This one was hard, because I'm pretty sure there's an error in the transcription. The proof says that the proposition can be derived from Lemma 7 of the previous section, but I see no obvious way to get it from that. OTOH, I can easily see how to get it from Lemma 11.
The first 7 corollaries are great. The guy derives the inverse square law of gravity for circular orbits, assuming Kepler's law (T^2 ~ R^3). Awesome.
Then Corollary 9 threw me through a loop. The translators used the term "mean proportional" rather than "geometric mean." But I sorted it out.
This book is a hard slog, but worth it.