CHAP. IV.

OF THE DIVISION OF SIMPLE QUANTITIES.

33. WHEN it is required to separate a number into two, three, or more equal parts, it is done by means of Division, by which we are enabled to determine the magnitude of one of the parts. If we wish for instance to divide the number 12 into three equal parts, we find by division that each of these parts is 4.

34. In this operation, the number to be divided is called the dividend; the number of equal parts sought is called the divisor; the magnitude of one of these parts is called the quotient. Thus in the foregoing example,

12 is the dividend,

3 is the divisor,
4 is the quotient.

35. From hence it follows, that if we divide a number by 2, or into two equal parts, one of the parts, or the quotient, taken twice, will exactly make the number proposed. In like manner if a number be divided by 3 or into three equal parts, any one of the parts taken three times will give the same number again; and generally the multiplication of the quotient by the divisor gives the dividend.

36. It is for this reason that division is given as a rule for finding a number or quotient, such that, being multiplied by the divisor, the result will produce the dividend. For example, if 35 is to be divided by 5, or into five equal parts, we must find such a number as, being multiplied by 5, will produce 35, viz. 7.

37. The

37. The dividend we may therefore consider as a product, of which one of the factors is equal to the divisor, and the other factor to the quotient. So that if 63 be to be divided by 7, we must seek such a product as, taking 7 for one of its factors, the other factor multiplied by it will give exactly 63. Now 7 times 9 is such a product, and consequently 9 is the quotient, which we obtain by dividing 63 by 7.

38. If in general it be required to divide the product ab by a, it is evident that the quotient will be b, because a being multiplied by b gives the dividend ab. It is equally clear that if ab be divided by b, the quotient will be a.

And in general in all the instances of division, if the dividend be divided by the quotient, we shall obtain again the divisor; the same as 24 divided by 4 gives 6, and divided by 6 gives 4.

39. As the whole therefore consists in representing the dividend by two factors, the one equal to the divisor, and the other to the quotient, the following examples will be easily understood. I say first, that the dividend abc divided by a gives bc; for a multiplied by bc gives abc; also abc divided by b gives ac, and abc divided by ac gives b. Also, 12mn divided by 3m gives 4n; for 3m multiplied by 4n gives 12mn. But if the same number be divided by 12, the quotient will be mn.

40. Since any number, as a, may be expressed by la, it is evident that if it be divided by 1, the quotient will be the same number a; but if, on the contrary, the same number a or 1 a be divided by a, the quotient will be 1.

41. It does not always happen that the dividend can be considered as the product of two factors, of which one is equal to the divisor, and in such case the division cannot be made in the manner above described,

For

For example, when we have 24 to divide by 7; it is clear at first sight that the number 7 is not a factor of 24, for 7 times 3 only gives 21, which is less; and 7 times 4 gives 28, which is greater than 24. But it is at least evident from this, that the quotient must be more than 3, and less than 4. In order therefore to determine it exactly, another species of numbers must be employed, called fractions, of which we shall treat in succeeding chapters.

42. Before we come to the use of fractions, it is customary to confine oneself to the number which approaches the nearest to the true quotient, paying attention however to the remainder; thus we say, the sevens in 24 go three times, and the remainder is 3; because three times 7 is only 21, and consequently 3 less than 24. The following is an example of the same nature;

6) 34 (5

30

4.

that is to say; the divisor is 6, the dividend 34, the quotient is 5, and the remainder 4.

43. We must observe, however, the following rule, in those instances where there is a remainder, viz.

That when the divisor is multiplied by the quotient, and to the product is added the remainder, we ought to obtain the dividend. This is the rule for proving the division, and ascertaining the correctness of our calculation. As in the last example, if we multiply 6 × 5, and to the product 30 add the remainder 4, we shall obtain 34, the dividend.

44. It is, lastly, necessary to observe here, with regard to the signs and, that if +ab be divided by +a, the quo+ tient will be evidently +6. But if it be proposed to divide +ab by a the quotient will be -b, because a multiplied by-b gives +ab. If the dividend be-ab, and it is pro

posed

posed to divide it by the divisor +a, the quotient will also be -b, because it is -b which, when multiplied by +a, gives -ab. Lastly, if it be proposed to divide-ab by-b, the quotient will be +a, because the dividend —ab is the product of +a into b.

45. Thus it appears that division is governed with respect to the signs and by the same rules as have been before laid down in Multiplication, viz.

+by+gives+; +by-gives

—by+gives—; -by-gives+

or that like signs give +, and unlike signs give –

46. Thus if 18pq be divided by -3p, the quotient will be -6q; also-30xy divided by +6y gives-5 x; and-54 abc by -96 gives +6ac.

OF THE DIFFERENT METHODS OF CALCULATING
COMPOUND QUANTITIES.

CHAP. V.

OF THE ADDITION OF COMPOUND QUANTITIES.

47. WHEN we have two or more formulas composed of several terms to add together, we frequently do nothing more than signify the addition by signs, placing each formula between two parentheses, and connecting them with the others by means of the sign +. If we wish, for example, to add together the formulas

a+b+c and d+e+f

we state the sum in this manner,

(a+b+c) + (d+e+f)

48. It will be easily seen that this is not addition itself, but merely the sign of it. In order however actually to perform addition, we have only to omit the parentheses; for since the number d+e+f is to be added to the other, it is clear that this must be done by first adding to it +d then +e and then +f; which gives the sum

a+b+c+d+e+f.

49. We should proceed in the same way if any of the - terms were affected by the sign -: we should join them one to another, with the sign which precedes each of them.