Abstract
context: no traces of ancient sources providing a context of similar problems, but: connection to non-trivial theorems/methodological devices with “potential” for future mathematical theories:
-
(i)
Diameters in configuration of tangent circles, theorem of Menelaus (→ points of similarity (projective geometry))
-
(ii)
Arithmetical progression (→ complete induction)
-
(iii)
Capture infinity using a quasi-mapping onto natural number progression
possible sources: lost monograph by Archimedes, with intermediate transmission stages (controversial, see below).
means: beyond Elements, but for the most part strictly “orthodox”; unusual means: nucleus form of complete induction (Props. 16, 18).
method: synthesis.
format: monograph in miniature form, lemmata, main theorems, corollaries.
history and reception: Liber assumptorum (deteriorated form, see below).
embedding in Coll. IV: tangent circles: Props. 8–10 (Apollonian problem, connection to Prop. 10 especially close); motif “chords and circles”: Props. 2–6, 11 and 12; motif “commensurable versus incommensurable straight lines in circle configuration”: Props. 2 and 3, motif “progression towards infinity”: Props. 19–21, 30; motif “association with Archimedes”: Props. 19–22, 30, 35b, 42–44.
purpose: illustration of plane synthetic geometry, monographic style: Archimedean. Through the connection with Archimedes (in style, if not in person), Props. 13–18 form a bridge to the second part of Coll. IV, specifically to Props. 19–22, which are indeed by Archimedes.
literature: Heath (1921, II, pp. 371–377); Buchner (1824), (arbelos via classical geometry, and via analytical geometry), Casey (1882, pp. 95–112) (involutions, limit processes), Hofmann (projective geometry, Zweiecke in: Hofmann (1990, II, pp. 146–164); see also Hofmann (1990, I, pp. 273–281)). The alternative treatments are interesting for a comparison in terms of methodology.
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Notes
- 1.
Latin translation in: Heiberg, Archimedes, Opera omnia, Vol. II, 510–525.
- 2.
Archimedis Opera Omnia Vol. II p. 514 Heiberg.
- 3.
Jones (1986a, pp. 538–539), for example, denies any connection with Archimedes for the Lib. ass. One of his reasons is the above-mentioned low level of sophistication in the Lib. ass. In my opinion, this low level could perhaps be explained as the result of progressive deterioration in transmission, and need not speak against an ultimate provenance of the material from Archimedes. Another point he makes is that there is no connection to the content of other works by Archimedes. In my opinion, this observation, too, can be relativized in its weight by pointing to such treatises as the Sand reckoner. I also think there is more coherence to Props. 13–18 than Jones’ remarks on p. 539 op. cit. and Hofmann (1990, I, pp. 146ff.) suggest.
- 4.
Cf. translation, beginning of Props. 13–18.
- 5.
See the list of indications pointing towards a larger extension of the original treatise below.
- 6.
The proof is exactly analogous to a group of proofs by Pappus in Coll. VII (64, 118, 128, and 130 Hu; Jones 1986a, # 118, 184, 195, and 198).
- 7.
Cf., e.g., Heath (1921, I, pp. 240–241) and Knorr (1986, pp. 185–186).
- 8.
See below, remarks on Prop. 13.
- 9.
As noted in the translation, the connection between the word “arbelos” and the cobbler’ tool is not securely established.
- 10.
Prop. 13 holds also for circles that touch internally, even though it is given only for circles that touch externally. On Hultsch’ reading of Prop. 14, the proposition uses the version for internally touching circles. Another explanation for the step in Prop. 14, one that does not imply that Pappus left a gap in the argument in the arbelos treatise is that the reference in Prop. 14 is rather to elementary lemmata for which Pappus gives a proof in Coll. VII 102ff. This is the explanation I preferred in the notes to the translation of Prop. 14. Independently from the question of completeness of the argument as given by Pappus, I think it is quite possible that the treatise from which Prop. 13 ultimately stems was more extensive and contained a greater number of preliminary lemmata, dealing with touching circles and points of similarity.
- 11.
In the configuration of Prop. 13, E is a point of similarity for the three circles concerned. A general theorem for such points was provided by G. Monge, according to Hilbert and Cohn-Vossen (1932, pp. 120–121). Compare also the contributions by Hofmann and Casey in the literature list above. As pointed out in the remarks on the mathematical context for Props. 7–10, Props. 13 and 14 are connected to Prop. 10, and this may be one of the reasons why Prop. 10 was formulated by Pappus the way it is.
- 12.
210, 6 Hu.
- 13.
210, 8 + app. Hu.
- 14.
211, #1 Hu, (Ver Eecke 1933b, p. 160, #7).
- 15.
E.g. Prop. 104 in Coll. VII, p. 828 Hu, (Jones 1986a, p. 234 # 166). Note the connection to Prop. 8, and recall that the lemma invoked here was not presented in the source for Prop. 8, but in Pappus’ commentary to Apollonius’ Tangencies. The geometrical situation for the arbelos theorem is connected to the Apollonian problem and its theoretical framework. Hultsch’ explanation (cf. above, translation) involves an auxiliary construction, and reference to Prop. 13, converse. Configurations 1, 2 could alternatively appeal to Lib. ass. I, cf. Archimedes, Opera Omnia II, p. 510–512 Heiberg.
- 16.
Compare QP 22and QP 23 in relation to QP 24.
- 17.
In Hultsch’ edition, the sequence of the resulting configurations is permutated: configuration 2 in Prop. 14 yields configuration 3 in Prop. 15. This re-numbering is, of course, of no consequence for the mathematical content of Prop. 15. The manuscript A has three diagrams. The first one concerns configuration 1, building on configuration 1 from Prop. 14, the second concerns the limit case when the second semicircle is replaced by a tangent to the first one (see appendix Hu p. 1227f.), and the third concerns configuration 2. There is no diagram for configuration 3 in A. See part I, text and translation, with notes.
- 18.
The fact that the configuration and the theorem relate to the theory of points of similarity means that Prop. 16 encapsulates at least the potential for a very deep insight, even if we do no longer have direct access to the actual mathematical context for such discussions in antiquity.
- 19.
It is perhaps worth noting that even though the proof as transmitted explicitly appeals to Prop. 14, *, Prop. 17 could be independent from 13–16, because the result from within Prop. 14 could easily be proved ad locum. Also, Prop. 17 uses a special case that appears to have been added in within Prop. 14 precisely with a view to Prop. 17. For it is not used anywhere else within the arbelos treatise.
- 20.
P. 828 Hu, (Jones 1986a, p. 234, # 166); Hultsch and Ver Eecke prefer here, as in Prop. 14, step 1, a reduction to an extended version of the converse for Prop. 13.
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Sefrin-Weis, H. (2010). Plane Geometry, Archimedean. 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_6
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