402 THE SCIENTIFIC MONTHLY 



attraction, especially gravitational, and its application to the solar 

 system. He made potential a real and valuable instrument of analysis 

 and discovery, working out many of its properties and! applying it in 

 various directions. The famous second order differential equation 

 Y 2 y=0, which is satisfied by every gravitational arrangement of 

 matter, has been used as a substitute for the simpler expression of the 

 Newtonian law of attraction and is especially interesting at the present 

 time, since it may be regarded as the Newtonian analogue of the 

 Einstein law. Laplace was the first to attempt a complete explanation 

 of the motions of the bodies of the solar system or at least to formulate 

 methods which could be applied for this purpose and his Mecanique 

 Celeste remained the standard work of reference in this subject for a 

 century. The theory of the development of the solar system from a 

 primeval nebula which goes by his name and which was independently 

 set forth somewhat earlier by Kant, has never been entirely rejected, 

 although its supporters have often changed it almost beyond recognition 

 while retaining his name in connection with their work. Its funda- 

 mental idea consists of the contraction of matter under gravitational 

 attraction, but few now believe that the matter can produce planets 

 and satellites by the throwing off of concentric rings as he supposed. 

 What he did for celestial mechanics, he also achieved for the subject of 

 probability, his Theorie Analytique des Probabilites being the classic 

 treatise in which the whole is put on a sound basis and developed in 

 various directions. It must be added that he gave many theorems and 

 results in pure analysis but in most cases these were invented to solve 

 physical problems. 



Of the remaining names in this period, Legendre was essentially 

 an analyst, his work being mainly in the theory of numbers, which 

 few mathematicians of this time altogether left alone, in integral 

 calculus and ellipitic functions, his treatises on these subjects being 

 still consulted. Monge and Poncelet may be regarded as the founders 

 of modern descriptive and projective geometry respectively. Fourier 

 in his Theorie Analytique de la Chaleur enunciated the theorem for 

 the expansion of a function in a periodic form which has had sueh! 

 immense value in the discussion of all periodic phenomena, and has 

 now a literature of its own. Poisson's work in attraction is on similar 

 lines to that of Laplace, whose natural successor he seems to be. 



It is conventient to view the progress made in the nineteenth and 

 twentieth centuries from two points of view: one, the development of 

 the three great branches of mathematics, geometry, analysis and their 

 applications to other studies; the other, the development of new ideas 

 which have applications in many branches of mathematics. While it 

 may be said that the former is more particularly characteristic of the 



