1. I have already defined simple motion to be that, in which the several points taken in a moved body do in several equal times describe several equal arches. And therefore in simple circular motion it is necessary that every strait line taken in the moved body be always carried parallel to itself; which I thus demonstrate.
First, let A B (in the [first figure]) be any strait line taken in any solid body; and let A D be any arch drawn upon any centre C and radius CA. Let the point B be understood to describe towards the same parts the arch B E, like and equal to the arch A D. Now in the same time in which the point A transmits the arch A D, the point B, which by reason of its simple motion is supposed to be carried with a velocity equal to that of A, will transmit the arch B E; and at the end of the same time the whole A B will be in D E; and therefore A B and D E are equal. And seeing the arches A D and B E are like and equal, their subtending strait lines AD and BE will also be equal; and therefore the four-sided figure A B D E will be a parallelogram. Wherefore A B is carried parallel to itself. And the same may be proved by the same method, if any other strait line be taken in the same moved body in which the strait line A B was taken. So that all strait lines, taken in a body moved with simple circular motion, will be carried parallel to themselves.
Coroll. I. It is manifest that the same will also happen in any body which hath simple motion, though not circular. For all the points of any strait line whatsoever will describe lines, though not circular, yet equal; so that though the crooked lines A D and B E were not arches of circles, but of parabolas, ellipses, or of any other figures, yet both they, and their subtenses, and the strait lines which join them, would be equal and parallel.
Coroll. II. It is also manifest, that the radii of the equal circles A D and B E, or the axis of a sphere, will be so carried, as to be always parallel to the places in which they formerly were. For the strait line B F drawn to the centre of the arch B E being equal to the radius A C, will also be equal to the strait line F E or C D; and the angle B F E will be equal to the angle A C D. Now the intersection of the strait lines C A and B E being at G, the angle C G E (seeing B E and A D are parallel) will be equal to the angle D A C. But the angle E B F is equal to the same angle D A C; and therefore the angles C G E and E B F are also equal. Wherefore A C and B F are parallel; which was to be demonstrated.
If circular motion be made about a resting centre, and in that circle there be an epicycle whose revolution is made the contrary way, in such manner that in equal times it make equal angles, every strait line taken in that epicycle will be so carried, that it will always be parallel to the places in which it formerly was.
2. Let there be a circle given (in the second figure) whose centre is A, and radius A B; and upon the centre B and any radius B C let the epicycle C D E be described. Let the centre B be understood to be carried about the centre A, and the whole epicycle with it till it be coincident with the circle F G H, whose centre is I; and let B A I be any angle given. But in the time that the centre B is moved to I, let the epicycle C D E have a contrary revolution upon its own centre, namely from E by D to C, according to the same proportions; that is, in such manner, that in both the circles, equal angles be made in equal times. I say E C, the axis of the epicycle, will be always carried parallel to itself. Let the angle F I G be made equal to the angle B A I; I F and A B will therefore be parallel; and how much the axis A G has departed from its former place A C (the measure of which progression is the angle C A G, or C B D, which I suppose equal to it) so much in the same time has the axis I G, the same with B C, departed from its own former situation. Wherefore, in what time B C comes to I G by the motion from B to I upon the centre A, in the same time G will come to F by the contrary motion of the epicycle; that is, it will be turned backwards to F, and I G will lie in I F. But the angles F I G and G A C are equal; and therefore A C, that is, B C, and I F, (that is the axis, though in different places) will be parallel. Wherefore, the axis of the epicycle E D C will be carried always parallel to itself; which was to be proved.
Coroll. From hence it is manifest, that those two annual motions which Copernicus ascribes to the earth, are reducible to this one circular simple motion, by which all the points of the moved body are carried always with equal velocity, that is, in equal times they make equal revolutions uniformly.
This, as it is the most simple, so it is the most frequent of all circular motions; being the same which is used by all men when they turn anything round with their arms, as they do in grinding or sifting. For all the points of the thing moved describe lines which are like and equal to one another. So that if a man had a ruler, in which many pens' points of equal length were fastened, he might with this one motion write many lines at once.
Properties of simple motion.
3. Having shown what simple motion is, I will here also set down some properties of the same.