“While I am thus celebrating a triumph over the motions of Mars, and fetter him in the prison of tables and the leg-irons of eccentric equations, considering him utterly defeated, it is announced in various places that the victory is futile, and war is breaking out again with full force. For while the enemy was in the house as a captive, and hence lightly esteemed, he burst all the chains of the equations and broke out of the prison of the tables. That is, no method administered geometrically under the direction of the opinion of ch. 45 was able to emulate in numerical accuracy the vicarious hypothesis of chapter 16 (which has true equations derived from false causes)...
“...I send new reinforcements of physical reasoning in a hurry to the scattered troops and old stragglers, and informed with all diligence, stick to the trail without delay in the direction whither the captive has fled.” (p.508)

“I shall begin by seeking out the distances of several places on the eccentric where the evidence was most trustworthy.”

We can find the distances of Mars from the sun at various positions along its eccentric through non-acronychal observations with the law of sines. We'll go through one example, and then build up all of the distances that Kepler develops.

Here is the example from page 515. Follow along with the color diagram, taken from a God's eye view above the plane of the eccentric. Yellow is the sun, blue the earth, and red Mars. Kepler:

“Second, I shall prove the same thing at parts closer to aphelion. On 1589 April 5 at 11^{h}33^{m} Mars was observed at 7°31'10" Scorpio with latitude 1°28'13" N. It was near the meridian, and consequently there were no horizontal variations. The mean longitude was concluded to be 7^{s}9°46'8". And the aphelion was at 4^{s}28°51'8". Therefore the mean anomaly was 70°55'0" which corresponds to an equated anomaly of 61°17'35", by the vicarious hypothesis. And so the eccentric position was 0°8'43" Scorpio. The sun's position was 25°52'43" Aries, its distance from the earth 100,560, the angle at the earth 11°38'27", at the planet 7°22'27".”

“Therefore, the distance of Mars from the sun was 158,090.”

This uses the law of sines: 100,560 : Sin(7°22'27") = Mars-Sun distance : Sin(11°38'27") Mars-Sun distance = 100,560 × Sin(11°38'27") / Sin(7°22'27") = 158,090.

The same technique is applied to distances all along the path of the planet: at 87° and −87°, at 71° and −71°, at 43° and −43°, at 12° and −12°, at 113° and −113°, and at 162° and −162°. He always finds that lengths on opposite sides of the line of apsides are the same. Here they are: