Area-Time

The time required for the planet to traverse equal lengths of arc is measured by its distance from the sun. In this animation, the color of the lines from the center corresponds to the color of the arc along which it moves. Try to get a sense of the amount of time per section of arc, and see if the distance from the sun is a measure of that time.

Do you see a problem with this? If the circle were divided into 15° segments as it is here, then the distance from the sun at the start of a 15° arc would supposedly measure its speed for the entire 15°. But, the everywhere-present principle of gravitation requires a constant change of speed, not a series of abrupt changes.

We'll have to make the arcs smaller!

Here the arcs are cut up in 5° increments, but isn't it still false to imply that the speed is constant anywhere, even along an arc this small?

Why not follow Archimedes and, using an infinite number of such lines, make small triangles out of them. Since the triangles all have (nearly) equal bases, why not use the sum of their area to measure the sum of the distances?

“It therefore seemed to me I could conclude that by computing the area I would have the sum of the infinite distances, not because the infinite can be traversed, but because I thought that the measure of the faculty by which the collected distances mete out the times is contained in this area, so that we would be able to obtain it by knowing the area without an enumeration of least parts.”(p.419)

“Thus the area becomes a measure of the time or mean anomaly corresponding to the arc, since the mean anomaly measures the time.” (p.420)