Hipparchus

100

Hipparchus : biography

190 BC –

Modern speculation

Hipparchus was in the international news in 2005, when it was again proposed (as in 1898) that the data on the celestial globe of Hipparchus or in his star catalog may have been preserved in the only surviving large ancient celestial globe which depicts the constellations with moderate accuracy, the globe carried by the Farnese Atlas. There are a variety of mis-steps , "Farnese Atlas Celestial Globe, Proposed Astronomical Origins", 2005. in the more ambitious 2005 paper, thus no specialists in the area accept its widely publicized speculation., "Epoch of the Constellations on the Farnese Atlas and their Origin in Hipparchus’s Lost Catalog", Journal for the History of Astronomy, May, 2005 versus Journal for the History of Astronomy, February, 2006.

Lucio Russo has said that Plutarch, in his work On the Face in the Moon, was reporting some physical theories that we consider to be Newtonian and that these may have come originally from Hipparchus;Lucio Russo, The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had To Be Reborn, (Berlin: Springer, 2004). ISBN 3-540-20396-6, pp. 286–293. he goes on to say that Newton may have been influenced by them.Lucio Russo, The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had To Be Reborn, (Berlin: Springer, 2004). ISBN 3-540-20396-6, pp. 365–379. According to one book review, both of these claims have been rejected by other scholars.Mott Greene, "The birth of modern science?" Review of The Forgotten Revolution, Nature 430 (5 August 2004): 614.

A line in Plutarch’s Table Talk states that Hipparchus counted 103049 compound propositions that can be formed from ten simple propositions; 103049 is the tenth Schröder–Hipparchus number and this line has led to speculation that Hipparchus knew about enumerative combinatorics, a field of mathematics that developed independently in modern mathematics.

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 of the orbits of the Moon and Sun. He tabulated values for the chord function, which gives the length of the chord for each angle. He did this for a circle with a circumference of 21600 and a radius (rounded) of 3438 units: this circle has a unit length of 1 arc minute along its perimeter. He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord of an angle equals twice the sine of half of the angle, i.e.:

chord(A) = 2 sin(A/2).

He described the chord table in a work, now lost, called Tōn en kuklōi eutheiōn (Of Lines Inside a Circle) by Theon of Alexandria (4th century) in his commentary on the Almagest I.10; some claim his table may have survived in astronomical treatises in India, for instance 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.Toomer, "The Chord Table of Hipparchus" (1973).

For his chord table Hipparchus must have used a better approximation for π than the one from Archimedes of between 3 + 1/7 and 3 + 10/71; perhaps he had the one later used by Ptolemy: 3;8:30 (sexagesimal) (Almagest VI.7); but it is not known if he computed an improved value himself.

But some scholars do not believe Arayabhatta’s Sin table has anything to do with Hipparchus’s chord table which does not exist today. Some scholars do not agree with this hypothesis that Hipparchus constructed a chord table. Bo. C Klintberg states " With mathematical reconstructions and philosophical arguments I show that Toomer’s 1973 paper never contained any conclusive evidence for his claims that Hipparchus had a 3438′-based chord table, and that the Indians used that table to compute their sine tables. Recalculating Toomer’s reconstructions with a 3600′ radius — i.e. the radius of the chord table in Ptolemy’s Almagest, expressed in ‘minutes’ instead of ‘degrees’ — generates Hipparchan-like ratios similar to those produced by a 3438′ radius. It is therefore possible that the radius of Hipparchus’s chord table was 3600′, and that the Indians independently constructed their 3438′-based sine table." Reference: Hipparchus’s 3600′-Based Chord Table and Its Place in the History of Ancient Greek and Indian Trigonometry, Bo C. Klintberg, Indian Journal of History of Science 40 (2):169-203 (2005)