cumference; if BC, CE, &c. be joined, the equilateral and Book IV. equiangular quindecagon will be inscribed: Which was requi red. COR. If, from what has been faid of the pentagon, right lines be drawn through the divisions of the circle, tangents to the fame, an equilateral and equiangular quindecagon will be described about the circle; or a circle may be inscribed or de scribed about the quindecagon. A multiple is a magnitude of a magnitude; the greater of the less, when the less measures the greater. HI. Ratio is a certain mutual habitude of magnitudes of the fame kind, according to quantity. IV. Magnitudes have proportion to each other; which, being multi V. Magnitudes have the same ratio to each other, viz. the first to the second, and third to the fourth, when there are taken any equimultiples of the first and third, and likewife any equimultiples of the second and fourth; if the multiple of the first be equal to the multiple of the second, then the multiple of the third will be equal to the multiple of the fourth; if greater, greater; and, if less, less. VI. Magnitudes which have the fame proportion are called Proportionals. VII. 1 When, of equimultiples, the multiple of the first exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth; the first to the second is faid to have a greater ratio than the third to the fourth. VIII. When three magnitudes are proportionals, the first has to the third a duplicate ratio of what it has to the second. ΧΙ. When four magnitudes are proportional, the first has to the fourth a triplicate ratio of what it has to the second; and always one more in order as the proportionals shall be extend ed. XII. Homologous magnitudes, or magnitudes of a like ratio, are such whose antecedents are to the antecedents and confequents to the consequents in the fame ratio. XIII. له Alternate ratio is the comparing the antecedent with the antecedent, and confequent with the confequent. XIV. Inverse ratio is, when the confequentis taken as the antecedent, and compared with the antecedent as a consequent. XV. Compounded ratio is, when the antecedent and confequent, ta ken as one, are compared with the confequent itself. XVI. Divided ratio is, when the excess, by which the antecedent exceeds the consequent, is compared with the consequent. XVII. Converse ratio is, when the antecedent is compared with the excess by which the antecedent exceeds the confequent. XVIII. Ratio of equality is when there are taken more than two magnitudes in one order, and a like number of magnitudes in another order, comparing two to two, being in the fame ratio; it shall be in the first order of magnitudes, as the first is to the last, so, in the second order of magnitudes, is the first to the laft. XIX. Ordinate proportion is, the ratio being, as in the last, as the antecedent is to the confequent, in the first order of magnitudes; fo > Book V. X so is the antecedent to the consequent in the second order of magnitudes; and as the consequent is to any other, so is the confequent to any other. XX. Perturbate proportion is, when there are three or more magnitudes, and others equal to them in number, taken two and two in the fame ratio; in the first order of magnitudes, as the antecedent is to the consequent, so, in the second order of magnitudes, is the antecedent to the consequent; and, as in the first order, the consequent is to some other, so, in the fecond order, is some other to the antecedent. E AXIOMS. I. QUIMULTIPLES of the fame, or of equal magnitudes, are equal to each other. II. These magnitudes that have the same equimultiples, or whose equimultiples are equal, are equal to each other, I PROP. I. THEOR. F there be any number of magnitudes, equimultiples of a like number of magnitudes, each of each, whatever multiple any one of the former magnitudes is of its correspondent one, the same multiple are all the former magnitudes of all the latter. A 1 Let AB, CD, be magnitudes, equimultiples of E, F, whatever multiple AB is of E, and CD of F, the fame multiple AB, CD, together, is of E, F, together. For, let the magnitudes in AB, equal to E, be AG, GB; and the magnitudes in CD, equal to F, be CH, HD; then AG, CH, are equal to E, F; and BG, HD, likewise equal to E, F; therefore, as often as AB contains E, and CD, F, fo often AB, CD, contains E, F: Wherefore, if there are, &c. PROP. II. THEOR. the first be the same multiple of the second, as the third is of the fourth; and if the the fifth be the fame multiple of the second, that the sixth is of the fourth; then shall the first, added to the fifth, be the fame multiple of the second, that the third, added to the fixth, is of the fourth. Let the first AB be the fame multiple of the second C, that Book V. the third DE is of the fourth F; and let the fifth BG be the fame multiple of the second C, that the sixth EH is of the fourth F; then AG will be the fame multiple of C that DH is of F. t For, because AB is the fame multiple of C that DE is of F, there are as many magnitudes in AB equal to C, as in DE, equal to F. For the fame reason, there are as many magnitudes in BG equal to C, as there are in EH, equal to F; therefore there are as many magnitudes in AG equal to C, as there are in DH equal to Fa. Wherefore, &c. a. I. PROP. III. THEOR. If IF the first be the fame multiple of the fame multiple of the second, that the third is of the fourth, and there be taken equimultiples of the first and third, then will the magnitudes so taken be equimultiples of the Second and fourth. Let the first A be the fame multiple of the second B that the third C is of the fourth D; and let EF, GH, be equimultiples of A, C ;then EF is the fame multiple of B, that GH is of D. For, let the magnitudes in EF, equal to A, be FK, KE; and the magnitudes in GH, equal to C, be HL, LG; then there are as many magnitudes equal to B in FE, as there are magnitudes equal to Din GH2; wherefore FE is the fame multiple of Ba that GH is of D. Wherefore, &c. PROP. IV. THEOR. IF the first have the same ratio to the fecond that the third has to the fourth, then shall also the equimultiples of the first have the fame ratio to the equimultiple of the second that the equimultiple of the third has to that of the fourth. Let there be four magnitudes, A, B, C, D, fuch, that A is to Bas C to D. Let E, F, be taken the fame multiples of A, C; and G, H, the fame multiples of B, D; then E is to Gas F is to H. For, take K, L, any equimultiples of E, F, and M, N, any equimultiples of G, H; then Kis the fame multiple of A that Lis of C. For the fame reason, M is the a 3. same multiple of B that N is of D2; but, because A is to Bas C is to D2 if K be equal to M, L will be equal to N; if great. |