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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 those connected 
with speculations on one 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 propucts 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. 
584.) The same principle was the basis of Laplace’s investigation of 
double refraction, of which (“ Sur la Loi de Ja Réfraction 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 
