5. If there be four magnitudes, and if any like multiples whatever be taken of the first and third, and any whatever of the second and fourth ; and if, according as the multiple of the first is greater than the multiple of the second, equal to it, or less, the multiple of the third is also greater than the multiple of the fourth, equal to it, or less: then the first of the magnitudes is said to have to the second the same ratio that the third has to the fourth. * 6. Magnitudes which have the same ratio are called proportionals: and equality or identity of ratios constitutes proportion or analogy. When magnitudes are proportionals, the relation is expressed briefly by saying, that the first is to the second as the third to the fourth, the fifth to the sixth, and so on.t 7. When of the multiples 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 not greater than that of the fourth; then the first is said to have to the second a greater ratio than the third has to the fourth ; and, on the contrary, the third is said to have to the fourth a less ratio than the first has to the second. 8. When there are three [ or more magnitudes of the same * It is plain that the first and second magnitudes must be of the same kind; and that the third and fourth must also be of the same kind compared with one another, though they may differ in kind from the first and second. A similar remark is applicable in respect to the next two definitions. The properties of proportionals may in many instances be familiarly illustrated by means of numbers. Thus, to illustrate this definition, let us take the numbers 10, 4, 15, and 6, and it will be seen that the ratio of the first to the second is the same as the ratio of the third to the fourth. For, if the first and third be multiplied by 2, and the second and fourth by 5, the multiples of the first and second are each 20, and those of the others each 30: but if instead of 2 and 5, the multipliers 3 and 8 be employed, the multiples of the first and third are respectively less than those of the second and fourth ;-& relation which is reversed, if the multipliers 4 and 9 be used. The operations will stand thus: 10, 4, 15, 6. 10, 4, 15, 6. 10, 4, 15, 6. 3, 8, 3, 8. 4, 9, 4, 9. 20, 20, 30, 30. 30, 32, 45, 48. 40, 36, 60, 54. In this definition, it must be understood that the relations hold, not merely for particular multiples, but for all multiples whatever. This will be exemplified by the numbers 7, 4, 9, 6, which would appear to exhibit equal ratios, if the first and third were multiplied by 3, and the second and fourth by 4; but not if the first and third be multiplied by 5 and the others by 8: and by the 7th definition it will appear, that 7 has to 4 greater ratio than 9 has to 6. This consideration must be kept in view in every elementary application of the theory of proportion. | Thus, 6, 9, 8, 12, 10, 15, are proportionals, the ratios of 6 to 9, of 8 to 12, and of 10 to 15 being all equal ; and we briefly express this relation by saying, that 6 is to 9, as 8 to 12, and as 10 to 15. We also usually write proportionals in succession with two dots between the terms of the ratios, and four between the others : thus 6:9::8:12: : 10 · 15. | It is plain that proportion, being the equality of ratios, consists in three terms at least, as with fewer terms there could not be two or more ratios. kind, such that the ratios of the first to the second, of the second to the third, and so on, whatever may be their number, are all equal ; the magnitudes are said to be continual proportionals. * 9. The second of three continual proportionals is said to be a mean proportional between the other two. + 10. When there is any number of magnitudes of the same kind, greater than two, the first is said to have to the last the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on to 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 ratios of A to B, B to C, and C to D. # 11. When three magnitudes are continual proportionals, the ratio of the first to the third is said to be duplicate of the ratio of the first to the second, or of the second to the third. 12. When four magnitudes are continual proportionals, the ratio of the first to the fourth is said to be triplicate of the ratio of the first to the second, of the ratio of the second to the third, or of the ratio of the third to the fourth. S rest means. · Hence, in continual proportionals, each of the terms, except the first and last, is used as the consequent of one ratio, and the antecedent of another ; while in other proportionals, each term is used only as an antecedent or consequent, and not as both. It may be proper to remark, that, by writers on arithmetic and algebra, continual proportionals are often called quantities in geometrical progression. This however, is improper, as they have nothing in their nature to connect them with geometry more than with arithmetic. The following numbers are continual proportionals, and may serve the learner for illustration : 3, 6, 12, 24; 54, 18, 6, 2; 4, 6, 9; 64, 48, 36, 27. + In any series of magnitudes, the first and last are called extremes, and the When, however, we speak simply of a mean proportional, it is always to be understood as defined above. # The meaning of compound ratio, -an expression which has been introduced to prevent circumlocution, will be better understood by the learner by studying proposition F of this book, the 18th proposition of the Supplement, and the 23d of the sixth book, than by any illustration that could be given in this place. $ In continual proportionals, by their own nature, and that of compound ratio, the ratio of the first to the third is compounded of two equal ratios; and the ratio of the first to the fourth, of three equal ratios ; and hence we see the reason and the propriety of calling the first duplicate ratio, and the second triplicate. It is plain, that on similar principles, the ratio of the first to the fifth would be said to be quadruplicate of the ratio of the first to the second, the second to the third, &c. ; and thus we might form other similar terms at pleasure. The terms subduplicate, subtriplicate, and sesquiplicate, which are sometimes employed by mathematical writers, are easily understood after the explanations given above. Io continual proportionals, the ratio of the first term to the second is said to be subduplicate of the ratio of the first to the third, and subtriplicate of that of the first to the fourth. Again, if there be four continual proportionals, the ratio of the first to the fourth is said to be sesquiplicate of the 13. In proportionals, the antecedent terms are called homologous to one another, as also the consequents to one another. * 'Geometers make use of the following technical words to denote different modes of deriving one proportion from another, by changing either the order or the magnitudes of the terms.' 14. Alternately : † this word is used when there are four proportionals of the same kind: and it is inferred, that the first has the same ratio to the third, which the second has to the fourth ; or that the first is to the third, as the second to the fourth : as is shown in the 16th proposition of this book. 15. By inversion: when there are four proportionals, and it is inferred, that the second is to the first, as the fourth to the third. Prop. B. Book V. 16. 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 V. 17. By division : when there are four proportionals, 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 V. 18. By conversion : when there are four proportionals, 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 V. 19. Ex æquo, or ex æquali (scil. distantiá), from equality of distance : when there is any number of magnitades more than two, and as many others, which, taken two in the one rank, and two in the other, in direct order, have the same ratio ; and it is inferred that the first has to the last of the first rank the same ratio of the first to the third : or, which amounts to the same, the ratio which is compounded of another ratio and its subduplicate, is sesquiplicate of that ratio. * Thus, if A :B::C:D::E: F; A, C, E are bomologous to one another; and B, D, F to one another. | Instead of the simple term, alternately, there is usually given permutando, alternando, by permutation, or alternately. The proposition, however, which is expressed in substance in this definition, is generally cited by the simple word, alternately, to the exclusion of the Latin terms, permutando and alternando, the use of which the taste of the present time properly rejects. For the same reason, in the next four definitions, the words, invertendo, componendo, dividendo, and convertendo, are omitted. The substance of the five preceding definitions may be exhibited briefly in the following manner, the signs + and - denoting addition and subtraction, as has been explained already, at the beginning of the second book : Let A : B::C: D; or let 15: 6 :: 10: 4: then, Alternately, A:C:: B: D; and 15: 10:: 6 : 4: By in version, B: A:: D: C; and 6: 15 : ; 4: 10: By composition, A+B: B :: C+D: D; and 21: 6 : : 14: 4: By division, A-B: B :: CD:D; and 9: 6:: 6 : 4: By conversion, A: A-B::C:C_D; and 15: 9:; 10: 6. ratio which the first of the other rank has to the last.* This is demonstrated in the 22d proposition of this book. 20. Ex æquo, inversely; when there are three or more magnitudes, and as many others, which taken two and two in a cross order, have the same ratio ; that is, when the first magnitude is to the second in the first rank, as the last but one is to the last in the second rank; and the second to the third of the first rank, as the last but two is to the last but one of the second rank, and so on; and it is inferred, as in the preceding definition, that the first is to the last of the first rank, as the first to the last of the other rank. t This is proved in the 23 proposition of this book. AXIO MS. 1. LIKE multiples of the same, or of equal magnitudes, are equal to one another. I 2. Those magnitudes of which the same, or equal magnitudes, are like multiples, are equal to one another. S 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude. I PROP. I. THEOR. IF any number of magnitudes be like multiples of as many others, each of each ; what multiple soever any one of the first is of its part, the same multiple shall all the first magnitudes be of all the others. Let any number of magnitudes AB, CD be like multiples of Thus, in the two ranks of magnitudes, A, B, C, D; P, Q, R, S, if A is to B, as P to Q; B to C, as Q to R; and C to D as R to S, the infer. ence, ex æquo, is that A is to D as P to S. The two ranks, 6, 9, 15, 12, and 10, 15, 25, 20, afford an example in numbers; as do also 21, 49, 56, 35, 14, and 15, 35, 40, 25, 10. + Thus, in the two ranks of magnitudes, A, B, C, D, E; P, Q, R, S, T, if A is to B as S to T, B to C as R to S, C to D as Q to R, and D to E as P to Q; then it is inferred, ex æquo inversely, that A is to E as P to T. The two ranks, 9, 12, 8, 4, and 18, 9, 6, 8, and the two, 24, 36, 18, 16, 60, and 12, 45, 40, 20, 30, afford examples in numbers. The learner will perceive, that, in both this definition and the foregoing, the ratio of the first to the last magnitude in the first rank, and the ratio of the first to the last in the second rank, are compounded of equal ratios. Or, if equals be multiplied by the same, the products are equal. This axiom might be derived from the second axiom of the first book. $ Or, if equals be divided by the same, the quotients are equal. || If the teacher and student wish to save time, the Supplement to this book may be used instead of what remains of the book itself. A B H as many others E, F, each of each ; whatever multiple AB is of & 미 GB; and CD into CH, HD, equal each of them to F: the number therefore, of the magnitudes CH, HD is equal (hyp.) to the number of the others AG, GB. And because AG is equal to E, and CH to F, therefore ? 비 AG and CH together are equal (I. ax. 2.) to E and F together. For the same reason, GB and HD together are equal to E and F together: wherefore, as many magnitudes as are in A B equal to E, so many are there in AB and CD together equal to E and F together. Therefore, whatever multiple AB is of E, the same multiple are AB and CD together, of E and F together: and the same demonstration would hold, if the number of magnitudes were greater than two. Therefore, if any number, &c. D PROP. II. THEOR. D A B+ 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 is AG, the first together with the fifth, the same multiple of C the second, that DH, the third together with the sixth, is of F the fourth. Because AB is the same multiple of C, that DE is of F; there are as many magnitudes in AB ili equal to C, as there are in DE equal to F. In like manner, as many as there are in BG equal to C, so many are there in EH equal to F. As many, then, as are in the whole AG equal to C, so many are there in the whole DH equal to F: therefore AG is the same multiple of C, that DH is of F; that is, AG, the first and fifth together, is the same multiple of the second C, that DH, the third and sixth together, is of the fourth F. If, therefore, &c. Cor. From this it is plain, that, if any number of magnitudes be multiples of another C; and as many others be the same multiples of F, each of each ; the whole of the first magnitudes is the same multiple of C, that the whole of the last is of F. G H |