Accordingly, in the admirable researches of MM. Dulong and Petit upon the cooling of bodies, it was assumed that the rate of cooling of the hot body was represented by the excess of F(_θ_ + _t_) above F(_θ_); where F represented some mathematical _function_, that is, some expression obtained by arithmetical operations from the temperatures _θ_ + _t_ and _θ_; although what these operations are to be, was left undecided, and was in fact determined by the experiments. And the result of their investigations was, that the function is of this kind: when the temperature increases by equal intervals, the function increases in a continued geometric proportion[34\4]. This was, in fact, the same law which had been assumed by Newton and others, with this difference, that _they_ had neglected the term which depends upon the temperature of the surrounding space. [Note 34\4: The formula for the rate of cooling is _ma^(θ + t) - ma^θ_, where the quantity _m_ depends upon the nature of the body, the state of its surface, and other circumstances.--_Ann. Chim._ vii. 150.] 18. This law falls in so well with the best conceptions we can form of the mechanism of cooling upon the supposition of a radiant fluid caloric, that it gives great probability to the scale of temperature on which the simplicity of the result depends. Now the temperatures in the formulæ just referred to were expressed by means of the _air thermometer_. Hence MM. Dulong and Petit justly state, that while all different substances employed as thermometers give different laws of thermotical phenomena, their own success in obtaining simple and general laws by means of the air thermometer, is a strong recommendation of that as the _natural scale of heat_. They add[35\4], 'The well-known uniformity of the principal physical properties of all gases, and especially the perfect identity of their laws of dilatation by heat, [a very important discovery of {353} Dalton and Gay Lussac[36\4],] make it very probable that in this class of bodies the disturbing causes have not the same influence as in solids and liquids; and consequently that the changes of bulk produced by the action of heat are here in a more immediate dependence on the force which produces them.' [Note 35\4: _Annales de Chimie_, vii. 153.] [Note 36\4: _Hist. Ind. Sc._ b. x. c. ii. sect. 1.]