IV. the third the duplicate ratio of that Magnitudes are said to have a ratio which it has to the second. to one another, when the less can XI. be multiplied so as to exceed the When four magnitudes are continual other. proportionals, the first is said to V. have to the fourth the triplicate raThe first of four magnitudes is said to tio of that which it has to the se have the same ratio to the second, cond, and so on, quadruplicate, which the third has to the fourth, &c. increasing the denomination when any equimultiples whatsoever still by unity, in any number of of the first and third being taken, and proportionals. any equimultiples whatsoever of the Definition A, to wit, of compound second and fourth; if the multiple of ratio. the first be less than that of the se- When there are any number of magcond, the multiple of the third is also nitudes of the same kind, the first less than that of the fourth ; or, if is said to have to the last of them the multiple of the first be equal to the ratio compounded of the ratio that of the second, the multiple of which the first has to the second, the third is also equal to that of the and of the ratio which the second fourth; or, if the multiple of the has to the third, and of the ratio first be greater than that of the se- which the third has to the fourth, cond, the multiple of the third is and so on unto the last magnitude. also greater than that of the fourth. For example, if A, B, C, D, be four VI. magnitudes of the same kind, the Magnitudes which have the same ra- first A is said to have to the last D tio are called proportionals. the ratio compounded of the ratio N.B. When four magnitudes are pro- of A to B, and of the ratio of B to • portionals, it is usually expressed C, and of the ratio of C to D; or, • by saying, the first is to the se- the ratio of A to D is said to be cond, as the third to the fourth.' compounded of the ratios of A to VII. B, B to C, and C to D. When of the equimultiples of four And if A has to B the same ratio magnitudes (taken as in the fifth de- which E has to F; and B to C, the finition), the multiple of the first is same ratio that G has to H; and greater than that of the second, but C to D, the same that K has to L; the multiple of the third is not then, by this definition, A is said greater than the multiple of the to have to D the ratio compounded fourth ; then the first is said to of ratios which are the same with hare to the second a greater ratio the ratios of E to F, G to H, ani than the third magnitude has to K to L: And the same thing is to the fourth ; and, on the contrary, be understood when it is more the third is said to have to the briefly expressed, by saying, A has fourth a less ratio than the first has to D the ratio compounded of the to the second, ratios of E to F, G to H, and K VIII. to L. Analogy, or proportion, is the simi. In like manner, the same things being • litude of ratios.' supposed, if M has to N the same IX. ratio which A has to D; then, for Proportion consists in three terms at shortness sake, M is said to have le28t. to N, the ratio compounded of the X. ratios of E to F, G to H, and K When three magnitudes are propor to L. tionals, the first is said to have to a XII. XVIII. In proportionals, the antecedent terms Es æquali (se, distantia), or ex æquo, are called homologous to one ano- from equality of distance; when ther, as also the consequents to one there is any number of magnitudes another. more than two, and as many • Geometers make use of the follow. others, so that they are proportion ing technical words to signify als when taken two and two of • certain ways of changing either each rank, and it is inferred, that • the order or magnitude of pro the first is to the last of the first portionals, so as that they continue rank of magnitudes, as the first is still to be proportionals. to the last of the others : Of this XIII. ' there are the two following kinds, Permutando, or alternando, by per- • which arise from the different or mutation, or alternately. This • der in which the magnitudes are word is used when there are four taken two and two.' proportionals, and it is inferred, XIX. that the first has the same ratio to Ex æquali, from equality. This term thc third, which the second has to is used simply by itself, when the the fourth: or that the first is to first magnitude is to the second of the third, as the second to the the first rank, as the first to the sefourth: As is shewn in the 16th cond of the other rank; and as the prop. of this 5th book. second is to the third of the first XIV. rank, so is the second to the third Invertendo, by inversion : When there of the other ; and so on in order, are four proportionals, and it is in- and the inference is as mentioned ferred, that the second is to the in the preceding definition; whence first, as the fourth to the third. this is called ordinate proportion. Prop. B. Book 5. It is demonstrated in 28d Prop. XV. Book 5. Componendo, by composition; when XX. there are four proportionals, and it Ex æquali, in proportione perturbata, is inferred, that the first, together seu inordinata, from equality, in with the second, is to the second, perturbate or disorderly proporas the third, together with the tion, This term is used when fourth, is to the fourth. 18th Prop. the first magnitude is to the second Book 5. of the first rank, as the last but one XVI. is to the last of the second rank; Dividendo, by division; when there and as the second is to the third of are four proportionals, and it is in- the first rank, so is the last but two ferred, that the excess of the first to the last but one of the second above the second, is to the second, rank; and as the third is to the as the excess of the third above the fourth of the first rank, so is the fourth, is to the fourth. 17th Prop third from the last to the last but Book 5. two of the second rank; and so on XVII, in a cross order: And the inference Convertendo, by conversion ; when is as in the 18th definition. It is there are four proportionals, and it demonstrated in the 23d. Prop. of is inferred, that the first is to its ex- Book 5. cess above the second, as the third to its excess above the fourth. Prop. E Book 5. a less. AXIOMS. 1. greater than the same multiple of EQUIMULTIPLES of the same, or of equal magnitudes, are equal to one IV. another. That magnitude of which a multiple II. is greater than the same multiple Those magnitudes of which the same, of another, is greater than that or equal magritudes, are equimul- Other magnitude. IIL. PROP. I. THEOR. If any number of magnitudes be equimultiples of as many, each of ench; 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, to F, therefore AG and CH together CD be equimultiples of as many are equal to (Ax. 2. 5.) E and F toothers E, F, each of each ; whatso- gether: For the same reason, beever multiple AB is of E, the same cause GB is equal to E, and HD to multiple shall AB and CD together F; GB and HD together are equal be of E and I toether. to E and F together. Wherefore, as Because AB is the many magnitudes as are in A B equal same multiple of E that to E, so many are there in AB, CD, CD is of F, as many A, together equal to E and F together. magnitudes as Therefore, whatsoever multiple AB AB equal to E, so many B is of E, the same multiple is AB and are there in CD equal CD together of E and F together. to F. Divide AB into Therefore, if any magnitudes, how magnitudes equal to E, many soever, be equimultiples of as viz. AG, GB; and CD many, each of each, whatsoever mulinto CH, HD, equal tiple any one of them is of its part, each of them to F: The the same multiple shall all the first number therefore of the D magnitudes be of all the other : ‘For magnitudes CH, HD, • the same demonstration holds in any shall be equal to the ' number of magnitudes, which was number of the others, AG, GB : And ' here applied to two.' Q. E. D. because AG is equal to E, and CH are in PROP. II. THEOR. If the first magnitude be the same multiple of the second thal the third is of the fourth, and the filth the same multiple of the second that the sixth is of the fourth, then shall the first together with the fijih 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 mul- third is of F the fourth ; and BG th. tiple of C the second, that DE the fifth, the same multiple of cth If, B second, that EH multiple of C, that DH is of F; that the sixth is of F DI is, AĞ the first and fifth together, is the fourth: Then A the same multiple of is AG the first, E the second C, that D together with the DH the third and A. fifth, the same sixth together is of Et multiple of C the the fourth F. G C H E second, that DH therefore, the first bé KI third, together with the sixth, is of the same multiple, F the fourth. &c. Q. E. D. Because AB is the same multiple H' dar of C, that DE is of F; there are as COR. From this many magnitudes in AB equal to C, “it ie plain, that if as there are in DE equal to F: In • any number of magnitudes AB, BG, like manner, as many as there are in GH, be multiples of another C'; and BG equal to C, so many are there in as many DE, EK, KL be the same EH equal to F: As many, then, as ' multiples of F, each of each ; the are in the whole AG equal to C, so ' whole of the first, viz. AH, is the many are there in the whole DH samé multiple of Ć, that the whole equal to F; therefore AG is the same • of the last, viz. DL, is of F.' 6 PROP. III. THEOR. If the first be the same multiple of the second, which the third is of the fourth ; and if of the first and third there be taken equimultiples, these shall be equimultiples, the one of the second, and the other of the fourth. Let A the first, be the same mul. therefore of the magnitudes EK, KF, tiple of B the second, that C the third shall be equal to the number of the is of D the fourth; and of A, C let others GL, LH: And because A is the equimultiples ÉF, GH be taken: the same multiple of B, that C is of Then EF is the same multiple of B, D, and that EK is equal to A, and that GH is of D. GL to C; therefore EK is the same Because EF is the same multiple multiple of B, that GL is of D : For of A, that GH is of C, there are as the same reason, KF is the same many magnitudes in Éf equal to A, multiple of B, that LH is of D; and so, if there be more parts in EF, GH equal to A, C: Because, therefore, HI the first EK is the same multiple of the second B, which the third GL is K of the fourth D, and that the fifth KF is the same multiple of the second B, which the sixth LH is of the fourth E' A B G C D D; EF the first, together with the fifth, is the same multiple (2. 6.) of the second B, which GH the third, u are in GH equal to C: Let EP be together with the sixth, is of the divided into the magnitudes EK, KF, fourth D. If, therefore, the first, &c. each equal to A, and GH into GL, &c. Q. E. D. LH, each equal to C: The number PROP. IV. THEOR. If the first of four magnitudes has the same ratio to the second which the ihird has to the fourth; then any equimultiples whatever of the first and third shall have the same ralio to any equimultiples of the second and fourth, viz. the equimultiple of the first shall have the • • same ratio to that of the second, which the cquimultiple of the third • has lo that of the fourth.' Let A the first, have to B the se- have been taken certain equimultiples cond, the same ratio which the third M, N; if therefore K be greater than C has to the fourth D; and of A and M, L is greater than N: and if equal, C let there be taken any equimultiples equal; if less, less. (5. Def. 5.) And K, whatever E, F; and of B and D any L, are any equimultiples whatever of equimultiples whatever G, H: Then E, F; and M, N any whaterer of E has the same ratio to G, which F G, H: As therefore E is to G, so is has to H. (5. Def. 5.) F to H. Therefore, if Take of E and F any equimultiples the first, &c. Q. E. D. whatever K, L, and of G, H, any equimultiples whatever M, N : Then, Cor. Likewise, if the first has the same ratio to the second, which the third has to the fourth, then also any equimultiples whatever of the first and third have the same ratio to the second and fourth : And in like maniner, the first and the third have the same ratio to any equimultiples whatever of the second and fourth. Let A the first, have to B the se. Κ Ε Α B G M cond, the same ratio which the third C has to the fourth D, and of A and L F C DH N C let E and F be any equimultiples whatever; then E is to B, as F to D. Take of E, F any equimultiples whatever K, L, and of B, D any equimultiples whatever G, H; then it may be demonstrated, as before, that K is the same multiple of A, that L is of C: And because A is to B, as C is to D, and of A and C certain because E is the same multiple of A, equimultiples have een taken, viz. that F is of C; and of E and F have K and L; and of B and D certain been taken equimultiples K, L; there- equimultiples G, H; therefore, if K fore K is the same multiple of A, that be greater than G, L is greater than L is of C; (3. 5.) For the same rea- H; and if equal, eqnal; if less, less : son, M is the same multiple of B, (5. Def. 5.) And K, L are any equithat N is of D: And because, as A multiples of E, F, and G, H any is to B, so is C to D, (Hypoth.) and whatever of B, D : as therefore E is of A and C have been taken certain to B, so is F to D: and in the same equimultiples K, L; and of B and D way the other case is demonstrated. |