vi 
I 
192 THE ELEMENTS DOWN TO PLATO’S TIME 
1 Therefore the lune, having been proved equal to the triangle, 
can be squared. 
‘In this way, assuming that the outer circumference of 
the lune is that of a semicircle, Hippocrates easily squared 
the lune. 
‘ Next after this he assumes (an outer circumference) greater 
than a semicircle (obtained) by constructing a trapezium in 
which three sides are equal to one another, while one, the 
greater of the parallel sides, is such that the square on it is 
triple of the square on each one of the other sides, and then 
comprehending the trapezium in a circle and circumscribing 
about (= describing on) its greatest side a segment similar 
to those cut off from the circle by 
the three equal sides.’ 
F 
[Simplicius here inserts an easy 
proof that a circle can be circum 
scribed about the trapezium. 1 ] 
‘ That the said segment [bounded 
by the outer circumference BAGI) 
ircumterence 
in the figure] is greater than a 
semicircle is clear, if a diagonal 
be drawn in the trapezium. 
‘For this diagonal [say BC], 
subtending two sides [BA, AC] of 
the trapezium, is such that the 
square on it is greater than double 
the square on one of the remain 
ing sides.’ 
D 
B 
[This follows from the fact that, AG being parallel to 
BD but less than it, BA and DG will meet, if produced, in 
a point F. Then, in the isosceles triangle FAG, the angle 
FAC is less than a right angle, so that the angle BAG is 
obtuse.] 
‘ Therefore the square on [BD] the greatest side of the trape 
zium [= 3 GD 2 by hypothesis] is less than the sum of the 
squares on the diagonal [BG] and that one of the other sides 
1 Heiberg (Philologies, 48, p. 340) thinks that the words sal on fuv 
'7Tfpi\r]<p6t](TeT(U kvkXco to Tpane^iov beitjeis [oonov] Stxoropijans rhs rod Tpcme£iov 
yonins (‘Now, that the trapezium can be comprehended in a circle you 
can prove by bisecting the angles of the trapezium ’) may (without ovras— 
F omits it) be Eudemus’s own. For on pev , . . forms a natural contrast 
to on St pci(.ov ... in the next paragraph. Also cf. p. 65. 9 Diels, rnvuov