394 THE SCIENTIFIC MONTHLY 



schools, they set up schools of their own which developed mainly on 

 algebraical lines. One name stands out prominently, that of Alkarismi 

 — who may be regarded perhaps as the founder of modern algebra — 

 the name of the subject itself comes from him. But he used no com- 

 plete symbolism, and he and his successors developed it mainly from 

 the arithmetical point of view. In this school the modern names of the 

 trigonometrical functions appeared although little was done in the way 

 of development. 



Chinese science seems to have started about the same time as that 

 of Greece, but to have had little or no connection with our western 

 science until the sixteenth century. Whatever mathematics the Japanese 

 had probably came from China. It would appear at first sight that the 

 earliest developments arose independently in all civilized nations about 

 the same time, speaking broadly, but our knowledge is so scanty that 

 we can only say that the records nearly all date back to similar periods 

 in each case. Whether there is anything significant in this fact must be 

 left to conjecture. 



With this brief sketch this era may be thought of as closed in so 

 far as the development of science and particularly mathematical science 

 is concerned. It witnesses a real beginning in the study of geometry 

 and algebra, and to a much less extent, of physical principles. A 

 system of logical reasoning was discovered which, in its main outlines, 

 still forms the basis of all deduction at the present day. The properties 

 of the simpler geometrical figures had been studied and committed to 

 writing. The advance of arithmetic was hindered by a poor numerical 

 notation, but the foundations were laid for future development by the 

 introduction of the Hindu symbolism. Algebra had started to emerge 

 from the rhetorical form of discussion into the more terse abbreviations 

 which we now use. Astronomy never failed to have exponents, though 

 many who doubtless desired to increase their knowledge were restrained 

 by the superstition of the age and by ecclesiastical authority, which 

 attempted to dictate the thoughts as well as the actions of men. In 

 mechanics, Archimedes seems to have stood almost alone, largely, per- 

 haps, because no scientific method in dealing with the fundamental 

 problems of nature had yet become current amongst the learned men. 



It is difficult to sum up in a phrase the reasons for the scientific 

 hiatus which occurred during the following three or four centuries. 

 The revival of learning which sprang up towards the close of the eighth 

 century was almost barren of progress in mathematics. The schools 

 established by Charlemagne, while teaching mathematics, developed in- 

 terest in other directions. Reverence for authority appears to be the 

 basis of nearly all the learning of the age and there is no more stifling 

 attitude of mind for progressive evolution of ideas. We may attribute 



