in the second, as many times smaller, as the number used in the multiplication or division indicates.

PROPOSITION II. As many times as the denominator of a fraction is made larger or smaller, the numerator remaining unchanged, so many times the value of the fraction is made smaller or larger.

For, the denominator being the number by which the unit is divided, as many times as this number is multiplied, so many times the unit is divided into more parts; and therefore, the parts becoming as many times smaller, an equal number of them represents a value as many times smaller; that is to say, the value of the fraction is as many times smaller, and inversely, when the denomiuator is divided by a number, the unit is divided by a number as many times smaller than this divisor indicates ; therefore, the parts become as many times larger, and the value of the fraction becomes as many times larger; all under the supposition: that an equal number of these parts be taken before and after the operation.

cause the 7 is divided by a number 13 times larger than 18;

1

or we have, 7 X

13 times smaller than

18 × 13

4*

18

because the 7 is divided into parts 9 times larger ;

PROPOSITION III. When the numerator and denominator of a fraction are both multiplied or divided by the same number, the value of the fraction remains unchanged.

This is an evident consequence of the combination of the two preceding propositions, which show the effect of the multiplication and division upon the numerator and the denominator, to be exactly opposite, and therefore, when performed with the same number, they exactly compensate each other; that is to say: as many times as the value of the fraction becomes larger or smaller, by the multiplication or division of the numerator of the fraction, so many times it becomes again smaller or larger, by the multiplication or division of the denominators.

where the mutual destruction of the effect, of the two operations, is self-evident.

The two first propositions solve directly all multiplication or division of fractions by whole numbers, in a double manner; for we have, evidently, every time, the choice between two operations, each of which may, according to the case, present a preference in application.

The third proposition will evidently furnish us the means to reduce fractions from one denominator to certain other ones, in order to obtain the fractional parts expressed so as to be adapted to cer

tain purposes in the operations of arithmetic, with out changing their value.

39. The investigations of § 37, have shown fractions to be equivalent to the product of a whole number into certain quantities expressed in parts of the unit; when thus representing quantities of different values or kinds, they have different denominators; their numerators therefore cannot be taken into one sum, or difference, without previous appropriate changes. By the third of the foregoing propositions, we have obtained means to make such changes, without altering the value of the fractions. The aim of such a change, must evidently be to obtain the same denomination for both, or all the fractions, whose sum or difference is desired.

We have seen in multiplication, that it is indifferent which of the two factors is multiplier or multiplicand, this shows that equal denominators may he obtained for two fractions, by multiplying the denominators together; if therefore, the numerators of the two fractions are also multiplied, each alternately by the denominator of the other, the value of the fraction will remain unchanged, according to the third proposition above; and if more fractions are concerned, considering the first result as one, and operating upon it in conjunction with another, exactly in the same way as before, and so on to the end, a result is evidently obtained, that applies to any number of fractions. This furnishes us with the following general rule.

To reduce fractions to a common denominator; multiply the numerator and denominator of each fraction by all the denominators except its own; then all the fractions will have the same denominator, and the numerators will be such that the value of the fractions will not be changed.

we evidently obtain step, by step, the following results:

Here quantities of the same kind, are evidently obtained, say equal parts of the unit, only in different quantities; for, according to what has been seen above, these fractions might be thus written :

$40. It is evident from the above, that fractions cannot be reduced to any denominator indiscriminately, as the new denominator must be a multiple or a quotient, of the former denominator.

If it should become necessary to take whole numbers under the same consideration, it will easily be judged, from what has been said, that they must be considered as having the denominanator, 1, and such indeed they are, for the unit is their measure as to quantity, like any other denominator in a fraction,

For every whole number whatsoever, must be considered as multiplied by 1, really to be a quantity if it was multiplied by 0, it would be said not to be at all, as 0, denotes the absence of all quantity; and if multiplied by any other number, the product would be another number.

§ 41. The continued multiplication of all the denominators evidently leads into large numbers, both for the numerators and the denominators, which it is desirable to avoid wherever possible; this will be the case when some of the denominators are products of the same number with different numbers, or have what is called, common factors; these are therefore not necessary to be repeated in the continued product of the denominators, which furnishes the new denominator, as the above example already shows, where 2 and 8, are products of 2, the first by 1, the second by 4.

The following problem and its solution, which will best be explained immediately by an example, will lead to this result.

PROBLEM. To find the smallest number which will be divisible by several other given numbers.

Solution. Write the numbers after each other,