164 Prof. Challis on Objections recently made to 



remarks on the parts respectively performed by reasoning and 

 by experiment or observation, in establishing the truth of the re- 

 sults obtained in applied mathematics, not having seen in modern 

 scientific productions sufficiently precise information as to the 

 relation between these two parts. 



(1) The experimental results obtained by Galileo and Atwood 

 respecting the vertical descent of bodies acted upon by gravity, 

 and the experimental proof that the vertical acceleration of a 

 projectile is independent of the motion in a curvilinear path, and 

 and that its horizontal motion is uniform, establish the laws 

 which govern constant accelerating forces. It was absolutely 

 necessary to ascertain these laws by experiment before any abs- 

 tract reasoning respecting accelerating force was possible. But 

 the laws relating to constant accelerating forces having thus 

 become known, it can be proved by means of Taylor's theorem, 

 or some equivalent process, that an accelerating force, whether 

 variable or constant, estimated in a given direction, is quantita- 

 tively expressed by the second differential coefficient of the func- 

 tion which gives the distance at any time of the accelerated par- 

 ticle from a fixed plane perpendicular to that direction. Taylor's 

 theorem is legitimately employed for this purpose on the general 

 principle that abstract calculation is comprehensive of all concrete 

 physical relations. This, in fact, is the proper a priori argument 

 for the truth of that general symbolic expression for accelerative 

 force. 



On the same expression the whole of Physical Astronomy 

 depends, as well as every department of applied science which 

 requires the calculation of motions produced by accelerative 

 forces. The satisfactory comparison of the results of calculation 

 so made with observed facts constitutes the argument a poste- 

 riori for the truth of the analytical symbol for accelerative force, 

 and with so much the greater evidence as the number of such 

 comparisons is greater. The analytical proof of Kepler's laws 

 is a very important part of such evidence, inasmuch as it involves 

 as a corollary the demonstration of the law of gravity, which 

 could not be given by observation alone, nor by calculation alone, 

 but requires that the two processes be combined. In short, the 

 discovery by geometrical or analytical reasoning that a variable 

 force is expressible by the second differential coefficient of space 

 with respect to time, specially characterizes the Newtonian epoch 

 of physical science. 



(2) A perfect fluid being defined by the properties of mutual 

 pressure, and of easy separability, of contiguous parts, it is pos- 

 sible to demonstrate, for fluid at rest, the law of equal pressure 

 in all directions from a given point, exclusively by reasoning 

 founded on these properties. (See ' Principles of Mathematics 



