Classics in the History of Greek Mathematics pp 81-109 | Cite as

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

## 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## Preview

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## Bibliographical Note

*AGM*= A. Szabó: 1969,*Anfänge der griechischen Mathematik*, Budapest and Munich/Vienna.Google Scholar

## Notes

- 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.“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.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.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.
*AGM*, III.1: ‘Der Beweis in der griechischen Mathematik’, esp. pp. 244–246.Google Scholar - 6.
*AGM*, III.3: ‘Der Ursprung des Anti-Empirismus und des indirekten Beweisverfahrens.’Google Scholar - 7.See the remark by W. C. Kneale, cited in note 2.Google Scholar
- 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.‘Transformation’ (see note 1), pp. 45–48;
*AGM*, pp. 292f.Google Scholar - 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.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.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.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.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.
*AGM*, III. 6–9, 13, 17, 21–23, 26.Google Scholar - 16.
*AGM*, III. 18, 20, 24, 25.Google Scholar - 17.Proclus,
*In Euclidem*, ed. G. Friedlein, Leipzig, 1873, p. 283.Google Scholar - 18.
*AGM*, III. 19.Google Scholar - 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.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.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.
*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.See Heath,
*HGM*I, pp. 221 f, 327–329.Google Scholar - 24.
*Ibid*., pp. 198f.Google Scholar - 25.Proclus,
*In Euclidem*, p. 213.Google Scholar - 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.I do not here claim that Szabo presumes to maintain this.Google Scholar
- 28.
*Prior Analytics*II, 25.Google Scholar - 29.
*In Euclidem*, p. 66.Google Scholar - 30.
*Evolution of the Euclidean Elements*, ch. VI/IV.Google Scholar - 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.
*EEE*, ch. III/III–IV.Google Scholar - 33.On Archytas, see Heath,
*HGM*, I, pp. 213–216.Google Scholar - 34.This view is prominent in B. L. van der Waerden,
*Science Awakening*, Groningen, 1954, ch. 5, esp. p. 115.Google Scholar - 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.Cited by Porphyry,
*In Ptolemaei harmonica*(cf. H. Diels and W. Kranz,*Fragmente der Vorsokratiker*, 6. ed., Berlin, 1951, 47*B*2).Google Scholar - 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.
*AGM*, III. 20. We return to the use of motion in the Greek geometry below.Google Scholar - 39.See my
*EEE*, ch. II and note 31 above.Google Scholar - 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 certain*results*(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.
*AGM*, III. 10, 13, esp. pp. 329, 341 f.Google Scholar - 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.
*AGM*, III. 20, 29.Google Scholar - 44.
*In Euclidem*, pp. 185–187.Google Scholar - 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.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.Heath,
*Euclid*, I, pp. 224ff.Google Scholar - 48.Such a recasting of the proof was recommended by Russell; cf. Heath,
*ibid*., pp. 249ff.Google Scholar - 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.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.
*In Euclidem*, pp. 188–190, 249f (citing Pappus).Google Scholar - 52.
*Posterior Analytics*, I, 13.Google Scholar - 53.For a review of the
*Elements*and its relation to the pre-Euclidean studies, see my*EEE*, ch. IX.Google Scholar - 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.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.
*In Euclidem*, pp. 178–184.Google Scholar - 57.Cf. Szabó,
*AGM*, III. 17, 20, cf. 21–26.Google Scholar - 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.Proclus,
*In Euclidem*, pp. 77–78.Google Scholar - 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*1005*b*11–20), but yet indemonstrable (*ibid*., 1006*a*5–15). 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.See the contributions cited in note 55.Google Scholar
- 62.See note 60.Google Scholar
- 63.One may note that the attempt, as by Szabó, to assign Euclid a dialectical motive for both hisGoogle Scholar
- 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.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.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.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.
*a*31–35. 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.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
- 70.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.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.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.For a survey of these constructions, see Heath,
*HGM*, I, ch. VII and the items cited in note 46 above.Google Scholar - 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 - 75.
- 76.Aristotle set out a doctrine of the “priority” of principles in the
*Posterior Analytics***I**, 6–7.Google Scholar - 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 - 77.I. Toth, ‘Das Parallelproblem im Corpus Aristotelicum’,
*Archive for History of Exact Sciences***3**, (1967), 249–422.Google Scholar - 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
- 79.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 - 80.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 - 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 - 82.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 - 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 - 84.See, for instance, the definitions given in
*Spiral Lines*and*Conoids and Spheroids*(prefaces).Google Scholar - 85.See my article cited in note 82 above and also ‘Archimedes and the spirals’,
*Historia Mathematica***5**, (1978), 43–75.Google Scholar - 86.Among the Archimedean treatises only
*Quadrature of the Parabola*adopts this usage.Google Scholar - 87.
- 88.Pappus,
*Collection*VII, ed. Hultsch, pp. 680–682.Google Scholar - 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