Geometry (hendese) is a word meaning "the measurement of the earth." The efforts of the Egyptians to redetermine the boundaries of their fields, boundaries lost each year when the Nile overflowed, led them to develop expertise in the practical branch of geometry. Astronomy, too, a science that itself stood in need of geometry, was among the disciplines held in high esteem in ancient societies.
The Greeks learned this science from the Egyptians, who had advanced its practical (amelî) branch, and took it up on a theoretical (nazarî) footing. The most celebrated of these theoretical works is Euclid's Elements, written around 300 BC and read continuously down to the present day. Geometry is divided into two branches: Euclidean geometry and non-Euclidean geometry. The problems treated in the Elements drew the interest not only of those occupied with mathematics but of philosophers as well [1]. It is even reported that the words "Let no one ignorant of geometry enter here" were inscribed at the entrance to Plato's Academy [2]. Euclid composed the Elements in thirteen books (where "book" may be understood as a "section"). At the beginning of Book I he sets out the definitions, postulates, and axioms; he then proceeds to present his theorems together with their proofs. Philosophers have expended considerable effort on the definitions, postulates, and axioms given at the opening of the work. The Egyptians' way of handling problems may be described as induction (istikrâ), whereas Euclid proves his theorems by the method of deduction (ta'lîl). Euclid's first three postulates cannot be derived from experience [3]. The fourth postulate is somewhat puzzling, and the fifth is considerably more intricate than those that precede it. After the postulates, he lists five axioms. According to Euclid, the essential difference between postulates and axioms is this: the postulates pertain specifically to geometry, whereas the axioms express more general principles. Furthermore, whereas one who denies the postulates may still reason comfortably about the other sciences, one who denies the axioms will find himself without any foundation in intellectual matters. Postulates and axioms express principles that may be accepted without the need of proof, and Euclid rests all his subsequent proofs upon these unproven principles.
Euclid's Elements was translated into Arabic during the reign of the Caliph Hārūn al-Rashīd, and commentaries were written upon it. Geometry, which took its place in the curriculum of the Ottoman madrasas, was studied from Arabic texts, Arabic being the lingua franca of the age. Among these texts were Nasīr al-Dīn al-Tūsī's Tahrīr al-Usūl li-Uqlīdis, Shams al-Dīn al-Samarqandī's Ashkāl al-Taʾsīs, and the commentary that Qāḍīzāde al-Rūmī wrote upon al-Samarqandī's work [4]. Geometry also came under discussion in the teaching of works pertaining to the science of kalām (theology). Among the scholars and intellectuals of the late Ottoman period there were also those who learned geometry through private lessons [5]. In Mevzûât al-Ulûm, the encyclopedic work in which he set out his classification of the sciences, Taşköprülüzâde explains the benefits of geometry as follows: "It sharpens one's perception and opens the mind. It endows a person with penetration and authority, thereby strengthening his thought. Furnished with diverse sciences and profound knowledge, he prevails in the arena of those who pose questions and surmounts difficult and arduous passages. He wins the contest among his peers and is singled out among scholars. For it is unanimously agreed that, with respect to demonstration and proof, geometry is the mightiest of the sciences" [6].
The translation of the Elements into Turkish was carried out by Hüseyin Rıfkı Tâmânî (1750?–1817), Chief Instructor of the Imperial Military Engineering School (Mühendishâne-i Berrî-i Hümâyûn), together with Selim Efendi, an English engineer who had converted to Islam; working from an English translation, they published it under the title Tercüme-i Usûlü'l-Hendese [7]. Tâmânî took certain liberties in his translation and voiced his objections here and there. In the book, however—presumably out of a concern that it serve a practical purpose—these objections were kept within certain limits. Practical aims evidently governed the choice of the source text as well, for the translation selected was the English version prepared by John Bonnycastle, one of the instructors at the Royal Military Academy. Ali Rıza Tosun, who has written his doctoral dissertation on Tâmânî's translation, observes: "If we consider that the Elements has still not been fully translated into modern Turkish, can we comfortably claim that we are today far ahead of the 1790s?"
In 2019, Ali Sinan Sertöz published Öklid'in Elemanları, the first complete rendering of all thirteen books of the Elements into modern Turkish, issued by TÜBİTAK in its popular-science series [8]. Whether a gap of more than two centuries between the first Turkish Elements and the first complete one is properly an occasion for satisfaction, or for a rather different sentiment, is a question the reader is perhaps best left to adjudicate.
References and Notes
[1] This remark, attributed to the German mathematician and philosopher Frege (1848–1925), brings out the place of geometry within philosophy: "A philosopher who has nothing to do with geometry is only half a philosopher, and a mathematician with no element of philosophy in him is only half a mathematician. These disciplines have estranged themselves from one another to the detriment of both." James Robert Brown, Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures, New York, 2008, p. xi.
[2] This is reported by John Philoponus, who lived in Alexandria in the sixth century, by the twelfth-century Byzantine writer John Tzetzes, and by others.
[3] Postulates: 1) A straight line may be drawn from any point to any other point. 2) A finite straight line may be extended continuously in the same direction as far as one wishes. 3) For any given point and length, a circle may be drawn having that point as its center and the given length as its radius. 4) All right angles are equal to one another. 5) If a straight line intersects two other straight lines in such a way that, when the two straight lines are extended far enough, they meet on the same side as that on which these angles lie. Axioms: 1) Things equal to the same thing are equal to one another. 2) If equals are added to equals, the sums are equal. 3) If equals are subtracted from equals, the remainders are equal. 4) Things that coincide with one another are equal to one another. 5) The whole is greater than the part. Stephen F. Barker, Philosophy of Mathematics, trans. Yücel Dursun, Ankara, 2003, p. 38.
[4] For a detailed account of the geometry texts taught in the Ottoman madrasas, see Cevat İzgi, Osmanlı Medreselerinde İlim, vol. I (Mathematical Sciences), Istanbul, 1997, pp. 274–329.
[5] One of these scholars was Ahmed Cevdet Pasha. Cevat İzgi, op. cit., p. 273.
[6] Taşköprülüzâde, Mevzûâtü'l-Ulûm (Encyclopedia of the Sciences), rendered into modern Turkish by Mümin Çevik, vol. I, Istanbul, 1975, p. 302.
[7] Ali Rıza Tosun, Hüseyin Rıfkı Tâmânî ve Elementler Çevirisi, Ankara, 2010. The book is the published version of his doctoral dissertation. The PDF of the thesis can be accessed here; the translated book can be accessed here.
[8] Euclid, Öklid'in Elemanları, trans. Ali Sinan Sertöz, TÜBİTAK Popüler Bilim Kitapları, Istanbul, 2019 (xiii + 691 pp.), ISBN 978-605-312-328-6. This edition constitutes the first complete Turkish translation of all thirteen books, prefaced by an extended discussion of the Elements and of the approach adopted in the translation.
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