74 Fundamental Principle of the Higher Calculus 



Art. VIII. — The Fundamental Principle of the Higher Calculux 

 demonstrated by the method of Indeterminates. 



(Communicated by Mr. Stiles French.) 



The following extracts from a recent commentary on Newton's 

 Principia,* contain a view of the principle presented in Lemma I. 

 Book I., which may interest mathematical readers. 



"The first section of the Principia," the commentator remarks, 

 "comprehends the substance of the method of Exhaustions of the 

 Ancients, and also of the Modern Theories, variously denominated 

 Fluxions, Differential Calculus, Calculus of Derivations, Functions. 

 &;c. he. Like them it treats of the relations which indefinite quan- 

 tities bear to one another, and conducts in general by a nearer route 

 to precisely the same results." 



On the 55th page of the commentary, he gives an accoimt of the 

 steps, by which he was led to a tlieory of these quantities, which he 

 considers as " divested of all the metaphysical obscurities and incon- 

 sistencies, which render the methods above enumerated so objection- 

 able as to their logic." 



"Having engaged," says he, "to write a Commentary upon the 

 Principia, we naturally sought to be satisfied as to the correctness of 

 the method of Prime and Ultimate Ratios. The more we endeav- 

 ored to remove objections, the more they continually presented them- 

 selves; so that after spending many months in tlie fruitless attempt, 

 we had nearly abandoned the work altogether ; when suddenly, in 

 examining the method of Indeterminate Coefficients in Dr. Wood's 

 Algebra, it occurred that the aggregates of tlie coefficients of die like 

 powers of the indefinite variable, must be separately equal to zero, 

 not because the variable might be assumed equal to zero, (which it 

 never is, although it is capable of indefinite diminution,) but because 

 of the different powers being essentially different from, and forming 

 no part of one another. 



" From this a train of reflections followed, relative to the treat- 

 ment of homogeneous definite quantities in other branches of Alge- 



* "A Commentary on Newton's Principia, with a supplementary volume. De- 

 signed for the use of students at the Universities. By J. M. F. Wright, A. B. laic 

 scholar of Trinity College Cambridge, author of Solutions of the Cambridge Prob- 

 lems, &c. &.C, In two volumes, pp. 45S and 415. London, 1828." 



