Abstract
It is well known fact that there are two definitions of proportion in Euclid's Elements: Book V, def. 5 and Book VII, def. 20. In the present paper we show that three different interpretations of definition V.5 can be given as modern notation is used: two of them arise from different readings of the definition itself, the third is a negation of disproportion (V, def. 7).References
Aaboe A. (1968), Matematyka w starożytności, tł. R. Ramer, PWN, Warszawa.
Acerbi F. (2003), Drowning by Multiples. Remarks on the pifth Book of Euclid's Elements, with Special Emphasis on Prop. 8, Archive for History of Exact Sciences 57, ss. 175-242.
Archimedes, On the Sphere and Cylinder, [w:] [Heath 1912].
Artmann B. (1991), Euclid's Elements andits Prehistory, Apeiron, XXIV (4), ss. 1-47.
Arystoteles, Kategorie, [w:] Arystoteles, Dzieła wszystkie, 1.1, tł. K. Leśniak, PWN, Warszawa 1990.
Baron M. (1969), The Origins ofthe Infinitesimal Calculus, Pergamon Press, Oxford.
Baszmakowa I.G. (1975), Grecja starożytna, [w:] Historia matematyki, t. 1, A.P. Juszkiewicz (red.), tł. St. Dobrzycki, PWN, Warszawa, ss. 64-115.
Baszmakowa I.G. (1975), Kraje hellenistyczne i imperium rzymskie, [w:] Historia matematyki, t. 1, A.P. Juszkiewicz (red.), PWN, tł. St. Dobrzycki, Warszawa, ss. 116-167.
Batóg T. (2000), Dwa paradygmaty matematyki, Wydawnictwo Naukowe UAM, Poznan
Berggren J.L. (1984), History of Greek Mathematics: A Survey of Recent Research, Historia Mathematica 11, ss. 394-410.
Błaszczyk P. (2007), Eudoxos versus Dedekind, Filozofia Nauki.
Borsuk K., Szmielew W. (1972), Podstawy geometrii, PWN, Warszawa.
Boyer C.B. (1964), Historia rachunku różniczkowego i całkowego i rozwój jego pojęś, tł. S. Dobrzycki, PWN, Warszawa.
Bourbaki N., (1966), Historical Note, [w:] [Bourbaki 1966(a)], ss. 406-416; cytowane za: Liczby rzeczywiste, [w:] N.
Bourbaki, Elementy historii matematyki, tł. S. Dobrzycki, PWN, Warszawa 1980, ss. 186-197.
Bourbaki N. (1966(a)), General Topology Part 1, Addison-Weseley, Reading, Massachussetts.
Capmski M., Cutland N. J. (1995), Nonstandard Methods for Stochastic Fluid Mechanics, World Scientific, Singapore 1995.
Claphan Ch., Nicholson J. (2005), The Concise Dictionary of Mathematics, Oxford University Press, Oxford.
Cohen, L.W., Ehrlich G. (1963), The Structure ofthe Real Number System, Van Nonstrand, Princeton, New Jersey.
Conway J. (1994), The Surreals and the Reals, [w:] [Ehrlich 1994], ss. 93-103.
Coxeter H.S.M. (1967), Wstęp do geometrii dawnej i nowej, tł. R. Kransodębski, PWN, Warszawa.
Dedekind Richard (1872), Stetigkeit und irrationale Zahlen, Friedrich Viewing und Son, Braunschweig 1960; wydanie pierwsze: Braunschweig 1872.
Dedekind R. (1876), Gessamelte Werke, t. III, Friedrich Vieweg und Son, Braunschweig 1932, cytowane za: Z korespondencji z Rudolfem Lipschitzem, tł. R. Murawski, [w:] Filozofia matematyki, opr. R. Murawski, Wydawnictwo Naukowe UAM, Pozna! 1994, ss. 149-154.
Edwards C.H. (1979), The Historical Development ofthe Calculus, Springer, New York.
Ehrlich P. (1994), Real Numbers, Generalizations ofthe Reals, and Theories of Continua, Kluwer, Dordrecht 1994.
Euklides, Elementy, [w:] [Heath 1956].
Fowler D.H. (1999), The Mathematics of Plato's Academy A New Reconstruction, Clarendon Press, Oxford.
Fowler D.H. (1992), An Invitation to ReadBook X o f Euclid's Elements, Historia Mathematica 19, ss. 233-264.
Goldblatt R. (1998), Lectures on the Hyperreals, Springer, New York 1998.
Goldstein J.A. (2000), A Matter of Great Magnitude: The Conflict over Arithmetization in 16th-, 17th- and 18thCentury English Editions of Euclid's Elements Books I Through VI (1561-1795), Historia Mathematica 27, ss. 36-53.
Grattan-Guiness I. (1996), Numbers, Magnitudes, Ratios, and Proportions in Euclid s Elements: How Did He Handle Them, Historia Mathematica 23, ss. 355-375.
Hale B. (2000), Reals by Abstraction, Philosophia Mathematica 3, vol. 8, ss. 100-123.
Hale B. (2003), Real Numbers, Quantities, and Measurement, Philosophia Mathematica 3, vol. 10, ss. 304-323.
Heath T.L. (1912), The Works of Archimedes. Edited in Modern Notation with Introductory Chapters by T.L. Heath with a Supplement The Method ofArchimedes. Recently Discovered by Heiberg, Dover (reprint), New York 1953.
Heath T.L. (1956), Euclid. The Thirteen Books of the Elements, vol. I-III, Cambridge Umversity Press, Cambridge.
Heath T.L. (1981), A History of Greek Mathematics, 1.1, prom Thales to Euclid, Dover, New York.
Hilbert D. (1930), Grundlagen der Geometrie, Teubner, Leipzig 1930; wydanie pierwsze: Leipzig 1899.
Hilbert D. (1980), poundations of Geometry, translated by L. Unger from the tenth German Edition, Open Court, La Salle, Illinois.
Joyce D.E. (1997), Euclid s Elements, http://aleph0.clarku.edu.
Kline M. (1972), Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York.
Knorr W.R. (1996), The Wrong Text of Euclid: On Heiberg s Text and its Alternative, Centaurus 38, ss. 208-276.
Kordos M. (1994), Wykłady z historii matematyki, WSiP, Warszawa.
Kostin W. (1952), Podstawy geometrii, tł. J. Turkowska, PZWS, Warszawa.
Kulczycki S. (1973), Z dziejów matematyki greckiej, PWN, Warszawa.
Lindstram T. (1988), An Invitation to Nonstandard Analysis, [w:] N. Cutland, Nonstandard Analysis and its Applications. Cambridge University Press, Cambridge, ss. 1-105.
Mioduszwski J. (1996), Ciagłość. Szkice z historii matematyki, WSiP, Warszawa.
Murawski R. (2002), Współczesna filozofia matematyki, PWN, Warszawa.
Nikolic M. (1974), The relation between Eudoxos' theory ofproportions andDedekind's theory of cuts, [w:] R.S. Cohen, J.J. Stachel, M.W. Wartofsky (eds.) ForDirk Struik, D. Reidel Publishing Company, Dordrecht, ss. 225-243.
Rosenfeld B.A. (1988), A History of Non-Euclidean Geometry, Springer, New York.
Sieklucki K. (1978), Geometria i topologia. Część I. Geometria, PWN, Warszawa.
Stein H. (1990), Eudoxos and Dedekind: on the Ancient Greek Theory ofRatios and its Relation to Modern Mathematics, Synthese 84, ss. 163-211.
Struik D.J., Krótki zarys historii matematyki. Do kośca XIX wieku, tł. P. Szeptycki, PWN, Warszawa 1960.
Weyl H. (1949), Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton.
Widomski J. (1996), Ontologia liczby. Wybrane zagadnienie z ontologii liczby w starożytności i średniowieczu, Wydawnictwo Uniwersytetu Jagielloinskiego, Kraków 1996.
Więsław W. (1997), Matematyka ijej historia, Nowik, Opole.