312 Review of the Prineipia of Newton. 



given the direction and initial velocity, and by the 39th pro- 

 position the altitude from which a body falling by virtue of 

 the variable centripetal force would acquire the same veloci- 

 ty. This, by the same proposition, will ever be proportion- 

 al to the nascent increment, fluxion, or differential of the 

 curvelinear trajectory, (see page 336 of the last volume of the 

 American Journal,) this nascent increment is an element, on 

 the quantity and direction of which, together with the radius 

 vector depends the paracentric velocity, and the generation 

 of the area. The fluxion of the area and of the time, is 

 therefore to be expressed in terms of those quantities, the in- 

 tegral of which will be the whole area, or time, correspond- 

 ing to any altitude, or vice versa, the time being given the al- 

 titude will be given. If we put the distance of the moving 

 body from the center of force = x, its distance at the com- 



Q 



mencement of the motion, = a, and z ==— , the fluxional for- 

 mula of Newton, for the area corresponding to any elapsed 



Qx- . 



time from the beginning, is — — - — — where v ABFD 

 & & 2-/ ABFD— z* 



is an area, or as a right line expressive of the velocity as ob- 

 tained in the 39th proposition. In a similar manner, the 

 area generated in a circle, of a radius, equal to the initial 



QX« 2 r 

 distance, will be expressed by 2 — ,___ — _ and that 



Q, X a, X' 

 of the arc itself by x% ,== 3 These are th& 



fluxonial expressions of the quadratures of curves, and there- 

 fore easily integrated. There will result the angular space, 

 or position of the body, and its distance from the center of 

 force, and from this a point in the trajectory, and all its 

 points or loci, may be expressed in terms of the radius vec- 

 tor, and the functions of an arc of a circle, and the curve 

 will be algebraical, when any sector of a circle can be ex- 

 pressed by a finite equation, equal to the sectoreal area, 

 generated by the initial radius vector, otherwise it will be. 

 transcendental. 



The celebrated Bernouilli, so often mentioned, some twen- 

 ty or thirty years after the publication of the Prineipia, pro- 

 duced what he called an analytical solution of this great 

 problem, or more properly, he converted Newton's geometri- 



