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 not greater than the multiple of the fourth; then the first is said to have to the second a greater ratio than the third magnitude 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. VIII. When there is any number of magnitudes greater than two, of which the first has to the second the same ratio that the second has to the third, and the second to the third the same ratio which the third has to the fourth, and so on, the magnitudes are said to be continual proportionals. IX. When three magnitudes are continual proportionals, the second is said to be a mean proportional between the other two. X. When there is any number of magnitudes of the same kind, 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 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: B:: E: F; and B: C:: G: H, and C: D:: K: L, then, since by this definition A has to D the ratio compounded of the ratios of A to B, B to C, C to D; A may also be said to have to D the ratio compounded of the ratios which are the same with 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 a ratio compounded of the same ratios, which compound the ratio of A to D; that is, a ratio compounded of the ratios of E to F, G to H, and K to L. XI. If 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. "Thus, if A be to B as B to C, the ratio of A to C is said to be dupli"cate of the ratio of A to B. Hence, since by the last definition, "the ratio of A to C is compounded of the ratios of A to B, and B "to C, a ratio, which is compounded of two equal ratios, is dupli̟"cate of either of these ratios." XII. If 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, or of the ratio of the second to the third, &c. "So also,if there are five continual proportionals; the ratio of the first "to the fifth is called quadruplicate of the ratio of the first to the "second; and so on, according to the number of ratios. Hence, a "ratio compounded of three equal ratios is triplicate of any one of "those ratios; a ratio compounded of four equal ratios quadrupli"cate," &c. XIII. 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 signify certain ways of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals. XIV. Permutando, or alternando, by permutation, or alternately; this word is used when there are four proportionals, 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: See Prop. 16. of this Book.. XV. Invertendo, 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. A. Book 5. XVI. 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. XVII. Dividendo, 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 5. XVIII. Convertendo, 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. D. Book 5. XIX. Ex æquali (sc. distantia), or ex æquo, 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. XX. 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. XXI. Ex æquali, in proportione perturbata, seu inordinata: 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, or inverse, order; and the inference is as in the 19th 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. If any number of magnitudes be equimultiples of as many others, each of each, what multiple soever any one of the first is of its part, the same multiple is the sum of all the first of the sum of all the rest. Let any number of magnitudes A, B, and C be equimultiples of as many otheus, D, E, and F, each of each A+B+C is the same multiple of D+E+F, that A is of D. Let A contain D, B contain E, and C contain F, each the same number of times, as, for instance, three times. Then, because A contains D three times, For the same reason, A=D+D+D. B=E+E+E; C=F+F+F. Therefore, adding equals to equals (Ax. 2. 1.), A+B+C is equal to D+E+F, taken three times. In the same manner, if A, B, and C were each any other equimultiple of D, E, and F, it would be shewn that A+B+C was the same multiple of D+E+F. Therefore, &c. Q. E. D. COR. Hence, if m be any number, mD+mE+mF=m(D+E+F). For mD, mE, and mF are multiples of D, E, and F by m, therefore their sum is also a multiple of D+E+F by m. PROP. II. THEOR. If to a multiple of a magnitude by any number, a multiple of the same magnitude by any number be added, the sum will be the same multiple of that magnitude that the sum of the two numbers is of unity. Let A=mC, and BnC; A+B=(m+n)C. For, since A=mC, A=C+C+C+ &c. C being repeated m times. For the same reason, B=C+C+ &c. C being repeated n times. Therefore, adding equals to equals, A+B is equal to C taken m+n times; that is, A+B=(m+n)C. Therefore A+B contains C as oft as there are units in m+n. Q. E. D.. COR. 1. In the same way, if there be any number of multiples whatsoever, as A=mE, B=nE, C=pE, it is shewn, that A+B+C= (m+n+p)E. COR. 2. Hence also, since A+B+C=(m+n+p)E, and since A= mE, B=nE, and C=pE,mE+nE+pE=(m+n+p)E. PROP. III. THEOR. If the first of three magnitudes contain the second as oft as there are units in a certain number, and if the second contain the third also, as often as there are units in a certain number, the first will contain the third as oft as there are units in the product of these two numbers. Let A=mB, and BnC; then A=mnC. Since B=nC, mB=nC+nC+&c. repeated m times. But nC+nC &c. repeated m times is equal to C (2. Cor. 2. 5.), multiplied by n+n +&c. n being added to itself m times; but n added to itself m times, is n multiplied by m, or mn. Therefore nC +nC+ &c. repeated m times innC; whence also mB = mnC, and by hypothesis AmB, therefore AmnC. Therefore, &c. Q E. D. = PROP IV. THEOR. If the first of four magnitudes has the same ratio to the second which the third has to the fourth, and if any equimultiples whatever be taken of the first and third, and any whatever of the second and fourth; the multiple of the first shall have the same ratio to the multiple of the second, that the multiple of the third has to the multiple of the fourth. Let A B C D, and let m and n be any two numbers; mA: nB:: mC: nD. า Take of mA and mC equimultiples by any number p, and of nB and nD equimultiples by any number q Then the equimultiples of mA, and mC by p, are equimultiples also of A and C, for they contain A and C as oft as there are units in pm (3. 5), and are equal to pmA and pmC. For the same reason the multiples of nB and nD by q, are qnB, qnD, Since, therefore, A: B:: C: D, and of A and C there are taken any equimultiples, viz. pmA and pmC, and of B and D, any equimultiples qnB, qnD, if pmA be greater than qnB, pmC must be greater than qnD (def. 5 5.); if equal, equal; and if less, less. But pmA, pm Care also equimultiples of mA and mC, and qnB, qnÐ are equimultiples of nB and nD, therefore (def. 5. 5.), mA': nB :: mC : nD. Therefore, &c. Q. E. D. COR. In the same manner it may be demonstrated, that if A: B:: C: D, and of A and C equimultiples be taken by any number m, viz. mA and mC, mA: B:; mC: D. This may also be considered as included in the proposition, and as being the case when n=1. PROP. V. THEOR. If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other; the remainder is the same multiple of the remainder, that the whole is of the whole. Let mA and mB be any equimultiples of the two magnitudes A and B, of which A is greater than B; mA-mB is the same multiple of A-B that mA is of A, that is, mA-mB=m(A-B) Let D be the excess of A above B, then A-BD, and adding B to both, A=D+B. Therefore (1. 5.) mA=mD+mB; take mB from both, and mA-mBmD; but D=A-B, therefore mA-mB=m(A -B). Therefore, &c. Q. E. D. P |