Thabit ibn Qurra's Generalization of the Pythagorean Theorem

Thabit ibn Qurra's Generalization of the Pythagorean Theorem Aydin Sayili Tsis, Vol. 51, No. 1. (Mar., 1960), pp. 35-37. Stable URL: http:/Ainks.jstor.org/sici Isis is currently published by The University of Chicago Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and.Gonditions of Use, available at http://www. jstor.org/about/terms. html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www. jstor.org/journals/ucpress.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. \ A bes DAVILA, À: HS 1 Pak U\S\ The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take http://www.jstor.org Thu Jan 17 02:27:04 2008

The History of Science Society Thâbit ibn Qurra's Generalization of the Pythagorean Theorem Author(s): Aydin Sayili Source: Isis, Vol. 51, No. 1 (Mar., 1960), pp. 35-37 Published by: The University of Chicago Press on behalf of The History of Science Society Stable URL: http://www.jstor.org/stable/227603 Accessed: 08/10/2008 07:48 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ucpress. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. The University of Chicago Press and The History of Science Society are collaborating with JSTOR to digitize, preserve and extend access to Isis. http://www.jstor.org

Generalization Qurra's Th^abit of the Pythagorean Theorem By Aydin Saysli * INAN Istanbul manuscript collection of scientific and philosophical articles, there are several monographs by Thabit ibn Qurra, ninth-century mathematician and scientist. One of them is on the "Socratic proof" of the Pythagorean theorem.' This article is a letter written by the author to a friend who was somewhat disappointed with the "Socratic proof" of the Pythagorean theorem,as this proof deals with a special case only, namely, that of an isosceles right triangle, and he wished Thabit to communicate to him the general proof. Thabit's answer constitutes an elaborate treatment and an important contribution to the subject, although the request itself apparently did not envisage such a comprehensive answer. The so-called Socratic proof refers to the well-known passage in Plato's Menon, but this work is not specifically mentioned in Thabit's article. Thabit conceives the question as one requiring the reconstruction of the "Socratic general proof," or as necessitating the establishment of a new proof which is in the spirit of the "Socratic special proof." Thus the fact that Euclid's Elements contains a general proof of the theorem does not make the question a superfluousone. Thabit ibn Qurra calls his method, which is in accord with the Socratic one, the method of reduction and composition, or the method of reduction to triangles and rearrangementby juxtaposition. He gives two different proofs for the general case, i.e., for the relation between the squares constructed on the sides of any right triangle, and both these proofs constitute examples of the method of reduction and composition. The first proof is shown in Figure 1. The triangle is ABC; AA'BB' and DFB'D' are the squares of the two right sides, and ACDE is the square of the hypothenuse. The first two squares are obtained by adding triangles 1 and 2 to the shaded polygon, and the square of the hypothenuse is obtained by adding the same polygon to triangles 3 and 4. But, as all these triangles are equal by construction,the sum of the first two squares must be equal to the latter square. That Thabit had given such a proof was known to us from another source, i.e., from Al Nayrizi's (ca. 900 A.D.) commentary of Euclid's Elements,2 but apparently the rest of the information contained in the present article is new. * Ankara University, Turkey. This paper was read at the December 1956 meeting of the History of Science Society. 1 The volume is in the Library of Ayasofya Museum and is registered under the number 4832. Ritter gives a description of this manuscript collection which is apparentlyover nine hundred years old (H. Ritter, Schriften Ja' 35 qiTbib;n Ishaq al-Kindi's in Stambuler Bibliotheken, Archiv Orientalni, 1932, 4: 363); the article in question occupies pp. 39a-41a. Another copy of this article exists in Cairo (H. Suter, Die Mathematikerund Astronomen der Araber, 1900, p. 37). 2 Codex Leidensis, 399, 1: Euclid's Elementa ex interpretationeAl Hadschdschadsch

AYDIN SAYILI E D D~4 A h/ 3 F C B FIG. 1 B A t A B FIG. 2 B C B C FIG. 3 The second proof, shown in Figure 2, is based on the fact that when, from the whole figure given, the three shaded triangles are deducted, the squares on the right sides are obtained, while the square of the hypothenuse results when from the same total figure the three triangles on the cornerswith heavily drawn sides are subtracted. As, however, all these triangles are equal to each other by construction, the sum of the two former squares is thus seen to be equal to the latter square. Such proofs are familiar today and are found in certain text books. Their origin is sometimes traced to late ninth-century Indian mathematicians.3 Such proofs may therefore have been found independently by other mathematicians also. The more important contributions of Thabit to the subject concern the further generalization of the theorem to any triangle. After giving the above-mentionedtwo proofs, Thabit compliments his friend for seeking a comprehensiveknowledge of things and adds that the generalization achieved by the proofs given above may not be considered sufficient. One could wish, for example, he says, not to restrict oneself either to squares or to right triangles. In the first case, one would show, as Euclid has done, that the sum of any similar figures similarly located on the right sides of any right triangle is equal to a similar figure similarly placed upon the hypothenuse.4 In the latter case, on the other hand, one would wish to generalize the theorem to any triangle whatsoever. To achieve this latter generalization, Thabit draws from the vertex A of any triangle ABC two lines forming with the base the angles AB'B and AC'C, both equal to A. (See Fig. 3.) He then states without proof that the sum of the squares of the sides AB and AC is equal to the rectangle (BB' + CC') X BC; he adds that the proof can easily be obtained with the help of the Elements. This theorem seems to have been rediscovered only recently.5 The theorem is reducible to or derivable from the Euclidean propositions II, 12 and II, 13,6 cumncommentariisAl Narisil, ed. R. 0. Besthorn and J. L. Heiberg, part 1, 1893, pp. 184-188; Anaritii in decent libros priores Elementorem Euclidiis comrmentarii,ed. Maximilianus Curtze, Euclidis Opera Omnia, ed. I. L. Heiberg and H. Menge, Supplement, Leipzig 1899, pp. 84-86. See also J. Tropfke, Geschichte der Elementar Matheinatik, 1923, 4: 143, 149. The Arabic text and Turkish translation of this article of Thabit ibn Qurra, together with its explanation and analysis, have appeared in print in Belleten., quarterly journal of the Turkish Historical Society: see, Aydin Saylli, "Sabit ibn Kurra'nin Pitagor Teoremini Tamimi," Belleten, 1958, 22: 527-549. 3 W. Lietzmann, Der Pythagoreische Lehrsats, Stuttgart, 1953, p. 24. Earlier editions of this book appeared in 1930 and 1937. 4 Euclid, Book VI, proposition 31. 5 Leitzmann, op. cit., 1937, pp. 38-40, 1953, pp. 46-47. The earlier edition of 1930 of his book does not contain this theorem. 6 See, A. Sayill, Belleten, 1958,22: 528-530.

THABIT & THE PYTHAGOREANTHEOREM and it is probable therefore that it was on these that Thabit based his proof. Thabit's article contains references to various special cases, e.g., that in which A is a right angle (B' and C' coinciding); those in which B or C is a right angle; the case when A<60' (C' and/or B' being situated outside the segment BC); and the case in which A>90? and BB' + CC'<BC. Thabit goes on to say that, should one wish to be in possession of a still more general theorem, i.e., one in which neither the type of triangle nor the shape of the figures drawn on its sides is restricted, then one may say that the sum of any similar figures similarly drawn upon two sides of any triangle is equal to a figure whose ratio to the similar figure similarly drawn upon the third side is the same as the ratio of BB' + CC' to BC.7 In concluding, Thabit remarks that our knowledge is perfect when it combines the most general and comprehensivewith the special and the particular; for, he says, in our purely general knowledge the knowledge of the particular cases exists only potentially. He also states that in the course of instruction one has to follow a procedurein which there is a gradual increase in generalization and comprehensiveness,and adds that the reason why Socrates mentioned only the proof of a special case was that the person he was teaching was a beginner in the subject and not an advanced student. Proof of the Theorem AB2+AC2 = BC'2+2BA.AC.cosA =BC2+ BA.AC (cosC'+cosB') A =BC2+BA.AC. FC'FAB' ;and fromthe similarity of triangles AB'B and ABC BC BA we have AB= = AB CA' -BBC AB2+AC2 = BC2+ / ~~~B c' F C A.AC(FC'+FB') = BC(BC+C'B') = BC(BB'+CC'). In the case where A>900, the same proof would naturally hold with an alteration made for the sign of cosA. Thus the proof is quite general. 7Thabit makes Ino reference to Pappus' generalized Pythagorean theorem (see, e.g., Euclid, Elements [3 vol., 2nd edn., republished in 1956], I, 349-368, especially p. 366; or Selections Illustrating the History, of Greek Mathematics [translated by Ivor Thomas, 2 vol., 1939-1941], II, 575-579. Cf. Dr. E. Smith, History of Mathematics [2 vol., 1923-1925], II, 289, or Lietzmann, 1937,