third quadrilateral ADCB Hippokrate ̄s starts from a semicircle FGD’ with diameter FC’D’
and center C’. He bisects C’D’ at H and draws HJ perpendicular to C’D’. He then deter-
mines B’ on the semicircle FGD’ and E’ on HJ by using a so-called neusis (verging) construc-
tion in which B’E’ is placed in such a way that it “verges” toward D’ and so that SQ(B’E’) is
3/2 × SQ(B’C’). He draws B’K parallel to D’F and connects C’ to B’ and to E’. He extends
C’E’ to meet B’K at A’, and connects D’ to E’ and A’. Then A’D’ is equal to B’C’, which is
equal to C’D’. So A’B’C’D’ is the desired quadrilateral. Simplicius does not discuss this neusis
construction so we can only conjecture how Hippokrate ̄s carried it out. One might think of
the verging argument as a matter of marking a line or ruler LN at a point M so that
SQ(LM) = 3/2 × SQ(B’C’), then moving the line around until a position is found in which L
lies on the circumference of the semicircle FGD’, M lies on HJ and the line passes through
D’. The construction can also be carried out using a fairly complicated application of
areas.
Having squared these three lunes, Hip-
pokrate ̄s shows how for any circle A’B’C’-
D’E’F’ one can construct a square equal to
it plus a lune AGCB which is constructed as
follows. ABCDEF and A’B’C’D’E’F’ are
taken as circles with center O arranged as
in figure 4 with A’B’C’D’E’F’ a regular
hexagon, with SQ(AD) = 6 × SQ(A’D’),
and with the circular segment AGC similar
to the segment a on A’B’. Scholars in gen-
eral have been dubious about whether
Hippokrate ̄s believed or even claimed that
he had squared the circle when he had
shown how to square any circle plus a par-
ticular lune and how to square particular
members of three classes of lunes into
Hippokrates of Khios, 1 © Mueller
Hippokrate ̄s of Khios, 2 © Mueller
HIPPOKRATE ̄S OF KHIOS
