196 THE ELEMENTS DOWN TO PLATO’S TIME 
line [AW] of length such that the square on it is 1-| times the 
square on AK between the circumference of the semicircle and 
CD in such a way that it will verge {yevtLv) towards B ’ [i.e. if 
produced, will pass through B\ This is a problem of a type 
which the Greeks called revertis, inclinationes or vergings. 
Theoretically it may he regarded as the problem of finding 
a length (x) such that, if F be so taken on CD that BF = x, 
BF produced will intercept between CD and the circumference 
of the semicircle a length EF equal to . AK. 
If we suppose it done, we have 
EB.BF=AB.BC = AK 2 ; 
or x [x 4- . a) = a 2 (where AK = a). 
That is, the problem is equivalent to the solution of the 
quadratic equation 
x 2 + ax = a 2 . 
This again is the problem of ‘ applying to a straight line 
of length V . a a rectangle exceeding by a square figure and 
equal in area to a 2 ’, and would theoretically be solved by the 
Pythagorean method based on the theorem of Eucl. II. 6. 
Undoubtedly Hippocrates could have solved the problem by 
this theoretical method; but he may, on this occasion, have 
used the purely mechanical method of marking on a ruler 
or straight edge a length equal to V|. AK, and then moving 
it till the points marked lay on the circumference and on Cl) 
respectively, while the straight edge also passed through B. 
This method is perhaps indicated by the fact that he first 
places EF (without producing it to B) and afterwards 
joins BF. 
We come now to the last of Hippocrates’s quadratures. 
Eudemus proceeds:] 
‘Thus Hippocrates squared every 1 (sort of) lune, seeing 
that 1 (he squared) not only (1) the lune which has for its outer 
1 Tannery brackets irdvra and dnep mi. Heiberg thinks (l.c, p. 348) 
the ivording is that of Simplicius reproducing the content of Eudemus. 
The wording of the sentence is important with reference to the questions 
(1) What was the paralogism with which Aristotle actually charged 
Hippocrates? and (2) What, if any, was the justification for the charge ? 
Now the four quadratures as given by Eudemus are clever, and contain in 
themselves no fallacy at all. The supposed fallacy, then, can only have 
consisted in ¿n assumption on the part of Hippocrates that, because he