n8 THE POPULAR SCIENCE MONTHLY 



differential calculus are in position to appreciate the value of mathe- 

 matical symbols for the purpose of centralizing and intensifying 

 thought. 



It is true that some of the roads which mathematical thought has 

 made through great difficulties have been practically abandoned and 

 that the popularity of many of the others has changed from time to 

 time. Among the former we may class the results of investigations 

 recorded at the beginning of the oldest extensive mathematical work 

 that has been deciphered, viz., the formulas relating to unit fractions 

 which are found in the nearly four thousand-years-old work of 

 Ahmes. A subject which appears to have been placed at the very 

 beginning of advanced mathematical instruction four thousand years 

 ago is now entirely abandoned in our courses, except when the history 

 of the development of the science is under consideration. While 

 mathematics presents a number of other roads which are now of interest 

 only to the historian, yet there are also many which have been known 

 for centuries and which have been pursued with profit and pleasure 

 by great minds in all the civilized nations. The latter class includes 

 all the longer ones leading gradually to points of view from which the 

 connection between many natural phenomena may be clearly discerned. 



The intellectual heights reached by means of a long series of con- 

 nected mathematical theorems do not always reveal their greatest lesson 

 to the first explorers. For instance, the large body of facts relating 

 to conic sections, developed by Apollonius and other Greek geometers, 

 became a much greater glory to the human mind through the discovery, 

 nearly two thousand years later, that the bodies of the solar system de- 

 scribe conic sections. Such experiences in the past tend to justify the 

 fact that a large number of men are devoting their lives to the discovery 

 of abstract results irrespective of applications, and they tend to explain 

 why the largest prize (about twenty-five thousand dollars) ever offered 

 for a mathematical theorem is being offered for a theorem in number 

 theory, which is not expected to have any application to subjects out- 

 side of pure mathematics. 



There seems to be a general impression abroad to the effect that 

 mathematics and the ancient languages constituted the main parts of 

 the curriculums of our colleges and universities a century or two ago. 

 As regards mathematics this is quite contrary to fact, as may be 

 seen from a few historical data. Less than two centuries ago the 

 students in Harvard College began the study of arithmetic in their 

 senior year. In fact, no knowledge of any mathematics was required 

 to enter Harvard before 1803, and it was not until 1816 that the whole 

 of arithmetic was required for entrance. In other American institu- 

 tions the mathematical situation was generally worse, and in Europe 

 the improvements were not very much earlier. It is during compara- 



