differential calculus are in position to appreciate the value of mathematical 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 connected 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 describe 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 outside 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 curriculum 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 institutions the mathematical situation was generally worse, and in Europe the improvements were not very much earlier. It is during compara-
