2. If we divide each of the equal parts AC, CD, DE, EB into two equal parts, the whole, AB, will be divided into eight equal parts; and we see that AC = ; AD = ; AE = ; AB = ; Now, we saw in 1, that AE = of the integer, and we have just shown that it is equal to . Hence = ; but would be got from by multiplying its terms (numerator and denominator) by 2. Hence we infer generally that multiplying the terms of any fraction by 2 does not alter its value. In like manner it may be shown that multiplying the terms of a fraction by any whole number does not alter its value. Hence it follows conversely, that dividing the terms of a fraction by a whole number does not alter the value. Hence we have the following important and fundamental theorem:—Two transformations can be made on any fraction without changing its value; namely, its terms can be either multiplied or divided by any whole number, and in either case the value of the new fraction is equal to the value of the original one. 3. If we take any number, such as 3, and multiply it by any whole number, the product is called a multiple of 3. Thus 6, 9, 12, 15, &c., are multiples of 3; but 10, 13, 17, &c., are not, because the multiplication of 3 by any whole number will not produce them. Conversely, 3 is a submultiple, or measure, or part of 6, 9, 12, 15, &c., because it is contained in each of these without a remainder; but not of 10, 13, 17, &c., because in each case it leaves a remainder. 4. If we consider two magnitudes of the same kind, such as two lines AB, CD, and if we suppose that AB is equal to of CD, it is evident, if AB be divided into 3 equal parts, and CD into 4 equal parts, that one of the parts into which AB is divided is equal to one of the parts into which CD is divided. And as there are 3 parts in AB, and 4 in CD, we express this relation by saying that AB has to CD the ratio of 3 to 4; and we denote it thus, 3 : 4. Hence the ratio 3 : 4 expresses the same idea as the fraction . In fact, both are different ways of expressing and writing the same thing. When written 3 : 4 it is called a ratio, and when a fraction. In the same manner it can be shown that every ratio whose terms are commensurable can be converted into a fraction; and, conversely, every fraction can be turned into a ratio. From this explanation we see that the ratio of any two commensurable magnitudes is the same as the ratio of the numerical quantities which denote these magnitudes. Thus, the ratio of two commensurable lines is the ratio of the numbers which express their lengths, measured with the same unit. And this may be extended to the case where the lines are incommensurable. Thus, if a be the side and b the diagonal of a square, the ratio of a : b is