406 THE SCIENTIFIC MONTHLY 



mathematicians to examine thoroughly the foundations of the calculus. 

 It consists of "all those finer and deeper questions relating to the 

 number system, the study of the curve, surface and other geometrical 

 notions, the peculiarities that functions present with reference to dis- 

 continuity, oscillation, differentiation and integration, as well as a 

 very extensive class of investigations whose object is the greatest pos- 

 sible extension of the processes, concepts, and results of the calculus." 

 (J. Pierpont, Bull, Amer, Math. Soc, 1904, p. 147) . 



Ever since the invention of the calculus, the relations between two 

 or more variables which contained also their derivatives and known 

 as differential equations, have been continually brought before the eyes 

 of the mathematician by the physicist, owing to the fact that the 

 simplest symbolic statement of nearly all physical problems has been 

 in the form of one or more differential equations. For finding the 

 derivatives or integrals which arose in his work, definite rules were 

 generally available even if they demanded much calculation. But he 

 failed to find such rules for most of his differential equations, and in 

 fact they do not exist. The pure mathematicians, led by Cauchy, took 

 up the question from other points of view asking, in particular, the 

 nature of the function which is defined by a differential equation. This 

 is naturally an extension of the theory of functions and the methods 

 of the latter opened the way. But the questions are so difficult that 

 only a particular form, known as the linear, has made any considerable 

 progress; this form, however, does embrace a large number of the 

 functions whose properties had been examined. In our own generation 

 the subject has been extended by the consideration of equations in 

 which integrals also occur; these again are necessary in certain physical 

 problems. 



Most of the progress which has been made in applied mathematics 

 will be treated in the article on Physics, but in addition to the remarks 

 at the close of this article some few words may be said of those 

 branches in the development of which mathematics plays the larger 

 part. The chief of these is the dynamics of a system of particles and 

 rigid bodies. W. R. Hamilton and C. G. J. Jacobi, in the second 

 quarter of the century, put the equations of motion of all such systems 

 into forms which not only permitted of remarkable generalisation, but 

 indicated new methods of integration which opened out research into 

 the general properties of such systems. The later work has been 

 mainly developments and applications of these methods. The par- 

 ticular branch of this subject known as celestial mechanics has been 

 continued on the practical side by extended theories of the motions of 

 the planets and the moon and on the theoretical side by investigations 

 into the general problem of three or more bodies. In the former 

 numerous writers have continued with increasing accuracy the work of 



