Chapter 59

Now that we have the true, elliptical path of the planet determined in chapter 58, the question remains of how to apply mathematics to this means of generation − how can we know where the planet will be? How can the distances from the planet to the sun be added without the errors discussed in earlier chapters where the planet moved in a circle? Kepler introduces several protheorems to get to his demonstration, some of which will be presented here.

I, II, III
Protheorem I states that the same ratio holds, between the perpendiculars to the ellipse and the circle, no matter where on the circle they are. The ratios between the red-orange and green lines is the same at different points on the figure.

Protheorem II states that the ratio of the area of the ellipse to the area of the circle is the same as this ratio between lengths in I.

Protheorem III tells us that this same ratio holds between the red-orange and green areas.

"If the circle be divided into any number of equal arcs by perpendiculars such as these, the ellipse is divided into unequal arcs, whose ratio [to the arcs of the circle] is greatest near the vertices and least in the middle positions."

At the middle longitudes, the ellipse is nearly as long as the circle, while at the apsides, it is significantly shorter.

V

The entire elliptical circumference is approximately the arithmetic mean between the circle on the greater diameter and the circle on the smaller diameter.

See chapter 48 for this one.

The gnomons of squares divided proportionally are to one another as the squares.

The red gnomon on the left is larger than that on the right, in the same ratio as the purple square on the left is larger than that on the right.

VII

If from the end of the shorter semidiameter on the circumference of an ellipse, a line equal to the longer semidiamter be extended, ending at the longer semidiamter, the distance between that point of intersection and the center is the side of a square equal to the gnomon that the square of the longer semidiameter places about the square of the shorter semidiamter.

The square on the hypothenuse, with the purple square of the shorter semidiameter removed, leaves the red gnomon, the area of the side on the diamter. This technique is the way you can find the focus of an ellipse you are given.

VIII

If a circle be divided into any number (or an infinity) of parts, and the points of division be connected with some point within the circumference of the circle other than the center, and also be connected with the center, the sum of the lines drawn from the center will be less than the sum of those from the other point.

Here, the purple line connecting two points opposite the center is shorter than the sum of the red lines connecting those points via the focus. This came up in chapter 40 as an objection to using area to measure the sum of distances: the sum of the distances is larger than the area of the circle.

If, on the other hand, instead of the lines from the point other than the center, those lines be taken which are bounded by perpendiculars drawn from that point to the lines which are drawn to the center − that is, if, in the terms of ch. 39 and 57, the diamteral distances are taken in place of the circumferential ones − then their sum will equal the sum of those drawn from the center.

Rather than using the red lines here, if the yellow-green and blue-green ones are used, their sum is a diameter.

X

The area of the ellipse is also not suited to measuring the sum of distances.

With these preliminaries completed, I shall now proceed to the demonstration.

If an ellipse divided by perpendiculars dropped from equal arcs of the circle, as in protheorem 4 above, the points of division of the circle and the ellipse be connected to the point that was found in protheorem 7, I say that those that are drawn to the circumference of the circle are the circumferential distances, while those that are drawn to the circumference of the ellipse are the diametral, which are established at an equal number of degrees from the apsides of the epicycle.

This means of constructing the ellipse is that from chapter 58. By construction, NM = KT. By circumferential and diametral distances, look at the figure on the left.

I say that NK is the circumferential distance αδ (this was proven in ch. 2) and NM is the diametral distance ακ.

So you can see that while δ is on the circumference of the epicycle, κ is on the line cutting its diameter in this diagram first introduced in chapter 39. Remarkably, in using the construction of chapter 58, the distances from the sun to points on the ellipse (NM), can be added to get the same area as that of the circle. This means that we can use area to measure the sum of distances without reservation!
XII

Again, it is also clear from the same that

The area of the circle, both as a whole and in its individual parts, is the genuine measure of the sum of the lines by which the arcs of the elliptical planetary path are distance from the sun's center.

XIII

Some would raise the objection that equal elliptical arcs should be taken, rather than the equal circular arcs presented here.

The reply is made that the arc of the ellipse on which the times are measured by the area AKN should by all means be divided into unequal parts, with those near the apsides being smaller.

Kepler realizes that were he not to use unequal arcs, the planet would be made to move too fast at the apsides. This protheorem, along with the remaining XIV and XV are complex, and need to be read through themselves.

There are some matters which no mind, however gifted, can present in such a way as to be understood in a cursory reading. There is need of meditation, and a close thinking through of what is said. (p.591)

Confirmation

When Kepler tries out this hypothesis, he finds that, instead of being up to 5½' off like the puff-cheeked hypothesis,

when NM was so applied as to end on KL, then when the equated anomaly MNA was applied to the mean anomaly AKN, it agreed exactly with the vicarious hypothesis.

Here is an animation of the vicarious hypothesis superposed upon the true elliptical orbit of chapter 58. It would be useful to look back to chapter 58 first. The vicarious hypothesis is drawn in purple. There are three purple dots and a circle. The circle is the path, the two dots on the line of apsides the center and the equant, and the small spot on the path is the planet's position as given by the vicarious hypothesis. The true values are on the left, and the match is perfect: the sun sees the planet as given in chapter 58 at the same sidereal longitude as the vicarious hypothesis would place it. Since it is hard to see, another diagram with exaggerated eccentricity is on the right.