has to L; then, by this Definition, A is said to have to Ď the Book V. ratio compounded of ratios which are the fame with the ratios of E to F, G to H, and K to L. and the fame thing is to be understood when it is more briefly expreffed, 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 fame things bang fuppofed, if M has to N the fame ratio which A has to D, then, for shortness fake, M is faid 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 homologous to one another, as alfo the confequents to one another. Geometers make use of the following technical words to fignify • certain ways of changing either the order or magnitude of pro'portionals, so as that they continue ftill to be proportionals.' XIII. Permutando, or Alternando, by Permutation, or alternately; this word is used when there are four proportionals, and it is infer- See N. red, that the first has the fame ratio to the third, which the fecond has to the fourth; or that the first is to the third, as the fecond to the fourth. as is fhewn in the 16th Prop. of this 5th Book. XIV. Invertendo, by Inversion; when there are four proportionals, and it is inferred, that the second is to the firft, as the fourth to the third. Prop. B. Book 5th. XV. Componendo, by Compofition; when there are four proportionals, and it is inferred, that the first together with the second, is to the fecond, as the third together with the fourth, is to the fourth. 18th Prop. Book 5th. XVI. Dividendo, by Divifion; when there are four proportionals, and it is inferred, that the Excess of the first above the second, is to the second, as the Excefs of the third above the fourth, is to the fourth. 17th Prop. Book 5th. XVII. Convertendo, by Converfion; when there are four proportionals, and it is inferred that the firft is to its Excefs above the fecond, as the third to its Excefs above the fourth. Prop. E. Book 5th. H Ex aequali (fc. distantia,) or, ex aequo, from equality of distance; when there is any number of magnitudes more than two, and as many others, so 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 aequali, from equality; this term is used fimply 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, fo is the fecond to the third of the other; and fo on in order, and the inference is as mentioned in the preceding Definition; whence this is called Ordinate Proportion. It is demonstrated in 22d Prop. Book 5th. XX. Ex aequali, in proportione perturbata, feu inordinata, from equality, in perturbate or diforderly 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, fo is the last but two to the last but one of the fecond rank; and as the third is to the fourth of the first rank, fo is the third from the last to the last but two of the second rank; and fo on in a cross order. and the inference is as in the 18th Definition. It is demonstrated in 23d Prop. of Book 5th. AXIOM S. I. EQUIMULTIPLES of the fame, or of equal magnitudes, are equal to one another. II. Thofe magnitudes of which the fame, or equal magnitudes, are equimultiples, are equal to one another. III. A multiple of a greater magnitude is greater than the fame multiple of a lefs. * 4. Prop. Lib. 2. Archimedis de fphaera et cylindro. IV. That magnitude of which a multiple is greater than the fame multiple of another, is greater than that other magnitude. PROP. I. THEOR. any number of magnitudes be equimultiples of as many, each of each; what multiple foever any one of them is of its part, the fame multiple fhall 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 fame multiple shall AB and CD together be of E and F together. A GE B Because AB is the fame multiple of E that CD is of F, as many magnitudes as are in AB equal to E, fo 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 fhall 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 to a E and F together. for the fame reason, becaufe 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, fo many are there in AB, CD together equal to E and F together. Therefore whatfoever multiple AB is of E, the fame multiple is AB and CD together of E and F together. Book V. 2. Ax. 2. 1. C F H D Therefore if any magnitudes, how many foever, be equimultiples of as many, each of each, whatsoever multiple any one of them is of its part, the fame multiple fhall all the first magnitudes be of all the other. for the fame Demonftration holds in any number ' of magnitudes, which was here applied to two.' Q. E. D. Book V. PROP. II. THEOR. F the first magnitude be the fame. multiple of the fecond that the third is of the fourth, and the fifth the fame multiple of the fecond that the fixth is of the fourth; then shall the first together with the fifth be the fame multiple of the fecond, that the third together with the fixth is of the fourth. Let AB the first be the fame multiple of C the second, that DE the third is of F the fourth; and BG the fifth the fame multiple of C the fecond, that EH the fixth is of F the fourth. Then is AG the first together with the fifth the fame multiple of C the fecond, that DH the third together with the fixth is of F the fourth. Because AB is the fame multiple of C, that DE is of F; there are as many magnitudes in AB equal to C, as there are in DE equal to F. in like manner, as many as there are in BG equal to C, fo many are there in EH equal to F. as many B D A E d H HF then as are in the whole AG equal to C, fo many are there in the whole DH equal to F. therefore AG is the fame multiple of C, that DH is of F; that is, AG the first and fifth together, is the fame multiple of the fecond C, that DH the third and fixth together is of the fourth F. If therefore the firft be the fame multiple, &c. Q. E. D. COR. From this it is plain, that if any B • multiples of another C; and as many DE, D A E K H CLF PROP. III. THEOR. [F the first be the fame multiple of the fecond, which the third is of the fourth; and if of the first and third there be taken equimultiples, these fhall be equimultiples the one of the fecond, and the other of the fourth. Let A the first be the fame multiple of B the second, that C the third is of D the fourth; and of A, C let the equimultiples EF, GH be taken. then EF is the fame multiple of B, that GH is of D. F K H Book V. Because EF is the fame multiple of A, that GH is of C, there are as many magnitudes in EF equal to A, as are in GH equal to C. let EF be divided into the magnitudes EK, KF, each equal to A, and GH into GL, LH, each equal to C. the number therefore of the magnitudes EK, KF, fhall be equal to the number of the others GL, LH. and because A is the fame multiple of B, that C is of D, and that EK is equal to A, and GL to C; therefore EK is the fame multiple of B, that GL is of D. for the fame reafon KF is the fame multiple of B, that LH is of D; and fo, if there be more parts in EF, GH equal to A, C. because therefore the first EK is the fame multiple of the second B, which the third GL is of the fourth D, and that the fifth KF is the fame multiple of the second B, which the fixth LH is of the fourth D; EF the first together with the fifth is the fame multiple of the fecond B, which GH the third together with the a. 2. 5If therefore the firft, &c. Q. E. D. fixth is of the fourth D. E AB G C D H 3 |