by continually subtracting unity, we should have the following series of negative numbers

17. All these numbers, whether positive or negative, are known by the name of integers, and are so called to distinguish them from broken or fractional numbers, and from many other kinds of numbers, of which we shall hereafter treat. For instance, 50 being greater by an entire unit than 49, we may easily understand, that there may be an infinity of intermediate numbers between 49 and 50, all greater than 49, and yet less than 50. We have only to imagine two lines, the one 50 feet long, and the other 49, and we can easily conceive that an infinite number of lines may be drawn, all longer than 49 feet, and yet shorter than 50.

18. It is of the highest importance throughout the whole of Algebra, that a proper idea should be formed of these negative quantities. I shall here content myself with observing that all such formulas as

+1−1+2−2+3-3, &c.

are equal to 0 or nothing, and that +2 - 5 is equal to -3. For if any one have two crowns and owe 5, he has not only nothing, but he still owes 3 crowns; in the same manner

as {

19. The same rule is to be observed, when, generally, numbers are represented by letters, for+a—a is always equal to 0. Therefore if we would determine the value of +a-b, two cases are to be considered; first, when a is greater than b; we must then subtract b from a, and the remainder is the value sought, before which the sign + is prefixed or understood: the second, where a is less than b; for we must then subtract a from b, and the remainder will be a negative number, with the sign - before it.

CHAP. III.

OF THE MULTIPLICATION OF SIMPLE QUANTITIES.

20. WHE

HEN two or more equal numbers are to be added together, the manner of expressing their sum may be abridged. For example,

21. From this we may form an idea of Multiplication, and it must be observed that

22. If therefore it be proposed to multiply a number expressed by a letter by another number, we need only write that number before the letter, thus

&c.

a

a multiplied by 201 are expressed {20%

b multiplied by 30

23. It is equally easy to multiply these products again by other numbers, thus

(*) These numbers, 2, 3, 5, &c., prefixed to any unknown quantities, are termed the coefficients of such quantities; thus in the quantities 3 a, 4 b, 7 x, 3, 4, and 7 are called the coefficients of a, b, and a respectively: when no number is prefixed, unity is understood as the coefficient.

And these products may be multiplied again by other numbers at pleasure.

24. When the multiplier is represented also by a letter, it must be placed immediately before the letter representing the multiplicand: so that if b is to be multiplied by a, the product is thus written ab; in like manner pq will be the product of q multiplied by p; and if that product be multiplied by a we shall obtain apq.

25. It is indifferent in what order the letters thus connected together in multiplication be placed, for ba is the same as ab, or in other words b multiplied by a is the same thing as a multiplied by b. For if we take any two known quantities, as 3 and 4, for the value of a and b respectively, we see immediately that 3 multiplied by 4 is the same thing as 4 multiplied by 3.

26. It will be observed, that we have denominated by the name of product the quantity resulting from the multiplication of two or more numbers; it should also be observed that the individual numbers or letters themselves are called the factors.

27. As yet we have only considered positive numbers, and it is impossible to doubt that the products so formed must be themselves positive: i. e. that + a multiplied by +b must necessarily produce +ab. It will be necessary however to examine the product of +a into b, and of a into -b.

28. Let us begin by multiplying —a by 3, i. e. +3. Now since a may be considered as a debt, it is clear, that if we take this debt 3 times it will be 3 times as great, and consequently that the product sought is -3a. In like manner therefore, if we multiply -a a by +b, we shall obtain ab. We arrive therefore at this conclusion, that if a positive quantity be multiplied by a negative one, the product will be negative; and we may receive it as a general rule, that + by +

pro

produces or plus, and on the contrary, that + by by+, produces - or minus.

—, or

29. It remains now to consider the case in which is multiplied by - for example, -α by b. It is evident, first, that, with respect to the letters, the product will be ab: but it is as yet uncertain whether the sign + or the sign is to be prefixed; all that we know is, that it must be one or the other. Now I say that it cannot be the sign because -a by +b produces - ab, and -a by -b cannot produce the same result; but it must produce a contrary one, that is +ab. Wherefore we have this rule, that duces the same as + by +.

multiplied by

pro

30. The rules that we have just explained may be more briefly expressed in the following theorem.

LIKE OR EQUAL SIGNS MULTIPLIED TOGether give +.
UNLIKE OR CONTRARY SIGNS GIVE

31. Thus if it is required to multiply together the four letters, a-b-c+d, the operation would be as follows; + a by-b produces-ab; this product by -c gives +abc; and lastly, the whole multiplied by +d gives +abcd.

32. The difficulties with regard to the signs being removed, we have only to consider how we ought to multiply together numbers which are themselves the products of other numbers. For example, if it were required to multiply ab by cd the product would be abcd, from which it appears that ab is first multiplied by c, and that product again by d. But if we wished to multiply 5ab by 3cd, we might indeed write 5ab 3 cd: at the same time, as the order of the letters to be multiplied is of no consequence, it will be better and is more customary to place all the known quantities before the letters, and to express the product thus 5 × 3abcd or 15 abcd, or, generally, to collect and multiply together all the coefficients, and to place them before the letters or unknown quantities, which

last quantities, for the sake of uniformity, are generally placed in the order in which they stand in the alphabet. In like manner, if we had to multiply 12pqr by 7xy we should obtain 84 pqrxy. (')

(") From this rule for the multiplication of algebraical quantities, it would appear that if a were to be multiplied by itself, we should write a a, and if multiplied again by itself we should write a aa. It may be necessary, therefore, here to observe, that this inconvenience is avoided by writing over the letter a the numbers 2, 3, 4, &c. to denote the number of times that it is taken, thus for aa we write a2, for aaa, as, &c. This observation appears to be required in this place, but the general subject, which is of the highest importance in algebra, will be found treated of at length in succeeding chapters.