without depending on any other known author as an intermediary. His text is notable for its
introductory material on bloodletting, as well as for its lack of literary pretension.
CHG vv.1– 2 passim; G. Björck, “Griechische Pferdeheilkunde in arabischer Überlieferung,” Le monde
oriental 30 (1936) 1–12; McCabe (2007) 245–258.
Anne McCabe
Hippokrate ̄s of Khios (440 – 400 BCE)
In his history of geometry, P mentions Hippokrate ̄s of Khios together with T-
K as distinguished people in geometry, and tells us that Hippokrate ̄s was
the first person said to have written elements of geometry (In Eucl. p. 66.4–8 Fr.). E
tells us that Hippokrate ̄s reduced the problem of constructing a cube twice the size of a
given one to the finding of two mean proportionals between two given straight lines x and y
(In Arch. 88.17–23), and Proklos says that he was the first person to reduce outstanding
geometric questions to other propositions (In Eucl. p. 212.24–213.11 Fr.). It appears that he
also concerned himself with questions of natural philosophy, since A tells us (Mete.
1.6 [342b35–343a20]) that those around Hippokrate ̄s and also his pupil A gave
an account of the tail of a comet as an optical illusion and explained the rareness with
which one appears; Aristotle also implies (1.8 [345b10–12]) that they also considered the
Milky Way to be an illusion.
However, the most mathematically interesting material relating to Hippokrate ̄s concerns
quadrature. In the Physics (1.2 [185a14–17]), Aristotle mentions an attempt to square the
circle “by means of segments” as a false inference from true geometrical principles. The
ancient commentators on the Physics passage all attribute this attempted quadrature to
Hippokrate ̄s of Khios. The commentators express uncertainty about what the quadrature
was, but it is now generally accepted that S provides our best information about it
in his comment on the Aristotle passage (In Phys. = CAG 9 [1882] 60.22–68.32) in which
Simplicius adds his own explanatory material to E’ reworking of Hippokrate ̄s’
argument. The argument has several problematic features, but I shall discuss only the three
major ones after giving my own formulation of Hippokrate ̄s’ quadratures. The first prob-
lematic aspect of the argument is Hippokrate ̄s’ assumption that:
If a and b are similar segments of circles on the bases α and β, then a is to b as the
square with side α [SQ(α)] is to SQ(β).Simplicius implies that Hippokrate ̄s proved this principle, as he could have, from an
equivalent of E’s Elements 12, prop. 2, according to which circles are to one another as
the squares on their diameters. However, Simplicius also says that Hippokrate ̄s proved this
proposition, which Euclid proves by the method of exhaustion, a method almost certainly
unavailable to Hippokrate ̄s.
Hippokrate ̄s applies his principle in squaring three “lunes,” the shaded plane figures in figures
1, 2, and 3a, which are contained by arcs of two circles. In particular, in figure 1, the lune ABCD
is contained by the semicircle ACB and the arc ADB, which is similar to the arcs cut off by
the equal straight lines AC and CB; in figure 2 arc ADCB is greater than a semicircle, DC is
parallel to AB, and SQ(AB) = SQ(AD) + SQ(DC) + SQ(CB); and in figure 3a ADCB is less
than a semicircle, DC is parallel to AB and equal to AD and CB, and such that AC and DB
intersect at E with SQ(AE) = 3/2 × SQ(AD). In order to carry out the construction of this
HIPPOKRATE ̄S OF KHIOS