Abstract
Given Halley’s attentiveness to Apollonius’s text, his attentiveness to those “certain concrete vestiges,” as he puts it in his epistolary introduction, which hint at Apollonius’s thought regarding the book of determinate problems, given his apparent desire to enter Apollonius’s mind, Halley’s project, contrary to what was said above, begins to look purely historical after all. And one could even accept this identity as a historian without denying Halley’s identity as a mathematician. For, certainly, while his reconstruction demanded a considerable degree of mathematical ingenuity, such as one might expect from the new Savilian professor of geometry, one need not assume the reconstruction was only a pretense for Halley to exhibit his mathematical prowess or to explore new mathematical themes, Apollonian or not, that might flow from Conics, Book VII. Yet the fact remains that Halley did come to the project as a mathematician and scientist and did so willingly, as I have already stressed. So the truth is, while Halley’s project should not be viewed as purely mathematical, it should not be viewed as purely historical either: we must look for the character of Halley’s relationship with texts of the past, like Apollonius’s Conics, in the middle ground between historical sensitivity and rigor and mathematical insight and interest.
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Notes
- 1.
“Le célèbre astronome Halley, très-versé dans la connaissance de la geométrie des Grecs, traduisit de l’arabe, comme on sait, le Traité de la Section de raison, at rétablit celui de la Section de l’espace et le VIIIe livre des Coniques d’Apollonius. L’énigme des Porismes devait naturellement lui offrir de l’attrait.”
- 2.
Maqāla fı̄ tamām kitāb al-makhrūṭāt.
- 3.
His full name was Abū (Ali Al-Ḥasan ibn al-Ḥasan ibn al-Haytham. “Alhacen” is apparently a corruption of Al-Ḥasan.
- 4.
There is no reason to believe Halley knew of Ibn al-Haytham’s work and very good reason to believe that he did not. At present, only a single manuscript of Ibn al-Haytham’s Completion of the Conics exists in Manisa, Turkey, and that came to light only around 1970.
- 5.
I am using Hogendijk’s (1985) groupings here: each grouping represents a series of thematically related problems. Hogendijk divides the book into 11 such problem sets.
- 6.
Translation from Hogendijk (1985), p. 134.
- 7.
The existence of this difference makes the question of which reconstruction of Book VIII is the truer reconstruction somewhat moot.
- 8.
See note 28 above.
- 9.
For this aspect of Zeuthen’s historical outlook, see Lützen and Purkert (1994).
- 10.
Halley was well aware of the advantages these afforded him. An explicit expression of this can be found, for example, in the opening paragraph of Halley’s (1694) “Method of finding the Roots of Aequations Arithmetically,” which was appended to Newton’s Universal Arithmetick: “The principal Use of the Analytick Art, is to bring Mathematical Problems to Aequations, and to exhibit those Aequations in the most simple Terms that can be…/The Antients scarce knew any Thing in these Matters beyond Quadratick Aequations. And what they writ of the Geometrick Construction of solid Problems, by the Help of the Parabola, Cissoid, or any other Curve, were only particular Things design’d for some particular Cases. But as to Numerical Extraction, there is every where a profound silence; so that whatever we perform now in this Kind, is entirely owning to the Inventions of the Moderns” (translation by Joseph Raphson and corrections by Samuel Cunn, London, 1720).
- 11.
See Boyer (1956), especially, chapters VI and VII.
- 12.
The problem is this. Suppose four lines are given fixed in position and that from a point lines are drawn at given angles to each of these four given lines. Then the locus of points for which the rectangle formed by two of the drawn lines has a given ratio to the rectangle formed by the other two lines will be a conic section given in position.
- 13.
From vol. 1, p. 81 of Newton (1962).
- 14.
I do not mean archaeological in the Foucauldian sense, but in the somewhat pejorative sense of digging up old bones from dead ages.
- 15.
“Quamvis de scientiis Mathematicis, hac nostra et superiore aetate, praeclare meruerint Viri eruditi, qui Algebram Speciosam, Arithmeticam Infinitorum, nuperamque Fluxionum doctrinam adinvenerunt et excoluerunt: nihil tamen inde Veterum gloriae detrahitur, qui Geometriam ad eam provexere perfectionem…/”
- 16.
Halley refers at this point to the set of lemmas in Newton’s Principia from Part V of Book I, which I mentioned above.
- 17.
I am leaving out some details, which can be found in the translation itself.
- 18.
By Conics I.16 and the definitions following it, this is equal to the rectangle contained by the axis and the difference of the axis and its latus rectum, all of which are given since the axis and latus rectum are given.
- 19.
See above.
- 20.
See, for example, the note 153 problem 13 in the translation.
- 21.
I might emphasize, however, that Halley’s consistency of approach is particularly striking given the fact that when he worked on the Cutting-off of a Ratio and the Cutting-off of an Area, he was still working closely with David Gregory. But David Gregory died in 1708. So Halley’s work on the extant books of the Conics and the reconstruction of Book VIII was done largely on his own. This suggests that Gregory’s influence on Halley’s general approach to the Greek mathematical works was either minimal or so complete that Halley continued in the same vein after Gregory’s death. The latter does not seem very likely. Halley’s approach to the works of Apollonius then was truly his own, even if it was in perfect sympathy with Gregory’s.
- 22.
In particular, concerning the positions of the bright stars Aldebaran, Sirius, and Arcturus.
- 23.
That is, a sight used with the naked eye.
- 24.
Later Halley’s relations with Hevelius soured somewhat as did Halley’s enthusiasm for Hevelius’s techniques. See Ronan (1969), chap. 4, and Cook (1998), chap. 4 for detailed accounts of this chapter in Halley’s career. Some of Halley’s correspondence with Hevelius can be found in MacPike (1932).
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Fried, M.N. (2012). Chapter 5 Halley’s Dialogue with the Past. In: Edmond Halley’s Reconstruction of the Lost Book of Apollonius’s Conics. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0146-9_5
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