The same reasoning may be applied to every other example of division in which there is a remainder, and in this case the quotient is composed of two parts; one, consisting of whole units, while the other cannot be obtained until the concrete or material units of the remainder have been actually divided into the number of parts denoted by the divisor; without this it can only be indicated by supposing, a unit of the dividend to be divided into as many parts as there are units in the divisor, and so many of these parts, as there are units in the remainder, taken to complete the quotient required.

5. In general, when we have occasion to consider quantities less than unity, we suppose unity divided into a certain number of parts, sufficiently small to be contained a certain number of times in these quantities, or to measure them. In the idea thus formed of their magnitude there are two elements, namely, the number of times the measuring part is contained in unity, and the number of these parts found in the quantities.

A nomenclature has been made for fractions, which answers to this manner of conceiving and representing them,

That which results from the division of unity

into 2 parts is called a moiety or half,

and so on, adding after the two first, the termination th to the number, which denotes how many parts are supposed to be in unity.

Every fraction then is expressed by two numbers; the first, which shows how many parts it is composed of, is called the numerator, and the other, which shows how many of these parts are necessary to form an unit, is called the denominator, because the denomination of the fraction is deduced from it. Five sixths of an unit is a fraction, the numerator of which is five, and the denominator six.

The numerator and the denominator together are called the two terms of the fraction.

Figures are used to shorten the expression of fractions, the

denominator being written under the numerator, and separated from it by a line,

53. According to the meaning attached to the words, numerator and denominator, it is plain, that a fraction is increased, by increasing its numerator, without changing its denominator; for this last, as it shows into how many parts unity is divided, determines the magnitude of these parts, which continues the same, while the denominator remains unchanged; and by augmenting the numerator, the number of these parts is augmented, and consequently the fraction increased. It is thus, for instance, that exceeds, and that exceeds 1.

It follows evidently from this, that by repeating the numerator 2, 3, or any number of times, without altering the denominator, we repeat, a like number of times, the quantity expressed by the fraction, or in other words multiply it by this number; for we make 2, 3, or any number of times, as many parts, as it had before, and these parts have remained cach of the same value. The fraction, then, is the triple of and 10 the double of •

A fraction is diminished by diminishing its numerator, without changing its denominator, since, it is made to consist of a less number of parts than it contained before, and these parts retain the sa ne value. Whence, if the numerator be divided by 2, 3, or any number, without the denominator being altered, the fraction is made a tike number of times smaller, or is divided by that number, for it is made to contain 2, S, or any number of times less parts than it contained before, and these parts remain of the same value. Thus is a third of and is half of 19.

54. On the contrary, a fraction is diminished, when its denominator is increased without changing its numerator; for then more parts are supposed in an unit, and consequently they must be smaller, but, as only the same number of them are taken to form the fraction, the amount in this case must be a less quantity than in the first. Thus is less than, and than 4.

Hence it follows, that if the denominator of a fraction be multiplied by 2, 3, or any number, without the numerator being changed, Arith.

5

the fraction becomes a like number of times smaller, or is divided by that number, for it is composed of the same number of parts as before, but each of them has become 2, 3, or a certain number of times less. The fraction is half of, and the third of. A fraction is increased when its denominator is diminished without the numerator being changed; because, as unity is supposed to be divided into fewer parts, each one becomes greater, and their amount is therefore greater.

Whence, if the denominator of a fraction be divided by 2, 3, or any other number, the fraction will be made a like number of times greater, or will be multiplied by that number; for the number of parts remains the same, and each one becomes 2, 3, or a certain number of times greater than it was before. According to this, is triple of and g the quadruple of

It may be remarked, that to suppress the denominator of a fraction is the same as to multiply the fraction by that number. For instance, to suppress the denominator 3 in the fraction is to change it into 2 whole ones, or to multiply it by 3.

55. The preceding propositions may be recapitulated as follows; the numerator, the fraction is

By multiplying }

By dividing

By multiplying}

By dividing

the denominator, the fraction is

multiplied. divided.

56. The first consequence to be drawn from this table is, that the operations performed on the denominator produce effects of an inverse or contrary nature with respect to the value of the fraction. Hence it results, that, if both the numerator and denominator of a fraction be multiplied at the same time, by the same number, the value of the fraction will not be altered; for if, on the one hand, multiplying the numerator makes the fraction 2, 3, &c. times greater, so on the other, by the second operation, the half or third part &c. of it is taken; in other words, it is divided by the same number, by which it had at first been multiplied. Thus is equal to, and is equal to 19.

57. It is also manifest that, if both the numerator and denominator of a fraction be divided, at the same time, by the same number, the value of the fraction will not be altered; for if, on the one hand, by dividing the numerator the fraction is made 2, 3, &c.

times smaller; on the other, by the second operation, the double, triple, &c. is taken; in short it is multiplied by the same number, by which it was at first divided. Thus the fraction is equal to, and is equal to 4.

58. It is not with fractions as with whole numbers, in which a magnitude, so long as it is considered with relation to the same unit, is susceptible of but one expression. In fractions on the contrary, the same magnitude can be expressed in an infinite number of ways. For instance, the fractions

in each of which the denominator is twice as great as the numerator, express, under different forms, the half of an unit. The fractions ttt. I fr, IT, &c.

•

4

5

9

of which the denominator is three times as great as the numerator, represent each the third part of an unit. Among all the forms, which the given fraction assumes, in each instance, the first is the most remarkable, as being the most simple; and, consequently, it is well to know how to find it from any of the others. It is obtained by dividing the two terms of the others by the same number, which, as has already been shown, does not alter their value. Thus if we divide by 7 the two terms of the fraction, we come back to ; and, performing the same operation on, we get 3.

59. It is by following this process, that a fraction is reduced to its most simple terms; it cannot, however, be applied, except to fractions, of which the numerator and denominator are divisible by the same number; in all other cases the given fraction is the most simple of all those, that can represent the quantity it expresses. Thus the fractions,,, the terms of which cannot be divided by the same number, or have no common divisor, are irreducible, and, consequently, cannot express, in a more simple manner, the magnitudes which they represent.

60. Hence it follows, that to simplify a fraction, we must endeavour to divide its two terms by some one of the numbers, 2, 3, &c; but by this uncertain mode of proceeding it will not be always possible to come at the most simple terms of the given fraction, or at least, it will often be necessary to perform a great number of operations.

If, for instance, the fraction

42

were given, it may be seen at

once, that each of its terms is a multiple of 2, and dividing them by this number, we obtain 12; dividing these last also by 2, we obtain. Although much more simple now than at first, this fraction is still susceptible of reduction, for its two terms can be divided by 3, and it then becomes 4.

If we observe, that to divide a number by 2, then the quotient by 2, and then the second quotient by 3, is the same thing as to divide the original number by the product of the numbers, 2, 2, and 3, which amounts to 12, we shall see that the three above operations can be performed at once by dividing the two terms of the given fraction by 12, and we shall again have 4.

The numbers 2, 3, 4, and 12, each dividing the two numbers 24 and 84 at the same time, are the common divisors of these numbers; but 12 is the most worthy of attention, because it is the greatest, and it is by employing the greatest common divisor of the two terms of the given fraction, that it is reduced at once to its most simple terms. We have then this important problem to solve, two numbers being given, to find their greatest common divisort.

61. We arrive at the knowledge of the common divisor of two numbers by a sort of trial easily made, and which has this recommendation, that each step brings us nearer and nearer to the number sought. To explain it clearly, I will take an example.

Let the two numbers be 637 and 143. It is plain, that the greatest common divisor of these two numbers cannot exceed the smallest of them; it is proper then to try if the number 143, which divides itself and gives 1 for the quotient, will also divide the number 637, in which case it will be the greatest common divisor sought. In the given example this is not the case; we obtain a quotient 4, and a remainder 65.

Now it is plain, that every common divisor of the two numbers, 143 and 637, ought also to divide 65, the remainder resulting from their division; for the greater, 637, is equal to the

† What is here called the greatest common divisor, is sometimes called the greatest common measure.