Abstract
If there is some uncertainty about Halley’s birth date, we are really in the dark about Apollonius’s. The most one can say is that Apollonius of Perga was born around the middle of the third century B.C.E. and died sometime at the beginning of the second. Even this is only a matter of inference, based partly on a comment by Apollonius’s late commentator and editor, Eutocius (c. 480–540 C.E.)—himself relying on yet another source, a certain Heraclius—that Apolloniuswas born “… in the time of Ptolemy Euergetes… ” whose reign we know to have been from 246 to 221 B.C.E.
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- 1.
The comment by Eutocius is from his Commentary on the Conics (Heiberg, Apolloniii Pergaei quae graece exstant cum commentariis antiquis, II, p.168). More information may be found in Fried and Unguru (2001) and Toomer (1970).
- 2.
Little is known for sure about Geminus’ life, and the time of his birth ranges from the 1st century B.C.E. to the 1st century C.E. In citing the earlier date, I am following Evans and Berggren (2006) here.
- 3.
Eutocius, Commentary, p.170.
- 4.
These are what we would call the two branches of the hyperbola. A hyperbola, for Apollonius, is only one of the branches. What this distinction says about Apollonius’s general approach to the conics is discussed in Fried (2004).
- 5.
These symptōmata are given, in a somewhat simplified form, in appendix 1, as are some of the other basic Apollonian terms.
- 6.
The “limits of possibilities” are, roughly speaking, the conditions under which a problem has a solution at all or more than one solution. See below.
- 7.
I am taking advantage of the English translation by R. Catesby Taliaferro in Apollonius (1998).
- 8.
We shall use the shorthand, from now on, “sq.AK” for “the square on AK” and “rect.AK, KG” for the “rectangle contained by the sides AK, KG.”
- 9.
That is, ΔE will bisect all chords of the ellipse drawn parallel to certain given line, in this case, line HZ. Earlier in Book I, in proposition I.7, Apollonius shows that any line such as ΔE, whether or not the plane ΔEΛ cuts sides AB and AΓ of the axial triangle, will be a diameter in this sense. Also note that while HZ is perpendicular to BΓ extended, it is not necessarily perpendicular to the plane of the axial triangle ABΓ, and, therefore, lines such as ΛM are not necessarily perpendicular to the diameter ΔE. If HZ is perpendicular to the plane ABΓ, then lines ΛM will be perpendicular to the diameter and the diameter will be an axis.
- 10.
See Netz (1999).
- 11.
Naturally, I have in mind here particularly H. G. Zeuthen’s (1886), Die Lehre von den Kegelschnitten im Altertum, where Apollonius’s geometrical presentation is thought to hide an algebraic mode of thought. Indeed, Zeuthen’s point of view dominated the historiography of Greek mathematics at least until the 1970s; it was wholly adopted, for example, by Thomas L. Heath and Bartel L. van der Waerden. But besides abundant material in the Conics, and elsewhere in Greek mathematics, which is awkward to explain by means of Zeuthen’s thesis, to say the least, it is dubious that Apollonius should have one mode of presentation and another, conceptually distinct, mode of thought. That said, a divide between presentation and thought is perfectly possible in the case of Halley, when he is occupied in the reconstruction of an ancient text.
- 12.
Halley probably would have read Apollonius’s Conics, I–IV from a text based on Federigo Commandino’s famous edition of 1566. Commandino’s edition included not only the first four books of the Conics but also Eutocius’s commentary and Serenus’s On the Section of a Cylinder and On the Section of a Cone. The latter were included with Halley’s reconstruction of Conics, Book VIII.
- 13.
Book IV also fits this description. In fact, its inclusion in the course of elements is not completely self-evident. The point is discussed at length in Fried and Unguru (2001), chapter III.
- 14.
It has been often said that Book V concerns normals to conic section, but this characterization is moot.
- 15.
Toomer (1990), vol. I, p. 382. In general, all translations from the Arabic Books V–VII are from Toomer (1990).
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Fried, M.N. (2012). Chapter 2 Apollonius’s Conics . 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_2
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