1-10 HISTORY OF THERMOTICS. 



the thermometrical scale of licat according to tlie expansion of liquids 

 (which, is the measure of temperature here adopted), was constructed 

 with a reference to Newton's law of radiation of heat ; and thus the 

 law is necessarily consistent with the scale. 



In any case in which the parts of a body are unequally hot, the 

 temperature will vary continuously in passing from one part of the 

 body to another ; thus, a long bar of iron, of which one end is kept 

 red hot, will exhibit a gradual diminution of temperature at successive 

 points, proceeding to the other end. The law of temperature of the 

 parts of such a bar might be expressed by the ordinates of a curve 

 which should run alongside the bar. And, in order to trace mathe- 

 matically the consequences of the assumed law, some of those pro- 

 cesses would be necessary, by which mathematicians are enabled to 

 deal with the properties of curves ; as the method of infinitesimals, or 

 the differential calculus ; and the truth or falsehood of the law would 

 be determined, according to the usual rules of inductive science, by a 

 comparison of results so deduced from the principle, with the observed 

 phenomena. 



It was easily perceived that this comparison was the task which 

 physical inquirers had to perform ; but the execution of it was de- 

 layed for some time ; partly, perhaps, because the mathematical pro- 

 cess presented some difficulties. Even in a case so simple as that 

 above mentioned, of a linear bar with a stationary temperature at one 

 cud, partial differentials entered ; for there were three variable quan- 

 tities, the time, as well as the place of each point and its temperature. 

 And at first, another scruple occurred to M. Biot when, about 1804, 

 he undertook this problem. 1 "A difficulty," says Laplace, 2 in 1800, 

 " presents itself, which has not yet been solved. The quantities of 

 heat received and communicated in an instant (by any point of the 

 bar) must be infinitely small quantities of the same order as the excess 

 of the heat of a slice of the body over that of the contiguous slice ; 

 therefore the excess of the heat received by any slice over the hetot 

 communicated, is an infinite!}' small quantity of the second order; 

 and the accumulation in a finite time (which depends on this excess) 

 cannot be finite." I conceive that this difficulty arises entirely from 

 an arbitrary and unnecessary assumption concerning the relation of 

 the infinitesimal parts of the body. Laplace resolved the difficulty by 

 further reasoning founded upon the same assumption which occasioned 



1 Biot,, Trait e de Phys. iv. p. G69. 2 Laplace, Mem. List, for 1809, p. 332. 



