Abstract
This chapter is devoted to such information on the lost Conics VIII as can be derived from extant sources. The most important of these are the extant part of the Conics,and the Collection of Pappus of Alexandria (fl. 320, DSB X,293–304) which will be discussed in §4.2. The available evidence will only enable us to draw conclusions of a very general nature. However, the subject-matter of Conics VIII was probably not identical to that of the reconstruction which E. Halley published in 1710 (§4.3). A discussion of the Arabic sources will be postponed till §4.4, because these sources do not give new information about the contents of Conics VIII.
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Halley translated the preface to Conics VII in accordance with his own suppositions about Book VIII. After: all of this is very useful in many kinds of problems, he interpolated especially in their diorismoi (praecipueque in eorum Stoptaµoiç); the problems in conic sections… which occur (among those that will be mentioned) in my translation are in his translation determinate conic problems (problematis conicis determinatis), that is, problems requiring a diorismos.The passage the eighth Book, that is the last Book of it in my translation is (the eighth Book), which is by way of an appendix (qui loco appendicis est), in his translation (Conics, tr. Halley, 99).
Min Abulûniyûs ilâ Atâlûs. Salâmun `alayka. Qad wajjahtu ilayka bi-l-magâlati l-sâbi`a min kitâbi l-makhrûtât ma`a kitâbi hadhâ. Wa fi hâdhihi l-magâla ashyâ’u kathiratun gharibatun hasanatun fi amri l-agtâr wa-l-ashkâli Had tu`malu `alayhâ, mufassilatun (?). Wa jami`u dhâlika azimu l-manfa`a fi ajnâsin kathiratin min al-masdil wa-l-hâja ilayhi shad idatun fimâ yaga`u min al-masdil fi quta`i l-makhrûtâti llati dhakarnâ mimmd yajri dhikruhu wa-bayânuhu fi l-magâlati l-thâmina (f i 1-magâla h (A)) min hâdhâ 1-kitâb wa-hiya âkhiru magâlatin f ihi, wasa’ahrisu cala ta jilihâ ilayka (`alayka (0)) wa-l-saldm. O = ms. Oxford, Bodl. Marsh 667, 137a:4–7, A = ms. Aya Sofya 2762, 267a: 4–9.
The “figure” is the rectangle contained by the latus transversum and the latus rectum, compare §3.1.
The word Stoptaµ6S and its derivatives are translated in different ways in the Conics. Compare the preface to Book I mpbs Toùs Stoptaµoûy for the diorismoi (ed. Heiberg I,4:8) = f tandidi 1-qutü` in the definition of conics (O 5a:17, see note 2) i pds TE Tàç cruvOgaEtq Twv aTepswv Tôrzwv xai Tobç and toptißµoûç for the syntheses of solid loci and the diorismoi (ed. Heiberg I,4:11–12) = fi tarkibi ashkâlin wa-tafsilihS for the synthesis of propositions and their diversification (O 5a:19). TO Ss,tapi Stop rru iwv Osmprlµâ’rwv one on theorems related to diorismoi (ed. Heiberg I:4:24–25) = wa-fi 1-thâlitha minhâ ashkâlun `aid qismatin in the third (of the last four books) are propositions on division (O, 5a:24). Clearly the translator did not always understand what was meant. In the preface to Book IV (ed. Heiberg II,4; 0 70a) diorismos is translated as tagsim division (into cases). The preface to Book V mentions the knowledge of division (into cases) and the diversification of problems and their synthesis (macrifat tagsim wa-tafsili 1-masâ’il wa-tarkibihh) O 84a:5. Here the lost Greek text probably had something like for the diorismoi of problems and their synthesis.
The dissertation of A. Jones (Brown University) will include a study of all the lemmas in Book VII of the Collection.
Ed. Heiberg I,152:14–15. The lemma proves EB Ad. It follows that the areas of triangles EdB and AEB are equal. This is assumed in 1: 52.
Ed. Heiberg I,152:14–15. The lemma does not refer to I: 49.
Related to II:14?
The lemma gives an alternative proof of III:4:TdE bisects AB, so by the lemma HZ II AB, hence the areas of triangles AH and BZd are equal. I disagree with Heiberg’s suggestion III:8.
Related to III:35,36?
References in parentheses are to pages and line numbers in the translation of the Conics by Ver Eecke. Section d (Book V): lemmas 1, 2 and 8 refer to V:27 (385:15–18), 51 (423:27–29) and 55 (438:12–14), respectively. Lemmas 3–7 seem to refer to an interesting part of Book V missing from the extant text, but related to V:52–55. Lemmas 9, 10 do not seem to refer to the extant text either. Section e: Lemmas 1, 2 and 5 relate to VI:13 (497:7–8, 15–16, 7–8, respectively). Lemmas 3 and 4 are the converses of lemmas 1 and 2. Lemmas 6 and 7 relate to VI:18 (508:12–14). Lemmas 8 and 9 refer to VI: 29 (529:3–5, see Ver Eecke’s note 2) and 31(538:11, see Ver Eecke’s note 1), respectively. Lemmas 10 and 11 probably refer to a missing part of the text. They are related to lemmas 1, 3–7.
The lemma refers to an alternative proof of VII: 5 without the normal DH (notations as in tr. Ver Eecke, p. 556): If p’ is the latus rectum of diameter BI, then by I:49 p’ • BE = 2Bd • Be. By the lemma 2Bd • Be = 2(0E EA + AZ • ZA) = BZ2 + 4ZA2 = AT • ZA + 4ZA2. Since BE = ZA it follows that p’ = AT + 4ZA.
O lb:28, 2a:13 (see note 2) or ms. Aya Sofya 4832, 223b:27, 224a:5–6 (facsimile in Terzioglu, Vorwort).
For Ibn al-Nadim see EI2 III,895–896. The Arabic text of the quoted passage is in Fihrist, ed. Flügel, 267. I quote the translation in Dodge II,637 with slight changes.
The references are in a treatise by Al-Sijzi on the regular heptagon and the trisection of the angle (GAS V,331,8) and in an anonymous abstract of the Correction of the Conics of Abu Ja`far al-Khâzin (GAS V,307,4). For references to the manuscripts see Hogendijk, Heptagon, 312. For Ishâq ibn Hunayn see GAS V,272–273.
For Ibn al-Qifti see EI2 III,840. The Ta’rikh al-Hukamâ’ was written after 1227, and is extant in an epitome made in 1249 by Al-Zawzani. The Arabic text of the quoted passage is in Ibn al-Qifti, ed. Lippert, 61.
Arabic text: agülu: ammâ hâdhâ 1-magâla fa-ghayru mawjüdatin bal wujida ashkâluh5 bilâ musâdarât wa-lam yalam al-tarâjimu `alâ mâdhâ tadullu min al-masâ’il fa-ahmalühâ wa-baqiya 1-kit5b sabra magalâtin. Ms. Manchester, John Rylands 358 (382), 2a:14–15, New York, Columbia University Library, Plimpton Or. 302, lb:16–17. I am indebted to Professor A. I. Sabra (Cambridge, Mass.) for the reference to this passage.
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Hogendijk, J.P. (1985). Traces of Conics VIII. In: Ibn al-Haytham’s Completion of the Conics . Sources in the History of Mathematics and Physical Sciences, vol 7. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4059-2_4
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