Homogeneity in Eudoxus's Theory of ProportionAuthor(s): Ian MuellerSource: Archive for History of Exact Sciences, Vol. 7, No. 1 (29.XII.1970), pp. 1-6Published by: SpringerStable URL: http://www.jstor.org/stable/41133314 .

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Homogeneity in Eudoxus' s Theory of Proportion Ian Mueller

Communicated by B. L. van der Waerden

Another case is the theorem about proportion, that you can take the terms alternately; this theorem used to be proved separately for numbers, for lines, for solids, and for times, though it admitted of proof by one demon- stration. But because there was no name comprehending all these things as one - I mean numbers, lengths, times, and solids, which differ in species from one another - they were treated separately. Now, however, the pro- position is proved universally; for the property did not belong to the subjects qua lines or qua numbers, but qua having a particular character which they are assumed to possess universally.

(Aristotle, Posterior Analytics A.5.74al7-25.) In this passage Aristotle apparently contrasts the theory of proportion

which Eudoxus formulated in the second quarter of the fourth century B.C. and its predecessor or predecessors. In the latter the law Ά :B =C'D => A:C = B:D' was proved separately for different species, but Eudoxus found one term for all these species and gave one proof for all of them. The theory of Eudoxus has come down to us as book V of Euclid's Elements, and the general term used there is μέγεθθ5, 'magnitude/ In the above passage and in others where he touches on the same subject (Ibid. A.24.85a33-bl, Metaphysics M.2.1077a<M2), Aristotle mentions no term perhaps because when he wrote no term was yet fixed or perhaps because he preferred to restrict the term 'magnitude* to geo- metric objects.

The point which Aristotle stresses in our passage is Eudoxus's unification of the theory of proportion. Eudoxus picked out a genus or category into which all objects standing in ratios fall. The Eudoxian theory of proportion is not for Aristotle the study of ratios between numbers, lines, planes, etc. although it can, of course, be applied to such ratios. In itself, however, it is the study of ratios involving one kind of thing, which Aristotle does not name, but which I following Euclid will call magnitudes. To understand Aristotle's conception of these objects it is necessary to understand his conception of mathematical objects in general.

Aristotle's account of mathematical objects1 is not entirely clear, but the crucial notions which it involves are abstraction (άφ&ιρεϊν) and separation

1 For a fuller discussion of this topic and the important passages in Aristotle, see M. D. Philippe, "Άφαίρεσις, πρόσθεσις, χωρίζειν dans la philosophie d'Aristote," Revue Thomiste, XLVIII (1948), 465 - 79, and my essay "Aristotle on geometric objects," to appear in Archiv für Geschichte der Philosophie. 1 Arch. Hist. Exact Sei., Vol. 7

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2 I. Mueller:

(χωρίζειν). There are no mathematical objects which actually exist separated from sensible objects, as the Platonists maintain. However, the mathematician is able to conceive such objects by abstracting from - i.e., leaving out of con- sideration - the sensible properties of sensible objects. He treats these con- ceptions as separately existing things even though they are not. Thus, although the objects of the theory of proportion are not " substance which is neither number nor points nor spatial magnitude nor time" (Metaphysics M.2.1077all-12, Ross translation), the mathematician reasons as if it were. He therefore reasons in terms of a very general notion of quantity or magnitude.

Commentators on book V of the Elements place great emphasis on the dis- tinction between theorems which presuppose the homogeneity of the four magni- tudes involved and theorems which do not.2 Two magnitudes are homogeneous if they are capable of having a ratio to one another: in Euclidean terms, if they "are capable, when multiplied, of exceeding one another" (V.def.4, Heath translation). Thus triangles and squares are homogeneous; triangles and lines are not. The law of alternation presupposes homogeneity, whereas V.I 7 (A +B:B = C +D:D => A:B = C:D) is true even if A, B are lines and C, D triangles. As we shall see, Euclid distinguishes very carefully between the two kinds of laws. However, Aristotle's conception of the theory of proportion completely obliterates the distinction. For him the theory involves abstraction from all differences between magnitudes. Thus the objects of the theory are necessarily homogeneous.

The question arises whether Aristotle's account of the theory of proportion is a philosophically biased misinterpretation or whether it reflects a feature of Eudoxus's original theory. Prima facie the latter alternative seems more likely, since Aristotle probably entered the Academy at a time when Eudoxus was its leader.3 In any case, the work of Eudoxus must have been studied carefully in the Academy while Aristotle was there. Moreover, although Eudoxus had apparently heterodox ideas about Platonic forms,4 he was trained by Plato in philosophy; and probably his ideas about mathematics included a Platonic emphasis on the abstractness of mathematics. Indeed, book V itself has been taken as evidence of Plato's influence on Eudoxus.5

The remarks in the previous paragraph establish the plausibility of the supposition that Eudoxus construed the theory of proportion in the way Ari- stotle did, and that therefore Eudoxus did not worry about the question of homogeneity of terms. In the remainder of this paper I want to provide positive evidence for this supposition by showing that book V contains traces of having been reworked by Euclid on the basis of an original in which questions of homo- geneity were ignored.

2 See, for example, F. Beckmann, "Neue Gesichtspunkte zum 5- Buch Euklids," Archive for History of Exact Sciences, IV (1967), 1 - 144.

3 See, for example, I. During, Aristotle in the Ancient Biographical Tradition (Göteborg, 1957), ΡΡ· 159 - 60.

4 Aristotle, Metaphysics, A.9.99la8-19- 5 O. Becker, "Eudoxos- Studien I. Eine voreudoxische Proportionlehre und inre

Spuren bei Aristoteles und Euklid," Quellen und Studien zur Geschichte der Mathematik, Astronomie, und Physik, II. Β (1933), 328.

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Eudoxus's Theory of Proportion 3

I begin by recalling the development of the theory of proportion for numbers in book VII of the Elements. First Euclid proves laws having to do with the relation of part or parts to addition and subtraction (VII. 5-8). Then he proves what we might call alternation for part or parts (VI 1. 9, 10). Then he proves the basic laws of subtraction and addition for proportion, using respectively 7, 8 and 5, 6:

VII.ll A+C:B+D = C:D =φ A:B = A + C:B +D (or, A:B = C:D

^A-C:B-D = A:B).

VII.12 A1:B1 = -.-=Au:Bn^A1:B1 = (A1 + ...+AJ:(B1 + -..+BJ.

Finally Euclid uses 9 and 10 to prove alternation. There are, of course, fundamental differences between the theory of pro-

portion in book V and the arithmetic theory just described. However, there are also significant correlations between the two theories. VII. 5 and VII. 7 correspond respectively to :

V.I A1 = tnB1&...&An = mBn^A1 + ...+An = tn(B1 + .-.+Bn).

V.5 A=mC&B = mD ^A-B = m{C-D).

And VII.ll, 12, 13 are identical with V.19, 12, 16 respectively. V.16 (alternation) is proved differently from its correlate VI I.I 3, which is to be expected on account of the different notions of ratio involved. V.I and 5 are derived exactly as their analogues are; and the derivation of V.I 2 from V.I corresponds closely to that of VII.12 from VII.5 and 6. However, V.19 is not derived from V.5 as VII.ll is from VII. 7 and 8. Indeed, V.5 is never used in the Elements at all. V.19 is derived instead from V.I 7 (A +B:B = C+D:D ^A:B = C:D) as follows:

Suppose A+C:B+D = C:D.

Then A + C:C=:B+D:D. (V.16) So A:C = B:D. (V.I 7)

Hence A: B = C:D. (V.16) So A : Β = A + C : Β + D . (V. 1 1 [transitivity]) .

Quite clearly, V.I 7 is derivable from V.19 as well:

Suppose A +B:B = C+D:D. Then A +B:C+D = B:D. (V.16)

So A:C = A+B:C+D. (V.19) Thus A:C = B:D. (V.ll)

So A:B = C:D. (V.16).

Euclid derives V.I 7 directly from propositions V.I and 2. Similarly he derives V.18 (A:B = C:D => A +B:B = C +D:D) from V.ll, 14, 17 although it would

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4 I. Mueller:

be directly derivable from V.I 2 using alternation and transitivity, just as V.I 7 is derivable from V.I 9·

There is no mystery about why Euclid does not derive V.I 7 and 18 from V.I 2 and 19 as suggested above. For 17 and 18 are true even if A and Β are not homogeneous with C and D; but V.I 2, 16, 19 presuppose homogeneity. So, for anyone who pays attention to presuppositions of homogeneity in laws of pro- portions, the alternative proofs of 17 and 18 suggested above would fail to establish those laws in their full generality. Euclid's avoidance of these proofs is the clearest indication we have of his paying attention to presuppositions of homogeneity.

However, there is clear evidence that book V is derived from an original by an author - presumably Eudoxus - who did not pay attention to such pre- suppositions. For, first of all, the proofs of 17 and 18 contain logical deficiencies. The proof of 17, as Beckmann has pointed out,6 involves an outright fallacy. In proving 18, Euclid makes the notorious assumption of the existence of a fourth proportional. And although Eudoxus perhaps made use of this assumption in other proofs (e.g., in a proof of XII. 2), the assumption does not occur else- where in book V. Moreover, the assumption of a fourth proportional is illegitimate if two of the magnitudes involved are numbers. For it is fallacious to infer A:B = m:k for some k from the assumption Α. Β Φ m: η. Thus, if Eudoxus interpreted his theory in the way Aristotle did, he could not legitimately have proved V.I 8 as Euclid does. Finally, the assumption of a fourth proportional is a nonconstructive assumption of the kind which mathematicians often make in looking for proofs. But, as Becker has made clear,7 book V except for the proof of 18 is thoroughly constructive, even though the definition of propor- tionality lends itself directly to nonconstructive inferences using the assumption : 'Either A:B<C:D or A:B = C:D or A:B>C:D.' Thus both 17 and 18 give clear indications of being reworked in their proofs.

A second reason to believe 17 and 18 were originally proved using alternation is the fact that they are given right after alternation is proved even though 17 presupposes only 1 and 2 and neither presupposes alternation. Moreover, when we reconstruct the original proof of 19, we use two of the four otherwise unused propositions of book V - 5 and 6 (mA- nA = (m- n)A)s plus the law 'A>B^C>D&A:B = C:D => A -C>B-D,' which is equivalent to a third unused proposition - V.25 (A >B ^C>D&A :B = C:D => A +D> Β +C). The remaining unused proposition in this book, V.23, is a relatively advanced law (A:B = E:F&B:C = D:E =$> A:C = D:F) which may have been included for its own sake. Certainly its presence in book V is not nearly so mysterious as that of the other three. It remains to give the proof of V.I 9- I first prove the equivalent of V.25 in the formulation

I. C>D =ï(A+C:B+D = C:D ^A>B). 6 „Neue Gesichtspunkte . ..," 83- 7 "Eudoxos- Studien IV. Das Prinzip des ausgeschlossenen Dritten in der griechi-

schen Mathematik," Quellen und Studien zur Geschichte der Mathematik, Astronomie, und Physik, III.B (1936), 371-75-

8 V.6 is stated in a less specific way by Euclid, but this formulation states what Euclid actually proves.

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Eudoxus's Theory of Proportion 5

I shall give the proof of this theorem in modern notation. In Euclidean style the proof is quite complicated, but everything in it is elementary. The proof is perhaps more elementary than the proof of V.8 and certainly simpler than the proof reconstructed by Becker for that theorem in the pre-Eudoxian theory of proportion he reconstructs.9

Suppose C>D and A^B. Let kA > C and / (C - D) > (k + 1 ) C. ( V.def .4)

Thus jC -jD >(k + i)C. (V.5)

Obviously / > k + 1

and jC -kC>jD +C

i.e., (j-k)C>jD+C. (V.6) So (j-k)C-C>jD.

Let m be the least integer such that

(j-k)C-C^mD. (V.def. 4)

Obviously m > / but also (/ - k)C>mD because if (j - k)C^mD

(j - k)C -C <mD -D = {m - ')D and m is not least. However (/ -k)C -mD ^ C < kA <(m - (j -k)) A =mA - (j - k) A

[V.6]^mB-(j-k)A. Thus we have (j -k)C -mD <mB - (j - k)A

and (/ - k)A +(/ - k)C <mB +mD Thus (j -k)(A +C) <m(B +D) (V.I) and (j -k)C>mD. SoA+C:B+D + C:D™ (V.def. 5)

Now the proof of V.I 9 is straightforward.

Suppose A +C:B+D = C:D.

Case 1 . Suppose m (A + C) > η (Β -f D) Then mC>nD (V.def. 5) s.ndm(A+C):n(B+D)=mC:nD (V.4) i.e., mA +mC:nB+nD = mC:nD. (V.I, 7, 11)

Thus mA > nB .

9 "Eudoxos- Studien I ..." 311 - 33. 10 I would like to thank John Snapper for the original version of this proof,

which I have modified here.

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6 I. Mueller: Eudoxus's Theory of Proportion

Case 2. Suppose m {A + C) = η (Β + D) .

Then mC = nD ( V.def . 5) and mA +mC = nB+nD. (V.I)

Subtracting equals from equals gives mA = nB .

Case 3 . Suppose η (Β + D) > m (A + C) .

As in case 1 we have

m(A +C):n(B+D)=mC:nD.

Using V.7 porism,11 we get

n(B+D):tn(A +C)=nD:mC. Now we proceed as in case 1 .

This reconstruction of the original Eudoxian theory of proportion supports the following hypothesis about the development of the treatment of ratios in the fourth century. Prior to Eudoxus the Greeks had no sound general theory of proportion but only a theory in which certain laws had to be proved by sub- division into special cases according to the objects involved in a proportion. Eudoxus produced a general theory of proportion, construing the theory in a Platonic-Aristotelian way as the study of purely abstract entities which even- tually were called magnitudes. Because of the way he construed the theory, he did not concern himself with considerations of homogeneity; all objects of the theory of proportion were for him homogeneous. He therefore proved the crucial propositions of book V in the following order : 1, 5, 25 (or the equivalent I. above), 12, 19, 16, 17, 18.

Mathematicians after Eudoxus concerned themselves with questions of homogeneity. Someone, presumably Euclid, realized that considerations of homogeneity rendered the proofs of 17 and 18 undesirable. He therefore changed those proofs to the ones which we find in the Elements. However, he also saw that 19 could be proved simply from 18 and 16, enabling him to omit the com- plicated proof of 25. But, in accordance with his custom throughout the Elements, Euclid did not drop the now unnecessary 5, 6, and 25. Rather, he retained 5 and 6 in their original place and added 25 to the end of the book, deriving it straightforwardly from 19.

11 This porism is almost universally rejected as an interpolation. It is, however, used by Euclid in proving V.20, 21, 23, 24. Thus, if it is an interpolation, Euclid, and presumably Eudoxus, must have considered the law Ά:Β =C:D =Φ Β: A =D:C to be an obvious fact which did not need proof. See C. Thaer, Die Elemente von Euklid (Leipzig, 1933), Π, 69-

Department of Philosophy The University of Chicago

(Received March 3, 1970)

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