80 Fundamental Principle of the Higher Calculus, &i'C. 



The limits of this article will not permit us to exhibit the manner in 

 which this principle of homogeneity, is applied to finding the Indefi- 

 nite Differences or Differentials of Complex Algebraic, or of Trans- 

 cendental Functions. The reader, we think, will easily perceive the 

 great clearness which must result from, its use in every process in- 

 volving the consideration of the limiting ratios of quantities. 



How far the praise of being the original discoverer of the relations 

 of this principle to the purposes of the Higher Calculus, is due to 

 Mr. Wright, we shall not attempt to decide. We would, however, 

 remark that the views of Carnot in his notice of the method of Inde- 

 terminates of Descartes,* though not fully developed or given in the 

 appropriate notation, seem in some measure to have anticipated those 

 above expressed. After establishing the fundamental principle of that 

 method by the same considerations as are employed by Mr. Wright, 

 Carnot says " let there be an equation of only two terms A + B x = 0, 

 in which the first term is constant and the second susceptible of being 

 rendered as small as we please : this equation cannot subsist unless 

 the terms A and B x are each, separately, equal to zero. We may 

 lay it down then as a general principle, and as an immediate corollary 

 of the method of Indeterminates, that if the sum or difference of two 

 pretended quantities is equal to zero, and if one of the two may he sup- 

 posed as small as we please, while the other contains nothing arbitrary, 

 these two pretended quantities will be each separately equal to zero. 



" This principle alone is sufficient to resolve by common algebra all 

 questions within the province of the infinitesimal analysis. The res- 

 pective processes of the two methods, simplified as they may be, are 

 absolutely the same." And again, in conclusion, after applying it to 

 some examples, "We thus see that the method of Indeterminates 

 furnishes a rigorous demonstration of the infinitesimal calculus. It 

 were to be desired, perhaps, that this course had been pursued in 

 arriving at the differential and integral calculus ; it would have been 

 as natural as that which was taken and would have prevented all 

 difficulties." 



*" Reflexions sur la metaphysique dii oalcul infinitesimal." Seconde edition, 

 page 150. 



