to be founded on rational principles, and supported by demonstration? I wish to warn you earnestly against this: receive no rule in any department of mathematics, the truth of which is not evident to your own understanding; and, therefore, in strict accordance with common sense. Multiplication is merely a short way of doing addition, and addition may always replace it: you have only to write the multiplicand down the proposed number of times, and to add all up. The sum is what in multiplication is called the product; but how could a sum of money be written down £19 19s. 11 d. times ? Even in common reduction, similar absurdities are to be met with in the books. If you wish to convert pounds into shillings, you are told to multiply the pounds by 20, and sometimes, which is worse, to multiply them by 20s. But if you multiply pounds by 20, you get-not shillings-but 20 times as many pounds, as is obvious: what you really do, is to multiply the number denoting how many pounds by 20; because there must be 20 times that number of shillings. The following is the rule for multiplying a compound quantity by a number : RULE I. When the Multiplier is not greater than 12. Put the multiplier under the quantity of least denomination: multiply that quantity by it, and divide the product by the number that expresses how many of such quantities make 1 of the next denomination : put down the remainder, and carry the quotient to the product arising from the multiplication of the next term,—and so on till all the terms have been multiplied. When the multiplier is greater than 12, and is yet such as to admit of being formed from factors, multiply by each factor in succession, as in short multiplication. The table of factors at the end will be found very useful in enabling us to tell at a glance whether any number not exceeding 100,000, can be decomposed into factors, within the limits of the multiplication table; and if so, what the factors are. £ 8. d. 23 14 Multiply £23 14s. 7ąd. by 7. Putting the multiplier 7 under the farthings, and multiplying them by the 7, the product is 21 farthings; and dividing 21 by 4, the number of farthings in a penny, we get 5 and 1 over; so that in 21 farthings there are 5 pence and 1 farthing: we put down the one under farthings, and carry the 5 pence to the 49, the pence product; which gives 54 pence, or 4s. 6d: we put down the 6d., and carry the 4 to the shillings' product, and thus get 102 shillings, or £5 2s.; and putting down the 2s., we carry the 5 to the pounds' product. The complete product is thus £166 2s. 6d. 166 26 Suppose the multiplier had been 105, then, seeing by the table that 1057 × 5 × 3; after the multiplication by 7, as above, we should have again multiplied by 5, and then by 3, as in the margin; from which we see that 105 times £23 14s. 7ąd. is £2491 17s. 9ąd. RULE II.-When the Multiplier exceeds 12, and is not divisible into factors, each less than 13. s. d. 23 14 7 323143 166 2 830 12 7 2491 17 9 Take that number in the table which is nearest to the proposed multiplier, whether greater or less, and use the factors of this number. To the final product add, if the number be less, and subtract from it if the number be greater, the product arising from multiplying the given quantity by the difference between the multiplier and number taken from the table: the result will obviously be the complete product required. £ 3. d. 23 14 7 x3 For example, if the multiplier of the sum above had been 107 instead of 105, we should still have taken 105, and have used the factors of it, as just shown; but to the product by these factors, we should have added twice the multiplicand; we should thus have got 105 times the sum and twice the sum, that is 107 times the sum as proposed. If the multiplier had been 109, then from 112 times, that is from 8 x 7 x 2 times, we should have subtracted 3 times the original multiplicand as in the margin, and should thus have found £23 14s. 73d. × 109 £2586 16s. 42d. The following statements are left for the learner to verify after the manner now shown: (1.) £3 18s. 6d. × 6 = £53 11s. Od. 63 = £9346 2s. 3ąd. 189 17 1329 27 22 2658 0 4 71 3 114 2586 16 4 (5.) 15 mil. 3 fur. 2 per. 4 yds. × 75 1153 mil. 6 fur. 4 per. 3 yds. = Division of Compound Quantities.-Division of concrete quantities may be viewed under two aspects, accordingly as the divisor is itself a concrete quantity or merely an abstract number. If you have to divide by a concrete quantity, your object is to find how many times the smaller quantity—the divisor-is contained in the larger-the dividend. But if you have to divide by an abstract number, you then seek to divide the proposed quantity into as many equal parts as there are units in the divisor. These, you see, are two different objects; and precision and accuracy of thought require that you should bear in mind the distinction. When you divide one concrete quantity by another, your quotient is, of course, an abstract number: but when you divide a concrete quantity by an abstract number, your quotient is also a concrete quantity of the same kind. You will remember that I am not here writing a book on Arithmetic exclusively with a view to mercantile practice. I am endeavouring to prepare you for a course of mathematical study; and I therefore wish you to cultivate habits of thought and reflection—to know what you are actually about, and not to feel contented by merely following a rule. I shall not insist upon any marked departure from the customary forms of expression, in the practical directions for working an example; but I do insist upon accuracy of thought, whatever want of precision in language custom may authorize. To divide a compound quantity by a number, the rule is this: RULE.-Commence with the highest denomination, and take the proposed part of it; reduce what is over to the next denomination, and carry the result to the next term of the dividend; take the proposed part of the sum, reducing what is over, and carrying as before; and so on, to the end. £ s. d. 7)22 15 9 3 5 14+f. Thus, if the 7th part of £22 15s. 9d. be required, we find it as in the margin: the 7th part of £22 is £3 and 208. over: this, carried to the 15s., gives 35s.; the 7th part of which is 5s., and there is nothing to carry: the 7th part of 9d. is 1d.; and 2d., or 8 farthings, over; the 7th part of which is 1 farthing, and 4 of a farthing; hence, the 7th part of the proposed sum is £3 5s. 14d. + 4ƒ. If the divisor exceed 12, we must proceed, upon the same principle, by long division, unless the divisor can be decomposed into convenient factors; when the operation need to consist only of successive steps like the single step above. When the divisor, instead of being an abstract number, is a concrete quantity, of the same kind as the dividend, the rule is as follows: RULE.-Reduce both dividend and divisor to the lowest denomnation found in either, and then perform the division exactly as in the case of mere numbers: the quotient will denote the number of times the smaller quantity is contained in the greater. For example, let it be required to divide £63 78. by £13 2s. 3d. Then, as pence is the lowest denomination that occurs, we reduce both quantities to pence, and then divide as in 38 13 £ s. d. 2 3 £ S. 20 63 7 262 12 3147) 1267 15204(1 2616 the margin the quotient shows that the smaller sum is contained in the larger between 4 and 5 times: it is contained in it 4 times and a fractional part of a time, represented by 31, which is nearly another time, but not quite. You may shorten the work a little by reducing the two quantities-not to pence, but to three-pences-as shown below; observing, that as 4 three-pences make 1 shilling, we multiply the number of shillings by 4, and take in the 1 threepence. From this mode of working, we should conclude that the dividend contains the divisor 4 times and a part of a time, denoted by the fraction, which differs from the former fraction only in appearance-not in value; for if we wish to express that one number is to be divided by another, we may, as you are aware, do so by writing the latter below the former, or by writing twice, three times, &c. the latter below twice, three times, &c. the former; as is pretty obvious, since the quotient of dividend and divisor is the same, whatever number both be multiplied by; and you see that the upper and lower numbers of the first fraction are only those of the second, each multiplied by 3. £ 8. d. 13 2 3 20 £ 8. 63 7 20 262 1267 4 1049) 5068(4 4196 872 There is no room in this treatise for many examples. I shall here give you two. The first is to show that £33 15s. 6d. divided by 13 gives £2 11s. 6d. for quotient; the second is to show that the same sum divided by £13 gives 2311 for quotient. In working the second example, you had better reduce to sixpences, not to pence.* Fractions. What has preceded suffices to convey a general, and, I hope, a pretty accurate notion of the arithmetic of integral quantities. I am now to show how the fundamental operations are to be applied to fractions. I have found it impossible to avoid all allusion to fractions in the foregoing part of the subject, because they force themselves upon our notice even when operating upon integers; but the arithmetic of fractions remains to be explained, and, indeed, the formal definition of a fraction to be given. In strictness, a fraction is a part of a whole-that is, it is less than the quantity of which it is said to be a fraction. Thus, 1, 3, 4, &c., are strictly fractions-proper fractions. The first denotes a third part of unit, or 1, the second a fifth part of 2, the third a forty-third part of 26, &c., each part being less than one whole. But, 3, 4, &c., are also called fractions, though four-thirds, seven-fifths, sixty-four forty-thirds, &c., are all greater than one whole, as is plain; fractions such as these, where the upper number, called the numerator, is not less than the lower, called the denominator, are said to be improper fractions. You will readily see why these terms, numerator and denominator, are so applied: the upper number enumerates, or states the number of parts of that particular denomination indicated by the For a great variety of instructive examples in all the rules of arithmetic, as well as for a comprehensive view of the theory, see the "Rudimentary Treatise on Arithmetic," published by Mr. Weale. lower number. Thus, means three of the parts called fourths: if it were of £1, then, since one-fourth is 5s., three-fourths, or , would be 15s., and so on. Instead of reading this fraction three-fourths, we may, if we please, say three divided by four. Three pounds divided by the number 4, is evidently the same as three-fourths of one pound; and any fraction may be viewed in either of these two ways-thus it is matter of indifference whether you call 4, five-sevenths, or 5 divided by 7: a moment's reflection will convince you that five-sevenths of anything, is the same as a seventh part of five such things; for a seventh part of one of them added to a seventh part of another, then again this sum increased by a seventh part of another, and so on, till a seventh part of each of the five has been taken, and all these sevenths added,-I say it is plain, that in this way we get 5 times a 7th part of one-that is, five-sevenths of it as the result of all five divided by 7. The fractional notation is perfectly general-any number may be expressed in it; a whole number, or an integer, as well as a fraction properly so called. Thus 6, 8, &c., may be written f, f, &c.; and it is sometimes convenient to write integers this way. Here the denominator is unit, or 1; but you may express an integer in the form of a fraction with any denominator you please. Thus, if you choose 7 for denominator, the two numbers, 6, 8, may be written 42, 56, as is evident: you have only to multiply the number by the chosen denominator, and to place the factor, thus used as a multiplier, underneath -that is, as a divisor. The numerator and denominator are called the terms of the fraction; and when an integer is united to a fraction, the whole is called a mixed number. Thus, 2, 3, &e., are mixed numbers. To reduce a Mixed Number to an Improper Fraction. The rule is this: multiply the integer by the denominator of the fraction; add the product. to the numerator, and put the denominator underneath. Thus, 2}=}, 34=36; for 2 is evidently, and §+}=}. In like manner 3 is, and 4+4=25; and so on. Here are other examples: 53=28; 43=Y; 7V=Y; 12&=440, &c. &c. To accomplish the contrary purpose-that is, To reduce an Improper Fraction to a Mixed Number, You have only to perform the division indicated by the denominator, and to annex to the quotient the fractional correction as in common division. Thus, 25, 39=47, 368, and so on. To reduce Fractions with Different Denominators, to others of the same Value with Equal Denominators. This is one of the most important operations in the arithmetic of fractions; for till fractions appear with a common denominator, they can neither be added to nor subtracted from one another: the reduction of fractions to a common denominator is thus a preliminary indispensably necessary to the application to them of the first rules of arithmetic. The operation is also useful in enabling us to discover at a glance which of two fractions, however nearly equal, is really the greater. Thus, of the two fractions, 35, 49, we see in a moment which is the greater, because their denominators are the same; but you could not so readily and confidently state which is the greater of and; yet the former are 욕 only these reduced to a common denominator-the values are the same. The rule for reducing fractions to a common denominator is as follows: RULE.-Multiply each numerator by the product of the denominators of all the other fractions; we shall thus get the numerators of the changed fractions. Multiply all the denominators together; the product will be the common denominator belonging to each changed numerator. For example: in order to reduce the fractions,,, to others of the same value with the same denominator, we proceed as follows: 2×5×7 70 1×3×7 21 the new numerators; 3x5x7=105 the common denominator; therefore the equivalent fractions, changed in form as required, are, 103, 105. If you compare these with the original fractions, you will see that they each arise from multiplying numerator and denominator of the former by the same number. Thus, 70 2 X 35 ·; and it is obvious that one number divided by another (in this case 2 by 3), 105 3 x 35 is the same as 35 times the former divided by 35 times the latter, or any number of times the former divided by the same number of times the latter. If you have any doubt of this, just consider, if you had to divide 28. among 3 people, whether the share of each would not be the same as if you had to divide 35 times 2s.—that is, 70s.-among 35 times 3 people—that is, 105 people. It is plain that, in either case, each would get a third part of 2s., or two-thirds of ls.; or, to view the matter more generally, it is self-evident that if you multiply any quantity by a number, and then divide by the same number, you virtually leave the quantity, as to value, untouched; for multiplication and division by the same number, are two operations which mutually neutralise one another: we may, therefore, always multiply numerator and denominator of a fraction by any number, without changing the value of the fraction. The rule just given will always effect the object proposed by it; but not always in the shortest way. In particular cases it will be desirable to proceed differently. Thus, if the fractions,,, are to be changed into equivalent ones with a common denominator, you see, by looking at the denominators, that the thing may be brought about without interfering with the middle fraction at all: you have only to multiply the terms of the first fraction by 2, and those of the third by 3,to get the desired result-the changed fractions being found in this way to be,, . If you had applied the rule, the new fractions would have been 34, 38, 38, forms far less simple than those above, although the same in value; they would be got by multiplying the terms of the simpler fractions, each by 6. In bringing fractions to a common denominator, you should always be on the look out for the simplest multiplier of the terms of each that will accomplish the object, and use the rule only as matter of necessity—that is, only when simpler multipliers than the rule supplies do not present themselves. Suppose you had, 3, 3, do you not see, from a glance at the denominators, that if the first be multiplied by 3, the second by 4, and the third by 8, that the products will be all alike? Multiply, then, the terms of the first fraction by 3, those of the second by 4, and those of the third by 8, and you will get the following-viz., 2, 4, ¡¡, for equivalent fractions with a common denominator. The rule would have given you these— 144, 112, 124, which, although equal to, are far less simple than the former. The smallest number that can be a common denominator of a row of fractions is evidently the smallest number that is divisible by each of the given denominators: it is called the least common multiple of those denominators. There is a rule for finding the least |