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give the rule. Give the proof. Make an example in Alligation Alternate; perform and explain it, and give the rule. Give the proof. How many answers may you have to examples in Alligation Alternate ? How do you proceed when one quantity is limited? when several quantities are limited? when the entire quantity is limited?

What is ARITHMETICAL PROGRESSION? What is an Arithmetical Series? When is a series increasing? when decreasing? Give examples of each. How many things are to be considered in a series? How many must be known, that the rest may be found? To what is the last term of a series equal? Show why? How do you find the common difference? Explain. How the number of terms? Explain. How the sum of the series? Explain.

What is GEOMETRICAL PROGRESSION? a Geometrical Series? What is the constant multiplier called? When the ratio is greater than unity will the series be increasing or decreasing? What will it be when the ratio is less than one? How many things are to be considered in a Geometrical Series? How many must be known, that the rest may be found? What is your rule for finding the last term? explain it. Your rule for finding the ratio? explain it. Your rule for finding the sum? explain it.

What are ANNUITIES? What is a Certain Annuity? a Perpetuity? a Life Annuity? a Pension? What is the amount of an annuity? the present worth? When are annuities in arrears? How are they gen erally computed? Give your rule for finding the amount of an annuity at simple interest. Illustrate it by an example. Give your rule for finding the amount of an annuity at compound interest. Illustrate it by an example. How do you use the table? How do you find the present worth of an annuity?

510. MISCELLANEOUS EXAMPLES.

1. of a number exceeds by 20; what is that number?

2. The sum of two numbers being and one of them being

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the difference between 10 and 24, what is the other?

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3. From the product of the sum and difference of 3.6 and 2.24, take the difference between the squares of 3.6 and 2.24.

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5. A can mow 2 acres in a day, B 24 acres, and C 3 acres; what is the smallest number of acres that will give a number of whole days' work for either?

6. What is the longest rail that will exactly fence either side

of a lot of ground, the sides being 56 ft., 42 ft., 63 ft., 77 ft., and 49 ft.?

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9. A quotient being 95ğ, and the divisor .33 of 302411, what is the dividend?

10. Bought a piece of cloth for $75.30, sold of it to one person, and of the remainder to another; what is the value of the unsold part at 10% advance upon the cost?

11. An aeronaut ascends at the rate of 44 miles an hour for 40 minutes, after which he maintains the same elevation; if his balloon is driven east 7 miles during the first hour from the time of his starting, and in an opposite direction at the rate of 10 miles an hour for the remaining time, how far from his starting-point in a straight line is he at the end of 5 hours?

12. What is the weight of a bale of cotton cloth, containing 13 pieces, 42 yards to the piece, every 3 yards weighing 1 lbs.?

13. A trader bought apples at $1.62 per barrel, and immediately sold them at $2.25, making $234.37; how many barrels were bought?

14. Divide 380+20+5+ (8169) × 5 by 7.5.5+

9.93-1.8

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15. The sum of three numbers is 554; two of them are 141 and 2411; what is the third?

16. Suppose a dividend to be 241.3, and the quotient .127; what was the divisor?

17. When the ice upon a pond is 10 inches thick, what will be the value of the ice taken from one acre of the pond at ct. per lb., 1 cu. ft. containing 583 lbs.?

18. A can do of a piece of work in 4 days, B of it in 5

days, and Cof it in 2 days; in what time will all together dc the whole ?

19. 1+1+1+of a certain number increased by 31+ 10504; what is the number?

20. A man left of his property, wanting $2000, to his daughter, of it to his son, and the balance to his widow, whose share was $500 more than that of his daughter; what was the share of each? Ans. Widow's, $7250; daughter's, $6750; son's $3500. 21. Of what number is the part?

22. A can do a piece of work in 1 hours, A and B in 48 min.; in what time can B do it alone?

23. A cubic inch of infusorial earth in Germany, weighing 220 grains, was found to consist of 41 billions of infusoria; what was the weight of one skeleton?

24.

Mr. JAMES H. EATON,

To 40000 ft. pine boards,

“15625 “walnut,

2875"scantling,

"234 thousand shingles,

CAMBRIDGE, Sept. 1, 1864.

TO FRANK CALDwell, Dr.
@ $12.00 per M.

"23 planks, 12 ft. long, 11 in. by 3 in.,*

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"12 sticks timber, 29 ft. long, 10 in. by 12 in., @ Required the cost of the above?

13.62" Ans. $887.79

25. A merchant buys gloves at 3s., and sells them at 4s. 6d.;

what does he gain in laying out 50 £?

26. Divide 52 into two such parts that equal of the other.

of one part shall

Ans. 12 and 40.

27. A person being asked the time of day, answered that it was between two and three o'clock, and that & of the time past two equalled of what it wanted of three. Required the time.

28. There is a pole 18 ft. long, standing in such a position that of the part in the mud is equal to of the part in the water, and to of the part in the air; how many feet are there in each

element?

Ans. 3 ft. in the mud, 4 ft. in the water, and 10 ft. in the air.

*See Note, page 141.

29. Suppose the water to rise so that of the part in the mud is equal to of the part in the water, how many feet are there in each element?

Ans. 3 ft. in the mud, 6 ft. in the water, and 9 ft. in the air. 30. Butter being worth 25 cents per lb., if g of a lb. will pay for of a dozen eggs, how many eggs will be required to pay for 6 lbs. of raisins, 7 lbs. of which cost 98 cents?

31. A, B, and C shared $102 between them, so that A had $17 more than B, and B had $20 more than C; what had each?

32. The captain of a ship at sea finds by his chronometer, at 12 o'clock at noon, that it is 45 m. past 8 o'clock in the evening at London; what is his longitude?

33. When it is 10 A. M., in X, which is 44° 15′ 2′′ W. long., what is the time in Y, which is 8° 4' 40" E. long.?

34. When it is half past 10 o'clock, P. M., of Dec. 31, 1865, in Albany, N. Y., what is the time in Constantinople? (See Art. 206.) 35. What is the exact time from Nov. 19, 1858, 18 min. of 4, P. M., to Apr. 9, 1863, 10 min. past 2, P. M.?

Ans. 4 yrs. 140 d. 22 h. 28 min.

36. Paid 64 cents a pound for 3200 lbs. of pork; on the sale of of it I gained $8}; with the money received I purchased 74 tons of plaster; how many cords of wood, of which 3 cords cost $22, would be worth 1 ton of the plaster?

37. If $1239 were paid for harvesting the wheat on a lot of land 400 rods long, 350 rods wide, what should be paid for harvesting the oats upon a lot 500 rods long, 450 rods wide, the cost of harvesting oats being as much as for harvesting wheat?

38. A coal dealer purchased 500 tons of coal at $7.50 per long ton, paid $1 per ton for freighting, and sold it for $11 by the short ton; what per cent. did he gain? Ans. $4419%.

39. When gold sells at 59% advance, how much can be bought with $100 of good bank bills?

40. A city collector has .8% for collecting taxes; he pays into the treasury $94625.64 after deducting his commission; what was the whole sum collected?

41. An army of 50000 men besieged a city for three months,

during the first month, 5% were lost by sickness and desertion; at the beginning of the second month an accession was made to their force equal to 20% of what remained; during the second month they lost 25% of the men, and during the last month 30% of what then remained were lost and detailed for service elsewhere; how many were left?

42. A grocer imported 75 galls. of oil, which cost him $2 a gall. and a duty of 10%. Suppose 5 galls. to leak out, for what must he sell the remainder per gall. to gain 10% on the money spent?

43. If you buy figs at the rate of 9 lbs. for $1.50, and sell them at the rate of 10 lbs. for $2, what do you gain per cent.?

44. Pedrick & Closson sold at auction,

2 mattresses,

at $16.00, which cost $13.50.

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1 bureau, at 18.00.

1 stove, at 17.00.

What were their net proceeds from the above sales?

45 I found, on going to Rand & Tyler's dry goods store, that they had that morning marked up their goods 15%; what did I save by purchasing the following goods the day before?

18 yds. blk silk, at $1.12.

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46. Paid 84 cents a gallon for a cask containing 27 galls. of kerosene, 10% of which leaked out; if the remainder was sold 25% on the gallon higher than it cost, what was the gain or loss on the money invested?

47. Sold 6 sewing machines at $72 each. On two of them I

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