Classics in the History of Greek Mathematics pp 115-137 | Cite as

# Construction as Existence Proof in Ancient Geometry

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## Abstract

The title of this essay is borrowed from a modern mathematical historian; its tag line is taken from an ancient philosopher. Their shared interest in questions dealing with existence has given rise to a familiar thesis about ancient geometry: that its constructions were intended to serve as proofs of the existence of the constructed figures. I propose here to examine that thesis, to argue its weakness as a *historical* account of ancient geometry, and to offer an alternative view of the role of problems of construction: ^{1} that constructions, far from being assigned a specifically existential role, were not even the commonly adopted format for treating of existential issues when these arose: that some central questions relating to existence were handled through postulates or tacit assumptions, rather than through explicit constructions; that, by contrast, when constructions were given, the motive lay in their intrinsic interest for the ancient geometers. On this basis I will maintain that preconceptions based on modern theories have interfered in the modern effort to interpret ancient mathematics, thus attaching to the existential view of constructions a greater credence than the ancient evidence could justify.

## Keywords

Equilateral Triangle Regular Polygon Existence Proof Existential Issue Ancient Tradition## Preview

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## Notes

- 1.Throughout my discussion here the word “problem” will be used only in the technical sense it has within the ancient mathematical literature: as that form of geometric proposition which seeks the construction of a figure in a specified relation to certain designated, or “given,” magnitudes. The formal distinction between problems and theorems will be elaborated below. Note that the term “proposition” in this context has a more restricted sense than it does in logic.Google Scholar
- 2.An account of the Platonic philosophy of mathematics is provided by A. Wedberg,
*Plato’s Philosophy of Mathematics*(Stockholm: Almqvist Wiksell, 1955 ). A commentary on the mathematical passages from Aristotle’s works appears in T.L. Heath,*Mathematics in Aristotle*( Oxford: Clarendon Press, 1949 ).Google Scholar - 3.Perhaps the most ambitious effort to portray ancient mathematics as a response to early dialectical critiques, particularly those of the Eleatics, is by A Szabó,
*The Beginnings of Greek Mathematics*(Dordrecht, Neth.: Reidel, 1978 [tr. of the German ed., 1969]); see also note 62 below.Google Scholar - 4.Proclus,
*In Primum Euclidis Elementorum Librum Commentarii*, G. Friedlein, ed. (Leipzig: Teubner, 1873 ) 233–235; cf. the translation by G. Morrow, Proclus:*A Commentary on the First Book ofEuclid’s Elements*( Princeton, N.J.: Princeton University Press, 1970 ) 182–183.Google Scholar - 5.It is possible that Proclus has a more limited pedagogical intent here, rather than the general view of existence I ascribe to him. In either event, he can hardly be representing any view Euclid would have maintained. If the existence of triangles is at issue, as background to the theory of congruence, pedagogically it would be important to direct the student toward the
*general*case, rather than to the special case of equilateral triangles. A construction could be effected by assuming any three noncollinear points as given, joining them by lines two by two (*via*post. 1), and then proving that the resulting figure is a triangle (*via*def. 19). Of course, we here depend on the existence of three such points; this is not postulated by Euclid, a salient gap in his axiomatic scheme (cf. 1. Mueller,*Philosophy of Mathematics and Deductive Structure in Euclid’s Elements*[Cambridge, Mass.: MIT Press, 1981] 15). On the other hand, his construction of the equilateral triangle in I 1 also depends on implicit postulates, specifically the intersection of two given circles (cf. Mueller,*Philosophy*27–28); objections of a different sort were lodged against this theorem by ancient critics (cf. Proclus,*In Euclidem*214–215). Ultimately, then, the existence of triangles is more nearly a matter for postulate than Proclus appreciates.Google Scholar - 6.Die geometrische Construction als ‘Existenzbeweis’ in der antiken Mathematik”
*Mathematische Annalen*47 (1896) 222–228. A characteristic favorable use is by O. Becker,*Mathematische Existenz (Jahrbuch für philosophische und phänomenologische Forschung*8, Halle a. d. S., 1927) 130–133. Perceptive criticisms of the thesis, however, have been made by Mueller,*Philosophy*14–16; and by A. Frajese, “Sur la signification des postulats euclidiens”*Archives internationales d’histoire des sciences*4(15) [1951] 383–392.Google Scholar - 7.The ancient distinction between problems and theorems will be discussed below.Google Scholar
- 8.Zeuthen, “Construction” 223 (my translation).Google Scholar
- 9.
*Ibid*. 224.Google Scholar - 10.
*Ibid*. 225. VI 27 establishes the necessity of the condition (a/2)2! b for solving the problem in VI 28 of the “defective” case of the application of areas; in Zeuthen’s algebraic formulation, this entails finding x such that x2—a•x+b=0 (Euclid’s formulation is entirely geometric). The Archimedean example in an appendix to*Sphere and Cylinder*II 4 will be discussed below. It effects a third-order analogue to the construction from Euclid just cited, namely (again in Zeuthen’s algebraization) to find x such that x3—a • x2+b2 • c=0, requiring the condition that b2 • c!9 4a3/27.Google Scholar - 11.
*Ibid*. 225–226. Zeuthen proposes to explain Euclid’s postulate via his need to guarantee the existence of points of intersection of lines in certain configurations (cf. I 44). But a simpler account is possible: that in his theorems on the sum of angles of triangles, to secure his theorem on the angles formed by the transversal to two parallel lines (I 29), Euclid recognized the need for precisely this step, and that ultimately unable to prove it as a theorem, he enunciated it as the postulate. A similar route seems to have led to the insertion of post. 6 (that two lines cannot enclose a space), assumed in I 4. The modern editors consider this postulate to be an interpolation (cf. Mueller,*Philosophy*31–32).Google Scholar - 12.
*Ibid*. 226–228. Zeuthen’s argument that the early restriction to perpendicular sections of right cones enables a direct correspondence between the geometric formation of these curves and their expression via second-order coordinate relations is technically neat. I believe, however, that he exaggerates the ancients’ reluctance to accept pointwise and mechanical constructions, and that his insight can be applied toward a more plausible view than his of the early studies of the conics; see my*The Ancient Tradition of Geometric Problems*(Basel/Stuttgart/Boston: Birkhäuser) ch. 3 (forthcoming).Google Scholar - 13.It is discussed prominently by Proclus, who cites many precedents among philosophical and mathematical writers. For instance, he divides the “things (following) from the first principles” (râ 7ro rcov eXcov) into problems and theorems (
*In Euclidem*77), and then discusses the views on this distinction advocated by Speusippus, Menaechmus and others (77–81). In his remarks on Posidonius’ views (80) he distinguishes between the theoretic and problematic “proposition”Google Scholar - 14.In fact, each problem will consist of a section executing the construction, followed by a section proving that the figure has the stated property. This latter section is, in effect, a theorem, and if the preceding construction is viewed simply as an extended
*protasis*, then the problem itself becomes a theorem. The conversion of theorems to problems, while possible, is not always so straightforward. Problems of locus (i.e., to identify the figure whose elements all satisfy a stated property) seem to form an intermediate class, since the proposition requires producing a construction, yet its enunciation is typically in the format of a theorem.Google Scholar - 15.In a constructivist view, the assertions of possibility and constructibility would be equivalent.Google Scholar
- 16.For a wide-ranging discussion of this method, see J. Hintikka and U. Remes,
*The Method of Analysis*(Dordrecht, Neth.: Reidel, 1974). It is a prominent interest in my study of the Greek geometric tradition (cited in note 12 above).Google Scholar - 17.Eutocius’ text and commentary appear in J. L. Heiberg’s edition of Archimedes,
*Opera Omnia*2nd ed. (Leipzig: Teubner, 1915) III 130–152. For discussions of the construction, see E. J. Dijksterhuis,*Archimedes*(New York: Humanities Press, 1957) 195–200; and my*Ancient Tradition*ch. 5.Google Scholar - 18.I here set in more algebraic form what Archimedes frames in a strictly geometric manner.Google Scholar
- 19.The problem in its initial formulation is always solvable; but in its more general form there can arise values of the given terms for which a solution cannot be constructed, so that a
*diorism*is then called for.Google Scholar - 20.The relation of the analysis and the synthesis is usually so close that the latter is sometimes omitted as “obvious” (tpaveed) or provided only in outline; cf. Diocles’ treatment of Archimedes’ problem on the division of the sphere in On
*Burning Mirrors*, G. Toomer, ed. (Berlin/Heidelberg/New York: Springer-Verlag, 1976) 86; and the problems on the regular solids presented by Pappus in the*Collection*III 48–52 (F. Hultsch, ed. [Berlin: Weidmann, 1876–78] I 144, 146, 148, 154, 162).Google Scholar - 21.
*Collection*Book VII, preface (ed. Hultsch, II. 636). The passage is discussed in detail by Hintikka and Remes, Method ch. II.Google Scholar - 22.
*Collection*V 29 (ed. Hultsch, I 382): to mark off an arc from a given circle such that the segments of the two tangents drawn from its endpoints shall be less than a given line segment. One of the ancient theories of proportion depends on the lemma: given three homogeneous magnitudes, to find a magnitude which shall be less than the first, greater than the second, and commensurable with the third; for references to the ancient texts and discussion, see my “Archimedes and the Pre-Euclidean Proportion Theory”*Archives internationales d’histoire des sciences*28 (1978) 183–244.Google Scholar - 23.I would estimate that about 300 geometric problems are extant in the works of Euclid, Archimedes, Apollonius, Pappus and the minor writers and commentators. Inclusion of Euclid’s
*Data*, which adheres to an alternative problematic format would add 94 more. Further, in the area of arithmetic,*all*of the propositions (amounting to almost 300) in the extant ten of the thirteen books of Diophantus’*Arithmetica*are problems. These figures can be considered only a bare sampling of the scope of the problematic literature in antiquity. We know, for instance, of whole treatises on loci by Euclid, Apollonius, Eratosthenes, and others, which are no longer extant; cf. the preface and commentary by Pappus in Collection Book VII, a résumé of which is given by T. L. Heath.*A History of Greek Mathematics*(Oxford: Heath. 1921 ) II 399–427.Google Scholar - 24.E. g. Book I 54–60; II 44–53; VI 28–33; the constructions of normals in V 58–63, which could easily be set in the form of problems, happen here to be framed as theorems (cf. the edition by H. Balsam.
*Des Apollonios von Perga sieben Bücher über Kegelschnitte*[Berlin: Reimer, 1861]); most of the propositions of the lost eighth book were problems associated with the theorems of the seventh (cf. VII, pref.)Google Scholar - 25.Cf. Pappus’ accounts of their work, cited in note 23 above.Google Scholar
- 26.
*Collection*IV 36–42 (ed. Hultsch. I 272–280). The technique of*neusis*, that is, construction via marked-ruler, is prominent among the researches by Archimedes and his successors; cf. my discussion in*Ancient Tradition*chs. 5 and 6.Google Scholar - 27.The existential variant is common in modern discussions of axiomatics: cf., for instance, D. Hilbert,
*Foundations of Geometry*ch. I (English ed., La Salle, Ill.: Open Court, 1971, based on the 10th German ed.). Mueller contrasts the expressions of Euclid and Hilbert,*Philosophy*14–15.Google Scholar - 28.The pseudo-Euclidean
*Catoptrics*prop. 29 provides a striking illustration. It is a problem, “it is possible that there be constructed a mirror such that many faces appear in it, some greater, some smaller, some nearer, some farther, some with the right on the right and the left on the left, others with the left on the right and the right on the left” (*Euclidis Opera*, J. L. Heiberg, ed. [Leipzig: Teubner, 1895] VII 338), which concludes a sequence of theorems (prop. 16–28) on the placement, size and orientations of images seen in plane, convex and concave mirrors. The problem can hardly have an existential role relative to those theorems; instead, it recapitulates their results in a manner suggestive of their practical implementation. Thus, the issue here is not the existence of this mirror, but the actual manner of its construction.Google Scholar - 29.Euclid orders the solids in the sequence 4, 8, 6, 20, 12 (relative to the number of faces); presumably, this follows the order of the lengths of the edges of the solids inscribed in the same sphere, where the tetrahedron has the longest edge, the octahedron a shorter one, and so on, until the dodecahedron with the shortest. The comparison of these lengths forms the content of the last theorem in the book (XIII 18), so that this scheme is appropriate. By contrast, other rationales for the ordering–e.g., according to the number of vertices, or according to the relative volumes of the solids inscribed in the same sphere–are not relevant to the subjects actually examined in the book.Google Scholar
- 30.The Greek technical language could presumably permit the coining of terms like TevTevywvdeSeov. The naming of the Archimedean semiregular solids continues the Euclidean pattern by following the number of faces: e.g., 8-hedron, 14-hedron (three forms), 26-hedron (two forms), 32-hedron (three forms), 38-hedron, 62-hedron (two forms), and 92-hedron; cf. Pappus,
*Collection*V 19 (ed. Hultsch, I 352–354 ).Google Scholar - 31.Mueller elucidates these constructions by supplying analyses (
*Philosophy*ch. 7). Such a procedure, which doubtless was followed by the ancients in working out these constructions, assumes the prior recognition of their qualitative description (cf. 254–255).Google Scholar - 32.
*Euclidis Opera*(ed. Heiberg) VII 80.Google Scholar - 33.For instance, prop. 39: “if a magnitude is set at right angles to the base plane,CHRW(133) it shall always be seen equal when transposed according to a disposition parallel to its original one” (ed. Heiberg, VII 84).Google Scholar
- 34.
*Ibid*. 104.Google Scholar - 35.Contrast the example of
*Cat*. prop. 29 (cited in note 28 above) where the problem serves to summarize and apply the results in the preceding theorems.Google Scholar - 36.The problem of dividing an arbitrarily given angle into n equal parts in general reduces to solving a relation of order n. The problem of division for all n, however, is transcendental. The ancients produced constructions utilizing the quadratrix and Archimedean spiral (cf. Pappus,
*Collection*IV 45–46), but classed it among the “linear” problems, that is, not solvable via the Euclidean techniques or*via*conic sections (ed. Hultsch, 284); cf. 54, 270 for Pappus’ remarks on the tripartite division of problems, discussed by Heath,*History*I 218–220 and in my*Ancient Tradition*ch. 8.Google Scholar - 37.
*Euclidis Opera*, ed. Heiberg, VII 106. The phrase “sometimes whole” (*pote holon*) does not make particularly good sense; we should have expected “sometimes one-third” (*pote triton*) in this context. The phrase is absent from the medieval Latin translation (reproduced by Heiberg, 107), which is usually in complete literal agreement with the Greek. Theon’s recension omits an analogue of prop. 49, doubtless for its mere duplication of the result of prop. 48.Google Scholar - 38.On the hybrid character of locus problems, see note 14 above.Google Scholar
- 39.Such appears to be the motive of problems in the pseudo-Euclidean
*Catoptrics*(cf. note 28 above), and is patently the case in the Heronian*Catoptrics*prop. 11–18 (in*Heronis Opera*, W. Schmidt, ed. [Leipzig: Teubner, 1900] II 336–364); e.g., prop. 16: “in a certain conveniently located window in a house, to place a mirror in the house through which people will appear as they come from the opposite direction in the streets and will be seen by those in a certain given place in the house” (352).Google Scholar - 40.
- 41.
*In Aristotelis Analytica Posteriora*, M. Wallies, ed. (*Commentaria in Aristotelem Graeca*XIII, Berlin: Reimer, 1909 ) 112.Google Scholar - 42.
*Archimedis Opera*, ed. Heiberg, III 230.Google Scholar - 43.In
*Spiral Lines*prop. 18, Archimedes shows that the subtangent corresponding to the position of the radius vector terminating one full turn of the spiral equals the circumference of the circle with radius equal to that radius vector. Strictly speaking, this does not rectify the circular arc, since it depends on the construction of the spiral and its tangent. Archimedes nowhere claims this to be a solution of the problems of the rectification and quadrature of the circle, but some of the commentators seem to view it as such; cf. Iamblichus, cited by Simplicius, In*Aristotelis Physica*ed. H. Diels, I 60 (alternative version reproduced in I. Thomas,*Greek Mathematical Works*LCL [Cambridge, Mass.: Harvard University Press, 1939–41] I 334 ).Google Scholar - 44.The same assumption is made in Euclid’s proof that a magnitude in a given ratio to a given magnitude is itself given (
*Data prop*. 2). The editor, R. Simson, adds the phrase “and ifCHRW(133) a fourth proportional can be found” to the enunciation, both here and in all subsequent appearances, since the proof in fact makes this assumption (*The Elements of Euclid*[Glasgow: Robert and Andrew Foulis, 1762] 360, 454–455.). But neither Euclid nor the writers in the later analytic tradition seem conscious of the need for such a qualification.Google Scholar - 45.O. Becker has indicated the discrepancy between these appeals to the assumption of the fourth proportional and views of the ancients’ adherence to a constructivist position; see his “Eudoxos Studien II”
*Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik*2, Abt. B (1933) 369–387.Google Scholar - 46.Zeuthen (“Construction” 226) maintains that the existence of this point follows from Euclid’s definition of the circle as “a plane figure contained by a single lineCHRW(133)” (Def. 15). But in fact, its existence depends on postulates of continuity as well. A discussion of the use of such postulates, particularly in the form proposed by Dedekind, for establishing incidence relations of lines and circles appears in T. L. Heath,
*The Thirteen Books of Euclid’s Elements*2nd ed. (Cambridge: Cambridge University Press, 1926) I 234–240.Google Scholar - 47.This suggests that, whatever the apparent axiomatic ideal of the
*Elements*might be, in execution Euclid does not articulate*every*assumption introduced into his proofs, but only those whose character is sufficiently nonobvious as to attract attention. I think this applies, for instance, to the case of the parallel postulate; cf. note 11 above.Google Scholar - 48.For a résumé of the argument, see Heath,
*History*II 206–213.Google Scholar - 49.I present support for this view in my “
*Archimedes and the Elements” Archive for History of Exact Sciences*19 (1978) 238–239.Google Scholar - 50.This can be done for some theorems of maxima, for instance, the fact that of all triangles inscribed in a given circle, the equilateral triangle has the greatest area (cf. my
*Ancient Tradition*ch. 3); this example is related to the mathematical passage in Plato’s*Meno*, to be discussed below.Google Scholar - 51.Cf. note 36 above.Google Scholar
- 52.A counterexample is presented by O. Toeplitz and H. Rademacher as background to their demonstration of the isoperimetric theorem for the circle; cf. their
*The Enjoyment of Mathematics*(Princeton, N.J.: Princeton University Press, 1966) [transl. from the German edition of 1933] chs. 21–22.Google Scholar - 53.A very effective way of retrieving the philosophical background of such writers as Pappus would be to attempt a source analysis of those passages where they adopt specific philosophical positions. In the case of Pappus’ account of the method of analysis, for instance, an indirect dependence on Aristotelian | passages can be perceived (cf. my
*Ancient Tradition*ch. 8). Little of this sort has been done, however, and the fragmentary nature of the evidence might severely limit the scope of one’s findings.Google Scholar - 54.Cf. the line quoted from
*Generation and Corruption*II 10 at the beginning of this paper. There Aristotle seeks to explain the reproductive cycle in living things: since nature strives for the best, and since being is better than nonbeing, and since individual living things necessarily have finite life spans, the species attains a form of eternal existence through the perpetual cycle of generations.Google Scholar - 55.Cf.
*Posterior Analytics*I 10 (76a32–37): What is denoted by the first (terms) and those derived from them is assumed; but, as regards their existence [lit.: that they are], this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight (line) is, or what a triangle is (must be assumed); and the existence of [lit.: that it is possible to take] the unit and of magnitude must also be assumed, but the rest must be proved. [from Heath,*Mathematics in Aristotle*50–51]Google Scholar - 56.Cf. Mueller,
*Philosophy*14–15.Google Scholar - 57.For a survey, see R. S. Bluck,
*Plato’s Meno*(Cambridge: Cambridge University Press, 1961). I discuss the technical aspects suggested by the passage in*Ancient Tradition*ch. 3.Google Scholar - 58.Heath points out this difficulty (
*History*I 3 03). The passage is labelled as a “*diorismos*” by I. Thomas in his reproduction of the text (*Greek Mathematical Works*I 394; cf. 397n).Google Scholar - 59.I reconstruct such a proof in
*Ancient Tradition*ch. 3.Google Scholar - 60.Cf. Proclus,
*In Euclidem*212–213; so also Eutocius, following Eratosthenes, in his commentary on Archimedes’*Sphere and Cylinder*(*Archimedis Opera*, ed. Heiberg, III 88 ).Google Scholar - 61.The mathematical views contained within these philosophies are investigated by Ian Mueller in his
*Coping with Mathematics*(*The Greek Way*) (Chicago: Morris Fishbein Center for the Study of the History of Science and Medicine (Publication No. 2), 1980 ); this inquiry is continued in his “Geometry and Scepticism” in M. Burnyeat et al., ed.,*Scepticism and Science*( Cambridge: Cambridge University Press, 1982 ).Google Scholar - 62.See, in particular, A. Szabó,
*Beginnings of Greek Mathematics*(cited in note 3 above) sect. III. I review and criticize some of his positions in my “Early History of Axiomatics”*Pisa Conference (1978) Proceedings*, ed. J. Hintikka et al. (Dordrecht, Neth.: Reidel, 1981) I 145–186.Google Scholar - 63.
*In Euclidem*233–234; cited in notes 4 and 5 above.Google Scholar - 64.This impasse is noted by Mueller in his “Geometry and Scepticism.” It may also underlie Aristotle’s remark that the geometer qua geometer does not have to address questions which violate the principles of his science (
*Physics*I 2, 185a14–17;*Sophistical Refutations*11, 171b7–18).Google Scholar - 65.Cf. scholium no. 50 (
*Euclidis Opera*, ed. Heiberg, V 135): “it is proved from this that indivisible lines do not exist, if indeed it is possible to bisect the proposed side.” The existence of incommensurable magnitudes is cited in scholia to Book X as evidence against the atomists; cf. no. 26 (ad X 1): “that there is not a least magnitude, as the Democriteans say, is also proved via this theorem, if in fact it is possible to take a magnitude smaller than any given magnitude” (436).Google Scholar - 66.Cf. schol. no. 1 to Book X (415–416), where the infinite divisibility of magnitude is cited as the cause underlying the nature of incommensurable magnitudes. Geminus, cited by Proclus (
*In Euclidem*278), claims, with reference to I 10, that the assumption of the continued divisibility of magnitude underlies the construction of the bisection of the line segment, rather than conversely: “that every magnitude shall be continually divided, and shall never result in an indivisibleCHRW(133) is provable; but it is an axiom (¬ξgtμα) that every continuous (magnitude) is divisible.” Eudemus (cited by Simplicius,*In Physica*, ed. Diels, I 55) maintains that it is the principle of the infinite divisibility of magnitude which is violated by Antiphon’s fallacious circle-quadrature (cf. Heath,*History*I 222 ).Google Scholar - 67.One would not wish to deny that Euclid’s inclusion of statements of the “common notions” (κοιlαi Ollοιαι) prefacing Book I, and perhaps also his formulations of some of the definitions, were influenced by philosophical discussions of the first principles. There are interesting parallels to Aristotelian remarks on “axioms,” as cited by Proclus (In Euclidem 193–194); for discussion, see Heath, Euclid’s Elements I 221–222 and Mathematics in Aristotle 50–57. But the question of the authenticity of the “common notions” is vexed, and Mueller observes that, in any case, the Euclidean listing seems not even intended to be complete (Philosophy 35–36).Google Scholar
- 68.The problematic format. “to
*find*CHRW(133),” adopted throughout Euclid’s so-called “arithmetic books” (VII-X) differs from that in the other books (where an overtly constructive terminology is used), and may thus reflect a sensitivity to the kinds of ontological issues raised by Plato. This is a point raised by H. Mendell in his current studies of the Platonic and Aristotelian mathematical philosophies.Google Scholar - 69.For a review of the intuitionist position, one may consult M. Black,
*The Nature of Mathematics*(repr., Paterson, N.J.: Littlefield, Adams, 1959); and W. and M. Kneale,*The Development of Logic*(Oxford: Clarendon Press, 1962) ch. XI.Google Scholar - 70.The formalist position is reviewed by S. Körner,
*The Philosophy of Mathematics*(London: Hutchinson, 1960); cf. also the works by Black and Kneale Kneale cited in the preceding note. Mueller contrasts Hilbert’s view with the Euclidean approach (*Philosophy*ch. 1).Google Scholar - 71.Zeuthen, “Construction” 223. The drawing of a line from a given point and equal to a given line is effected immediately via standard compasses; Euclid’s construction reveals that his third postulate (i.e., the drawing of a circle of given center and radius) is equivalent to construction
*via collapsing*compasses. Thus, one must show that constructions via the one instrument are possible also via the other. For further remarks, see Mueller,*Philosophy*15–16, 24–25; and Heath,*Euclid*I 246.Google Scholar - 72.The advanced treatises of Archimedes and Apollonius are not even axiomatic
*in form*; although they of course develop their subject matters according to a strict deductive sequence of theorems, they presuppose the entire elementary literature, the “data,” and other technical materials. To be sure, postulates are stated at the beginning of Archimedes’*Sphere and Cylinder*Book I, and also Apollonius’*Conics*Book I; but these enunciate only those few special principles applied in the treatise, without attempting to provide a complete listing of all assumptions made therein. Euclid’s*Optics*and Archimedes’*Plane Equilibria*Book I list their initial postulates in a manner which at first suggests an axiomatic intent. But the works themselves freely admit assumptions not covered in the lists, so that the formal execution of the project, if indeed it was intended to be axiomatic, is seriously flawed (cf. P. Suppes, “Limitations of the Axiomatic Method in Ancient Greek Mathematical Sciences,”*Pisa Conference Proceedings*, ed. J. Hintikka et al. [Dordrecht, Neth.: Reidel, 1981] I 197–213).Google Scholar - 73.These include the continuity assumptions, discussed in the section above.Google Scholar
- 74.The constructive aspect of geometry seems to have been a basic concern for Zeuthen in his own mathematical studies. See the biography and bibliography prepared by M. Noether in
*Mathematische Annalen*83(1921) 1–23 (esp. 9–11). The constructive, even “inductive,” approach advanced by Zeuthen would appear to contrast with the more abstract, deductive position on existence advocated by Hilbert.Google Scholar - 75.For instance, Zeuthen insists that the ancients could not have considered the use of conic sections for solving cubic relations to afford any practical advantage, but that the introduction of conics and other special curves was for their use in extending geometric theory. He does admit, however, that “practical execution [of the simpler constructions] is not ruled out, where these actually are of use” (“Construction” 222–223).Google Scholar
- 76.Diocles,
*On Burning Mirrors*props. 4–5, 10 and 12 present pointwise constructions of the parabolic and “cissoid” curves (see citation in note 20 above); Diocles’ pointwise construction of the “cissoid” is also paraphrased by Eutocius (*Archimedis Opera*, ed. Heiberg, III 66–68). On Anthemius, see G. Huxley,*Anthemius of Tralles*(Cambridge, Mass.: Harvard University Press, 1959) for the texts, translation and commentary on fragments dealing with the pointwise construction of elliptical and parabolic burning-mirrors.Google Scholar - 77.
*Collection*III 7 (ed. Hultsch, I 54). An interpolation in Eutocius’ commentary on Archimedes mentions a mechanical device invented by Isidore of Miletus for drawing parabolas (*Archimedis Opera*, ed. Heiberg, III 84), and in his commentary on Apollonius, Eutocius indicates the practical utility of pointwise constructions for the conic curves (*Apollonii Opera*, J.L Heiberg [Leipzig: Teubner, 1893] II 230–234 ).Google Scholar - 78.Cf. in particular the methods of Hero, Nicomedes, Pappus and Eratosthenes reported by Eutocius (
*Archimedis Opera*III 58–96) and Pappus (*Collection*III 7–10 [ed. Hultsch, I 56–68]). Contexts of their practical application are evident from the treatments by Hero in the Mechanics (I, 9; Heronis Opera, L. Nix, ed. [Leipzig: Teubner, 1900] II 23) and the Belopoeica (Greek and Roman Military Treatises: Texts, E. W. Marsden, ed. [Oxford: Oxford University Press, 1971] 40–41), and are stated in the account Eutocius draws from Eratosthenes (*Archimedis Opera*III 90). These include any situations where weights or volumes are to be scaled up in proportion, as in the design of military engines or the building of ships.Google Scholar - 79.Preface to Book V; for translations and discussion of the prefaces, see T.L. Heath,
*Apollonius of Perga*(Cambridge: Cambridge University Press, 1896) lxix-lxxiv.Google Scholar - 80.
*Collection*VII, preface (ed. Hultsch, II 634); emphasis mine.Google Scholar - 81.Many of the theorems in Book III, for instance, deal with relations of the products of segments of chords and tangents to the conic sections. Such relations turn out to be critical for the solution of certain conic problems, such as the drawing of conics which are to pass through given points or have given lines as tangents.Google Scholar
- 82.Construction” p. 222: We have inherited geometric construction from the Greeks. In particular, such constructions as can be effected via compass and straight edge serve as excellent exercises in our schools and are indispensable for practitioners.Google Scholar
- 83.On the conception of mathematical entities as “abstracted” from substances, see the passages cited by Heath, in
*Mathematics in Aristotle*64–67.Google Scholar - 84.Cited by Pappus in
*Collection*IV 31 (ed. Hultsch, I 254).Google Scholar - 85.
*In Analytica Posteriora*, ed. Wallies, 105; reproduced by Heiberg in*Apollonii Opera*II 106.Google Scholar - 86.Cf. the passage on the circle-quadrature cited from Philoponus in note 41 above.Google Scholar
- 87.Cf. the studies of Mueller, cited in note 61 above.Google Scholar
- 88.The Neoplatonist element is evident throughout Proclus’ work, not only in his commentary on Euclid, but also in his commentaries on Plato. Pappus’ philosophical inclinations are most clearly seen in his commentary on Euclid’s Book X (edited from the Arabic manuscript by G. Junge and W. Thomson [Cambridge, Mass.: Harvard University Press, 1930]); cf. also my
*Ancient Tradition*ch. 8.Google Scholar - 89.In the preface to the
*Method*, for instance, Archimedes insists on the distinction between heuristic treatments, like that in accordance with his “mechanical method,” and formally acceptable geometric proofs. His technique in this work makes prominent use of manipulations of infinitesimal magnitudes (e.g., the line segments which comprise a plane figure). Criticizing difficulties in the concept of indivisible magnitudes was a major issue for Aristotle (as in Physics Book VI), who already had a wealth of material to draw from in the debates surrounding the doctrines of the pre-Socratic atomists, and left a legacy for future contention among the Epicureans and Stoics. But in the Method, Archimedes has not a single word to contribute to these debates. Apparently, the philosophical coherence of his method was irrelevant for its heuristic efficacy. In this, I take him to be representative of the division between technical and philosophical interests in the ancient mathematical field.Google Scholar - 90.In Proclus’ view the omission follows from the need to apply higher curves, like the conchoids, quadratrices, spirals, or other such “mixed lines,” which lie outside the domain of the Elements (
*In Euclidem*272). Pappus ascribes the belated efforts on these problems to the ancients’ early unfamiliarity with the theory of the conics and the difficulty of drawing these curves (*Collection*, ed. Hultsch, I 54, 272). In neither case is there any question of the existence of the solving entities.Google Scholar - 91.See the passages from Pappus and Apollonius cited in notes 79 and 80.Google Scholar