SECT. I.] INTRODUCTION. 25 



and their intensity is rigorously proportional to the sine of the 

 angle which the direction of each ray makes with the element of 

 surface. 



If we could observe the changes of temperature for every in 

 stant at every point of a solid homogeneous mass, we should dis 

 cover in these series of observations the properties of recurring 

 series, as of sines and logarithms ; they would be noticed for 

 example in the diurnal or annual variations of temperature of 

 different points of the earth near its surface. 



We should recognise again the same results and all the chief 

 elements of general analysis in the vibrations of elastic media, in 

 the properties of lines or of curved surfaces, in the movements of 

 the stars, and those of light or of fluids. Thus the functions ob 

 tained by successive differentiations, which are employed in the 

 development of infinite series and in the solution of numerical 

 equations, correspond also to physical properties. The first of 

 these functions, or the fluxion properly so called, expresses in 

 geometry the inclination of the tangent of a curved line, and in 

 dynamics the velocity of a moving body when the motion varies ; 

 in the theory of heat it measures the quantity of heat which flows 

 at each point of a body across a given surface. Mathematical 

 analysis has therefore necessary relations with sensible phenomena ; 

 its object is not created by human intelligence; it is a pre-existent 

 element of the universal order, and is not in any way contingent 

 or fortuitous ; it is imprinted throughout all nature. 



21. Observations more exact and more varied will presently 

 ascertain whether the effects of heat are modified by causes which 

 have not yet been perceived, and the theory will acquire fresh 

 perfection by the continued comparison of its results with the 

 results of experiment ; it will explain some important phenomena 

 which we have not yet been able to submit to calculation ; it will 

 shew how to determine all the therm ornetric effects of the solar 

 rays, the fixed or variable temperature which would be observed at 

 different distances from the equator, whether in the interior of 

 the earth or beyond the limits of the atmosphere, whether in the 

 ocean or in different regions of the air. From it will be derived 

 the mathematical knowledge of the great movements which result 

 from the influence of heat combined with that of gravity. The 



