PLATO
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points, lines, triangles, squares, &c., as objects of pure thought.
If we use a diagram in geometry, it is only as an illustration ;
the triangle which we draw is an imperfect representation
of the real triangle of which we think. Constructions, then,
or the processes of squaring, adding, and so on, are not of the
essence of geometry, but are actually antagonistic to it. With
these views before us, we can without hesitation accept as
well founded the story of Plutarch that Plato blamed Eudoxus,
Archytas and Menaechmus for trying to reduce the dupli
cation of the cube to mechanical constructions by means of
instruments, on the ground that ‘ the good of geometry is
thereby lost .and destroyed, as it is brought back to things
of sense instead of being directed upward and grasping at
eternal and incorporeal images ’. 1 It follows almost inevitably
that we must reject the tradition attributing to Plato himself
the elegant mechanical solution of the problem of the two
mean proportionals which we have given in the chapter on
Special Problems (pp. 256-7). Indeed, as we said, it is certain
on other grounds that the so-called Platonic solution was later
than that of Eratosthenes; otherwise Eratosthenes would,
hardly have failed to mention it in his epigram, along
with the solutions by Archytas and Menaechmus. Tannery,
indeed, regards Plutarch’s story as an invention based on
nothing more than the general character of Plato’s philosophy,
since it took no account of the real nature of the solutions
of Archytas and Menaechmus; these solutions are in fact
purely theoretical and would have been difficult or impossible
to carry out in practice, and there is no reason to doubt that
the solution by Eudoxus was of a similar kind. 2 This is true,
but it is evident that it was the practical difficulty quite as
much as the theoretical elegance of the constructions which
impressed the Greeks. Thus the author of the letter, wrongly
attributed to Eratosthenes, which gives the history of the
problem, says that the earlier solvers had all solved the
problem in a theoretical manner but had not been able to
reduce their solutions to practice, except to a certain small
extent Menaechmus, and that with difficulty ; and the epigram
of Eratosthenes himself says, ‘ do not attempt the impracticable
1 Plutarch, Quaest. Conviv. viii. 2. 1, p. 718 R
2 Tannery, La géométrie grecque, pp. 79, 80.