298 
PLATO 
* yes ’ or ‘ no ’ for an answer, but leading up to the geometrical 
construction of V 2. Starting with a straight line AB 2 feet 
long, Socrates describes a square ABCD upon it and easily 
shows that the area is 4 square feet. Producing the sides 
AB, AD to G, K so that BG, DK are equal to AB, AD, and 
completing the figure, we have a square of side 4 feet, and this 
square is equal to four times the original square and therefore 
has an area of 16 square feet. Now, says Socrates, a square 
8 feet in area must have its side 
greater than 2 and less than 4 feet. 
The slave suggests that it is 3 feet 
in length. By taking N the 
middle point of DK (so that AN 
is 3 feet) and completing the square 
on AN, Socrates easily shows that 
the square on AN is not 8 but 9 
square feet in area. If L, M be 
the middle points of GH, HK and 
GL, CM be joined, we have four 
squares in the figure, one of which is ABCD, while each of the 
others is equal to it. If now we draw the diagonals BL, LM, 
MD, DB of the four squares, each diagonal bisects its square, 
and the four make a square BLMD, the area of which is half 
that of the square A GHK, and is therefore 8 square feet; 
BL is a side of this square. Socrates concludes with the 
words; 
‘ The Sophists call this straight line (BD) the diameter 
(diagonal); this being its name, it follows that the square 
which is double (of the original square) has to be described on 
the diameter.’ 
The other geometrical passage in the Meno is much more 
difficult, 1 and it has gathered round it a literature almost 
comparable in extent to the volumes that have been written 
to explain the Geometrical Number of the Republic. C. Blass, 
writing in 1861, knew thirty different interpretations; and 
since then many more have appeared. Of recent years 
Benecke’s interpretation 2 seems to have enjoyed the most 
1 Meno, 86 e-87 c, 
2 Dr. Adolph Benecke, Ueber die geometrische Hypothesis in Platon's 
Menon (Elbing, 1867). See also below, pp. 802-3.