Digest Number 2164

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Ancient Mathematical Treasties from Hindu Dharma

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 "eqation".

 

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 sectoin on

"sulba-sutras". These sutras mention the precise measurements of the

"yajnavedi" (sacrifical 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 resorcefulness, 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 astology.

 

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.