Leonid Zhmud The Origin of the History of Science in Classical Antiquity

History 1, 221f.; Mueller, I. Aristotle and the quadrature of the circle, Infinity
and continuity, 146–164; Knorr. AT, 26f.
167
Lloyd, G. E. R. The alleged fallacy of Hippocrates of Chios,
Apeiron 20 (1987)
103–128.

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4. Early Greek geometry according to Eudemus
203
According to Eudemus, Hippocrates, being an expert in geometrical con-
structions, 1) was the first to apply the method of reduction (@pagwg2, one of
the methods foreshadowing analysis) to complex constructions; 2) in particu-
lar, was the first to reduce the problem of doubling the cube to finding two mean
proportionals in continuous proportion between two given magnitudes; 3) dis-
covered the quadrature of lunes; and 4) was the author of the first
Elements.
168
Proclus defines @pagwg2 as a “transition from a problem or a theorem to an-
other that, if known or constructed, will make the original proposition evident”
and identifies it with the method Hippocrates employed to solve the problem of
doubling the cube (
In Eucl., 212.24–213.11). The problem itself was con-
nected, of course, not with the demand of the Delphic oracle to double the altar
on Delos, but with Pythagorean mathematics.
169
The Pythagoreans solved the
problem of squaring a rectilinear figure by finding the mean proportional
x be-
tween two lines (
x =
√
ab). This problem had to be followed, naturally, by that
of finding two mean proportionals between two lines. It is to this latter problem
that Hippocrates reduced the duplication of the cube using the method of @pa-
gwg2.
170
The attempt at squaring a circle followed, in turn, from the squaring of the
rectangular figure considered in book II of the
Elements.
171
Of course, Hippo-
crates could not solve the problem of squaring a circle. He succeeded, however,
in squaring three lunes – figures limited by two circular arcs.
172
According to
Simplicius,
In book II of the
History of Geometry Eudemus says the following: “The quadra-
tures of lunes, which were considered to belong to an uncommon class of prop-
ositions on account of the close relation (of lunes) to the circle, were first inves-
tigated (ëgráfhsan) by Hippocrates, and his exposition was thought to be in
correct form.
173
It is worth noting that Eudemus refers here to an opinion of specialists, and not
to that of Aristotle, who erroneously believed that Hippocrates pretended to
have solved the problem of squaring a circle. Eudemus also points out that the
solution offered by Hippocrates was of a general character:
Eudemus, however, in his
History of Geometry says that Hippocrates did not
demonstrate the quadrature of the lune on the side of a square (only), but gen-
erally, as one might say. For every lune has an outer circumference equal to a
168
1) Eutoc.
In Archim. De sphaer., 88.18–23 (from Eratosthenes); 2) fr. 133, 140;
3) Procl.
In Eucl., 213.7–11; 4) ibid., 66.4f. = fr. 133.
169
Heath.
History 1, 200f.; Knorr. AT, 23f.
170
See Saito K. Doubling the cube: A new interpretation of its significance for early
Greek geometry,
HM 22 (1995) 119–137.
171
Neuenschwander. Beiträge, 127; cf. above, 199 n. 150.
172
The fact that, using compasses and a ruler, one can square only five types of closed
circular lunes was demonstrated only in the last century.
173
Fr. 140, p. 59.28–60.2 Wehrli, transl. by T. Heath.

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Chapter 5: The history of geometry
204
semicircle or greater or less, and if Hippocrates squared the lune having an outer
circumference equal to a semicircle and greater and less, the quadrature would
appear to be proved generally.
174
Simplicius further adds that he is going to give a literal quotation from Eude-
mus on the squaring of lunes, expanding, for clarity’s sake, his “brief proofs in
the ancient manner” (ibid., 59.26). Interestingly, it is precisely where Eudemus
promised “to deal with the quadratures of lunes at length and to go through
them” (ëpì pléon âyømeqa te kaì diélqwmen, ibid., 60.1) that Simplicius
characterizes his account as brief. In fact, Eudemus gave a concise description
of some of Hippocrates’ demonstrations and quoted some others verbatim,
which, from Simplicius’ point of view, was still not complete enough.
175
The
reconstruction of Hippocrates’ solution reported by Eudemus has already given
rise to a vast literature, and there is no point in considering it here in detail.
176
I
would like to note only that the considerable length of the text that Eudemus de-
voted to the squaring of lunes (about 4.5 pages of a Loeb format) allows us to
estimate the length of his