HISTORIA MATHEMATICA 22 (1995), 119-137 Doubling the Cube: A New Interpretation of Its Significance for Early Greek Geometry KEN SAITO Faculty of Letters, Chiba University, Yayoi-cho, lnage-ku, Chiba, 263 Japan It is widely known that Hippocrates of Chios reduced the problem of doubling the cube to the problem of finding two mean proportionals between two given lines. Nothing, however, is known about how this reduction was justified. To answer this question, propositions and patterns of arguments in Books VI, XI, and XII of the Elements are examined. A reconstruction modelled after Archimedes' On Sphere and Cylinder, Proposition II-1, is proposed, and its plausibility is discussed. © 1995 AcademicPress, Inc. I1 est admis qu'Hippocrate de Chio a r6duit la duplication du cube au probl~me de la recherche de deux moyennes proportionnelles entre deux segments donn6s, mais rien n'est certain quant ~ la justification de cette r6duction. Pour r6pondre ~ cette question, sont exami- n6es les propositions comme les argumentations caract6ristiques des livres VI, XI, et XII des Elements. Une reconstruction sera propos6e d'apr~s De la sphere et du cylindre, proposition II-1 d'Archimbde, et son caract6re plausible sera discut6. © 1995 AcademicPress, Inc. Es ist wohlbekannt, daft Hippocrates von Chios das Problem der Verdoppelung des Wtirfels auf das Problem der Auffindung von zwei mittleren Proportionalen zwischen zwei gegebenen Geraden zurtickfiihrte. Jedoch ist nichts bekannt tiber die Rechffertigung dieser Reduktion. Um diese Frage zu beantworten, werden S~itze und Strukturen von Argumenten aus den Btichern VI, XI, und XII der Elemente untersucht. Eine Rekonstruktion wird dann vorgeschla- gen, die sich auf Satz II-1 in Archimedes Uber Kugel und Zylinder stiitzt, und ihre Plausibilitat wird diskutiert. © 1995 Academic Press, Inc. MSC 1991 subject classifications: 01A20, 51-03. KEY WORDS: Theory of proportion, Euclid, Archimedes. 1. INTRODUCTION The problem of doubling the cube is one of the central problems in Greek mathematics. Since it attracted many mathematicians of the period, we possess exceptionally rich ancient materials on this problem. As is well known, Hippocrates of Chios (ft. ca. 440 B.C.) is said to have originated the tradition of investigating cube duplication. Several sources credit him with reducing this problem to another, that of finding two mean proportionals between two given lines. For example, in his commentary to Archimedes' On Sphere and Cylinder (hereafter SC), Eutocius cites Eratosthenes: And it was sought among the geometers in what way one could double the given solid, keeping it in the same shape, and they called this sort of problem the duplication of the cube. And when they all puzzled for a long time, Hippocrates of Chios first conceived that if, for two 119 0315-0860/95 $6.00 Copyright © 1995 by Academic Press, Inc. All rights of reproduction in any form reserved. 120 KEN SAITO HM 22 given lines, two mean proportionals were found in continued proportion, the cube will be doubled. Whence he turned his puzzle into another no less puzzling) Hereafter we call this transformation of cube duplication into the problem of finding two mean proportionals "Hippocrates' reduction." It was in this reduced form that the solution of cube duplication was sought during antiquity. The traditions of these ancient studies have recently been so thoroughly studied by Knorr [9, 11-153] that it seems as if nothing further could be teased out of the extant materials. In this article, however, we concentrate on a problem about which almost all the ancient sources are strangely taciturn: how cube duplication was reduced to finding two mean proportionals between two given lines. Strangely enough, in contrast to the abundance of solutions for inserting two mean proportionals, we have little testimony on how Hippocrates' reduction was proved in antiquity. No extant text explains how Hippocrates arrived at this reduction. It would therefore seem worth- while to try to understand the kind of arguments he was likely to have made. 2 Throughout this paper, we will be less concerned with the heuristic context that motivated Hippocrates' work than with the justification of the discovery, the proof of the equivalence of cube duplication to finding two mean proportionals. Our concern will lead us to propose a new interpretation of the status of the theory of proportion in the early stage of Greek mathematics, where we will highlight a group of propositions which may have served as a tool for problem-solving (including that of cube duplication) for the mathematicians of the period. 2. THE PROBLEM AND ITS REDUCTION First, let us briefly analyze the problem of doubling the cube, taking into account ancient testimonies and modern studies on its reduction. Cube duplication is a problem that appears to have been originally stated as follows: [CD] To find a cube which is twice as large (in size, not side) as a given one. Fairly early on, however, this problem was studied in the following generalized form: [CDx] To find a cube whose ratio to a given cube equals the ratio of two given lines (not necessarily in the ratio 2:1) (see also [8, 23]). This generalized problem was then reduced to the following problem: 1 [5, 3:88]. We cite Knorr's translation in [9, 147]. Knorr persuasively asserts the genuineness of Eratosthenes' accounts. See [8, 17-24] and [9, 131-153]. 2 It might be doubted if Hippocrates really proved the truth of his reduction, since Eratosthenes simply says that Hippocrates "first conceived" the truth of the reduction. Therefore, the reader with maximum reserve should read the phrase "Hippocrates' proof (or justification) of his reduction" in this paper as "the first rigorous proof of Hippocrates' reduction." This proof was surely found very soon, if not by Hippocrates himself, since it was concerned with such an important problem as cube duplication. Archytas already contrived a method to insert two mean proportionals [5, 3:84-88; 9, 100-110], so that we may assume that the proof was already available to Archytas. It seems to us that the attribution of both the discovery and the proof of the "reduction" to Hippocrates himself fits better with other testimonies which rank him as one of the great mathematicians of the period. HM 22 DOUBLING THE CUBE 121 [CDxr] To find two mean proportionals between two given lines. Thus given two lines a, b, one must find x, y, such that a : x = x : y = y : b. The last problem, [CDxr], was usually investigated by the ancients under the name of cube duplication, although strictly speaking, this only covers the case b = 2a. We have a few sources that demonstrate the truth of Hippocrates' reduction, and they are based on the use of the concept of triplicate ratio (as used in a proposition like Elements XI-33), or the compounding of ratios. We can confirm this in Diocles [15, 102], Pappus [6, 1:66-68] (see also [9, 90-91]) and Philoponus [18, 104-105] (for translation, see [9, 20]). We cannot, however, regard these argu- ments as reflecting the first proof of Hippocrates' reduction for reasons that will be explained below. Some of the modern reconstructions follow the same lines as these authors of late antiquity. For example, Knorr explains the equivalence of cube duplication [CDx] and its reduction [CDxr] as follows: If for any two given lines, A and B, we can insert the two mean proportionals, X and Y, then A : X = X : Y = Y: B. Thus, by compounding the ratios, one has (A : X) 3 = (A : X)(X : Y)(Y : B), that is, A 3 : X 3 = A : B. Thus, X will be the side of a cube in the given ratio (B : A) to the given cube (m3). [8, 23] In this passage, Knorr is less interested in restoring Hippocrates' line of thought than in convincing modern readers of the truth of Hippocrates' reduction. Indeed, he does not pretend to restore any ancient argument. Using modern notations, he assumes the equivalence of triplicate ratio with the operation of compounding the same ratio three times, 3 and he further assumes that the triplicate ratio of two lines is equal to the ratio of the cubes on them. The truth of the former, though evident to us, is not explicitly proved in the extant Elements, while the latter is a special case of Elements XI-33. 4 In earlier studies [12; 13], we have argued that the concepts of duplicate and triplicate ratio, as well as that of compound ratio, appear to have been relatively late arrivals in the corpus of Greek mathematics, and that the same holds true for Proposition XI-33. If this thesis is correct, then these notions were certainly not available to Hippocrates when he formulated his reduction of cube duplication. It would, therefore, be worthwhile to search for a proof without these notions. ~ In this context, we should note another way to formulate cube duplication which must have been clear to Greek geometers from the time of Hippocrates. The problem [CDx] can easily be represented in the following more geometric form: [CDxg] Given a rectangular prism on a square base, to find a cube equal to it. 3 Knorr is far from the first to make this assumption. For example Heath, in his account of Hippocrates' reduction, states that "Hippocrates could work with compound ratios" [3, 1: 200]. 4 Another modern reconstruction in [1, 57ff.] is based on the corollary of XI-33. 5 In this respect, two modern reconstructions [16, 119; 14, 128] are to be noted. These authors do not resort to multiplicate and compound ratios, but use instead more fundamental theorems. However, as they did not intend to offer historical reconstructions, they offered no textual evidence for these arguments.