But the oldest testimony, both to the existence of the definition, and the meaning of the disputed word, is Eutocius, lu his commentary on book li. prop. 5 (of Torelli, 4 of preceding editors). He here cites, expressly from the elements, the definition as given ; and adds, as the explanation of TnAeolant, that it is the number which, by multiplica tion, turns the consequent into the antecedent. This number, he says, gives name to the ratio, and he cites Nicomnachus and Hertel as under standing it in the same way. But, ho goes on to say, the word is more properly Laken when this number is an integer.
Leaving now out of view ;what Euclid really did write, wo shall proceed to consider the subject of composition of ratios, so as to supply what, on any supposition, must be acknowledged to be wanted in the elements. The notion of a ratio is easily and almost necessarily con nected with the idea of alteration in that ratio. We cannot express a ratio without two magnitudes, the first of which, altered in the ratio given, becomes the second. If we want to alter in the ratio of to q, this is easily done when the quantity to be altered is 1.; for then the process is only writing Q instead of P. But if the quantity be A, then B must be found, so that A and n shall have the same ratio as r and Q.
If it be a numerical ratio which we consider, say that of 3 to 5, alteration of any number in that ratio implies that we change all its threes into fires, and any remaining fraction of three into the same fraction of five. Alteration of any magnitude, say a length, In that ratio implies that, choosing any length as a measure, we alter every three such lengths which the given magnitude contains into five, and every fraction of three into the same fraction of five. This amounts to changing tho number or magnitude into five-thirds of what it was ; and generally,alteration in the ratio of a to b (numbers) is nothing but multiplication by —4 a 31 Take a magnitude A, alter it in the ratio of P to Q : say that it then becomes n ; that is, A is to n as r to Q. Tako the magnitude we left off with, B, alter it in the ratio of it to s, making it c. Take c, alter it in the ratio of v to w, making it D. Then at three processes, by three successive alterations dictated by given ratios, we have changed A into D, or have altered A in the ratio of A to n. Say that the ratio of A to D is more simply expressed by that of at to N. Then, if we begin with
A, and alter it at once in the ratio of u to N, we change it into n, pro ducing the same effect as if we had successively altered in the ratios of r to c1,11 to s, and v to w. Hence the ratio of at to x is properly said to be compounded of the ratios of r to Q, 1/ to s, and v to w,: it dictates the alteration which will produce at once the effect of the three alterations prescribed by the three other ratios. In like manner, we say an to addition, that 10 is compounded of 6 and 4 ; for addition of 10 is equivalent to the addition of 6 and of 4. In multiplication we say that 24 is compounded of 6 and 4. And generally, the com pound should be defined as that which produces the united effect of all the components, when both components and compound are used in the same way. Euclid, vi. 23, is now more than a mere addition ,to the phraseology of geometry. The parallelograms A n e n and (1 11 [the reader may draw the figuro for himself ] being mutually equiangular, it tells tin that if we take any magnitude and alter it in the ratio of A c to s c, and then alter the result in the ratio of A n to E r, the change thus made at two steps might be made in one by altering the eriginal magnitude in the ratio of the area A li C n to the area This process applies equally to commensurable and incommensur able ratios; but in the former case of course the arithmetical substitute for composition of ration is easy. We want to compound the ratios of ma to n and of a to b, all four being integer numbers : it being known that every commensurable ratio is expressible by the ratio of two integer numbers. 'fake any magnitude is and alter it in the ratio of me to n : it becomes manths of P. Alter this in the ratio of a to b : we have then /paths of n-tnths of r, or treamtha of r, which would also be obtained by altering r in the ratio of a x ea to b x n. Hence composition of nume rical ratios in performed by multiplication of the antecedents for an antecedent and of the conaequents for a consequent. The process then is merely equivalent to that of the multiplication of fractions. If a and — were called the quautuplicities (ireauaireart) of the ratios, then a the quantuplieity of the compound ratio is the product of the quantu plleities of the components, as in the definition (be it Euclid's or no) which is found in the manuscripts of the elements.