84 



powers of x, which is convergent when x < 2c. The convergeney, 

 however, becomes very slow when x aj)proaches the limit 2c ; and 

 the series does not point out the law according to which f(x) or y 

 approaches its extreme value as x approaches 2e. When the con- 

 stant term in the second member of the preceding equation is omit- 

 ted, the equation may be integrated in finite terms ; and consequently 

 the variables can be separated in the actual equation, so that/(;r) 

 can be expressed explicitly by means of definite integrals. In this 

 way the author has obtained/(2c — x)—f(x) in finite terms, so that 

 the numerical value of f(x) may readily be obtained from x=c to 

 x — 2c, after it has been calculated from the series from x = Q to x=c : 

 and between these limits the series is very convergent, being ulti- 

 mately a geometric series with a ratio — . The author has also in- 

 vestigated a series proceeding according to ascending powers of c — x, 

 which converges more rapidly than the former when x approaches c. 

 By the use of these two series, f(x) may be calculated by means of 

 series which are ultimately geometric series, with ratios ranging 

 from to £. 



The unsymmetrical form of the trajectory, and the largeness of 

 the deflection produced by the moving body, come out from the in- 

 vestigation. By means of the numerical values of f(x) the author 

 has drawn a figure representing the trajectory for four different ve- 

 locities. The expression for the central deflection, however, becomes 

 infinite when x becomes equal to 2c, which shows that it is neces- 

 sary to take into account the inertia of the bridge ; although, if the 

 bridge be really light, the solution obtained when the inertia of the 

 bridge is neglected may be sufficiently exact for the greater part of 

 the body's course. 



On Hegel's Criticism of Newton's Principia. By Dr. Whewell. 



Parts of Hegel's Encyclopaedia are here examined with the purpose 

 of testing the value of his philosophy, not of defending Newton. 

 Hegel says that the glory due to Kepler has been unjustly transferred 

 to Newton ; confounding thus the discovery of the laws with the 

 discovery of the force from which the laws proceed, in which latter 

 discovery Kepler had no share. Hegel pretends to derive the New- 

 tonian "formula" from the Keplerian law, thus; — by Kepler's law, 



. . . A 3 

 A being 1 the distance, and T the periodic time, — - is constant : but 



A 



Newton (Hegel says) calls —universal gravitation, whence universal 



gravitation is inversely as A 2 : — a most absurd misrepresentation of 

 the course of Newton's reasoning. In the same manner Hegel criti- 

 cises, and utterly misrepresents Newton's explanation, for the ellip- 

 tical orbit, of the body's approaching to and receding from the centre; 

 and of the reason why the body moves in an ellipse. Hegel also 

 offers his own explanation of Kepler's laws from his own h priori 

 assumptions. He says that the motion of the heavenly bodies is not 

 a being pulled this way or that, as is imagined by the Newtonians ; 

 they go along, as the ancients said, like blessed gods. 



