378 THE MATHEMATICAL PRINCIPLES [BOOK IT to a certain limit, then again swifter near the circumference, certainly neither the sesquiplicate, nor any other certain and determinate ratio, can obtain in them. Let philosophers then see how that phenomenon of the sesquiplicate ratio can be accounted for by vortices. PROPOSITION LIII. THEOREM XLI. Bodies carried about in a vortex, and returning- in the same orb, are of the same density with the vortex, and are moved according to the same law with the parts of the vortex, as to velocity and direction oj motion. For if any small part of the vortex, whose particles or physical points preserve a given situation among each other, be supposed to be congealed, this particle will move according to the same law as before, since no change is made either in its density, vis insita, or figure. And again ; if a congealed or solid part of the vortex be of the same density with the rest of the vortex, and be resolved into a fluid, this will move according to the same law as before, except in so far as its particles, now become fluid, may be moved among themselves. Neglect, therefore, the motion of the particles among themselves as not at all concerning the progressive motion of the whole, and the motion of the whole will be the same as before. But this motion will be the same with the motion of other parts of the vortex at equal distances from the centre; because the solid, now resolved into a fluid, is become perfectly like to the other parts of the vortex. Therefore a solid, if it be of the same density with the matter of the vortex, will move with the same motion as the parts thereof, being relatively at rest in the matter that sur rounds it. If it be more dense, it will endeavour more than before to re cede from the centre ; and therefore overcoming that force of the vortex, by which, being, as it were, kept, in equilibrio, it was retained in its orbit, it will recede from the centre, and in its revolution describe a spiral, re turning no longer into the same orbit. And, by the same argument, if it be more rare, it will approach to the centre. Therefore it can never con tinually go round in the same orbit, unless it be of the same density with the fluid. But we have shewn in that case that it would revolve accord ing to the same law with those parts of the fluid that are at the same or equal distances from the centre of the vortex. COR. 1. Therefore a solid revolving in a vortex, and continually going round in the same orbit, is relatively quiescent in the fluid that carries it. COR. 2. And if the vortex be of an uniform density, the same body may revolve at any distance from the centre of the vortex. SCHOLIUM. Hence it is manifest that the planets are not carried round in corporeal vortices ; for, according to the Copernican hypothesis, the planets going