TABLE OF MULTIPLIERS.

1. Radius of a circle X6-28318531-Circumference.
2. Square of the radius of a circle X3.14159265=Area.
3. Diameter of a circlex3.14159265-Circumference.
4. Square of the diameter of a circle X0-78539816—Area.
5. Circumference of a circle X0 15915494-Radius.
6. Circumference of a circleX0-31830989-Diameter.
7. Square root of area of a circle X0-56418958-Radius.

8. Square root of area of a circle X1·12837917=Diameter.

9. Radius of circle X1-73205081-Side of inscribed equilateral triangle
10. Side of inscribed equilateral triangleX0-57735027=Radius of circle.
11. Radius of a circle X1-41421356 Side of inscribed square.

12. Side of inscribed square X0-70710678 Radius.
13. Square of radius of a sphereX12-56637061=Surface
14. Cube of radius of a sphereX4-18879020 Volume.
15. Square of diameter of a sphereX3.14159265-Surface.
16. Cube of diameter of a sphereX0-52359878-Volume.
17. Square of circumference of a sphereX0-31830989 Surface.
18. Cube of circumference of a sphereX0.01688686 Volume.
19. Square root of surface of a sphereX0-28209479=Radius.
20. Square root of surface of a sphereX0-56418958=Diameter.
21. Square root of surface of a sphere X1-77245385-Circumference.
22. Cube root of volume of a sphereX0-62035049=Radius.
23. Cube root of volume of a sphere X1-24070098=Diameter.
24. Cube root of volume of a sphereX3-89777707=Circumference.
25. Radius of a sphereX1.15470054-Side of inscribed cube.
26. Side of inscribed cubeX0-86602540-Radius.

PROBLEM XI.-To find the volume of a frustum of a pyra

mid, or of a cone.

RULE.

Find a mean proportional between the area of the two bases, to which add the sum of the bases, and multiply the result by one-third the altitude of the frustum.

EXAMPLES.

1. Suppose a cistern in the form of a frusturn of a

cone, to be 9 feet deep, having for diameters 8 feet and 10 feet. How many cubic feet will it contain? 102 × 0.7854-100 × 0.7854-area of une base 6a × 0.7854 36×0·7854= 66 other "

60×0-7854= {

mean proportion between

bases.

196 × 0.7854=Sum.

And 1960-7854x of 9-461-8152 cubic feet, for its volume.

2. Suppose a measure to be in the form of a frustum of a regular cone. If its top diameter is 6 inches, and the bottom diameter 9 inches, and it is 12 inches deep, how many cubic inches will it contain? and how many beer gallons of 282 cubic inches each?

PROBLEM XII.—To find the area of an ellipse.

NOTE.-A line drawn through the centre of an ellipse is called its diameter. The longest diameter is called the transverse diameter; the shortest is called the conjugate diameter. Thus AB is the transverse diameter, and CD is the conjugate diameter.

The area of an ellipse may be found by this

RULE.

Multiply the product of the transverse and conjugate

diameters by 0.7854.

EXAMPLES.

1. How many square feet in the surface of an elliptical pond, whose transverse diameter is 100 feet, and conju gate diameter 60 feet?

Ans. 100 × 60 ×0·7854-4712·4 square feet.

2. How many square inches is an elliptical table whose transverse diameter is 5 feet 3 inches, and conjugate diam. And how many square feet?

eter 3 feet 6 inches?

144. 1. Suppose I purchase $1200 worth of goods, of which is on a credit of 3 months, on a credit of 6

months, and the remaining on a credit of 9 months How much ready money ought to pay the purchase, in terest being 7 per cent.? Ans. $1159-64, nearly. 2. In the above example, by the principles of equation of payments, how much credit ought I to have on the whole sum of $1200? Ans. 6 months.

5. Now, what is the present worth of $1200 due at the end of 6 months, interest being 7 per cent.?

Ans. $1159-42, nearly.

4. I employed A and B to ditch my meadow. A was to receive 87 cents per rod, and B was to have 112 cents per rod; each wrought until his wages amounted to $50. What was the amount of ditch dug by both? Ans. 10137.

5. Three merchants, A, B, and C, enter into partnership A advances $1200, B $800, and C $600. A leaves his money 8 months, B 10 months, and C 14 months in the business. They gain $500. What is the share of

6. A and B have the same income; A saves

of his;

but B, by spending $120 per annum more than A, at the end of 10 years finds himself $200 in debt. What was the income? Ans. $500.

7. Suppose a book to contain 365 pages, averaging 40 lines of 10 words each on each page. How many words would the book contain ? Ans, 146000 words.

8. There are 31173 verses in the Bible; how many days will it require to read it through, if 30 verses are read daily? Ans. 1039 days.

9. After expending of my money, and of the re mainder, I had remaining $72; how much had I at first? Ans. $128.

10. If I sell cloth at $1.50 per yard, and gain 25 per cent., how ought I to have sold it so as to lose 20 per cent.? Ans. $0.96. 11. Sold cloth at $1.50 per yard, and gained 25 per What should I have lost per cent., if I had sold it at $0.96 per yard? Ans. 20 per cent. 12. If I buy cloth at $1.20 per yard, how must I sell it so as to gain 25 per cent. ? Ans. $1.50.

cent.

13. A merchant has to make the following payments at three different periods: $2832 in 3 months, $2560 in 9 months, and $1450 in 16 months. The creditor wishes to receive the whole sum of $6842 at once. When ought the payment to be made? Ans. In 8 months.

14. A father gives to his five sons $1000, which they are to divide according to their ages, so that each elder son shall receive $20 more than his next younger brother. What is the share of the youngest ? Ans. $160.

15. A company of 90 persons consists of men, women, and children The men are 4 in number more than the women, the children 10 more than the adults. How many men, women, and children, are there in the company?

16. The common school fund for the state of New-York was $1975093 15 in 1843, and during the same year there were in the state, 677995 children between the ages of 5 and 16 years. How much would the above fund amount to per scholar? Ans. $2.91, nearly.