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Can someone give a history of coordinates? When did coordinates first appear, which was the first ever coordinate system? In which field were they used?

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The coordinates on a surface (sphere) first appear in the Greek astronomy, in the very beginning of Hellenistic period (~300s BC). Since almost no sources from that time survive, it is speculated that one purpose was to make celestial globes (One of them is credited to Archimedes. This is attested by some Roman authors, but nobody describes it in detail). Hipparchus is said to make the first star catalog, so he must have used some coordinates in it, unfortunately most works of Hipparchus are lost. (There is a conjecture that he used some mixture of ecliptic and equatorial coordinates).

At the same time Eratosthenes used coordinates in geography. Unfortunately no works of Eratosthenes survive. We know about the work of Hipparchus and Eratosthenes from later sources which cite them.

Our main source on Greek astronomy is the book of Ptolemy, Mathematical Synthesis (better known its Arabic name Almagest), which is written in the second century AD. In it he describes the work of his predecessors, in particular Hipparchus.

Ptolemy and other Greek astronomers made use of 3 different coordinate systems for the sky: ecliptic (to describe the motion of Sun, Moon and planets), equatorial (to describe stars) and horizontal (to describe everything as seen by the observer), and the familiar to us geographic coordinates, which are similar to equatorial coordinates in the sky. All these coordinate systems are still widely used in astronomy and geography, almost without change.

On the pure mathematical side, Apollonius used some rudimentary coordinates in his work on conic section. There is no apparent connection of his work with contemporary work with spherical coordinates in astronomy and geography. He was a contemporary of Eratosthenes, slightly younger.

Remark. The first of the two ecliptic coordinates, longitude, comes from Babylon. The Greeks took from Babylonians the idea to divide the ecliptic into 12 "signs" and each sign into 30 degrees. So if we include the coordinate on a circle to coordinate systems, then the priority should be probably assigned to Babylonians. Neugebauer in his book "The exact sciences in antiquity" estimates that division of the circle into signs and degrees was introduced between 500 and 400 BC.

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