APPENDIX I

ON THE NOTION OF ΜΕΣ0ΤΗΣ IN GREEK MATHEMATICS

In the Greek mathematical writers the term μεσότης is ambiguous. It signifies, first, the middle or middle space in general, as in the dictum ascribed to the Pythagorean Occelus (Diels-Kranz 48, 8): ή τριάς πρώτη συνέστησεν άρχήν, μεσότητα και τελευτήν (and cf. Aristotle, De Caelo 268al0, cited by Diels-Kranz 58, 17). Then, more specifically, μεσότης can signify a mean in the commonly accepted sense of that term in English, i.e. the middle term in a continuous three-term progression, the mid-point between ex-tremes. Thus, in English we say: "If three numbers are members of an arithmetic progression, the middle one is the arithmetic mean" (Van Nosfrond's Scientific Encyclopedia, 3rd ed. [Princeton, N. J., 1958], p. 1310, s.v. "Progression"). Likewise the Greek math-ematician can say that eight is the μεσότης between six and twelve in an harmonic progression (Diels-Kranz 44A24, line 32). This is obviously the sense in which Plato uses the term in Timaeus 36A. However, the notion of mean or middle term is more often ex-pressed by the Greek terms τό μέσον and ό μέσος δρος (e.g. Diels-Kranz 47B2 [I, 436, lines 10-11] and 58, 2 lines 28-30). Thus Plato uses τό μέσον for the mean in describing the nature of proportion (άναλογία) at Timaeus 31C-32A.

The second, and perhaps more common, significance of μεσότης is "a proportion" or "progression". Burnet remarks that "the Greek word does not mean only or even primarily the arithmetical mean: it is the oldest word for a proportion of any kind and however determined" (The Ethics of Aristotle [London, 1900], pp. 69-70). In this sense μεσότης becomes closely identified with άναλογία, though the latter term expressed primarily the notion of a geomet-

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APPENDIX I 345

ric proportion, as Iamblichus mentions in his commentary on Nicomachus' Introduction to Arithmetic (ed. H. Pistelli [Leipzig, 1894], p. 100, lines 15-18): "One should mention first that the an-cients applied the term άναλογία primarily to the geometric, but now it is applied more broadly to all the other μεσότητες generi-cally". Modern commentators, following the lead of G. H. F. Nesselmann (Geschichte der Algebra, Vol. I [Berlin, 1842], 210-212) accept this as historically accurate. Cf. J. Gow, A Short History of Greek Mathematics (Cambridge, Eng., 1884; reprint N.Y., 1923), p. 93, n. 1; T. L. Heath, The Thirteen Books of Euclid's Elements, 2nd. ed. revised (N.Y., 1956), II, 292-293. Thus M. L. D'Ooge, in a note on Nicomachus' Introduction to Arithmetic II, 21.2 (cited according to the edition of R. Hoche [Leipzig, 1866]), remarks that:

Two Greek words, άναλογία and μεσότης ... may be translated "proportion", and Nicomachus points out that an άναλογία is, strictly speaking, a combination of ratios (like 1, 2, 4, i.e. in his classification only the "geometrical proportions", γεωμετρικαν άναλογίαι or μεσότητες). Properly then an arithmetical progression of three or four terms (e.g. 1, 2,3, or 1,2,3,4) should not be called an άναλογία, but in practice it is so called. Nesselmann states that originally άναλογία was applied only to geometrical proportions and μεσότης to the other two, the harmonic and arithmetic, but that in later usage the distinction of terms vanished. It is certain that Nicomachus uses them indiscrim-inately of all three types ... (Nicomachus of Gerasa: Introduction to Arithmetic [N.Y., 1926], p. 264, n. 2).

This close identification of μεσότης and άναλογία must have taken place early, for according to Theon of Smyrna (Expositio rerum mathematicarum ad legendum Platonem utilium, ed. E. Hiller [Leipzig, 1878], p. 106, lines 15-20) the Peripatetic Adrastus at the beginning of the second century B.C. found it necessary to explain (as Iamblichus did later) that άναλογία properly refers to geo-metric proportion, though "more broadly the other μεσότητες are also called άναλογίαι by some people". Cf. Heath, op. cit., II, 292. The first three types of μεσότης — the arithmetic, geo-metric, and harmonic — are commonly traced back to Pythagoras himself, though Archytas and Hippasus are credited with changing the name of the last from "sub-contrary" to "harmonic" (Iam-blichus, ed. Pistelli, p. 100, lines 15-25); several other types of

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346 APPENDIX I

μεσάτης were known to Plato and his contemporaries {ibid., pp. 101, lines 1-5; 116, 1-4; 118, 19ff.); and the number of μεσάτητες was finally brought to ten by two later Pythagoreans, Myonides and Euphranor (ibid., 116, 4-6).

It is clear, then, that as practically synonymous with άναλογία, μεσάτης can only signify a "proportion" (i.e. a three-term progres-sion) and not a "mean'''; and when it has this sense of a three-term progression or "proportion", μεσάτης is distinguished from "the mean" as the whole from the part. In fact Nicomachus calls atten-tion to this distinction when he remarks {Intro, to Arith. II. 27.1 ; see D'Ooge, p. 278, n. 2) that μεσάτης (in the sense of a three-term progression) is well named because each type of μεσάτης is deter-mined by the specific nature of its μέσος ορος or mean, not by its extremes (άκρα). And Philoponus, commenting on this remark (ed. Hoche [Berlin, 1867], Book II, p. 32, XXVII, 1, lines 5-7), ex-plains clearly: "And for this reason the whole relationship of the three terms (ή δλη των γ ορων σχέσις) is called μεσάτης, from this fact that the μέσος [δρος] (mean or middle term) has the determining influence over the whole relationship of the terms."

Since μεσάτης is ambiguous, then, its significance in each case must be gathered from the particular context, so that it cannot be translated uncritically as "the mean" at every occurrence. For μεσάτης may signify in mathematics the whole relationship of two extremes joined by a mean, i.e. a "proportion". And applied analogously to physical realities, μεσάτης will obviously suggest a disposition in which extremes of any kind are balanced in a mean, i.e. a state of equilibrium among opposing factors.

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