Pappus, Plato and the Harmonic Mean

Pappus, Plato and the Harmonic MeanAuthor(s): Malcolm BrownSource: Phronesis, Vol. 20, No. 2 (1975), pp. 173-184Published by: BRILLStable URL: http://www.jstor.org/stable/4181963 .

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Pappus, Plato and the Harmonic Mean MALCOLM BROWVN

There is a well known construction by Pappus of the three means, anrthmetic, geometric and harmonic, which produces all of them in the same semicircle.' But Pappus is only rehearsing the

argument, not claiming credit for it; his intent is to expose an error on the part of its author, whom he leaves nameless. The argument is defective, Pappus says, in omitting to prove that the third line it constructs is the harmonic mean between the given lines. Heath and Bulmer-Thomas have shown2 how the argument can be supplemented by a few steps so that its third mean is indeed shown to be the harmonic mean between the given lines. Unhappily, such supplementing does not meet Pappus' challenge in its own form. For by using some extra steps to say what the anonymous author leaves unargued, one shows only that the geometrical length atrived at is the right one; but in the process he confirms the thing Pappus is complaining of, namely that the argument is incomplete as it stands, so to speak has the wrong logical length.

The first purpose of this paper is to vindicate the argument in Pappus' source in the more fundamental way, by showing that it is complete as it stands. The key to doing this will be a mathematical reconstruction of an alternative definition of the third mean, the definition which stands behind the archaic term "subcontrary," which Archytas tells us is another name for it. The second purpose will be to do some historical reconstruction, to restore some of the context of this argument. Since all of our authorities ascribe the theory of the three means to early Pythagoreans, in the era before Archytas, it is natural to ask whether this defining construction of all three conies from that source. Some arguments will be given for answering in the affirmative, and it will be shown how from this and other evi- dence from Pythagorean writers between Archytas and Pappus, a better understanding can be reached of how Pappus' own error is a natural one. If both the logical and the historical reconstructions are

I Greek Mathematical WVorks, ed. and tr. I. Thomas, Loeb Classical Library (Cambridge, USA, 1939, 1941) 2, pp. 568-571. 2 T. L. Heath, Hist. ol Greek Math. (Oxford, 1921) 2, pp. 364-365; I. Bulmer- Thomas, see Thomas of note 1 above, p. 571 note a.

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right, this early Pythagorean way of systematizing the three means will be seen to furnish an extra bit of background for our understanding of these same means in the Timaeus.

I

First let us review the construction of the "someone" (rn) and the particulars of Pappus' criticism. (For convenience let us simply refer to the anonymous author whose argument Pappus rehearses and criticizes as "the author.") The author begins by laying out the pair of given magnitudes which are the extremes to be middled all three ways. He sets them out as a pair of lines AD and DG end to end, thus in effect adding the two extremes to form the sum-line ADG. Let us call the extremes a and b, accoiding to the usual modern conventions, a > b. Now, after bisecting the sum-line ADG at point E, he constructs a semicircle on the whole, as in fig. 1. Clearly he has already constructed

the arithmetic mean (let us refer to this as M.), since EG, or indeed any of the semicircle's i adii, being equal to half the sum of the extremes (i.e. (a + b)/ 2), has the required length. Now the second mean is easily constructed by means of a line perpendicular to ADG and intersecting the semicircle at B. The reasoning is straightforward and standard3 by which DB is shown to be the geometric mean (Mg) be tween the given extremes (i.e. V~ab). After joining EB with a radius (thus EB, like EG and EA, will also be Ma), the construction continues by dropping a perpendicular onto EB from D, thus determi- ning the poing Z as in fig. 2:

A D D G E

fig. 1

3 Euclid preserves two constructions, those of II.14 and VI.13, which are essentially the same as the author's for the first two means here. Quite likely neither one was new in Euclid's time, though Heath wavered on this question (Excursus I to his edition of Book II makes all of it early Pythagorean, but Math. in Aristotle, p. 193 makes 11.14 new in Euclid's time).

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B

z

A E D G

fig. 2 (see I. Thomas, ed., Greek Mathematical Works, 2, p. 569)

We are now ready for both the argument's conclusion and Pappus' criticism. The author claims to have already produced all three of the required means, the arithmetic mean EG (or EB), the geometric mean DB, and the harmonic mean BZ. Pappus concedes him the first two, but charges him with falling short of proving the third:

"But how ZB is a harmonic mean, or between what kind of lines, he did not say, but ornly that it is a third proportional (6ovov 8 6'nt 'pLr'- &vaXoyOv E'TLv) to EB, BD." (I.Thomas, ed., op. cit., 2, p. 571.)

Pappus' complaint is clearly of a logical order: the author has failed to complete his argument, by failing to show the fact that BZ, in addition to being third proportional in relation to the other two means, also has the "harmonic" relation to the two extremes directly. "He does not know (&yvo&v)," Pappus continues, "that from EB, BD, BZ, which are in geometrical proportion [i.e. BZ is the third proportional to the other two means, in that order] the harmonic mean is formed." (ibid.) We must not see anything "epistemic" in the alleged failure to know here. In all probability the author knew as well as Pappus did that his BZ, as a matter of mathematical fact, had the right size relations to the given lines. That is not really at issue. Pappus is charging a logical or argumentative error, namely that the subject argument unaccountably stops short of showing that the "harmonic" relation holds between BZ and the given lines. Indeed it does not so much as relate the constructed line to the original givens at all; rather, it is content to express the third mean, as we should say, as a function of the other two. Thus the author "officially" does not know his conclusion, even if in fact he knew it as well as Pappus and Heath and Bulmer-Thomas, and in his Alexandrian way Pappus appears to be entirely right.

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II

Let us proceed to vindicate the author against Pappus. We may begin by noticing that nothing prevents the author's having held that the third mean was defined by its relation to the other two, in a given order. There is even something positive to recommend this possibility: Iamblichus' reviews an interpretation of the third mean according to which it is precisely the meaning of its being "subcontrary" that it relates this way to the other two means, taken in series. It being a non-Nicomachean view of the meaniing of the third mean's "subcontrariety," Iamblichus is unsympathetic to it. But the important thing for the present argument is that this position is there in the Pythagorean tradition to be unsympathetically reacted to by Iam- blichus. For the position refuted by lamblichus is precisely that which our author's argument needs to be fully vindicated: that the third mean is essentially (i.e. is defined by being) subcontrary to the other two means in a certain order, where subcontrariety is further explained as entailing the proportional sequence Ma, Mg, Third. Yet rather than conclude at this stage that the author is indeed working from precisely this definition of the third mean, of which there are surviving echoes in the late Pythagorean tradition, let us take a fresh start, and produce a more genieral construction of the mean called "subcontrary." The idea will be both to give this notion of subcontrariety a more general mathematical setting, and to locate an early source independent of the Pythagorean tradition, on which to build.

We propose to ourselves to find the third mean between a given paii of magnitudes whose Ma and Mg we have already determined. Thus the problem is, given Ma and Mg between a pair of extremes, to find the third mean in that same interval. We first construct the two known means as line segments AB, BC sharing a terminal point B. Let us postpone proving that Ma > Mg, which is necessarily true unless the extremes originally given were equal; for the moment let us simply assume it, so that in our construction AB > BC. An angle will have been formed, as in fig. 3. Except that ABC must not lie along a single straight line, no restriction need be made on how this angle is constructed; it may be of any size, acute right or obtuse. Next, join the end points A, C, thus forming the triangle ABC. Now thirdly, construct an angle at C equal to the one at A, and draw up the

4 lamblichus, In Nicomachi arithmeticam introductionem, ed. H. Pistelli (Leipzig, 1894), 110.26-111.1.

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B DA

B Let BA = Ma

\ \ / o BC = 1\1

6H\ ,(thus, since '-I> Mg, BA > BC)

fig. 3

liie so formed till it intersects BA at some point D. It is easy to show that this new angle, BCD, will fall inside the original triangle ABC, the basic step in the argument being the very elemen- tary (and probably pre-Euclidean also) theorem6 thatin any triangle the angle subtending the lesser side is itself lesser. Since the two triangles are equiangular, they are similar, so that the sides about their equal angles are pr-oportional. In other words, what this construction effects is the construction of a third proportional to two lines originally set out. BD is that third proportional. And since AB and BC were set out as the Ma and Mg we started from, BD will be the third proportional to those two means, in that order.

The relatively early (over half a millennium before Pappus) and independent source from whom we may draw to connect this generalized construction to the idea of subcontrariety is Apollonius of Perga. Apollonius uses the term6 "subcontrary" to designate the relation, in an inessential way generalized, between the pair of triangles in our construction (fig. 3's triangles CBD and ABC), which is also the relation of the author's pair of triangles (fig. 2's DBZ, DBE). Two special facts about Apollonius' use of the term are noteworthy. First he takes it for granted that this technical term is sufficiently familiar and intelligible that it can designate the "subcontrary section" of the cone. Secondly, when he uses the notion in his own argument, he supplies an explanatory paraphrase (a rou-Ti=t) identical

I 1. Bulmer-Thomas, in a recent paper reviewing the whole question of what is original in Euclid's Elements, concedes all of the propositions enunciated in Bk 1, where this proposition is proved, to authors before Euclid: "Ci6 che e originale e ci6 che b derivato negli Elementi di Euclide," Atti 1973, Accademia .Nazionale dei Lincei, Quaderno N. 184, Rome, 1973, pp. 49-64. See esp. pp. 58-59. n The LSJ entry on the mathematical meaning of S'cvavrEoq was contributed ly Heath, LSJ s.v. 4b. The only two examples listed are those whose background is here being argued to be the same: the examples are from Pappus and Archytas.

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to the step taken in our generalized construction of the third, or "subcontrary," mean: "that is, so that the angle AKG is equal to the angle ABC" (see fig. 4).

Now admittedly Apollonius' application of the term is less restricted than that of either of the two cases otherwise similar. That is, while the author's argument (fig. 2) requires both that the "inside, but backwards" triangle be a right triangle, and that the two triangles share two vertices, and while the other construction sketched (fig. 3) has the second of these requirements only, Apollonius' pair of triangles are under neither restriction. His triangles need not either be right nor share more than one vertex. Yet it is clear that the differences in generality are inessential to the fundamental thing named by the term "subcontrary," the "inside, but backwards" relation of the smaller to the larger similar triangle. For that depends only upon what is common to all the cases, namely that the smaller triangle have its angle equal, but oppositely placed, to the angle of the larger.

It is worth noting that, although it does not affect the basic notion of "subcontrariety," the differences in generality among the constructions do correspond to differences in the mathematical motiva- tions of the arguments. Apollonius does not need the sides of his triangles to give three terms in geometric series for what he is proving about the subcontrary conic section (that it is a circle). Nor does anything in his argument depend upon the "oppositely placed" angle's being right. Our two arguments, on the other hand, must both have the three-term geometric series of its triangles' sides. Is there anything gained in the author's restriction to right triangles? Yes there is, and the advantage is Pythagorean in the definite sense that it

A

G

K

B C

fig. 4 Conica I.5 (after Great Books of the Western World, vol. 11, p. 608)

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makes the Pythagorean theorem directly applicable to getting a needed result, in a single step. It is the very result, in fact, which we postponed consideration of above, namely that in all nontrivial cases Ma is necessarily greater than Mg. One need only recall that in the author's construction the three means, arithmetic, geometric and subcontrary are arranged as successive sides of the pair of triangles, in this case nrght triangles. Thus since Ma is the hypotenuse of the larger triangle and Mg one of its legs, by the Pythagorean theorem it follows at once that Ma > Mg. Repeating the same argument in the subcontrary triangle, one gets the relation Mg > Mh (Mh for harmonic mean). Thus the overall inequality M. > Mg > Mh is available at once, and based on a thoroughly Pythagorean argument. It may be of significance that, in a proportionalized form, this serial relation of the three means is just what Archytas goes on to say about the three means in Fr. 2, as soon as he has finished defining them. That is, if Archytas knew the argument of the author, he would have had a sound, and characteristicallv Pythagorean, argument by which to formi a logical connection between the two parts of his remarks, (1) the definitions of the three means, and (2) their arrangement in order of size.7 Perhaps it is no accident that this is the same fragment in which he calls the third mean "subcontrary."

Let us return to the matter of vindicating the author's argument, reinforced by this technical meaning of "subcontrariety." It will be seen that, although Archytas offers no alternative definition behind his alternative name "subcontrary" for the third mean, this background definition is reconstructable with the aid of Iamblichus and Apollonius, and of Archytas himself. And the reconstruction matches

7A third point of contact between our author and Archytas is the "Archytan Procedure" by which Tannery showed that he set up converging series of rational approximations to pure quadratic surds. I have discussed this and its connection with Plato's Theaetetus in an article in J. [Iist. of Philosophy 7 (1969) 370 ff. The condition now stated as Eucl. X, 1 - namely that the remainder after a given stage of an antanairesis process must be less than half the remainder at its immediately preceding stage - must be satisfied by the Archytan Procedure if it is to be rigorous in identifying its irrational Mg between successive pairs of Ma, Mh. But this condition is seen to be satisfied directly from the author's diagram. For the starting interval is the difference between the given extremes, i.e. a-b. And the interval remaining after the first stage is Ma-Mh, or the line ZE (see fig. 2), which is already constructed in his figure. But the size (a-b)/2 is also there in the figure, and is the line ED. The condition is therefore satisfied at once, because the hypotenuse of a right triangle is necessarily greater than either leg.

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rather neatly the argument which Pappus attributes to his "someone." We need only suppose that the author aimed at showing the third mean in its archaic form, that is as "subcontrary" to the other two means, to vindicate his argument as it stands. For in this case its relation to the other two means is what defines it as the third of the standard three means.

III

Can we advance any reasonable historical hypothesis to explain the difficulty of reconstructing the subcontrary definition of the third mean, thereby enabling us to see how Pappus' having over- looked it was a natural mistake? It is not sufficient to appeal to the length of the time interval between pre-Archytan Pythagoreans and Pappus, though it is over half a millennium. Many importantly similar pieces of mathematics did in fact survive that interval, even without benefit of attachment to canonized works like those of Plato or Euclid. Philolaus B 6, on the musical proportion, and Archytas' own fragment (from his On Music) are cases in point, and Pappus seems to have had available another rich source of this sort of material in the form of Eratosthenes' On Means, two books, part of the so- called Treasury of Analysis of which Pappus writes.8 There is nothing to prevent a specimen's surviving incompletely, as it appears to have done in our case, the argument itself intact, but a key part of its mathematical motivation missing.

There is a very early sign of the process of obscuring the "subcontrary" detinition of the third mean, in the same fragment from Archytas which gives it that name. For it is he who, after introducing this mean under the name "subcontrary," goes on to say (and repeat) that "we call" it something else, namely "harmonic." lamblichus, when reviewing the early history of Pythagorean tesearch into means, makes this shift by Archytas into a turning-point of the development. He sets off the era beginning with Archytas and Eudoxus and ending in Eratosthenes from its predecessor by reference to exactly this shift in the understanding of the third mean. From the same report we learn that Eudoxus took the name "subcontrary" and suggested that it be used to designate the fourth of his six means rather than,

8 For a discussion of the Treasury of Analysis, see Heath, History 2, pp. 399-427. The reference to Eratosthenes On Means, which of course could be Pappus's source for the author's argument, is on p. 401 of Heath.

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as previously, the third. Indeed if we press a detail of Iamblichus' report here, we may even get an indication that the old "subcontrary triangles" construction behind the third mean had already been eclips- ed by Eudoxus' time. For the reported reason why Eudoxus proposed his new use of the term was that his fourth mean is "peculiarly" (tMwo: lamblichus, p. 101.9) subcontrary. Is the third mean not peculiarly so? Indeed on one very common understanding of it, it is not. There is much neo-Pythagorean worry about this, in the form of puzzlement over just what the "contrariety" of the third mean as against the arithmetic mean alone, can amount to.9 Why would any contrariety one found relating the third mean to the arithmetic mean alone fail to make the latter as contrary (or subcontrary) as the former? Possibly it was just this embarrassment which induced Eudoxus to propose shifting the term to the fourth mean, which could be so designated peculiarly. But whether or not this was his reason, or indeed whether or not he even had a reason, the net result of this shift in designation of the term within the theory of means will have been to advance the process of obscuring the pre-Archytan definition of the third mean. Thus it would not be surprising if, with so many intervening causes tending to obscure the old definition, and with Apollonius' use of the term in its archaic sense appearing in a seemingly unrelated context, by Pappus' time it would have been easy to miss the point of an argument which relied on that archaic construction.

IV

In section II above, it was remarked that if Archytas knew this bit of pre-Archytan mathematics, he would have had an easy and characteristically Pythagorean way of connecting the two parts of his statement about the three means, and both in turn with the so- called Archytan procedure (p. 179, with n. 7). We may now venture to say something parallel about Archytas' friend Plato, and his

9 There is difficulty, apart from that revealed in Iamblichus' controverting the view of "some" that the third mean is subcontrary to both others. Nico- machus is sure it is the arithmetic mean to which the third has this relation, but he offers no explanation of the "sub-" prefix, only of the "contrariety," thus has no answer to the difficulty which seems to have bothered Eudoxus. In his explanation (II, xxiii, 6) he even drops the prefix and describes its relation as kvav'icaq only. Philoponus (Eis to deuteron tes Nikomachou Arith- metikes Eisagoge, pt. III, R. Hoche ed., p. 25) leaves the matter confused when he describes the arithmetic mean as ivavm'dw to the geometric mean; a few pages later he makes it 6ncv=(ciq to the third (p. 29).

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Timaeus. Plato's enthusiasm for the theory of three means is clear from a pair of passages in that dialogue, where he defines all of them. Tim. 31 C-32 A, which defines what we have called Mg, echoes one third of Archytas' Fr. 2, and Tim. 36 A B, which defines both Ma and Mh, echoes the remaining two thirds of the same fragment. The geometric mean and the proportion which it characterizes is Plato's way of bonding the components of the world's body to one another and to itself. The remaining pair of means are Plato's way of bonding the units of the world soul to one another and to itself. Now the specimen of mathematics which we have been examining, which defines the third mean in terms of its relation to the other two, gives an easy way of binding these two sorts of binding to one another. Thus the question naturally arises whether Plato may have had this higher level of unification in mind in this part of the Timaeus. Let us examine this possibility biiefly before concluding.

Suppose we take quite strictly the extremely general terms in which Plato formulates the scope of the unification his cosmology is seeking as a foundation. The binding or unifying relation must achieve a sort of ultimacy (6rXt La'ro: 31 C 3), so that its result should be the unification of all things (tv 7xa'wx oarxL: 32 A 6-7). The forrmal conditions of the somatic and psychic unifications, then, are to be special cases of this foundational principle of cosmic unification. Now it is true, and possibly known to Plato, that the basic relation among the three means expressed by our pre-Archytan argument does provide a way of directly unifying this pair of unifiers with one another and with this relation.There is some extra interest in carrying through such an extra binding operation in that the rule according to which it is done is called, in this same context, tuXoyfatsaoca (Timaeus Locrus, 96 B 5) and in fact exhibits a close formal analogy to syllogistic's fundamental operation.'0

Let us set out in order the relation with which Plato begins (the somatic proportion) and the relation with which our author connects the three means (the subcontrary relation):

'? Gilbert Ryle's interpretation of the Timaeus Locrus, in this journal, vol. 10 (1965), reprinted in Ryle, Collected Papers (London, 1971), 1, pp. 72-88, makes it an extract from Timaeus written by Aristotle during Plato's lifetime, the same work as was entitled in a late report To& &x ro5i TL,LcEOU K xmi9v 'ApXu-r'CEv. It is certainly true that this work's version of the construction of world's body and soul (95 A-96 C) is closer to Archytas' Fr. 2 than Plato's own dialogue is, especially in that it compares (1) with (3) before beginning the construction (95 B 4-C 1).

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(1) the somatic proportion a Mg Mg : b (= Tim. 32 A 1-2)

(2) the subcontrary proportion Ma Mg: Mg: b (-our author)

It is significant that the rule for combining proportions does not permit us to combine these two in their present form. Rather, for an ex aequali operation to be allowed on these, an operation must be performed first which inverts (or converts) the order of terms within each of the ratio pairs on one line. If one is also to have all the terms appear either in a descending or ascending order of size, he will also have to transpose the entire ratios (term-pairs) to the reverse side of the proportion sign. Thus doing this preliminary operation on (1) amounts to this:

(1) a : Mg : : Mg : b (1') b: Mg:: Mg: a (by conversion and transposition)

Now (1') does "syllogize" with (2), as can be seen from this schema:

(1') b : Mg:: Mg: a (2) Mg : Mh : : Ms : Mg (from (2), by transposition)

(3) b: Mh:: Ma: a, or, in decreasing order of size, (3') a : Ma : : Mh : b

The rule of combination, AL' taou, or ex aequali, is defined in Book V of Euclid (Def. 17, aliter) this way, W &?Xc- Ai4L4 T&JV &Xp(&)V XOM' U'rn-

tatprmv scov ,ukawv. This is not the place to go into the formal analogy between this "cancellation of middles" operation and syllogistic.11 What should be noted, however, is that (3) is the proportion Plato uses to determine the arrangement of the world soul, Tim. 36 A B. It should also be noted that the first thing Plato does in the text of Titn. 32 A, after presenting the definition (1) above is to invert and transpose it to form (1'). The text runs: 6ocomp so6 7rp&ov7 p6p auso, T05To aU,T6 7rp6q r6 F-aXT'OV [=1j .t 7Z'XLV 06&L4 [1], trtl - 6 aXao-o 11 The work of F. Sommers and D. Massie in formalizing syllogistic has had recourse to arthimetical notions very similar to this pre-Aristotelian pattern of "cancellation of the middles." The historical tracing of antecedents of Ari- totle's own notions in syllogistic to systems of reckoning, and its mathematical terms, was begun by B. Einarson (Amer. J. of Philology, 1936). But he did not include the so called Di' isou rule; further research in this direction might reveal more of the historical roots of the Organon. I have read a paper "An Ancestor of Syllogism in thLe Timaeus" before the American Philos. Assn. (Pacafic Division) March. 1975.

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7tpog T& ligaOV, T6 1160V 7tp'9 'r6 7 prov. (32 A 1-3). And for this unification process to go tllrough (1') and (2) to (3), it is strictlyr required that "the middle [Mg] become first and last" as the next clause of text says that they do (32 A 3-4). Moreover an anomaly is removed if one supposes that the purpose of transforming (1) into (1') is precisely to prepare to combine it with the subcontrary relation (2). For (1') by itself does not have the feature claimed in A 3-4; its middle (Mg) has not exchanged positions with first and last terms. On the other hand if one takes (1') in connection with (2) the text's claim is saved. For what stands first in (1) stands - and must stand - last in (1'), as must its last become first. And what stands as the middle of both (1) and (1') stands - and must stand - first and last in (2). Thus the subcontrary relation allows one to bind and unify Plato's bond for the world's body to that of the world's soul.12

In summary, then, this paper has worked out two reconstructions. Mathematically it reconstructed a definition of the third mean which, if supplied to the argument which Pappus reviews and criticizes, vindicates it as it stands. Although this conception of the third mean gives every appearance of being identifiable with the one which we know from Archytas was of early date, the mathematical reconstruction was kept independent of the history of the matter, as next presented. The historical reconstruction followed up the suggestions found both in Pythagorean writers and independently in Apollonius, and concluded that the argument is to be dated to a time before Archytas and Plato. Taking both reconstructions as firm, it was then shown how they might be helpful in reading the world's body and world's soul passages of Timaeus as parts of a more fundamental unity. '3

Brooklyn College, City University of New York

It is worth notice that by the same rule of reckoning Plato's two proportions (1) and (3) combine at once to form (2), and (2) combines the same way with (3) to yield (1). The last of these might be taken to be a mathematical trace of the idea that although in one way body precedes soul L(1) precedes (3)1, in another way the order is reversed. Compare Tim. 34 B 10-35 A 1. 's I am grateful to the City University of New York Faculty Research Foundati- on for a grant supporting this research, I am grateful to Ian Mueller, Carl Boyer and a reader for Phronesis for valuable criticisms of an earlier draft of this paper.

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Pappus, Plato and the Harmonic Mean