74 Fundamental Principle of the Higher Calculus 
Arr. VIIL—The Fundamental Principle of the Higher Calculus 
- demonstrated by the method of Indeterminates. 
(Communicated by Mr. Strues FRENCH.) 
Tue following extracts from a recent commentary on Newton’s 
Principia,* contain a view of the principle presented in Lemma I. 
k 1., 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, ee Calculus, Calculus of Derivations, Functions, 
&e e 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 account of the 
steps, by which he was led to a theory 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 eae sought to be satisfied as to the correctness of 
e and Ultimate Ratios. The more we endeav- 
ored to remove sje the more they continually presented them- 
selves; so that after spending many months in the fruitless attempt, 
we had nearly abandoned the work altogether; when suddenly, in 
examining the method of Indeterminate Coefficients in Dr. Wood’s 
Algebra, it oceurred that the aggregates of the coefficients of the 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- 
* «© 4 Commentary on Newton’s Principia, with a oy gcrone? volume. De- 
signed for the use of students at the Universities. By J. M. F. Wright, A.B late 
scholar of Trinity College Cambridge, author of Solutions of the amare’ Prob- 
lems, &c. &c. In two volumes, pp. 458 and 415. 
