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Ancient Versions of two Trigonometric Lemmas

Published online by Cambridge University Press:  11 February 2009

Wilbur Knorr
Wilbur Knorr
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Stanford University
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Extract

To justify certain steps of the computation developed in his Sand-Reckoner, Archimedes cites (without proof) the following inequalities relative to the sides of right triangles:

if of two right-angled triangles, (one each of) the sides about the right angle are equal (to each other), while the other sides are unequal, the greater angle of those toward [sc. next to] the unequal sides has to the lesser (angle) a greater ratio than the greater line of those subtending the right angle to the lesser, but a lesser (ratio) than the greater line of those about the right angle to the lesser.

That is, with reference to the two right triangles ABG, DEZ (Fig. 1), where AG equals DZ and the angle at B is greater than that at E, ZE:GB < angle B:angle E < DE:AB.

Type
Research Article
Information
The Classical Quarterly , Volume 35 , Issue 2 , December 1985 , pp. 362 - 391
Copyright
Copyright © The Classical Association 1985

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References

1 Opera, ed. Heiberg, , 2nd ed., ii. 232Google Scholar.

2 In modern terms this result may be expressed as

sin a:sin b < a:b < tan a:tan b,

for angles a, b (a> b).

3 On the Sizes and Distances of Sun and Moon, props. 4, 7, 12; cf. Aristarchus of Samos, ed. Heath, , pp. 366–9, 376–81, 390–1Google Scholar for text and translation and technical notes.

4 It appears in Zenodorus' writing on isoperimetric figures, versions of which are transmitted by Pappus and Theon; see references for texts C and F in Section II below. Theodosius assumes the lemma in Spherics 3.11; cf. text E.

5 Prop. 8; see text A.

6 Ptolemy, , Syntaxis, 1, ch. 10Google Scholar; see text G.

7 Hultsch, F. has attempted a comparison of three of the variants in Pappi Collectionis quae supersunt, iii. 1234–5Google Scholar; cf. also notes 20 and 23 below.

8 In addition to his extensive commentaries, Theon produced new recensions of the Euclidean works, in particular, of the Elements, the Data, and the Optics. (A useful résumé of Theon's work is given by Toomer, G. J. in his article, ‘Theon of Alexandria’ [Dictionary of Scientific Biography, vol. 13, 1976, pp. 320–5]Google Scholar.) Commentaries on Apollonius' Conics include those of Pappus (in Book 7 of the Collection) and Eutocius (early 6th century a.d.; edited by Heiberg in Apollonii Opera, ii). Eutocius also produced commentaries on parts of the Archimedean corpus (see Heiberg, Archimedis Opera, iii). The Prolegomena of Heiberg's editions of Euclid provide extensive reviews of Theon's editorial manner; his findings are summarised by Heath, T. L., Euclid's Elements (2nd ed., Cambridge, 1926, 1, ch. V)Google Scholar. But Heiberg's effort to explicate the older tradition of Apollonius falters, I believe, on a misconception of the nature of Pappus' commentary and so underestimates the significance of Eutocius' role; see my ‘Hyperbola-Construction in the Conics, Book II’, Centaurus, 25 (1982), 253–91Google Scholar. Similarly, Heiberg's notion that the extant recensions of Archimedes' works appeared only in the generation after Eutocius is a basic error that calls into question much of his account of the older tradition of Archimedes. My current studies of the text of the Dimension of the Circle (forthcoming in Archive for History of Exact Sciences) indicate that Theon and his followers modified significantly the text of this, and perhaps other, Archimedean works.

9 Greek texts appear in Appendix 1. Full source references for these and the other versions are stated in Section II below. A key to my numbered subdivisions of these texts appears in App. 3.

10 The heading of E reproduces the line from Theodosius, ' text (Spherics 3. 11)Google Scholar which it explicates. It is not part of the scholium itself in the Theodosius MSS collated by Heiberg, but it does appear as a heading in the alternative version edited by Hultsch from a Munich MS, Monac. 301 (see note 20 below for reference). Heiberg includes this MS in his apparatus and designates it by M, as I shall do in the following notes. As lines E: 5, 7 and 14 reveal, M sometimes preserves a better text than do the versions in the Theodosius MSS.

11 For E:5 I follow the text of Hultsch's M. Heiberg's Theodosius MSS offer a variety of partly garbled readings. The lines I set off by dashes are in Heiberg: ‘for it is right, while the (angle) D is acute; therefore the (angle) under ADG is obtuse, while AD is greater than ED’. For comparing the relative sizes of AD and ED, the magnitudes of angles D (that is, BDE) and ADG are irrelevant, so that the reading in M is preferable. Adopting it, we render unnecessary Tannery's change of the last ‘therefore’ (ara) to ‘while’ (de). Juxtaposition with G indicates that the entire step set off by dashes could be considered evident, so that its appearance in E might have resulted from an editor's addition.

12 E:7 follows the text of M. In Heiberg's collation one reads: ‘let it pass through (hêiketô) as the (circle) ΕΘΖ’. Hultsch (p. 420n) judges the term hêketô to be unusual, perhaps idiosyncratic for ‘let it be drawn’. I suspect an error in the oral transmission of a phrase like echetô hôs (‘let it hold as’), which one finds in some mathematical texts (cf. Archimedes, Dim. Circ., prop. 1). But passages lending support to the MS reading in E:7 can be cited (see my remarks in App. 2). Referring to the point Θ Heiberg's text entails completion of the full circle, rather than just an arc. This aspect of the construction would distinguish version E from all other variants of the tangent and chord lemmas; thus, I find possible a corruption from an original reading ‘ΗΕΖ’ (cf. ‘ΕΖΗ’ in M).

13 A:6 is inserted from the text of A in Euclid's Optics, prop. 8 (I have altered the lettering to conform with the diagram of E). This happens to be the only extant Greek version of the tangent lemma that preserves this step in correspondence with G:8 of the chord lemma. But the medieval Hebrew translation of Theodosius preserves a form of the scholium which includes this step (see note 21 for reference). Thus, its absence from E appears to be due to scribal omission.

14 G:9–10 appear as a scholium in Heiberg's manuscripts of the Syntaxis. But comparison with Theon's commentary (H) indicates that these lines were originally part of Ptolemy's text (see Section III below). The steps also appear in the parallel version J (see Table I).

15 F:8 is inserted from a scholium (F) to Pappus' Collection, where I have altered the lettering to conform with E. This scholium betrays a close textual affinity with E, as will be seen in Section III below. This step is included in the Hebrew version of the scholium (see notes 13 and 21). The words in parentheses follow a restoration by Hultsch, apparently based on considerations of parallelism with the preceding line; it appears well conceived, in view of the ‘therefore’ which opens line F:8.

16 This parenthesised portion of E:14 appears in M but is absent from Heiberg's other manuscripts. Correspondingly, M also omits the line “for ED…to BD” at the end.

17 In a remark at the end of version H, Theon reveals his responsibility for the proof of the arc-sector theorem which appears at the end of Book 6 of Euclid's Elements. He thereby provided a valuable clue, exploited by Heiberg, for distinguishing between the Theonine and pre-Theonine recensions of Euclid. An alternative version of the theorem is preserved in Book 5 of Pappus' Collection and provides unusual evidence for a form of proportion theory alternative to Euclid's. I display its association with one of Archimedes' proofs and argue a Eudoxean provenance in ‘Archimedes and the pre-Euclidean Proportion Theory’, Archives internationales d'histoire des sciences 28 (1978), 183–244Google Scholar.

18 Sphere and Cylinder, 1, posts. 1–2 and prop. 1; cf. Opera, ed. Heiberg, , i. 8, 10Google Scholar. A proof along these modified lines was proposed by Hultsch as possibly the one intended in Pappus' assumption of the lemma; cf. op. cit., p. 311 n. and text F below.

19 That Archimedes' Dim. Cir. is earlier than his Sph. Cyl. is argued in my ‘Archimedes and the Elements’, Archive for History of Exact Sciences 19 (1978), 211–90CrossRefGoogle Scholar.

20 The scholium has also been edited by Hultsch, F. in Scholien zur Sphaerik des Theodosios, Abhandlungen der k. Sächsischen Gesellschaft der Wissenschaften, phil.-hist. Cl., Leipzig, 10, no. 5 (1887), pp. 381–446 (esp. pp. 440–1)Google Scholar; it appears also among the set of four scholia edited from the Munich manuscript Monac. 301 by Hultsch, , in “ΛΗΜΜΑΤΑ ΕΙС ΤΑ СΦΑΙΡΙΚΑ. Reste einer verloren geglaubten Schrift”, Jahrbücher für classische Philologie 29 (1883), 415–20 (esp. 417)Google Scholar.

21 Additional versions of the tangent lemma are held in the medieval Arabic, Latin and Hebrew recensions of Theodosius' Sphaerica. By comparison with the ancient versions, the medieval tradition of the lemma reveals considerable fluidity. One can perceive, nevertheless, its ultimate grounding in an ancient Theodosius scholium in the same tradition as E. I will occasionally note here one version in particular, in the Hebrew translation of Theodosius, made from an Arabic intermediary. (I have consulted the Bodleian MSS Hunt. 16, f. 107r and Heb. d. 4, f. 59v–60r, and designate it K in Section IV below.) As the detailed analysis of these documents is likely to be of greater interest to medievalists than to classicists, however, I prefer to undertake that discussion elsewhere.

22 Ibid., p. 1235. The criterion of ‘elegance’ is of course to a large extent subjective; Hultsch here seems to take it as a synonym for ‘brevity’. The notion, moreover, that elegance increases over time is dubious at best. Finally, his assumption that Theon's version is a transcript of Zenodorus' text is quite implausible, as the discussion of text C will make clear below.

23 See his “ΛΗΜΜΑΤΑ ΕΙС ΤΑ СΦΑΙΡΙΚΑ” (cited in note 20 above). Note that Hultsch's principal speculation here, that the lemma is a vestige of a very ancient compilation, can be sustained, albeit for reasons different from those he proposes (cf. Section IV below).

24 Björnbo, A. A., ‘Studien über Menelaos' Sphärik’, Abhandlungen zur Geschichte der mathematischen Wissenschaften 14 (Leipzig, 1902), 113–14Google Scholar. In Commentaires…de Théon d'Alexandrie sur l'Almageste, ii. 357 n., Rome cites Hultsch's edition of the scholia (1887) for remarks on readings, but not on the provenance of the text.

25 The interested reader may apply to the author for copies of the complete texts with translations.

26 That Theon sometimes reproduces his sources virtually verbatim is revealed in his use of Pappus' commentaries on Ptolemy as a resource for his own; examples illustrating this aspect of his procedure are included in my study of Archimedes' Dimension of the Circle, cited in note 8 above. But in such instances, one would presume Theon is transferring materials from documents not accessible to his students. In the case of his commentary on Ptolemy, however, they surely would have access to the basic text. Or does Theon's procedure here indicate that they did not in fact have access to the entire work, but rather only to those portions of it expounded by Theon?

27 It would appear that the commentaries were associated with a lecture course on Ptolemy's system of astronomy; cf. Rome, , Commentaires…, pp. lxxxiii, 317Google Scholar.

28 Heiberg gives the lines in his apparatus with the observation ‘mg. pro scholio B et…C’; Ptolemaei Opera, i. 44n. The codices B and C are, respectively, Vat. gr. 1594 (9th cent.) and Marc. gr. 313 (10th cent.), which with codex A (Par. gr. 2389, 9th cent.) are the oldest of the six codices collated by Heiberg in his edition.

29 This aspect of the relations of the versions of the isoperimetric writings is not emphasised in the principal discussions; cf. Hultsch, , Pappi Collectio, iii. 1189–90Google Scholar; Rome, op. cit., ii. 355–6n.; Mogenet, J., Introduction à l'Almageste, pp. 37–9Google Scholar; Busard, H. L. L., ‘Der Traktat De isoperimetris’, Mediaeval Studies 42 (1980), 61–2CrossRefGoogle Scholar. I examine this issue in a paper in progress on the isoperimetric versions of Pappus, Theon and the anonymous author. But aspects of the question appear in my study of Archimedes' Dimension of the Circle (cited in note 8 above).

30 I present this evidence in my paper on the Dimension of the Circle (see preceding note).

31 Theon introduces this section of his commentary with an explicit reference to Zenodorus: ‘We shall now make the proof of these things in epitome from the things proved by Zenodorus in the (book) On isoperimetric figures’ (Commentary on 1.3; ed. Rome, ii. 355). This had led Hultsch and others to assume that Theon has directly exploited a writing by Zenodorus. But the case for his dependence on Pappus is clear, and this provides the appropriate basis for discerning the relations among the three isoperimetric writings. I undertake this project in the paper in progress cited in note 29.

32 The relevant passages are cited and discussed in my paper on the isoperimetric writings (cf. notes 29 and 31). Many of them can be identified in the annotations provided by Hultsch to his translation of Theon's version; cf. Pappi Collectio, iii. 1190–1211.

33 Book 6 of Pappus' Collection bears the heading: ‘containing the resolutions of difficulties in the small astronomizing (locus)’; cf. the edition of Hultsch, ii. 474.

34 Ibid., p. 1143n.; cf. also Mogenet, op. cit., p. 38 and Busard, op. cit., pp. 61–2. But Hultsch elsewhere admits the possibility that the Pappus reference is to a commentary on the minor corpus; cf. “ΛΗΜΜΑΤΑ”, op. cit., p. 415.

35 Pappus' Book 6 includes lemmas to propositions in Theodosius' Spherics, Autolycus' On the Moving Sphere, Theodosius' On Days and Nights, Aristarchus' On the Sizes and Distances of Sun and Moon, Euclid's Optics and Phaenomena. To these some editors add Euclid's Data and the pseudo-Euclidean Catoptrica, Autolycus' Risings and Settings of the Fixed Stars, Hypsicles' Ascensions, and Menelaus' Spherics (cf. ibid., p. 475n., citing Fabricius), presumably through comparison with the intermediate curriculum in the Arabic astronomical tradition. An overview of the minor writings in spherics is given by Neugebauer, O., History of Ancient Mathematical Astronomy, iv D 3, 1Google Scholar; 3, 3; 3, 6; Neugebauer views the tradition of a ‘little’ astronomical corpus as the invention of modern bibliographers and doubts that any standard collection of this sort existed in ancient times (ibid., pp. 768–9). He thus maintains that one does not know which of Theon's commentaries is referred to by the isoperimetric writer (in text D). Neugebauer's caution on this point is salutary, even if his dismissal of the ancient evidence is a bit cavalier.

36 Collection, ed. Hultsch, , i. 310Google Scholar. A scholium to the Optics, prop. 8 in Theon's recension (our text B) reads: ‘In the 11th theorem of the 3rd book of the Spherics you will find outside [sc. in the margin] a scholium which you may compare with the present demonstration’ (Euclidis Opera, ed. Heiberg, , vii. 261Google Scholar); Heiberg notes that the Optics manuscripts which hold this scholium also include Theodosius' Spherics. Although this scholium appears to be due to a scribe after the 10th century, one would hardly suppose that the ancient commentators were incapable of the same kind of cross-referencing. A procedure like this is suggested in my comments here on texts E and F.

37 As indicated, G:9–10 appear as a scholium in the Ptolemy manuscripts (see note 28 above). But the fact that their analogues appear in J:9–10 and E:8–9 (see note 14) supports the view, already argued by comparison with Theon's version in H:9–10, that they were indeed part of Ptolemy's text.

38 It is far from being a trivial insight, one may note, to perceive how a proof of the tangent lemma might be refashioned into a proof of the chord lemma, or conversely.

39 For Hultsch's view, see his “ΛΗΜΜΑΤΑ”, pp. 415–16.

40 Source references to the Elements and other standard treatises are a common form of interpolation in mathematical writings, and are typically bracketed by the modern editors. Hultsch includes over a dozen scholia of this sort in an appendix to his edition of Pappus, (Collection, iii. 1173–86)Google Scholar, and many others within the body of the text. Inserting such references is a common practice of the later commentators, as one can see in the numerous citations of Apollonius appearing in Eutocius' versions of theorems by Archimedes, Dionysodorus and Diodes (cf. Archimedes, , Opera, ed. Heiberg, , 2nd ed., iii. 134, 138, 142, 144, 154, 168, 170Google Scholar). In this way, Eutocius' text has come to include anachronisms, since these geometers lived before or just at the time of Apollonius, and so were either unable or unlikely to have cited his Conics.

41 See the quoted passage corresponding to note 1 above.

42 According to Theon, ‘a pragmateia of chords is proved by Hipparchus in 12 books, and also by Menelaus in 6’ (Commentary on Ptolemy's Book 1, ad 1, 10, ed. Rome, ii. 451). In his note Rome argues that Theon does not here intend to transmit the exact title of these works, but only a general reference to their subject matter. For a compilation of testimonia to Menelaus' work, see Björnbo, op. cit., pp. 4–10.

43 On the medieval versions of the Spherics, see Björnbo, op. cit., pp. 10–16.

44 See, e.g. Pans MS BN 9335, f. 53v. I here follow the text given by M. Krause in his edition of the Arabic translation of Menelaus by Manṣûr, Abû Naṣr, Die Sphärik von Menelaos (Berlin, 1936), 239n.Google Scholar; cf. also Björnbo, op. cit., pp. 116–17. Krause cites the analogous passage from the Hebrew (op. cit., p. 106). Note that in speaking of it as a ‘scholium’ I wish only to indicate that it is not the type of text usually found in the body of ancient mathematical treatises. Nevertheless, one need not exclude the possibility that Menelaus himself inserted remarks like this in his Spherics. As Björnbo observes (op. cit., p. 117n.), even if the passage is by a later Greek or Arabic scholar, it would surely be founded on ‘authentic reports’.

45 Hebrew: ‘in the first figure of the figures’. Both versions are difficult, however, since it is the eleventh theorem of Book 3 which they ought to mean. A confusion of this kind is not likely in rendering the Arabic, but is easily understood in the light of the expected Greek base: ‘in the eleventh’ would be written ⋯ν τ⋯ι ιαω, so that by missing an iôta this would be read as ⋯ν τ⋯ι ιαω, that is, ⋯ν τῷ πρώτῳ, ‘in the first’. This would indicate that the passage was indeed held within the Greek recension of the Spherics used by the Arabic translator.

46 Hebrew: ‘the universal (kôlêl) book’. Krause cites an alternative rendering from the Arabic of al-Harawî: ‘his book on the universal (kullîya) production (sinϲa)’ (op. cit., p. 107), which reads well as a literal rendering for katholou pragmateia (ibid., p. 239n.). In Halley's Latin edition, based primarily on the Hebrew, the phrase becomes liber de principiis universalibus (cf. Björnbo, op. cit., p. 117n.).

47 Hebrew: ‘and already he uses this there’. A confusion between first- and third-person forms is quite common in Arabic-based translations.

48 Hebrew: ‘and a necessity great of benefit makes it necessary’.

49 Hebrew: ‘universal (kôlêl)’; cf. note 46. Presumably, the reference is to a formal geometric demonstration.

50 Björnbo, op. cit., p. 117n. Cf. Krause, op. cit., p. 239n.

51 The standard view on this lost work derives from Tannery and Heiberg; cf. Heath, T. L., A History of Greek Mathematics (1921), ii. 192–3Google Scholar; and Heiberg, J. L., Apollonii Opera (1893), ii. 133–7Google Scholar. Marinus cites this work for its definition of the term ‘given’ (sc. as ‘ordered’); but he also cites the Neuses for the same definition. Thus, although Heiberg seeks to assign other fragments of a general nature (e.g., items taken from Proclus' commentary on Euclid's Book 1) to the lost Pragmateia, that need not be the case. For our information on the Neuses makes its strictly technical nature quite clear (cf. the discussion of Pappus' account in Heath, ibid., pp. 189–92).

52 In the course of one series of data on simultaneous settings, Hipparchus remarks, ‘for each of the things said is proved geometrically (dia tôn grammôn) in the general (katholou) treatises (pragmateiais) compiled by us about these matters’ (In Arati et Eudoxi Phaenomena, ed. Manitius, C., 1894, p. 150Google Scholar; cf. Björnbo, op. cit., p. 69n. and Heath, op. cit., p. 258). This reference to a combined numerical and geometric study of the problems associated with spherical astronomy invites comparison with Theon's reference to the treatise on chords (see note 42). In neither passage does it seem that pragmateia is used as the title, although the manner of the reference lends itself to being construed in that way. An ambiguity of this sort might have affected the reference to Apollonius' work in the passage from Menelaus.

53 This is the view of Björnbo, op. cit., pp. 116–17; cf. also Heath, op. cit., pp. 252–3.

54 The Arabic text is given by Krause, op. cit., p. 107.

55 Krause cites the alternative Arabic recension, ibid., p. 107n.

56 This part of the text is not cited by Krause; I have translated from the manuscript, MS Leiden Or. 399, 2, f. 105r.

57 Arabic: iḍϲâf. Menelaus, like Ptolemy after him, adopted the expression ‘chord of the double arc’ for what one now signifies as ‘double of the sine’; cf. Björnbo, op. cit., p. 89n.

58 Arabic: ṣaḥîḥa. Presumably, the term is included here as a reminder that Menelaus' inequality for the chords relates to the doubles of the arcs GH, DE in Theodosius' result.

59 For a synopsis of these theorems, see Heath, op. cit., pp. 250–1, 272–3. In Theodosius' theorem the ratio of two arcs marked off along intersecting great circles is found to be less than the ratio of the diameters of corresponding parallel circles; in Menelaus' theorem the ratio of the sines of these arcs is found to equal the ratio of the product of these same diameters to the product of two others. Further, Menelaus' diagram is more general, including Theodosius' as a special case. The arcs along one of the great circles can represent the segments of the zodiacal signs in the ecliptic, while the arcs along the other represent the corresponding segments along the equator; the latter are proportional to the rising times of the associated oblique arcs.

60 For text, see Theodosius, ed. Heiberg, pp. 193–4.

61 Archimedes, , Plane Equilibria, 1.7Google Scholar. Pappus, , Collection 5, prop. 12, ed. Hultsch, , i. 336–40Google Scholar; and Commentary on Ptolemy 6.7, ed. Rome, pp. 254–8.

62 ‘Archimedes and the Pre-Euclidean Proportion Theory’, Archives internationales d'histoire des sciences 28 (1978), 183–244Google Scholar.

63 My translation from the text of Hett, W. S., Aristotle: Problems (LCL, Cambridge, Mass./London, 1970), i. 330–2Google Scholar. Related problems on varying length of shadows arise in 15.9 and 10.

64 Optics, prop. 4: ‘of equal distances along the same straight line, those at greater distance appear smaller’; that is, with reference to Fig. 6, if LE, EZ were equal, then arc AB would be less than arc BG. A slightly different configuration is given in prop. 7, and an indirect proof based on either could establish the lemma of the Problems. An alternative direct proof could take this form: DZ < DE since DE is opposite the greater angle (the obtuse angle DZE) in triangle DEZ; similarly, DE < DL (cf. Euclid, , Elements 1.19Google Scholar). Since DE is bisector of angle LDZ, LE:EZ = LD:DZ (6.3). Since LD > DZ, it follows that LE > EZ (5.14), as claimed. For a somewhat different form of the proof, see Heath, T. L., Mathematics in Aristotle (Oxford, 1949), 260–1Google Scholar.

65 To prove the general case for commensurable arcs AB, BG, one can introduce their common measure and show through finite repetition of the equal case that LE:EZ > arc AB: arc BG. The incommensurable case can be obtained by an indirect proof based on this result. This manner of dealing with proportions appears in a few theorems of Archimedes and Theodosius; I propose its pre-Euclidean origin and give a detailed account of the technique in the paper cited in note 62. If any such method was tried for the lemma, our sources do not attest it. A simpler alternative is possible merely by introducing the arc, centred on D with radius DE, between the lines LD, DZ. By considering the triangles and sectors formed, as one does in the proofs of the tangent lemma, one obtains the desired inequality. Note that the angle at Z need not be right, as one assumes in the tangent lemma.

66 Similar metrical interests underlie the uses of the bisector theorem (6.3) by Archimedes (Dim. Circ., prop. 3) and Aristarchus (Sizes and Dist., prop. 7). Problems 15.7 helps motivate another proposition of the Optics. It asks why the division between light and dark of the moon at its half phases appears straight, when it is in fact a great circle. In Optics, prop. 22, Euclid proves that the appearance of a circle, when viewed along the line of its plane, is as a straight line. Examples like these from the Problems thus suggest the scientific contexts for the Euclidean theorems.

67 See note 23.

68 Björnbo, op. cit., pp. 128–33.

69 Toomer, G. J. proposes a view along these lines, with a promised later elaboration, in his Diodes: On Burning Mirrors (1976), 162Google Scholar.