What is ADDITION? How is the sign plus written, and what does it denote? How is the sign of equality written? How are numbers written in Addition? Propose an example in Addition, and perform it on the board, giving the reason for each step of the process. How may Addition be proved? What is SUBTRACTION? What is the Minuend?-the Subtrahend?-the Remainder? What other terms are applied to the Remainder? What is the form of the sign minus, and its use? How are numbers written in Subtrac tion? If any figure of the Subtrahend exceeds the one above it, what may be done? Show that the true result may thus be obtained. How is Subtraction proved? Propose an example in Subtraction, and perform it on the board, giving the reason for each step.

What is MULTIPLICATION? What is the Multiplier ?the Multiplicand?—the Product? What are the factors of a number? What is a prime number?-a composite number? What is the sign of Multiplication? How is multiplication performed? How proved? How many decimals are pointed off in the product? Explain the reason for this. How may any number be multiplied by 10, 100, 1000, &c.? What other abbreviations may be adopted? How may the product be obtained in a single line? Propose and explain an example in Multiplication.

What is DIVISION? What is the Divisor?-the Dividend? -the Quotient?-the Remainder? What is a Fraction? What is the Numerator?-the Denominator? How may the remainder of any division be expressed in the form of a fraction? How is division performed? How many decimals are contained in the quotient? How is division proved? What abbreviations may be adopted? How is the true quotient figure obtained? What is a multiple ?—a common multiple-a sub-multiple? How is the least common multiple found? How may it be found by a table of prime fac tors? What is a common divisor, or common measure? How is the greatest common divisor found? How may it be found by a table of prime factors? Propose and explain an example in Division ;-in finding the least common multiple; the greatest common measure.

In how many different ways may FRACTIONS be regarded?

Show the application of each. What are the terms of a fraction? What is a proper fraction?—an improper fraction?-a mixed number?-a compound fraction?-a complex fraction? How may we reduce an improper fraction to a whole or mixed number?-a whole number to a fraction having any given denominator?—a mixed number to an improper fraction?-a compound fraction to a simple one?— a fraction to a decimal?-a decimal to a fraction?-a fraction to its lowest terms?-two or more fractions to a common denominator? Propose and explain an example of each reduction. Propose and explain examples in Addition, Subtraction, Multiplication, and Division of Fractions.

What are CIRCULATING DECIMALS, and whence do they arise? What is the repetend ?—the finite part? How may we reduce infinite decimals to fractions? How may we divide by any number of 9's? How, when all but the units' figure are 9's? Give an example of each division, and explain the principle on which it depends.

What are COMPOUND NUMBERS? How are the operations on them performed? Propose and explain an example in reducing higher denominations to lower ;-lower denominations to higher;-lower denominations to the fraction of a higher; to the decimal of a higher. Propose and explain an example in Compound Addition ;-Compound Subtraction; Compound Multiplication;- Compound Division. What are Duodecimals? Perform examples in Multiplication and Division of Duodecimals, and explain each principle involved.

Prove an example in each of the simple rules, by CASTING OUT THE NINES. Show the application of the principle to errors of transposition.

How may business operations in which Compound Numbers are involved, be frequently abbreviated? Give an example? Propose examples of abbreviations in multiplying by any number of 9's;-in multiplying and dividing by 5;by 25;-by 75;-by 125;-by 375;-by 625;—by 875;— in multiplying by any number within 12 of 100, 1000, &c.; -in squaring a number ending in 5;-in multiplying two numbers in which the tens are alike and the sum of the units is 10;-in finding the product of two numbers, one of

which is as much above, as the other is below, a certain number of tens;-in multiplication, when one figure of the multiplier is an aliquot part of some of the remaining figures.

What is PERCENTAGE? Give and explain an example of the application of percentage to Commission;-Insurance ;Taxes;-Stocks;-Gain and Loss ;-Duties;-Simple Interest;-Compound Interest ;-Discount. What is the distinction between Simple and Compound Interest? What is the Principal?-the Rate?-the Amount? What is the usual rate? What is the Bank Rule for computing interest? -the Decimal Rule? Explain the principles on which each is founded. Give an example of an Account Current. How do we find the rate, when the principal, interest, and time are given?—the time, when the principal, interest, and rate are given ?—the principal,—the time, rate and interest being given the principal, the time, rate and amount being given? What is the usual mode of computing discount?

Propose and explain examples in EQUATION OF PAYMENTS, and AVERAGE.

What is RATIO? What is a proportion? What are the terms of a proportion called? What are the antecedents ?— The consequents ?-The extremes ?-The means? What is said of the product in a proportion? Prove the fact. When one of the extremes, and the two means are given, how may the other extreme be found? How may the antecedents and consequents be diminished? State the RULE OF THREE, and solve a question by it. Explain the distinction into multipliers and divisors ;—into cause and effect. What simple rule is founded on these distinctions? How is the statement made when the terms are fractional? Why? What is Arbitration of Exchange? What is the Chain Rule? Show the connection of cause and effect, in the Chain Rule. Propose and explain examples in Proportion, and Arbitration of Exchange.

What is the object of the rule of FELLOWSHIP? Propose and explain examples to which the rule is applicable.

What is ALLIGATION? How may any desired mixture be made, when there is no limit ?-when the whole quantity is limited?—when one or more of the ingredients is limited?

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 example. 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 determine 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

found? Illustrate the rule on the board.

How is the com

number of terms

mon difference found, the extremes and 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 number 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 extreme, and number of terms being given ;—the ratio, when the extremes and number of terms are given. How may we insert any number of mean proportionals between two given numbers? Explain the analogy between Arithmetical and Geometrical Progression.

What is HARMONICAL PROPORTION ?-Harmonical Progression? 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