Review of the Principia of Newton. 311 



Art. VI. — Review of the Principia of Newton. 



(Continued from Vol. XII, page 338.) 



In a review, which has for its objects, some account of the 

 inventions and discoveries of the greatest genius and inven- 

 tor, that has ever appeared on earth, and which aims at truth, 

 by asserting the rights due to an illustrious progenitor, in op- 

 position to plagiaries and pretenders, it would evidently be a 

 dereliction of its purpose, not to exhibit some of the greater 

 and more marvellous productions of our author, however lit- 

 tle they may be understood or appreciated by the readers of 

 the present times. 



We k have now come to that which has been denominated 

 the most noble problem, {problema nobilissimum,) viz! the 

 41st of the principia, we may add the epithet of the most 

 universal in the whole theory of motion. It is thus enunci- 

 ated in the translated work of our author. 



" Supposing a centripetal force of any kind, and granting 

 the quadrature of curvilinear figures, it is required to find as 

 well the trajectories in which bodies will move, as the times 

 of their motions in the trajectories found." 



The form of expression here used, viz. the quadrature of 

 curvilinear figures, would not be very intelligible to those not 

 well versed in the history of mathematics, neither would that 

 used in the 39th proposition, where the velocity of a falling 

 body is proved to be as a right line, whose power is as the 

 area Of a certain curve ; but in modern phraseology, the 

 quadrature of curves is the integral, or fluent of an express- 

 ion, involving only one variable, the function of the abscissa, 

 into its differential, which constitutes the differential of the 

 area ; and by the power the ancient mathematicians meant 

 the square or second power. 



The analysis of this proposition would be impossible with- 

 out the aid of the physical discoveries previously made in the 

 principia, viz. the laws of motion, the composition and reso- 

 lution of motion, the uniform and equable description of 

 areas, and the more profound principles of the 39th and 40th 

 propositions. We are hence led to the precepta or princi- 

 ples of the analysis, which consist in finding the angular 

 space passed over, and the altitude or distance of the body 

 from the centre of force, after any elapsed time. There are 



