300 PLATO 
described, is equal to the given area, that area is inscribed in 
the form of a triangle in the given circle. 1 
In order, therefore, to inscribe in the circle an isosceles 
triangle equal to the given area (X), we have to find a point E 
on the circle such that, if ED be drawn perpendicular to AB, 
the rectangle AD . DE is equal to the given area X (‘ applying ’ 
to AB a rectangle equal to X and falling short by a figure 
similar to the ‘ applied ’ figure is only another way of ex 
pressing it). Evidently E lies on the rectangular hyperbola 
1 Butcher, after giving the essentials of the interpretation of the 
passage quite correctly, finds a difficulty. ‘ If’, he says, ‘ the condition ’ 
(as interpreted by him) ‘ holds good, the given yw/nW can be inscribed in 
a circle. But the converse proposition is not true. The ycopiW can still 
be inscribed, as required, even if the condition laid down is not fulfilled; 
the true and necessary condition being that the given area is not greater 
than that of the equilateral triangle, i. e. the maximum triangle, which 
can be inscribed in the given circle.’ The difficulty arises in this way. 
Assuming (quite fairly) that the given area is given in the form of a rect 
angle (for any given rectilineal figure can be transformed into a rectangle 
of equal area), Butcher seems to suppose that it is identically the given 
rectangle that is applied to AB. But this is not necessax-y. The termi- 
nology of mathematics was not quite fixed in Plato’s time, and he allows 
himself some latitude of expression, so that we need not be surprised to 
find him using the phrase ‘ to apply the area (ywpioj/) to a given straight 
line ’ as short for 4 to apply to a given straight line a rectangle equal (but not 
similar) to the given area ’ (cf. Pappus vi, p. 544. 8-10 pg ndv to boQiv 
7rapa rr)v Sodeiaav Trapa^dWeadai eWelnov tit pay a>va>, ‘ that it is not every 
given (area) that can be applied (in the form of a rectangle) falling short 
by a square figure’). If we interpret the expression in this way, the 
converse is true; if we cannot apply, in the way described, a rectangle 
equal to the given rectangle, it is because the given rectangle is greater 
than the equilateral, i. e. the maximum, triangle that can be inscribed in 
the circle, and the problem is therefore impossible of solution. (It was 
not till long after the above was written that my attention was drawn to 
the article on the same subject in the Journal of Philology, xxviii, 1908, 
pp. 222-40, by Professor Cook Wilson. I am gratified to find that my 
interpretation of the passage agrees with his.)