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On the Early History of Axiomatics: The Interaction of Mathematics and Philosophy in Greek Antiquity

  • Wilbur Richard Knorr
Part of the Boston Studies in the Philosophy of Science book series (BSPS, volume 240)

Abstract

The manner of the origins of deductive method both in philosophy and in mathematics has exercised the thoughts of many notable scholars. Unique to the ancient Greek tradition were conscious efforts to comprehend the process of thinking itself, and these inquiries have since developed into the philosophical subfields of logic, epistemology and axiomatics. At the same time, the pre-Euclidean Greek mathematicians turned to the problem of organizing arithmetic and geometry into axiomatic systems, in effect setting a precedent for the field of mathematical foundations. Did these two movements in the history of thought arise independent of each other, through some extraordinary coincidence? It would appear far more likely that this common adoption of deductive method resulted through interaction between the fields of philosophy and mathematics. But if this is so, to which shall we attribute priority?

Keywords

Common Notion Deductive Method Indirect Argument Plane Equilibrium Book Versus 
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Bibliographical Note

  1. AGM = A. Szabó: 1969, Anfänge der griechischen Mathematik, Budapest and Munich/Vienna.Google Scholar
  2. EEE = W. Knorr: 1975, The Evolution of the Euclidean Elements, Dordrecht.Google Scholar
  3. HGM = T. L. Heath: 1921, A History of Greek Mathematics, 2 vol., Oxford.Google Scholar

Notes

  1. 1.
    A. Szabó: ‘Transformation of mathematics into deductive science and the beginnings of its foun‑dation on definitions and axioms’, Scripta Mathematics 27 (1964), 27–48A, 113–139 (p.137).Google Scholar
  2. 2.
    “I do not think we should assume that mathematicians cannot use a logically valid pattern of rea­soning in their work until some philosopher has written about it and told them that it is valid. In fact we know that this is not the way in which the two studies, logic and mathematics, are related.” – W. C. Kneale in his commentary to A. Szabó, ‘Greek dialectic and Euclid’s axiomatics’ (in I.Lakatos (ed.), Problems in the Philoso|phy of Mathematics, Amsterdam, 1967, pp. 1–27), p.9. While acknowledging that the Eleatics were the first to make conscious use of indirect arguments in dialectic, N. Bourbaki deems it most probable that the mathematicians of the same period had already availed themselves of the same method in their own work (Éléments d’histoire des mathématiques, Paris, 1969, p. 11). The “internalist” position is advocated strongly by A. Weil: “[the question is] what is and what is not a mathematical idea. As to this, the mathematician is hardly inclined to consult outsiders. ... The views of Greek philosophers about the infinite may be of great interest as such; but are we really to believe that they had great influence on the work of Greek mathematicians? ... Some universities have established chairs for “the history and philosophy of mathematics”: it is hard for me to imagine what those two subjects can have in com­mon.” – ‘History of mathematics: why and how’, (pp. 6–7): lecture delivered at the International Congress of Mathematicians held in Helsinki, August 15–23, 1978.Google Scholar
  3. 3.
    In addition to the essay cited in note 1, Szabó has expounded his views on the rise of deductive method in the following: ‘Deiknymi als mathematischer Terminus für beweisen’,Maia 10 (N.S.) (1958), 106-131; ‘Anfänge des euklidischen Axiomensystems’, Archive for History of Exact Sciences 1, (1960), 37-106; ‘Der älteste Versuch einer definitorisch-axiomatischen Grundlegung der Mathematik’, Osiris 14, (1962), 308–369. These and other essays have been reworked as the basis of his Anfänge der griechischen Mathematik, Budapest/Munich, 1969 (esp. its third part, ‘Der Aufbau der systematisch-deduktiven Mathematik’). It is this last-named work to which I will most frequently refer here (to be cited as AGM). The first and second parts of AGM are concerned with the pre-Euclidean study of incommensurability and the terminology of early proportion theory, respectively. These will not be discussed here, but many points have been examined by me in The Evolution of the Euclidean Elements, Dordrecht, 1975.Google Scholar
  4. 4.
    P. Tannery, Pour l’histoire de la science helléne, Paris, 1887, pp. 259–260. His view was elabo­rated by F. M. Cornford (1939) and J. E. Raven (1948), but has been discounted by most scholars since; for references and discussion, see W. Burkert, Lore and Science in Ancient Pythagoreanism, Cambridge, Mass., 1972, pp. 41–52, 285–289 and my EEE, p. 43. On the pre-Euclidean “foun­dations crisis”, see H. Hasse and H. Scholz, ‘Die Grundlagenkrisis der griechischen Mathematik’, Kant-Studien 33 (1928), 4–34. This view has been much modified and criticized, as by B. L. van der Waerden (1941) and H. Freudenthal (1963); see my EEE, pp. 306–313.Google Scholar
  5. 5.
    AGM, III.1: ‘Der Beweis in der griechischen Mathematik’, esp. pp. 244–246.Google Scholar
  6. 6.
    AGM, III.3: ‘Der Ursprung des Anti-Empirismus und des indirekten Beweisverfahrens.’Google Scholar
  7. 7.
    See the remark by W. C. Kneale, cited in note 2.Google Scholar
  8. 8.
    For texts from the ancient Babylonian tradition, see O. Neugebauer, Mathematical Cuneiform Texts, New Haven, 1945. For the Egyptian tradition, see A. B. Chace et al., The Rhind Mathematical Papyrus, Oberlin, Ohio, 1927–29.Google Scholar
  9. 9.
    ‘Transformation’ (see note 1), pp. 45–48; AGM, pp. 292f.Google Scholar
  10. 10.
    AGM, III.3: I render Szabo’s anschaulich as “graphical”; other possibilities are “illustrative”,“visual”, “perceptual”, or even “intuitive”. I will understand him to refer to a kind of demon­stration based on concrete perceptual acts, such as setting out numbered objects or constructing suitably illustrative diagrams.Google Scholar
  11. 11.
    Of course, the ancient Babylonian geometers did not do this either. In this sense they too mightbe viewed as using “non-graphical” approaches in geometry. As far as the Greek classical tradi­tion is concerned, we should beware pushing this point on the non-graphical or abstract nature of the discipline too far. For within it the production of the appropriate diagram was always an inte­gral part of the proof. Proclus, for instance, includes “exposition” (ekthesis) and “construction” (kataskeue) as proper parts of any demonstration (see note 65 below) and the term for “diagram” (diagramma) could actually serve as a synonym for “theorem” (see my EEE, ch. III/II). Indeed, the availability of the diagram must have eased considerably the burden of the awkward geometric notation used by the Greeks and perhaps explains why they never saw fit to overhaul it.Google Scholar
  12. 12.
    See my ‘Problems in the interpretation of Greek number theory’, Studies in the History and Philosophy of Science 7 (1976) 353–368.Google Scholar
  13. 13.
    On this much-discussed aspect of Plato’s epistemology, see, for instance, F. M. Cornford,‘Mathematics and dialectics in the Republic VI–VII’, (1932), repr. in R. E. Allen (ed), Studies in Plato’s Metaphysics, London, 1965.Google Scholar
  14. 14.
    For a discussion of the relevant passages, see T. L. Heath, Mathematics in Aristotle, Oxford,1949, ch. IV, esp. pp. 64–67 and J. Barnes, Aristotle’s Posterior Analytics, Oxford, 1975, p. 161.Google Scholar
  15. 15.
    AGM, III. 6–9, 13, 17, 21–23, 26.Google Scholar
  16. 16.
    AGM, III. 18, 20, 24, 25.Google Scholar
  17. 17.
    Proclus, In Euclidem, ed. G. Friedlein, Leipzig, 1873, p. 283.Google Scholar
  18. 18.
    AGM, III. 19.Google Scholar
  19. 19.
    The fragment is preserved by Simplicius, In Aristotelis Physica, ed. H. Diels, Berlin, 1882,pp. 60–68. See T. L. Heath, A History of Greek Mathematics, Oxford, 1921, I, pp. 182–202.Google Scholar
  20. 20.
    Notably, Quadrature of the Parabola and Sphere and Cylinder I. The same format of exposition is followed in several of the discussions in Pappus’ Collection IV and V.Google Scholar
  21. 21.
    As we shall develop below, neither are the works of Archimedes, save for Plane Equilibria I, accurately viewed as efforts at axiomatization.Google Scholar
  22. 22.
    AGM, pp. 450–452. Given the great importance of the Hippocrates-fragment, Szabó’s discus­sion of it in AGM (part III) is remarkably slender, and what he does say inconsistent. On the one hand, Szabó insists on the deductive, even axiomatic, form of Hippocrates’ work: the “Elements” attributed to him by Proclus must surely have had some sort of foundations (pp. 309f, 342). Szabó, of course, wishes to assert the nascent axiomatic form of early Greek geom­etry as a mark of Eleatic influence. Yet he elsewhere stresses the non-axiomatic nature of Hippocrates’ study of the lunules; for instance, Hippocrates’ “beginning premisses” are lem­mas, whose proofs are given or assumed, not “first-principles” in the Aristotelian axiomatic sense (pp. 330f). Presumably, if the mathematicians had advanced too far in this direction so early, one might propose them as rivals to the Eleatics in the initiation of foundational inquiries. Similarly, Szabó points out that Hippocrates does not use indirect reasoning in the fragment (p. 33 1) – without, we should observe, recognizing the inappropriateness of such reasonings for this subject-matter; while he leaves the issue open, for want of explicit documentation, as to whether Hippocrates knew of this method, he clearly wishes to deny to the mathematicians pri­ority in its use. Yet his criticism of Toeplitz’ | reconstruction of a direct “inductive” proof of the area of the circle is based on the possibility that Hippocrates could have applied some form of the “exhaustion method”. This latter technique, ascribed to Eudoxus and applied throughout Elements XII, is especially characterized by its use of indirect reasoning. Szabó’s intent here is to attack Toeplitz’ thesis of the “Platonic reform” of geometry, for Szabó views the adoption of formal methods in geometry as a direct response to the Eleatics much before Plato. It is clear that Szabo has shifted his positions on the interpretation of Hippocrates’ work, without regard to consistency, in order to gain a favorable argumentative stance as context recommends.Google Scholar
  23. 23.
    See Heath, HGMI, pp. 221 f, 327–329.Google Scholar
  24. 24.
    Ibid., pp. 198f.Google Scholar
  25. 25.
    Proclus, In Euclidem, p. 213.Google Scholar
  26. 26.
    On the nature of this method and the extensive scholarship on it, see J. Hintikka and U. Remes, The Method of Analysis, Dordrecht, 1974,esp. ch. I.Google Scholar
  27. 27.
    I do not here claim that Szabo presumes to maintain this.Google Scholar
  28. 28.
    Prior Analytics II, 25.Google Scholar
  29. 29.
    In Euclidem, p. 66.Google Scholar
  30. 30.
    Evolution of the Euclidean Elements, ch. VI/IV.Google Scholar
  31. 31.
    On the early history of the study of incommensurables, see my EEE, ch. II. As far as the earli‑est discoveries are concerned, I argue a dating not much before 420 B.C. and a method involv­ing a form of “accidental” discovery rather than any deliberate formal proof (e.g., as in the indirect argument based on the odd and even). The possibility – indeed, the likelihood —– that these early discoveries were accidental in some such way undercuts any attempt, such as Szabo’s, to use the formal character of the study of incommensurables as a debt to the Eleatics (see AGM, I, 12; III, 2).Google Scholar
  32. 32.
    EEE, ch. III/III–IV.Google Scholar
  33. 33.
    On Archytas, see Heath, HGM, I, pp. 213–216.Google Scholar
  34. 34.
    This view is prominent in B. L. van der Waerden, Science Awakening, Groningen, 1954, ch. 5, esp. p. 115.Google Scholar
  35. 35.
    The fragment is preserved in Latin translation in Boethius, De institutione musica; see my dis‑cussion in EEE, ch. VII/I.Google Scholar
  36. 36.
    Cited by Porphyry, In Ptolemaei harmonica (cf. H. Diels and W. Kranz, Fragmente der Vorsokratiker, 6. ed., Berlin, 1951, 47B2).Google Scholar
  37. 37.
    Eutocius, In Archimedem, in the commentary to Sphere and Cylinder II, 2; cf. Archimedes, Opera, ed. J. L. Heiberg, III, Leipzig, 1915, pp. 84–88 and Heath, HGM, I, pp. 246–249.Google Scholar
  38. 38.
    AGM, III. 20. We return to the use of motion in the Greek geometry below.Google Scholar
  39. 39.
    See my EEE, ch. II and note 31 above.Google Scholar
  40. 40.
    AGM, III. 30, esp. pp. 446f. Becker develops his view in ‘Die Lehre vom Geraden und Ungeraden ...’, Quellen und Studien, 1936, 3:B, pp. 533–553. While here subscribing to Becker’s reconstruction, Szabo had earlier dismissed it on the grounds that knowledge of cer­tainresults (such as the properties of odd and even numbers presented in Elements IX) does not in itself justify ascribing to the Pythagoreans comparable formal proofs. As elsewhere Szabó here adopts inconsistent positions as context demands (cf. note 22 above).Google Scholar
  41. 41.
    AGM, III. 10, 13, esp. pp. 329, 341 f.Google Scholar
  42. 42.
    AGM, III. 17–20 (on the postulates), 21–25 (on the axioms). Heath covers this material in con‑siderable detail in Euclid’s Elements, 2. ed., Cambridge, 1926, I, ch. IX. We will take up the question of Euclid’s relation to Aristotle on the classification of the “first-principles” in the sec­ond part of this paper.Google Scholar
  43. 43.
    AGM, III. 20, 29.Google Scholar
  44. 44.
    In Euclidem, pp. 185–187.Google Scholar
  45. 45.
    Indeed, it is accepted by P. Bernays in his commentary to the paper by Szabó cited in note 2 above. S. Demidov also raises this view in his commentary on the present paper.Google Scholar
  46. 46.
    Archytas: see note 37 above. Eudoxus et al.: Heath, HGM, I, p. 255 and I. Thomas, History of Greek Mathematics, London, 1939, I, pp. 262–266, 388. Archimedes, Spiral Lines, preface, definitions (preceding prop. 12), and prop. 1, 2, 12 and 14. Motion-generated curves (e.g., “quadratrix”, “conchoid”): Heath, op. cit., pp. 238–240, 260–262. Related to this is the solution of problems by means of neuses (“inclinations”, that is, constructions involving a sliding ruler); see Heath, op. cit., pp. 235–238, 240f, and my article, ‘Archimedes’ Neusis-Constructions in Spiral Lines’, Centaurus 22 (1978), pp. 77–98.Google Scholar
  47. 47.
    Heath, Euclid, I, pp. 224ff.Google Scholar
  48. 48.
    Such a recasting of the proof was recommended by Russell; cf. Heath, ibid., pp. 249ff.Google Scholar
  49. 49.
    For instance, the congruence of circular segments subtending equal arcs in equal circles would follow from an “exhaustion” proof based on the triangular case.Google Scholar
  50. 50.
    Archimedes: Plane Equilibria I, Axiom 4 and prop. 9, 10 (aliter);Conoids and Spheroids, prop. 18. Pappus employs superposition in his proof of the proportionality of arcs and sectors in equal circles (Collection V, 12).Google Scholar
  51. 51.
    In Euclidem, pp. 188–190, 249f (citing Pappus).Google Scholar
  52. 52.
    Posterior Analytics, I, 13.Google Scholar
  53. 53.
    For a review of the Elements and its relation to the pre-Euclidean studies, see my EEE, ch. IX.Google Scholar
  54. 54.
    A. Seidenberg, ‘Did Euclid’s Elements, Book I, Develop geometry axiomatically?’ Archive for History of Exact Sciences 14 (1975), 263–295.Google Scholar
  55. 55.
    The literature on the Euclidean and Aristotelian divisions of the first-principles is rather large. In addition to the studies by Heath and Szabó cited in note 42 above and to Barnes’ notes on the Posterior Analytics (cf. note 14), the following may be considered: H. D. P. Lee, ‘Geometrical method and Aristotle’s account of first principles’, Classical Quarterly 29 (1935), 113–124; B. Einarson, ‘On certain mathematical terms in Aristotle’s logic’, American Journal of Philology 57 (1936), 33–54, 151–172; K. von Fritz, ‘Die APXAI in der griechischen Mathematik’, Archiv für Begriffsgeschichte 1 (1955), 13–103. J. Hintikka discusses this ques­tion in his paper in the present volume. Recent contributions include B. L. van der Waerden, ‘Die Postulate und Konstruktionen der frühgriechischen Geometrie’, Archive for History of Exact Sciences 18 (1978), 343–357. On constructions and existence-proofs, see note 70 below.Google Scholar
  56. 56.
    In Euclidem, pp. 178–184.Google Scholar
  57. 57.
    Cf. Szabó, AGM, III. 17, 20, cf. 21–26.Google Scholar
  58. 58.
    For instance, opening Plane Equilibria I, Archimedes “postulates” properties of | equilibrium and centers of gravity. Proclus would prefer that he have used “axiom” in this context (In Euclidem, p. 181).Google Scholar
  59. 59.
    Proclus, In Euclidem, pp. 77–78.Google Scholar
  60. 60.
    Surely it is this wide applicability which accounts for the designation of these principles as “common”, both by Aristotle and by Euclid; cf. the passages discussed by Heath, Mathematics in Aristotle, pp. 53–57, 201–203. Admittedly, Aristotle also describes the “axioms” as impossible to be mistaken about (Metaphysics 1005b11–20), but yet indemonstrable (ibid., 1006a5–15). Apparently this has given rise to the alternative view that these “common notions” are “common to all men”, as suggested in Heiberg’s rendering of Euclid’s koinai ennoiai as communes animi conceptiones (Szabó also subscribes to this view; cf. AGM, III. 25).Google Scholar
  61. 61.
    See the contributions cited in note 55.Google Scholar
  62. 62.
    See note 60.Google Scholar
  63. 63.
    One may note that the attempt, as by Szabó, to assign Euclid a dialectical motive for both hisGoogle Scholar
  64. 64.
    I will review below the familiar thesis of Zeuthen, that constructions served the role of exis‑ tence proofs in the ancient geometry. One should note well that the modern conception of proofs of existence in mathematics and logic has been greatly influenced by developments in the fields of analysis and set theory in the late nineteenth century. There is thus a real danger of interpreting early geometry anachronistically in the case of such questions.ostulates” and his “common notions” would impute to him an even more extreme dialectical position than that suggested by Aristotle; for this would make Euclid’s reason for articulating the “common notions” the desire to oblige those extremist critics who might challenge not only the “postulates” (some of which might, after all, appear to permit of proof) but also the “axioms” which seem self-evident.Google Scholar
  65. 65.
    Proclus gives an account of the formal subdivision of a theorem and its proof: In Euclidem, pp. 203–205; cf. Heath, Euclid, I, pp. 129–131.Google Scholar
  66. 66.
    Indeed, there are many such steps in the ancient geometry which passed by without being then recognized as tacit assumptions; cf. the discussion by Becker cited in note 72.Google Scholar
  67. 67.
    Typically, auxiliary elements may be introduced in the “construction” (kataskeue) later in the proof; cf. Hintikka’s discussion of auxiliary constructions in the book cited in note 26.Google Scholar
  68. 68.
    a31–35. I have since come upon the article by A. Gomez–Lobo, ‘Aristotle’s hypotheses and the Euclidean postulates’, Review of Metaphysics 30 (1977), 430–439 in which the same view is argued in detail.Google Scholar
  69. 69.
    These postulates assert (1) that the line segment connecting two given points may be drawn; (2) that a given line segment may be indefinitely extended in a straight line; and (3) that the circle of given center and radius may be drawn. The remaining two postulates do not involve construction as such: (4) that all right angles are equal; and (5) that if two lines are cut by a third such that the interior angles made on one side of the third are less than two right angles, then the given lines, extended sufficiently far, will meet on that side of the third.Google Scholar
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    The primary statement of this thesis is by H. G. Zeuthen, ‘Geometrische Konstruktion als Existenzbeweis ...’, Mathematische Annalen 47, (1896), 272–278. It has since been elaborated by O. Becker, Mathematische Existenz, Halle a. d. S., 1927; A. D. Steele, | ‘Ueber die Rolle von Zirkel und Lineal in der griechischen Mathematik’, Quellen und Studien 3:B, (1936), 288–369; and E. Niebel, ‘... die Bedeutung der geometrischen Konstruktion in der Antike’, Kant-Studien, Ergänzungsheft, 76, Cologne, 1959. I criticize Zeuthen’s view in EEE, ch. III/II (cf. also note 64 above). Van der Waerden and Seidenberg also deny the existential sense of Euclid’s postulates (cf. notes 54 and 55 above).Google Scholar
  71. 71.
    Even proofs of incommensurability do not assume the form of negative existence theorems (i.e., “there exist no integers which have the same ratio as given magnitudes”). In this regard, the closest one comes to an existential expression is in Elements X, 5–8; e.g., “commensurable magnitudes have to each other the ratio which a number has to a number” (X, 5). Cf. also X, Def. 1: “magnitudes are said to be commensurable when they are measured by the same meas­ure, but incommensurable if none can become (genesthai) their common measure”. In such instances an existential statement is hardly to be avoided. Even so, it is remarkable how these formulations emphasize the properties of assumed magnitudes, rather than the existence of magnitudes having these properties.Google Scholar
  72. 72.
    These issues are examined by O. Becker, ‘Eudoxos-Studien I–IV’, Quellen und Studien B:1–3, 1933–36 (esp. II and III).Google Scholar
  73. 73.
    For a survey of these constructions, see Heath, HGM, I, ch. VII and the items cited in note 46 above.Google Scholar
  74. 74.
    For instance, Hippocrates’ third lunule (Heath, ibid., p. 196) and Archimedes’ construction in Spiral Lines, prop. 5, are both effected by neusis, even though a “Euclidean” construction is possible; this is discussed in my article, cited in note 46 above.Google Scholar
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    Pappus, Collection IV, 36; cf. my article on neusis (note 46 above).Google Scholar
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    Aristotle set out a doctrine of the “priority” of principles in the Posterior Analytics I, 6–7.Google Scholar
  77. 76a.
    In the Method Archimedes reduces the problem of the volume of cylindrical section to that of the area of the parabolic segment. Doubtless, he knew of the equivalence of the problems of finding the area of the parabola and that of the spiral; but this notion is used in neither of the treatises devoted to these problems and does not seem to have appeared before Cavalieri’s Geometria Indivisibilibus (1635). The reduction of the area of the spiral to the volume of the cone is employed in Pappus’ Collection IV, 22 in a treatment which appears to stem from Archimedes (see my article cited in note 85). The center of gravity of the paraboloid was solved by Archimedes in a lost work On Equilibria and applied in the extant work On Floating Bodies; see my discussion in ‘Archimedes’ lost treatise on centers of gravity of solids’, Mathematical Intelligencer 1, (1978), 102–109. In this instance, it seems that Archimedes used an independent summation-procedure based on the same expressions proved in Spiral Lines and Conoids and Spheroids.Google Scholar
  78. 77.
    I. Toth, ‘Das Parallelproblem im Corpus Aristotelicum’, Archive for History of Exact Sciences 3, (1967), 249–422.Google Scholar
  79. 78.
    It is not even clear whether among the Eleatics the arguments on the nature of being were intended as a possible basis for a cosmological system, or whether they were merely a negative device for the criticism of other cosmologies.Google Scholar
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    See my EEE, ch. VII. Theaetetus’ role in the foundation of number theory was | argued by H. G. Zeuthen, ‘Sur les connaissances géométriques des Grecs avant la réforme platonicienne’, Oversigt Dansk. Videns. Sels. For., 1913, pp. 431–473.Google Scholar
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    I present the reconstructed “Eudoxean” theory in ‘Archimedes and the Pre-Euclidean Proportion Theory’, Archives internationales d’histoire des sciences, 28 (1978), 183–244. The anthyphairetic theory was first proposed by Zeuthen and by Toeplitz, later elaborated by O. Becker, ‘Eudoxos-Studien I’ (see note 72). I review and modify his reconstruction of this theory in EEE, ch. VIII/II-III and Appendix B.Google Scholar
  82. 81.
    See, for instance, the circle-quadratures of Antiphon and Bryson (Heath, HGM, I, pp. 220–226). Democritus conceived solids as somehow constituted of parallel indivisible plane sections (ibid., pp. 179f). The latter has been construed (I believe, mistakenly) to be the basis of an ancient infinitesimal analysis; cf. S. Luria, ‘Die Infinitesimaltheorie der antiken Atomisten’, Quellen und Studien 2:B, (1933), 106–185.Google Scholar
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    On Archimedes’ early works and their dependence on pre-Euclidean sources, see my ‘Archimedes and the Elements’, Archive for History of Exact Sciences 19, (1978), 211–290 and the article cited in note 80.Google Scholar
  84. 83.
    For a review of criticisms of this work, see E. J. Dijksterhuis, Archimedes, 1957, ch. IX and my articles cited in notes 80 and 82 above.Google Scholar
  85. 84.
    See, for instance, the definitions given in Spiral Lines and Conoids and Spheroids (prefaces).Google Scholar
  86. 85.
    See my article cited in note 82 above and also ‘Archimedes and the spirals’, Historia Mathematica 5, (1978), 43–75.Google Scholar
  87. 86.
    Among the Archimedean treatises only Quadrature of the Parabola adopts this usage.Google Scholar
  88. 87.
    See Apollonius, Conics I, preface and Hypsicles, Elements XIV, preface.Google Scholar
  89. 88.
    Pappus, Collection VII, ed. Hultsch, pp. 680–682.Google Scholar
  90. 89.
    In certain fields outside the area of formal geometry notable advances were made in later antiquity: for instance, plane and spherical trigonometry, mathematical astronomy and number theory (in the Diophantine tradition). Each of these was associated with techniques of practical computation, rather than theoretical geometry, and none was systematized along the lines of Euclid, Archimedes and Apollonius. The non-axiomatic character of much ancient mathematics and mathematical science is examined by P. Suppes in his contribution to this volume and by F. Medvedev in his commentary.Google Scholar

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