PROPOSITION 16. PROBLEM. To inscribe an equilateral and equiangular quindecagon in a given circle. Let ABCD be the given circle: it is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD. Let AC be the side of an and let AB be the side of an Then, of such equal parts as the whole circumference E ABCDF contains fifteen, the arc ABC, which is the third part of the whole, contains five, and the arc AB, which is the fifth part of the whole, contains three; therefore their difference, the arc BC, contains two of the same parts. Bisect the arc BC at E; [III. 30. therefore each of the arcs BE, EC is the fifteenth part of the whole circumference ABCDF. 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 for 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 also, as for 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, that the third has to the fourth, when any equimultiples whatever of the first and the third being taken, and any equimultiples whatever of the second and the 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, and 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, and 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. When four magnitudes are proportionals it is usually expressed by saying, the first is to the second as the third is to the fourth. 7. When of the equimultiples of four magnitudes, taken as in the fifth definition, the multiple of the first is greater than the multiple 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 has to the fourth; and 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. 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. [The second magnitude is said to be a mean proportional between the first and the third.] 11. When four magnitudes are continued 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 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 which is 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 that E has to F; and B to C the same ratio that G has to H; and C to D the same ratio 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 that A has to D; then, for the sake of shortness, 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 said to be 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 the magnitude of proportionals, so that they continue still to be proportionals. 13. Permutando, or alternando, by permutation or alternately; when there are four proportionals, and it is inferred that the first is to the third, as the second is to the fourth. V. 16. 14. Invertendo, by inversion; when there are four proportionals, and it is inferred, that the second is to the first as the fourth is to the third. V. B. 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. V. 18. 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. V. 17. 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 is to its excess above the fourth. V. E. 18. Ex æquali distantia, 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. This term is used simply by itself, when the first magnitude is to the second of the first rank, as the first is to the second of the other rank; and the second is to the third of the first rank, as the second is to the third of the other; and so on in order; and the inferenco is that mentioned in the preceding definition. V. 22. 20. Ex æquali in proportione perturbatâ 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 the second is to the third of the first rank, as the last but two is to the last but one of the second rank; and the third is to the fourth of the first rank, as the last but three is to the last but two of the second rank; and so on in a cross order; and the inference is that mentioned in the eighteenth definition. V. 23. AXIOMS. 1. 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. |