(alike in all their parts) are a straight line, a circle, and a cylindrical helix1;
(11) (in the note on I. 10) on the question whether a line is made up of indivisible parts (ἀμερῆ), as affecting the problem of bisecting a given straight line2;
(12) (in the note on I. 35) on topical, or locus-theorems3, where the illustration of the equal parallelograms described between a hyperbola and its asymptotes may also be due to Geminus4.
Other passages which may fairly be attributed to Geminus, though his name is not mentioned, are the following:
(1) in the prologue, where there is the same allusion as in the passage (8) above to a remark of Aristotle that it is equally absurd to expect scientific proofs from a rhetorician and to accept mere plausibilities from a geometer5;
(2) a passage in the prologue about the subject-matter, methods, and bases of geometry, the latter including axioms and postulates6;
(3) another on the definition and nature of elemenis7;
(4) a remark on the Stoic use of the term axiom for every simple statement (ἀπόφανσις ἁπλῆ）8;
(5) another discussion on theorems and problems9 in the middle of which however there are some sentences by Proclus himself10.
(6) another passage, in connexion with Def. 3, on lines including or not including a figure (with which cf. part of the passage (4) above)11;
(7) a classification of different sorts of angles according as they are contained by simple or mixed lines (or curves)12;
(8) a similar classification of figures13, and of plane figures14;
(9) Posidonius' definition of a figure15;
(10) a classification of triangles into seven kinds16;
(11) a note distinguishing lines (or curves) producible indefinitely or not so producible, whether forming a figure or not forming a figure (like the “single-turn spiral”
)17;
(12) passages distinguishing different sorts of problems18, different sorts of theorems19, and two sorts of converses (complete and partial)20;
(13) the definition of the term “porism”
as used in the title of Euclid's Porisms, as distinct from the other meaning of “corollary”
21;
(14) a note on the Epicurean objection to I. 20 as being obvious even to an ass22;
(15) a passage on the properties of parallels, with allusions to
1 Proclus, p. 251, 2-11.
2 ibid. pp. 277, 25-279, 11.
3 ibid. pp. 394, 11-395, 2 and p. 395, 13-21.
4 ibid. p. 395, 8-12.
5 ibid. pp. 33, 21-34, 1.
6 ibid. pp. 57, 9-58, 3.
7 ibid. pp. 72, 3-75, 4.
8 ibid. p. 77, 3-6.
9 ibid. pp. 77, 7-78, 13, and 79, 3-81, 4.
10 ibid. pp. 78, 13-79, 2.
11 ibid. pp. 102, 22-103, 18.
12 ibid. pp. 126, 7-127, 16.
13 ibid. pp. 159, 12-160, 9.
14 ibid. pp. 162, 27-164, 6.
15 ibid. p. 143, 5-11.
16 ibid. p. 168, 4-12.
17 ibid. p. 187, 19-27.
18 ibid. pp. 220, 7-222, 14; also p. 330, 6-9.
19 ibid. pp. 244, 14-246, 12.
20 ibid. pp. 252, 5-254, 20.
21 ibid. pp. 301, 21-302, 13.
22 ibid. pp. 322, 4-323, 3.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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