A characteristic of Greek social life is responsible both for the next phase of the development of mathematical thought and for the misapprehension of its nature by so many moderns. "When Archytas and Menaechmus employed mechanical instruments for solving certain geometrical problems, 'Plato,' says Plutarch, 'inveighed against them with great indignation and persistence as destroying and perverting all the good that there is in geometry; for the method absconds from incorporeal and intellectual or sensible things, and besides employs again such bodies as require much vulgar handicraft: in this way mechanics was dissimilated and expelled from geometry, and being for a long time looked down upon by philosophy, became one of the arts of war.' In fact, manual labor was looked down upon by the Greeks, and a sharp distinction was drawn between the slaves who performed bodily work and really observed nature, and the leisured upper classes who speculated, and often only knew nature by hearsay. This explains much of the naïve dreamy and hazy character of ancient natural science. Only seldom did the impulse to make experiments for oneself break through; but when it did, a great progress resulted, as was the case of Archytas and Archimedes. Archimedes, like Plato, held that it was undesirable for a philosopher to seek to apply the results of science to any practical use; but, whatever might have been his view of what ought to be in the case, he did actually introduce a large number of new inventions" (Jourdain, _The Nature of Mathematics_, pp. 18-19). Following the Greek lead, certain empirically minded modern thinkers construe geometry wholly from an intellectual point of view. History is read by them as establishing indubitably the proposition that mathematics is a matter of purely intellectual operations. But by so construing it, they have, in geometry, remembered solely the measuring and forgotten the land, and, in arithmetic, remembered the counting and forgotten the things counted.
Arithmetic experienced little immediate gain from its new association with geometry, which was destined to be of momentous import in its latter history, beyond the discovery of irrationals (which, however, were for centuries not accepted as numbers), and the establishment of the problem of root-taking by its association with the square, and interest in negative numbers.