PRINCIPLE OF LEAST ACTION. 189 



of being dependent on a metaphysical principle of which 

 he did not show the necessary truth.* Huyghens ar- 



* The theoretical principles here glanced at, are tho^e connected 

 with speculations on on of the most curious points presented by the 

 theory of light; which, perhaps, it may be desirable briefly to explain. 

 Ptolemy had shown that when light is reflected from any surface, the 

 law of reflexion, or equality of angles, is precisely that which causes 

 light to pass from any one point in its course, before incidence to any 

 other in its reflected course, by the shortest path and in the least time, 

 its velocity being uniform and equal before and after reflexion. 



Fermat extended the same principle, called the " principle of least 

 time," to the case of refraction according to the law of sines, provided 

 we suppose the velocity diminished in the denser medium : that is, he 

 showed that the sum of the times, or of the spaces DIVIDED by the veloci- 

 ties, is a minimum. 



Huyghens, adopting the theory of waves, deduced from it the law 

 of the sines; and as, in conformity with that theory, the velocity 

 must be diminished in the denser medium, on this theory the principle 

 of " least time " applies to the case of refraction, and that of reflexion 

 also easily follows as a particular case. 



On the other hand, on the molecular theory, the law of refraction is 

 deduced on the principle of attraction, which the molecules undergo 

 in the medium, and it is a necessary consequence that the velocity 

 must be increased in the denser medium. Maupertuis, on these prin- 

 ciples, attempted an analogous investigation ; but here it was neces- 

 sary to adopt, not the principle of "least time," but that of " least 

 action," or that the sum of the PRODUCTS of the spaces and velocities is 

 a minimum ; and, on this view, the law of the sines equally results as 

 that which fulfils the condition. 



This refers to ordinary refraction : when the same inquiry was ex- 

 tended to double refraction, or to the extraordinary ray, more complex 

 considerations were introduced. This subject is fully discussed by 

 Dr. Young in his Life of Fermat. ( Works, ed. Peacock, vol. ii. p. 

 684.) The same principle was the basis of Laplace's investigation of 

 double refraction, of which (" Sur la Loi de la Refraction Extraordi- 

 naire, &c.," Journal de Physique, 1809) Dr. Young produced his well- 

 known refutation in the Quarterly Review for the same year. 



In the case of ordinary refraction, the investigation is very simple. 

 As it is not clearly stated, as far as we are aware, in any elementary 

 treatise, it may be satisfactory to some readers to have it briefly put 

 before them. 



Let any lengths, respectively, of the incident and refracted rays be 



