rule with all the examples before the questions are asked that are an nexed to each rule. When the student commences a rule, the teacher should commence questioning him, and explain the nature and principles of the rule, requiring the student to give reasons for all the operations of his work. What is simple subtraction? A. Taking a less number from a greater. What are the two given numbers called? A. Minuend and subtrahend. What is the number called that arises from the operation of the work? A. Difference or remainder. What does the difference or remainder show? A. It shows how much the minuend is greater than the subtrahend. What does minuend signify? A. A number to be lessened by another. What does subtrahend mean? A. A number to be taken from a greater. How do you place the given numbers in subtraction? A. So that units stand under units, tens under tens, &c. How do you proceed in the work? A. Commence at the right hand, and take the figures in the subtrahend from those directly above them in the minuend, placing the difference in each place directly under. When the lower figure in any place exceeds that directly above it, what must be done? A. Add ten to the upper figure, and take the lower figure from the amount, placing the difference directly under, and then add one to the next lower figure of the subtrahend. Why does not this adding ten to the figure in the minuend and then one to the next lower figure of the subtrahend, alter the true difference between the given numbers? A. Because it is adding equals to both the given numbers, and adding equals to both, the difference between two numbers must ever remain the same; and we add equals to both, because ten in an inferiour column, (which we add to the minuend,) is only equal to the one which we add to the next superiour column of the subtrahend.— How do you prove subtraction ? A. By adding the difference to the subtrahend, and if the amount equals the minuend, the work is right; because it is evident, that it can take no more than the difference between two numbers to make the less equal to the greater. Can subtraction be made to prove itself? A. Yes. How is that done? A. By subtracting the difference from the minuend, and if it leave a number equal to the subtrahend the work is right, because it is evident, that if the difference between two numbers be taken from the greater, it reduces it equal to the less. What is subtraction of decimals? A. Taking something that is less from that which is greater, where one or both of the given sums contain parts of an integer. How do you place decimals for subtracting? A. Tenths under tenths, hundredths under hundredths; or in federal money, which is the same, dimes under dimes, cents under cents, &c. What are integers in federal money? A. Dollars. What are dimes? A. Tenths. What are cents? A. Hundredths. What are mills? A. Thousandths. What is the decimal expression called when dimes, cents and mills are taken together? A. Thousandths of a dollar. How do you subtract decimals? A. The same as in whole numbers. Why do you subtract the same? A. Because ten in an inferiour column or denomination, is equal to one in the next superiour, the same as in whole numbers. Why do we call dimes and cents, cents only? A. We call them all cents, because the left hand figure in cents expresses tens in cents, which is the same as dimes. When your subtrahend contains denominations which are not named in the minuend, what do you do? A. Supply the vacant denominations of the minuend with ciphers, then place the subtrahend under, so that cents may occupy the place of cents, and mills the place of mills. If you are required to subtract dimes from dollars, what do you first do? A. Join two ciphers to the right hand of the minuend, placing a separatrix between them and dollars; then place your dimes under the left hand cipher joined to the minuend, and to the dimes join a cipher, and they will stand as cents, and then subtract. How do you point off in subtraction of decimals? A. Place the separatrix in the result, directly below the separatrix in the given numbers. Why is federal money introduced under the head of decimals? A. Because dimes, cents and mills form a decimal expression of which a dollar is the integer. SIMPLE MULTIPLICATION, Is repeating one of two numbers as often as the other contains a unit; or, it is the shortest method of performing addition, where the same number is to be repeated a given number of times. The two given numbers are called multiplicane and multiplier. The multipli cand is the number to be repeated. The multiplier is the repeater, or number by which you multiply. The number produced by the operation of the work, is called the product. This is the most useful rule in practical arithmetick. When the price of one is given, by this rule, we obtain the price of any number, or quantity; when length and breadth are given, by it, we find the area or surface; in reduction, it affords the greatest facility in reducing higher denominations to lower, and its principles are advantageously applied in all practical business of buying and selling.. NOTE.—The multiplicand and multiplier taken together, are called factors or substitutes. Multiplication is denoted by this character, x; thus, 6x3=18, which signifies, that the product of 6 multiplied by 3, is 18. NOTE. No pains should be spared by the student in making himself master of the following table. The task is easy if persevered in; but when the student suffers himself to pass over it superficially, no time afterwards spent in work is scarcely sufficient to make it familiar to his mind. A good knowledge of it will greatly facilitate his progress, and save him the trouble of repeatedly reviewing the same work detect mistakes. The student should be required mentally to answer the fol owing questions. What will 3 apples come to, at 3 cents each? What will 8 bushels of wheat cost, at 8 shillings a bushel? 8 times 8 are how many? What will 10 bushels of oats come to, at 2 shillings a bushel? What are 9 yards of calico worth, at 3 shillings a yard? 3 times 9 are how many ? What must you pay for 12 cows, at 12 dollars each? What will 4 cows come to, at 10 dollars each? What will 5 cows? what will 6 cows? what will 7 cows? what will 8 cows? what will 10 cows? What will 11 ploughs cost, at 11' dollars each ? What will 9 hogs cost, at 9 dollars each? A pound contains 20 shillings; how many shillings in 2 pounds? in 3 pounds? in 4 pounds? in 5 pounds? How many are twice 20? how many are 3 times 20? 4 times 20? 5 times 20? Twelve pence make 1 shilling; how many pence in 2 shillings? in 3 shillings? in 4 shillings? in 5 shillings? in 6 shillings? in 7 shillings? in 8 shillings? in 9 shillings? in 10 shillings? in, 11 shillings? in 12 shillings? How many are twice 12? 3 times 12? 4 times 12? 5 times 12? 6 times 12? 7 times 12? 8 times 12? 9 times 12? 10 times 12? 11 times 12? 12 times 12? CASE I-When the multiplier is a single, significant figure. RULE-Place the multiplier under the right hand figure of the multiplicand, and draw a line underneath. First; multiply the right hand figure of the multiplicand by the multiplier; and when the product does not exceed 9, place it directly under; but if the product exceed 9, place down the right hand figure of the product; and add the left to the product of the next figure of the multiplicand, and so proceed, till you have multiplied all the figures of the multiplicand by the multiplier; remembering to set down the whole product of the left hand figure. METHODS OF PROOF.-1st. Make the multiplicand a multiplier, and if it produce the same result, the work is right. 2nd. Multiplication may be proved by addition. Write the multiplicand down as many times as the multiplier expresses a unit; then add, and if the same result be produced, the work is right. 3d. Multiplication may be proved by subtraction. From the product, subtract the multiplicand as many times, as the multiplier expresses a unit; and if it diminish it to nothing, the work is right. 4th. Lastly, multiplication may be proved by division. Divide the product by either of the factors, and if the operation produce the other, the work is right. Although this is the best method of proof, yet it should be omitted, till the student has become acquainted with division. Other methods of proof might be given; but these are introduced on account of their being the most simple, and best suited to our purpose in illustrating the principles of this rule; and showing the relation which it bears to the other simple rules. 4 1. Multiply 8 by 4. EXAMPLES. We place the multiplier under the multiplicand, as our rule directs. Then we say 4 times 8 are 32, placing the product under; we then have 32 for a product, which is 8, 4 times repeated. DEMONSTRATION.-That this is a short way of performing addition, is very evident: for we arrive at the result at once, which, in addition, requires the multiplicand 8, to be set down four times and added; thus, 8181818=32. 1st. EXAMPLE-Proved according to the first method. Multiplicand, 4 What was before the multiplicand, Multiplier, 8 now becomes the multiplier. We now Product, say 8 times 4 are 32, the same product 32 as was produced by the first operation. DEMONSTRATION. It is plain, that it can make no difference, whether 8 be repeated 4 times, as, 4 times 8 are 32; or 4 repeated 8 times, as, 8 times 4 are 32, the same result is produced. 1st. EXAMPLE. Proved according to the 2nd method. DEMONSTRATION. Here, we set down the multiplicand 4 times 8 8 8 8 because in the example, our multiplier expresses units; we then add, and the result clearly demonstrates, that when the same number is repeated by multiplying, it may be repeated by adding, and con sequently proves the work. But proof by addition is a more tedious method than proof by multiplication, and is therefore only introduced to show the relation between the two rules. It is also plain, that it can make no difference, whether we add the multiplicand as many times as our multiplier expresses a unit; thus, 8+8+8+8=32; or, add the multiplier as many times as our multiplicand expresses a unit, thus, 4+4+4+4+4+4+4+4=32; the same result, you perceive, must be produced. 32 Amount. 1st. EXAMPLE.-Proved according to the 3d method. Multiplicand, 8 DEM.-You have just learned, that multiplicaMultiplier, 4 tion is a short way of performing addition. You Carried up. |