[1] Azure transparent spheres conceived by the ancients to surround the
earth one within another, and to carry the heavenly bodies in their
revolutions.
[2] Book I., Prop. i. The areas which revolving bodies describe by radii
drawn to an immovable centre of force do lie in the same immovable
planes and are proportional to the times in which they are described.
Prop. ii. Every body that moves in any curve line described in a plane
and by a radius drawn to a point either immovable or moving forward with
a uniform rectilinear motion describes about that point areas
proportional to the times is urged by a centripetal force directed to
that point.
Prop. iii. Every body that, by a radius drawn to another body, howsoever
moved, describes areas about that centre proportional to the times is
urged by a force compounded out of the centripetal force tending to that
other body and of all the accelerative force by which that other body is
impelled.
[3] If the periodic times are in the sesquiplicate ratio of the radii,
and therefore the velocities reciprocally in the subduplicate ratio of
the radii, the centripetal forces will be in the duplicate ratio of the
radii inversely; and the converse.
[4] _i.e._, showing convexity when in such a position as that, to an
observer on the earth, a line drawn between it and the sun would subtend
an angle of _90_° or thereabouts.