80 Fundamental Principle of the Higher Calculus, &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+ Bx = 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 ma 
Jay 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 be 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.” 
***Réflexions sur la métaphysique du calcul infinitésimal.” Seconde édition, 
page 150. 
