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.