- Archimedes and the measurement of the circle: a new interpretation.
Arch. History Exact Sci. 15 (20 (1975/76), 115-140.
In the third proposition of the 'Dimensio Circuli', Archimedes established 3 1/7 as an upper bound, 3 10/71 as a lower bound for the ratio of the perimeter to the diameter of the circle. Eutocius remarks that Apollonius derived bounds even more accurate. I shall argue in the present study that Archimedes himself introduced such refinements, yielding bounds at least twenty times more accurate than those in the 'Dimensio Circuli'. A close examination of these computations will affirm that the classical Greek mathematics, dominated by an interest in theoretical geometry, yet included expertise in practical arithmetic.
- Archimedes' neusis-constructions in spiral lines.
Centaurus 22 (2) (1978/79), 77-98.
In his theorems on the properties of the tangents drawn to the spirals (Spiral Lines, prop. 18-20), Archimedes depends on the determination of certain points by means of neuses (i.e., "vergings", or "inclinations"), as specified in earlier theorems, prop. 5-9. This dependence has earned him the disapproval of critics, both ancient and modem. Pappus, for instance, objects to his use of a "solid neusis," when plane methods ought to suffice, and proceeds to effect the construction by means of intersecting conic sections instead (Collection IV, 36, 52-54). In modem times, P Tannery and T L Heath each proposed alternative constructions in which only plane methods are used. The present study will examine Archimedes' theorems on the tangents and the nature of the neusis-constructions on which they rely. We will then consider the alternative proofs presented by Tannery and Heath to determine whether these succeed in avoiding the objections posed by Pappus. We will next show how to construct by plane methods alone a neusis satisfying conditions slightly weaker than those imposed by Archimedes, but still adequate for the purposes of SL, 18. The availability of such an alternative construction will assist in considering these questions: why did Archimedes choose to employ the stronger version of the neusis constructions, when in the context of the problem of the spiral these were formally inappropriate? what was the ancient view of the appeal to "non-constructive" assumptions in geometric proofs ? how was Archimedes' work related to the investigations of neuses by his younger contemporaries, Apollonius and Nicomedes? Finally, we will seek to explain a gap in Pappus' discussion: given that his construction of the solution via intersecting conic sections is itself classified as a "solid" technique, how did he propose to avoid the objection he raises against Archimedes?
- Archimedes' lost treatise on the centers of gravity of solids.
Math. Intelligencer 1 (2) (1978/79), 102-109.
The present paper is one of a series developing out of my study of the chronological ordering of the Archimedean corpus ("Archimedes and the 'Elements'," to appear late this year). This issue had been obscured by several implausible ideas introduced by Heiberg, reproduced by Heath (10, p. xxxii; 9, II, p. 22) and perpetuated uncritically in the secondary literature ever since. For instance, viewing the 'Dimension of the Circle' as late and the 'Method' as early, as Heath did, how could one avoid confusion on the development of Archimedes' work? In discovering Heath's error and reversing his judgment on these two works, I hit upon a criterion by which to divide the writings in the corpus into an "early group" and a "mature group." The former, including 'Dimension of the Circle' and parts of 'Quadrature of the Parabola', apply techniques characteristic of Euclid's 'Elements': most notably, convergence is one-sided, a magnitude being approximated by a sequence of inscribed rectilinear figures. By contrast, the mature treatises invariably apply two-sided convergence, in which the magnitude is compressed between sequences of circumscribed and inscribed figures whose difference can be made arbitrarily small. Further, these works frequently employ the "Archimedean axiom" on convergence, instead of the Euclidean bisection principle (Euclid's 'Elements' X, I). Through this criterion the placement of the mechanical writings becomes clearer. For instance, by virtue both of their content and of their use of the Euclidean convergence technique, the two books 'On Plane Equilibria' (establishing the principle of equilibrium and the centres of gravity of the parallelogram, the triangle, the trapezium and the parabolic segment) can be associated with the "early group." As said above, the 'Method' ought to be set near the end of the corpus, so to be viewed as a retrospective on the "mechanical method" by which Archimedes had initially studied the problems presented in the mature treatises. Indeed, the theorems in the mature writings frequently betray signs of their preliminary investigation through the mechanical manipulation of indivisibles. The two books 'On Floating Bodies' also must be viewed as late; for the second book requires the volume of segments of paraboloids, as given in 'Conoids and Spheroids', a work known to be the latest of the extant geometric treatises. This revised chronology can form the basis for addressing questions on the development of Archimedes' techniques such as these: what were the methods and motives by which Archimedes advanced beyond Euclid's 'Elements'? what did the mechanical procedures contribute, not only to the heuristic study of the problems, but also to the elaboration of formal techniques? how did the scholastic formalism of the geometers at Alexandria influence Archimedes' style of proof? Moreover, I have been led to the view that Archimedes depended on certain pre-Euclidean works, not the Euclidean edition of the 'Elements', as his source for proportion theory and other materials. I argue further that Pappus' alternative treatments of the sphere and the spirals (Collection IV and V), cast according to the techniques of the "early group," ought to be accepted as close paraphrases of early Archimedean versions, no longer extant. I propose now to examine Archimedes' study of the centres of gravity of solids. His treatise on this subject, now lost, can be placed within the "mature group," following 'Conoids and Spheroids', but preceding 'Floating Bodies' and the 'Method'. From clues preserved in these three extant works I shall seek to identify the theorems it contained and the manner of their proof, and then to provide in outline two of them, on the centres of gravity of the cone and the paraboloid.
- Archimedes and the spirals: the heuristic background.
Historia Math. 5 (1) (1978), 43-75.
In his work, 'The Method', Archimedes displays the heuristic technique by which he discovered many of his geometric theorems, but he offers there no examples of results from Spiral Lines. The present study argues that a number of theorems on spirals in Pappus' Collectio are based on early Archimedean treatments. It thus emerges that Archimedes' discoveries on the areas bound by spirals and on the properties of the tangents drawn to the spirals were based on ingenious constructions involving solid figures and curves. A comparison of Pappus' treatments with the Archimedean proofs reveals how a formal stricture against the use of solids in problems relating exclusively to plane figures induced radical modifications in the character of the early treatments. ... We have argued that Pappus' discussion of spirals can be taken as a guide to understanding the development of Archimedes' study from its heuristic stage to its formal stage. The book Spiral lines contains internal indications of stages in its composition. While great emphasis was indeed set on formal rigor, geometers like Archimedes made free use of heuristic devices, such as the introduction of mechanical notions and the manipulation of indivisibles. A review of Pappus' theorems on the spiral, accepted as an extract from early Archimedean studies, shows how a different set of heuristic devices - a modification of the technique of indivisibles and the introduction of auxiliary solids - formed the basis of Archimedes' discoveries on the spiral.
- Archimedes and the 'Elements': proposal for a revised chronological ordering of the Archimedean corpus.
Arch. Hist. Exact Sci. 19 (3) (1978/79), 211-290.
With the rediscovery by J L Heiberg of the manuscript of Archimedes' Method and its publication by him in 1907, a suspicion held for centuries by mathematicians and mathematical historians was at last corroborated by direct evidence: Archimedes had indeed applied in his discovery of geometric theorems a heuristic technique of analysis, although the highly formal synthetic proofs given in their demonstration concealed this fact. Disappointingly little effort has since been made to search for other indications of growth in the composition of Archimedes' treatises. In particular, no one has attempted to exploit the relative chronology determined by J Torelli in 1792 for a major part of the corpus. Despite the potential of even this partial ordering for a deeper understanding of Archimedes' work, no subsequent edition, translation or commentary has retained it as the organizing principle. In the present reexamination of this question of Archimedean chronology, I will (1) show how tenuous are the foundations of the orderings now in standard usage; (2) present a chronological order which differs from the standard accepted sequence in certain important respects, notably, in the position of the Dimension of the Circle and the Method; and (3) distinguish an "early group" of studies from the "mature group", employing as my criterion the predominantly Euclidean character of the techniques used in the former. I believe that the results of my inquiry have broader implications, as in reinterpreting as Archimedean certain contributions preserved by such later authors as Hero and Pappus.
- The hyperbola-construction in the Conics, Book II: ancient variations on a theorem of Apollonius.
Centaurus 25 (4) (1981/82), 253-291.
In the fourth proposition of Book II of Apollonius' Conies one finds a method of constructing the hyperbola of which a point and both asymptotes are given. But a close examination of this proposition in comparison with three other versions of the construction extant from ancient authors reveals that the version in the Conies is incorrectly established, that it was not included within the early editions of that work, but entered its tradition via late interpolation. Its composer and the agent of its insertion into the Conies proves to be Eutocius of Askalon, the early sixth-century commentator on the works of Apollonius and Archimedes. Eutocius appears to have framed his solution with the assistance of a prior text closely resembling one preserved in the Collection of Pappus of Alexandria (early fourth century). Nevertheless, one extant form of the construction may be viewed as a fragment of the ancient geometrical analysis which flourished under Apollonius and his colleagues toward the close of the third century B.C. The variant forms of this construction thus provide an unusual case history of the working methods of the editors and commentators of late antiquity. One is surprised by the low level of expertise demonstrated by Pappus, as well as the high degree of responsibility Eutocius had for the production of the extant manuscript tradition of the Conies. Further, one perceives however dimly something of the nature and objectives of the older tradition and the form of the fourth-and early third-century studies from which it developed.
- Techniques of fractions in ancient Egypt and Greece.
Historia Math. 9 (2) (1982), 133-171.
In the ancient Egyptian and Greek arithmetical traditions, computations with fractions typically resorted to the mode of unit-fractions. The present study examines the principal documents extant from these traditions to determine the computing techniques there used. Recent writers have debated the alleged aesthetic criteria underlying the Egyptian computations; by contrast, B L van der Waerden has stressed the purely technical character of these computations. Although the Greek evidence is rarely brought to bear on this question, the present study seeks to show that this evidence strongly upholds van der Waerden's position. In itself the Greek tradition is notably ambivalent, on the one hand revealing continuity with the more ancient Egyptian technique of unit-fractions, but on the other hand making frequent use of the general manner of fractions, equivalent to that familiar in modern school arithmetic. Through comparison of the various stages of the ancient computational tradition, one can discern different strata corresponding to differences in the time and manner of contact between the Greek and the Egyptian traditions.
- Observations on the early history of the conics.
Centaurus 26 (1) (1982/83), 1-24.
In the preceding paper on the variant texts of the hyperbola-construction certain questions were raised concerning the early history of the study of the conic sections. In particular, the traditional views of the contributions of Menaechmus in the mid-fourth century and of Aristaeus at the beginning of the third were perceived to present difficulties. I now propose to take up these questions and to indicate some new ideas, unavoidably speculative and novel, for resolving these difficulties. To Menaechmus is due a solution of the problem of cube-duplication, in one form utilizing the intersection of a hyperbola with a parabola, in another the intersection of two parabolas. On the basis of the report of Eutocius to this effect, Menaechmus has thus been credited with a form of the theory of the conic sections. This appears affirmed by a line in Eratosthenes? epigram on the cube-duplication in which the "Menaechmean triads" are cited; for these would naturally refer to the three conic sections allegedly studied by him, the hyperbola, the parabola and the ellipse. Moreover, we know that about the same period or earlier the geometers Archytas and Eudoxus had initiated studies in solid geometry to which such a study by Menaechmus could easily be related: Indeed, the natural philosopher Democritus had posed a problem in connection with the parallel sectioning of the cone. Surely, then, the mid-fourth century was an environment in which the geometric study of such sections had its origins.
- On the transmission of geometry from Greek into Arabic.
Historia Math. 10 (1) (1983), 71-78.
In the context of a relatively unfamiliar tradition, one cannot with confidence whether a given document is representative or singular. This is what makes especially interesting the Arabic geometric fragment edited by J P Hogendijk . Indeed, one might say, if with a bit of hyperbole, that documents of this kind have revolutionary implications for the study of the transmission of Greek geometry into Arabic. The text in question is an angle-trisection preserved under the name of Ahmad (ibn Musa) ibn Shakir, a ninth-century patron of mathematical science active in the flourishing circle then at Baghdad. His method and others like it (such as the one received under the name of his colleague, Thabit ibn Qurra) were familiar to geometers in the 10th century, notably al-Sijzi. As Hogendijk notes, however, the latter was of the opinion that the Arabic scholars originated their methods of angle-trisection , save perhaps for the "ancient method" which is implied in the Archimedean 'Lemmas'. But as comparison reveals at once, the methods of Ahmad and Thabit are in fact identical with that preserved by Pappus in 'Collection', Book IV. We thus have a clear instance where an Arabic document produced as a translation from the Greek was subsequently perceived to be an original contribution due to an Arabic geometer. Is this a bizarre exception or, to the contrary, an example of a type of misunderstanding which might have occurred with some frequency? And if the latter, then with what degree of frequency? Of course, we cannot address this issue until we gain a better grasp of this class of geometric materials, and it may well be that too little now survives to actually permit a firm determination. Nevertheless, we have real grounds for caution. Whenever we come upon a technical treatment securely within the compass of Greek methods and format, we must seriously consider whether that treatment was modelled after a prototype in Greek, even when no such Greek document is now known.
- "La croix des mathématiciens": the Euclidean theory of irrational lines.
Bull. Amer. Math. Soc. (N.S.) 9 (1) (1983), 41-69.
For the modern reader Euclid's Tenth Book is by far the most intimidating portion of the 'Elements', by virtue of its enormous length and the obscurity of its techniques and motives. To approach this material, one requires a key, of the sort the Flemish mathematician-engineer Simon Stevin boasted to possess almost four centuries ago:
After we had viewed and reviewed the Tenth Book of Euclid treating of incommensurable magnitudes, and also had read and reread several commentators on the same, of whom some judged it for the most profound and incomprehensible matter of mathematics, others that these are most obscure propositions and the cross of mathematicians, and beyond this I persuaded myself (what folly doesn't opinion cause men to commit?) to understand this matter through its causes, and that there are in it none of the difficulties such as one commonly supposes, I have taken it upon myself to describe this treatise.Stevin's ploy, by which "this whole affair is easy and without difficulty", involved the expression of Euclid's propositions via a calculus of surd quantities, and more recent commentaries, such as those by Heath and by Junge, follow suit in the application of algebraic modes for explaining this material. But for the historically minded reader the issue of interpretation has been complicated by this, for the originators of this theory cannot have had such algebraic modes at hand in their formulation. The project of elucidating the motives underlying Euclid's geometric form of the theory has largely eluded even the best of the modern accounts. I here propose to offer a view of the geometric problems on which the structure of Euclid's theory is built. This view fills out the details of a sketch I presented in my study of the pre-Euclidean geometry a few years ago and this treatise. I will show how the essential idea of the theory emerges through consideration of the Euclidean constructions of regular plane and solid figures, next trace out the development of this idea in the theory of Book X, and close with some thoughts on the nature of the formal project embraced in this book. It will first be necessary to introduce a few basic notions drawn from the early phases of the study of incommensurable magnitudes.
- Archimedes' Dimension of the circle: a view of the genesis of the extant text.
Arch. Hist. Exact Sci. 35 (4) (1986), 281-324.
Of all the works in the Archimedean corpus, none has been more widely studied from ancient and medieval times to the present day than the short tract on the measurement of the circle. It was cited frequently by the ancient mathematical commentators, was known in the Latin Middle Ages in translations both from the Arabic and directly from the Greek, and remains prominent in all general discussions of Archimedes' geometry to this day. The popularity of this work is easily understood. Its technical demands are slight, requiring little beyond the theory in Book XII of Euclid's 'Elements' and the resolve to wade through the details of an extended computation. Yet a close reading reveals unexpected subtleties, and for all its brevity it represents well the objectives and methods evident throughout the corpus. Scholars have long noted, however, that the extant version of the 'Dimension of the Circle' is at best an edited extract from the original composition. Its second proposition, for instance, employs a result proved in the third, in violation of the formal deductive ordering of demonstrations, and fails to distinguish between exact and approximate values for the constants of measurement. It lacks the theorem on circular sectors which ancient commentators cite from their version of this Archimedean work. Its first proposition, moreover, includes a terminological error (reading tomeus, 'sector', for tmêma, 'segment'), while the modern Archimedes editor, J L Heiberg, has described its brevity of exposition as "so negligent as to betray the hand of an excerptor, rather than that of Archimedes." Virtually unnoticed in the interpretation of this work is the potential insight to be gained through its comparison with two alternative versions preserved by the commentators Pappus and Theon. I propose now to make these comparisons and explore their implications, in the effort to determine the provenance of the version extant within the Archimedean corpus.
- The practical element in ancient exact sciences.
Synthese 81 (3) (1989), 313-328.
When ancient mathematical treatises lack expositions of numerical techniques, what purposes could ancient mathematical theories be expected to serve? Ancient writers only rarely address questions of this sort directly. Possible answers are suggested by surveying geometry, mechanics, optics, and spherics to discover how the mathematical treatments imply positions on this issue. This survey shows the ways in which these ancient theoretical inquiries reflect practical activity in their fields. This account, in turn, suggests that the authors may have intended their theorems not to predict, but to explain phenomena. We may then consider what kind of explanations they were seeking.
- On Archimedes' construction of the regular heptagon.
Centaurus 32 (4) (1989), 257-271.
For admirers of the Archimedean construction of the regular heptagon, the appearance of Jan Hogendijk 's recent edition  of the unique extant text in Arabic has been highly welcome. Since the Greek prototype is lost, the only surviving witness of the Archimedean work is based on the Arabic translation made by Thabit ibn Qurra in the 9th century, as reedited in the 18th century under the title 'Book of the construction of the circle divided into seven equal parts, by Archimedes'. The text, which has suffered obvious corruption (for instance, only the last two of its eighteen propositions bear at all on the heptagon construction), is transcribed by Hogendijk, with translation, commentary, and ample documentation of later Arabic efforts on the problem. With access now to a correct text, one learns of certain inaccuracies that had been earlier introduced in C Schoy's translation (1926), and thence propagated by the principal commentators on it, J Tropfke (1936), E J Dijksterhuis (1956), and M Clagett (1970). One is thus inclined to receive with full seriousness the provocative claim made by Hogendijk, that the attribution of the construction to Archimedes is mistaken, or at least, that "the question of Archimedes' authorship of the construction [is] unsettled" [1984, 213]. I wish here to review the objections that Hogendijk raises against the construction, and to make the effort in defence of its authenticity.
- John of Tynemouth alias John of London: emerging portrait of a singular medieval mathematician.
British J. Hist. Sci. 23 (3)(78) (1990), 293-330.
In 1953 Marshall Clagett presented a preliminary scheme of the medieval Latin versions of Euclid's 'Elements'. Since then a considerable body of these texts has become available in critical editions, thanks to Clagett's labours on the Archimedean tradition and H L L Busard's work on the Euclidean versions. Further, Busard, M Folkerts, R Lorch and C Burnett have scrutinized the pivotal 'second' version of Adelard of Bath, and have thereby exposed a diversity of text forms that spells real complications for the effort to establish its provenance and use. In his recent overview of the medieval Euclidean tradition, Folkerts displays not only how these studies have filled out and expanded upon Clagett's initial framework, but also how they have compelled rethinking of some basic issues, such as on the source relations and authorship of the various versions.
- On a medieval circle quadrature: De circulo quadrando.
Historia Math. 18 (2) (1991), 107-128.
Among several treatments of circle quadrature known from the Middle Ages is one with the title 'De circulo quadrando'. I discuss this text and an alternative version (incipit: Quelibet media proportionalia), their relation to each other, and the question of their authorship. The pattern of their circulation suggests connections between the masters of geometry at Oxford and Paris in the 13th century.
- On the principle of linear perspective in Euclid's Optics.
Centaurus 34 (3) (1991), 193-210.
In the classical optics of Euclid and his ancient commentators, the apparent magnitude and position of an object are described only in terms of the angle formed by its boundary with the eye. The propositions of the Optics thus embrace the project of describing and explaining geometrically the appearances of objects in specified relations to the observer. This method of "angular" (or "natural") perspective differs from the linear perspective introduced in the Renaissance as a technique for reconstructing the visual space presented to the observer. There has thus arisen a debate as to the implications of the difference between these two perspective systems.
- When circles don't look like circles: an optical theorem in Euclid and Pappus.
Arch. Hist. Exact Sci. 44 (4) (1992), 287-329.
In the 'Optics' Euclid proves that "the wheels of chariots sometimes appear circular, sometimes oval. His claim falls within a much older tradition of painting that recognised how circular objects like wheels, shields, and the like, when viewed obliquely, take on an elongated appearance. A quite remarkable example has recently been unearthed in the excavations of the royal tombs at Vergina in Macedon, which, if really housing the remains of Philip II, father of Alexander the Great, would date from 336 B.C., or about a generation or so before Euclid. In a mural from one of the tombs, where Pluto is shown abducting Persephone as terrified attendants draw back helpless, the illusion of our almost being run down by the god's chariot is fostered by the elongated appearance of the wheels, the one on the right being enlarged to enhance the angular effect. Although carefully drawn, the wheel appears somewhat asymmetrical, being narrower at the bottom that at the top. Presumably, the technique of foreshortening is pragmatic, in imitation of the appearance of objects actually inspected, rather than being based on an established technique for depicting such appearances, even less on a geometric awareness of the theoretical figure.
- The wrong text of Euclid: on Heiberg's text and its alternatives.
Centaurus 38 (2-3) (1996), 208-276.
Over a century ago two young academics engaged in a brief debate. One, a Gymnasiallehrer at Altona in Germany, M Klamroth, was a specialist in medieval Arabic mathematics. The other, J L Heiberg, only five years beyond his doctoral dissertation at Copenhagen, had already completed a new critical edition of the Greek text of Archimedes and was in the process of issuing a new critical edition of the Greek text of Euclid. The debate between them was short, only one paper each, but the outcome was significant. For it settled, once and for all, as it appears, the issue of what one should take as the definitive text of Euclid's 'Elements'. No one since Heiberg determined the matter in 1884 has seriously scrutinized the arguments he deployed against Klamroth's alternative view in 1881. My reason for reviving the issue is that I believe Klamroth was right and Heiberg was wrong. If that is the case, we have been consulting, and continue to consult, the wrong text in our efforts to interpret the Euclidean tradition.
- On Heiberg's Euclid.
Intercultural transmission of scientific knowledge in the middle ages: Graeco-Arabic-Latin, Berlin, 1996, Sci. Context 14 (1-2) (2001), 133-143.
In two articles published in 1881 and 1884, the young academics, Martin Klamroth and Johan L Heiberg, engaged in a brief debate on the textual choices that should govern the publication of a new critical edition of Euclid's 'Elements'. This short debate seemed to settle the problem in Heiberg's favor as to what should be taken as the definitive text of Euclid's 'Elements'. But the issue ought to be considered once again for there are good reasons for the claim that Klamroth was right, and that Heiberg was wrong. If so, we have been consulting, and continue to consult, the wrong text in our efforts to interpret the Euclidean tradition. In order to substantiate this claim, the textual issue debated by Klamroth and Heiberg is rehearsed again, and the principal reasons brought forth by Heiberg against Klamroth's position are reconstructed. Specimens from three broad areas of evidence - structural, linguistic, and technical - will be considered. They reveal how the medieval tradition of the text advocated by Klamroth displays textual superiority to the Greek tradition promoted by Heiberg. Such a reorientation of the texts has the potential to change significantly our understanding of ancient mathematics.