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What does PERMUTATION show? Give an example, and explain the rule. What is COMBINATION ? Propose an example, and explain the rule by which it is solved.

What is INVOLUTION? What is the product obtained by Involution called ? What is the root ?—the 2d power ?—the 5th power ?—the index, or exponent? What other names are given to the 2d and 3d powers ?—Why? What is the effect of adding the exponents of two powers of the same number ?—of subtracting the less exponent from the greater ? -of multiplying the exponent by any number? Explain each case.

What is EVOLUTION ? What is a radical sign ?-a fractional exponent? What is indicated by the numerator of a fractional exponent ?—by the denominator ? What is a rational number ?-a surd ? What is meant by the extraction of the square root ? Explain the mode in which it is done, and perform an example. To what is the square of any number equivalent ? Prove that this is true by an ex. ample. To what are the areas of similar figures proportional ? What relation exists between the sides of a rightangled triangle? What is a mean proportional ? How is it found? What is the mean proportional between 3 and 4 ? Mention some of the properties of

square numbers. What is the CUBE Root of a number? How may we determine the number of figures that any cube root will contain ? Explain by blocks, the formation of cubes, and apply the principle to the extraction of cube roots. How may the trial divisors, after the first, be readily found ? Mention some of the properties of cubes.

When the exponent of a power can be resolved into factors, how may the root be extracted ? Show the application of this rule, by extracting the 8th root of 13. Apply the general rule by extracting the 5th root of 101621504799.

What is ARITHMETICAL PROGRESSION ? By what other name is it called? What are the extremes ?-the means? -the common difference? What is an ascending series? -a descending series? What is required, in order to de. termine an equi-different series? When one extreme, the common difference, and the number of terms are given, how may the other extreme, and the sum of all the terms be

How may

found ? Illustrate the rule on the board. How is the common difference found, the extremes and number of terms being given ? How is the rule deduced ? The extremes and common difference being given, how do we find the number of terms? Why does this process give us the num. ber of terms ?

What is GEOMETRICAL PROGRESSION ? By what other names is it called? What is the ratio ? Propose examples, and explain on the board, the process for finding one of the extremes, when the other extreme, the ratio, and number of terms are given ;-the sum of the terms,—the ratio, one ex. treme, and number of terms being given ;—the ratio, when the extremes and number of terms are given. we insert any number of mean proportionals between two given numbers ? Explain the analogy between Arithmetical and Geometrical Progression.

What is HARMONICAL PROPORTION ?-Harmonical Progres. sion? Why are they so called ? Two terms of a harmonical progression being given, how are the remaining terms found? How may we insert any number of harmonical means between two given numbers ?

What is an Annuity ?-an annuity certain ?-an annuity contingent ?-an annuity in possession ?-an annuity in reversion ? How do we find the amount of an annuity in arrears ?—the present worth of an annuity certain ?the present worth of a perpetual annuity ?-of an annuity in reversion ?

What is the commercial signification of ExCHANGE ? What is a Bill of Exchange ? Illustrate the operation of a Bill of Exchange, in the payment of debts. Is the bill generally transmitted directly?' Who is the drawer ?—the drawee ?the payee ?—the acceptor ?—the holder ?—the indorsee? What two kinds of indorsement are there, and what is the distinction between them? How is a bill accepted ? If payable after sight, what should be done on its acceptance ? What is the true par of exchange? What is the peculiarity in the nominal par with England ? What is the course of exchange, and by what is it caused ? What is the limit to the premium on the true par of exchange? What is a protest? What is meant by Domestic Exchange? What term

is applied to Domestic Bills of Exchange? State the value of the old currencies of different sections of the Union.

When are numbers said to be PRIME TO EACH OTHER? How may we tell by inspection, whether a number is divisible by 2, 3, 4, 5, 6, 9, 10, 11, or 25 ? How may we determine whether a given number is a prime number? How is a number resolved into its prime factors ? How do we discover all the divisors of a number? What is the rule for determining the number of divisors? How may the resolution into prime factors be applied in the reduction of fractions to their lowest terms ?

What is the cause of NUMERICAL APPROXIMATIONS ? Propose and explain examples in Addition and Subtraction of Circulating Decimals. In contracted multiplication and division ;-in division of circulating decimals. What is a con. tinued fraction ? Explain the mode of finding approximate values for a continued fraction? How may any square root be resolved into a continued fraction? How may the extraction of roots of the higher powers be contracted ?

What is the Julian Year? Explain the distinction between the Old and New Style? What is the Dominical Letter?. How is it found for any Julian Year ?—for any Gregorian Year? How may we find the day of the week, corresponding to any given day of the month in any year proposed ?

How may we find the Area or a TRIANGLE ?-of a rectangle ?—of a trapezoid ?--of any plain surface bounded by straight lines ?-of a circle ?-of an ellipse ?—the surface of a sphere ?-of a cylinder? What is the rule for finding the solid contents of a cylinder ?—of the frustrum of a cone? -of a sphere? How do we find the dimensions of a cask ? -the carpenters' tonnage of a vessel ?

How do we find the SPECIFIC GRAVITY of a body ?-the distance at which objects may be seen, at sea or on level ground ?—the distance of a gun or thunder cloud ?—the pressure of water against a bank?

Repeat the Table of AvoiRDUPOIS Weight; Troy Weight ;-Apothecaries' Measure ;-Dry Measure ;-Liquid Measure ;-Long Measure ;-Time Measure ;--Square Measure ;-Cubic Measure ;--Cloth Measure ;-Chain Measure.

MISCELLANEOUS EXAMPLES.

1726

1. If the multiplicand is 7, and the product 2, what is the multiplier ?

2. The dividend is 1, and the quotient 8; what is the divisor?

3. The dividend is 6), and the quotient 183; what is the divisor?

4. The sum of two numbers is 7}, and one of the numbers is 4.759; what is the other ?

5. The difference of two numbers is 133, and the greater number is 29.43; what is the less ?

6. The sum of two numbers is 7 y, and their difference is 63

}; what are the numbers ? 7. What number must be subtracted from 397 to leave

? 8. What number must be added to 236 to make 47.432?

9. What number must be multiplied by 37, and the product divided by 7.365 to give 8f as a quotient ?

10. What is the difference between it and 46 of 7 T. 15 cwt. 3 qr. ?

11. What is the difference between 28 miles, and 27 m. 7 fur. 39 r. 5 yd. 2 st. 11.9 in. ?

12. Reduce 4 of 9 m. 7 fur. 39 r. 5 yd. 2 ft., to inches.

13. If from a purse containing £35 7s. 11d., I pay each of 15 laborers, £1 9s. 8 d., how much will be left?

14. Find the sum, the difference, and the product of 874.91, and 42 s.

15. If 29 men can build 47} rods of wall in 97 days, how much can 15 men build in 244 days ?

16. A grocer sold 17 cwt. 3 qr. 17 lb. of sugar, at 63 cents a pound, receiving in exchange 29 barrels of flour at $11 per barrel, and the balance in money. How much money did he receive ?

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17. From of 3 T. 17 cwt., subtract i of 7 T. 3 cwt. 1 qr. 18 lb.

18. What will be the freight of 173 cwt. for 894 miles, if $7.63 be paid for carrying 11 tons, 9} miles ?

19. How many raisins at 8} cents a pound, must be given in exchange for 163 gal. 2 qt. 3 gi. of wine, at $1.125 per gallon?

20. Bought 16 cwt, 3 qr. 16 lb. of rice, at $4.00 per cwt. and 9 cwt. 2 qr. 5 lb. of pearl barley, at $4.374 per cwt. How much would be gained on the whole, by selling each at 44 cents a pound?

21. If 63yd. of broadcloth cost $255, at what price must it be sold per yd. in order to gain $25,50 ?

22. A hogshead of sugar at $7.00 per cwt. cost $43.75. What did it weigh?

23. How much shalloon that is & yd. wide will line 145 yards of cloth that is 14 yd. wide ?

24. What is the price of 16 boxes of raisins, each hold. ing 93 lb. at 9 cents per lb. ?

25. How much money that is 9 per cent. below par, will pay a debt of $187.50 ?

26. If 9 men mow 10 acres of grass in a day, how much will 11 men mow in 21 days.

27. In what time will $150 gain $6.37} at 6 per cent. simple interest ?

28. When molasses is 314 cents a gallon, how many hogs. heads, each holding 97 gal. 3 qt. can I buy for $366.564 ?

29. What will be the price of 7 bales of sheeting, each bale containing 9 pieces, and each piece measuring 308 yd., if 26 yd. cost. $2.921 ?

30. Bought 39} bushels of potatoes for $12.871. At what price per bushel must they be sold, in order to gain 15

per cent. ?

31. What is the interest of $9431 for 3 yr. 7 mo. 13 dy. at 7 per cent. ?

32. How much may a man spend per day, whose income is $500 a year, after deducting per cent. for taxes ?

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