Book IV. 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 equilate. ral and cquiangular quindecagon shall be described about it: And likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circum. Icribed about it. THE Ε L Ε Ε TS A I. ' II. greater is measured by the less, that is, 'when the greater III. · Ratio is a mutual relation of two magnitudes of the fame See N. • kind to one another, in respect of quantity.' IV. V. the second, which the third has to the fourth, when any e- of, Book V. or, if the multiple of the first be greater than that of tho second, the multiple of the third is also greater than that of VI. ally expressed by saying, the first is to the second, as the VII. fifth definition) the multiple of the first is greater than that VIII. IX. X. have to the third the duplicate ratio of that which it has to XI. See X. When four magnitudes are continual proportionals, the first is said to bave to the fourth the triplicate ratio of that which it Definition A, to wit, of compound ratio. the first is said to have to the last of them the ratio com- magnitude. kind, the first A is said to have to the last D the ratio com- And And if A has to B, the fame ratio which E has to F; and B Book V. to C, the same ratio that G has to H; and C to D, the same N the same ratio which A has to D; then, for shortness fake, XII. one another, as also the consequents to one another. XIII. Permutando, or alternando, by permutation, or alternately ; See N, this word is used when there are four proportionals, and it XIV. and it is inferred, that the second is to the first, as the fourth XV. als, and it is inferred, that the first, together with the fe• XVI. it is inferred, that the excefs of the first above the second, is XVII. als, Book V. als, and it is inferred, that the first is to its excefs above the second, as the third to its excess above the fourth Prop. E. book 5. XVIII. stance ; when there is any number of magnitudes more than XIX. when the first magnitude is to the second of the first rank, book 5 XX. quality, in perturbate or disorderly proportion †; this term is E QUIMULTIPLES of the same, or of equal magnitudes, are equal to one another. II. Those |