than the radius which accompanies it ; and, on the contrary, 

 were that point of circumference really to revolve about the 

 body's axis, accompanied by the radius-vector, the indispensable 

 condition of such a revolution would be the setting of the radius- 

 vector free from its attachment to the orbit's centre, and then its 

 inner point, so set free, would be seen moving away from the 

 centre to the outside of the orbit, and back again, just as often 

 as the body turned round on its axis, whether that happened 

 once only, or twice, or a thousand times during a single revolu- 

 tion in its orbit. 



But the radius-vector is not a material vinculum capable of 

 being thus detached and moving away from the centre. It is 

 merely a mathematical or imaginary line, forming the shortest 

 straight line between the revolving body's centre and the centre 

 of the orbit ; and this character it preserves at every point all 

 round the orbit's circumference. But these points of circumfe- 

 rence remain invariably at the same distance from the centre ; 

 and the centre or axis of the revolving body has precisely the 

 same relation to that centre as the point of the orbit which it 

 may occupy for the time, so that its radius-vector is successively 

 identical with, or occupies precisely the same place as, the radii 

 one after another of the points which it passes, and is conse- 

 quently invariably perpendicular to tlie planet's line of motion, 

 or the orbit which the planet's axis describes ; and round that 

 axis it could not revolve without losing its perpendicularity, and 

 ceasing to be the radius-vector, or, simply, could not revolve at 

 all. If a material vinculum, acting as a radius-vector, were to 

 be set free from attachment to the orbit's centre, it would then 

 become virtually an outlying arm of the planet itself, partaking 

 of all its movements, whether of orbital revolution or rotation ; 

 whereas, on the contrary, the true radius-vector, by the necessity 

 of its nature as such, takes part in the orbital revolution only, 

 and is never, and cannot be, in the slightest degree affected by 

 the planet's rotation on its own axis. The true immaterial radius- 

 vector remains invariably, in all circumstances, the shortest 

 straight line between the revolving axis and the immovable 

 centre round which it revolves. It follows of course the forward 

 movement of the revolving axis, but never does nor can revolve 

 about it, any more than the centre itself of the orbit, which is 

 certainly fixed in its place and immovable. If, then, the inva- 

 riable rectangular radius-vector cut invariably the same point of 

 the planet's circumference, that of itself is the most direct, the 

 surest, and the most incontrovertible proof that the revolving 

 body does not turn round on its own axis ; for to say that it 

 does so, in such circumstances, is no less preposterous than to up- 

 hold that the immovable centre of the orbit leaves its place, and 



