Abstract
The earliest surviving trigonometric tables used to compute numerical values for geometrical magnitudes occur in Ptolemy’s Almagest (composed in the 2nd century AD). Current historical evidence cannot fix with precision the exact origins of such trigonometric tables. However, it seems likely that the Greek astronomer Hipparchos of Nicaea (2nd Century BC) was among the first, if not the very first to compute the ratio of chord to radius for a series of central angles in a circle, and to set the example of their use in astronomy for Ptolemy’s later work. By comparison, geometrical methods for the determination of magnitudes are considerably older, and have become highly formalized no later than the end of the 4th century BC, in Euclid’s Elements. This raises questions with regard to the comparative advantages of trigonometry over the older geometrical methods, and the particular emphasis that they received in the context of Greek mathematical astronomy.
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Ibid., p. 484 [H322]. Ptolemy gives the dates relative to the ruling years of the Roman emperor at the time. Thus he gives T 1 as “fifteenth year of Hadrian, Tybi 26/27 in the Egyptian calendar, 1 equinoctial hour after midnight, at about Gemini 21°.” The rendition into modern dates is included in Toomer’s translation.
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As I have shown elsewhere (Yavetz 2010), a similar difficulty emerges already in Aristarchus of Samos’s computation of the sizes and distances of the sun and moon. All four quantities can be found by a single construction, which is much simpler in principle than Aristarchus’s complicated geometrical argument leading to the upper and lower bounds of the desired results. In practice, however, the distance from the earth to the sun must be drawn around twenty meters long on a good flat surface, in order to consistently obtain close results to the mean values between Aristarchus’s boundaries.
References
Pedersen, O. (1974). A survey of the Almagest (pp. 273–283). Odense University Press.
Toomer, G. J. (translator) (1998). Ptolemy’s Almagest X.7 (pp. 484–498). Princeton: Princeton University Press.
Yavetz, I. (2010). Wandering stars and ethereal spheres: Landmarks in the history of Greek astronomy (Hebrew) (pp. 157–160). Jerusalem: Kineret, Zmora-Bitan, Dvir/Publishing House Ltd/The Hebrew University Magnes Press.
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Yavetz, I. (2017). Trigonometry, Construction by Straightedge and Compass, and the Applied Mathematics of the Almagest . In: Lenhard, J., Carrier, M. (eds) Mathematics as a Tool. Boston Studies in the Philosophy and History of Science, vol 327. Springer, Cham. https://doi.org/10.1007/978-3-319-54469-4_4
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