PLATO 
287 
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.