Commentary on Proposition XIII.1¶
Back to Proposition XIII.1.
The first five propositions are in the nature of lemmas, which are required for later propositions but are not in themselves of much importance.
It will be observed that, while the method of the propositions is that of Book II., being strictly geometrical and not algebraical, none of the results of that Book are made use of (except indeed in the Lemma to XIII. 2, which is probably not genuine). It would therefore appear as though these propositions were taken from an earlier treatise without being revised or rewritten in the light of Book II. It will be remembered that, according to Proclus (p. 67, 6), Eudoxus “greatly added to the number of the theorems which originated with Plato regarding the section” (i.e. presumably the “golden section”); and it is therefore probable that the five theorems are due to Eudoxus.
That, if AB is divided at C in extreme and mean ratio, the rectangle 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 (sq. on CD) = 5 (sq. on AD).
The figure shows at once that □ CH = □ HL, so that □ CH + □ HL = 2 (□ CH) = □ AG. Also sq. HF = (sq. on AC) = rect. AB, BC = CE. By addition, (gnomon MNO) = sq. on AB = 4 (sq. on AD); 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. 11.
In this figure SR = AC + $\frac{1}{2}$ AB, by construction; and we have therefore to prove that (sq. on SR) = 5 (sq. on AR).
This is obvious, for (sq. on SR) = (sq. on RB) = sum of sqs. on AB, AR = 5 (sq. on AR).
The MSS. contain a curious addition to XIII. 1—5 in the shape of analyses and syntheses for each proposition prefaced by the heading: “What is analysis and what is synthesis. “Analysis is the assumption of that which is sought as if it were admitted <and the arrival> by means of its consequences at something admitted to be true. “Synthesis is an assumption of that which is admitted <and the arrival> by means of its consequences at something admitted to be true.” There must apparently be some corruption in the text; it does not, in the case of synthesis, give what is wanted. B and V have, instead of “something admitted to be true,” the words “the end or attainment of what is sought.”
The whole of this addition is evidently interpolated. To begin with, the analyses and syntheses of the five propositions are placed all together in four MSS.; in P, q they come after an alternative proof of XIII. 5 (which alternative proof P gives after XIII. 6, while q gives it instead of XIII. 6), in B (which has not the alternative proof of XIII. 5) after XIII. 6, and in b (in which XIII. 6 is wanting, and the alternative proof of XIII. 5 is in the margin, in the first hand) after XIII. 5, while V has the analyses of 1—3 in the text after XIII. 6 and those of 4—5 in the same place in the margin, by the second hand. Further, the addition is altogether alien from the plan and manner of the Elements. The interpolation took place before Theon’s time, and the probability is that it was originally in the margin, whence it crept into the text of P after XIII. 5. Heiberg (after Bretschneider) suggested in his edition (Vol. v. p. lxxxiv.) that it might be a relic of analytical investigations by Theaetetus or Eudoxus, and he cited the remark of Pappus (v. p. 410) at the beginning of his “comparisons of the five [regular solid] figures which have an equal surface,” to the effect that he will not use “the so-called analytical investigation by means of which some of the ancients effected their demonstrations.” More recently (Paralipomena zu Euklid in Hermes XXXVIII., 1903) Heiberg conjectures that the author is Heron, on the ground that the sort of analysis and synthesis recalls Heron’s remarks on analysis and synthesis in his commentary on the beginning of Book II. (quoted by an-Nairīzī, ed. Curtze, p. 89) and his quasi-algebraical alternative proofs of propositions in that Book.
To show the character of the interpolated matter I need only give the analysis and synthesis of one proposition. In the case of XIII. 1 it is in 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; and let $AD = \frac{1}{2}AB$, that is, AD = $\frac{1}{2}$ AB. I say that (sq. on CD) = 5 (sq. on AD). (Analysis.) “For, since (sq. on CD) = 5 (sq. on AD),” and (sq. on CD) = (sq. on CA) + (sq. on AD) + 2 (rect. CA, AD), therefore (sq. on CA) + 2 (rect. CA, AD) = 4 (sq. on AD). But rect. BA . AC = 2 (rect. CA . AD), and (sq. on CA) = (rect. AB, BC).
Therefore (rect. BA, AC) + (rect. AB, BC) = 4 (sq. on AD), or (sq. on AB) = 4 (sq. on AD): and this is true, since AD = $\frac{1}{2}$ AB.
(Synthesis.) Since (sq. on AB) = 4 (sq. on AD), and (sq. on AB) = (rect. BA, AC) + (rect. AB, BC), therefore 4 (sq. on AD) = 2 (rect. DA, AC) + sq. on AC.
Adding to each the square on AD, we have (sq. on CD) = 5 (sq. on AD).