34 A History ofMathematics
Fig. 1TheMenoargument. The large square has side 4 feet (area 16 square feet), the four small squares have side 2 feet (area
4 square feet). The four diagonals form a square of area 8 square feet.at http://classics.mit.edu /Plato/meno.html.) What is done in this dialogue is a good introduction
to Greek mathematics—or the kind which is considered ‘typical’, the classics if you like. The
other kinds, referred to in our opening quotation, will be discussed in the next chapter. Although
the problem and the solution would easily have been available to the Egyptians or Babylonians a
thousand years earlier, what seems suddenly to be new is the appeal to argument and discussion.
The philosophical point of the dialogue is an idea about ‘knowing’. Socrates has a strange theory
that the truths which we know have not been learned but were always present in our minds and
we simply bring them to consciousness or ‘remember’ them. (He is referring to a particular kind of
truth—knowledge about triangles or the Good, not mere facts like ‘it’s raining’.) With this aim, he
calls over a supposedly ignorant slave-boy, and asks him how, if you are given a square of side 2 feet,
you can construct a square twice the size. Since the original square has an area of 4 square feet,
then the one you construct must have an area of 8. The slave-boy suggests squares of side 4 feet
(wrong, because its area is 16 square feet) and 3 feet (again wrong, area 9 square feet). He then
becomes perplexed, and admits to not knowing. Socrates then—we assume—draws the figure
shown (Fig. 1), and continues to ‘find out’ from the boy that it contains the answer to the problem.
His arguments are given in Appendix A to this chapter, and are conceptually quite simple.
- Each of the four squares in the diagram is 2×2 square feet, and so has an area 4 square feet.
- So each of the eight triangles (half of a square) has an area of 2 square feet.
- Now look at the square made by the four diagonals. It consists of four triangles, so its area must
be 4×2 or 8 square feet. - It is therefore twice the area of the( 2 ×2 square feet)square we started off with so it is the
‘square of double the size’ we were looking for.
It is easy to find fault with the way the dialogue is conducted: Socrates is in fact leading the
witness inadmissibly, putting the answer which he himself knows into the boy’s mouth, and then
claiming that he has done nothing of the kind. However, more purely mathematical objections
arise, and they relate to some key ideas about the nature of Greek mathematics. In particular,
a question which Socrates does not deal with—which is interesting given the precise use of numbers
like ‘2 feet’, ‘3 feet’, and so on—is what the length of the diagonal (the side of the ‘eight-foot square’)
is. Today, we would say that, since the area is 8, the side must be√
8 =2.828..., which is not a