'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 naive 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.

The Greeks had only subtracted smaller numbers from larger, but the
Arabs began to generalize the process and had some acquaintance with
negative results, but it was difficult for them to see that these
results might really have significance. N. Chuquet, in the fifteenth
century, seems to have been the first to interpret the negative numbers,
but he remained a long time without im