a b, and is to be divided by the divisor + a, b; because it is b, which, multiplied by Lastly, if we have to divide the dividend

dend — a b is the product of — a by + b.

56. With regard, therefore, to the signs + and —, division admits the same rules that we have seen applied in multiplication, viz.

or in a few words, like signs give plus, unlike signs give minus. 57. Thus, when we divide 18 p q by

-6q. Further;

3p, the quotient is

5 x, and

30xy, divided by + 6y, gives
-54 a b c, divided by 9 b, gives + 6 ac;

for in this last example, -96, multiplied by +6 a c, makes — 6 × 9 a b c, or 54 ab c. But we have said enough on the division of simple quantities; we shall therefore hasten to the explanation of fractions, after having added some farther remarks on the nature of numbers, with respect to their divisors.

CHAPTER VI.

Of the Properties of Integers with respect to their Divisors.

58. As we have seen that some numbers are divisible by certain divisors, while others are not; in order that we may obtain a more particular knowledge of numbers, this difference must be carefully observed, both by distinguishing the numbers that are divisible by divisors from those which are not, and by considering the remainder that is left in the division of the latter. For this purpose let us examine the divisors;

2, 3, 4, 5, 6, 7, 8, 9, 10, &c.

59. First, let the divisor be 2; the numbers divisible by it are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, &c. which, it appears, increase always by two. These numbers, as far as they can be continued, are called even numbers. But there are other numbers, namely,

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, &c.,

which are uniformly less or greater than the former by unity, and which cannot be divided by 2, without the remainder 1; these are called odd numbers.

The even numbers are all comprehended in the general expression 2 a; for they are all obtained by successively substituting for a the integers, 1, 2, 3, 4, 5, 6, 7, &c., and hence it follows that the odd numbers are all comprehended in the expression 2 a + 1, because 2a+1 is greater by unity than the even number 2 a.

60. In the second place, let the number 3 be the divisor, the numbers divisible by it are,

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and so on;

and these numbers may be represented by the expression 3 a; for 3 a divided by 3 gives the quotient a without a remainder. All other numbers, which we would divide by 3, will give 1 or 2 for a remainder, and are consequently of two kinds. Those which, after the division leave the remainder 1, are;

1, 4, 7, 10, 13, 16, 19, &c.,

and are contained in the expression 3 a + 1; but the other kind, where the numbers give the remainder 2, are;

2, 5, 8, 11, 14, 17, 20, &c.,

and they may be generally expressed by 3a + 2; so that all numbers may be expressed either by 3 a, or by 3 a + 1, or by 3 a +2. 61. Let us now suppose that 4 is the divisor under consideration; the numbers which it divides are;

4, 8, 12, 16, 20, 24, &c.,

which increase uniformly by 4, and are comprehended in the expression 4 a. All other numbers, that is, those which are not divisible by 4, may leave the remainder 1, or be greater than the former by 1; as

1, 5, 9, 13, 17, 21, 25, &c.,

and consequently may be comprehended in the expression 4 a + 1: or they may give the remainder 2; as

2, 6, 10, 14, 18, 22, 26, &c.,

and be expressed by 4 a+ 2; or, lastly, they may give the remainder 3; as

3, 7, 11, 15, 19, 23, 27, &c.,

and may be represented by the expression 4 a + 3.

All possible integral numbers are therefore contained in one or other of these four expressions;

4a, 4a+1, 4a + 2, 4 a + 3.

62. It is nearly the same when the divisor is 5; for all numbers which can be divided by it are comprehended in the expression 5 a, and those which cannot be divided by 5, are reducible to one of the following expressions :

5a1, 5a +2, 5a + 3, 5a + 4;

and we may go on in the same manner, and consider the greatest divisors.

63. It is proper to recollect here what has been already said on the resolution of numbers into their simple factors; for every number among the factors of which is found,

2, or 3, or 4, or 5, or 7,

or any other number, will be divisible by those numbers. For example; 60 being equal to 2 × 2 × 3 × 5, it is evident that 60 is divisible by 2, and by 3, and by 5.

64. Further, as the general expression a b c d is not only divisible by a, and b, and c, and d, but also by

ab, ac, ad, bc, bd, cd, and by

abc, abd, a cd, bed, and lastly by
abcd, that is to say, its own value;

it follows that 60, or 2 × 2 × 3 × 5, may be divided not only by these simple numbers, but also by those which are composed of two of them; that is to say, by 4, 6, 10, 15; and also by those which are composed of three of the simple factors, that is to say, by 12, 20, 30, and lastly by 60 itself.

65. When, therefore, we have represented any number, assumed at pleasure, by its simple factors, it will be very easy to show all the numbers by which it is divisible. For we have only, first, to take the simple factors one by one, and then to multiply them together two by two, three by three, four by four, &c. till we arrive at the number proposed.

66. It must here be particularly observed; that every number is divisible by 1; and also that every number is divisible by itself; so that every number has at least two factors, or divisors, the number

itself and unity; but every number, which has no other divisor than these two, belongs to the class of numbers, which we have before called simple, or prime numbers.

All numbers, except these, have, beside unity and themselves, other divisors, as many be seen from the following table, in which are placed under each number all its divisors.

1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 62 6

P.P.P. P. P.

P. P.

P.

P.

67. Lastly, it ought to be observed, that 0, or nothing, may be considered as a number which has the property of being divisible by all possible numbers; because by whatever number a we divide 0, the quotient is always 0; for it must be remarked that the multiplication of any number by nothing produces nothing, and therefore 0 times a, or 0 a, is 0.

CHAPTER VII.

Of Fractions in general.

68. WHEN a number, as 7 for instance, is said not to be divisible by another number, let us suppose by 3, this only means, that the quotient cannot be expressed by an integral number; and it must not be thought by any means that it is impossible to form an idea of that quotient. Only imagine a line of 7 feet in length, no one can doubt the possibility of dividing this line into 3 equal parts, and of forming a notion of the length of one of those parts.

69. Since therefore we may form a precise idea of the quotient obtained in similar cases, though that quotient is not an integral number, this leads us to consider a particular species of numbers, called fractions or broken numbers. The instance adduced furnishes an illustration. If we have to divide 7 by 3, we easily conceive the quotient which should result, and express it by; placing the divisor under the dividend, and separating the two numbers by a stroke or line.

70. So, in general, when the number a is to be divided by the

a

b

number b, we represent the quotient by and call this form of expression a fraction. We cannot, therefore, give a better idea of a fraction, than by saying that we thus express the quotient resulting from the division of the upper number by the lower. We must remember also, that in all fractions the lower number is called the denominator, and that above the line the numerator.

71. In the above fraction,, which we read seven thirds, 7 is the numerator, and 3 the denominator. We must also read, two thirds;, three fourths;, three eighths;, twelve hundredths; and, one half.

a

a

72. In order to obtain a more perfect knowledge of the nature of fractions, we shall begin by considering the case in which the numerator is equal to the denominator, as in 2. Now, since this expresses the quotient obtained by dividing a by a, it is evident that this quotient is exactly unity, and that consequently this fraction - is equal to 1, or one integer; for the same reason, all the following fractions, 4, 3, 4, 5, 6, 7, 8, &c.

2
29

are equal to one another, each being equal to 1, or one integer.

73. We have seen that a fraction, whose numerator is equal to the denominator, is equal to unity. All fractions therefore, whose numerators are less than the denominators, have a value less than unity. For, if I have a number to be divided by another which is greater, the result must necessarily be less than 1; if we cut a line, for example, two feet long, into three parts, one of those parts will unquestionably be shorter than a foot; it is evident then, that is less than 1, for the same reason, that the numerator 2 is less than the denominator 3.