# Who invented the angles

## On the history of trigonometry

Angle measurement with the Jacob's stick

As early as 150 BC In BC Hipparchus, one of the greatest astronomers of antiquity, calculated a tendon table, but it was lost. He was also familiar with stereographic projection. One came from Menelaus (around 100 AD) *Sphere*which, however, has not been preserved in the original, only in an Arabic translation. Finally, Claudius Ptolemy (around 140 AD) in the later so-called *Almagest* the calculus of tendons and the trigonometry of the Greeks summarized in an overview.

Later trigonometry was further developed by the Arabs and Indians. Best known of the mathematicians in Eastern Arabic is Al Battani (died 929 AD), who already gives the side cosine theorem for the spherical triangle. Through the Persian Nassir al-Din al-Tusi (1201 - 1274) the flat and spherical trigonometry was expanded to some extent.

In Europe, Johann Müller from Königsberg in Franconia, who called himself Regiomontanus after his hometown and lived from 1436 - 1476, made trigonometry an independent branch of mathematics. He uses the sine as the only angle function and has dealt with plane trigonometry completely and spherical trigonometry to a large extent. A detailed sine value table also comes from him.

A great advance in the more convenient use of trigonometric formulas was brought about by Napier's (1550-1617) invention of the logarithms. They were then converted to base 10 by Briggs (1556-1630). At the same time triangulation and with it practical trigonometry in land surveying was introduced by Snellius (1580 - 1626).

Modern spherical geometry was founded by Leonhard Euler (1707 - 1783). He is the creator of today's representation and writing. The concept of the triangle was later expanded by Carl Friedrich Gauß (1777 - 1855), who introduced the im *Euler's triangle* included restrictions on the sides and angles. Gauß and Friedrich Wilhelm Bessel (1784 - 1846) also gave practical trigonometry its modern form by setting up precisely developed methods of calculation. Finally, the concept of the spherical triangle was conceived by Eduard Study (1862-1931) in the sense of group theory.

Cartography has developed along with geodesy and geography. The first surviving hand drawings of maps come from the *Geographical Hyphegesia* of Ptolemy, who later also *Cosmographia* was called. In the Middle Ages, many world maps were made by the monks, but they show numerous errors and misunderstandings. The first, still very flawed globe comes from Martin Behaim (1459 - 1506) from Nuremberg. Later, after a large number of coastal maps, the first usable world maps appeared. The greatest advance was the invention of the nautical chart by Gerhard Mercator (1512 - 1594). After that, many card designs are found and refined.

A script for a lecture on trigonometry can be found on this page.

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