CHORD
application for Android (Schrausser, 2023): Famous table of chord lengths according to Ptolemy's Almagest (1515, fol. 7r ff.) converted into decimal values and calculated in comparison using the sine function, see Halma (1813, p. 38 ff.), Heiberg (1898, p. 48 ff.) or Toomer (1984, p. 57 ff., 1998, res.), c.f. chords_time
tables, Schrausser (2024), respectively.
Chord lengths
Chord lengths
Sixtieth is the average interpolation number to be added to length
Lengths
This is equivalent in terms of content to distance
In the absence of trigonometric sine functions, however, no calculation was made with distance parameters
were used and interpolated to the corresponding angle values of expansion:
Figure 3.
Chord parameters
Chord length values
Differences chords
, chords_tab
or chords.xlsx
tables.
Using this method along with methods for parallax determination, Ptolemy was able to determine e.g. Moon's distance and radius quite accurate:
"We have explained in the Almagest [...] that the least distance of the Moon is 33 earth radii, and its greatest distance 64 earth radii [...] ", (Goldstein, 1967, p. 7).
Goldstein, B. R. (1967). The Arabic Version of Ptolemy’s Planetary Hypotheses. Transactions of the American Philosophical Society, 57(4), 3-55. DOI:10.2307/1006040
Halma, N. (1813). Composition mathématique de Claude Ptolémée. Traduite pour la première fois du grec en français, sur les manuscrits originaux de la Bibliothèque Impériale de Paris, par M. Halma; et suivie des notes de M. Delambre, ... A Paris, chez Henri Grand, libraire, Rue Saint-André-des-Arcs, N° 51. (Mathematical composition of Claude Ptolemy. Translated for the first time from Greek into French, on the Original Manuscripts of the Imperial Library of Paris...) https://gallica.bnf.fr/ark: /12148/bpt6k62056165?rk=21459;2
Heiberg, J. L. (1898). Claudii Ptolemaei Opera quae exstant omnia. Syntaxis Mathematica. Vol. 1. 1. Lipsae: In aedibus B. G. Teubneri. https://archive.org/details/pt1claudiiptolemaei01ptoluoft/page/n561/mode/2up
Ptolemaeus, C. (1515). Almagestum CL. Ptolemei Pheludiensis Alexandrini astronomorum principis: Opus ingens ac nobile omnes Caelorum motus continens. Felicibus astris eat in lucem: Ductu Petri Liechtenstein Coloniensis Germani. Anno Virginei Partus, 1515, Die 10. Ia. Venetiis ex officina eiusdem litteraria. (Almagest of CL. Ptolemy Pheludiens, head of the Alexandrian astronomers: A great and noble work containing all the movements of the heavens...) DOI:10.3931/e-rara-206
Schrausser, D. G. (2023). Schrausser/Ptolemy-s-table-of-chords: Calculator (v3.5.7). Zenodo. DOI:10.5281/zenodo.7948117
———. (2024). Ptolemy’s Table of Chords: Implications Considered and Discussed. OSF Open Science Framework, 05/24. DOI:/10.31219/osf.io/dv4nz
Toomer, G. J. (1984). Ptolemy's Almagest. Duckworth, London & Springer, New York. DOI:10.2307/631776
———. (1998). Ptolemy’s Almagest. Revised. Princeton, NJ: Princeton University Press. https://www.amazon.de/Ptolemys-Almagest-Ptolemy/dp/0691002606