DEVELOPMENT OF VEDIC MATHEMATICS

He becomes richer by adding the finer qualities of both Lakshmi and Saraswathi to himself. His bad qualities have been subtracted from him by the kind look of Shiva. Since his good qualities have increased, admiration's from others have multiplied for him. He divides the hearts of scholars and keeps them with him. i.e., all these scholars appreciate him. These are the achievements of good-hearted person, who churned the ocean of mathematics and he excels all scholars. 

This is a verse by the author in praise of Mathematics and Mathematicians. Since Mathematics is none other than the logic, wherever and whenever Mathematical sciences advance, the growth of Physical sciences follows. This is a fact established by history. In order to assess the state of scientific development of any country at a given time, generally it may be adequate to study the state of mathematics of the time. We can infer the state of other sciences from that. Just as Mathematics, is for Physical sciences, so is logic for philosophical knowledge. Wherever logic develops, math too flourishes there. In fact, there is only one difference between math and logic. 
Logic is expressed in the local language, whereas math is expressed in numbers and lines. Ancient Indians, who had a spectrum of sciences for their credit, valued mathematics with the same reverence. 
Here is an evidence for this, form a work called Vedanga Jhyotisha, which belongs to a period older than 1200 B.C.  

Just as the natural feather on the head of a peacock and just as gems on the head of a divine serpent, Jyotisha - the science of astronomy is placed on the head of the other sastras - so says the well known Vedanga Jyotisha. 

Another version of this sloka reads

infact these versions do not make much difference, for the terms Jyotisha and Ganita are almost synonyms.

In Bhagavatgita the lord says - 

is the meaning given by Sri Sankaracharya. 

are the basic forms of 

Means addition,

means subtraction. Simplified addition is multiplication and simplified subtraction is division. Whatever may be the extent of development, mathematics can never go beyond these four operations, to be more precise, beyond these two operations. 

Hence, we can easily infer that Kaala and Ganita are not different from each other. Jyotisha is the science of Kaala. Hence, Jyotisha and Ganita can not be different. 

Hence we can write: 

 Time = Astronomy = Mathematics

But you can ask one question. 

Time is represented by numbers, where as jyotisha deals with geometrical figures. Then, how can these two be equal ? 

When these two are not equal, how can we say 

 Time = Astronomy = Mathematics

Your question may be correct only to some extent. Astronomy is the science of time and so, some may not like to say that 

time = Astronomy even then, they cannot deny the equations. 

Why not ?

This is because, mathematics is a science in which there are two important and unseperable branches, viz., 

1. Mathematics of numbers & 

2. Mathematics of space. 

The maths of time deals with numbers and the maths of space deals with lines, whether straight or curved.  

Since time and space are inseparable, you can not separate these two branches of mathematics. Since the most ancient and the most modern concepts accept that, in the ultimate, time and space are not different, these two branches of mathematics also cant be different, of course , in the ultimate. 

The “ultimate” is thoroughly discussed in the Vedas from different angles and hence, the Veda had to deal with the numbers, the space and the mathematics which embodies both of them. 

Stray numbers can be used and utilized by even primitive societies, but numbers as a system can be utilized only by a developed society. In fact, history proves to us that, in any period of time, where the numbers are used in a more systematic way, the better is the civilization of that society as a whole. 

Even though we are not exactly sure of what a Vedic period is, whatever it may be, we can estimate its civilization basing on the number systems available in the Vedas.

Pythagoras, the celebrated philosopher and mathematician of 3rd century B.C, tried to evolve a numbering system, to count the particles of a sand in a given jar of sand and wrote his thesis - “The calculus of sand”. 

But, unfortunately, he could not develop a perfect decimal system of numbers, because he could not think of “Zero” at that time. 

But thousands of years before that, we find a full-fledged decimal system of numbers in the Veda mantras:

This is not a rare or strange reference from Veda. 

“ Oh Lord Agni! Prostrations to you once, twice, thrice, four times five times, ten times, hundred times upto thousand times and unlimited number of times”. Here we find a definite pattern of progression of numbers. In the well known Chamakadhyaya of Krishna Yajurveda, the mantra 

gives two sequences of numbers 

  1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31 

and

  4, 8,12, 16, 20, 24, 28, 32, 36, 40, 44, 48. 

The first one is a sequence of odd numbers from 1 to 31 for the second one is not a sequence of simple even numbers. 

We can observe that these sequences follow the formula.

x being the member of the 1st sequence and y , of the 2nd sequence.

Eg: 1 + 3 = 4; 3 + 5 = 8 upto the 12th place. 

Thus we find an intricate and perfect systematic use of numbers in the Vedas. Added to this, perfect discretions of cosmological events are found in the Vedas. 

The famous Nakshatreti Prakarana of Yajurveda, the cosmological details given in Rigveda, are more than enough to prove this.

As we entered the post Vedic Ancient Literature, there are four important places, where we find extensive use of mathematical formulas. 

 these are the six Auxiliaries to Veda - so say the wise ones. 

1. Of these, the above referred Sulba Sutras are an integral part of 
    Kalpa. 

2. Vedanga Jyotisha is the essence of jyotisha, the science of time 
    and the science of cosmological bodies. 

3. Chandas is the science of meters of letters.

4. Tantra is a science and art which tries to simplify Vedic rituals, 
    for the benefit of the less privileged beings. 

Of these four – 

1. Sulba Sutras use maths in the context of the 

2. Jyotisha uses maths, at every step. 

3. Chandas uses maths, when it has to deal with the permutations 

    and combinations of the meters and

 4. Tantra uses maths when it translates the divine energies into 

     geometrical figures called Yantras. 

So, to understand Vedic Mathematics, we have to study all the four subjects, referred above. But studying them directly is not a simple thing because they are not direct textbooks of mathematics. But a lot of mathematics is used there. So, to make mathematics simpler and easier to understand, the later scholars started dividing mathematics into several sub-branches, thus - 

PROMINENT VEDIC MATHEMATICIANS 

Naarada, Kapila, Bodhayaana, Aapastamba, Lagadhaa etc., are said to be the prominent mathematicians of the ancient periods. 

While 

     

Aaryabhatta of 5th Century A.D  

    Varaahamihir of 6th Century A.D 

   Brahmagupta of 7th Century A.D

Sreedhara of 8th Century A.D  

 Mahaaveera of 9th Century A.D

Bhaaskara of 12th Century A.D

are some of the very prominent mathematicians of medieval periods.  

These masters have developed wonderful methods of teaching mathematics to the youngsters. But, in the process of doing so, unfortunately, it seems that they had deviated from the age-old Vedic techniques, which were prevalent during the Ancient periods.  

In the recent past Jagadguru Bhaarati Krishna Teertha of Dwaraka Sankaraachaarya Peetha, revived 16 fundamental Vedic statements of mathematics and proved how these 16 formulas could be used for almost all branches of mathematics. 

His writings were published under the title “ Vedic Mathematics” in 1965 for the first time and eye brows of international mathematicians rose up at once, since the methods annunciated in the Vedic mathematics are unbelievable, simple and perfect, as compared to today’s so called advanced western mathematics. Hence Bhaarati Krishna Teertha’s Vedic Mathematic is acquiring more and more importance, internationally.