By Dr. S. K. Uma, Professor and Head, Department of Mathematics, Sir M Visvesvaraya Institute of Technology
Gaṇeśa Daivajña is one of the great luminaries of post –Bhāskara period who is responsible for upholding the great tradition of Indian astronomy and Mathematics through his very creative and popular works. Gaṇeśa has simplified the procedures of computations of planetary positions and eclipses by completely dispensing with trigonometric functions which are otherwise laborious by the traditional methods. For this reason even today his works are famous among pañcāṅga makers in most parts of India.
Dates and place of Gaṇeśa
Gaṇeśa Daivajña’s father was the famous astronomer Keśava Daivajña and his mother name was Lakṣmī.. He was born in 1507 CE (śaka 1429) at a place called Nandigrāma on the Western seacoast. Gaṇeśa’s teacher in astronomy was his father himself while Keśava’s guru was Vaijanātha.
Works of Gaṇeśa Daivajña
Gaṇeśa composed several important works in the fields of mathematics, astronomy, astrology, etc. among which his astronomical treatise Grahalāghava is the most famous. In fact, the remarkable popularity of the Grahalāghava surpassed that of his father’s Graha Kautuka which was truly an important text in its own right.
Ganeśa’s other works are: Laghu- and Bṛhat-Tithi Cintāmaṇi, a commentary on Bhāskara’s Siddhānta Śiromaṇi, a commentary on Bhāskara’s Līlavatī called Buddhi Vilāsinī Vivāha vṛndāvanatīkā, Srāddha nirṇaya etc. Gaṇeśa himself mentions another work of his, Parva nirṇaya. Ganeśa’s commentary on Līlavatī called Buddhivilāsinī is an extremely useful text to understand the rationales for formulae and methods used by Bhāskara II and his predecessors.
Special feature and popularity of Grahalāghava
Among Gaṇeśa’s works, the Grahalāghava is the most popular astronomical text. It is believed that he composed this text when he was just 13 years old. The epoch of Grahalāghava is March 19, 1520 CE.(J), Monday. In this text the positions of planets are given for the newmoon day of Pālguṇa of śaka 1441 corresponding to March 19, 1520 CE.(Julian). Gaṇeśa has made computation of planetary positions and eclipses very easy by simplifying the procedures which is otherwise laborious by the methods explained in the texts Sūryasiddhānta, Brahmagupta’s Brahmaspuṭasiddhānta and Khaṇḍakhādyaka, Bhāskara’s Siddhāntaśiromaṇi etc. Some of the special features of his simplification in Grahalāghava are given below:
- To avoid handling of large number for ahargaṇa(the number of civil days elapsed since a chosen epoch), he has introduced a cycle(cakra) of 4016 days which is approximately 11 solar years. This modified ahargaṇanever exceeds 4016 and hence handy and also numerical error in multiplication etc. can be avoided.
- For the purpose of making the procedures simpler for pañcāṅgamaking and for the benefit of beginners in astronomy, who may be ignorant of trigonometry, Gaṇeśa has completely avoided the sine (jyā) and cosine (koṭijyā ) functions by adopting good approximations for these trigonometric functions.
- The dropping of trigonometric ratios has not seriously affected the accuracy of the results; on the contrary he has simplified procedures greatly. Gaṇeśa has given reasonably very good and mathematically justifiable approximations for trigonometric functions. Gaṇeśa highlighting the special feature of his text, declares, “The earlier preceptors very proudly rose to their peak of their fame even though only in few places they did calculations without using sine But I have made the calculations of the entire siddhānta(astronomy) simple(lāghava) by omitting sineetc. throughout. I have my intelligence(and knowledge) enriched only from their works and hence I do not exhibit pride(of my achievement)”
The Grahalāghava is considered as the most comprehensive, exhaustive and easy to use text among the karaṇa works(hand-books) on Indian astronomy.
The Grahalāghava carries with it very useful and authoritative commentaries by reputed astronomers like Gangādhara (1586 CE.), Mallāri ( 1602 CE.) and Viśvanātha (around 1612 CE.) Gangādhara’s commentary on the Grahalāghava is called Manoramā. Viśvanātha has given a large number of examples in his commentary to illustrate the methods of the Grahalāghava.
The Grahalāghava is extensively used by the pañcāṅga -makers particularly in Maharashtra, Gujarat, and northern part of Karnataka, the Hyderabad Deccan region of Andhra Pradesh and by the Deccanis of Vāraṇāsi, Gwalior and Indore. It is pointed out that even the government almanacs published at Indore and Gwalior used the Grahalāghava and the Tithicintāmaṇi of Gaṇeśa Daivajña.