258 
THE DUPLICATION OF THE CUBE 
Then AR . RG = PR 2 , 
or (a + x){r — x) = y 2 . 
Also, by similar triangles, 
PR:RG = 80:0G 
= OG : OB ; 
(1) 
or 
y 
r—x 
(2) 
From the equation (1) we obtain 
x 2 + y 2 + ax 
T = ^ 5 
a + x 
and, by multiplying (1) and (2), we have 
by (a + x) = ry 2 , 
whence, substituting the value of r, we obtain, as the locus of 
P, a curve of the third degree, 
h{a + x) 2 — y{x 2 + y 2 + ax). 
The intersection [M) of this curve with the axis of y gives 
Oil/ 3 = a 2 b. 
As a theoretical solution, therefore, ‘ Plato’s ’ solution is 
more difficult than that of Menaechmus. 
(£) Eratosthenes. 
This is also a mechanical solution effected by means of 
three plane figures (equal right-angled triangles or rectangles) 
which can move parallel to one another and to their original 
positions between two parallel rulers forming a sort of frame 
and fitted with grooves so arranged that the figures can 
move over one another. Pappus’s account makes the figures 
triangles, 1 Eutocius has parallelograms with diagonals drawn ; 
triangles seem preferable. I shall use the lettering of Eutocius 
for the second figure so far as it goes, but I shall use triangles 
instead of rectangles. 
1 Pappus, hi, pp. 56-8.