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Note. — Cot.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, anil 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, anil cube the giren sides, diameters, altitudes, or circumferences, and the cube root of the fourth term of the proportion is lie answer required.

549. 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 giren solid, anil 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 t wo 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. 37 6.

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 J 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-f- inches.

8. If a pipe 1£ 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 31£ 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 mu1berry-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 produet 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-f-.

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 100 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 ^ 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. 450001b.

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 ouincunx order.

22. Four men, A, B, C, and D, bought a. grindstone, the diameter of* which was 40 inches and the place for the shaft 4 inches in diameter. It was agreed that A should grind off his share first, then in turn B, C, and D. Required how many inches each man will grind off from the semidiameter, providing they each paid the same sum.

Ans. A, 2.651in.; B, 3.137in.; C, 4.064in.; and D, 8.148in.

23. I have a board whose surface contains 49£ square feet;! the board is 1£ inches thick, and I wish to make a cubical box of it. Required the length of one of its equal sides.

Ans. 36 inches.

24. A carpenter has a plank 1 foot wide, 22£J feet long, and 2£ inches thick; and he wishes to make a box whose width shall be twice its height, and whose length shall be twice its width. Required the contents of the box.

Ans. 5719 cubic inches.

25. If a ball, 3 inches in diameter, weigh 4 pounds, what will be the weight of a ball that is 6 inches in diameter?

Ans. 321bs.

26. If a globe of gold, one inch in diameter, be worth $ 120, what is the value of a globe 3£ inches in diameter?

27. If the weight of a well-proportioned man, 5 feet 10 inches in height, be 180 pounds, what must have been the weight of Goliath of Gath, who was 10 feet 42 inches in height? Ans. 1015.1+1b.

28. If a bell, 4 inches in height, 3 inches in width, and £ of an inch in thickness, weigh 2 pounds, what should be the dimensions of a similar bell that would weigh 2000 pounds?

Ans. 3ft. 4in. high, 2ft. 6in. wide, and 2£in. thick.

29. What are the two mean proportionals between 56 and 12096? Ans. 336 and 2016.

30. Having a small stack of hay, 5 feet in height, weighing lcwt., I wish to know the weight of a similar stack that is 20 feet in height. Ans. 64cwt.

31. If a man dig a small square cellar, which will measure 6 feet each way, in one day, how long would it take him to dig a similar one that measured 10 feet each way? .

Ans. 4.629-f- days.

32. If an ox, whose girth is 6 feet, weighs 6001b., what is the weight of an ox whose girth is 8 feet? Ans. 1422.2-f-1b.

33. Four women own a ball of yarn, 5 inches in diameter. It is agreed that each shall wind off her share from the ball. How many inches of its diameter shall each wind off?

Ans. First, .43-f- inches; second, .ol-\- inches; third, .82-Linches; fourth, 3.149-)- inches.

34. John Jones has a stack of hay in the form of a quadrangular pyramid. It is 16 feet in height, and 12 feet wide at its base. It contains 5 tons of hay, worth $ 17.50 per ton. Mr. Jones has sold this hay to Messrs. Pierce, Row, Wells, and Northend. As the upper part of the stack has been injured, it is agreed that Mr. Pierce, who takes the upper part, shall have 10 per cent. more of the hay than Mr. Rowe; and Mr. Rowe, who takes his share next, shall have 8 per cent. more than Mr. Wells; and Mr. Northend, who has the bottom of the stack, that has been much injured, shall have 10 per cent. more than Mr. Wells. Required the quantity of hay, and how many feet of the height of the stack, beginning at the top, each receives.

Ans. Pierce receives 27^^cwt. and 10.300-)-feet in height; Rowe, 24£?8cwt. and 2.493 feet; Wells, 22£f JcwU and 1.666 feet; Northend, 252%cwt. and 1.474 feet.

PROGRESSION, OR SERIES.

554. A Series is a succession of numbers that depend on one another by some fixed law.

The numbers constituting a series are called its terms; of which the first and last are called extremes, and the other terms the means.

ARITHMETICAL PROGRESSION.

555. Arithmetical Progression, or Progression By Difference, is a series that increases or decreases by a constant number, called the common Difference.

The series is said to be an ascending one when each term

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