Abstract
Ptolemy's main scientific interest was the mathematical description of the phenomena of nature. In pure mathematics he did but little work, and what he wrote had but little value. Thus Simplicius tells us that the gifted Ptolemy in his book On Dimension showed that there are not more than three dimensions; for dimensions must be deter-minate, and determinate dimensions are along perpendicular straight lines, and it is not possible to find more than three straight lines at right angles one to another, two of them determining a plane and the third measuring depth; therefore, if any other were added after the third dimension, it would be completely immeasurable and undetermined In libr. de caelo i, 1; trans. Thomas ii, 411).
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- 1.
The principal works on the history of Greek logistics are the papers by Klein (1934) and Vogel (1936). The importance of astronomical calculations and tables for the development of logistics is hinted at by Wussing, 1965, p. 197.
- 2.
A Greek treatise called Introduction to the Almagest and extant in at least 24 MSS has been examined by Mogenet (1956) who ascribed it to the Byzantine mathematician Eutocius (ca. A.D. 500). It deals with the isoperimetric problem, and multiplication, division, and root extraction in the sexagesimal system. In the Middle Ages there were several introductions to the so-called Algo-rismus de minutiis which was taught as a separate course in the universities and dealt with both decimal and sexagesimal fractions.
- 3.
See to the following Ideler (1812), Delambre (1817, ii, 36 ff.), Cantor (1892, 350 f.), Braunmuhi (1900, i, 10-30) and Aaboe (1964, 101 ff.).
- 4.
A reconstruction by G. J. Toomer of Hipparchus' trigonometrical table is to be published in Centaur us, vol. 18.
- 5.
The various classical solutions based on other methods are given by Thomas, i, pp. 346-363.
- 6.
Cf. Tropfke, 1928, p. 466 f. In fact, the inequality was used (without proof) by Aristarchus in his book On the Sizes and Distances of the Sun and Moon (see Heath 1913 p. 333), by Archimedes in The Sand-Reckoner (transl. Heath p. 226), and by Menelaus (see Bjornbo, 1902, p. 112). But there is no evidence that it was proved until Ptolemy provided the proof given above.
- 7.
For a brief survey of Ptolemy's trigonometry, see Czwalina (1927). A short, general account of the early history of both plane and spherical trigonometry with much emphasis on Muslim and Indian contributions has been given by Kennedy (1969). Luckey (1940) deals in greater detail with spherical trigonometry among the Arabs.
- 8.
Ptolemy's dependence on Menelaus appears from the fact that the proofs found in the Almagest are abbreviated versions of those found in Leiden MS 930 containing an Arabic translation of the Sphaerica made about A.D. 1007 by al-Biruni's teacher Abu Nasr Mansur (see Suter, 1900, pp. 81 and 225). This was pointed out by Bjornbo (1902, p. 88). Abu Nasr Mansur's contributions to spherical trigonometry and astronomy have been examined by C. Jensen (1971).
- 9.
This is the accepted opinion of historians like Zeuthen, Cantor, v. d. Waerden, and others. The view that the discovery of incommensurables provoked a crisis in the history of Greek ma- thematics has recently been challenged by H.-J. Waschkies, Eine neue Hypothese zur Entdeckung der inkommensurablen Grossen durch die Griechen, TruesdelFs Archive for History of Exact Sciences, 7, (1970-71), pp. 325-353.
- 10.
Another but less accurate way of dealing with a continuously variable geometrical model would be to construct a mechanical instrument simulating the behaviour of the model. This led to various types of planetaria of which some at least were invented in Antiquity by Archimedes {Works, transl. Heath, p. xxi), Heron (Drachmann 1971), and others. It seems that such instru- ments were not used for computations, unlike the Mediaeval aequatorea which were real analogue computers; their prototype is the device described by Proclus {Hyp. Ill, 4, p. 73), cf. note 12.1, page 355.
- 11.
Here one might quote Euler (1748, I, 4) who gives the following definition: Functio quantitatis variabilis est expressio analytica quomodocunque composita ex ilia quantitate variabili et numeris seu quantitatibus constantibus, continuing with a classification of functions based on their ana- lytical expressions. This concept went into many text-books, and was popularized by VEncy- clopedie (vol. XIV, Lausanne et Berne 1779, p. 855). Montucla (Histoire des MathSmatiques III, nouv. ed. Paris, An X (1802), p. 265) gave an even more restrictive definition of a function as an expression algebrique.
- 12.
It is customary to mention Dirichlet (1837) as the originator of this concept of function: Man denke sich unter a und b zwei feste Werthe und unter x eine veränderliche Grösse, welche nach und nach alle zwischen a und b liegenden Werthe annehmen soil. Entspricht nunjedem x ein einziges, end-liches 7, und zwar so, dass, wdhrend x das Intervall von a bis b stetig durchlauft, y = f(x) sich eben-falls allmahlich verandert, so heisst y eine stetige oder continuirliche Function von xfiir dieses Inter-vall. Es ist dabei gar nicht nd'thig, dass y in diesem ganzen Intervalle nach demselben Gesetze von x abhangig sei, ja man braucht nicht einmal an eine durch mathematische Operationen ausdruckbare Abhangigkeit zu denken. It is worth noticing, however, that already S. F. Lacroix (1797) came very near to the new idea: Toute quantite dont la valeur depend d'une ou de plusieurs autres quantite's, est dite fonction de ces dernier es, soit qu'un sache ou qu*on ignore par quelles operations il faut passer pour remonter de celles-ci a la premiere.
- 13.
In this sense the function seems to have been introduced in India where it appears already in the Surya Siddhanta; see Kennedy (1969) p. 346.
- 14.
Thus the 13th century, anonymous Theorica planetarum: Et sciendum est, quod maiores sunt equaciones argumenti centra epicicli existente in opposite* augis ecentrici quam in auge, et differencia, que est inter has equaciones argumenti centra epicicli existente in auge et in opposito augis, dicitur equacio diuersitatis dyametri circuit brevis.
- 15.
This seems, in general, to have escaped the attention of historians of mechanics. Thus R. Dugas, Hlstoire de la mecanique, Neuchatei, 1950, has a chapter on Sources Alexandrines in which Ptolemy is not mentioned. Neither does his name appear in the long excursus on Greek kinematics in M. Clagett, The Science of Mechanics in the Middle Ages, Madison, 1959, 163-184, although the author is aware of the role played by astronomy in the development of kinematics.
- 16.
That the idea of functional relationships was utilized outside the field of astronomy has been demonstrated by Schramm (1965) who examined such relationships in Ptolemy's Optics, partic- ularly in connection with the law of optical refraction (Schramm also stressed the importance of astronomical tables, but only in the context of Muslim astronomy). Sambursky (1963 p. 76-78) has drawn attention to a passage in Johannes Philoponos' commentary to Aristotle's De generatione et corruptione describing (in qualitative terms) the variation of a function of two variables.
- 17.
Dividing this number by 60 (the radius of the circle) we find 1; 24,51,10 which is precisely equal to the value of \/2 known from Babylonian mathematics, in particular from the tablet YBC 7289. See Neugebauer and Sachs, 1945, p. 42.
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Pedersen, O. (2011). Ptolemy as a Mathematician. In: Jones, A. (eds) A Survey of the Almagest. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-0-387-84826-6_3
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