GBANTHAM ADDEESS. 287 



may be permitted to compare the career of those great men. 

 But Columbtis did not invent the mariner's compass, as Newton 

 did the instrument which guided his course and enabled him 

 to make his discoveries, and his successors to extend them by 

 closely following his directions in using it. Nor did the 

 compass suffice to the great navigator without making any 

 observations ; thoiigh he dared to steer without a chart ; while 

 it is certain that by the philosopher's instrument his dis- 

 coveries were extended over the whole system of the universe, 

 determining the masses, the forms, and the motions of all its 

 parts, by the mere inspection of abstract calculations and 

 formulas analytically deduced.* 



The two great improvements in this instrument which have 

 been made, the Calculus of Variations by Euler and Lagrange, 

 the method of Partial Differences by D'Alembert, we have 

 every reason to believe were known, at least in part, to 

 Newton himself. His having solved an isoperimetrical 

 problem (finding the line whose revolution forms the solid of 

 least resistance) shows clearly that he must have made the co- 

 ordinates of the generating curve vary, and his construction 

 agrees exactly with the equation given "by that calculus. f 



* The investigation of the masses and figures of the planets from their 

 motions by Newton the discovery by Laplace of peculiarities in those 

 motions never before suspected, a discovery made from the mere inspection 

 of algebraical equations without leaving their study are as if Columbus 

 had never left his cabin. 



t The differential equation of the curve deduced by help of the calculus 

 of variations is of this form : 

 y d f 



_ 



dx 

 Which may be reconciled with the equation in the commentary to the 



Schol. of Prop. XXXIV. lib. II. If p = |^, the equation becomes 



el x 



c(l + p 2 } 2 

 y = -- 3 - . T. Simpson, in his general solution of isoperimetrical 



problems (' Tracts/ 1757), gives a method which leads precisely to the 

 above result derived from the calculus of variations, see p. 104. See, too, 

 Emerson's ' Fluxions," where we see his near approach to the calculus. 



