154 PHILOSOPHICAL TRANSACTIONS. [aNNO 1704. 



planets, as is most probable) to describe their respective courses : to pass over 

 these cases I say, yet even in those cases in which it returns into itself, and 

 completes its orbit, some of its eccentricities are so great, that about d and e 

 (fig. 2) the curve becomes convex towards the sun ; and therefore the planet 

 would require a centrifugal force from the sun, in order to describe this part of 

 its orbit; while at the same time both in places that are nearer to and farther 

 from A and b, there ought to be a centripetal force towards the sun. That is, 

 it must be granted, that the circumsolar bodies must move by such a law, that 

 at equal distances from the sun, in the one a centripetal, and in the other a 

 centrifugal force must take place ; which it is easy to perceive is very different 

 from the known laws of nature. And though none of the planets has so great 

 an eccentricity; yet since it is known to geometricians, that if all the species of 

 a figure, beyond a certain limit, are unfit for performing a natural effect, then 

 the remaining species of that figure within the limit cannot be admitted as 

 proper to perform the same effect ; it therefore follows, that this curve of 

 Cassini's, must necessarily be rejected out of astronomy ; not only for the 

 reasons alleged in Prop. 8. lib. 3, of the Elements of Astronomy, viz. that it 

 neither agrees with celestial observations because of the shortness of the less 

 axis, nor do physical reasons correspond with it, since for the description of 

 this there is required a centripetal force towards the sun, very different from 

 that which nature employs; but also because of its utterly impossibility. For 

 it is impossible that any species of this figure can be described by a planet, so 

 that the angles at a focus different from the sun may be proportional to the 

 times; for thus the area described by the radius vector could not be proportional 

 to the times. For the angle at one focus being increased by equal increments, 

 the cotemporary increments of the area at the other cannot become equal ; as 

 of late I had hastily imagined. 



In the last two figures (2 and 3) the greatest breadth of the figure is found, 

 if with the centre c a circle be described through the foci ; for this will cut the 

 curve in l, l, the points required. And the greatest ordinate kl is a third pro- 

 portional to the right lines gp and fd in the first of these, or a fourth propor- 

 tional to GP, GA, AP in both of them. 



DE remaining, the ordinate pp from the focus, is equal to the less semiaxis 

 c D, when the less axis is to the distance of the foci, as the side of a square to 

 its diagonal. If the distance of the foci be greater than in this ratio, then fp 

 will exceed cd. 



