$3. It remains to resolve the case in which is multiplied by -; or, for example, a by-b. It is evident, at first sight, with regard to the letters, that the product will be ab; but it is doubtful whether the sign+, or the sign, is to be placed before the product; all we know is, that it must be one or the other of these signs. Now I say that it cannot be the sign for a by +b gives a b, and a by-b cannot produce the same result as a by+b; but must prodce a contrary result, that is to say,+ ab; consequently we have the following rule : — multiplied by produces +, in the same manner as + multiplied by +.* -: * It is a subject of great embarrassment and perplexity to learners to con. ceive how the product of two negative quantities should be positive. This arises from the idea they receive of the nature of multiplication as explained and applied in arithmetic, where positive quantities only are employed. The term is used in a more enlarged sense when negative quantities are concerned, as may be shown without making use of letters. If I wished to multiply, for instance, 9— 5 (or 9 diminished by 5) by 3, I should first find the product of 9 by 3 or 27. But this is evidently taking the multiplicand too great by 5, and of course the product too great by 3 times 5; I accordingly write for the product 27 — 15, equivalent to 12, which is the product that would arise from first performing the subtraction indicated by the sign, and using the result as the multiplicand. Thus, Let us now take for the multiplier the quantity 7-4, which is equivalent to 3. We multiply, in the first place, by 7, in the manner that we have just done by 3, and the result is 63-35. But as the multiplier is 7 diminished by 4, multiplying by 7 must give 4 times too much. Accordingly we take 4 times the multiplicand, or 36—20, and subtract this from 63-35, or 7 times the multiplicand. Now in making this subtraction it is to be observed that the subtrahend 36 — 20 is 36 diminished by 20, and if we subtract 36 we take away too much by 20, and must therefore add this latter quantity. Consequently the true product will be 63 — 35 — 36 + 20, equivalent to 12, as be fore. Thus this mode of proceeding gives the same result as that obtained by first performing the subtractions indicated in the latter term of the multiplicand and multiplier. The several steps in each case are as follows: Multiplicand 9-5 which is equal to Multiplier 4 7-4 which is equal to 3 34. The rules which we have explained are expressed more briefly as follows: Like signs, multiplied together, give +; unlike or contrary signs Thus we see that 7 or +7 by 5 gives 35, and and - 4 by -- 4 by +9 gives-36, 5 gives +20. The same general reasoning will apply when let ters are used instead of numbers. Multiplicand a- b C- d We say in this case that, when we multiply a by e we take the multiplicand too great by b, and must therefore diminish the result a c by the product of b by c or b c. So also in multiplying the two terms of the multiplicand by c, we have taken the multiplier too great by d, and must therefore diminish the result a c-be by the product of a- b by d, or a db d. But if we subtract the whole of a d, we subtract too much by bd; bd must accordingly be added. The rule for negative quantities here illustrated is not necessary where mere numbers are employed, because the subtraction indicated may always be performed. But this cannot be done with respect to letters which stand for no particular values, but are intended as general expressions of quantities. The truth of the rule may be shown also when applied to quantities taken singly. We say that multiplying one quantity by another is taking one as many times as there are units in the other, and the result is the same, whichever of the quantities be taken for the multiplicand. Thus multiplying 9 by 3 is taking 9 three times, or, which is the same thing, taking 3 nine times (Arith. 27). But in arithmetic, quantities are always taken affirmatively, that is additively. When therefore we take 9 or + 9 three times additively, or + 3 nine times additively, the result will evidently be additive or +27. When on the contrary one of the factors is negative, as for instance, in multiplying 5 by +3; in this case, — 5 is to be taken three times additively, and — 5 added to 5 added to - 5 is clearly — 15. So also if we consider + 3 as the multiplicand, then +3 is to be taken five times subtractively; now 3 taken subtractively once (or which is the same thing 3 × -1) is equivalent to — 3, taken subtractively twice is -6, three times is 9, five times is 15. But, when the multiplicand and multiplier are both negative, as in the case of mul. tiplying 5 by -4; here a subtractive quantity is to be taken subtractively, that is, we are to take away successively a diminishing or lessening quantity, which is certainly equivalent to adding an increasing quantity. Thus if we take away - 5 once, we augment the sum with which it is to be connected by+5; if we take away-5 twice, we make the augmentation + 10; if four times, +20; that is, 5x4 is equivalent to +20, - give. Thus, when it is required to multiply the following numbers; +a,b,c,+d; we have first + a multiplied by b, which makes-ab; this by-c, gives + abc; and this by +d, gives + abcd. 35. The difficulties with respect to the signs being removed, we have only to shew how to multiply numbers that are them'selves products. If we were, for instance, to multiply the number a b by the number c d, the product would be a bc d, and it is obtained by multiplying first ab by c, and then the result of that multiplication by d. Or, if we had to multiply 36 by 12; since 12 is equal to 3 times 4, we should only multiply 36 first by 3, and then the product 108 by 4, in order to have the whole product of the multiplication of 12 by 36, which is consequently 432. 36. But if we wished to multiply 5 ab by 3 cd, we might write 3cdx5 ab; however, as in the present instance the order of the numbers to be multiplied is indifferent, it will be better, as is also the custom, to place the common numbers before the letters, and to express the product thus: 5x Sabcd, or 15 abcd; since 5 times 3 is 15. So if we had to multiply 12pqr by 7 xy, we should obtain 12 x 7 pqr xy, or 84 p qr xy. L CHAPTER IV. Of the nature of whole numbers or integers, with respect to their factors. 37. We have observed that a product is generated by the multiplication of two or more numbers together, and that these numbers are called factors. Thus the numbers a, b, c, d, are the factors of the product abcd. 38. If, therefore, we consider all whole numbers as products of two or more numbers multiplied together, we shall soon find that some cannot result from such a multiplication, and consequently have not any factors; while others may be the products of two or more multiplied together, and may consequently have two or more factors. Thus, 4 is produced by 2x2; 6 by 2×3; 8 by 2 × 2 × 2; or 27 by 3 x 3 x3; and 10 by 2 x 5, &c. 39. But, on the other hand, the numbers, 2, 3, 5, 7, 11, 13, 17, &c., cannot be represented in the same manner by factors, unless for that purpose we make use of unity, and represent 2, for instance, by 1 x 2. Now the numbers which are multiplied by 1, remaining the same, it is not proper to reckon unity as a factor. All numbers therefore, such as 2, 3, 5, 7, 11, 13, 17, &c. which cannot be represented by factors, are called simple, or prime numbers; whereas others, as 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, &c. which may be represented by factors, are called compound numbers. 40. Simple or prime numbers deserve therefore particular attention, since they do not result from the multiplication of two or more numbers. It is particularly worthy of observation that if we write these numbers in succession as they follow each other thus ; 2, 3, 5, 7, 11, 15, 17, 19, 28, 29, 31, 37, 41, 43, 47, &c. we can trace no regular order; their increments are sometimes greater, sometimes less; and hitherto no one has been able to discover whether they follow any certain law or not. 41. All compound numbers, which may be represented by factors, result from the prime numbers above mentioned; that is to say, all their factors are prime numbers. For, if we find a factor which is not a prime number, it may always be decomposed and represented by two or more prime numbers. When we have represented, for instance, the number 30 by 5 x 6, it is evident that 6 not being a prime number, but being produced by 2 × 3, we might have represented 30 by 5 × 2 × 3, or by 2 × 3 × 5; that is to say, by factors, which are all prime numbers. 42. If we now consider those compound numbers which may be resolved into prime numbers, we shall observe a great difference among them; we shall find that some have only two factors, that others have three, and others a still greater number. We have already seen, for example, that 43. Hence it is easy to find a method for analysing any number, or resolving it into its simple factors. Let there be pro posed, for instance, the number 360; we shall represent it first by 2 x 180. Now 180 is equal to 2 x 90, and So that the number S60 may be represented by these simple factors, 2 × 2 × 2x 3x3x 5; since all these numbers multiplied together produce 360. 44. This shews, that the prime numbers cannot be divided by other numbers, and on the other hand, that the simple factors of compound numbers are found, most conveniently and with the greatest certainty, by seeking the simple, or prime numbers, by which those compound numbers are divisible. But for this, division is necessary; we shall therefore explain the rules of that operation in the following chapter. CHAPTER V. Of the Division of Simple Quantities. 45. WHEN a number is to be separated into two, three, or more equal parts, it is done by means of division, which enables us to determine the magnitude of one of those parts. When we wish, for example, to separate the number 12 into three equal parts, we find by division that each of those parts is equal to 4. The following terms are made use of in this operation. The number, which is to be decompounded or divided, is called the dividend; the number of equal parts sought is called the divisor; the magnitude of one of those parts, determined by the division, is called the quotient; thus, in the above example; 12 is the dividend, 3 is the divisor, and 4 is the quotient. 46. It follows from this, that if we divide a number by 2, or into two equal parts, one of those parts, or the quotient, taken twice, makes exactly the number proposed; and, in the same |