Salih Zeki and Asâr-ı Bâkiye

Erdal İnönü

In commemoration of the hundred fortieth anniversary of the birth of Salih Zeki (1864-1921), the last renown Ottoman mathematician, a summary and evaluation of his book on the development of mathematics in Islam, Asâr-ı Bâkiye [Athar-i Baqiya]is presented.

Originally, the book was planned to have four volumes, dealing separately with trigonometry, arithmetic and algebra, astronomy and geometry. Unfortunately, only the first two volumes have been printed (1913). Recently, the science historians at Ankara University transcribed these two volumes into modern Turkish and, my presentation is based on this edition completed in 2004.

In his preface, Salih Zeki explains his aim in writing this book as "indicating the additions brought by Eastern mathematicians to the old edifice of Greek mathematics, and establishing clearly the levels of development at which this legacy was transmitted to Western scientists."

The first volume on trigonometry starts with explaining how the old astronomers, including Ptolemaios (c.100-c.170) dealt with angles, using only the chords, and a relevant theorem of Menelaus (c.100). He continues with the introduction of the concepts and tables of trigonometric terms, such as sines by Thabit ibn Qurra (836-901), followed by Jabir al-Battani (c.858-929) and tangents by Abu'l-Wafa al-Buzjani (940-998). Through a detailed analysis of various astronomical or trigonometric tables, he makes it clear that the science of trigonometry is essentially the creation of Eastern mathematicians (mostly Islamic, following a beginning in India). I may add that this conclusion is shared by the statements of Western historians, such as Paul Tannery (in 1893) and Colin Ronan (in 1983).

The second volume relates in detail how the present number characters (both Arabic and Latin) evolved from Indian numerals through the book Kitab fi el Hisab el-Hindi by al-Khwarizmi (c. 780-c. 850) which was written to teach the people how to use numbers in their daily lives. There were two kinds of numerals in al-Khwarizmi's book, both taken from Indian mathematics. One of them, through the translations Liber Abaci, 1202 by L.Fibonacci (1170-post 1240) and Summa di Arithmetica, Geometria (1494) by L.Pacioli (c. 1445-c.1514) led to the development of Latin numerals, and the other, through the works of Islamic mathematicians gave rise to Arabic numerals.

Salih Zeki presents several arguments to refute the claim made by some Western historians that the Indian-Arabic numerals were first invented in ancient Greece by the followers of Pythagoras (c.570-c.490). Finally, he gives two examples to show that decimal fractions were first used by Islamic mathematicians; one is the decimal expression of ? by Jamshid Ghiyath al-Din al-Kashi (d. 1429) obtained in Samarkand around 1420, which was valid to the sixteenth decimal point, and the other is the decimal expression used for the trigonometric tables prepared by the astronomer Taqi al-Din (d.1585) in Istanbul, around 1570.

As a conclusion, I wish to express the view that, Salih Zeki's Asâr-ı Bâkiye, both by documenting attempts to clarify the priorities in the development of mathematics, and in providing a detailed explanation of many interesting computations by ancient mathematicians, remains an important reference work for historians of mathematics all over the world.

Key words: Mathematics, Islam, trigonometry, decimal fractions, Salih Zeki, Asâr-ı Bâkiye, Athar-i Baqiya; Anahtar kelimeler: Matematik, İslam, trigonometri, ondalık kesirler, Salih Zeki, Asâr-ı Bâkiye.