The Use of Decimal Fractions in Trigonometry and Astronomy by Taqî al-Dîn
European historians have for long believed that Turkish scholars did not contribute to the medieval exact sciences. The reason was that historical studies about Turkish scholars were fairly inadequate. However, in recent years, research made on the works by al-Harezmî, Abd al-Hamid ibn Türk, al-Farabî, Avicenna, Ulugh Beg and Ali al-Qushji altered this conviction, though did not abolish it completely.
In this article, the contributions of the Turkish mathematician and astronomer Taqî al-Dîn ibn M'aruf (1521-1585) to the use of decimal fractions is studied. Founder of the Istanbul Observatory (1575), Taqî al-Dîn observed the celestial bodies (especially the sun, the moon and the stars) for about five years. He learned the decimal fractions from Ghiyath al-Dîn Jamshid al-Kashi's work, Miftah al-Hisab (1427). He improved al-Kashi's studies on fractions and prepared zijs, in which he used the decimal fractions instead of the sexagesimal ones. According to Taqî al-Dîn, al-Kashi's knowledge on the decimal fractions was limited. He could only perform the four basic operations such as addition, substraction, multiplication and division with these fractions. However, he did not use them in trigonometry or astronomy.
Why did Taqî al-Dîn use decimal fractions in trigonometry and astronomy? Taqî al-Dîn's main purpose was to make the trigonometrical and astronomical calculations easier and simpler. He wrote an arithmetical treatise called Bugya al-Tullab where he introduced two different calculation systems, namely al-Hisab al-Hindi (the decimal system) and al-Hisab al-Müneccim (the sexagesimal system), that were familiar to Ottoman mathematicians and astronomers. After describing in detail the concept of "decimal fraction" in the ninth chapter, he showed through examples the way to operate the four basic operations and duplications as well as to take their halves and square roots. He also explained the way to convert the sexagesimal fractions into the decimal fractions or vice versa. However, Taqî al-Dîn did not introduce any symbols to distinguish the integers from the decimal fractions. He transcribed for example, the number 532,876 in the form of 5 hundreds 3 tens 2 ones 8 one tenths 7 one hundredths 6 one thousandths or in the form of 532876 one thousandths; that is, the forms used by Taqî al-Dîn were verbal.
In Bugya al-Tullab, Taqî al-Dîn pointed out that the sexagesimal system used by astronomers in spherical astronomy to determine the positions of the heavenly bodies was not adequate for the calculations. The reason was that in the sexagesimal system doing multiplications and divisions was rather troublesome and these operations were time consuming for the astronomers. Moreover, the sexagesimal multiplication table which was similar to the decimal one was not very efficient. He argued that the arithmetical calculations as well as the preparation and use of astronomical tables would be made more easily through decimal fractions.
Taqî al-Dîn decimalized the fractions of trigonometrical functions, angles and arcs. In Sidra al-Muntaha, he assumed the radius of the circle as 10 units and not 60 or 1. Moreover, he divided one of the two surfaces of Zat al-cayb (an astronomical instrument used at Istanbul observatory) into 10 equal parts.
He put his theoretical knowledge into practice in his zij called al-Teshil (1580) where he minutely defined the application of decimal fractions to trigonometry and astronomy. In this zij based on observations made at the Istanbul Observatory, all fractions were decimalized.
In Carida al-Durar, a zij dated 1584, Taqî al-Dîn calculated a sine-cosine table and a tangent-cotangent table where he took the radius of the trigonometrical circle as 10 units and decimalized the fractions of trigonometrical functions. If he had assumed the radius as 1 unit and not 10 units, he would have calculated the trigonometrical tables that we are using today.
In Europe , the first treatise introducing the decimal fractions was De Thiende (1585) written by the Dutch mathematician Simon Stevin and published in Leiden . In this 32- page booklet, although Stevin suggested signs designating the decimal fractions and used decimal notations for weights and measures, there is no evidence that he applied them to trigonometry or astronomy. Therefore, it may be well stated that decimalization in these fields started with Taqî al-Dîn, one of the distinguished scholars of the 16th century Ottoman Turkey.