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Ancient Mathematical Treatises
(HinduDharma: Jyotisa)

Jyotisa, as we have seen, consists of three sections. There was a scholarly man in the Matha who was particularly learned in this science. We wished to honour him with a title and decided upon "Triskandha-Bhaskara". "Skandha" literally means a big branch springing from the trunk of a tree. The three skandhas of Jyotisas are : siddhanta, hora and samhita.

The siddhanta-skandha deals with arithmetic, trigonometry, geometry and algebra. The higher mathematics developed by the west in later centuries is found in our ancient Jyotisa.

Arithmetic, called "vyakta-ganita" in sanskrit, includes addition, subtraction, multiplication and division. "Avyakta-ganita" is algebra. "Jya" means the earth and "miti" is method of measurement. "Jyamiti" evolved with the need to measure the sacrificial place :"geometry" is derived from this word. The "geo" in geography is from "jya". There is a mathematical exercise called "samikarana" which is the same as "equation".

The sixth Anga of the Vedas, Kalpa ( I will speak about it later ), has a great deal to do with the fifth, that is Jyotisa. Kalpa has a section on "sulba-sutras". These sutras mention the precise measurements of the "yajnavedi" (sacrificial altar). The character of the yajnabhumi is called "cayana". The sulba- sutras deal with a number of cayanas like, for instance, the one shaped like Garuda. They tell us how to construct a brick-kiln ---the number of bricks required for the cayana of such and such shapes. The siddhanta-skandha is used in all this.

There is an equation in the Apastamba sulba sutras which could not be proved until recently. Westerners had thought it to be faulty merely as they could not solve it. Now they accepted it as right. That Indians had taken such great strides in mathematics, thousand of years ago has caused amazement in the West. There are a number of old equations still to be solved.

Our sastras mention branches of mathematics like "rekhaganita, "kuttaka", "angapaka", etc. "Avyakta-ganita" is also called "bijaganita".

Eight hundred years ago there lived a great mathematician called Bhaskaracarya. An incident in his life illustrates how relentless destiny is. Bhaskaracarya had a daughter called Lilavati. The great astrologer that he was, he found that she had "mangalya-dosa" in her horoscope, but he felt confident that he could change his daughter's destiny, as foreshadowed by the stars, with his ingenuity and resourcefulness, as an astrologer. He decided to celebrate Lilavati's marriage during a lagna in which all the planets would be in positions favourable to the bride. This should, he thought, ensure that Lilavati would remain a "dirgha-sumangali".

In those days there were no clocks as we have today. A water-pot was used to measure time. It consisted of an upper as well as a lower part. The water in the upper receptacle would trickle down through a hole into the lower container. The lower part was graduated according to the unit of time then followed ---nazhikai (nadika), one sixtieth of a day or 24 minutes. So the time of day was calculated by observing the level of the water in the lower container. ("Water-clock" and "hour-glass" are English names for such an apparatus. Since water evaporates quickly sand was used instead. )

According to the custom then prevailing, Lilavati's marriage was to be celebrated when she was still a child. On the appointed day, she sat beside the water--clock and bent over it fascinated by the apparatus. As she fumbled around a pearl from her nose--stud got loosened and fell into the apparatus lodging itself in its hole. The flow of water into the lower receptacle was reduced. So what the clock indicated as the hour fixed for the marriage was not the right one---the auspicious hour had passed. Nobody including Lilavati, had noticed the pearl dropping into the water-clock. When they came to know about it, it was too late. They realised that destiny could not be overcome.

Later Bhaskaracarya wrote a mathematical treatise and named it "Lilavati" after his daughter. The father taught his widowed daughter mathematics and she became highly proficient in the subject. Lilavati deals with arithmetic, algebra, etc. It is a delightful book in which the problems are stated in verse as stories. Bhaskaracarya also wrote the Siddhanta-Siromani which deals with how the positions and movement of the heavenly bodies are determined.

We learn the text of an edict in the Pracinalekhamala that a Gurjara (Gujarat) king had made an endowment to popularise the works of Bhaskaracarya.

Parts 7, 8, 9 and 10 of Euclid's Geometry are believed to be lost. All the 12 books on mathematics in Sanskrit are still available. "Making additions several times is multiplication; carrying out subtraction several times is division. " We remain ignorant of such easy methods of calculations dealt with in our mathematical texts.

Varahamihira lived several years before Bhaskaracarya, that is about 1, 500 years ago. He wrote a number of treatises including the Brhat-Samhita and the Brhajjatika. The first is a digest of many sciences, its contents being a wonderful testimony to the variety of subjects in which our forefathers has taken strides. Brhajjatika is all about astrology.

Aryabhata, famous for his Aryabhatiya-Siddhanta, also lived 1, 500 years ago. The vakya--ganita now in use is said to be based on his Siddhanta. Varahamihira and Aryabhata are much acclaimed by mathematicians today.

All these books on mathematics also deal with the movements of the celestial bodies. There are seven "grahas" according to the ancient reckoning--the five planets and the sun and the moon. Rahu and Ketu are called "chaya -grahas" (shadow planets) and their orbits are opposite of the sun's and the moon's.

"Hindu Dharma" is a book which contains English translation of certain invaluable and engrossing speeches of Sri Sri Sri Chandrasekharendra Saraswathi MahaSwamiji (at various times during the years 1907 to 1994).
For a general background, please see here