THE BEGINNINGS OF CERTITUDE 57 



unto that of Europe after the long sleep of Christendom. The 

 more clearly we are able to trace any especial development, 

 however, the more we discover it to be genetic and even, rather 

 than saltatory or precipitate. 



Be this as it may, we know that by the time of Thales and 

 Pythagoras methods of geometrical reasoning had taken deep 

 hold. They stirred to a singular enthusiasm as well. Professor 

 Benn remarks illuminatively upon the lively emotions that 

 must have been excited among an intelligent people when 

 squaring and cubing, the construction and equivalence of 

 figures and the like, was just as strange and wonderful, and in 

 another way much more rare, than the electric light dynamos 

 and their kindred, a generation ago. There is a story of how 

 Pythagoras sacrificed a hecatomb, a hundred oxen, in honour 

 of his discovery of the theorem of three squares that is, as to 

 the equivalence of a square on the hypothenuse of a right- 

 angled triangle. It is a little doubtful, for Pythagoras was a 

 vegetarian and opposed to the shedding of blood ; but the 

 same story is told of Thales so we may take our choice. A 

 mathematical training came to be considered a mark and even 

 necessity of culture. 



We do not ordinarily think of Plato as a careful reasoner, 

 and the puerile phantasies with which his pages are strewn 

 do not give us a very high idea of his powers of mind. Never- 

 theless, his contributions to geometry were not few, and when 

 he set up his academy at Athens, it will be remembered that 

 he wrote over the door : 



" Let no one enter here who knows not geometry." 



By the time the Ptolemies had set up their line of Greek 

 kings in Egypt, the science had attained a high dignity. We 

 may recall the rebuke of Euclid when the king asked for an 

 easier path to learning than through mastery of his everlasting 

 theorems : 



" There is no royal road to mathematics." . 



The power of the new weapons found its highest exemplar 

 in Archimedes, greatest mathematician of antiquity. It was 

 while solving a geometrical problem, at past seventy, that 

 he was struck down by a Roman soldier. Spite of all the 

 mechanical marvels which he devised, we know that he selected 



