subject was treated, we have in Eucl. XIII. at least a partial recapitulatio the contents of the treatise of Aristaeus. After Euclid, Apollonius wrote on the comparison of the dodecahe and the icosahedron inscribed in one and the same sphere. This we learn from Hypsicles, who says in the next words following those a Aristaeus above quoted: "But it is proved by Apollonius in the se edition of his Comparison of the dodecahedron with the icosahedron that, a surface of the dodecahedron is to the surface of the icosahedron [insc in the same sphere], so is the dodecahedron itself [i.e. its volume] to icosahedron, because the perpendicular is the same from the centre of sphere to the pentagon of the dodecahedron and to the triangle of icosahedron." BOOK XIII. PROPOSITIONS. PROPOSITION 1. If a straight line be cut in extreme and mean r square on the greater segment added to the half of the is five times the square on the half. For let the straight line AB be cut in extreme an ratio at the point C, and let AC be the greater segment; let the straight line AD be produced in a straight line with CA, and let AD be made half of AB; I say that the square on CD is five times the square on AD. For let the squares AE, DF be described on AB, DC, and let the figure in DF be drawn ; let FC be carried through to G. Now, since AB has been cut in extreme and mean ratio at C, therefore the rectangle AB, BC is equal to the square on AC. [vi. Def. 3, VI. 17] L N M P H A K G And CE is the rectangle AB, BC, and FH the: on AC; therefore CE is equal to FH. And, since BA is double of AD, while BA is equal to KA, and AD to AH, therefore KA is also double of AH. But, as KA is to AH, so is CK to CH; therefore CK is double of CH. But LH, HC are also double of CH. Therefore KC is equal to LH, HC. But AE is equal to the gnomon MNO; therefore the gnomon MNO is also quadruple of AP; therefore the whole DF is five times AP. And DF is the square on DC, and AP the square on D therefore the square on CD is five times the square on DA Therefore etc. Q. E. D. The first five propositions are in the nature of lemmas, which are requi for later propositions but are not in themselves of much importance. It will be observed that, while the method of the propositions is that Book II., being strictly geometrical and not algebraical, none of the results that Book are made use of (except indeed in the Lemma to XIII. 2, which probably not genuine). It would therefore appear as though these propositi were taken from an earlier treatise without being revised or rewritten in light of Book II. It will be remembered that, according to Proclus (p. 67, Eudoxus "greatly added to the number of the theorems which originated w Plato regarding the section" (i.e. presumably the "golden section"); and i therefore probable that the five theorems are due to Eudoxus. That, if AB is divided at C in extreme and mean ratio, the rectan AB, BC is equal to the square on AC is inferred from vi. 17. AD is made equal to half AB, and we have to prove that whence, adding the sq. on AD to each, we have (sq. on CD) = 5 (sq. on AD). The result here, and in the next propositions, is really seen more readily by means of the figure of II. II. In this figure SR=AC+ AB, by construction; and we have therefore to prove that (sq. on SR) = 5 (sq. on AR). S A R K "What is analysis and what is synthesis. "Analysis is the assumption of that which is sought as if it were a < and the arrival > by means of its consequences at something adm be true. "Synthesis is an assumption of that which is admitted and the by means of its consequences at something admitted to be true." There must apparently be some corruption in the text; it does no case of synthesis, give what is wanted. B and V have, instead of "so admitted to be true," the words "the end or attainment of what is sou The whole of this addition is evidently interpolated. To begin w analyses and syntheses of the five propositions are placed all together MSS.; in P, q they come after an alternative proof of x111. 5 (which alt proof P gives after XIII. 6, while q gives it instead of XIII. 6), in B (w not the alternative proof of XIII. 5) after XIII. 6, and in b (in which wanting, and the alternative proof of XIII. 5 is in the margin, in the fir after XIII. 5, while V has the analyses of 1-3 in the text after XII those of 4-5 in the same place in the margin, by the second hand. the addition is altogether alien from the plan and manner of the The interpolation took place before Theon's time, and the probability it was originally in the margin, whence it crept into the text of P after Heiberg (after Bretschneider) suggested in his edition (Vol. v. p. lxxx it might be a relic of analytical investigations by Theaetetus or Eudo he cited the remark of Pappus (v. p. 410) at the beginning "comparisons of the five [regular solid] figures which have an equal s to the effect that he will not use "the so-called analytical investiga means of which some of the ancients effected their demonstrations.' recently (Paralipomena zu Euklid in Hermes XXXVIII., 1903) Heibe jectures that the author is Heron, on the ground that the sort of analy synthesis recalls Heron's remarks on analysis and synthesis in his com on the beginning of Book II. (quoted by an-Nairizi, ed. Curtze, p. 89) quasi-algebraical alternative proofs of propositions in that Book. To show the character of the interpolated matter I need only analysis and synthesis of one proposition. In the case of XIII. I substance as follows. The figure is a mere straight line. Let AB be divided in extreme and mean ratio at C, AC being the greater segment; D A and let I say that (Analysis.) "For, since AD AB. (sq. on CD) 5 (sq. on AD)," and (sq. on CD) = (sq. on CA) + (sq. on AD) + 2 (rect. CA, AD) therefore But and (sq. on CA) + 2 (rect. CA, AD) = 4 (sq. on AD). rect. BA.. (sq. on CA) = (rect. AB, BC). and therefore (sq. on AB)="(rect. BA, AC) + (rect. AB, BC), Adding to each the square on AD, we have (sq. on CD) 5 (sq. on AD). = PROPOSITION 2. If the square on a straight line be five times the squar a segment of it, then, when the double of the said segment is in extreme and mean ratio, the greater segment is the remain part of the original straight line. For let the square on the straight line AB be five ti the square on the segment AC of it, and let CD be double of AC; I say that, when CD is cut in extreme and mean ratio, the greater segment is CB. Let the squares AF, CG be described on AB, CD respectively, let the figure in AF be drawn, and let BE be drawn through. Now, since the square on BA is five times the square on AC, AF is five times AH. Therefore the gnomon MNO is quadruple of AH. A M H N K E F B And, since DC is double of CA, therefore the square on DC is quadruple of the square on that is, CG is quadruple of AH. But the gnomon MNO was also proved quadruple of therefore the gnomon MNO is equal to CG. And, since DC is double of CA, while DC is equal to CK, and AC to CH, therefore KB is also double of BH. |