Greek Geometrical Analysis

Abstract

THIS article offers an interpretation of the passage in Pappus describing geometrical analysis (Collection, vii., Preface, pp. 634-6 in Hultsch; full text also in Thomas, Greek Mathematical Works, II, pp. 596-9, Loeb). The principal difficulty is that Pappus appears to give two different accounts of the direction of the analysis. (i) as an upward movement to prior assumptions from which an initial assumption follows (ev IL'V y&'p -T 0oV0F.XUxcC iT4. xCXX0UtV (FVOAsLv). (ii) as a downward movement of deduction from an initial assumption (8vTv. npo6fX-%toc). Two interpretations have been offered. The first accepts (ii) as the proper formulation of what the Greeks called geometrical analysis, and explains (i) as merely an alternative way of describing (ii). The second accepts (i) as the proper formulation, and explains (II) as merely an alternative way of describing (i). The first is the commonly accepted interpretation. Lucid and detailed accouats of the method on this interpretation are given by Heath (The Thirteen Books of Euclid's Elements, I, pp. I37-142), Robinson (Mind, N.S. XLV, I936, pp. 464-473), and Cherniss (Review of Metaphysics, IV, i9gi, pp. 4I4-4I9). The method is one of assuming to be true a geometrical proposition which it is required to prove, or assuming a geometrical problem to be solved, and, by analysis, deducing con-sequences until one reaches either a proposition known independently to be true, or a construction which it is possible to satisfy, or a proposition known to be false, or a construction which it is impossible to satisfy. In the first case, the last step in the analysis becomes the first in the synthesis, which repeats the steps of the analysis in reverse order until the original assumption is reached and so proved. In the second case one may conclude, without resort to the synthesis, that the original assumption is false, or the solution impossible. A requirement of the method is that the implications at each step are reciprocal. The second part (ii) of Pappus' account seems, quite clearly, to be describing this method. So also do the definitions of analysis and synthesis interpolated in Euclid XIII (see Heath, op. cit. p. I 38). Moreover, as Robinson emphasizes, there are excellent examples of its use in Archimedes, in Pappus, in the alternative proofs of XIII, i-g, interpolated in Euclid, and elsewhere (loc. cit. pp. 469-472; see also Heath, op. cit., I, pp. I4I-142). That this, and no other method is what the Greek geometers called analysis is, says Robinson (Plato's Earlier Dialectic,