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The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.
On Triangles, by Regiomontanus
This is the title page of On Triangles by Regiomontanus (Johannes Müller) (1436-1476). Although the work was written in 1464, it was not published until 1533. The page can be translated as follows:
A complete translation and analysis of On Triangles is available in Barnabas Hughes, trans., Regiomontanus on Triangles (Madison: University of Wisconsin Press, 1967).
This page contains theorems 12 and 13 of Book II of On Triangles:
Theorem 12: If the perpendicular is given and the base and the ratio of the sides are known, then each side is known.
Theorem 13: If each of the segments of the base is known and also the ratio of the sides, then each side and the perpendicular may be found.
There is a calculation in the margin by an owner of this book. It would be interesting to work out the details.