Construction. Let AC be the side of an equilateral triangle inscribed in the circle (IV. 2); and AB the side of an equilateral and equiangular pentagon inscribed in the same (IV. 11). A Proof. Therefore 1. If such equal parts as the whole circumference ABCDF contains fifteen, the circumference ABC contains five, it being the third part of the whole, and 2. The circumference AB contains three parts, it being the fifth part of the whole; therefore 3. Their difference BC contains two of the same parts. Bisect BC in E (III. 30); therefore 4. BE, EC are, each of them, the fifteenth part of the whole circumference ABCD; therefore if the straight lines BE, EC be drawn, and straight lines equal to them be placed round in the whole circle (IV. 1), an equilateral and equiangular quindecagon will be inscribed in it." Q.E.F. And in the same manner as was done in the pentagon, if through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon will be described about it; and likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it. BOOK V. DEFINITIONS. 1. A LESS magnitude is said to be a part of a greater magnitude, when the less measures the greater; that is, "when the less is contained a certain number of times exactly in the greater." 2. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, "when the greater contains the less a certain number of times exactly." 3. "Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity." 4. Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other. 5. The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth; or, if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth. 6. Magnitudes which have the same ratio are called proportionals. N.B. "When four magnitudes are proportionals, it is usually expressed by saying, the first is to the second, as the third to the fourth." L 7. 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. 8. 'Analogy, or proportion, is the similitude of ratios.” 9. Proportion consists in three terms at least, 10. 66 When three magnitudes are proportionals, the first is said to have to the third, the duplicate ratio of that which it has to the second. 11. When four magnitudes are continual proportionals, the first is said to have to the fourth, the triplicate ratio of that which it has to the second, and so on, quadruplicate, &c., increasing the denomination still by unity, in any number of proportionals. Definition A, to wit, of compound ratio. When there are any number of magnitudes of the same kind, the first is said to have to the last of them 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 has to B the same ratio which E has to F; and B to C the same ratio that G has to H; and C to D the same that K has to L; then, by this definition, A is said to have to D the ratio compounded of ratios which are the same with the ratios of E to F, G to H, and K to L. And the same thing is to be understood when it is more briefly expressed 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 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 the ratio compounded of the ratios of E to F, G to H, and K to L. 12. 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 that they continue still to be proportionals :" 13. 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; as is shown in Prop. 16 of this Fifth Book. 14. 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. B. Book V. 15. 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. Prop. 18. Book V. 16. 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. Prop. 17. Book V. 17. 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. E. Book V. 18. Ex æquali (sc. distantiâ), or ex æquo, from equality of distance; when there is any number of magnitudes more than two, and as many others such 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.” 19. 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 Prop. 22. Book V. 20. Ex æquali in proportione perturbatâ seu inordinatâ, 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 order; and the inference is as in the 18th definition. It is demonstrated in Prop. 23. Book V. AXIOMS. EQUIMULTIPLES of the same, or of equal magnitudes, are equal to one another. 2. Those magnitudes, of which the same or equal magnitudes are equimultiples, are equal to one another. 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. PROPOSITION 1.-Theorem. If any number of magnitudes be equimultiples of as many, each of each; 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, CD be equimultiples of as many others E, F, each to each. Then whatsoever multiple AB is of E, the same multiple shall AB and CD together be of E and F together. *Prop. 4. Lib. II. Archimedis de sphæra et cylindro. |