Abstract
The transition to the general angle division implies a transition from solid to linear geometry. This is explicitly mentioned by Pappus. Some linear problems will arise from generalizing a plane or solid problem, and this is the case for the angle division. The connection to problems of the second, and even the first kind remains transparent; in fact, Props. 37/38 is an analogue to IV, 10/11. Pappus makes no attempt to single out, via diorismos, which of the general cases would become solid or plane. The introductory sentence to Props. 35–38 even suggests that Pappus thought the character of a problem is sufficiently established when an analysis leads to conic sections: the problem is then taken to be solid in general. Again, this has consequences for the evaluation of analysis as a technique to determine the appropriate level of a problem/theorem in 42–44. Apparently, it is limited in power and application, and regularly only used “after the fact,” i.e., to subject existing arguments to critical evaluation.
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Notes
- 1.
Recall cube multiplication, Prop. 24: if Pappus thought it is analogous to general angle division, i.e., linear as a result of generalization, he was mistaken. He is, however, correct in his assessment that the general angle division is “linear.”
- 2.
For singling out the plane and solid cases of general angle division, one would need an instrument comparable in power to Galois theory. As mentioned in the introduction to 31–34, Gauss was able to single out the plane cases. I know of no attempt for solid cases.
- 3.
Procl in Eucl. 272 Friedlein associates Nicomedes with a systematic study of the properties of the quadratrix. So Nicomedes is a possible source for Props. 35–41; cf. also Iambl. apud Simpl. in Cat. 192 Kalbfleisch, 65b Brandis. Within the present commentary, we cannot explore the hypothesis that Props. 35–41 are in fact taken from Nicomedes’ book on the quadratrix. But that is at least a plausible possibility. Quite a number of connections between 25–27 and 35–41 can be detected, beyond the use of the symptoma of the quadratrix.
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Sefrin-Weis, H. (2010). General Angle Division. 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_10
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