388 THE SCIENTIFIC MONTHLY 



material fed into it. It is the purpose of the mill to work up the raw 

 material and the better the machinery the more finished and more 

 varied will be the product. 



To write a connected story of mathematical development within the 

 limits of a brief article on a consistent plan without making it a mere 

 catalogue of names and results is a difficult, perhaps impossible, task 

 and no attempt is made here to accomplish it. If we try to lay stress 

 on the workers rather than on what they achieved, in one period we 

 encounter schools which developed particular subjects, in another, 

 outstanding figures with or without influence on contemporary de- 

 velopment and in still another numerous investigators who contributed 

 in varying degrees to the advances made in their age. On the 

 other hand if the development of ideas be the basis, parallel develop- 

 ments followed independently by different schools sometimes occur, 

 at another time general methods of treatment seem to pervade and 

 again we may find some fundamental advance made, the effect of 

 which is not felt for many years. Consequently the plan which seems 

 to fit best any particular period has been adopted for that period. 

 Another difficulty consists in assigning the relative values to either men 

 or ideas, about which probably no two persons will agree. There is, 

 however, one stumbling block which is peculiar to mathematics. The 

 very names themselves of many important branches of pure mathe- 

 matics convey no meaning to the majority of scientific readers who are 

 not trained mathematicians and to whom alone this article is intended 

 to appeal. An attempt at brief definition has sometimes been made, 

 but it can at best only give a partial view of the subject even with 

 concrete illustrations of its significations. In the general outline, the 

 historical development has been followed, but the methods of carrying 

 it forward have varied with different periods. When a choice of names 

 mentioned has to be made, a rough guide has been furnished in the 

 earlier periods by selecting those who have taken the step forward 

 which has rendered the subject capable of expansion or application 

 by others as judged in the light of present knowledge. In the nine- 

 teenth and present centuries to carry out this method has proved to be 

 beyond the ability of the writer; consequently, in most cases personal 

 mention has only been incidental and the names of many of those who 

 have done great service are missing. Fortunately, the history of mathe- 

 matics has received much attention in articles and separate volumes and 

 to them the reader who is interested in obtaining fuller information is 

 referred. 



The earliest traces in the form of written records have come to us 

 from the Babylonians, mainly in the form of clay tablets which appear 



