THE HISTORY OF MATHEMATICS 403 



first half of the nineteenth century and the latter of the succeeding 

 period, it would give a false idea of the method of progress to regard 

 these as anything more than general tendencies. But the difficulty (men- 

 tioned earlier,) of conveying an understanding of the advances made 

 in pure mathematics, even by one much more familiar with them than 

 the writer, occurs in an exaggerated form in attempting a chronicle 

 of the work of the past century. The task is far easier in applied 

 mathematics because most of us have some acquaintance with the 

 problems, and the ideas to be conveyed are not so far away from our 

 every day experience. Consequently, in the former, I can do little 

 more than point to a sign post here and there, occasionally indicate the 

 course of the road which has been followed, or describe by an 

 analogy or an example a result which has been obtained. 



The older geometry which consisted in the investigation of figures 

 which could be generated under some simple descriptive definition like 

 lines, circles, or conic sections, was greatly extended by Descartes' 

 invention of analytical geometry. In the latter an algebraic statement 

 of the properties gave rise to various classes of curves which could be 

 ordered according to the forms of algebraic statement. Their proper- 

 ties could be investigated with much greater ease. The way was opened 

 also for the consideration of the different kinds of curves or surfaces 

 which possessed some definite general property; the properties com- 

 mon to a given class of curves or surfaces; the relations which may 

 exist between theorems in analysis and geometry; and so on. When 

 the methods of the calculus were added, the range of investigation was 

 again widely extended through its facility for dealing with the proper- 

 ties of tangents and curvature. The new results obtained were un- 

 doubtedly instrumental in stimulating investigation from the more 

 purely geometrical point of view. The names of Desargues, Monge, 

 and Poncelet have been already mentioned as the creators of the sub- 

 ject of projective geometry; their work was continued and, during the 

 third decade of the ninetenth century, developed into a separate branch 

 by Moebius, Pliicker, Steiner and a host of writers who have followed 

 them. Simple illustrations of the idea involved, are that of map- 

 making in which we represent portions of the spherical surface of the 

 earth on a plane, or that of the shadow of a figure cast by a ray of 

 light. These simple ideas have been generalised by considering the 

 common properties of figures which are projected according to some 

 given law and more generally by correspondences between two or 

 more figures. Another development is that of transformations which 

 leave properties unchanged. It will be seen at once that measure- 

 ments of actual lengths of lines or metric properties, as they are 

 called, are not those which would ordinarily be unchanged by projec- 



