Raiders of the Lost  (Continued)

"Socrates: They say that the soul of man is immortal…."

From August 16, 2012—

In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not ,  at this point
in the dialogue, the diamond in Jowett's picture.

An 1892 figure by Jowett illustrating Plato's Meno—

A more correct version, from hermes-press.com —

Socrates: He only guesses that because the square is double, the line is double.Meno: True.

 

Socrates: Observe him while he recalls the steps in regular order. (To the Boy.) Tell me, boy, do you assert that a double space comes from a double line? Remember that I am not speaking of an oblong, but of a figure equal every way, and twice the size of this-that is to say of eight feet; and I want to know whether you still say that a double square comes from double line?

[Boy] Yes.

Socrates: But does not this line (AB) become doubled if we add another such line here (BJ is added)?

[Boy] Certainly.

Socrates: And four such lines [AJ, JK, KL, LA] will make a space containing eight feet?

[Boy] Yes.

Socrates: Let us draw such a figure: (adding DL, LK, and JK). Would you not say that this is the figure of eight feet?

[Boy] Yes.

Socrates: And are there not these four squares in the figure, each of which is equal to the figure of four feet? (Socrates draws in CM and CN)

[Boy] True.

Socrates: And is not that four times four?

[Boy] Certainly.

Socrates: And four times is not double?

[Boy] No, indeed.

Socrates: But how much?

[Boy] Four times as much.

Socrates: Therefore the double line, boy, has given a space, not twice, but four times as much.

[Boy] True.

Socrates: Four times four are sixteen— are they not?

[Boy] Yes.

As noted in the 2012 post, the diagram of greater interest is
Jowett's incorrect  version rather than the more correct version
shown above. This is because the 1892 version inadvertently
illustrates a tesseract:

A 4×4 square version, by Coxeter in 1950, of  a tesseract—

This square version we may call the Galois  tesseract.