Thread by @matemagikern

I just finished the book Heavenly mathematics. Here are nine things I learned, and one thing I still wonder: #mathreads
1. The word trigonometry comes from the book “Trigonometria” by Bartholomew Pitiscus in 1600.

2. To describe the position of planets and stars on the celestial sphere, one needs a coordinate system. There are (at least) three different ones, using in turn the celestial equator, the suns trajectory (the ecliptic) or the horizon, as its base. -->

Determining the coordinates requires solving spherical triangles – hence the need for spherical trigonometry. Translating between the coordinate systems was one of the primary tasks of ancient astronomers.

3. The problems facing ancient astronomers were determining sides of spherical triangles, where the sides are arcs, not planar triangles, where the sides are straight lines. But our common trigonometric functions were still used. How come? Because Menelaos theorem shows how...

... a theorem of planar triangles, can be translated into a theorem of spherical triangles, using the fact that certain ratios of line segments, can be replaced by the sines of the corresponding arcs.

4. The sine function was invented in India, some time after the Greeks invented the chord (where chord v in our modern terminology is equal to 2r sin (v/2)).

5. Trigonometry was first used for navigation by Venetian merchant ships in the 14th century.

6. The French mathematician Albert Girard (1595-1632) was one of the first to use the abbreviations sin, tan and sec.

7. Did you know that trigonometry was used to solve religious problems? The arabic calendar is lunar. For muslims to know when to fast, they (or rather, their astronomers) needed to keep track of the phases of the moon. Finding the direction of prayer (Kaba) also required trig.

8. You may know that John Napier invented the logarithm – the tool that turns a difficult multiplication problem into a simple addition. But did you know that the motivation behind Napier’s logarithms was to simplify trigonometric calculations? In fact...

...Napier’s logarithmic tables, didn’t actually include pure logarithms, but logarithms of sines!

9. Curiously enough, the first rigorous proof of Euler’s polyhedral formula (V – E + F = 2) was a piece of spherical trigonometry! (Legendre)

Question: So it seems that ancient astronomers were well aware of the fact that the sum of the angles in a spherical triangle always exceeds 180 degrees. Yet, non-Euclidean geometries were not invented/found until the 19th century. How come?