Memoir of Mr Gregory. 229 



in which the more recent improvements of the calculus might 

 be embodied. 



Since the beginning of the century, the general aspect of 

 mathematics has greatly changed. A different class of pro- 

 blems from that which chiefly engaged the attention of the 

 great writers of the last age has arisen, and the new re- 

 quirements of natural philosophy have greatly influenced the 

 progress of pure analysis. The mathematical theories of heat, 

 light, electricity, and magnetism, may be fairly regarded as 

 the achievement of the last fifty years. And in this class of 

 researches an idea is prominent, which comparatively occurs but 

 seldom in purely dynamical enquiries. This is the idea of dis- 

 continuity. Thus, for instance, in the theory of heat, the con- 

 ditions relating to the surface of the body whose variations of 

 temperature we are considering, form an essential and peculiar 

 element of the problem ; their peculiarity arises from the dis- 

 continuity of the transition from the temperature of the body 

 to that of the space in which it is placed. Similarly, in the 

 undulatory theory of light, there is much difficulty in deter- 

 mining the conditions which belong to the bounding surfaces 

 of any portion of ether ; and although this difficulty has, in 

 the ordinary applications of the theory, been avoided by the 

 introduction of proximate principles, it cannot be said to have 

 been got rid of. 



The power, therefore, of symbolizing discontinuity, if such 

 an expression may be permitted, is essential to the progress 

 of the more recent applications of mathematics to natural 

 philosophy, and it is well known that this power is intimately 

 connected with the theory of definite integrals. Hence the 

 principal importance of this theory, which was altogether 

 passed over in the earlier collection of examples. 



Mr Gregory devoted to it a chapter of his work, and noticed 

 particularly some of the more remarkable applications of de- 

 finite integrals to the expression of the solutions of partial 

 differential equations. It is not improbable that in another 

 edition he would have developed this subject at somewhat 

 greater length. He had long been an admirer of Fourier's 

 great work on heat, to which this part of mathematics owes 

 so much ; and once, while turning over its pages, remarked to 



