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GROWTH OF MATHEMATICAL SCIENCE 19 

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 labour 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, 
hazy, and dreamy 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 with 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 the case, he did 
actually introduce a large number of new inventions. 

We will not consider further here the development of 
mathematics with other ancient nations, nor the chief problems 
investigated by the Greeks ; such details may be found in some 
of the books mentioned in the Bibliography at the end. The 
object of this chapter is to indicate the nature of the science of 
geometry, and how certain practical needs gave rise to investi- 
gations in which appears an abstract science which was worthy 
of being cultivated for its own sake, and which incidentally 
gave rise to advantages of a practical nature. 

There are two branches of mathematics which began to be 
cultivated by the Greeks, and which allow a connection to be 
formed between the spirits of ancient and modern mathematics. 

The first is the method of geometrical analysis to which 
Plato seems to have directed attention. The analytical method 
of proof begins by assuming that the theorem or problem is 
solved, and thence deducing some result. If the result be 
false, the theorem is not true or the problem is incapable of 
solution : if the result be true, if the steps be reversible, we 
get (by reversing them) a synthetic proof ; but if the steps be 
not reversible, no conclusion can be drawn. "We notice that 
the leading thought in analysis is that which is fundamental in 
algebra, and which we have noticed in the case of Ahmes : 
the calculation or reasoning with an unknown entity, which is