Notices respecting New Books. 137 



the I may be neglected in comparison of »?. But it cannot be ad- 

 mitted that two arcs, however great, which differ by a quantity 0, 

 have t!ie same cosines independent!}' of the value of W. The fallacy 

 of the reason assigned for neglecting 1, will be apparent, by putting 

 the sum of the series under this other form, 



m 4- 1 . . ni 

 cos — ^ — d sin — 





 cos - 



2 



which does not give the same result as before, when 1 is neglected 

 in comparison of »i. I have adverted to this error, because in con- 

 sequence of it, Lagrange exhibits to view a discontinuous function, 

 the possibility of doing which, may well be called in question. It is 

 not necessary to inquire how the reasoning may be conducted, if 

 this step be corrected, because the second Researches are in prin- 

 ciple the same as the first, and are not liable to a similar objection. 

 In these he has elaborately, yet strictly shown, as far as I have been 

 able to follow the reasoning, that the motions he is in search of, are 

 not subject to any law of continuity ; — that the motions, for instance, 

 at a given instant, in a column of fluid stretching between two given 

 points, cannot be given generally by any known line or function. 

 He supposes, therefore, that they will be given, by a neto set of 

 functions, neither algebraical, transcendental, nor mechanical, but 

 discontinuous per se, and by this property of discontinuity distin- 

 guished from every other. This definition has been admitted by all 

 subsequent writers. But it deserves to be considered in what sense, 

 and to what extent an investigation of this nature can demonstrate 

 any property of functions. The science of quantity is a perfect sci- 

 ence ; it needs not the aid of any other, and exists prior to its ap- 

 plications to questions of nature, and independently of them. When 

 in the applications, any form or property of functions is arrived at 

 by the operations that are performed, it will always be possible to 

 arrive at the same, by abstracting from the physical question, and 

 performing the same operations by pure analytical reasoning. For 

 in the applications, we are, in general, concerned about time, space, 

 force, and matter, — ideas of a totally dissimilar kind, but possessing 

 this in common, that we can conceive of them as consisting of parts, 

 and in virtue of this common quality, after establishing a unit for 

 each, wc are able to express their observed relations numerically, 

 or by lines or letters the representatives of numbers. All subse- 

 quent reasoning is then conducted according to the rules of analysis, 

 and cannot possess a greater generality in regard to the modes of 

 expressing quantity, than the operations conducted by those rules 

 admit of. If an attempt be made to prove the existence of discon- 

 tinuous functions by pure analysis, it vvill be impossible to succeed, 

 because, as Lagrange says, " the principles of the Differential and 

 Integral Calculus, depend on the consideration of variable alge- 

 braical functions, and it does not appear, that we can give more ex- 

 tent to the conclusions drawn from these principles, than the nature 

 N. S. Vol. 7. No. 38. Fc/j. 18.30. T of 



