A less magnitude is said to be a part of a greater magni- Book V. tude, when the less measures the greater ; that is, when the less is contained a certain number of times exactly in the greater.' II. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less; that is, when the greater contains the less a certain number of times exactly.' III. Ratio is a mutual relation of two magnitudes of the same See N. kind to one another, in respect of quantity.' IV. Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other. v. The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth ; or, if the multiple of the first Book V. be greater than that of the second, the multiple of the third is also greater than that of the fourth. VI. tionals. •N. B. When four magnitudes are proportion- VII. When of the equimultiples of four magnitudes (taken as in | the fifth definition), the multiple of the first is greater than that of the second, but the multiple of the third is VIII. IX X. to have to the third the duplicate ratio of that which it XI. See N. When four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that Definition A, to wit, of compound ratio. kind, the first is said to have to the last of them the and so on unto the last magnitude. For example, if A, B, C, D, be four magnitudes of the same kind, the first A is said to have to the last D the ratio compounded of the ratio of A to B, and the ratio of B to C, and of the ratio C to D, or, the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D: And if A has to B the same ratio which E has to F; and Book V. B to C, the same ratio that G has to H; and C to D; the same that K has to L; then, by tbis definition, A is said to have to D the ratio compounded of ratios which are the same with the ratios of E to F, G to H, and K to L: And the same thing is to be understood when it is more briefly expressed by saying, A has to D the ratio com pounded of the ratios of E to F, G to H, and K to L. In like manner, the same things being supposed, if M has to N the same ratio which A has to D; then, for short- XII. to one another, as also the consequents to one apother. signify certain ways of changing either the order or XIII. This word is used when there are four proportionals, and See N. XIV. als, and it is inferred, that the second is to the first, as XV. tionals, and it is inferred, that the first, together with the XVI. and it is inferred, that the excess of the first above the XVII. tionals, and it is inferred, that the first is to its excess Book V. above the second, as the third to its excess above the XVIII. stance; when there is any mumber of magnitudes more this there are the two following kinds, which arise from XIX. self, when the first magnitude is to the second of the first XX. equality in perturbate or disorderly proportion.* This AXIOMS. I I. , EQUIMULTIPLES of the same, or of equal magnitudes, are equal to one another. * 4 Prop. lib. 2. Archimedis de sphæra et cylindre. Book V. II. ' Those magnitudes of which the same, or equal magnitudes, are equimultiples, are equal to one another. III. A multiple of a greater magnitude is greater than the same multiple of a less. IV. That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude. PROP. I. THEOR. If any number of magnitudes be equimultiples of as many, each of each ; what multiple soever any one of them is of its part, the same multiple shall all the first magnitudes be of all the other. Let any number of magnitudes AB, CD be equimultiples of as many others E, F, each of each; whatsoever multiple AB is of E, the same multiple shall AB and CD together be of E and F together. Because AB is the same multiple of E that CD is of F, as many magnitudes as are in AB equal to E, so many are there in CD equal to F. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into CH, HD equal each of them to F: The number therefore of the magnitudes CH, HD, shall G be equal to the number of the others AG, GB: And because AG is equal to E, and CH to F, therefore AG and CH together are equal toa E B * Ax. 2.1. and F together: For the same reason, because GB is equal to E, and HD to F; GB and HD together are equal to E and F together. Wherefore as many magnitudes as are in AB equal to E, so many are there in AB, CD together equal Hp to E and F together. Therefore, whatsoever multiple AB is of E, the same multiple is AB and CD together of E and F together. Therefore, if any magnitudes, how many soever, be equimultiples of as many, each of each, whatsoever multiple any one of them is of its part, the same multiple shall all the first magnitudes be of all the other: 'For the same |