (B D)?, (A C) is to (B C), and (C E)” is to (C D)? Likewise the larger square, of which A C is one of the equal sides, is to the smaller square, of which B C is one of the equal sides, as (A C)? is to (B C)?. 540. All circles (Art. 143) are to each other as the squares of their diameters, semidiameters, or circumferences. The circumference of a circle is the line which bounds it; and the diameter is a line drawn through the center, and terminated by the circumference; as A B and C D. Then, the larger circle, of which A B is the diameter, is to the smaller, of which C D is the diameter, as (A B)’ is to (CD)', &c. 541. To find the side, diameter, or circumference of any surface, which is similar to a given surface. State the question as in Proportion, and square the given sides, diameters, or circumferences, and the square root of the fourth term of the proportion will be the answer required. Thus, if 12 feet be the length of a side of a triangle whose area is 72 square feet, the length of the corresponding side of a A D similar triangle whose area is 32 square feet would be found as follows: 72 : 32 :: 122 144 : 64; v 64 = 8 feet, length required. 542. To find the area of any surface which is similar to a given surface. State the question as in Proportion, and square the given sides, diameters, or circumferences, and the fourth term of the proportion will be the answer required. Thus, if 72 square feet be the area of a triangle of which 12 feet is one of the sides, the area of a similar triangle of which the corresponding side is 8 feet would be found as follows: 122 = 144 : 82 = 64 :: 72 sq. ft. : 32 sq. ft., area required. 543. To find the side of a square equal in area to any given surface. Find the square root of the given area, and that root will be the side of the area required. 544. A sphere is a solid bounded by a continued convex surface, every part of which is equally distant from the point within called the centre. The diameter of a sphere is a straight line passing through the centre, and terminated by the surface; as A B. B с 545. A Cone is a solid having a circle for its base, and tapering uniformly to a point, called the vertex. The altitude of a cone is its perpendicular height, or a line drawn from the vertex perpendicular to the plane of the base, as B C. The diameter of its base is a straight line drawn through the centre of the plane of the base from one side of the circle to the other; as A D. 546. Spheres are to each other as the cubes of their diameters, or of their circumferences. Similar cones are to each other as the cubes of their altitudes, or of the diameters of their bases. All similar solids are to each other as the cubes of their homologous or corresponding sides, or of their diameters. D 549. NOTE. Cor.es and other solids are said to be similar when their correponding parts are in direct proportion to each other. 547. To find the contents of any solid which is similar to a given solid. State the question as in Proportion, and cube the given sides, diam. eters, altitudes, or circumferences, and the fourth term of the proportion is the answer required. 548. To find the side, diameter, altitude, or circumference of any solid, which is similar to a given solid. State the question as in Proportion, and cube the gren sides, diameters, altitudes, or circumferences, and the cube root of the fourth term of the proportion is the answer required. To find the side of a cube that shall be equal in solidity to any given solid. Find the cube root of the contents of the given solid, and that root will be the side of the cube required. 550. To find a mean proportional (Art. 333) between any two numbers. Find the square root of the product of the two numbers, and that root will be the mean proportional required. 551. To find two mean proportionals between two given numbers. Find the cube root of the quotient of the greater of the two numbers divided by the less. The product of the less number by that root will be the least mean proportional, and the quotient of the greater number by the same root will be the other mean proportional. 552. To find any two numbers, whose sum and product are given. From the square of half the sum of the two numbers subtract their product, and the square root of the remainder will equal half the difference of the two numbers, which added to half their sum will give the larger, and subtracted from half their sum will give the smaller, of the numbers required. 553. To find any two numbers, when their sum and the difference of their squares are given. The difference of their squares divided by the sum of the numbers will give their difference; and half of their difference added to half of their sum will give the larger, and half of their difference subtracted from half of their sum will give the smaller, of the numbers required. EXAMPLES 1. A certain general has an army of 141376 men. How many must he place in rank and file to form them into a square ? Ans. 376. 2. If the area of a circle be 1760 yards, how many feet must the side of a square measure to contain that quantity ? Ans. 125.857+ feet. 3. If a line 144 feet long will reach from the top of a fort to the opposite side of a river 64 feet wide, on whose brink it stands, what is the height of the fort ? Ans. 128.99+ 4. A certain room is 20 feet long, 16 feet wide, and 12 feet high ; how long must a line be to extend from one of the lower corners to the upper corner farthest from it ? Ans. 28.28ft. 5. A certain field is 40 rods square; what must be the length of one of the equal sides of another field that shall contain only one fourth as much area ? Ans. 20 rods. 6. The areas of two similar triangular-shaped fields are 60 and 90 acres, and a side of the former is 66 rods. Required the corresponding side of the latter ? 7. If a lead pipe i of an inch in diameter will fill a cistern in 3 hours, what should be its diameter to fill it in 2 hours ? Ans. .918+ inches. 8. If a pipe 14 inches in diameter will fill a cistern in 50 minutes, how long would it require a pipe that is 2 inches in diameter to fill the same cistern? Ans. 28m. 7{s. 9. If a pipe 6 inches in diameter will draw off a certain quantity of water in 4 hours, in what time would it take 3 pipes of four inches in diameter to draw off twice the quantity ? Ans. 6 hours. 10. The first term of a proportion is 40, and the fourth term 90. Required a mean proportional between them. Ans. 60. 11. In a pair of scales a body weighed 314 pounds in one scale, and only 20 pounds in the other scale. Required its true weight. Ans. 25 pounds. 12. I wish to set out an orchard of 2400 mulberry-trees, so that the length shall be to the breadth as 3 to 2, and the distance between any two adjacent trees 7 yards. How many trees must there be in the length, and how many in the breadth ; and on how many square yards of ground will they stand ? Ans. 60 in length ; 40 in breadth ; 112749 sq. yd. 13. The sum of two persons’ages is 50 years, and their product is 600 years. What are their ages ? Ans. Of the one, 20 years; of the other, 30 years. 14. Two ships sail from the same port ; one goes due north 128 miles, the other due east 72 miles; how far are the ships from each other? Ans. 146.86+. 15. There are two columns in the ruins of Persepolis left standing upright; one is 70 feet above the plane, and the other 50; in a straight line between these stands a small statue, 5 feet in height, the head of which is 100 feet from the summit of the higher, and 80 feet from the top of the lower column. Required the distance between the tops of the two columns. 16. The sum of two numbers is 44, and the square of their difference is 16. Required the numbers. Ans. 24 the larger number; 20 the smaller. 17. A tree 80 feet in height stands on a horizontal plane; at what height from the ground must it be broken off, so that the top of it may fall on a point 40 feet from the bottom of the tree, the end where it was broken off resting on the stump? Ans. 30 feet. 18. The height of a tree, growing in the centre of a circular island, 100 feet in diameter, is 160 feet; and a line extending from the top of it to the farther shore is 400 feet. What is the breadth of the stream, provided the land on each side of the water be level ? Ans. 316.6+ feet. 19. A ladder 70 feet long is so planted as to reach a win dow 40 feet from the ground, on one side of the street, and without moving it at the foot it will reach a window 30 feet high on the other side ; what is the breadth of the street ? 20. If an iron wire 1 of an inch in diameter will sustain a weight of 450 pounds, what weight might be sustained by a wire an inch in diameter? Ans. 45000lb. 21. A gentleman proposes to plant a vineyard of 10 acres. If he places the vines 6 feet apart, how many more can he plant by setting them in the quincunx order than in the square order, allowing the plat to lie in the form of a square, and no vine to be set nearer its edge than 1 foot in either case ? Ans. 1870 more in the quincunx order. |