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 be greater than that of the second, the multiple of the third is also greater than that of the fourth. VI. Magnitudes which have the same ratio are called proportionals. •N. B. When four magnitudes are proportionals, it is usually expressed by saying, the first is to the second, as the third to the fourth.' 6 VII. as in the fifth definition), the multiple of the first is VIII. IX. X. said to have to the third the duplicate ratio of that XI. See N. When four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on, quadruplicate, &c. increasing the denomination still by unity, in any number of proportionals. . Definition A, to wit of compound ratio. kind, the first is said to have to the last of them the fourth, 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 of the ratio of B to C, and of the ratio of 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 B to C the same ratio that G has to H; and C to D the same that K has to L; then, by this 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 compounded 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 shortness sake, M is said to have to N the ratio compounded of the ratios of E to F, G to H, and K to L. XII. In proportionals, the antecedent terms are called bomo logous to one another, as also the consequents to one or magnitude of proportionals, so that they continue XIII. nately. This word is used when there are four pro- fourth : as is shewn in the 16th Prop. of this fifth Book. XIV. Invertendo, by inversion; when there are four propor tionals, and it is inferred, that the second is to the first as the fourth to the third. Prop. B. Book 5. XV. Componendo, by composition; when there are four proportionals, and it is inferred that the first together with the second, is to the second, as the third together with the fourth, is to the fourth. 18th Prop. Book 5. XVI. Dividendo, by division; when there are four propor tionals, and it is inferred, that the excess of the first above the second, is to the second, as the excess of the third above the fourth, is to the fourth. 17th Prop. Book 5. XVII. Convertendo, by conversion; when there are four pro portionals, and it is inferred, that the first is to its excess above the second, as the third to its excess above the fourth. Prop. E. Book 5. XVIII. Ex æquali (sc. distantiâ), or ex æquo, from equality of distance ; when there is any number of magnitudes more than two, and as many others, such that they are proportionals when taken two and two of each rank, and it is inferred, that the first is to the last of the first rank of magnitudes, as the first is to the last of the others : Of this there are the two following kinds, which arise from the different order in which the magnitudes are taken, two and two.' XIX. Ex æquali, from equality. This term is used simply by itself, when the first magnitude is to the second of the first rank, as the first to the second of the other rank; and as the second is to the third of the first rank, so is the second to the third of the other; and so on in order: and the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonstrated in the 22d Prop. Book 5. XX. Ex æquali in proportione perturbatâ seu inordinatâ, from equality in perturbate or disorderly proportion*. This term is used when the first magnitude is to the second of the first rank, as the last but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank; and as the third is to the fourth of the first rank, so is the third from the last to the last but two of the second rank; and so on in a cross order: and the inference is as in the 18th definition, It is demonstrated in the 23d Prop. of Book 5. AXIOMS. I. EQUIMULTIPLEs of the same, or of equal magnitudes, are equal to one another. 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. each of each; what multiple soever any one of them is of 4 Prop. lib. 2. Archimedis de Sphæra et Cylindro. tiples 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 В! HT is equal to E, and CH to F, therefore AG and • 2 Ax. 1. CH'together are equal to * E and F together: D for the same reason, because GB is equal to E, 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 others : For • the same demonstration holds in any number of magnitudes, which was here applied to two.' Q. E. D. PROP. II. THEOR. . If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then shall the first together with the fifth be the same multiple of the second, that the third together with the sixth is of the fourth. Let AB the first be the same multiple of C the second, that DE the third is of F the fourth; and BG the fifth the same multiple of C the second, that EH the sixth is of F the fourth: then shall AG, the first together with the fifth, E BI Because AB is the same multiple of C that DE is of D A |