Classics in the History of Greek Mathematics (Boston Studies in the Philosophy of Science, Volume 240)

Classics in the History of Greek Mathematics (Boston Studies in the Philosophy of Science, Volume 240)

Language: English

Pages: 463

ISBN: 2:00281596

Format: PDF / Kindle (mobi) / ePub


The twentieth century is the period during which the history of Greek mathematics reached its greatest acme. Indeed, it is by no means exaggerated to say that Greek mathematics represents the unique field from the wider domain of the general history of science which was included in the research agenda of so many and so distinguished scholars, from so varied scientific communities (historians of science, historians of philosophy, mathematicians, philologists, philosophers of science, archeologists etc. ), while new scholarship of the highest quality continues to be produced. This volume includes 19 classic papers on the history of Greek mathematics that were published during the entire 20th century and affected significantly the state of the art of this field. It is divided into six self-contained sections, each one with its own editor, who had the responsibility for the selection of the papers that are republished in the section, and who wrote the introduction of the section. It constitutes a kind of a Reader book which is today, one century after the first publications of Tannery, Zeuthen, Heath and the other outstanding figures of the end of the 19th and the beg- ning of 20th century, rather timely in many respects.

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rectangle O(AQ, QB) “visible”. From this point of view, the significance of the two main auxiliary methods mentioned by Zeuthen is completely different. They are not methods of treating the lines and areas as general quantities in a way similar to modern algebra, but they are the means for transformation between “visible” and “invisible” forms of areas. The former is indispensable because it makes geometric intuition available, while the latter is adapted to the formal statement of results as

encountered disappears. 29 30 31 32 33 34 Mueller, op. cit., p. 50. Note that this claim has already been made by Arpád Szabó, in his Anfänge der griechischen Mathematik (1969). Mueller, op. cit., pp. 161–162. Ibid., pp. 168–170, 192–194. Tannery, op. cit., p. 274. Mueller, op. cit., pp. 300–302. 157 BOOK II OF EUCLID’S ELEMENTS 2. The double form in II 5, 6, II 9, 10 and II 4, 7 can be explained by the geometric interpretation. Their role is that of mutual complement in the geometric

points there will be two solutions.13 G F Z D A H B W C J BC = x AB = y GH = FB = ZW = b HB = BW = b xy = b2 = starting area Diagram 4 (based on Knorr, p. 72) In diagram 4, the square GFBH (b2) is taken as equal to the starting area. Points Z and D are on the rectangular hyperbola that has as its asymptotes the diameter of the circle BCJ and the tangent drawn at its end point, BAF, where the product of the horizontal coordinate and the vertical one (i.e. in the case of D, x and y) is

problems. Zeuthen, however, intends it as a historical account of the geometers’ view of their own technical efforts. Doubtless, the ancients recognized the logical connection between possibility and existence. But this of itself does not entail an existential intent behind their constructions. Indeed, this logical connection complicates the project of testing Zeuthen’s view, for in reading the ancient evidence, one might easily be led to impute an existential intent where it need not have been.

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 142 132 6 7 8 9 143 10 11 12 13 14 15 16 17 18 19 WILBUR R. KNORR I 1 also depends on implicit postulates, specifically the

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