Editing Hipparchus (section)

==Geometry, trigonometry and other mathematical techniques==
Hipparchus was recognized as the first mathematician known to have possessed a [[trigonometric table]], which he needed when computing the [[eccentricity (orbit)|eccentricity]] of the [[orbit]]s of the Moon and Sun. He tabulated values for the [[chord (geometry)|chord]] function, which for a central angle in a circle gives the length of the straight line segment between the points where the angle intersects the circle. He may have computed this for a circle with a circumference of 21,600 units and a radius (rounded) of 3,438 units; this circle has a unit length for each arcminute along its perimeter. (This was “proven” by Toomer,{{r|toomer1974-chordtable}} but he later “cast doubt“ upon his earlier affirmation.{{sfn|Toomer|1984|page=215}} Other authors have argued that a circle of radius 3,600 units may instead have been used by Hipparchus.{{r|klintberg2005}}) He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord subtended by a central angle in a circle of given radius {{mvar|R}} equals {{mvar|R}} times twice the [[sine]] of half of the angle, i.e.:

:<math>\operatorname{chord} \theta = 2R \cdot \sin\tfrac12\theta</math>

The now-lost work in which Hipparchus is said to have developed his chord table, is called ''Tōn en kuklōi eutheiōn'' (''Of Lines Inside a Circle'') in [[Theon of Alexandria]]'s fourth-century commentary on section I.10 of the ''Almagest''. Some claim the table of Hipparchus may have survived in astronomical treatises in India, such as the ''[[Surya Siddhanta]]''. Trigonometry was a significant innovation, because it allowed Greek astronomers to solve any triangle, and made it possible to make quantitative astronomical models and predictions using their preferred geometric techniques.{{r|toomer1974-chordtable}}

Hipparchus must have used a better approximation for [[pi|{{mvar|π}}]] than the one given by [[Archimedes]] of between {{frac|3|10|71}} (≈&thinsp;3.1408) and {{frac|3|1|7}} (≈&thinsp;3.1429). Perhaps he had the approximation later used by Ptolemy, [[sexagesimal]] 3;08,30 (≈&thinsp;3.1417) (''Almagest'' VI.7).

Hipparchus could have constructed his chord table using the [[Pythagorean theorem]] and a theorem known to Archimedes. He also might have used the relationship between sides and diagonals of a [[cyclic quadrilateral]], today called [[Ptolemy's theorem]] because its earliest extant source is a proof in the ''Almagest'' (I.10).

The [[stereographic projection]] was ambiguously attributed to Hipparchus by [[Synesius]] (c. 400 AD), and on that basis Hipparchus is often credited with inventing it or at least knowing of it. However, some scholars believe this conclusion to be unjustified by available evidence.{{r|synesius}} The oldest extant description of the stereographic projection is found in [[Ptolemy]]'s [[Planisphaerium|''Planisphere'']] (2nd century AD).{{r|neugebauer1949}}

Besides geometry, Hipparchus also used [[arithmetic]] techniques developed by the [[Chaldea]]ns. He was one of the first Greek mathematicians to do this and, in this way, expanded the techniques available to astronomers and geographers.

There are several indications that Hipparchus knew spherical trigonometry, but the first surviving text discussing it is by [[Menelaus of Alexandria]] in the first century, who now, on that basis, commonly is credited with its discovery. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) Ptolemy later used spherical trigonometry to compute things such as the rising and setting points of the [[ecliptic]], or to take account of the lunar [[parallax]]. If he did not use spherical trigonometry, Hipparchus may have used a globe for these tasks, reading values off coordinate grids drawn on it, or he may have made approximations from planar geometry, or perhaps used arithmetical approximations developed by the Chaldeans.