Page 22 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Unit 1: Number Systems

Notes
In the 9th C, Mahaviracharya (Mysore) wrote Ganit Saar Sangraha where he described the
currently used method of calculating the Least Common Multiple (LCM) of given numbers. He
also derived formulae to calculate the area of an ellipse and a quadrilateral inscribed within a
circle (something that had also been looked at by Brahmagupta). The solution of indeterminate
equations also drew considerable interest in the 9th century, and several mathematicians
contributed approximations and solutions to different types of indeterminate equations.
In the late 9th C, Sridhara (probably Bengal) provided mathematical formulae for a variety of
practical problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates
of travel, wages, and filling of cisterns. Some of these examples involved fairly complicated
solutions and his Patiganita is considered an advanced mathematical work. Sections of the
book were also devoted to arithmetic and geometric progressions, including progressions with
fractional numbers or terms, and formulas for the sum of certain finite series are provided.
Mathematical investigation continued into the 10th C. Vijayanandi (of Benares, whose
Karanatilaka was translated by Al-Beruni into Arabic) and Sripati of Maharashtra are amongst
the prominent mathematicians of the century.

The leading light of 12th C Indian mathematics was Bhaskaracharya who came from a long-line
of mathematicians and was head of the astronomical observatory at Ujjain. He left several
important mathematical texts including the Lilavati and Bijaganita and the Siddhanta Shiromani,
an astronomical text. He was the first to recognize that certain types of quadratic equations
could have two solutions. His Chakrawaat method of solving indeterminate solutions preceded
European solutions by several centuries, and in his Siddhanta Shiromani he postulated that
the earth had a gravitational force, and broached the fields of infinitesimal calculation and
integration. In the second part of this treatise, there are several parts relating to the study of
the sphere and its properties and applications to geography, planetary mean motion, eccentric
epicyclical model of the planets, first visibilities of the planets, the seasons, the lunar crescent
etc. He also discussed astronomical instruments and spherical trigonometry. Of particular
interest are his trigonometric equations: sin(a + b) = sin a cos b + cos a sin b; sin(a – b) = sin a
cos b - cos a sin b;
The Spread of Indian Mathematics
The study of mathematics appears to slow down after the onslaught of the Islamic invasions
and the conversion of colleges and universities to madrasahs. But this was also the time when
Indian mathematical texts were increasingly being translated into Arabic and Persian. Although
Arab scholars relied on a variety of sources including Babylonian, Syriac, Greek and some
Chinese texts, Indian mathematical texts played a particularly important role. Scholars such
as Ibn Tariq and Al-Fazari (8th C, Baghdad), Al-Kindi (9th C, Basra), Al-Khwarizmi (9th C.
Khiva), Al-Qayarawani (9th C, Maghreb, author of Kitab fi al-hisab al-hindi), Al-Uqlidisi Ibn-
Sina (Avicenna), Ibn al-Samh (Granada, 11th C, Spain), Al-Nasawi (Khurasan, 11th C, Persia),
Al-Beruni (11th C, born Khiva, died Afghanistan), Al-Razi (Teheran), and Ibn-Al-Saffar (11th
C, Cordoba) were amongst the many who based their own scientific texts on translations of
Indian treatises. Records of the Indian origin of many proofs, concepts and formulations were
obscured in the later centuries, but the enormous contributions of Indian mathematics was
generously acknowledged by several important Arabic and Persian scholars, especially in
Spain. Abbasid scholar Al-Gaheth wrote: “India is the source of knowledge, thought and insight”.
Al-Maoudi (AD 956) who travelled in Western India also wrote about the greatness of Indian
science. Said Al-Andalusi, an 11th C Spanish scholar and court historian, was amongst the most
enthusiastic in his praise of Indian civilization, and specially remarked on Indian achievements
in the sciences and in mathematics. Of course, eventually, Indian algebra and trigonometry
reached Europe through a cycle of translations, travelling from the Arab world to Spain
and Sicily, and eventually penetrating all of Europe. At the same time, Arabic and Persian
translations of Greek and Egyptian scientific texts become more readily available in India.
Contd...

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