GENERALIZATION OF PRINCIPLES. 353 



round by the motion of the eartli ; and obtained an erroneous result. 

 Kepler and Fermat attempted the same problem, and obtained solu- 

 tions different from that of Galileo, but not more correct. 



Even Newton, at an early period of his speculations, had an erro- 

 neous opinion respecting this curve, which he imagined to be a kind 

 of spiral. Hooke animadverted upon this opinion when it was laid 

 before the Royal Society of London in 1679, and stated, more truly, 

 that, supposing no resistance, it would be " an eccentric ellipsoid," that 

 is, a figure resembling an ellipse. But though he had made out the 

 approximate form of the curve, in some unexplained way, we have no 

 reason to believe that he possessed any means of determining the 

 mathematical properties of the curve described in such a case. The 

 perpetual composition of a central force with the previous motion of 

 the body, could not be successfully treated without the consideration 

 of the Doctrine of Limits, or something equivalent to that doctrine. 

 The first example which we have of the right solution of such a prob 

 lem occurs, so far as I know, in the Theorems of Huyghens concern- 

 ing Circular Motion, which were published, without demonstration, at 

 the end of- his Horologium Oscillatorium, in 1673. It was there as- 

 serted that when equal bodies describe circles, if the times are equal, 

 the centrifugal forces will be as the diameters of the circles ; if the 

 velocities are equal, the forces will be reciprocally as the diameters, 

 and so on. In order to arrive at these propositions, Huyghens must, 

 virtually at least, have applied the Second Law of Motion to the limit- 

 ing elements of the curve, according to the way in which Newton, a 

 few years later, gave the demonstration of the theorems of Iluyghens 

 in the Principia. 



The growing persuasion that the motions of the heavenly bodies 

 about the sun might be explained by the action of central forces, gave 

 a peculiar interest to these mechanical speculations, at the period now 

 under review. Indeed, it is not easy to state separately, as our present 

 object requires us to do, the progress of Mechanics, and the progress 

 of Astronomy. Yet the distinction which we have to make is, in its 

 nature, sufficiently marked. It is, in fact, no less marked than the dis- 

 tinction between speaking logically and speaking truly. The framers 

 of the science of motion were employed in establishing those notions, 

 names, and rules, in conformity to which all mechanical truth must be 

 expressed ; but what was the truth with regard to the mechanism of 

 the universe remained to be determined by other means. Physical As- 

 tronomy, at the period of which we speak, eclipsed and overlaid theo 

 VOL. I. 23 



