• DOI:10.1086/348837
• Corpus ID: 119868978

Thâbit ibn Qurra's Generalization of the Pythagorean Theorem

@article{Sayili1960ThbitIQ,
  title={Th{\^a}bit ibn Qurra's Generalization of the Pythagorean Theorem},
  author={Aydin M. Sayili},
  journal={Isis},
  year={1960},
  volume={51},
  pages={35 - 37}
}

• A. Sayili
• Published 1 March 1960
• Mathematics
• Isis

IN AN 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 superfluous one. 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 rearrangement by 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.