Abstract
Props. 31–34 are our only sources for the trisection of the angle via conics/solid loci in antiquity. Following up on the introduction of the problem in the meta-theoretical passage, Pappus uses the trisection as an exemplary argument to illustrate mathematics of the second, the solid kind. In his methodological portrait, it looks as though the dominant mode of argumentation in this field was analysis-synthesis, focusing on loci, and that Apollonius was the culminating figure for this discipline, although his work rested on earlier achievements (Aristaeus, inter alia) and did not completely supersede them. The arguments in Props. 31–34 are, together with Props. 42–44, Menelaus’ cube duplication as reported by Eutocius, and selected arguments from Konika V also our only surviving examples for a treatment of solid locus problems, and Props. 31–34 and 42–44 are the only analysis-based ones. As in the case of the symptoma-mathematics of motion curves, this uniqueness obviously makes Props. 31–34, presented here in their original context, most valuable sources for historians of mathematics, while also creating the problem that we cannot decide to what degree Pappus’ portrait, drawn up with a visible program in mind, is representative of the actual mathematical practice. His portrait should be carefully evaluated on its own terms and as a whole. As in the case of the symptoma-mathematics of motion curves, Pappus implicitly traces a developmental line, from the pre-Euclid-ean treatment of “solid” problems down to Apollonius and his reception. For the portrait of “solid” geometry, the majority of modern commentators agree with Pappus’ reconstruction, i.e., their assessment of the development of the ancient analytic treatment of conic sections is congruent with Pappus’ account. In addition, there is general agreement on the character of the historical layers detectable in Pappus’ report. What has not received enough scholarly attention thus far, and this is, again, parallel to the case of Props. 28 and 29, is the methodological emphasis. The portrait in Pappus stresses the practice of the technique of Greek geometrical analysis for solid loci, as a method of argumentation, in Props. 31, 33, and 34.
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- 1.
The angle trisection, though it may look like a very special isolated question, is indeed rather typical, even exemplary, for problems that can be solved via conics. As pointed out in the introduction to Props. 23–25, the two problems of trisecting the angle and doubling the cube already exhaust solid geometry, in the sense that any problem that can be solved by means of conics reduces to one of these two basic construction problems. The angle trisection is thus a fitting topic for an exemplary illustration of geometry of the solid kind.
- 2.
Cf. Eutocius In Arch. Sph. et cyl. 78–84 Heiberg.
- 3.
Specifically: Knorr (1986, pp. 128/129) on Prop. 34b; Knorr (1986, pp. 272–276) on Props. 31–33; Knorr (1986, pp. 282–284) on Prop. 34a; Knorr (1986, pp. 303/312) on neusis, Apollonius, and Nicomedes; Knorr (1986, pp. 321–328) on Apollonius and Aristaeus as contributors to 34a/34b.
- 4.
This passage addresses the testimony of Al-Sijzi and Al-Quhi. See Knorr (1989, pp. 247–372) for a comprehensive presentation of Arabic sources with connection to ancient angle trisections.
- 5.
Cf. Heath (1921, I, p. 235).
- 6.
Cf. introduction to Props. 26–29.
- 7.
Heath (1921, I, pp. 226–227).
- 8.
Cf. Heath (1921, I, pp. 235–237).
- 9.
Cf. Heath (1921, I, pp. 239–240); Procl. in Eucl. 272, 3–7 Friedlein; Cantor (1900, I, pp. 335–337).
- 10.
Cf. Heath (1921, I, pp. 240–241); see also Hogendijk (1981), and the remarks on Props. 31/32.
- 11.
A neusis of this type is subjected to analysis in Props. 42–44, to show that it is “solid.” Knorr (1986, pp. 186–187) draws the connection between the trisection via Lib. ass. VIII and Archimedes; so does Heath (1921, I, pp. 240, 241).
- 12.
Cantor (1900) draws the connection to the conchoid of the circle; Knorr (1986, 221ff.) argues that it is plausible that Nicomedes worked with conchoids, including the one on a circle, for his angle trisection. If Nicomedes indeed investigated the conchoid of the circle, it is a tempting possibility to speculate that Archimedes may have experimented with this curve and its properties as well. For Nicomedes seems, in general, to have taken Archimedean contributions as a basis for his own, analytically based contributions. But at present, we do not have enough “hard evidence” for such a thesis.
- 13.
This argument is not reported in Heath (1921), cf. Prop. 33.
- 14.
Cf. Heath (1921, I, pp. 243–244).
- 15.
Cf. Heath (1921, I, pp. 242–243); see also Jones (1986a, pp. 582–584).
- 16.
Cf. Sezgin (1974), and Knorr (1989, pp. 247–372) for information on Islamic mathematics, specifically on cube duplication and angle trisection, as well as Knorr (1983a, 1989, pp. 216–224) on the transmission of ancient angle trisections into Islamic culture.
- 17.
Cf. (Knorr 1989, pp. 267–275) (angle trisection according to Ahmed ibn Musa).
- 18.
Cf. Knorr (1989, pp. 277–291) for Ibn Qurra’s angle trisection.
- 19.
Cf. Knorr (1986, 197, #107).
- 20.
Cf. Knorr (1986, 185, #106); on Al-Sijzi's and Al-Quhi’s trisections cf. also Knorr (1989, pp. 293–309).
- 21.
Cf. Bos (1984, 2001) for a history of this project.
- 22.
It is doubtful whether Coll. IV could have been known in the Middle Ages to any significant degree. Unguru (1974) argues that a passage from Witelo’s Optics betrays knowledge of a substantial passage from Coll.VI. Commandino p. 95 C provides a plane argument, drawn from Witelo, in connection with the neuses discussed in Props. 42–44. Perhaps this is an indication that Witelo looked at Coll. IV as well, though other explanations are possible, also.
- 23.
Cf. Cantor (1900, II, pp. 81–82).
- 24.
Cf. Cantor (1900, II, p. 104/105).
- 25.
Bombelli ed. Bortolotti (1923, 1929, pp. 265–267).
- 26.
Vieta’s argument can be found in Vieta ed. Schooten (1646, pp. 240–257); it was first formulated in Vieta’s Supplementum Geometriae from 1593. Descartes (1637, pp. 396/397) (206–209 Latham/Smith) uses parabola and circle for the angle trisection. He also developed an instrument, a kind of compass, for the trisection. See also Descartes (1659, pp. 178 ff).
- 27.
Cf. Whiteside (1972, V, 426/428) (conchoid for angle trisection); 428–432 (neusis reduced to construction via conics, close connection to Prop. 31–33 and Props. 23/24); 458–464 (angle trisection, with explicit reference to Pappus (Prop. 32) ); cf. also the solution of cubic equations via neusis in 432 ff. (closely connected to the Archimedean neusis from lib. ass, VIII), and the summary remarks on solid neusis constructions pp. 454/456 and 474.
- 28.
According to Ver Eecke (1933b, XXXVIII, #1).
- 29.
Ver Eecke (1933b, XXXIX, #1), presents the matter somewhat differently.
- 30.
Cf. Jones (1986a, pp. 540–541) for the Apollonian construction of plane loci, and pp. 573–577 as well as Knorr (1989, pp. 94–100) on Menaechmus’ cube duplication via solid loci.
- 31.
This is the decisive, non-deductive step in the analysis. Con. II, 12 actually states the reverse: all points H on the hyperbola through D with asymptotes AB, BC will fulfill the above conditions for rectangles/parallelograms. Because we know this, thanks to Con. II, 12, we can conclude, for the purpose of the analysis, that H must lie on this hyperbola. This is not a logical derivation, but a prospective argument, if you will. We can conclude this way, because we know that the reverse, in the upcoming synthesis, will give us a valid deduction.
- 32.
Other, more complex and sophisticated constructions, using reconstructions of other possible Nicomedean conchoids have been suggested inter alia by Knorr (1986, 220ff).
- 33.
This is the only explicit diorismos in Coll. IV.
- 34.
Cf. Zeuthen (1886, pp. 280–282) for a reconstruction of a possible diorismos in terms of analysis of loci, Coll. VII, Prop. 72 (pp. 780–782 Hu, Jones (1986a, pp. 202–208), Heath (1921, II, pp. 412 and 413), and Knorr (1986, pp. 298–300) on Heraclius’ argument itself; Descartes (1637, pp. 387–389) (188–193 Smith/Latham) discusses the same problem, as a case where a problem with a cubic equation (“solid-looking”) can be reduced, with explicit reference to Pappus.
- 35.
Commandino (Co 102–103 E and 103–104 E) also provides a synthesis. It covers all possible cases.
- 36.
According to Jones, there are quite a few problems with the argument as presented by Pappus in Coll. VII; the lemma seems to contain several errors. This makes the task of reconstructing the original “Aristaeus” from here all the more difficult. For literature on Prop. 34a/b see the list given at the beginning of the exposition on Props. 31–34, and the footnotes to the section on attested ancient solutions.
- 37.
The image created in Coll. IV by the way Pappus presents the geometry of the solid kind agrees to a large degree with the portrait given by Zeuthen (1886).
- 38.
Non-deductive analysis step as in Prop. 31; because the reverse step, used in the synthesis, is a valid theorem, we can conclude, in the analysis, that B lies on that hyperbola. For if it does, the preceding steps of the analysis can be deduced.
- 39.
Diorismos, not given explicitly in Pappus’ text.
- 40.
As in the case of Props. 31 and 34a, this last step of the analysis is non-deductive; its validity rests on the fact that the converse is a valid theorem.
- 41.
Coll. VII, #237 Hu, Jones (1986a, I, pp. 365–369, # 316–317) constructs such a hyperbola. Its points B fulfill the conditions and proportions analyzed above. Therefore, we can conclude in the analysis that B lies on this hyperbola.
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Sefrin-Weis, H. (2010). Angle Trisection. In: Sefrin-Weis, H. (eds) Pappus of Alexandria: Book 4 of the Collection . Sources and Studies in the History of Mathematics and Physical Sciences. Springer, London. https://doi.org/10.1007/978-1-84996-005-2_9
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