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Revered Guruji used to say that he had reconstructed the sixteen mathematical formulae from the Atharvaveda after assiduous research and Tapas (austerity) for about eight years in the forests surrounding Sringeri. Obviously these formulae are not to be found in the present recensions of Atharvaveda. They were actually reconstructed, on the basis of intuitive revelation, from materials scattered here and there in the Atharvaveda.
India and the Scientific Revolution
The following contributions are
Multiplying by 12 - shortcut
Vedic Mathematics is the name given to the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to his research all of mathematics is based on sixteen Sutras or word-formulae. For example, 'Vertically and Crosswise` is one of these Sutras. These formulae describe the way the mind naturally works and are therefore a great help in directing the student to the appropriate method of solution.
Perhaps the most striking feature of the Vedic system is its coherence. Instead of a hotch-potch of unrelated techniques the whole system is beautifully interrelated and unified: the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. And these are all easily understood. This unifying quality is very satisfying, it makes mathematics easy and enjoyable and encourages innovation.
In the Vedic system 'difficult' problems or huge sums can often be solved immediately by the Vedic method. These striking and beautiful methods are just a part of a complete system of mathematics which is far more systematic than the modern 'system'. Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, direct and easy.
The simplicity of Vedic Mathematics means that calculations can be carried out mentally (though the methods can also be written down). There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one 'correct' method. This leads to more creative, interested and intelligent pupils.
Interest in the Vedic system is growing in education where mathematics teachers are looking for something better and finding the Vedic system is the answer. Research is being carried out in many areas including the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc.
But the real beauty and effectiveness of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most refined and efficient mathematical system possible.
The Vedic Mathematics Sutras
This list of sutras is taken from the book Vedic Mathematics, which includes a full list of the sixteen Sutras in Sanskrit, but in some cases a translation of the Sanskrit is not given in the text and comes from elsewhere.
This formula 'On the Flag' is not in the list given in Vedic Mathematics, but is referred to in the text.
The Main Sutras
The Sub Sutras
Try a Sutra
Mark Gaskell introduces an alternative
At the Maharishi School in Lancashire we have developed a course on Vedic mathematics for key stage 3 that covers the national curriculum. The results have been impressive: maths lessons are much livelier and more fun, the children enjoy their work more and expectations of what is possible are very much higher. Academic performance has also greatly improved: the first class to complete the course managed to pass their GCSE a year early and all obtained an A grade.
Vedic maths comes from the Vedic tradition of India. The Vedas are the most ancient record of human experience and knowledge, passed down orally for generations and written down about 5,000 years ago. Medicine, architecture, astronomy and many other branches of knowledge, including maths, are dealt with in the texts. Perhaps it is not surprising that the country credited with introducing our current number system and the invention of perhaps the most important mathematical symbol, 0, may have more to offer in the field of maths.
The remarkable system of Vedic maths was rediscovered from ancient Sanskrit texts early last century. The system is based on 16 sutras or aphorisms, such as: "by one more than the one before" and "all from nine and the last from 10". These describe natural processes in the mind and ways of solving a whole range of mathematical problems. For example, if we wished to subtract 564 from 1,000 we simply apply the sutra "all from nine and the last from 10". Each figure in 564 is subtracted from nine and the last figure is subtracted from 10, yielding 436.
This can easily be extended to solve problems such as 3,000 minus 467. We simply reduce the first figure in 3,000 by one and then apply the sutra, to get the answer 2,533. We have had a lot of fun with this type of sum, particularly when dealing with money examples, such as £10 take away £2. 36. Many of the children have described how they have challenged their parents to races at home using many of the Vedic techniques - and won. This particular method can also be expanded into a general method, dealing with any subtraction sum.
The sutra "vertically and crosswise" has many uses. One very useful application is helping children who are having trouble with their tables above 5x5. For example 7x8. 7 is 3 below the base of 10, and 8 is 2 below the base of 10.
The whole approach of Vedic maths is suitable for slow learners, as it is so simple and easy to use.
The sutra "vertically and crosswise" is often used in long multiplication.
Suppose we wish to multiply
We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. By the end of Year 8, I would expect all students to be able to do a "3 by 2" long multiplication in their heads. This gives enormous confidence to the pupils who lose their fear of numbers and go on to tackle harder maths in a more open manner.
All the techniques produce one-line answers and most can be dealt with mentally, so calculators are not used until Year 10. The methods are either "special", in that they only apply under certain conditions, or general. This encourages flexibility and innovation on the part of the students.
Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. Here is an example of doing this in a special method for long multiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by 92. 96 is 4 below the base and 92 is 8 below.
We can cross-subtract either way: 96-8=88 or 92-4=88. This is the first part of the answer and multiplying the "differences" vertically 4x8=32 gives the second part of the answer.
This works equally well for numbers above the base: 105x111=11,655. Here we add the differences. For 205x211=43,255, we double the first part of the answer, because 200 is 2x100.
We regularly practise the methods by having a mental test at the beginning of each lesson. With the introduction of a non-calculator paper at GCSE, Vedic maths offers methods that are simpler, more efficient and more readily acquired than conventional methods.
There is a unity and coherence in the system which is not found in conventional maths. It brings out the beauty and patterns in numbers and the world around us. The techniques are so simple they can be used when conventional methods would be cumbersome.
When the children learn about Pythagoras's theorem in Year 9 we do not use a calculator; squaring numbers and finding square roots (to several significant figures) is all performed with relative ease and reinforces the methods that they would have recently learned.
For many more examples, try elsewhere on this page, the Vedic Maths Tutorial
Mark Gaskell is head of maths at the Maharishi School in Lancashire
'The Cosmic Computer'
Saturday school for primary teachers at
19th May 2000 Times Educational Supplement (Curriculum Special)
copyright to the ACADEMY OF VEDIC MATHEMATICS
MATHS OR MAGIC?
A PEEP INTO VEDIC MATHEMATICS
INTRODUCTORY LECTURES ON VEDIC MATHEMATICS
DISCOVER VEDIC MATHEMATICS
VERTICALLY AND CROSSWISE
VEDIC MATHEMATICAL CONCEPTS OF SRI VISHNU SAHASTRANAMA STOTRAM
ISSUES IN VEDIC MATHEMATICS
THE NATURAL CALCULATOR
VEDIC MATHEMATICS FOR SCHOOLS BOOK 1
INTRODUCTION TO VEDIC MATHEMATICS
THE COSMIC COMPUTER COURSE
GEOMETRY FOR AN ORAL TRADITION
This book demonstrates the kind of system that could have existed before literacy was
widespread and takes us from first principles to theorems on elementary properties of
circles. It presents direct, immediate and easily understood proofs. These are based on
only one assumption (that magnitudes are unchanged by motion) and three additional
provisions (a means of drawing figures, the language used and the ability to recognise
valid reasoning). It includes discussion on the relevant philosophy of mathematics and is
written both for mathematicians and for a wider audience.
THE CIRCLE REVELATION
VEDIC MATHEMATICS FOR SCHOOLS BOOK 2
Astronomica; Applications of Vedic Mathematics
Vedic Mathematics, Part 1
We found this book to be well-written, thorough and easy to read.
INTRODUCTION TO VEDIC MATHEMATICS Part II
VEDIC MATHEMATICS FOR SCHOOLS BOOK 3
THE COSMIC CALCULATOR
TEACHERS MANUALS ELEMENTARY & INTERMEDIATE
TEACHERS MANUAL ADVANCED
FUN WITH FIGURES (subtitled: Is it Maths or Magic?)
Book review of 'Fun with Figures'
From 'inTouch', Jan/Feb 2000, the Irish National Teachers Organisation (INTO) magazine.
Vedic Maths is based on sixteen Sutras or principles. These principles are general in
nature and can be applied in many ways. In practice many applications of the sutras may be
learned and combined to solve actual problems. These tutorials will give examples of
simple applications of the sutras, to give a feel for how the Vedic Maths system works.
Use the formula ALL FROM 9 AND THE LAST FROM 10 to
For example 1000 - 357 = 643
We simply take each figure in 357 from 9 and the last figure from 10.
So the answer is 1000 - 357 = 643
And thats all there is to it!
This always works for subtractions from numbers consisting of a 1 followed by noughts: 100; 1000; 10,000 etc.
Similarly 10,000 - 1049 = 8951
For 1000 - 83, in which we have more zeros than
Try some yourself:
1) 1000 - 777 =
2) 1000 - 283 =
3) 1000 - 505 =
4) 10,000 - 2345 =
5) 10,000 - 9876 =
6) 10,000 - 1011 =
7) 100 - 57 =
8) 1000 - 57 =
9) 10,000 - 321 =
10) 10,000 - 38 =
Answers to exercise 1 Tutorial 1 < click
Using VERTICALLY AND CROSSWISE you do
Suppose you need 8 x 7
The answer is 56.
You subtract crosswise 8-3 or 7 - 2 to get 5,
That's all you do:
See how far the numbers are below 10, subtract
7 x 6 = 42
Here there is a carry: the 1 in the
1) 8 x 8 =
2) 9 x 7 =
3) 8 x 9 =
4) 7 x 7 =
5) 9 x 9 =
6) 6 x 6 =
Here's how to use VERTICALLY AND CROSSWISE
Suppose you want to multiply 88 by 98.
Both 88 and 98 are close to 100.
You can imagine the sum set out like this:
As before the 86 comes from subtracting crosswise:
This is so easy it is just mental arithmetic.
1) 87 x 98 =
2) 88 x 97 =
3) 77 x 98 =
4) 93 x 96 =
5) 94 x 92 =
6) 64 x 99 =
7) 98 x 97 =
Answers to Exercise 2 Tutorial 2 < click
Multiplying numbers just over 100.
103 x 104 = 10712
Similarly 107 x 106 = 11342
Again, just for mental arithmetic
Try a few:
1) 102 x 107 =
2) 106 x 103 =
3) 104 x 104 =
4) 109 x 108 =
5) 101 x123 =
6) 103 x102 =
Answers to exercise 3 Tutorial 2 < click
The easy way to add and subtract fractions.
Use VERTICALLY AND CROSSWISE
Multiply crosswise and add to get the top of the answer:
Subtracting is just as easy: multiply
Try a few:
Answers to Exercise 1 Tutorial 3 < click
A quick way to square numbers that end in 5 using
752 = 5625
75² means 75 x 75.
Similarly 852 = 7225 because 8 x 9 = 72.
1) 452 =
2) 652 =
3) 952 =
4) 352 =
5) 152 =
Answers to Exercise 1 Tutorial 4 < click
Method for multiplying numbers where the
32 x 38 = 1216
Both numbers here start with 3 and the last figures (2 and 8) add up to 10.
So we just multiply 3 by 4 (the next number up)
And we multiply the last figures: 2 x 8 = 16 to
And 81 x 89 = 7209
We put 09 since we need two figures as in all the other examples.
1) 43 x 47 =
2) 24 x 26 =
3) 62 x 68 =
4) 17 x 13 =
5) 59 x 51 =
6) 77 x 73 =
An elegant way of multiplying numbers using a simple pattern
21 x 23 = 483
This is normally called long multiplication but actually
There are 3 steps:
a) Multiply vertically on the left: 2 x 2 = 4.
And thats all there is to it.
Similarly 61 x 31 = 1891
6 x 3 = 18; 6 x 1 + 1 x 3 = 9; 1 x 1 = 1
Try these, just write down the answer:
1) 14 x 21
2) 22 x 31
3) 21 x 31
4) 21 x 22
5) 32 x 21
Multiply any 2-figure numbers together by mere mental arithmetic!
If you want 21 stamps at 26 pence each you can
easily find the total price in your head.
There were no carries in the method given above.,/p>
However, there only involve one small extra step.
21 x 26 = 546
The method is the same as above
So 21 stamps cost £5.46.
Practise a few:
1) 21 x 47
2) 23 x 43
3) 32 x 53
4) 42 x 32
5) 71 x 72
33 x 44 = 1452
There may be more than one carry in a sum:
Vertically on the left we get 12.
Then vertically on the right we get 12 and the 1
6) 32 x 56
7) 32 x 54
8) 31 x 72
9) 44 x 53
10) 54 x 64
Any two numbers, no matter how big, can be
multiplied in one line by this method.
Multiplying a number by 11.
To multiply any 2-figure number by 11 we just put
26 x 11 = 286
Notice that the outer figures in 286 are the 26
And the middle figure is just 2 and 6 added up.
So 72 x 11 = 792
Multiply by 11:
1) 43 =
2) 81 =
3) 15 =
4) 44 =
5) 11 =
77 x 11 = 847
This involves a carry figure because 7 + 7 = 14
Multiply by 11:
1) 11 x 88 =
2) 11 x 84 =
3) 11 x 48 =
4) 11 x 73 =
5) 11 x 56 =
234 x 11 = 2574
We put the 2 and the 4 at the ends.
We add the first pair 2 + 3 = 5.
and we add the last pair: 3 + 4 = 7.
Multiply by 11:
1) 151 =
2) 527 =
3) 333 =
4) 714 =
5) 909 =
Method for dividing by 9.
23 / 9 = 2 remainder 5
The first figure of 23 is 2, and this is the answer.
The remainder is just 2 and 3 added up!
43 / 9 = 4 remainder 7
The first figure 4 is the answer
and 4 + 3 = 7 is the remainder - could it be easier?
Divide by 9:
1) 61 / 9 = remainder
2) 33 / 9 = remainder
3) 44 / 9 = remainder
4) 53 / 9 = remainder
5) 80 / 9 = remainder
134 / 9 = 14 remainder 8
The answer consists of 1,4 and 8.
1 is just the first figure of 134.
4 is the total of the first two figures 1+ 3 = 4,
and 8 is the total of all three figures 1+ 3 + 4 = 8.
Divide by 9:
6) 232 = remainder
7) 151 = remainder
8) 303 = remainder
9) 212 = remainder
10) 2121 = remainder
842 / 9 = 812 remainder 14 = 92 remainder 14
Actually a remainder of 9 or more is not usually
Since the remainder, 14 has one more 9 with 5
Divide these by 9:
1) 771 / 9 = remainder
2) 942 / 9 = remainder
3) 565 / 9 = remainder
4) 555 / 9 = remainder
5) 2382 / 9 = remainder
6) 7070 / 9 = remainder
Return to Exercise 1 Tutorial 1
Return to Exercise 1 Tutorial 2
Return to Exercise 2 Tutorial 2
Return to Exercise 3 Tutorial 2
Return to Exercise 1 Tutorial 3
Return to Exercise 1 Tutorial 4
Return to Exercise 2 Tutorial 4
Return to Exercise 1 Tutorial 5
Return to Exercise 2a Tutorial 5
Return to Exercise 2b Tutorial 5
Return to Exercise 1 Tutorial 6
Return to Exercise 2 Tutorial 6
Return to Exercise 3 Tutorial 6
1) 6 r 7
Return to Exercise 1a Tutorial 7
1) 25 r 7
Return to Exercise 1b Tutorial 7
1) 714 r15 = 84 r15 = 85 r6
Return to Exercise 2 Tutorial 7
copyright to the
This little math trick will show you whether
So, this is how it works.
Let's look at 1234
Does 4 divide evenly into 1234?
For 4 to
divide into any number we have
If it is an odd number, there is no way it will go in evenly.
Now we know that for 4 to divide evenly into any
Back to the question...
4 into 1234, the solution:
Take the last number and add it to 2 times the second last number
If 4 goes evenly into this number then you know
Therefore 4 into 1234 does not go in completely.
Lets try 4 into 3436546
So, from our example, take the last number, 6 and add it to
6 + (2 X 4) = 14
Let's try one more.
4 into 212334436
6 + (2 X 3) = 12
Therefore 4 goes into 234436 evenly.
So what use is this trick to you?
So how does the 12's shortcut work?
12 X 7
The first thing is to always multiply the 1 of the twelve by the
Multiply this 7 by 10 giving 70. (Why? We are working with BASES here.
Now multiply the 7 by the 2 of twelve giving 14. Add this to 70 giving 84.
Therefore 7 X 12 = 84
Let's try another:
17 X 12
Remember, multiply the 17 by the 1 in 12 and multiply by 10
1 X 17 = 17, multiplied by 10 giving 170.
Multiply 17 by 2 giving 34.
Add 34 to 170 giving 204.
So 17 X 12 = 204
lets go one more
24 X 12
Multiply 24 X 1 = 24. Multiply by 10 giving 240.
24 X 12 = 288 (these are Seriously Simple Sums to do arent they?!)
In this section you will learn how to convert Kilos to Pounds, and Vice Versa.
Lets start off with looking at converting Kilos to pounds.
Step one, multiply the kilos by TWO.
86 x 2 = 172
Step two, divide the answer by ten.
172 / 10 = 17.2
Step three, add step twos answer to step ones answer.
172 + 17.2 = 189.2
86 Kilos = 189.2 pounds
50 Kilos to pounds:
Step one, multiply the kilos by TWO.
50 x 2 = 100
Step two, divide the answer by ten.
100/10 = 10
Step three, add step two's answer to step one's answer.
100 + 10 = 110
50 Kilos = 110 pounds
Here is a nice simple way to add hours and minutes together:
Let's add 1 hr and 35 minutes and 3 hr 55 minutes together.
What you do is this:
Now you want to add these two numbers together:
So we now have a sub total of 490.
What you need to do to this and all sub totals is
No matter what the hours and minutes are,
490 + 40 = 530
So we can now see our answer is 5 hrs and 30 minutes!
This is a shortcut to convert Fahrenheit to Celsius and vice versa.
you will get will not be an exact one, but it will
Fahrenheit to Celsius:
Take 30 away from the Fahrenheit, and then divide the answer by two.
So 74 Fahrenheit = 22 Celsius.
Celsius to Fahrenheit just do the reverse:
Double it, and then add 30.
30 Celsius double it, is 60, then add 30 is 90
30 Celsius = 90 Fahrenheit
Remember, the answer is not exact but it gives you a rough idea.
With a little practice, it's not hard to recall
First, there are 3 you should know already:
1/2 = .5
Starting with the thirds, of which you already know one:
1/3 = .333...
You also know 2 of the 4ths, as well, so there's only one new one to learn:
1/4 = .25
Fifths are very easy. Take the numerator (the number on top),
1/5 = .2
There are only two new decimal equivalents to learn with the 6ths:
1/6 = .1666...
What about 7ths? We'll come back to them
8ths aren't that hard to learn, as they're just
1/8 = .125
9ths are almost too easy:
1/9 = .111...
10ths are very easy, as well.
1/10 = .1
Remember how easy 9ths were? 11th are easy in a similar way,
1/11 = .090909...
As long as you can remember the pattern for each fraction, it is
Oh, I almost forgot! We haven't done 7ths yet, have we?
One-seventh is an interesting number:
1/7 = .142857142857142857...
For now, just think of one-seventh as: .142857
See if you notice any pattern in the 7ths:
1/7 = .142857...
Notice that the 6 digits in the 7ths ALWAYS stay in the same
If you know your multiples of 14 up to 6, it isn't difficult to,
For 1/7, think "1 * 14", giving us .14 as the starting point.
For 4/14, 5/14 and 6/14, you'll have to adjust upward by 1:
For 4/7, think "(4 * 14) + 1", giving us .57 as the starting point.
Practice these, and you'll have the decimal equivalents of
If you want to demonstrate this skill to other people, and you know
If they give you 96 divided by 7, for example, you can think,
This is a useful method for when travelling between imperial
The formula to convert kilometres to miles is number of (kilometres / 8 ) X 5
So lets try 80 kilometres into miles
80/8 = 10
multiplied by 5 is 50 miles!
40 / 8 = 5
5 X 5= 25 miles
Master Multiplication tables, division and lots more!
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By Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaj (1884-1960)
Book ref: ISBN 0 8426 0967 9 Published by Motilal Banarasidas
From the Introduction by Smti Manjula Trivedi 16-03-1965. An extract:
Revered Guruji used to say that he had reconstructed the sixteen mathematical formulae from the Atharvaveda after assiduous research and Tapas (austerity) for about eight years in the forests surrounding Sringeri. Obviously these formulae are not to be found in the present recensions of Atharvaveda. They were actually reconstructed, on the basis of intuitive revelation, from materials scattered here and there in the Atharvaveda.
From the Preface by the author Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaj
We may however, at this point draw the earnest attention of every one concerned to the following salient items thereof:
The Sutras (aphorisms) apply to and cover each and every part of each and every chapter of each and every branch of mathematics (including Arithmetic, Algebra, Geometry plane and solid, Trigonometry plane and spherical, Conics geometrical and analytical, Astronomy, Calculus differential and integral etc.) In fact, there is no part of mathematics, pure or applied, that is beyond their jurisdiction.
The Sutras are easy to understand, easy to apply and easy to remember, and the whole work can be truthfully summarised in one word Mental!
Even as regards complex problems involving a good number of mathematical operations (consecutively or even simultaneously to be performed), the time taken by the Vedic method will be a third, a fourth, a tenth, or even a much smaller fraction of the time required according to modern (i.e. current) Western methods.
And in some very important and striking cases, sums requiring 30, 50, 100 or even more numerous and cumbrous steps of working (according to the current Western methods) can be answered in a single and simple step of work by the Vedic method! And little children (of only 10 or 12 years of age) merely look at the sums written on the blackboard and immediately shout out and dictate the answers. And this is because, as a matter of fact, each digit automatically yields its predecessor and its successor! And the children have merely to go on tossing off (or reeling off) the digits one after another (forwards or backwards) by mere mental arithmetic (without needing pen or pencil, paper, slate etc.).
On seeing this kind of work actually being performed by the little children, the doctors, professors and other big-guns of mathematics are wonder-struck and exclaim: Is this mathematics or magic? And we invariably answer and say: It is both. It is magic until you understand it; and it is mathematics thereafter. And then we proceed to substantiate and prove the correctness of this reply of ours!
As regards the time required by the students for mastering the whole course of Vedic Mathematics as applied to all its branches, we need merely state from our actual experience that 8 months (or 12 months) at an average rate of 2 or 3 hours per day should suffice for completing the whole course of mathematical studies on these Vedic lines instead of 15 or 20 years required according to the existing systems of the Indian and also of foreign universities.
And we were agreeably astonished and intensely gratified to find that exceedingly tough mathematical problems (which the mathematically most advanced present day Western scientific world had spent huge amount of time, energy, and money on and which even now it solves with the utmost difficulty and that also after vast labour involving large numbers of difficult, tedious and cumbersome steps of working) can be easily and readily solved with the help of these ultra-easy Vedic Sutras (or mathematical aphorisms) contained in the Parisista (the appendix portion) of the Atharvaveda in a few simple steps and by methods that can be conscientiously described as mere mental arithmetic.
It is thus in the fitness of things that the Vedas include 1. Ayurveda (anatomy, physiology, hygiene, sanitary science, medical science, surgery etc.), not for the purpose of achieving perfect health and strength in the after-death future but in order to attain them here and now in our present physical bodies. 2.Dhanurveda (archery and other military sciences), not for fighting with one another after our transportation to heaven but in order to quell and subdue all invaders from abroad and all insurgents from within. 3. Gandharva Veda (the science of art and music) and 4. Sthapatya Veda (engineering, architecture etc. and all branches of mathematics in general). All these subjects, be it noted, are inherent parts of the Vedas i.e., are reckoned as spiritual studies and catered for as such therein.
Similar is the case with Vedangas (i.e., grammar, prosody, astronomy, lexicography etc.) which according to the Indian cultural conceptions, are also inherent parts and subjects of Vedic (i.e. religious) study.
From the Foreward by Swami Pratyagatmananda Saraswati Varanasi, 22-03-1965
Vedic Mathematics by the late Shankaracharya (Bharati Krsna Tirtha) of Govardhan Pitha is a monumental work. In his deep-layer explorations of cryptic Vedic mysteries relating especially to their calculus of shorthand formulae and their neat and ready application to practical problems, the late Shankaracharya shows the rare combination of the probing insight of revealing intuition of a Yogi with the analytic acumen and synthetic talent of a mathematician.
With the late Shankaracharya we belong to a race, now fast becoming extinct, of diehard believers who think that the Vedas represent an inexhaustible mine of profoundest wisdom in matters of both spiritual and temporal; and that this store of wisdom was not, as regards its assets of fundamental validity and value at least, gathered by the laborious inductive and deductive methods of ordinary systemic enquiry, but was direct gift of revelation to seers and sages who in their higher reaches of Yogic realisation were competent to receive it from a source, perfect and immaculate.
Whether or not the Vedas are believed as repositories of perfect wisdom, it is unquestionable that the Vedic race lived not as merely pastoral folk possessing a half or a quarter developed culture and civilisation. The Vedic seers were, again, not mere navel-gazers or nose-tip gazers. They proved themselves adepts in all levels and branches of knowledge, theoretical and practical. For example, they had their varied objective science both pure and applied.
Let us take a concrete illustration. Suppose in a time of drought we require rains by artificial means. The modern scientist has his own theory and art (technique) for producing the result. The old seer scientist had his both also, but different from these now availing. He had his science and technique, called Yajna, in which Mantra, Yantra, and other factors must co-operate with mathematical determinateness and precision. For this purpose, he had developed the six auxiliaries of the Vedas in each of which mathematical skill and adroitness, occult or otherwise, play the decisive role. The Sutras lay down the shortest and surest lines. The correct intonation of the Mantra, the correct configuration of the Yantra (in the making of the Vedi etc., e.g. the quadrate of a circle), the correct time or astral conjunction factor, the correct rhythams etc. All had to be perfected so as to produce the desired results effectively and adequately. Each of these required the calculus of mathematics. The modern technician has his logarithmic tables and mechanics manuals. The old Yajnik had his Sutras.
Pages from the history of the Indian sub-continent:
In all early civilizations, the first expression of mathematical understanding appears in the form of counting systems. Numbers in vey early societies were typically represented by groups of lines, though later different numbers came to be assigned specific numeral names and symbols (as in India) or were designated by alphabetic letters (such as in Rome). Although today, we take our decimal system for granted, not all ancient civilizations based their numbers on a ten-base system. In ancient Babylon, a sexagesimal (base 60) system was in use.
The Decimal System in Harappa
In India a decimal system was already in place during the Harappan period, as indicated by an analysis of Harappan weights and measures. Weights corresponding to ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500 have been identified, as have scales with decimal divisions. A particularly notable characteristic of Harappan weights and measures is their remarkable accuracy. A bronze rod marked in units of 0.367 inches points to the degree of precision demanded in those times. Such scales were particularly important in ensuring proper implementation of town planning rules that required roads of fixed widths to run at right angles to each other, for drains to be constructed of precise measurements, and for homes to be constructed according to specified guidelines. The existence of a gradated system of accurately marked weights points to the development of trade and commerce in Harappan society.
Mathematical Activity in the Vedic Period
In the Vedic period, records of mathematical activity are mostly to be found in Vedic texts associated with ritual activities. However, as in many other early agricultural civilizations, the study of arithmetic and geometry was also impelled by secular considerations. Thus, to some extent early mathematical developments in India mirrored the developments in Egypt, Babylon and China . The system of land grants and agricultural tax assessments required accurate measurement of cultivated areas. As land was redistributed or consolidated, problems of mensuration came up that required solutions. In order to ensure that all cultivators had equivalent amounts of irrigated and non-irrigated lands and tracts of equivalent fertility - individual farmers in a village often had their holdings broken up in several parcels to ensure fairness. Since plots could not all be of the same shape - local administrators were required to convert rectangular plots or triangular plots to squares of equivalent sizes and so on. Tax assessments were based on fixed proportions of annual or seasonal crop incomes, but could be adjusted upwards or downwards based on a variety of factors. This meant that an understanding of geometry and arithmetic was virtually essential for revenue administrators. Mathematics was thus brought into the service of both the secular and the ritual domains.
Arithmetic operations (Ganit) such as addition, subtraction, multiplication, fractions, squares, cubes and roots are enumerated in the Narad Vishnu Purana attributed to Ved Vyas (pre-1000 BC). Examples of geometric knowledge (rekha-ganit) are to be found in the Sulva-Sutras of Baudhayana (800 BC) and Apasthmaba (600 BC) which describe techniques for the construction of ritual altars in use during the Vedic era. It is likely that these texts tapped geometric knowledge that may have been acquired much earlier, possibly in the Harappan period. Baudhayana's Sutra displays an understanding of basic geometric shapes and techniques of converting one geometric shape (such as a rectangle) to another of equivalent (or multiple, or fractional) area (such as a square). While some of the formulations are approximations, others are accurate and reveal a certain degree of practical ingenuity as well as some theoretical understanding of basic geometric principles. Modern methods of multiplication and addition probably emerged from the techniques described in the Sulva-Sutras.
Pythagoras - the Greek mathematician and philosopher who lived in the 6th C B.C was familiar with the Upanishads and learnt his basic geometry from the Sulva Sutras. An early statement of what is commonly known as the Pythagoras theorem is to be found in Baudhayana's Sutra: The chord which is stretched across the diagonal of a square produces an area of double the size. A similar observation pertaining to oblongs is also noted. His Sutra also contains geometric solutions of a linear equation in a single unknown. Examples of quadratic equations also appear. Apasthamba's sutra (an expansion of Baudhayana's with several original contributions) provides a value for the square root of 2 that is accurate to the fifth decimal place. Apasthamba also looked at the problems of squaring a circle, dividing a segment into seven equal parts, and a solution to the general linear equation. Jain texts from the 6th C BC such as the Surya Pragyapti describe ellipses.
Modern-day commentators are divided on how some of the results were generated. Some believe that these results came about through hit and trial - as rules of thumb, or as generalizations of observed examples. Others believe that once the scientific method came to be formalized in the Nyaya-Sutras - proofs for such results must have been provided, but these have either been lost or destroyed, or else were transmitted orally through the Gurukul system, and only the final results were tabulated in the texts. In any case, the study of Ganit i.e mathematics was given considerable importance in the Vedic period. The Vedang Jyotish (1000 BC) includes the statement: "Just as the feathers of a peacock and the jewel-stone of a snake are placed at the highest point of the body (at the forehead), similarly, the position of Ganit is the highest amongst all branches of the Vedas and the Shastras."
(Many centuries later, Jain mathematician from Mysore, Mahaviracharya further emphasized the importance of mathematics: "Whatever object exists in this moving and non-moving world, cannot be understood without the base of Ganit (i.e. mathematics)".)
Panini and Formal Scientific Notation
A particularly important development in the history of Indian science that was to have a profound impact on all mathematical treatises that followed was the pioneering work by Panini (6th C BC) in the field of Sanskrit grammar and linguistics. Besides expounding a comprehensive and scientific theory of phonetics, phonology and morphology, Panini provided formal production rules and definitions describing Sanskrit grammar in his treatise called Asthadhyayi. Basic elements such as vowels and consonants, parts of speech such as nouns and verbs were placed in classes. The construction of compound words and sentences was elaborated through ordered rules operating on underlying structures in a manner similar to formal language theory.
Today, Panini's constructions can also be seen as comparable to modern definitions of a mathematical function. G G Joseph, in The crest of the peacock argues that the algebraic nature of Indian mathematics arises as a consequence of the structure of the Sanskrit language. Ingerman in his paper titled Panini-Backus form finds Panini's notation to be equivalent in its power to that of Backus - inventor of the Backus Normal Form used to describe the syntax of modern computer languages. Thus Panini's work provided an example of a scientific notational model that could have propelled later mathematicians to use abstract notations in characterizing algebraic equations and presenting algebraic theorems and results in a scientific format.
Philosophy and Mathematics
Philosophical doctrines also had a profound influence on the development of mathematical concepts and formulations. Like the Upanishadic world view, space and time were considered limitless in Jain cosmology. This led to a deep interest in very large numbers and definitions of infinite numbers. Infinite numbers were created through recursive formulae, as in the Anuyoga Dwara Sutra. Jain mathematicians recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite. Permutations and combinations are listed in the Bhagvati Sutras (3rd C BC) and Sathananga Sutra (2nd C BC).
Jain set theory probably arose in parallel with the Syadvada system of Jain epistemology in which reality was described in terms of pairs of truth conditions and state changes. The Anuyoga Dwara Sutra demonstrates an understanding of the law of indeces and uses it to develop the notion of logarithms. Terms like Ardh Aached , Trik Aached, and Chatur Aached are used to denote log base 2, log base 3 and log base 4 respectively. In Satkhandagama various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers. The operations are repeated to produce new sets. In other works the relation of the number of combinations to the coefficients occurring in the binomial expansion is noted.
Since Jain epistemology allowed for a degree of indeterminacy in describing reality, it probably helped in grappling with indeterminate equations and finding numerical approximations to irrational numbers.
Buddhist literature also demonstrates an awareness of indeterminate and infinite numbers. Buddhist mathematics was classified either as Garna (Simple Mathematics) or Sankhyan (Higher Mathematics). Numbers were deemed to be of three types: Sankheya (countable), Asankheya (uncountable) and Anant (infinite).
Philosophical formulations concerning Shunya - i.e. emptiness or the void may have facilitated in the introduction of the concept of zero. While the zero (bindu) as an empty place holder in the place-value numeral system appears much earlier, algebraic definitions of the zero and it's relationship to mathematical functions appear in the mathematical treatises of Brahmagupta in the 7th C AD. Although scholars are divided about how early the symbol for zero came to be used in numeric notation in India, (Ifrah arguing that the use of zero is already implied in Aryabhatta) tangible evidence for the use of the zero begins to proliferate towards the end of the Gupta period. Between the 7th C and the 11th C, Indian numerals developed into their modern form, and along with the symbols denoting various mathematical functions (such as plus, minus, square root etc) eventually became the foundation stones of modern mathematical notation.
The Indian Numeral System
Although the Chinese were also using a decimal based counting system, the Chinese lacked a formal notational system that had the abstraction and elegance of the Indian notational system, and it was the Indian notational system that reached the Western world through the Arabs and has now been accepted as universal. Several factors contributed to this development whose significance is perhaps best stated by French mathematician, Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. It's simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions."
Brilliant as it was, this invention was no accident. In the Western world, the cumbersome roman numeral system posed as a major obstacle, and in China the pictorial script posed as a hindrance. But in India, almost everything was in place to favor such a development. There was already a long and established history in the use of decimal numbers, and philosophical and cosmological constructs encouraged a creative and expansive approach to number theory. Panini's studies in linguistic theory and formal language and the powerful role of symbolism and representational abstraction in art and architecture may have also provided an impetus, as might have the rationalist doctrines and the exacting epistemology of the Nyaya Sutras, and the innovative abstractions of the Syadavada and Buddhist schools of learning.
Influence of Trade and Commerce, Importance of Astronomy
The growth of trade and commerce, particularly lending and borrowing demanded an understanding of both simple and compound interest which probably stimulated the interest in arithmetic and geometric series. Brahmagupta's description of negative numbers as debts and positive numbers as fortunes points to a link between trade and mathematical study. Knowledge of astronomy - particularly knowledge of the tides and the stars was of great import to trading communities who crossed oceans or deserts at night. This is borne out by numerous references in the Jataka tales and several other folk-tales. The young person who wished to embark on a commercial venture was inevitably required to first gain some grounding in astronomy. This led to a proliferation of teachers of astronomy, who in turn received training at universities such as at Kusumpura (Bihar) or Ujjain (Central India) or at smaller local colleges or Gurukuls. This also led to the exchange of texts on astronomy and mathematics amongst scholars and the transmission of knowledge from one part of India to another. Virtually every Indian state produced great mathematicians who wrote commentaries on the works of other mathematicians (who may have lived and worked in a different part of India many centuries earlier). Sanskrit served as the common medium of scientific communication.
The science of astronomy was also spurred by the need to have accurate calendars and a better understanding of climate and rainfall patterns for timely sowing and choice of crops. At the same time, religion and astrology also played a role in creating an interest in astronomy and a negative fallout of this irrational influence was the rejection of scientific theories that were far ahead of their time. One of the greatest scientists of the Gupta period - Aryabhatta (born in 476 AD, Kusumpura, Bihar) provided a systematic treatment of the position of the planets in space. He correctly posited the axial rotation of the earth, and inferred correctly that the orbits of the planets were ellipses. He also correctly deduced that the moon and the planets shined by reflected sunlight and provided a valid explanation for the solar and lunar eclipses rejecting the superstitions and mythical belief systems surrounding the phenomenon. Although Bhaskar I (born Saurashtra, 6th C, and follower of the Asmaka school of science, Nizamabad, Andhra ) recognized his genius and the tremendous value of his scientific contributions, some later astronomers continued to believe in a static earth and rejected his rational explanations of the eclipses. But in spite of such setbacks, Aryabhatta had a profound influence on the astronomers and mathematicians who followed him, particularly on those from the Asmaka school.
Mathematics played a vital role in Aryabhatta's revolutionary understanding of the solar system. His calculations on pi, the circumferance of the earth (62832 miles) and the length of the solar year (within about 13 minutes of the modern calculation) were remarkably close approximations. In making such calculations, Aryabhatta had to solve several mathematical problems that had not been addressed before, including problems in algebra (beej-ganit) and trigonometry (trikonmiti).
Bhaskar I continued where Aryabhatta left off, and discussed in further detail topics such as the longitudes of the planets; conjunctions of the planets with each other and with bright stars; risings and settings of the planets; and the lunar crescent. Again, these studies required still more advanced mathematics and Bhaskar I expanded on the trigonometric equations provided by Aryabhatta, and like Aryabhatta correctly assessed pi to be an irrational number. Amongst his most important contributions was his formula for calculating the sine function which was 99% accurate. He also did pioneering work on indeterminate equations and considered for the first time quadrilaterals with all the four sides unequal and none of the opposite sides parallel.
Another important astronomer/mathematician was Varahamira (6th C, Ujjain) who compiled previously written texts on astronomy and made important additions to Aryabhatta's trigonometric formulas. His works on permutations and combinations complemented what had been previously achieved by Jain mathematicians and provided a method of calculation of nCr that closely resembles the much more recent Pascal's Triangle. In the 7th century, Brahmagupta did important work in enumerating the basic principles of algebra. In addition to listing the algebraic properties of zero, he also listed the algebraic properties of negative numbers. His work on solutions to quadratic indeterminate equations anticipated the work of Euler and Lagrange.
Emergence of Calculus
In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was obliged to introduce the concept of infinitesimals - i.e. tatkalika gati to designate the infinitesimal, or near instantaneous motion of the moon, and express it in the form of a basic differential equation. Aryabhatta's equations were elaborated on by Manjula (10th C) and Bhaskaracharya (12th C) who derived the differential of the sine function. Later mathematicians used their intuitive understanding of integration in deriving the areas of curved surfaces and the volumes enclosed by them.
Applied Mathematics, Solutions to Practical Problems
Developments also took place in applied mathematics such as in creation of trigonometric tables and measurement units. Yativrsabha's work Tiloyapannatti (6th C) gives various units for measuring distances and time and also describes the system of infinite time measures.
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 chapters relating to the study of the sphere and it's 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, Syrian, 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 (10th C, Damascus, author of The book of Chapters in Indian Arithmetic), 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 (956 AD) 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 became more readily available in India.
The Kerala School
Although it appears that original work in mathematics ceased in much of Northern India after the Islamic conquests, Benaras survived as a center for mathematical study, and an important school of mathematics blossomed in Kerala. Madhava (14th C, Kochi) made important mathematical discoveries that would not be identified by European mathematicians till at least two centuries later. His series expansion of the cos and sine functions anticipated Newton by almost three centuries. Historians of mathematics, Rajagopal, Rangachari and Joseph considered his contributions instrumental in taking mathematics to the next stage, that of modern classical analysis. Nilkantha (15th C, Tirur, Kerala) extended and elaborated upon the results of Madhava while Jyesthadeva (16th C, Kerala) provided detailed proofs of the theorems and derivations of the rules contained in the works of Madhava and Nilkantha. It is also notable that Jyesthadeva's Yuktibhasa which contained commentaries on Nilkantha's Tantrasamgraha included elaborations on planetary theory later adopted by Tycho Brahe, and mathematics that anticipated work by later Europeans. Chitrabhanu (16th C, Kerala) gave integer solutions to twenty-one types of systems of two algebraic equations, using both algebraic and geometric methods in developing his results. Important discoveries by the Kerala mathematicians included the Newton-Gauss interpolation formula, the formula for the sum of an infinite series, and a series notation for pi. Charles Whish (1835, published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland) was one of the first Westerners to recognize that the Kerala school had anticipated by almost 300 years many European developments in the field.
Yet, few modern compendiums on the history of mathematics have paid adequate attention to the often pioneering and revolutionary contributions of Indian mathematicians. But as this essay amply demonstrates, a significant body of mathematical works were produced in the Indian subcontinent. The science of mathematics played a pivotal role not only in the industrial revolution but in the scientific developments that have occurred since. No other branch of science is complete without mathematics. Not only did India provide the financial capital for the industrial revolution (see the essay on colonization) India also provided vital elements of the scientific foundation without which humanity could not have entered this modern age of science and high technology.
Mathematics and Music: Pingala (3rd C AD), author of Chandasutra explored the relationship between combinatorics and musical theory anticipating Mersenne (1588-1648) author of a classic on musical theory.
Mathematics and Architecture: Interest in arithmetic and geometric series may have also been stimulated by (and influenced) Indian architectural designs - (as in temple shikaras, gopurams and corbelled temple ceilings). Of course, the relationship between geometry and architectural decoration was developed to it's greatest heights by Central Asian, Persian, Turkish, Arab and Indian architects in a variety of monuments commissioned by the Islamic rulers.
Transmission of the Indian Numeral System: Evidence for the transmission of the Indian Numeral System to the West is provided by Joseph (Crest of the Peacock):
Quotes Severus Sebokht (662) in a Syriac text describing the "subtle discoveries" of Indian astronomers as being "more ingenious than those of the Greeks and the Babylonians" and "their valuable methods of computation which surpass description" and then goes on to mention the use of nine numerals.
· Quotes from Liber abaci (Book of the Abacus) by Fibonacci (1170-1250): The nine Indian numerals are ...with these nine and with the sign 0 which in Arabic is sifr, any desired number can be written. (Fibonaci learnt about Indian numerals from his Arab teachers in North Africa)
Influence of the Kerala School: Joseph (Crest of the Peacock) suggests that Indian mathematical manuscripts may have been brought to Europe by Jesuit priests such as Matteo Ricci who spent two years in Kochi (Cochin) after being ordained in Goa in 1580. Kochi is only 70km from Thrissur (Trichur) which was then the largest repository of astronomical documents. Whish and Hyne - two European mathematicians obtained their copies of works by the Kerala mathematicians from Thrissur, and it is not inconceivable that Jesuit monks may have also taken copies to Pisa (where Galileo, Cavalieri and Wallis spent time), or Padau (where James Gregory studied) or Paris (where Mersenne who was in touch with Fermat and Pascal, acted as an agent for the transmission of mathematical ideas).
1.Studies in the History of Science in India (Anthology edited by Debiprasad Chattopadhyaya)
2.A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin: Studies in the history of mathematics, "Nauka" (Moscow, 1974), 220-222; 302.
3. B Datta: The science of the Sulba (Calcutta, 1932).
4.G G Joseph: The crest of the peacock (Princeton University Press, 2000).
5. R P Kulkarni: The value of pi known to Sulbasutrakaras, Indian Journal Hist. Sci. 13 (1) (1978), 32-41.
6. G Kumari: Some significant results of algebra of pre-Aryabhata era, Math. Ed. (Siwan) 14 (1) (1980), B5-B13.
7. G Ifrah: A universal history of numbers: From prehistory to the invention of the computer (London, 1998).
8. P Z Ingerman: 'Panini-Backus form', Communications of the ACM 10 (3)(1967), 137.
9.P Jha: Contributions of the Jainas to astronomy and mathematics, Math. Ed. (Siwan) 18 (3) (1984), 98-107.
9b. R C Gupta: The first unenumerable number in Jaina mathematics, Ganita Bharati 14 (1-4) (1992), 11-24.
10. L C Jain: System theory in Jaina school of mathematics, Indian J. Hist. Sci. 14 (1) (1979), 31-65.
11. L C Jain and Km Meena Jain: System theory in Jaina school of mathematics. II, Indian J. Hist. Sci. 24 (3) (1989), 163-180
12. K Shankar Shukla: Bhaskara I, Bhaskara I and his works II. Maha-Bhaskariya (Sanskrit) (Lucknow, 1960).
13. K Shankar Shukla: Bhaskara I, Bhaskara I and his works III. Laghu-Bhaskariya (Sanskrit) (Lucknow, 1963).
14. K S Shukla: Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya, Ganita 22 (1) (1971), 115-130.
15. R C Gupta: Varahamihira's calculation of nCr and the discovery of Pascal's triangle, Ganita Bharati 14 (1-4) (1992), 45-49.
16. B Datta: On Mahavira's solution of rational triangles and quadrilaterals, Bull. Calcutta Math. Soc. 20 (1932), 267-294.
17. B S Jain: On the Ganita-Sara-Samgraha of Mahavira (c. 850 A.D.), Indian J. Hist. Sci. 12 (1) (1977), 17-32.
18. K Shankar Shukla: The Patiganita of Sridharacarya (Lucknow, 1959).
19. H. Suter: Mathematiker
20. Suter: Die Mathematiker und Astronomen der Araber
21. Die philosophischen Abhandlungen des al-Kindi, Munster, 1897
22. K V Sarma: A History of the Kerala School of Hindu Astronomy (Hoshiarpur, 1972).
23. R C Gupta: The Madhava-Gregory series, Math. Education 7 (1973), B67-B70
24. S Parameswaran: Madhavan, the father of analysis, Ganita-Bharati 18 (1-4) (1996), 67-70.
25. K V Sarma, and S Hariharan: Yuktibhasa of Jyesthadeva : a book of rationales in Indian mathematics and astronomy - an analytical appraisal, Indian J. Hist. Sci. 26 (2) (1991), 185-207
26. C T Rajagopal and M S Rangachari: On an untapped source of medieval Keralese mathematics, Arch. History Exact Sci. 18 (1978), 89-102.
27. C T Rajagopal and M S Rangachari: On medieval Keralese mathematics, Arch. History Exact Sci. 35 (1986), 91-99.
28. A.K. Bag: Mathematics in Ancient and Medieval India (1979, Varanasi)
29. Bose, Sen, Subarayappa: Concise History of Science in India, (Indian National Science Academy)
30. T.A. Saraswati: Geometry in Ancient and Medieval India (1979, Delhi)
31.N. Singh: Foundations of Logic in Ancient India, Linguistics and Mathematics ( Science and technology in Indian Culture, ed. A Rahman, 1984, New Delhi, National Instt. of Science, Technology and Development Studies, NISTAD)
32. P. Singh: "The so-called Fibonacci numbers in ancient and medieval India, (Historia Mathematica, 12, 229-44, 1985)
33. Chin Keh-Mu: India and China: Scientific Exchange (History of Science in India Vol 2.)
Another view on Indian Mathematics:
Dr. David Gray writes:
"The study of mathematics in the West has long been characterized by a certain ethnocentric bias, a bias which most often manifests not in explicit racism, but in a tendency toward undermining or eliding the real contributions made by non-Western civilizations. The debt owed by the West to other civilizations, and to India in particular, go back to the earliest epoch of the "Western" scientific tradition, the age of the classical Greeks, and continued up until the dawn of the modern era, the renaissance, when Europe was awakening from its dark ages."
Dr Gray goes on to list some of the most important developments in the history of mathematics that took place in India, summarizing the contributions of luminaries such as Aryabhatta, Brahmagupta, Mahavira, Bhaskara and Maadhava. He concludes by asserting that "the role played by India in the development (of the scientific revolution in Europe) is no mere footnote, easily and inconsequentially swept under the rug of Eurocentric bias. To do so is to distort history, and to deny India one of its greatest contributions to world civilization."
The following essays can be found at http://india_resource.tripod.com/indianhistory.html
Development of Philosophical Thought and Scientific Method in Ancient india
Philosophical Development from Upanishadic Theism to Scientific Realism
History of the Physical Sciences in India
1. Math and Ethnocentrism
The study of mathematics in the West has long been characterized by a certain ethnocentric bias, a bias which most often manifests not in explicit racism, but in a tendency toward undermining or eliding the real contributions made by non-Western civilizations. The debt owed by the West to other civilizations, and to India in particular, go back to the earliest epoch of the "Western" scientific tradition, the age of the classical Greeks, and continued up until the dawn of the modern era, the renaissance, when Europe was awakening from its dark ages. This awakening was in part made possible by the rediscovery of mathematics and other sciences and technologies through the medium of the Arabs, who transmitted to Europe both their own lost heritage as well as the advanced mathematical traditions formulated in India.
George Ghevarughese Joseph, in an important article entitled "Foundations of Eurocentrism in Mathematics," argued that "the standard treatment of the history of non-European mathematics is a product of historiographical bias (conscious or otherwise) in the selection and interpretation of facts, which, as a consequence, results in ignoring, devaluing or distorting contributions arising outside European mathematical traditions." (1987:14)
Due to the legacy of colonialism, the exploitation of which was ideologically justified through a doctrine of racial superiority, the contributions of non-European civilizations were often ignored, or, as Joseph argued, even distorted, in that they were often misattributed as European, i.e. Greek, contributions, and when their contributions were so great as to resist such treatment, they were typically devalued, considered inferior or irrelevant to Western mathematical traditions.
This tendency has not only led to the devaluation of non-Western mathematical traditions, but has distorted the history of Western mathematics as well. In so far as the contributions from non-Western civilizations are ignored, there is the problem of accounting for the development of mathematics purely within the Western cultural framework. This has led, as Sabetai Unguru has argued, toward a tendency to read more advanced mathematical concepts into the relatively simplistic geometrical formulations of Greek mathematicians such as Euclid, despite the fact that the Greeks lacked not only mathematic notation, but even the place-value system of enumeration, without which advanced mathematical calculation is impossible. Such ethnocentric revisionist history resulted in the attribution of more advanced algebraic concepts, which were actually introduced to Europe over a millennium later by the Arabs, to the Greeks. And while the contributions of the Greeks to mathematics was quite significant, the tendency of some math historians to jump from the Greeks to renaissance Europe results not only in an ethnocentric history, but an inadequate history as well, one which fails to take into account the full history of the development of modern mathematics, which is by no means a purely European development.
2. Vedic Altars and the "Pythagorean theorem"
A perfect example of this sort of misattribution involves the so-called Pythagorean theorem, the well-known theorem which was attributed to Pythagoras who lived around 500 BCE, but which was first proven in Greek sources in Euclid's Geometry, written centuries later. Despite the scarcity of evidence backing this attribution, it is not often questioned, perhaps due to the mantra-like frequency with which it is repeated. However, Seidenberg, in his 1978 article, shows that the thesis that Greece was the origin of geometric algebra was incorrect, "for geometric algebra existed in India before the classical period in Greece." (1978:323) It is now generally understood that the so-called "Pythagorean theorem" was understood in ancient India, and was in fact proved in Baudhayana's Shulva Sutra, a text dated to circa 600 BCE. (1978:323).
Knowledge of mathematics, and geometry in particular, was necessary for the precise construction of the complex Vedic altars, and mathematics was thus one of the topics covered in the brahmanas. This knowledge was further elaborated in the kalpa sutras, which gave more detailed instructions concerning Vedic ritual. Several of these treat the topic of altar construction. The oldest and most complete of these is the previously mentioned Shulva Sutra of Baudhaayana. As this text was composed about a century before Pythagoras, the theory that the Greeks were the source of Geometric algebra is untenable, while the hypothesis that India was have been a source for Greek geometry, transmitted via the Persians who traded both with the Greeks and the Indians, looks increasingly plausible. On the other hand, it is quite possible that both the Greeks and the Indians developed geometry. Seidenberg has argued, in fact, that both seem to have developed geometry out of the practical problems involving their construction of elaborate sacrificial altars. (See Seidenberg 1962 and 1983
3. Zero and the Place Value System
Far more important to the development of modern mathematics than either Greek or Indian geometry was the development of the place value system of enumeration, the base ten system of calculation which uses nine numerals and zero to represent numbers ranging from the most minuscule decimal to the most inconceivably large power of ten. This system of enumeration was not developed by the Greeks, whose largest unit of enumeration was the myriad (10,000) or in China, where 10,000 was also the largest unit of enumeration until recent times. Nor was it developed by the Arabs, despite the fact that this numeral system is commonly called the Arabic numerals in Europe, where this system was first introduced by the Arabs in the thirteenth century.
Rather, this system was invented in India, where it evidently was of quite ancient origin. The Yajurveda Samhitaa, one of the Vedic texts predating Euclid and the Greek mathematicians by at least a millennium, lists names for each of the units of ten up to 10 to the twelfth power (paraardha). (Subbarayappa 1970:49) Later Buddhist and Jain authors extended this list as high as the fifty-third power, far exceeding their Greek contemporaries, who lacking a system of enumeration were unable to develop abstract mathematical concepts.
The place value system of enumeration is in fact built into the Sanskrit language, where each power of ten is given a distinct name. Not only are the units ten, hundred and thousand (daza, zata, sahasra) named as in English, but also ten thousand, hundred thousand, ten million, hundred million (ayuta, lakSa, koti, vyarbuda), and so forth up to the fifty-third power, providing distinct names where English makes use of auxillary bases such as thousand, million, etc. Ifrah has commented that:
By giving each power of ten an individual name, the Sanskrit system gave no special importance to any number. Thus the Sanskrit system is obviously superior to that of the Arabs (for whom the thousand was the limit), or the Greeks and Chinese (whose limit was ten thousand) and even to our own system (where the names thousand, million etc. continue to act as auxillary bases). Instead of naming the numbers in groups of three, four or eight orders of units, the Indians, from a very early date, expressed them taking the powers of ten and the names of the first nine units individually. In other words, to express a given number, one only had to place the name indicating the order of units between the name of the order of units immediately immediately below it and the one immediately above it. That is exactly what is required in order to gain a precise idea of the place-value system, the rule being presented in a natural way and thus appearing self-explanatory. To put it plainly, the Sanskrit numeral system contained the very key to the discovery of the place-value system. (2000:429)
As Ifrah has shown at length, there is little doubt that our place-value numeral system developed in India (2000:399-409), and this system is embedded in the Sanskrit language, several aspects of which make it a very logical language, well suited to scientific and mathematical reasoning. Nor did this system exhaust Indian ingenuity; as van Nooten has shown, Pingala, who lived circa the first century BCE, developed a system of binary enumeration convertible to decimal numerals, described in his Chandahzaastra. His system is quite similar to that of Leibniz, who lived roughly fourteen hundred years later. (See Van Nooten)
India is also the locus of another closely related an equally important mathematical discovery, the numeral zero. The oldest known text to use zero is a Jain text entitled the Lokavibhaaga, which has been definitely dated to Monday 25 August 458 CE. (Ifrah 2000:417-1 9) This concept, combined by the place-value system of enumeration, became the basis for a classical era renaissance in Indian mathematics.
The Indian numeral system and its place value, decimal system of enumeration came to the attention of the Arabs in the seventh or eighth century, and served as the basis for the well known advancement in Arab mathematics, represented by figures such as al-Khwarizmi. It reached Europe in the twelfth century when Adelard of Bath translated al-Khwarizmi's works into Latin. (Subbarayappa 1970:49) But the Europeans were at first resistant to this system, being attached to the far less logical roman numeral system, but their eventual adoption of this system led to the scientific revolution that began to sweep Europe beginning in the thirteenth century.
4. Luminaries of Classical Indian Mathematics
The world did not have to wait for the Europeans to awake from their long intellectual slumber to see the development of advanced mathematical techniques. India achieved its own scientific renaissance of sorts during its classical era, beginning roughly one thousand years before the European Renaissance. Probably the most celebrated Indian mathematicians belonging to this period was Aaryabhat.a, who was born in 476 CE.
In 499, when he was only 23 years old, Aaryabhat.a wrote his Aaryabhat.iiya, a text covering both astronomy and mathematics. With regard to the former, the text is notable for its for its awareness of the relativity of motion. (See Kak p. 16) This awareness led to the astonishing suggestion that it is the Earth that rotates the Sun. He argued for the diurnal rotation of the earth, as an alternate theory to the rotation of the fixed stars and sun around the earth (Pingree 1981:18). He made this suggestion approximately one thousand years before Copernicus, evidently independently, reached the same conclusion.
With regard to mathematics, one of Aaryabhat.a's greatest contributions was the calculation of sine tables, which no doubt was of great use for his astronomical calculations. In developing a way to calculate the sine of curves, rather than the cruder method of calculating chords devised by the Greeks, he thus went beyond geometry and contributed to the development of trigonometry, a development which did not occur in Europe until roughly one thousand years later, when the Europeans translated Indian influenced Arab mathematical texts.
Aaryabhat.a's mathematics was far ranging, as the topics he covered include geometry, algebra, trigonometry. He also developed methods of solving quadratic and indeterminate equations using fractions. He calculated pi to four decimal places, i.e., 3.1416. (Pingree 1981:57) In addition, Aaryabhat.a "invented a unique method of recording numbers which required perfect understanding of zero and the place-value system." (Ifrah 2000:419)
Given the astounding range of advanced mathematical concepts and techniques covered in this fifth century text, it should be of no surprise that it became extremely well known in India, judging by the large numbers of commentaries written upon it. It was studied by the Arabs in the eighth century following their conquest of Sind, and translated into Arabic, whence it influenced the development of both Arabic and European mathematical traditions.
Born in 598 CE in Rajastan in Western India, Brahmagupta founded an influential school of mathematics which rivaled Aaryabhat.a's. His best known work is the Brahmasphuta Siddhanta, written in 628 CE, in which he developed a solution for a certain type of second order indeterminate equation. This text was translated into Arabic in the eighth century, and became very influential in Arab mathematics. (See Kak p. 16)
Mahaaviira was a Jain mathematician who lived in the ninth century, who wrote on a wide range of mathematical topics. These include the mathematics of zero, squares, cubes, square-roots, cube-roots, and the series extending beyond these. He also wrote on plane and solid geometry, as well as problems relating to the casting of shadows. (Pingree 1981:60)
Bhaaskara was one of the many outstanding mathematicians hailing from South India. Born in 1114 CE in Karnataka, he composed a four-part text entitled the Siddhanta Ziromani. Included in this compilation is the Biijagan.ita, which became the standard algebra textbook in Sanskrit. It contains descriptions of advanced mathematical techniques involving both positive and negative integers as well as zero, irrational numbers. It treats at length the "pulverizer" (kut.t.akaara) method of solving indeterminate equations with continued fractions, as well as the so-called "Pell's equation (vargaprakr.ti) dealing with indeterminate equations of the second degree. He also wrote on the solution to numerous kinds of linear and quadratic equations, including those involving multiple unknowns, and equations involving the product of different unknowns. (Pingree 1981, p. 64)
In short, he wrote a highly sophisticated mathematical text that proceeded by several centuries the development of such techniques in Europe, although it would be better to term this a rediscovery, since much of the Renaissance advances of mathematics in Europe was based upon the discovery of Arab mathematical texts, which were in turn highly influenced by these Indian traditions.
The Kerala region of South India was home to a very important school of mathematics. The best known member of this school Maadhava (c. 1444-1545), who lived in Sangamagraama in Kerala. Primarily an astronomer, he made history in mathematics with his writings on trigonometry. He calculated the sine, cosine and arctangent of the circle, developing the world's first consistent system of trigonometry. (See Hayashi 1997:784-786) He also correctly calculated the value of p to eleven decimal places. (Pingree 1981:490)
This is by no means a complete list of influential Indian mathematicians or Indian contributions to mathematics, but rather a survey of the highlights of what is, judged by any fair, unbiased standard, an illustrious tradition, important both for its own internal elegance as well as its influence on the history of European mathematical traditions. The classical Indian mathematical renaissance was an important precursor to the European renaissance, and to ignore this fact is to fail to grasp the history of latter, a history which was truly multicultural, deriving its inspiration from a variety of cultural roots.
There are in fact, as Frits Staal has suggested in his important (1995) article, "The Sanskrit of Science", profound similarities between the social contexts of classical India and renaissance Europe. In both cases, important revolutions in scientific thought occurred in complex, hierarchical societies in which certain elite groups were granted freedom from manual labor, and thus the opportunity to dedicate themselves to intellectual pursuits. In the case of classical India, these groups included certain brahmins as well as the Buddhist and Jain monks, while in renaissance Europe they included both the monks as well as their secular derivatives, the university scholars.
Why, one might ask, did Europe take over thousand years to attain the level of abstract mathematics achieved by Indians such as Aaryabhat.a? The answer appears to be that Europeans were trapped in the relatively simplistic and concrete geometrical mathematics developed by the Greeks. It was not until they had, via the Arabs, received, assimilated and accepted the place-value system of enumeration developed in India that they were able to free their minds from the concrete and develop more abstract systems of thought. This development thus triggered the scientific and information technology revolutions which swept Europe and, later, the world. The role played by India in the development is no mere footnote, easily and inconsequentially swept under the rug of Eurocentric bias. To do so is to distort history, and to deny India one of it's greatest contributions to world civilization.
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