THE HISTORY OF MATHEMATICS 395 



this to a variety of causes any one of which may seem sufficient to ex- 

 plain it, but in reality, not one of them explains anything. It may per- 

 haps be best likened to one of those pauses which nature seems to de- 

 mand and in which breath is sought before taking the next forward step. 



The first sign of activity found expression in the Crusades under- 

 taken to free Jerusalem from Mohammedan control. Soon after this 

 time, that is, at the beginning of the twelfth century, several of the 

 modern universities were founded, partly as gatherings of students to 

 learn and discuss, and partly as developments of schools which had 

 been under the charge of the monasteries. The old Greek works on 

 mathematics were revived and the learning transmitted by the Arabs 

 began to be assimilated and taught. Systematic instruction and the 

 writing of treatises for the transmission of knowledge were begun. 

 These were mingled with courses on astrology, alchemy, and magic, 

 perhaps, we may conjecture, because the learned man had little chance 

 to earn his living except through supplying what was in common de- 

 mand, and perhaps also because the ancient learning gave little enough 

 scope for cultivation of the imagination. 



A century later Roger Bacon appears on the scene. It is difficult 

 to overestimate the man who, travelling over Europe and studying in 

 Paris and doubtless absorbing the learning of his time, becoming a 

 Doctor in Theology and probably a monk, could break away from his 

 training and absolutely reject the ideas and methods of his age. One 

 who could write that in order to learn the secrets of nature we must first 

 observe, that in order to predict the future we must know the past, must 

 certainly have had unusual clarity of vision. He taught too that mathe- 

 matics was the basis of all science, but he clearly recognized that it 

 could not replace experiment and knowledge of phenomena, but could 

 be used to great advantage in deducing results when the phenomena had 

 once been observed: the point of view is thoroughly modern. But his 

 three great works failed to have much influence on his times and seem 

 to have been forgotten for several hundred years. Like many of his 

 predecessors and followers in science, he suffered for his opinions. 



The real start of modern science opens, as has been mentioned 

 earlier, in the middle of the fifteenth century, and from this time on 

 there is no break in the sequence of scientific discovery. We have also 

 less need to relate its progress to that of the social order. While there 

 have been periods in which wars have seriously disturbed the ability 

 of nations to produce and to foster learning, there has been no time in 

 the period in which the whole of the civilized world was so far involved 

 in struggles that intellectual progress came to a stop everywhere. 

 Moreover, all attempts since 1450 to enslave the world, either physically 

 or spiritually, have ended in failure. It is true that five hundred years 

 is comparable with the periods of Greek, Roman, and Mohammedan 



