58 THE WORLD MACHINE 



as his chief title to fame the inscription of the cylinder within 

 a sphere, and it was this figure which in after years made known 

 his tomb to Cicero. 



The development which the science has taken, especially 

 since the introduction of the decimal system, the discovery of 

 logarithms, and the like has been extensive. Beside it the 

 mathematics of the ancients seem in many ways crude enough. 

 The management of their systems of notation, in multiplication, 

 in division, to say nothing of their struggles with fractions, would 

 be to us simply distracting. To-day the schoolboy solves 

 countless problems with an ease which would have filled Archi- 

 medes with breathless admiration. The mere introduction of 

 the zero must have simplified operations, as measured by their 

 rapidity, at least a dozen times. It seems incredible now that 

 its advantages should not have been earlier perceived. There 

 is some evidence that the Indian or decimal system was known 

 in Europe before the rise of the Arabs ; but it found no general 

 introduction until it was borrowed from the Arabs at about 

 the beginning of the thirteenth century. It does not appear 

 to have been in use among the Hindus earlier than perhaps 

 the fifth or sixth century. It was certainly not known either 

 to the Greek or Latin mathematicians. 



We gain a glimpse of the time from the fact that the clumsy 

 abacus was then in universal use, as it is among the Chinese 

 still. It consisted simply of beads or pebbles strung on parallel 

 rods. Our word " calculate " from calx, " a stone," is a survival 

 from this rude count ing- machine. They had no algebra even ; 

 although Aristotle and even old Ahmes employed letters to 

 represent indeterminate quantities, the -swifter use of algebraic 

 equations did not come in until some centuries later. Here, 

 as elsewhere, we perceive the exceeding slowness with which 

 new methods and new devices make their way. 



Despite the crudity of their devices, our respect for the 

 ancient mind is heightened rather than dulled by the applica- 

 tions which they made of the means at hand. In this later 

 time we have come to perceive that the knowledge of antiquity, 

 the range of its ideas, was far more extensive than had long 

 been supposed. The high development of Greek philosophy, 

 itself no doubt more or less a rescript of a yet more ancient 

 time, is sufficient evidence of the ancient's capacity for abstract 

 reasoning. In the pages that follow we shall find many acute 



