incapable of changing its place within the circle, and, conse- 

 quently, maintains invariably the same relation to every part of 

 the circle and its circumference, and to every body, axis, or ma- 

 thematical point, moving about on that circumference. At what- 

 ever point of the circumference the revolving axis may be; how- 

 ever much it may vary its relations to objects outside, its relation 

 to the centre is invariable, namely, it is always at the outer end 

 of the shortest straight line that can be drawn between it and the 

 centre, and which straight line, invariably perpendicular to the 

 circumference of a perfect circle, is its radius-vector. Each of 

 the radii also, whether considered as lines in the substance, or on 

 the flat surface of the circle, including the flitting radius-vector 

 itself, is by the same necessity of its nature as the radius of a 

 circle, just as incapable as the centre itself of changing its angular 

 relation to the circumference and the centre, and remains invari- 

 ably the shortest straight line between them at any given point 

 Whatever, therefore, can be truly predicated of any movement 

 or event taking place at the outer end of any one radius, would 

 be just as truly predicated of the same or a similar movement or 

 event if it were to happen at the circumference, at the outer end of 

 any other radius, or proportionally at the outer end of each of 

 them in succession. 



When a round body, as a globe or wheel, moves forward in 

 space, the direction of its line of motion is indicated by a line, 

 straight or curved as the case may be, passing through its axis, 

 parallel to, or concentric with, the road, rail, or mathematical line 

 traced by either side of its circumference ; and if it rotate or 

 turn round on its axis, the rotation is indicated by the passage 

 of every point of its circumference in succession past that line 

 of motion, and also past another line drawn perpendicularly to 

 the line of motion, and which perpendicular line, in the case of 

 orbital progression and revolution, is the radius-vector, virtually 

 identical with any one, and with all in their turn, of the fixed 

 radii of the orbit, and as incapable, by the necessity of its nature, 

 as any one of the fixed radii themselves, which it successively re- 

 presents, of turning round about on the axis of the revolving 

 body. If, on the contrary, no part of the revolving body's cir- 

 cumference pass the radius- vector and the line of motion, the 

 same respective points remaining continually in one with one or 

 other of these lines without any change of their relation to either 

 of the lines, then it cannot be more certain that the sun and the 

 moon revolve, or seem to revolve, about the whole heavens, than 

 that a body so remaining unchanged in all its relations to its own 

 line of motion and radius-vector, progresses without turning 

 round on its axis. The moon, however, by the admission and 

 declaration of all the astronomers of the last two centuries, re- 

 volves in her orbit precisely in these circumstances ; that is, with- 



