Vedic Science

100 years ago, Albert Einstein published three papers that rocked the world. These papers proved the existence of the atom, introduced the theory of relativity, and described quantum mechanics . Pretty good debut for a 26 year old scientist, huh? His equations for relativity indicated that the universe was expanding. This bothered him, because if it was expanding, it must have had a beginning and a beginner.

Since neither of these appealed to him, Einstein introduced a 'fudge factor' that ensured a 'steady state' universe,one that had no beginning or end. But in 1929, Edwin Hubble showed that the furthest galaxies were fleeing away from each other, just as the  Big Bang model predicted. So in 1931, Einstein embraced what would later be known as the Big Bang theory, saying, "This is the most beautiful and satisfactory explanation of creation to which I have ever listened." He referred to the 'fudge factor' to achieve a steady-state universe as the biggest blunder of his career.

As I'll explain during the next couple of days, Einstein's theories have been thoroughly proved and verified by experiments and measurements. But there's an even more important implication of Einstein's discovery.Not only does the universe have a beginning, but time itself, our own dimension of cause and effect, began with the Big Bang.

That's right -- time itself does not exist before then. The very line of time begins with that creation event. Matter, energy, time and space were created in an instant by an intelligence outside of space and time. About this intelligence, Albert Einstein wrote in his book "The World As I See It" that the harmony
of natural law "Reveals an intelligence of such superiority that, compared with it, all the systematic thinking and acting of human beings is an utterly insignificant reflection."*
He went on to write, "Everyone who is seriously involved in the pursuit of science becomes convinced that a spirit is manifest in the laws of the Universe-- a spirit vastly superior to that of man, and one in the face of which we with our modest powers must feel humble."* Pretty significant statement, wouldn't you say?Stay tuned for tomorrow's installment: "Bird Droppings on my Telescope."
Respectfully Submitted,Perry Marshall *Einstein quotes are from "Einstein and Religion: Physics and Theology" by Max Jammer

"SO WHAT'S SO SPECIAL ABOUT VEDIC MATHEMATICS?"

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.

The simplicity of the Vedic methods means that answers can be obtained in one line. Bharati Krsna Tirthaji, who rediscovered the system from the Vedic texts between 1911 and 1918, titled his book:"Vedic Mathematics or Sixteen Mathematical Formulae from the Vedas (For One-line Answers to all Mathematical Problems)". This means that the system is also a system of mental mathematics. Being a mental system students progress faster in terms of mental agility, capacity to hold ideas in the mind and to remember past impressions. They also develop flexibility in their methods of tackling problems and evolve their own strategies to handle situations not met before. All this also helps considerably in the study of other subjects, in personal growth and in everyday life.

The Vedic system is based on sixteen mathematical formulae. These formulae are given in word form, such as "Vertically and Crosswise" and "By One More than the One Before". They have a unifying effect: they describe principles or ways of using the mind and therefore help the student by giving direction to the mind. These formulae do not have to be learnt, they describe ways in which the mind operates and referring to them during study clarifies the work. As one Vedic Mathematics course participant once said "These formulae work the way my mind works". The mind is very subtle and without realising it we have all learnt various mental techniques- extending, combining, reversing and generalising ideas are simple examples of mental methods we use all the time and which are included in the Vedic formulae.
So the Vedic system is very special; but its real beauty and effectiveness 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.

SRI BHARATI KRSNA TIRTHAJI - More than a Mathematical Genius

Sri Bharati Krsna Tirthaji (hereafter "Bharati Krsna") lived from 1884 to 1960. He is said to have reconstructed the ancient system of Vedic Mathematics from certain Sanskrit texts which other scholars had dismissed as nonsense. He tells us that the Vedic system which he rediscovered is based on sixteen Sutras which cover all branches of mathematics, pure and applied. The methods he showed and the simple Sutras on which it is based are extraordinarily simple and easy and the whole system possesses a unity not found in conventional mathematical methods. It can hardly be doubted that Bharati Krsna's remarkable discoveries in mathematics will in time change the teaching of and approach to mathematics worldwide: but this was not his main interest in life. His life was devoted to helping those individuals he could and also helping to bring about world peace and spiritual renewal. This short article aims to show something of his character and life and is drawn mainly from an introduction by Mrs Manjula Trivedi in the book "Vedic Mathematics" by Bharati Krsna. She looked after Bharati Krsna in the last years of his life and afterwards took charge of the Foundation he set up in Nagpur in 1953- the Sri Vishwapunarniman Sangha.

According to Manjula Trivedi, Bharati Krsna "named as Venkatraman in his early days, was an exceptionally brilliant student and invariably won the first place in all subjects in all the classes throughout his educational career. . . at the age of just twenty he passed M.A. Examination in further seven subjects simultaneously securing the highest honours in all, which is perhaps the all-time record of academic brilliance. His subjects included Sanskrit, Philosophy, English, Mathematics, History and Science". In 1908 he was made first Principal of the newly started National College at Rajmahendri, a post he held for three years.

Having a "burning desire for spiritual knowledge, practice and attainment" he then spent many years at the most advanced studies with the Shankaracharya at Sringeri in Mysore and was given the name Bharati Krsna Tirtha when he was initiated into the order of Samnyasa at Benares in 1919. He later, in 1925, became a Shankaracharya (the highest religious title in India).He believed in the ancient Vedic tradition of all-round spiritual and cultural harmony, and his ambition for humanity was a world-wide cultural and spiritual renewal. People flocked to him in crowds and waited at his doors for hours. Granted an interview with him people felt that he immediately knew their need. Even when suffering from excessive strain he refused to take rest, continuing with his studies, talks, lectures and writings with unabated and youth like vigour and enthusiasm.

In 1958 Bharati Krsna went on a tour to America, addressing audiences in hundreds of colleges, universities, churches and other institutions. He also gave talks and mathematical demonstrations on television and gave some lectures in the UK on his way back to India, in May 1958.
Bharati Krsna wrote sixteen volumes on Vedic Mathematics, one on each Sutra, but the manuscripts were irretrievably lost. He said that he would rewrite them from memory but owing to ill-health and failing eyesight got no further than writing a book intended as an introduction to the sixteen volumes. That book "Vedic Mathematics" written with the aid of an amanuensis is currently available and is the only surviving work on mathematics by this most remarkable man.

THE VEDIC NUMERICAL CODE

In the Sanskrit language each consonant can be associated with a number. This
enables text to be translated into numbers and there are various advantages
in doing this. Some other languages have a similar method: Greek and Hebrew
for example.

There have been different forms of the Sanskrit code but Sri Bharati Krsna
Tirthaji, who reconstructed the ancient system of Vedic Mathematics, used a
particular form which he describes in his book. It seems this was
instrumental in deciphering the Sanskrit texts which were headed Ganita
Sutras (Mathematics) but which the western scholars of the late nineteenth century were unable to understand.
The code is as follows:

ka, Ta, pa and ya all denote 1;
kha, tha, pha and ra all represent 2;
ga, Da, ba, and la all stand for 3;
gha, dha, bha, and va all denote 4;
gna, Na, ma and sa all represent 5;
ca, TA and sha all stand for 6;
cha, tha, and Sa denote 7;
ja, DA and ha all represent 8;
jha and dha stand for 9; and
Ksa (or Ksudra) means Zero!

Those consonants above which are written with a capital letter, like TA,
are pronounced with the tongue initially in the cerebral position, i.e. the
tongue points up and then moves forward and down. In words with conjunct
consonants only the last consonant counts and vowels do not count at all.
Thus 'papa' is 11, 'mama' is 55 and 'mary' is 52 and so on.

This code was not used for concealment (it follows naturally from the
structure of the Sanskrit language) but to aid memory and to add historical
allusions, political reflections etc. to text. One striking example given by
Bharati Krsna is a hymn to the Lord Shri Krishna which translates into the
value of pi to 32 figures.

The hymn starts:

Gopibhagyamadhuvrata......

Here 'go' gives 3; pi gives 1; 'bha' gives 4; 'ya' gives 1; 'ma' gives 5;
'dhu' gives 9; 'ra' gives 2; TA gives 6.
Together we get 31415926, the first eight figures of pi (3.1415926...).

Thus by knowing the code and the hymn you know the value of pi to 32 figures!

THE VINCULUM AND OTHER DEVICES

The Vedic system uses a variety of methods for simplifying calculations. The vinculum is one of these as it allows us to remove some or all digits over five from a calculation so that only 0, 1, 2, 3, 4 and 5 are used.

The vinculum is a horizontal line written over a digit which thereby makes it negative. So 19 could be written as twenty minus one, i.e. 2(-1), where we write (-1) instead one with a bar over it (as we cannot write that in an email). This is quite natural as most people would, for example, add 19 by adding 20 and taking one away.

Similarly 38 would be written as 4(-2), that is 4 followed by a 2 with a bar over it, 98 becomes 10(-2), one hundred take away two, and 283 = 3(-2)3, three hundred, minus twenty, add three.

So we can have numbers which are partly positive and partly negative.

This increases flexibility as it means we have a choice whether or not to use the vinculum and in fact we can convert any number into a variety of vinculum forms using the vinculum device. So 46 for example could also be written as 5(-4), or as 1(-5)(-4). We use the form that is appropriate depending on what we want to do with the number.

In this way 0 and 1, which are particularly easy to work with, occur twice as often. This also means that in a calculation where the vinculum is used the positive and negative numbers tend to cancel each other out so that we can work with smaller numbers, and in fact, because of the flexibility the vinculum allows, we can choose to use forms of numbers that maximise this cancelling. So we have more control over a calculation: if you are finding a sine or cosine for example you can arrange to keep any carry figures to a minimum.

The use of the vinculum is optional though: all the marvellous Vedic methods can be carried out without using the it. But it can also simplify things a lot and so it is certainly worth experimenting with and getting familiar with working with a mixture of positive and negative digits, or even just using it occasionally when it is obviously an advantage.

We can also write numbers in different forms without using the vinculum. 46 for example can be written as 3/16 where the "1" is written as a subscript (and the oblique line is not written), so 3/16 means 30+16. And we can combine this with the vinculum to get yet more forms of the same number.

In division too answers can be expressed in a variety of ways. If you divide 20 by 7 you get 2 remainder 6. But 3 remainder (-1) is also a valid answer and so is 1 remainder 13 and so on.

This element of choice in the Vedic system leads to flexibility and creativity and so is very useful nowadays when these qualities are particularly important for developing the young mind.

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The article below was written by Andrew Nicholas at the request of the Chairman of SEAL (The Society for Effective Affective Learning) and was published in the July issue of their journal. This directly relates to Vedic Mathematics in that Vedic Mathematics presents mathematics in its simplest form.

IS KNOWLEDGE ESSENTIALLY SIMPLE?

Whenever I have understood something I have always found it to be simple. Others tell me that is what they have found, too. This leads me to think that perhaps knowledge IS simple, essentially. I'm talking about knowledge as the understanding of something, not just the gathering of facts.

There's another consideration which supports this idea. Major advances in science simplify things, and make them more readily comprehensible. Copernicus gave us a much simpler cosmological system than Ptolemy. He did so by putting the Sun at the centre of the universe, with the Earth and planets going round it in circles. This replaced Ptolemy's more complicated description of the motions of the heavenly bodies, placing the Earth at the centre of everything.

So if the most important advances in science are simplifications, what are the ultimate consequences of simplification following on simplification? Do you think it might be something fairly simple? That is what I am inclined to think. So here's a further reason for supposing that knowledge is essentially simple.

But what about these contexts one finds difficult, as I've been finding, recently, trying to set up a palm-top computer to send e-mail? However, as I,ve gone on it's become easier. Past experience has shown me that when I don't really understand something I can find it difficult, and then it gets easier as the understanding grows. Eventually I may even find it quite easy. At this point it seems I have connected with the underlying simplicity.

Is it possible that those subjects which are generally found to be difficult are awaiting THEIR 'Copernican revolutions' - to be followed, perhaps, by further simplifications? [I am assuming that the problem is not just the unnecessary use of jargon.]

So there you have it. I am suggesting that perhaps, when we really know something, it is found to be simple. Otherwise put, to know something is to connect with its essential simplicity, and to see its pattern as a whole.

But what is simplicity? I once heard Professor Karl Popper remark that simplicity is not itself a simple concept. That is not a view I take myself. However there's no doubt that there's a lot to simplicity.

Back to the suggestion that knowledge is essentially simple. There's a well-known phrase: it's easy when you know how. In many contexts this can be expressed as: it's simple once you understand it. Time and again people have told me that that is what they have found, but they have had to struggle and do a lot of work to get there.

We would be greatly helped if ways could be found of taking us quickly and easily to that understanding without having to go through the intervening struggle. THE IDEAL FORM OF STUDY WOULD BE ONE WHICH TAKES US DIRECTLY TO THE UNDERLYING SIMPLICITY.

Here is an example of what I have in mind. It's taken from my recent book on geometry, 'The Circle Revelation', which is written for the non-mathematician. It uses the basic principle, or fundamental theorem, that figures which are constructed in the same way are identical. This applies also to parts of figures. One consequence is that, if two sides of a triangle are equal, the angles facing them will be equal also, being constructed in the same way.

The aim underlying this book was to produce a system of geometry in which the methods are simple enough to be used in the context of an oral tradition, with no writing, just drawing figures and speaking. In consequence, the bulk of its contents can b presented to non-mathematicians as a short course, lasting little more than an our and a half, going from first principles to elementary properties of circles. This was not previously possible - at least, not in recorded history.

Geometry is built up step-by-step, revealing a steadily enlarging 'picture', into which the parts fit like pieces in a jigsaw puzzle. This system uses simple steps to enable the 'student' to appreciate larger and larger wholes - all connected by the golden thread of reason.

I have been asked what led me to formulate this system. I did so because I believed it to be possible and worth doing. The idea of finding such a system intrigued me. It was clearly not trivial.

What resulted was simple, but that was what I was working towards. In retrospect, I was convinced that there was an underlying simplicity, waiting to be found. It was like a guiding principle, drawing me on. Without that belief I would not have been able to formulate this system.

There are plenty of examples around of the consequences of following the belief that knowledge is complex. Had that been my belief I think I would have come to the same conclusion as Professor Popper, about simplicity not being simple itself.

However, there are also plenty of examples around illustrating connections with underlying simplicity, such as the wheel. If we valued brevity and directness and simplicity more there would be a lot more of them - and not just more wheels!

Dream awhile. Suppose it to be true that knowledge is essentially simple. Then the ideal follows, surely, of having studies formulated in this way - taking us swiftly and easily to the underlying simplicity at the heart of the whole subject. That would make life a lot easier and richer for us all.
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THE NINE POINT CIRCLE

Mark a point at the top of a circle and divide the circle into three equal parts by marking two more points on the edge. Similarly divide each of these three sections into three equal parts so that you have a circle marked with nine equally spaced points. Label the top point "9" and number the other points anticlockwise from 8 to 1. This figure is called the nine point circle (enneagram).

Our number system uses just nine digits and zero so this diagram represents number in general (the circle itself can be thought of as zero). All greater numbers are just combinations of these. This simple device has a great number of useful applications.

You can for example show each of the multiplication tables as a pattern on the circle. The multiples of 2 for example are 2,4,6,8,10,12,14,16,18,20 . . . Reducing these using digit sums we get 2,4,6,8,1,3,5,7,9,2 . . . So to get the pattern for the two times table we start at 2 on the nine point circle and draw a straight line to 4, then a line from 4 to 6 and so on until the pattern begins to repeat itself.

Any sequence of numbers can be reduced with digit sums and shown on the nine point circle: the sequence of square numbers, cube numbers, triangular numbers, the Fibonacci sequence and so on all have interesting patterns as do the digits in recurring decimal expansions of fractions. And then there is the sequence of prime numbers, the digits of pi etc.

By allocating a note of a specific pitch to each of the nine digits any pattern can be heard. Some nice tunes can be constructed in this way.

We can continue to number round the circle so that, as 10 follows 9, we write 10 next to the 1. Then we put 11 next to 2, 12 next to 3 and so on. We can continue indefinitely, going round and round the circle so that all whole numbers are represented. Next we note that the digit sum of 10 is 1 and 10 has been placed next to 1. Similarly the digit sum of 11 is 2 and 11 has been placed next to 2. And so on. This shows that all whole numbers can be basically related to one of the nine numbers and that every whole number can be reduced to its 'base number' by finding its digit sum. The number 1884, for example, has a digit sum of 3 and so will be located on the 3-branch of the nine point circle.

This also shows clearly that 1884 will have a remainder of 3 when divided by 9. And also that if the digit sum of a number is 9 then the number is ivisible by 9 and it is not divisible by 9 if the digit sum isnot 9.

Zero and the negative numbers can be included on the circle by counting backwards from 1: so zero goes at 9, -1 at 8 and so on. Where might 1.5 be placed on the circle? Perhaps half way between 1 and 2, which is opposite the 6 (and the digit sum of 1.5 is 6). And so on for other numbers with halves. Other fractions can also find their place in this system.

Possibilities multiply when you have two (or more) circles. Or when you use the space inside the circle as a second dimension.

If you are interested in the philosophical aspect of numbers the nine numbers can represent the steps in the creative process (on an individual or cosmic level) where one represents the creator and nine the full creation, with all the stages of the creative process in-between. "The science of numbers is a practical science which helps us in daily life and also helps us to understand the mysteries of the universe". These are the words of His Holiness Sankaracarya Santananda Saraswati in an interview in the 1970's, from which the following quotation is also taken. "The mathematical proposition which is given by philosophy is the collection of laws under which the universe works. The philosophical aspect of mathematics fits in perfectly well with the pattern of the universe." More of this interview, in which the qualities of the numbers are described in cosmic terms, can be found by clicking on the link in this paragraph in the copy of this newsletter on the www.vedicmaths.org web site (click on "Newsletter" on the home page).

Thus, though it is a very simple device, the nine point circle gives us new ways of looking at numbers and their relationships and meaning.

This issue's article:

This is written by Barbara Salmon who has been researching pattern in number and Vedic Mathematics for some years.

THE VEDIC TRIANGLE

The Vedic Triangle, contains the same numbers and number patterns as the Vedic Square (see Newsletter number 7), but laid out in the shape of a triangle. There are two main steps:-

Step 1 - create a multiplication triangle;
Step 2 - reduce the triangle to Digit Sums.

When treated as exercises, steps such as these offer a simple and interesting way to explore various aspects of number.

Step 1: CREATING A MULTIPLICATION TRIANGLE

We can create a multiplication triangle, by multiplying digits 123456789 x 987654321 (i.e. digits 1 to 9 as ascending and descending) using the 'Vertically and Crosswise' step sequence described in the original Vedic Mathematics Treatise (Chapter 3) by Sri Bharati Krsna Tirthaji. This gives us the product of every number by every other number, in the range 1 to 9. Following this step sequence automatically gives us the products, in the shape of a triangle.

Here is a small example of that particular step sequence, using the first two digits of the multiplier 12, and multiplicand 98:-

1 2 x The step sequence is:-
9 8 vertically 1 x 9 = 9;
------------ crosswise 1 x 8 = 8 then 9 x 2 = 18;
9: 8: 16 vertically 2 x 8 = 16.
18
-------------
1 1 7 6

( If we want the product of 12 and 98 we carry the 1 of 16 and 1 of 18 one column to the left to get 1176.)

9: 8: 16 The individual multiplication products now form the
18 the shape of a small triangle.

If we extend the multiplication from 12 x 98 to 123456789 x 987654321 we have a multiplication triangle as follows:-

1 2 3 4 5 6 7 8
9 8 7 6 5 4 3 2
--------------------------------------------------------
MULTIPLICATION TRIANGLE
--------------------------------------------------------
9: 8: 7: 6: 5: 4: 3: 2: 1: 2: 3: 4: 5: 6: 7: 8: 9
18:27:36:45: 54:63:72: 81:72:63:54: 45:36:27: 18
16:14:12:10: 8: 6: 4: 6: 8:10:12:14:16
24:32:40:48:56: 64:56:48:40:32:24
21:18:15:12: 9: 12:15:18:21
28:35:42:49:42:35:28
24:20:16:20:24
30:36:30
25

Viewed this way 'Vertically and Crosswise' is acting more like a 'modeling' tool for Number. The pattern of the Multiplicand and Multiplier together shape the pattern of the multiplication products.

Exercise: Create your own multiplication triangle - using squared paper (or lined paper overlain at right angles), or the 'table' option if you have a text processing package on your computer. Colour in all the products of the five times table using the same colour.

Step 2: REDUCE THE MULTIPLICATION TRIANGLE TO DIGITS SUMS

To reduce a number to its Digit Sum, simply add the digits of the original number together. Repeat the process if necessary until only one digit remains. For example, 48 is 4 + 8 = 12, which is 1 + 2 = 3. Usually 'Digit Sums' are used in arithmetic to cross-check results
e.g.: 359 x 257 = 92263
Digit Sum: 8 x 5 = 40 (reducing to 4 ) so the answer is probably correct. In this article, we are using Digit Sums to see how numbers interrelate.

VEDIC TRIANGLE

9: 8: 7: 6: 5: 4: 3: 2: 1: 2: 3: 4: 5: 6: 7: 8: 9:
9: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9:
7: 5: 3: 1: 8: 6: 4: 6: 8: 1: 3: 5: 7
6: 5: 4: 3: 2: 1: 2: 3: 4: 5: 6
3: 9: 6: 3: 9: 3: 6: 9: 3
1: 8: 6: 4: 6: 8: 1
6: 2: 7: 2: 6
3: 9: 3
7

Exercise: Copy out the triangle onto squared paper and colour in the numbers which correspond to the five times table products you highlighted in the Multiplication Triangle.

Are/were you able to see the following ? :-
* There are eight vertical pairs of numbers.
* One number (7) only appears once.
* Each vertical pair of numbers appears twice e.g.. 5 and 9.
* Each pair reduces to 5 e.g.. 5 + 9 = 14, which is 1 + 4 = 5.
* 7 is the Digit Sum for 5 squared i.e.. 25 which is 2 + 5 = 7.
* If we double 7 to make a pair, we get 14, which also reduces (1 + 4) to 5.
* If we add together all the pairs (including 2 x 7) we have a total of 45, which reduces (4 + 5) to 9.
* If we add together the digits of each pair of pairs e.g.. (1 + 4) + (1 + 4) they reduce to 1.
* Similarly if we double the 7:7 pair, again we have (7+7) + (7 + 7) = 28, which also reduces to one.
* If we add together all the 'pairs of pairs' we have 5 !

The recognition of pattern in number, as illustrated by the types of exercise described above, helps us to develop our mental agility, improvise solutions and think for ourselves.

You will see by visiting the Multiplication and Addition parts of the Creative Maths website, that 'Vertically and Crosswise' offers a broad range of multiplication styles (including the ones we commonly use today). The alternative styles were improvised by myself in March this year. Once the pattern was seen, - each style took a few minutes to develop. The only thing that is changing in each example, is the sequence of multiplication steps (which in all cases can be from left to right or right to left - or even a bit of both!) and the way we record the sums. The underlying principle is the same! This demonstrates great flexibility within what is undoubtedly a unified system - the system of Vedic Mathematics.

THE VEDIC SQUARE

The Vedic Square is a nine by nine square array of numbers formed by taking a multiplication table (up to nine times nine) and replacing each number by its digit sum. The digit sum is found by adding the digits in a number and adding again if necessary:
42 becomes 6 and
56 becomes 11 which becomes 2.
So the first row consists of 1,2,3,4,5,6,7,8,9 and the second row is
2,4,6,8,1,3,5,7,9 and so on.

Questions have been raised about this Square but it has not been easy to track down references. An Internet search revealed that it is used in Numerology and Astrology as well as for educational purposes. A reference to an article was also found: "Investigating the Vedic Square" by Nicola Woolf in a UK journal called Micro Math, Volume 11/3, autumn 1995.

There are some pages (119-125) on the Vedic Square in the book "Multicultural Mathematics" by David Nelson, George Gheverghese Joseph and Julian Williams (published by Oxford University Press, 1993). This describes various patterns that occur in the Square and in particular the patterns that are produced by joining the ones in the table (or the twos, threes etc.).

There is also a 2-page section in Book 3 of "The Cosmic Computer" course by Kenneth Williams and Mark Gaskell (published by Inspiration Books, 1998). This shows a method of drawing designs by choosing an angle of rotation (90 degrees when using square paper and 60 or 120 degrees when using triangular paper) and using a line in the Vedic Square to define a sequence of distances. So starting with the fourth row, say, of the Square (4,8,3,7,2,6,1,5,9) and using square paper you would choose a point on the paper and an initial direction and draw a line 4 units long. Then turn 90 degrees anticlockwise and go 8 units, then another 90 degrees and go 3 units, and so on. When reaching the end of a line in the Square you go back to the beginning and continue. A variety of very pleasing designs result from different choices of rotation angle and lines (rows, columns, diagonals etc.) in the Vedic Square.

Each of the ideas mentioned above could be developed further in a number of ways, leading to different or more advanced applications of the Vedic Square. So the Square is very useful for teaching purposes as many skills can be developed by using the square in different ways.

Many beautiful patterns in the Vedic Square have been uncovered by Barbara Salmon who has also found some marvellous relationships in the closely related Vedic Triangle. We hope to show something of this in a later Newsletter.

The origin of the Square still remains a mystery however. Dr George Joseph, an expert on the history of Indian Mathematics at Manchester University, is not aware of any Vedic origin and some references suggest the Square was used in the Muslim tradition.

"MATHS MANTRA"

The debate on whether Vedic mathematics should be accepted in mainstream academics as an annex to arithmetic, has been going on for a long time now. In fact the matter has become more contentious in the last couple of years as supporters of Vedic science have been organising workshops at schools to mobilise students towards the new mathematical pattern. The NCERT (National Centre for Education Research) however, has remained non-committal, expressing several doubts concerning its applicability.

Vedic mathematics in simple words is 'a one line formula to all arithmetical equations'. An authentic subject, it finds its origin in the Vedas. In the Atharvaveda, to be more specific, which is a field that deals with architecture and such technical sciences. However its original systematic programmer is Sri Jagadguru Shankaracharya from Govardhan Math who envisaged the arithmetical pattern. From the year 1965 when the books on Vedic mathematics were first published, the subject has been constantly besieged in controversies.

Motilal Banarsi Das, the pioneer publishers in the field of Indian culture and heritage have been staunch supporters of Vedic mathematics for a long time now. After starting modestly in 1903, the Indology publishers are now a brand name undisputedly attached to niche areas in pure sciences and pure social sciences like anthropology. Publisher and owner of Motilal Banarsi Das, R. P. Jain along with other supporters have been taking the lead to educate students in schools about Vedic mathematics, in Pune. Recently the publishers organised workshops in Vijay Vallabh School, MCA English School and Choksey School in the city area.

"Even though Vedic science originated in India, it is rather ironic that it is taught in most other countries, except India. In fact there is a separate centre for Vedic mathematics in Singapore. Considering this it is astonishing that Vedic mathematics is being dismissed so easily in India", says R. P. Jain, giving reasons for his publications long-standing support for Vedic science.

Jain also feels that the rejection has only come from scientists and mathematicians who have labelled the patterns as a 'bag of tricks' without higher applicability. He says the public response to the subject has been overwhelming with a record 20,000 copies on the subject being sold every year. "Its popularity can be compared to that of any best-selling book. The reason why we are supporting Vedic mathematics is because it has the potential to bring about a revolution in the field of maths. In Vedic mathematics there is no need to memorise tables. It does away with most of the complicated steps in an equation. Moreover it allows the brain to function in an objective manner, which helps one in making day-to-day decisions better. The pattern that we are using presently is linear and extremely difficult. The phobia that many experience towards this subject is partly due to that", explains Jain.

The most irksome factor according to Jain, is people rejecting the subject without knowing enough. The supporters of the science, as for now, would like to see the usage of Vedic mathematics as an appendix to the conventional pattern. Some experts agree upon it being an excellent method for cross-checking answers. It is also prescribed that students appearing for competitive exams study this field, as they stand to benefit immensely from Vedic mathematics.

Right now the publishers, going beyond the books, aim to promote the subject through regular workshops and seminars. "The demand should come more strongly from the public. Only then will the Government take it seriously" states Jain firmly. Anybody to volunteer for maths, at least now
VEDIC SOURCES OF 'VEDIC MATHEMATICS'

The above is the title of an article by Dr N. M. Kansara, Director of the
Akshardham Centre for Applied Research in Social Harmony in Akshardham,
Gandhingar, India.

In this article Dr Kansara examines the arguments that have been circulating
in India about the authenticity and validity of Bharati Krsna Tirthaji's
reconstruction of Vedic Mathematics. In particular he discusses the lost
volumes and the validity of the term 'Vedic' for Bharati Krsna's system. It
is good to see the negative arguments that have been put forward in the past
refuted in such a scholarly way.

Bharati Krsna says he wrote sixteen volumes on Vedic Mathematics and
according to Dr Kansara ...at the age of his 34th or 35th year and for the
next few years he was busy working on these Sutras, and he seems to have
definitely written, in school notebooks, all of his sixteen volumes treating
each of his sixteen Sutras in one independent volume ...

These notebooks were left with a devotee for a long period, Says Dr Kansara,
so that the house in which they were stored passed on to the devotee's son
who sold them to a German scholar for 80,000 rupees. Professor Vijaya Sane
tried over a long period to locate the German scholar but without success.

Regarding the validity of the term 'Vedic' as applied to Bharati Krsna's
system Dr Kansara says: "The VM Sutras contain some very common mathematical
terminology, which has so far been hardly examined from this point of view.
Dr Satyakama Varma has concluded in his research paper that though the term
employed in these Sutras cannot be claimed to be of the Vedic origin, yet
they are later synonyms of the equivalent original Vedic terms, that the
Vedic texts include much of the scientific and mathematical statements, which
can make a strong basis for such like Sutras, and that when the Jagad Guru
claims that he has adopted nothing but Vedic Mathematics, he is right in his
own way." "Thus, in point of the language the VM Sutras too are similar and
cannot be segregated as non-Vedic; they are as much Vedic as are the Srauta
Sutras and the Sulba Sutras so far as the point of their language is
concerned. And this is perhaps, because of their likelihood of being of the
yet untraced Sulba-sutra of the Atharvaveda. And it is in view of these
sutras being a part of the yet untraced Sulbasutra, and therefore belonging
to the Sthapatyaveda an upaveda, of the Atharvaveda, that we may regard them
as 'Vedic', which is general term denoting not merely the texts connected
with some Vedic Sakha, but not necessarily the Samhita and Brahmana only".

The word 'Veda' refers to actual Vedic texts, but its literal meaning is
'knowledge' and this latter is the meaning stressed by Bharati Krsna himself.
Dr Kansara says: "But says Dr V. S. Agrawala, this criticism loses all its
force if we inform ourselves of the definition of Veda given by BKTM himself
as quoted abobe. It is the whole essence of his assessment of Vedic tradition
that it is not to be approached from a factual standpoint, viz., that the
Vedas as traditionally accepted in India are the repository of all knowledge,
and hence they should be, and not what they are, in human possession. That
approach entirely turns the tables on all the critics, for the authorship of
Vedic Mathematics then need not be laboriously searched in the texts as
preserved from antiquity."
VEDIC MATRIX

I did this activity with a group of 13 year olds in a school in Massachusetts, United States. To generate the Vedic Matrix, I asked the students, each student was given a 9 by 9 multiplication matrix and asked to complete the table. If any of the products in cell was more than 9, students were to repeatedly add the digits until the sum was less than or equal to 9 and the result recorded in the corresponding cell on a separate matrix. For example, 8 x 7 = 56; 56 is greater than 9 so add 5 and 6 to get 11. Since 11 is greater than 9 add the digits, i.e., 1+1 = 2 so 2 is recorded in the cell. After generating the matrix, the students were challenged to find as many patterns as possible.

Discussion of the Patterns Identified
· In the 3rd row or column, 3 + 6 = 9 and in the 6th row 6 + 3 = 9.
· In any vertical or horizontal set of numbers, the sum of the first and last numbers is 9 (ignoring the last column and row). For example, in the second row they noticed that 2 +7 = 9; 4 +5 = 9; 6 + 3 = 9; and 8 + 1 = 9. These pairs of numbers can be written as ordered pairs: (1, 8), (2, 7), (3, 6), and (4, 5).
With the exception of the 9th row and column, the sum of the numbers in each column or row is 45 and that when you add the digits of the sum the result is 9 (e.g., 4 + 5 = 9). The sum of the numbers in the 9th row or column is 81 and the sum of the digit is 9.
· If you add the first and the last numbers in each row or column you get the following sequence of numbers: 10, 11, 12, 13, 14, 15, 16, 17, 18. When you add the digits you get 1, 2, 3, 4, 5, 6, 7, 8, and 9.
· The first four numbers generated in the 7th column and row, again ignoring the last number in the column and row, are odd numbers (i.e., 7, 5, 3, 1) while the next four numbers were even (i.e., 8, 6, 4).
· The numbers in the 1st row or column are the reverse of the numbers in the 8th row or column (without taking into account the 9th row and column). This is true for rows/columns 2 and 7; 3 and 6; 4 and 5.

The students were guided to arrive at the following relationships:
· The pair of numbers (1, 8), (2, 7), (3, 6), and (4, 5) from the matrix have some relationship with the nine times table. For example,
(1, 8): 1 + 8 = 9; 18 = 9 x 2, or 81 = 9 x 9
(2, 7) 2 + 7 = 9 27 = 9 x 3, or 72 = 9 x 8
(3, 6) 3 + 6 = 9 36 = 9 x 4, or 63 = 9 x 7
(4, 5) 4 + 5 = 9 45 = 9 x 5, or 54 = 9 x 6

There are twelve 3s and twelve 6s, twenty-one 9s, six 1s and six 8s, six 2s and six 7s, and six 4s and six 5s. Now let us examine some calculations using the above data.
(1, 8): (1 x 6) + (8 x 6) = 6 + 48 = 54 and 5 + 4 = 9
(2, 7): (2 x 6) + (7 x 6) = 12 + 42 = 54 and 5 + 4 = 9
(3, 7): (3 x 12) + (6 x 12) = 36 + 72 = 108 = 2(54) and 1 + 8 = 9
(4, 5): (4 x 6) + (5 x 6) = 24 + 30 = 54 and 5 + 4 = 9
For the 9s: 9 x 21 = 189 = 1+ 8 + 9 = 18; 1 + 8 = 9

One of the fascinating things about this activity is that opportunities exist for making numerous connections among different mathematics concepts at multiple grade levels.

Many students identified a variety of number patterns and their relationships. For example, some found that in the 3rd row, 3 + 6 = 9 and in the 6th row 6 + 3 = 9. Similarly, others noticed that in any vertical or horizontal set of numbers, the sum of the first and last numbers is 9 (ignoring the last column and row). For example, in the second row they noticed that 2 +7 = 9; 4 +5 = 9; 6 + 3 = 9; and 8 + 1 = 9. These pairs of numbers can be written as ordered pairs: (1, 8), (2, 7), (3, 6), and (4, 5).
Some students pointed out that the sum of the numbers in each column or row is 45 and that when you add the digits of the sum the result is 9 (e.g., 4 + 5 = 9). Others noted that the first four numbers generated in the 7th column and row, again ignoring the last number in the column and row, were odd numbers while the next four numbers were even. Figure 3 contains some samples of students' conclusions or observations.
One fundamental characteristic of the Vedic Matrix in terms of digit sums (Figure 2) is that if you do not count the 9, the 1 times row and the 8 times row are the reverse of each other. This is the same with the 2 times row and the 7 times row, 3 times row and the 6 times row, and the 4 times row and the 5 times row. Also, the appearance of the number 9 in many different forms in Vedic Matrix indicates a strong relationship between the matrix (Figure 2) and the nine times row. In the nine times row, the sum of the digits of each product is 9, explaining why the 9s column and row have all 9s. Another important connection is that the above mentioned pair of numbers from the matrix have some relationship with the nine times table. For example,
(1, 8): 1 + 8 = 9; 18 = 9 x 2, or 81 = 9 x 9
(2, 7) 2 + 7 = 9 27 = 9 x 3, or 72 = 9 x 8
(3, 6) 3 + 6 = 9 36 = 9 x 4, or 63 = 9 x 7
(4, 5) 4 + 5 = 9 45 = 9 x 5, or 54 = 9 x 6

Part II: Generating Shapes
An implied premise in the use of the Vedic Matrix mentioned earlier is that when connected, numbers form symmetrical shapes (Nelson, et al, 1993; Shan & Bailey, 1991). To investigate this premise, we asked our students to work in pairs. One student was responsible for connecting all the 1s, 2s, 3s, and 4s with a straight line while the other partner connected all the 5s, 6s, 7s, and 8s. To connect each number, the students placed a tracing paper on the final matrix (Figure 2) and marked off each number using a dot. Next, they connected all the dots with a straight line making sure that all the points are connected. We did a whole class demonstration on an overhead. After the demonstration, the students completed connecting the rest of the numbers and compared their shapes with their partners. Figure 4 shows the shapes of 1 and 8 when connected with a straight line.

Discussion of the Shapes
Before they could complete drawing all the shapes, many students noticed some important connections between each pair of numbers. For example, they found that the shapes of one and eight were reflections of each other. Similar observations were made about (2, 7), (3, 6), and (4, 5) as shown in Figure 5. Some of the students conjectured that two shapes are a reflection of each other provided that the sum of the numbers they represent equals 9. For instance, 1 and 8 are reflections of each other since 1 + 8 = 9. These observations and conjectures about reflective symmetry indicate that for each horizontal set of numbers, there is an identical vertical set of numbers and in each pair of numbers, one is the reverse of the other. In addition to the above observations, some of the students identified different geometric figures they found in their shapes such as triangles, rectangles, octagons, etc. We have included samples of students' responses as figure 6.

The more I look at the Vedic Matrix, the more patterns I find.

A DESCRIPTIVE PREFATORY NOTE ON THE ASTOUNDING WONDERS
OF ANCIENT INDIAN VEDIC MATHEMATICS

1. In the course of our discourses on manifold and multifarious subjects (spiritual, metaphysical, philosophical, psychic, psychological, ethical, educational, scientific, mathematical, historical, political, economic, social etc., etc., from time to time and from place to place during the last five decades and more, we have been repeatedly pointing out that the Vedas (the most ancient Indian scriptures, nay, the oldest "Religious" scriptures of the whole world) claim to deal with all branches of learning (spiritual and temporal) and to give the earnest seeker after knowledge all the requisite instructions and guidance in full detail and on scientifically- nay, mathematically- accurate lines in them all and so on.

2. The very word "Veda" has this derivational meaning, i.e. the fountain-head and illimitable store-house of all knowledge. This derivation, in effect, means, connotes and implies that the Vedas should contain within themselves all the knowledge needed by mankind relating not only to the so-called 'spiritual' (or other-worldly) matters but also to those usually described as purely "secular", "temporal", or "worldly"; and also to the means required by humanity as such for the achievement of all-round, complete and perfect success in all conceivable directions and that there can be no adjectival or restrictive epithet calculated (or tending) to limit that knowledge down in any sphere, any direction or any respect whatsoever.

3. In other words, it connotes and implies that our ancient Indian Vedic lore should be all-round complete and perfect and able to throw the fullest necessary light on all matters which any aspiring seeker after knowledge can possibly seek to be enlightened on.

4. It is thus in the fitness of things that the Vedas include (i) Ayurveda (anatomy, physiology, hygiene, sanitary science, medical science, surgery etc., 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; (ii) Dhanuveda (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; (iii) Gandharva Veda (the science and art of music) and (iv) 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.

5. Similar is the case with regard to the Vedangas (i.e. grammar, prosody, astronomy, lexicography etc., etc.,) which, according to the Indian cultural perceptions, are also inherent parts and subjects of Vedic (i.e. Religious) study.

6. As a direct and unshirkable consequence of this analytical and grammatical study of the real connotation and full implications of the word "Veda" and owing to various other historical causes of a personal character (into details of which we need not now enter), we have been from our very early childhood, most earnestly and actively striving to study the Vedas critically from this stand-point and to realise and prove to ourselves (and to others) the correctness (or otherwise) of the derivative meaning in question.

7. There were, too, certain personal historical reasons why in our quest for the discovering of all learning in all its departments, branches, sub-branches etc., in the Vedas, our gaze was riveted mainly on ethics, psychology and metaphysics on the one hand and on the "positive" sciences and especially mathematics on the other.

8. And the contemptuous or, at best patronising attitude adopted by some so-called Orientalists, Indologists, antiquarians, research-scholars etc., who condemned, or light-heartedly, nay; irresponsibly, frivolously and flippantly dismissed, several abstruse-looking and recondite parts of the Vedas as "sheer-nonsense"- or as "infant-humanity's prattle", and so on, merely added fuel to the fire (so to speak) and further confirmed and strengthened our resolute determination to unravel the too-long hidden mysteries of philosophy and science contained in India's Vedic lore, with the consequence that, after eight years of concentrated contemplation in forest-solitude, we were at long last able to recover the long lost keys which alone could unlock the portals thereof.

9. 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 lots of time, energy and money on and which even now it solves with the utmost difficulty and after vast labour and 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 Parishishta (the Appendix-portion) of the ATHARVAVEDA in a few simple steps and by methods which can be conscientiously described as mere "mental arithmetic".

10. Ever since (i.e. since several decades ago), we have been carrying on an incessant and strenuous campaign for the India-wide diffusion of all this scientific knowledge, by means of lectures, blackboard-demonstrations, regular classes and so on in schools, colleges, universities etc., all over the country and have been astounding our audiences everywhere with the wonder and marvels not to say, miracles of Indian Vedic Mathematics.

11. We were thus at last enabled to succeed in attracting the more than passing attention of the authorities of several Indian universities to this subject. And, in 1952, the Nagpur University not merely had a few lectures and blackboard-demonstrations given but also arranged for our holding regular classes in Vedic Mathematics (in the University's Convocation Hall) for the benefit of all in general and especially of the University and college professors of mathematics, physics etc.

12. And, consequently, the educationists and the cream of the English educated section of the people including the highest officials (e.g. the high-court judges, the ministers etc.,) and the general public as such were all highly impressed; nay, thrilled, wonder-struck and flabbergasted! and not only the newspapers but even the University's official reports described the tremendous sensation caused thereby in superlatively eulogistic terms; and the papers began to refer to us as " the Octogenarian Jagadguru Shankaracharya who had taken Nagpur by storm with his Vedic Mathematics", and so on!

1,2,3,4: PYTHAGORAS AND THE COSMOLOGY OF NUMBER

At the heart of vedic mathematics lies a principle that underscores most, if not all, of the ancient wisdom traditions, the conveying of knowledge through cryptic, highly compressed expressions, open to multiple levels of interpretation. A prime example of this is the teaching of the Greek mathematician and sage Pythagoras. According to his ancient biographers:

"In the Pythagorean school, knowledge was transmitted symbolically, through the use of cryptic statements and riddles, in which a small number of words was pregnant with multiple levels of interpretation. Students were required to find meaning in these enigmatic lessons, sometimes through questioning and dialogue, sometimes by meditating upon their many possible meanings." (1)

If this was true of Pythagorean teachings, it was even more significant in more ancient schools of knowledge; it was, after all, at these schools, in Egypt, Babylon, and elsewhere, that Pythagoras gained his knowledge. In the case of the Indian tradition, both in Vedic times and later in the Hindu and Buddhist periods, the term most commonly encountered for this kind of cryptic literature was the sutra or collection of sutras. While this is often translated as "aphorism," or "formula," the word comes from the Sanskrit root for "thread," a usage that persists in the modern word "suture." As doctors use sutures to sow us up after surgery, the ancient sutras tie together our knowledge and integrate our awareness. There is no better example than the teachings contained in Patanjali's Yoga Sutras whose terse expressions contain instructions for the development of higher states of consciousness. Similarly, all the principles of vedic mathematics are encapsulated in sixteen sutras, which, along with thirteen sub-sutras, provide the basis for all the operations described in "The Cosmic Computer" (2).
If vedic mathematics can be counted as part of vedic literature, its ultimate source is the Rg Veda. This is certainly not concise, consisting of over 10,000 verses, but, as His Holiness Maharishi Mahesh Yogi has explained, it has a unique structure in which the essence of the whole text is essentially contained in one highly compressed expression--its first word. "It is the purpose of all ciphers to invest a few signs with much meaning," Carlo Suarès tells us. "In the severity of its beginning, in its first chapter, in its first sequence of letter numbers, is the seed, and in the seed is the whole." (3)
Suarès is referring to the beginning of Genesis, in which the process of creation is described, using the symbolism of gematria, in which each letter is given a numerical value. (4) According to Maharishi, the Rg Veda also sets forth a cosmogony in its first word-- Agni, but using a purely linguistic symbolism based on the physiology of speech. The first letter, or sound, AAAAAA…, pronounced with the mouth and throat fully open, and thus with a fully open sound, represents the fullness of the unmanifest, unbounded Brahman. But the letter G, a full glottal stop, introduces the first boundary on the full openness of the sound AAAAAA…. As the wave value of a sub-atomic particle collapses onto a point value when observed, so the unity, or samhita, value of Brahman collapses onto a point and becomes the triadic value of rishi, devata and chhandas, observer, process of observing, and object of observation. From here the process of manifestation begins. As the full stream of manifestation emerges, it leads on to the fullness of creation, and this is represented by the syllable NI, the same name given to the leading tone in Indian music (Sa, Re, Ga, Me Pa, Dha, Ni.....). The details of the process, and the content of manifestation and evolution, are unfolded through the rest of the verses of Rg Veda and commented upon by the rest of Vedic literature, including vedic mathematics.
Unity, duality, diversity, wholeness. These are the mechanics of creation described in different symbolic formulations in different knowledge traditions. To find it in purely mathematical or numerical form we return to the Pythagorean tradition, and its most concise expression comes from his successor Plato. Considered the most Pythagorean of Platonic dialogues, the Timaeus begins with a question by Socrates: "One, two, three ? but where, my dear Timaeus, is the fourth of my guests of yesterday who were to entertain me today?" (5) Commentators usually ignore this statement, but, as we have seen, in ancient literature every expression is "pregnant with multiple levels of meaning." This is particularly true when dealing with numbers.

"He [Pythagoras] held that the ultimate substances of all things, material and immaterial, were numbers, which had two distinct and complimentary aspects. On the one hand, they had a spatial and dynamic existence, and, on the other, they were fundamental formulating principles which were purely abstract. Thus, for example, the monad was understood by the Pythagoreans both as the number one, which had physical properties that could be manipulated in nature, and as an idea, which embodied the original unity at the source of all creation." (6)

The fundamental formulating principles in the universe are those values of unity, duality, diversity and wholeness we have already encountered. In Pythagorean thought these principles are clearly expressed in the first four numbers. Furthermore, this symbolism can be interpreted in terms of the Quadrivium, the four Pythagorean mathematical disciplines: arithmetic, music, geometry and astronomy.
Arithmetic was seen as the study of the abstract essence of things. Thus each number had a cosmological, as well as mathematical, significance. The monad, manifest as the number one, denotes the primordial unity at the basis of creation. The transition from one to two, from the monad to the dyad, represents the first step in the process of creation--unity polarizing within itself becomes duality. Three, the triad, is the first true number. One contains the seed, and two introduces potential. Three brings number into being, causing the potential contained within the monad to manifest into its true expression, the world of plurality and multitude.
If one and two initiate creation, three and four complete the process. Therefore, the tetrad, four, represents completion. Everything in the universe, both natural and numerical, is completed in the progression from one to four as 1 + 2 + 3 + 4 = 10, which brings us to the decad, also known to the Pythagoreans as the tetraktys, and representing their most sacred symbol. The same sequence, from unity to multiplicity via duality and trinity, is expressed even more graphically in the simplest and most basic musical relationships, those expressed through the numbers 1,2,3,4. The simplest and most fundamental musical relationship is the octave, discovered by Pythagoras to be the 1:2 relationship, and by Joseph Saveur (1653-1716) many centuries later, to be the first relationship in the harmonic overtone series. The experience of the octave is of two notes that are the same and yet different, and these values, sameness and difference are the fundamental substances used by the Demiurge to create the World-Soul in the Timaeus. Further, the octave provides the boundary conditions within which the musical universes contained within scales are formed, the values of Do in Do, Re, Mi, Fa, So, La, Ti, Do. Of these intervals, the central ones are those found to be next in the overtone series, 2:3, known as the fifth and 3:4 known as the fourth. These values are found in the first four harmonics of the overtone series, first 1:2 (octave), then 2:3 (fifth) then 3:4 (fourth) recapitulating the octave at the next power of two. In four simple sounds the whole process of unity, duality, multiplicity and wholeness is presented to the awareness.
In subsequent centuries, the science of geometry was developed into a sacred form in which the same process is represented by the circle (unity), contrasted with the square (diversity), and reconciled in the squaring of the circle, in alchemical practice, and the development of the mandala in Eastern art and architecture. "The object of sacred geometry being to depict that fusion of opposites, the squared circle is therefore its first symbol. Temples and cosmological cities throughout antiquity were founded on its proportions." (7) For Pythagoras, the symbolism of wholeness (kosmos) and order (harmonia) extended beyond mathematical to astrological phenomena. A theoretical planet called the counter-earth was posited to bring the number of heavenly bodies in the Pythagorean firmament to ten, the perfect number, the number of the tetraktys. And over time, an association between planets and musical notes was developed and elaborated into the famous "music of the spheres," a beautiful image of the kosmos as a divine harmony.
Having seen its range of implications, it could almost be stated that the sequence 1,2,3,4 sums up, in a compressed symbolism, the whole range of Pythagoreanism. But if we delve deeper into Platonic thought, a further dimension is revealed. In one of his most potent allegories, known as the "Divided Line," Plato sets out his theories of ontology and epistemology, and again it is done in terms of the number four. In this analogy, Plato makes a distinction between the outer realm of the world, illuminated by the sun and the inner realm of the mind, illuminated by the Good. The Divided Line passage divides each of these realms into two further sections. Plato also deals with the state of mind in which the resultant four realms are apprehended, resulting in the following scheme:

Level Object Faculty Type of Knowledge

IV Forms dialectic transcendental cognition } internal
III mathematics thinking, scientific understanding } world
Etc. reasoning

--------------------------------------------------------------------------------

II physical sense common-sense belief } external
objects perception } world
I shadows illusory illusion (8)
perception

It can be seen from this scheme that within the subjective realm of the mind, Plato posits a level of knowledge higher than that which deals with mathematical objects through the processes of thinking and reasoning. This is the level of the forms and it is reached, Plato tells us, through the use of the "second phase" of the dialectic, a technique that, according to Jonathan Shear is similar to the practice of jñana yoga. (9) This again reflects the Pythagorean approach to mathematics, one that must, on some level at least, apply to vedic maths also:

"For Pythagoras, mathematics served as a bridge between the visible and invisible worlds. He pursued the discipline of mathematics not only as a way of understanding and manipulating nature, but also as a means of turning the mind away from the physical world, which he held to be transitory and unreal, and leading it to the contemplation of eternal and truly existing things that never vary. He taught his students that by focusing on the elements of mathematics, they could calm and purify the mind, and ultimately, through disciplined effort, experience true happiness." (10)

THE ABSOLUTE NUMBER

The Absolute Number, recently discovered and described by Maharishi Mahesh Yogi,fulfills the historical goals of mathematics by giving complete understanding of the entire range of orderliness and precision studied by mathematics. It provides a means of fathoming and perfectly quantifying the entire field of all the laws of nature governing the universe. Even more importantly, it provides a technology whereby everyone can live in full accord with these laws so that every aspect of life, individual and national, is lived mistake-free with complete coherence and harmony throughout the world.

----------

Maharishi’s Absolute Number is a central feature of his Vedic Mathematics. Its defining role is that it is a common basis and source for all aspects and concepts of modern mathematics.“The mathematical precision and order maintained in the universe and calculable through the utilization of mathematical structures and number systems that serve to express the precise theories of different disciplines of modern science can now be seen as having their common basis in the field of the Absolute Number—the unmanifest catalyst of all numbers, number systems, and mathematical structures that quietly initiates (from the unmanifest field of intelligence—the field of the Cosmic Mathematician) all number systems and mathematical structures and their expressions in the entire, ever-expanding universe.” (Maharishi Mahesh Yogi, 1996, p. 381)

Maharishi’s Absolute Number is, however, more than just another mathematical concept. “The most important characteristic of the Absolute Number in Vedic Mathematics,” Maharishi (1996) explains, “is that it is a meaningful living reality, not just a notion or concept, and therefore does not depend on the intellect” (p. 625). This means that Maharishi’s Absolute Number is a field that supports all aspects of modern, abstract mathematics but which is, at the same time, able to be directly experienced as the field of pure intelligence. The qualities of Maharishi’s Absolute Number of orderliness and mistake-free precision can spontaneously develop in each individual through the regular direct experience of this field.

In this way, Maharishi’s Absolute Number is both a timely answer to the weaknesses of modern mathematics described above and a natural completion of the knowledge initiated and developed by modern mathematics. It also has qualities leading to benefits that go far beyond anything possible through modern mathematics. For example, Maharishi’s Absolute Number has the organizing power necessary to be the foundation for a totally new concept in the defense of a nation which creates a level of invincibility so powerful that it even prevents the creation of an enemy. Maharishi (1996) explains,

“My Absolute Theory of Defence has its basis in invincibility, which is characterized by the self-referral dynamics of consciousness, the eternal, unbounded, unmanifest, neverchanging state of eternal Unity, which stands for the Absolute Number.” (p. 634). Maharishi’s Absolute Number also has the precision and order necessary to form the basis for Maharishi Master Management which provides perfect management by engaging the managing intelligence of Natural Law.

These excerpts are taken from the Introduction to the article: you can see the article in full at http://mum.edu/msvs/v07/number.pdf.

The main text on Maharishi’s Vedic Mathematics is “ Maharishi’s Absolute Theory of Defence” by Maharishi Mahesh Yogi. ISBN 81-7523-000-2. It is referred to extensively in the above article and is published by Age of Enlightenment Publications, 1996. This book can be purchased at http://www.maharishi.co.uk.

MATHEMATICS WITH SMILES: THE VEDIC WAY

Does your mind wobble when confronted by a mathematical challenge more forbidding then two plus two? Do you dream of becoming a Shakuntala Devi, rattling off answers to the most complicated sums in a fraction of a second? If the answer is "yes", you need Vedic mathematics. It is the doughty giant killer before whom numbers grovel and give up the ghost and predatory equations change into tabby cats.

This beautiful system was locked, unrecognized, in Vedic texts until Sri Jagadguru Shankaracharya from Govardhan Math reconstructed it. He wrote a book entitled "Vedic Mathematics" in 1958, which was published in 1965 from BHU.

Vedic mathematics is the easy and natural way to do maths. It helps increase speed, accuracy and analytical power and answers appear in one line. Try this for identifying the speed at which it works:

What's the square of 85?

Multiply the first digit 8 by its successor 9, The answer is 72.

Find the square of the second digit, 5, which is 25.

Now bring the two together. Bingo, the answer is 7225.

Don't believe it. Try it out with any 2-figure number ending in 5.

An introduction to Vedic mathematics is like entering Alice's wonderland, where logic is turned upside down. Division can be a process of multiplication and addition, and multiplication is by either cross subtraction or cross addition. The simplicity of approach exposes the top heavy processes of our logic-driven world. All the Vedic methods can be properly explained and they are more interrelated than the current methods: division for example is just multiplication reversed.

One other advantage of Vedic mathematics is that it offers choices. The same calculation can be done by different methods. This way, Vedic mathematics actually helps in holistic development of the brain and children become more creative, inventing their own methods and understanding what they are doing. There is also often a choice about whether to calculate from left to right or from right to left.

In conventional maths there is no way to multiply 88 by 98, for example, except by ‘long multiplication’, but the Vedic method, seeing the numbers are close to 100 uses the deficiencies, 12 and 2, of the numbers from 100:

-12 -2

88 × 98 = 86/24

88 – 2 = 86, or 98 – 12 = 86 for the left-hand part of the answer,

and 12 × 2 = 24 for the right-hand part.

Since 1965 the subject has been besieged in controversies with some Indian mathematicians calling it a fraud. Even though it originated in India, it is ironically taught in many other countries except India: in London and Singapore there are separate centres for Vedic mathematics.

Motilal Banarasidas, the premier publisher in the field of Indian Culture and Heritage have been staunch supporters of Vedic mathematics for a long time now. R. P. Jain of Motilal Banarasidas says the reason they are supporting Vedic mathematics is because it has the potential to bring about a revolution in the in the country in the field of maths. A very successful school course has been published by Motilal Banarasidas & developed in the UK, (covering their National Curriculum) “The Cosmic Calculator” by Kenneth Williams & Mark Gaskell (3 books + Teacher’s Guide & Answer Book) & made available at very low price. The subject has been promoted for social awareness through regular workshops at Schools & national level. Further publications are expected shortly and efforts are underway to establish World Academy for Vedic Mathematics, Nagpur headed by Dr. L M Singhvi.

Vedic Mathematics can really bring about self-confidence in our children, so that it becomes fun with figures and not something dreadful like a nightmare before exams. Get rid of this maths-phobia & aim much higher with promising results.

THE SIGN OF NINE (maybe a touch of Sherlock Holmes there!)

I have a method for multiplying any number with 9, 19, 29 etc. ie any number ending with a 9. It's very fast. You can use it for any multiplier ending with 9. 2 by 2 digits or 1 by 2 digits will take a few seconds. It is truly an 'at-sight' calculation. I came up with this method about 25 years ago in school but never thought of publishing or letting anyone know. Here it is...

A few meanings/tips:

If you put the number ending with 9 or 9 itself on the left (it should be since it's easier for the eyes to follow) then the number ending with 9 or 9 itself (as the case may be) is the multiplicand. The other number is the multiplier.

A complement is the difference of the number from 10.

Multiplying with single digit multiplicands can be done like lightning!

Say you want to multiply 29 x 6. You calculate this way - the first digit of the multiplicand is multiplied by the multiplier and the product is added to one less than the multiplier. This is the first part of the answer. The second part is the complement of the multiplier. So the calculation would be 2 x 6 is 12 and 12 + 5 is 17 and the complement of 6 is 4; so the answer is 174 and that works for all.

Isn't that fast?

Take another example, 49 x 7... so 28 + 6 is 34 and 3 so 343. (Always write the digits ending with 9 on the left. It is easier for the eye to follow and work out the calculation.)

39 x 7 is 21 + 6 which is 27 and the complement is 3; so 273.

Now this can be extended to two digits.

Say 29 x 15. You do the same thing 2 x 15 is 30 + 13 (in this case two less than 15) The way this is worked out is to see the first number of the multiplier. You subtract one more than the first number. So 15 has a 1, so you subtract 2. The last number is the complement of the last digit of the multiplier. So the calculation is (looking at the sum) 2 x 15 is 30 plus 13 is 43 and then 5. So the answer would be 435. Once you get used to the concept it is very fast.

So a sum like 39 x 25 would be calculated like this and fast ... you look at the sum and say, 75 + 22 is 97 and 5 so 975. With practice the two digit ones are also very fast. Concept wise one digit ones are actually the same since in 49 x 4 (4 is 04 so 1 more than 0 is 1) therefore it is one less than 4. In this case 196.

This works even for bigger numbers. Say 49 x 112. In this case it will be 4 x112 + 100 (12 less than 112 since 11 is the first number) which is 448 + 100 = 548 and the last number is 8. So the answer is 5488.

A few more examples,

9 x 34 = 0 x 34 + 30 is 30 and 6. So the product is 306. So for multiplication of nines go straight for the addition since the first part will always be a zero. So 9 x 45 is 45 + 40/5…855.

19 x 34 is 34 + 30/6 = 646

29 x 56 is 112 + 50/4 = 1624

49 x 6 is 24 + 5/4 =294

69 x 9 is 62/1

39 x 24 is 93/6

I have at present extended this concept of 'at sight' multiplication to multiplication with 6, 8 and numbers ending with 8. I think there is lots of scope in this field. Any takers…?

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THE VEDIC IDEAL
In the vedic system, the work is done mentally. This stems from the tradition being an oral one.

In practice, today, the initial problem or question is usually written down and the answer or solution also. The work being done mentally, a one-line answer results. This is the vedic ideal.

BACKGROUND

But what is this word ‘vedic’? It refers to an ancient period in India’s history. Tradition has it that the system of the vedas covered all branches of knowledge. Originally an oral tradition, it began to be written down around 1600 or 1700BC, according to western scholars. Over the next thousand years four vedas, as they were called, were recorded - rig-veda, yajur-veda, sama-veda and atharva-veda.

An appendix to this last contained a section headed ‘Ganita Sutras’, i.e. mathematical formulae, or principles. In the nineteenth century scholars began to look at it, but could make no sense of what they found there: statements such as, ‘In the reign of King Kamsa, famine, pestilence, and insanitary conditions prevailed.’

Then a brilliant south Indian scholar, Shri Bharati Krishna Tirthaji (1884-1960), began a detailed investigation. He concluded that the above statement about King Kamsa was a cryptic form of the decimal fraction for 1/17, using letters to represent single-digit numbers, much as we might use the letter A to represent 1, and B to represent 2, etc.

Having obtained one clue, further investigation led him to conclude that the whole of mathematics is based on 16 sutras, and he finally wrote 16 volumes on the topic.

Then events intervened. He was virtually forced into becoming a Shankaracharya. Hindu India has four of these top religious leaders a bit like having four Popes.

The upshot was that he left his beloved vedic mathematics alone for many years. Returning to the subject in the 1950‘s, it emerged that the 16 volumes had been lost. On realising this, he decided to re-write them all, and began by writing a book intended to introduce the whole series. Ill health stopped him from getting any further, and he died in 1960. This introductory book is now all that we have by him. It was first published in 1965.

DIFFERENT IDEAS ABOUT NUMBER

Western version

When measuring weight, the bigger the number, the greater the weight. Similarly for temperature, length, electric current etc. We are used to the idea that larger numbers are weightier.

Vedic version

In the vedic system, numbers are viewed differently. An analogy is telephone numbers, which we don’t associate with quantity. They are patterns of digits acting as addresses.

Similarly, when working to a base of ten (as we normally do), the vedic system deals with the single-digit numbers 1, 2, 3, 4, up to 9, together with the zero, arranged in different patterns. For example, we don’t divide by 52, we divide by 5, and take account of the 2 afterwards. This shift of focus eliminates the heaviness, or weight, associated with the common view of numbers. The vedic mathematician considers a number such as 52 as 5 and 2 in succession.

IS VEDIC MATHEMATICS CURRENTLY USED IN INDIA?

CAN YOU TELL US ABOUT DEVELOPMENTS THERE?

To answer the first question first, yes and no. It is used there to some extent. Here is a brief account of the developments.

Tirthaji died in 1960

Vedic Mathematics’ was published in 1965

Before going to India in 1981 I wrote to all Indian universities to find out what more was known about the subject. About 30% of them replied. No one could tell me anything more about it. Evidently the subject was being neglected. However, one or two letters pointed me to Tirthaji’s last residence and ashram in Nagpur. Visiting there, I was invited to return the following year to teach a fortnight’s course.

These days, the subject can be taught in schools, alongside the conventional system. Where this is done, I am told, the pupils have no problem with learning the two approaches side-by-side - the western and the vedic.

There is also a passionate debate raging about the status of Tirthaji’s system. Some argue that it is historically accurate, despite the lack of normal historical evidence. Others argue that, lacking evidence for its historical validity, it should be dismissed - despite the fact that, mathematically, it works.

My view (which I am not alone in holding) is that it is a reconstruction. At present we are unable to say for sure that it is historically accurate - nor to prove that it is not. This is because we are dealing with an oral tradition, and it is no surprise that written evidence may not be available.

WHAT IS THE POTENTIAL OF THE SYSTEM?

Tirthaji points out that it normally takes about 16 years to go from first steps in mathematics to a Degree in the subject. (e.g. from age 5 to age 21). But he states that with the vedic system the course in its entirety could be done in about two years! Of course, at present we don’t have all the material that’s needed available.

Needless to say, however, this would benefit everybody - not least those who are not interested in mathematics and would prefer to spend less time on it!

I think, myself, that once vedic mathematics begins to win general acceptance it will lead people to question other academic disciplines. Are rapid methods available in other subjects? If so, are they being used, and if not can they be developed?

MULTIPLICATION ON YOUR FINGER TIPS

I am presently conducting Workshops in Vedic Mathematics for school children. The following article on "Multiplication on Finger Tips" will appear in my compilation on Vedic Mathematics, which I hope to publish soon. During my Lectures, I found many children and parents fascinated by the method of doing multiplication of 9, 19, 29, 39, etc. on hand. I believe it will likewise generate interest in others also.

Multiplication Tables of 9, 19, 29, 39, etc. on hand. We have the answers at the tip of our fingers, literally. Isn't it wonderful. Even a child can give the answer for 6 x 79 within a few seconds by this method.

We have 10 fingers in our hands. Keep the palms with fingers spread turned towards your face.
Fig.1 shows the fingers numbered from 1 to 10, so the left thumb is 1 and the right thumb is 10.
The numbers shown against the fingers in Fig.1 are the values attached to those fingers and represent the multiplier.

For example in 9x3; 9 is the multiplicand and 3 is the multiplier.
Similarly in 19 x 6, 19 is the multiplicand and 6 is the multiplier.
Rules to remember:
1.. The folded finger represents the multiplier.
2.. Consider any finger to the left of the folded finger as representing the digit in the 10s place and those to the right represent the digit in units place.

(Eg.1) Consider the multiplication table for 9.
1.. Fold the thumb of left hand (to represent 1 in 9 x 1). Since all the remaining 9 fingers are to the right of the folded finger, we have 9 units.
Therefore 9 x 1 = 9. (See Fig. 2)

2.. Fold the index finger of the left hand (representing 2 in 9 x 2). Now we have the thumb to the left of folded finger and 8 units to the right. The thumb represents 1 Ten.
Therefore 9 x 2 = 18. (See Fig. 3)

3.. Fold the small finger of right hand (representing 6 in 9 x 6). Now we have five fingers to the left and 4 fingers to the right of folded finger. The five fingers to the left represent 5 tens. Hence 50. The four fingers to the right are 4 units.
Therefore 9 x 6 = 54. (See Fig. 4)

******************
(Eg.2) Multiplication Table of 19. In this case we proceed as explained above, but to every tens digit, we add the digit represented by the folded finger.
Thus 19 x 1 = 1/9 = 19

The folded finger represents 1. There is no tens to the left of the thumb of left hand therefore we get 1 in 10s place.

Therefore 19 x 1 = 1/9 = 19. (See Fig. 5)

TENS UNITS
1 x 1 9
Therefore 19 x 1 = 19
******************

Consider 19 x 2. We fold the index finger of left hand representing the multiplier 2. There is 1 ten represented by the thumb to the left of the folded finger. Hence, adding the 2 (multiplier) to 1, we get 3 in tens place.
There are 8 units to the right.
Therefore 19 x 2 = (2+1)/8 = 38 (See Fig. 6)

TENS UNITS
1 8
+ 2 x 1
=3
Therefore 19 x 2 = 38
*****************

19 x 6: The folded finger is multiplier 6. This value 6 is added to the 5 tens to the left of folded finger to get 11 tens. The four fingers on the right are the four units. Fig. 7

TENS UNITS
5 4
+ 6
=11
Therefore 19 x 6 = 114
******************

(Eg.3) Multiplication Table for 29:

The method is exactly as explained above, only we add twice the digit represented by the folded finger, (i.e. the multiplier) to the number of fingers to its left.

29. TABLE
The value of folded finger is doubled. 29 x 1: Here the multiplier is doubled and added to the tens on the left. Thus in 29 x 1, the folded finger represents multiplier 1. Multiply this by 2. We get 2 tens. The 9 fingers to the right represent 9 units. (See Fig. 8)
29 x 1
TENS UNITS
1 x 2 9
= 2
Therefore 29 x 1 = 29
******************

29 x 2: The multiplier 2 is doubled and added to the 1 ten represented by the thumb. We thus get (4+1) tens. The 8 fingers on the right gives 8 units. (See Fig. 9)
29 x 2
TENS UNITS
1 8
+ 2 x 2
=5
Therefore 29 x 2 = 58
*****************

29 x 7: The multiplier 7 is doubled and added to the 6 tens represented by the fingers to the left of folded finger. We thus have 20 tens. There are 3 units to the right. (See Fig.10)
TENS UNITS
6 3
+ 7 x 2
=20
Therefore 29 x 7 = 203
******************

In the case of multiplication table for 39, we multiply the digit represented by folded finger (i.e. the multiplier) by 3 and add it to the tens represented by fingers to the left of folded fingers.

39 x 4 = (4 x 3) + 3/6 = 156 (There are 3 tens to the left of folded finger and 6 units to the right)

Similarly 49 x 6 = (6 x 4) + 5/4 = 294 (There are 5 tens to the left of folded finger and 4 units to the right)

99 x 7 = (7 x 9) + 6/3 = 693 (There are 6 tens to the left of folded finger and 3 units to the right)

119 x 8 = (8 x 11) + 7/2 = 952 (There are 7 tens to the left of folded finger and 2 units to the right)

We can extend the above logic for multiplication tables of 8, 18, 28, etc. also on hand.

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ONLY A MATTER OF 16 SUTRAS

All articles published in India are critical of Vedic mathematics, but it seems appropriate to give a response based on the work carried out within the United Kingdom. I have taught Vedic mathematics for almost twenty years at an independent school in London.

Our associate schools also teach Vedic mathematics. Both teachers and students have found Vedic mathematics of great benefit because it has so many positive qualities. The most obvious of these are found through experience of working with the sutras or formulae.

What is Vedic mathematics really about? What is it about the subject that an English school should use it so thoroughly? Is it really Vedic and what was the purpose of Sri Bharati Krishna Tirthaji's book on the subject? In this account I attempt to answer some of these questions.

Vedic mathematics is based on sixteen sutras together with a similar number of sub-sutras. Each sutra provides a principle of mental working applicable to many diverse areas of mathematics. The word Veda generally has two meanings. The first is the collection of ancient Indian texts relating to both spiritual and secular knowledge. The second way that Veda is used is to describe true knowledge in the present which resides within peoples hearts or minds. This is the meaning which we have accepted. It is quite possible that Sri Tirthaji intuited the sutras from his deep understanding of the subject, the Vedas and the nature of the human mind.

If only this first meaning is accepted then the question as to whether Tirthaji's system is or is not Vedic becomes an almost insurmountable task. In my experience it is better to approach Vedic mathematics from the second meaning relating to natural laws working within the human psyche. This is a practical approach and certainly most of the work in the UK has followed this line.

The introduction to Vedic Mathematics indicates that during the early part of the 20th century Sri Tirthaji rediscovered or reconstructed Vedic mathematics from stray references within the appendix portions of the Atharvaveda. He evidently spent a large proportion of his life teaching the system but it was only shortly before he passed away that he set down an illustrative volume on the subject. This was published posthumously in 1965 and is the main source of all the serious study on the subject.

His book offers a snapshot of the sutraic system. Some of the sutras are applied to relatively elementary topics in arithmetic and algebra, giving rise to fast and easy methods of calculation. The really surprising aspect is contained within one of his introductions where he describes these few sutras as having jurisdiction over the whole of mathematics.

Years ago when I was first involved with various groups studying Vedic mathematics we all thought this statement outrageous and absurd. How could sixteen sutras apply to the whole of mathematics? Our view was strengthened by the text due to the paucity of explanation of some of the rules. For example, there is a sutra, Vyashti Samashti, which is mentioned only once in the text and even then it is given in relation to a very particular type of biquadratic equation.

As it turns out this sutra is fundamental to mathematics particularly in statistics and mechanics. It has countless applications because it describes a common mental process. When we first came across Vedic mathematics in London we were impressed by the methods of calculation. We found the sutras really brought the subject alive and we still find that all students feel more alive by practising Tirthaji's methods.

Once our enthusiasm was kindled we studied and practiced the whole of his book. We worked through every sum and read and re-read every word to try and make sense of the system. Over a period of years this work continued and we gradually began to see more and more applications to fast methods of calculation, algebraic manipulations and geometrical theorems. The next step was to consider topics within mathematics and simply ask, what sutra is working here? For example, what is the sutra working when you bisect an angle or when simplifying an irrational number? The elementary topics are fairly straightforward but what about more sophisticated mathematics?

You are given two circles drawn inside a larger circle so that they all touch each other. The area between the circles is the Arbelos or shoe-maker's knife. The problem is to construct further circles within the Arbelos as shown in the diagram. We worked through some of the conventional solutions to this. There was one particularly elegant solution that seemed the quickest and easiest method. It required transformations of a series of circles. Whilst looking at this solution it dawned on us that the sutra involved was none other than Transpose and adjust, one of the most common sutras in this Vedic system. These anecdotal instances help describe the nature of what we see as Vedic mathematics.

NUMERACY

In this chapter the acquisition of numeracy skills is discussed. The relative importance of numeracy and understanding and whether one has to be sacrificed in favour of the other is questioned.

As has been described in the preceding chapter, the Numeracy Strategy has been started using classroom methods which seem to be successfully used in other countries. These are not the only methods available. A dramatic increase in numeracy, and a consequent improvement in mathematical attainment at GCSE has been achieved in a UK secondary school by using 'Vedic Maths' This was reported in the Summer 2000 TES Mathematics Curriculum Special.

§3.1 Vedic Maths, Vedic Maths depends on identifying and using the appropriate 'processes' or 'sutra' which are to used to tackle calculations. The sutra are algorithms chosen not for their transparency to cognitive processes (in fact many are particularly opaque, so much so that they could well be used as the subject of interesting algebraic investigations for secondary GCSE maths pupils) but for their simplicity and memorability.

'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.'
(Gaskell, 2000)

In his article Gaskell describes the enthusiasm of children who find that they can do complicated arithmetic with speed and ease; children who go home and challenge their parents and beat them in races to do mental and pen and paper computations.
Williams (1984) has written a concise account of the Sutras used in the secondary mathematics course together with their applications, examples and brief explanations and proofs. Teaching materials used in the Maharishi School in Lancashire are being printed in India and will soon be available here for other schools. The sutras, are memorable phrases like 'All from nine and the last from ten', 'vertically and crosswise', 'transpose and apply', 'the first by the first and the last by the last' and are taught with initial explanations of why they work but are applied without the necessity for understanding how and why they work. This is similar to the way in which many pupils remember rules like 'cross-multiplication' and in equations 'change the side and change the sign' without consciously remembering or understanding why these rules work even if they did at first. The problem with this is that if a rule is forgotten or mis-remembered, it cannot be reconstructed because the understanding required is not there to do so. However, because the sutra are so memorable, this is rarely a problem.
In our search for and universal adoption of arithmetical algorithms which are cognitively transparent, have we done a major disservice not merely to those whose understanding necessarily remains at an instrumental level but also to those who, while understanding the processes involved, are forever encumbered by clumsy or messy algorithms when much neater ones are available and indeed used to be taught in the past? Is this why we fail in the numeracy stakes in the international maths 'olympiads'? We concentrate on making children understand what is going on when they calculate, instead of drilling them so that they can do arithmetic in their heads or on paper nearly as quickly as with a calculator. Perhaps this is why our children do so well in the understanding science (Shorrocks-Taylor, D. et al. 1998) sections and so badly in numeracy. We teach children to think and understand what they are doing, and therefore do it slowly and thoughtfully. Aught we really to worry about this or have our children acquired more useful cognitive skills rather than aptitude as quick and accurate calculators? Should we revert to earlier teaching methods and teach algorithms which concentrated more on speed, clarity and economy of effort than on cognitive transparency?
© 2001 Margaret Derrington

References:

Gaskell M (2000) Try a Sutra TES Maths Curriculum Special Summer 2000)

Shorrocks-Taylor, D. et al. (1998). An investigation of the performance of English pupils in the Third International Mathematics and Science Study (TIMSS).

Williams K (1984) Discover Vedic Mathematics, Inspiration Books

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THE DIVINE UNITY AND ZERO

In mathematics unity means the number 1. '1' is the life giving force for the whole number system, because, it is from '1' that the whole system of numbers sprung up. Without '1' there could not be numbers like 2, 3, 4, etc., and hence no mathematics at all! Hence '1' is called 'Unity' or 'The Absolute'. From '1' arose numbers 2, 3, 4, 5, etc., by just adding '1' to itself.

2 = 1 + 1
3 = 1 + 1 + 1

'1' can thus be compared to the life giving force 'God' , the 'primal Supreme Reality', because it is from this Pure Spirit that the whole universe with all its diverse forms is born. Just as the whole universe with all its varied forms arose out of that One Vital Force, similarly all the numbers with all their various properties were born out of '1'.

The Pure Spirit resides in each and every form of life. In the same way '1' resides in every number as a factor.

For example, 7 = 1 x 7, 18 = 1 x 18

Of course we cannot 'see' the '1' in the numbers, yet we know it is there. It is a hidden presence. Just as we cannot 'see' the God in every form of life even though we know that He is present everywhere, in the same way we do not see the factor '1' in every number.

The Infinite reality has limitless potentiality inherent within itself. Similarly the number '1' has unlimited potentiality to create different types of numbers. With the help of the various operations like +, - ,x, divide, etc., it can create positive, negative, rational, irrational and imaginary numbers. Yet all these numbers cannot exist without '1'.

Consider the fraction ½. (Half). Now we know that 'half' cannot exist without the 'whole'. Since '1' represents the whole, the idea of half is possible. Even the imaginary numbers arise from square root of '-1'. Thus '1' is 'All Pervasive' or the 'Absolute'.

Let us now look at the magical number 'Zero'. The symbol is '0'. In fact '0' is not an ordinary number. We can say that '0' is just a symbol to represent 'Nothing'. Robert Kaplan in his book titled ' The Nothing That Is' says, " If you look at zero you see nothing, but look through it and you will see the world." Zero is the small pivot on which swings the whole realm of mathematics. The invention of 'zero' by the ancient Indian mathematicians made arithmetic infinitely easier. It now forms part of the binary code which powers all computers.

It is because of '0' that the very idea of place value was possible. The pride of giving this to the world goes to India. With just 9 symbols and a '0', our ancient mathematicians could write any number as big as they wanted.

Zero brought variety to numbers. Consider '0' by itself. You have nothing. Put it next to '1'. You have '10'. Add as many zeroes as you want now. You have a staggeringly big number whose value you cannot even tell. We can proceed towards infinity as we add more and more zeroes next to '1'. This is why in Sanskrit zero is also called 'Anantha' the 'Infinite'. This is the power of '0'. Possessing no value itself, it is able to confer infinite value to '1' by just placing itself next to it repeatedly. '0' is thus able to confer infinite forms and values not only to '1' but also to other numbers.

Now we saw that '1' is related to the 'Supreme Reality' or the 'Pure Spirit'. To what is '0' related?. What is that inscrutable, inexplicable mighty power which is the cause of the multifarious creations in the universe, which makes the impossible possible and creates differences and distinctions?. It is 'Maya'.

Maya is defined as 'That which is Not'. Swami Swaroopananda has explained 'Maya'as follows. " One cannot say that she exists nor can one say that she does not exist. She is not separate from the Lord, nor is she one with the Reality. Hence Maya has been described as Shakti, the incomprehensible and mysterious power by which the Infinite projects the universe." Because of Maya, 'One' appears as 'Many'. There are many different types of life forms as well as inanimate forms due to this Maya Shakti.

The entire play of creations is made up of Maya. It is only because of Maya Shakti the Supreme becomes Ishwara and takes different forms, so we can worship him in his different forms. Remove Maya, all the infinite forms vanish and only the supreme reality is left behind.

Similarly 'Zero' is able to confer infinite values to 'unity'. Let us consider an example which brings out the power of '0'. Let us try to factorise the following expression.
x^4 + 64

We cannot find any solution or method at the outset. It seems impossible to factorise the given expression, since it is of the form 'a^2 +b^2 ' Let us now call '0' to our aid. Let us add '0' in the form '16x^2 - 16x^2' to the above expression.
We get,
x^4 + 16x^2 - 16x^2 + 64
= x^4 + 16x^2 + 64 - 16x^2
= (x^2 + 8)^2 - 16x^2
= (x^2 + 8 + 4x)(x^2 + 8 - 4x)

Thus we see how '0' has helped us in factorizing an expression which seemed impossible to factorise. In this way '0' can help us in solving many equations. This is the creative power of 'zero'.

'Zero' can be added in any form we want, and those forms always occur in pairs, one positive and another negative. Just as Maya manifests duality in all forms like, the good and the bad, strong and weak, young and old, likes and dislikes, light and dark, etc., 'zero' brings into focus the dual nature of any number. Remove Maya and duality vanishes. Similarly, it is due to '0' that negative numbers exist. Thus every number has its opposite number, like 2 and -2, 3 and -3 ½ and - ½.

Also, 2 + (-2) = 0, 3 + (-3) = 0, etc. Together these opposite numbers vanish into '0'.

Yet '0' has no sign nor any opposite. There is only one '0' like the one and only one 'Maya Shakti'. Maya shakti creates opposites, differences and varieties yet Maya shakti itself is unaffected by all these differences. Similarly '0' has given birth to opposite numbers but it has no opposite number. It is unique.

Thus '0' has infinite power. Like 'Maya shakti' '0' can also take infinite forms and vanish whenever it pleases.

Once Swami Chinmayanandaji was asked, "How is it that Lord Krishna had 64000 wives?" He replied thus. "The stories of Mahabharata and Ramayana and other epics should not be taken literally. Each story contains deep philosophical meanings. There are about 64000 types of thoughts in our minds. Lord Krishna represents the divine spark. He is the one who infuses life into each of these thoughts. Without life, there can be no thoughts. It is because of the presence of this 'life spark' the thoughts materialize. Gopis are the materialistic representation of these thoughts." Now '0' is like a thought and '1' is the 'Absolute'. When these two stand as one we have '10'. With the help of '1' and '0' we have infinite varieties of numbers. Just as thoughts become very powerful and materialize into matter due to the divine will, in the same way 'zeroes', which are valueless become very powerful due to the presence of 'one'. Thus '1' and '0' empower each other. Thus the cosmic dance of numbers go on.

GUESS WHICH SUBJECT LACKS SOUND FOUNDATIONS
(AND WHAT CAN BE DONE ABOUT IT)

PART 1: THE PROBLEM AND THE SOLUTION

It is widely believed that the foundations of mathematics were sorted out long ago. This is not so. The truth is that not one of the established systems can be shown or even trusted to have sound foundations. But theorems depend on the foundations; how far can we trust them if the foundations are unsound?
Morris Kline drew attention to this state of affairs in a book published in 1980, 'Mathematics: the loss of certainty'. Two quotations from it suffice to make the point.
(1) 'The disagreements about the foundations of "the most certain science" are both surprising and, to put it mildly, disconcerting. The present state of mathematics is a mockery of the hitherto deep-rooted and widely reputed truth and logical perfection of mathematics.'
(2) 'According to Homer, the gods condemned Sisyphus, king of Corinth, after his death to roll a big rock uphill, only to see it fall back to the bottom each time he neared the summit. He had no illusion that some day his labors would end. Mathematicians have the will and the courage that comes almost instinctively to complete and solidify the foundations of their subject. Their struggle too may go on forever, they too may never succeed. But the modern Sisyphuses will persist.'

Kline evidently suspected that the problem may never be solved - and may even be insoluble. I disagree, and show in this paper how the problem can be solved, giving as an example the foundations of a recently formulated system of Euclidean geometry.

The essence of the solution is this. It is the job of the foundations to state everything necessary for the study. If we can show that the proposed foundations are necessary for the study we are on firm ground. To complete the job it is also desirable to show that they are sufficient for the study, so far as is possible.

Surprisingly enough, the key first step has never been taken for any of the established systems of mathematics. It will now be shown how the above-mentioned new system of geometry satisfies the first of these steps fully, and the second in part.

But prior to taking a look at this, a discussion on the nature of Euclidean geometry helps to set the scene. Some of these simple points have far-reaching consequences, as this paper shows.

WHAT IS EUCLIDEAN GEOMETRY?
It arises in our experience of space. Objects around us have shapes, with surfaces and boundaries, and the figures of geometry can be thought of as simplified representations of these. Furthermore there are measurable aspects of figures: lengths, angles, areas and volumes. We call them magnitudes because what they have in common is magnitude (size). Euclidean geometry is a study of magnitudes in figures.

Geometry works with theorems, these being statements
which are then proved. Mostly, these statements relate directly to figures. Typically a theorem tells us that, e.g., Angle A equals angle B (angles A and B being illustrated in a figure), and a proof follows. Note that the underlined statement can be spoken. It can also be written in mathematical short-hand. That statements in mathematical short-hand can be spoken is often over-looked.

THE NEW SYSTEM
A good way of introducing this new system is by asking the question:
What do we really need in order to study Euclidean geometry?

The study is set in the context of an oral tradition. The original reason for doing so is discussed in 'Geometry for an Oral Tradition', but an immediate consequence is that it draws attention to the role of language in the study. Since we cannot do without it, formal acknowledgement is in order. And the same applies to anything else we cannot manage without. So a good first step is to note what else is indispensable.
Investigating this, I have found three requirements without which the study of Euclidean geometry is not possible in the context of an oral tradition. They are:
A language, in use.
A means of drawing figures.
The ability to recognize valid reasoning.

Lacking any one of these three the study cannot proceed. E.g. without a language it can neither be formulated nor explained. Indeed, without a language there can be no oral tradition. Again, figures are needed because geometry is a study of some of their properties.

But these three requirements are not sufficient for the study. For they make no commitment to any type of geometry, Euclidean or non-Euclidean. That is to say, they provide nothing to help get the study of geometry under way.

To get started we need at least one assumption in the form of an axiom. For theorems are proved using earlier theorems, and we need a starting point somewhere. Effectively, an axiom is a theorem which we do not prove.

Where can we find such an axiom? A helpful consideration is that Euclidean geometry has an everyday role as the geometry of our everyday experience. Indeed, this is where the study originates. It describes the spatial properties we are familiar with, as well as some we habitually overlook. This gives a clue as to where we might look for an axiom.

Euclidean geometry deals with space, and today we recognize that the latter is relative. It is no longer adequate to treat physical space as absolute, as is done by the accepted formulations of Euclidean geometry. This was acceptable in Euclid's day, but not now.

Now we are on the track of what is needed, for we will only be dealing with relative space if the principle of relativity is satisfied by our formulation of Euclidean geometry. And as it happens there is just one way of satisfying the principle of relativity which yields Euclidean geometry. Expressed physically, it is for objects to be unchanged in shape or size by motion. Expressed more mathematically it is for magnitudes to be unchanged on being moved around.

Thus, an axiom required for Euclidean geometry is:
magnitudes are unchanged by motion (i.e. on being moved around).
This is the fourth requirement of the study. Thus all four have been shown to be essential. There is also some evidence that they are sufficient for the study, as will be explained next.

SUFFICIENCY
Evidence that the Four Provisions are sufficient to prove the theorems of Euclidean geometry is provided in two ways:
[1] In 'Geometry for an Oral Tradition' they are used to develop theorems as far as elementary properties of a circle. [2] Equivalents or near-equivalents to Euclid's five Common Notions and five Postulates follow from the four provisions (see the Commentary to 'Geometry for an Oral Tradition'). This shows that the system is capable of proving some of the theorems, and suggests that it may be capable of as much as or more than the 'Elements'.

THE CONCEPT OF A PROVISION
In the book 'Geometry for an Oral Tradition' the term provision is introduced for what is provided. This is a concept of greater generality than that of an axiom. Each of the above-mentioned four requirements is a provision, but only the fourth is an axiom.

One might ask, what is the difference between an axiom and other provisions? Note that any proof whatever makes use of the first and third provisions, and the second comes in wherever there is a figure. But the axiom proposed here is needed only for Euclidean geometry. E.g. 'Geometry for an Oral Tradition' shows how it can be used to start off the theorems of Euclidean geometry. This points to an axiom being an assumption made for the purpose of a particular study. [The 'Shorter Oxford Dictionary' gives, for axiom 'a self-evident principle', pointing out that this goes back to Aristotle]

THE NEED FOR A COMMENTARY
Since 'Geometry for an Oral Tradition' does not follow a standard approach, a Commentary (of 51 pages) is included to discuss the issues arising, and to show that the approach is a valid one.

DEFINITIONS
The system will not work efficiently - or even not at all - unless the definitions are well-handled.

In everyday use there are lots of words used which are not normally defined because all concerned know what they mean. This system follows suit: words in common use need not be defined.

Definitions are given, however, of terms needed for the study, words such as theorem and angle and line, even though some of them are in common use. Two important definitions are those of magnitude and equality. With their aid three theorems can be proved which Euclid gives as axioms (Common Notions 1-3). For a suitable definition gives us something we can work with in proofs.

In general, if we do not define terms necessary for the study, more axioms are needed to fill the gap [definitions of words such as theorem and postulate excepted]. This study sets out to give all relevant definitions, so minimizing the number of axioms required.

There are a number of words in common use which have a bearing on geometry, such as circle, line, straight, angle, length, square, etc. As this list shows, when we begin the study we do have some knowledge of geometry; it makes sense to begin there, and then deepen and extend what we know. Giving a definition is part of the process.

This brings us to some illuminating points concerning definitions. There is a tradition in mathematics, passed on by Euclid, of giving definitions in sequence. Euclid probably presented the definitions in this way as a means of introducing them one at a time. This would explain why he avoids using a word until it is defined. Consequently, words defined earlier in the sequence may be used in later definitions, but not vice-versa.

Amongst mathematicians this procedure has resulted in a pretence (for mathematical purposes) that we do not know what a word means until it has been defined. This is unrealistic, and often untrue. In fact, knowing what an angle or a straight line is proves useful, enabling us to judge whether definitions proposed for them are acceptable or not.

But note further, that whereas later definitions can make use of earlier ones, the first definition has no other definition to draw on. Consequently, we need to start with something undefined.

The obvious source of undefined words is, of course, words in common use; for this is how speech works. But words not on the list of definitions are usually ignored, as though they do not exist. Mathematicians make use of them, and know what they mean, but do not define them. There is no thought that they are unknown because they are undefined. Language does not receive formal acknowledgement in mathematics today, but there are nevertheless rules or conventions governing its use.

A SUMMARY OF THE FOUR PROVISIONS APPROACH
That leading grammarian of the English language, Otto Jespersen, makes a telling comment:
'Educated people are apt to forget that language is essentially speech.' (Essentials of English Grammar)

By setting the study in the context of an oral tradition, the focus on language is a focus on speech. And in the light of Jespersen's comment this makes it relevant today.
So that is the first point: whereas mathematicians have always tacitly assumed a language to be available, in this study language is overtly acknowledged to have a key role, in its spoken form.
In everyday speech words in common use are rarely defined. This study follows suit: words in common use need not be defined.
Definitions are given, however, of words essential for the study (mathematical terms). Words such as theorem and axiom apart, every relevant definition gives us something to work on in proofs, making it possible to keep the number of axioms to a minimum.
A clear statement of the role of the foundations is required in order that we can ensure that the foundations proposed are necessary and sufficient for the purpose. The foundations are a statement of what is needed for the study, once the teacher and one or more willing pupils are assembled in a suitable place. [Teacher and pupil may be the same person, as in the case of a researcher.]
A generalization of the axiom concept, the provision, is introduced. It is used to include formally what has previously been tacitly assumed, including implicit acknowledgement of our relevant powers and faculties. Without these the study would not be possible, and axioms cannot provide for them.
The key steps are: (1) a statement of the four provisions proposed, and (2) demonstration that they are essential for the study. The Four Provisions are: [1] A language, in use. [2] A means of drawing figures; specifically, a plane, a pen, a straight edge, and a pair of compasses. [3] The ability to recognize valid reasoning. [4] The sole axiom; magnitudes are unchanged by motion (i.e. on being moved around).
The demonstration that these four are essential for the study can be summarized as follows: [1] In the absence of any one of the first three provisions the study is clearly not possible. [2] To ensure that the study is set in relative space the principle of relativity needs to be satisfied. There is only one way of doing so which yields Euclidean geometry, and that is the fourth provision.
That the Four Provisions are to some extent sufficient has been shown.
Finally, this being a non-standard approach, a Commentary is needed to demonstrate its underlying unity. (A 51-page Commentary is given in 'Geometry for an Oral Tradition')

CONCLUSIONS
The necessary and sufficient condition is a well-known one, but no attempt appears previously to have been made to apply it to the foundations in any branch of mathematics. The consequence is that the establishing of sound foundations has appeared to be beyond reach.

This paper shows how the situation can be rectified, giving foundations which are demonstrably essential for Euclidean geometry as an example

GUESS WHICH SUBJECT LACKS SOUND FOUNDATIONS
(AND WHAT CAN BE DONE ABOUT IT)

PART 2: COMMENTARY

FURTHER INSIGHTS INTO THE FOUR PROVISIONS SYSTEM
There are certain ideals underpinning the Four Provisions system, and to do it justice we need to look at them. They are features which distinguish this from other systems of Euclidean geometry - as well as enabling us to look at the approach from a slightly different point of view. There are three such ideals:

IDEAL 1 Connect the formulation of the study with what is being studied, wherever possible. These meeting points may be called points of contact between the formulation and the object of study. They include definitions of point, line, plane, circle ,triangle, parallels ,right angles, etc.

This is one reason why it is helpful to define all words needed for the study. For almost every one is a point of contact, all of which need to be acknowledged.

Another important point of contact is Provision Four, which is the condition for space to be relative, i.e. to satisfy the principle of relativity. Each of these points of contact helps to locate the study fully in our world.

IDEAL 2 Acknowledge whatever is essential for the study. This is the role of the foundations. Carrying this part of the programme through led to the four provisions discussed earlier.

IDEAL 3 (1) Keep the number of assumptions to a minimum. And then (2) show that this is the minimum number of assumptions on which the study can be based, and (3) show that these assumptions are essential for the study. We then have grounds for describing the assumptions as axioms of the study.
All of these steps have been carried through successfully in this study. And in the process an idea has emerged concerning the nature of an axiom which may be new, and has certainly not been put into effect before: an axiom is an assumption which can be shown to be essential for the study.

AN OBJECTION RAISED AND ANSWERED
An account of what may be called the 'Four Provisions approach' has now been given. Is there more to be said? Doubtless opponents of the system can think of a few things to say. Here is an example of what might be said by an advocate of the system widely regarded as leader in the field of Euclidean geometry,
David Hilbert's 'Foundations of Geometry' (1899):
Hilbertian Your definition of a straight line is not valid. You give,
Definition 13 That line which is uniquely specified given two points on it is said to be straight.
Now it is a well-established principle that a definition ought not to contain axioms and this definition consists of two axioms. As given by Hilbert they are:
Axiom1,1 For every two points A,B there exists a line 'a' that contains each of the points A,B
Axiom1,2 For every two points A,B there exists no more than one line that contains each of the points A,B
Need I say more? All your careful reasoning is worthless since you start off on the wrong foot.
Response To reply briefly, Hilbert gives 20 statements which he calls 'axioms', without proving that any of them are essential for the study. Nor does his approach allow them to be considered self-evident. Consequently his 20 statements are really so many assumptions, no reason being given for accepting them.

It sounds less impressive to say, 'Your definition is really a combination of two of the assumptions with which Hilbert begins his study' - which may be so but certainly does not amount to a flaw in the definition.

Furthermore, the use of 20 assumptions shows no respect for the principle that the number of assumptions should be kept to a minimum. It is a weakness in Hilbert's approach.

But there is more to be learnt from this example. Hilbert's first two axioms and the above definition of a straight line look at the same material, and present it in two slightly differing ways. We can use this material in proofs, whether we take it from the definition (Four Provisions approach) or from the two axioms (Hilbert's approach).

However, Hilbert pretends that we do not know what a point, a (straight) line, and a plane are (by leaving them undefined). In this way he turns his back on the origin of his axioms, turning them into assumptions: statements for which no justification is given.

But there is a further problem here. If point, (straight) line, and plane do not have their normal meanings, we cannot relate Hilbert's axioms to diagrams. This did not trouble Hilbert, who preferred not to have to deal with diagrams. In his system theorems are proved directly from the axioms, without reference to diagrams. He did use the latter 'for illustrative purposes', however, by assigning their ordinary meanings to point, (straight) line, and plane.

Hilbert's approach is heavy going, dispensing with diagrams as it does, and using a lot of axioms. But there is a more serious objection to it. For Euclidean geometry is a study of some properties of figures. Succinctly put, no figures, no Euclidean geometry. Thus, only when point, (straight) line, and plane have their normal meanings is this a study of Euclidean geometry. But it is an essential feature of Hilbert's system that point, (straight) line, and plane do not have their normal meanings. Either Hilbert's study is not a study of Euclidean geometry or else it is inconsistent.

Two objections to Hilbert's approach have now been pointed out, the first a serious one and the second a fundamental flaw. Both stem from his requirement that point, (straight) line, and plane be undefined and therefore unknown. But even apart from these, there is a fatal objection to his formulation: it is set in absolute space and is therefore anachronistic. It does not fit in with Newton's world-view, let alone Einstein's.

SOME OF NEWTON'S INSIGHTS ON GEOMETRY
Sir Isaac Newton's 'Principia' is set out in the same manner as Euclid's 'Elements', beginning with definitions and axioms and continuing with theorems. This is partly because of his subject-matter, dealing with objects (bodies) at rest and in motion. [The state of motion being the province of mechanics and the state of rest the province of geometry.] These two, including the study of forces acting on bodies, he calls 'universal mechanics'.

He points out that mechanics and geometry are interlinked, as the following two quotations from his Preface to the 'Principia' show:
(1)'…the description of right lines and circles, upon which geometry is founded, belong to mechanics.'
(2)'…geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.'
Newton explains that geometry deals with the state of rest. Effectively the 'Principia' extends the topic, embracing geometry (he took Euclid's theorems as given) and including the motion of objects (bodies). Thus it encompasses the states of rest and motion. This made it appropriate for him to write his book in the style of Euclid's 'Elements'.
One of the reasons for mentioning these points here is because since ancient times there has been some resistance to taking motion into account in geometry, as though it is not quite respectable. This is a pity, because it is vital to the subject. The states of rest and motion cannot be considered in total isolation from one another. Absolute space is one consequence of attempting to do so.
Another lesson from these two quotations, however, is that geometry relates to our experience of the world; it is not pure thought, unrelated to anything. Applied mathematics and physical sciences depend on their experiments, and topics of pure mathematics such as arithmetic and geometry draw on experience,

Finally, note that the 'Principia' deals with the relative motions of objects, and assumes that these are unchanged in shape or size by motion. That is to say, it contains Provision Four as a hidden axiom.
MORE ON SUFFICIENCY OF THE FOUR PROVISIONS
Are the Four Provisions sufficient to prove all valid theorems of Euclidean geometry?
Godel's Incompleteness Theorem tells us that there exist valid theorems which are not provable from the axioms. If it is applicable here, the answer is, no; the four provisions are not sufficient to prove all valid theorems of Euclidean geometry.
This raises the interesting prospect of finding a theorem of Euclidean geometry which cannot be proved using the Four Provisions, and which may turn out to be an illustration of Godel's Incompleteness Theorem.

REFERENCES
1) Mathematics: the Loss of Certainty, M.Kline, OUP 1980
2) Geometry for an Oral Tradition, A.P.Nicholas, Inspiration Books, 1999
3) The Thirteen Books of Euclid's Element's, Sir T.L.Heath, CUP, 1908
4) Essentials of English Grammar, O.Jespersen, Allen & Unwin, 1933
5) The Foundations of Geometry, translated from D.Hilbert's Grundlagen der Geometrie, Kegan Paul, 1st edition, London 1902
6) The Mathematical Principles of Natural Philosophy, translated by A Motte 1729, revised by F. Cajori, of Philosophiae Naturalis Principia Mathematica by Sir I Newton, 1686, University of California Press 1934

RESEARCH IN VEDIC MATH

In spring of 2003, Anand Pattabiraman conducted research on a topic called Vedic Math. Vedic Math was said to be a fast way to do arithmetic and he wished to see if this was true.
Ancient Indian mathematicians made numerous contributions to mathematics such as Vedic Math. The decimal system and the concept of zero were other major contributions. The advanced system of Vedic Math is believed to be described in the Parasista (Par - a - shish -tha), the appendix portion of the Atharvatheva, one of the 4 Veda books. The Vedic Math system was rediscovered in the 20th century by Jagadguru Swami Sri Bharati Krishna Tirthaj Maharaja.
For his ROGATE Research Project, Anand selected Vedic Math as the topic. He did some preliminary research and formulated his hypothesis. He found two primary and two secondary resources to complete his research. He designed and conducted an experiment. He collected and analyzed the data and finally created his presentation and report.
The hypothesis that Anand formulated was: Students who use Vedic Math are quicker and more accurate when doing computations. The primary resources that he used to research this hypothesis were an interview with Kenneth Williams, a Vedic Math scholar and mathematician, and an experiment involving six 6th grade students who were advanced in mathematics. The secondary resources were three websites online and a newspaper article.
As one of Anand's primary resources, he contacted Dr. Kenneth Williams, Vedic Math scholar and mathematician, and interviewed him on the topic of Vedic Math. Dr. Williams stated that Vedic Math can and does apply to all mathematical problems.
But how can Vedic Math apply to all mathematical problems when there are just 16 sutras (rules)? Dr. Williams explained "These seem to relate to the way the mind works and that is the reason why there are 16 sutras as there are just 16 ways that the mind can function and from that point of view, the Vedic math must cover all of math if these 16 functions cover all the ways in which we think, then it must be a complete system."
But could there be any disadvantages that could make teachers not want to teach Vedic Math to students? "Doing mental maths increases your brain power and your mental agility and creativity, it's really an advantage, there aren't really any disadvantages.", Dr. Williams told Anand.
But is Vedic Math right for everyone? Dr. Williams declared "Oh certainly, it should be used everywhere it's a much more coherent system, much easier to use, more flexible. Children who are slow at math find it easy and kids who are good at math find that they like it as well, everybody seems to like it. It has been demonstrated that people do get higher grades and win awards who do Vedic Math."
All of the information Anand Pattabiraman found online agreed with what Dr. Williams cited and had only positive opinions regarding Vedic Math. "The beautiful system of Vedic Mathematics is far more unified and direct than conventional mathematics.", Dr. Williams expounded, adding that "exceedingly tough mathematical problems …can be easily and readily solved with the help of these ultra-easy Vedic Sutras. The Sutras (aphorisms) apply to and cover ...every branch of mathematics …. In fact, there is no part of mathematics, pure or applied, which is beyond their jurisdiction."
Dr. Williams exclaimed "It is so fascinating, it has turned math-haters into math-lovers!" Anand Pattabiraman was very surprised by all this information.
For his experiment, Anand selected three out of the 16 sutras. These three sutras helped with adding and subtracting fractions, multiplying with numbers close to 100, and subtracting from numbers in the series 10, 100, 1000, etc. He created a test involving the uses of these three sutras. The test had 30 problems with ten problems for each of the sutras. Three 6th grade students were given this test and were supposed to use their own method of arithmetic. Three other 6th graders were taught the Vedic Math guidelines and then given the same test. During all the tests, the completion time was noted and the tests were scored for accuracy. In addition, the students who learned Vedic Math were also given a questionnaire to see their reaction to Vedic Math.

RESULTS

Control Group Time (minutes) Accuracy
Subject #1 8:26 30/30
Subject #2 13:53 28/30
Subject #3 14:32 24/30
Average 12:28 27/30

Experimental Group Time (minutes) Accuracy
Subject #1 6:36 27/30
Subject #2 11:33 29/30
Subject #3 14:58 25/30
Average 11:03 27/30

 

 

 

 

The average completion time of the control group, i.e. the kids who were not taught, was 12:28 minutes, while the experimental group, the kids who were taught Vedic Math, had an average of 11:03 (minutes). Therefore the experimental group was 11.5% faster than the control group.
The efficiency for each subject was calculated by dividing the %accurate answers by the time taken to complete the test. Then the average efficiency for each group was calculated (Control = 8.02, Experimental = 9.19). The conclusion is that the experimental group was still 15% more efficient than the control group, even though they had the same average score.
After the tests were taken and times recorded, children in the control group were taught the rules of Vedic Math (optional). It appeared that the only student who had scored 30/30 in this experiment belonged to the control group and she had already learnt 2 out of the 3 rules and used it in the test. This suggests that Vedic Math could be taught in other places, it is just not known by the title Vedic Math. This person should really have been part of the experimental group and could have boosted the accuracy of the experimental group even further.
At the end, Anand Pattabiraman conducted a survey on the kids who were taught Vedic Math. Some of the questions and answers are listed here. Two out of the three sixth graders agreed that the Vedic Math rules were useful it but the third student Anand attempted to teach said "Yeah, the ways of doing Vedic math were cool… but, it didn't really help me." This person didn't want to participate in the experiment. All three of the sixth graders thought it was cool and liked it. They were undecided whether it should be used in schools but one was sure, "Maybe, it will be sort of hard if kids don't know their basics." "Yes, it can help other kids solve problems easier, too." All of them agreed that they would remember some of it and use it again.
Anand concluded that his data and research support his hypothesis that Students who use Vedic Math are quicker and more accurate when doing computations.

How can this research be applied to new situations?
The Vedic Math system can be used in schools around the world.
The system will be easier for teachers to teach and students to use.
It can also be used to train math teams for competitions where speed and accuracy count.

Acknowledgement: Anand Pattabiraman would like to thank Mrs. D. Schulthes, Coordinator of the Discovery Programs at Tenafly Middle School, Tenafly, NJ, for her guidance and help and Dr. Kenneth Williams and the students who participated in this research.

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3-SPACE MATHEMATICS, SCIENCES AND TECHNOLOGIES
1. '3-space' with focus upon its linear dimensions makes 'mathematics, sciences and technologies' of this space being of 'linear features'.
2. Cube as representative regular body of 3-space with 'Volume of a cube' as dominant expression being domain fold of this as manifestation layer makes 'mathematics, sciences and technologies' of this space, being of 'physical content lump' features.
3. From minutest granule to the biggest solid, the whole range, being of same physical content features as well as of identical linear dimensional axes characteristics, as such within each point of 3-space stands inherently in-built a three dimensional frame so 'the mathematics, sciences and technologies' of this space are of 'macro state' features.
4. As the origin from which the axes of 3 dimensional frame remaining sealed and dormant at 0 value, so 'mathematics, sciences and technologies' of this space essentially remain to be of 'monad nature' of linear quantifiers.
5. The monad nature essentially being of features of 'monad' always coming into play 'as a whole', therefore 'the mathematics, sciences and technologies' of this space are to be of 'positive features', and conceptually, the sequential progression is to be of 'one directional flow' alone.
6. Being of one directional flow alone, as such, its processing potentialities are to be only of 'half range', and other half range is always to remain un covered by its processing systems.
7. The working out of second half of the range in terms of the first half of the range makes the processing of this space 'as of forced symmetries', sacrificing the zero and negative artifices attributes.
8. The working bias of 'forced symmetry' deserves to be compensated failing which the Reality is bound to go guised and its restoration is bound to be accepted as an impossibility.
9. The Reality so processed under forced symmetric acceptance is to deprive of not only of the existence of four and higher dimensional spaces but also the lower, negative, zero as well as one and two spaces as well have to remain beyond reach.
10. The range of 'two and one' spaces would get disguised underneath the processing steps of first and second of the 3 axes of three-dimensional frame of 3-space, and the processing in respect of one and two space, as such would be of those spaces within 3-space, and not as of their independent existence as 1-space and 2-space respectively.
11. Euclids and Descartes were duped, and all of us, not being conscious of this trap, are also the easy preys of it.
12. The answer and remedy for ensured escape is to first concentrate upon the Creator's space (4-space), and then the cube as representative regular body of 3-space be approached as creation manifesting along the four folds manifestation format supplied by Creator's space.
13. 'The cube' as manifested creation is of four folds, of which 1-space is playing the role of dimension and its content lump is manifesting as the dimension fold.
14. '2-space' is playing the role of boundary and its content lump is manifesting as the boundary fold.
15. '3-space' is playing the role of domain and its content lump is manifesting as the domain fold.
16. '4-space' is playing the role of origin and its content lump is manifesting as the origin fold.
17. A step ahead, the chase is to be of the way the transcendental base of Creator's space fulfills the Creator's space with solid quantifiers, and the cube has its distinct role but as along the transcendence range of five folds, the fifth in sequence being the role of 5-space itself otherwise lively at the base of the Creator's space.
18. As such, the sadkhas fulfilled with an intensity of urge to fully know and completely chase the different roles of '3-space' and hence 'the mathematics, sciences and technologies of 3-space' shall permit their mind to transcend following the way Creator's space is fulfilled with transcendental features and the way creations automatically attain self referral features.

TO CHASE THE MANIFESTATION OF CUBE
1. Vedic mathematics, sciences and technologies of 3-space, essentially are to chase the manifestations of solids and in particular the cubes, spheres and cones, prisms and pyramids with rectangular and polygon formats, and 'this specific shapes and forms features of manifestations of 3-space bodies' as such, as well may be designated as chase of CRYSTALS, DIAMONDS and GEMS, as Macro states.
2. Of these, the beginning shall be had with manifestation of CUBE.
3. The manifestation of cube as a set-up within geometric envelope stitched as of 8 corner points, 12 edges and 6 surfaces, in all, 26 geometric components is the first basic feature, which need be chased.
4. Each of the 8 corner points accepts the role of origin for a three-dimensional frame of half dimensions, and thereby, there is available, as many as 8 such frames, which together, shall be constituting a set of 4 three dimensional frames.
5. The coordination of 8 three dimensional frames of half dimensions as a set of 4 three dimensional frames of full dimensions is a feature which deserves to be chased with a focus as the cube as a representative regular body of 3-space shall be having only 3 dimensions and these three dimensions can supply at the most a set of six half dimensions, while the requirement of all the half dimensions for 8 corner points is of 8 x 3 =24 half dimensions.
6. The split for three dimensional frame, even as of three solid dimensions, shall be making available only a set of three pairs of three dimensional frames of half dimensions. It shall, as such be expecting one another pair of three-dimensional frames, and the same, in the circumstances may be hoped having been supplied by the 'space' itself. The supply of a pair of three-dimensional frames of half dimensions of solid order is to be there only from the transcendental worlds (5-space) of solid dimensional order.
7. The split of a three dimensional frame of solid dimensional order into a pair of such dimensional frames is to be there because of the availability of Creator's space at the origin.
8. It is this availability of the Creator's space at the origin which need be chased along its all the four spatial dimensions, and it is this chase within spatial dimensions shall be providing a format of a pair of orientations permitting inter change there of, because of which the split of a three dimensional frame into a pair of three dimensional frame shall also be permitting reversal as well as translation of them for their setting along the corner points of the cube.
9. It is this pairing of the corner points and simultaneously insertion of a pair of three dimensional frames of half dimensions obtained simultaneously with split of a three dimensional frame and also reversal translation from them to reach and to be established within so paired corner points of the cube, and thereby, making the set up of the cube ensured for its all the four pair of corner points as a real solid set up.
10. The above feature of stitching of the set up of a cube by pairing corner points and embedding three dimensional frames in all the 8 corner points further ensures the stitching for all the twelve edges of the cube as of di monad format with pair of parts of the edge of the cube to be supplied by two different three dimensional frames of corner points coordinated by the concerned edge.
11. This feature of pairing of 24 half dimensions into twelve edges, further in a sequence pairs twelve edges into six spatial frames and thereby is ensured the required six surface plates for the geometric envelope of the cube.
12. This sequential pairing in a pair of steps from 24 half dimensions to 12 edges to 6 surfaces, in that sequence and order or reverse thereof taking from six surfaces to 12 edges to 24 half dimensions to 8 three dimensional frame of half dimensions to 4 three dimensional frame of full dimensions are the sequential progressions which need be chased for replicating the Vedic mathematics, sciences and technologies.
13. For this exercise, another feature which need be taken account of is the exhaustive coverage for all the 8 corner points in terms of only 7 of the edges, to be designated as the manifested edges, while the remaining five edges to remain un manifested support.
14. This sequential progression for coordination of 8 corner points in terms of 7 edges, is to be of 3 fold orientations along the 3 dimensions springing out from each corner points. The chase along any of such orientations along any of the dimensions at its end reach at the 8th corner in that sequence and order, naturally would also provide reversal for such progression from first corner to 8th corner into 8th corner to 1st corner. This way, two fold progressions, for both of orientations for each edge would make the synthetic stitching of the geometric envelope along the edges to be of manifestation formats of the order of (-1) space playing the role of dimension for (+1) space, and as such the transition and transformation from macro states of edges to micro states of edges would be available. It is this availability and permissibility of transition and transformation from macro states to microstates for the edges would make '0-space' lively in the role of boundary fold for such manifestation. The 0-space in its role as of dimension fold for 2-space would inherently coordinate corner points as 0-space bodies with surfaces of the cube as 2-space bodies.
15. Such is the richness of the geometric envelope of the cube and replication of it, naturally, can be expected for supplying parallel richness for the mathematics, sciences and technologies of 3-space, provided the whole approach to the 3-space bodies is the way these avail their manifestation formats.
16. The simple progressions emerging from 2 corner points of an interval, 4 corner points of square and 8 corner points of cube, is just to sway away with it a forced symmetries by working up till half range intervals and covering second half parallel to it. The progression 2N, for its values N= 0, 1, 2, 3, 4 as manifestation layer (0, 1, 2, 3) is to be of the values 1, 2, 4 and 8. There is a jump over artifices '3' and '5, 6, 7'. This jump, would be a jump along third, fifth, sixth and seventh edges coordination of 8 corner points, and parallel to it would be a jump in respect of third, fifth, sixth and seventh geometries of 3-space.
17. The chase of seven edges coordination of a cube, would firstly reverse the orientations from that of first edge to that of third edge, and then further the circular orientation of first three edges would have reversal for it along the last three edges, namely, along fifth, sixth and seventh edges. It is this reversal of orientations firstly at third and secondly at fifth, sixth and seventh edges, which would go disguised for working only with half of interval. The forcing of symmetry by working with half interval is going to be at such heavy structural cost.
18. The seven geometries of 3-space, accept classification as 3 positive geometries, 3 negative geometries and one zero signature geometry. The neutrality of orientation for zero signature geometry is to cause slip with it being its own negative and there by there being four non positive and also four non negative geometries. The working with half interval, as such, in the context is bound to be at the cost of four out of seven geometries in all of three space.

MULTIPLYING BY NINE

For single digit multiplication we always mark the Number by folding the finger down as a marker (as I refer to it as the "Bent Finger" or the "Bend"). In this example 3 x 9, the third finger was bent. So the answer to the first digit is the # of fingers before the bend which is 2 and the second digit is the # of fingers after the bend which is 7, therefore the answer for 3 x 9 is 27.

For double digit multiplication we always mark the First Number by creating a space or "Vshape" (as I refer to it as the "Split"). In this example 38 x 9, a split is created between the third and fourth finger. The Second Number is mark by the Bend at the eighth finger. So the answer to the first digit is the # of fingers before the split which is 3 and the second digit is the # of fingers after the split and before the bend which is 4, and the third digit is the # of fingers after the bend which is 2, therefore the answer for 38 x 9 is 342.

Here is my new discovery on finger computation and the rule for the answer:

In this example 32 x 9, for the First Number "3" we marked it with a split created between the third and fourth finger. The Second Number "2" is marked by the Bend at the second finger. So the answer to the first digit is the # of fingers before the split which is 2 and the second digit is the # of fingers after the split and before the bend, so we have to also include the # of fingers on the left hand which is 1 (finger on left hand before the bend) + 7 (2 on left hand after the split and 5 on right hand) = 8, and the third digit is the # of fingers after the bend which is 8, therefore the answer for 32 x 9 is 288.

The rule is that when multiplying double digits by nine, using fingers only, you will always get a three digit answer. Please remember, you are counting from your left hand to your right hand with this rule. The first digit is the number of fingers standing before the split. The second digit is the number of fingers standing after the split and before the bent finger. In the case of a smaller second digit you will include the number of fingers standing before the bend on the hand or hands that were before the split. The third digit is the number of fingers standing after the bent finger.

Two more examples.

44 x 9 = 396
The split and the bend is at the fourth finger on left hand,
First digit is the 3 fingers standing before the split
Second digit is the 3 fingers before the bend plus 1 finger after the split on the left hand and the 5 fingers on the right hand.
Third digit is the 6 fingers after the bend ( left thumb and 5 fingers on right hand).

87 x 9 = 783
The split is at the eighth (middle) and ninth (ring) finger and bend is at the seventh (index) finger on the right hand.
First digit is the 5 fingers on the left hand and 2 fingers standing before the split on the right hand.
Second digit is the 5 fingers on the left hand and the 1 finger standing before the bend and 2 fingers after the split on the right hand .
Third digit is the 3 fingers after the bend on the right hand.

WHAT YOU WANTED TO KNOW ABOUT VEDIC / ANCIENT INDIAN MATHEMATICS….

THE 10 MOST OFTEN ASKED QUESTIONS….!

INTRODUCTION:
It is a well known fact that our country (Bharath) stands aloft with its rich cultural heritage since thousands of years. Our Ancient Seers & Scientists have made wonderful contributions in the field of Science & Technology also. In spite of these glorious achievements, our people including the so-called educated have a dismal knowledge about the Scientific & Technological achievements and do everything to curb the professional growth of these.
After nearly one and a half decades of dedicated research, I found that the following facts and truths about the Vedic / Ancient Indian Mathematics have to be brought to light for a proper understanding and appreciation of the same in a proper perspective.
The facts are given in a nutshell as under.

Q1. What is 'Vedic' or 'Ancient Indian Mathematics'?
Ans: 'Vedic Mathematics' basically consists of Mathematical concepts, methods and techniques embodied in the Vedic & Upanishadic literature. These are expressed in terms of Mantras and Sutras (Aphorisms), in subtle forms.
Ancient Indian Mathematics refers to post Vedic & medieval times up to the 19th century. Some of the greatest names are Aryabhata, Brahmagupta, Bhaskaracharya (I & II), Sridharacharya, Madhavacharya, Mahaveeracharya, Narayana Pandita etc., The mathematical concepts, methods, postulates, theorems and techniques are expressed in terms of Slokas; which are Lyrically beautiful, but mathematically precise! And also in terms of Sutras & coded literature.

Q2 What is the scope of VM/AIM?
Ans: The scope extends right from the fundamental Arithmetic to Algebra, Geometry Trigonometry, Calculus (Integral and Differential), Biquadratics, solution of Polynomials, Astronomy, Graph theory to some computer based numerical methods. The Nyaya Sastras & MIMAMSA are the veritable source of knowledge in the fields of NLP & NLU and Artificial Intelligence.

03. What is the relevance of VM / AIM in the present world of advanced mathematics and computer science and technology? Is it just historical?
Ans: VM / AIM present a novel, creative approach to understanding and solving the problems; right from the basic to the advanced level and open up new Vistas of knowledge.
a) They are chosen based on their superiority over the conventional methods on the following criteria.
i) Step size ii) Step length iii) Computational time
iv) Elegance and v) Novelty.
b) Incidentally, the Ancient Indian Scientists have discovered many mathematical concepts and techniques, centuries before their modern counter parts!

0 4. What is the methodology of presentation?
Ans: VM / AIM are expressed in terms of modern mathematical notations to make them user friendly and original references are provided to make them authentic.

Q 5. How stable are the methods and techniques?
Ans: They have stood the test of time for centuries and continue to inspire and challenge even now.

Q 6. Does one need to be very good in maths to learn this?
Ans: Not necessary. Students, with lesser aptitude, who used to score low marks, have found it very interesting and user friendly. The scores have improved very much even with little training.

Q 7. What is the feed back from the teachers?
Ans: In fact, they have wholeheartedly welcomed VM / AIM, since they are highly creative, ultra fast, give different dimension to teaching of maths. Reports confirm that, in about 50-60% of the time, the present syllabi can be effectively & efficiently covered! They strongly recommend the introduction of VM/AIM in the school curriculum.

Q 8. What about college students?
Ans: It is a boon to all the candidates of competitive exams. They can clear the paper in about 60-70% of the time, with 100% accuracy.
They can adopt many of these methods in their curriculum like calculus, computer based numerical methods etc. They can also work on the projects based on the application of VM/AIM.

Q 9. In spite of all these benefits, why is it not finding its place in the curriculum?
Ans: i) Apathy towards our own heritage.
ii) People at the helm of affairs have cynical and passive attitude towards VM/AIM; resulting in curbing of genuine researchers.
iii) Lack of organizational support.
iv) Authentic course material and syllabi have to be professionally designed.

Q 10. What is the future of VM/AIM?
Ans: a) The future is very bright. Already about 20 Universities like Birmingham, Philadelphia, Oslo, Zurich, Munchen, Sydney, Heidelberg, London School of Economics etc., have introduced VM/AIM in their curriculum.
b) Serious R & D works are going on.
c) The number of International Conferences on Ancient Indian Science and Mathematics are on steady increase.
d) In India also, barring, a very few organizations supporting R & D works, lot of individuals are doing R & D works on these.
e) There is a tremendous scope for India to lead the world in education to bring the best results.
Incidentally learning of VM/AIM removes the fear psychosis in maths and the whole world is looking towards India for its valuable contribution.
Incidentally, the Scientists have reported that, learning of VM/AIM enhances the right brain activities and makes children more creative.

Come! Let us collectively spread this wonderful Ancient Indian Mathematics through out the world….

MATH YOGA, By John Myers
Ancient Hindu sutras offer intriguing shortcut to solving complex modern
math problems
BOMBAY, INDIA - Math instructor Vivek Astunkar barely caps his pen before 12-year-old Janhavi Shah calls out "one, three, two, one, six, double zero!" She has, in mere seconds, correctly answered Astunkar's whiteboard challenge - multiply 1,120 by 1,180.

Unlike Dustin Hoffman in the film "Rain Man," Janhavi cannot instantly count toothpicks dropped to the floor. In fact, her nine classmates - all normal, middle-class Indian youngsters - confirm her results just seconds later.

They've used a shortcut from Vedic math, an alternative approach to calculation that Indian ascetics may have devised more than 3,500 years ago. These yogic math techniques are mental exercises that can waylay the fear of numbers, build confidence and enhance creativity, says Astunkar.

Vedic math is largely unknown in India, let alone the West. But that's changing. A local newspaper recently counted more than 40 Bombay schools using it. Astunkar, alone, has tutored thousands of students, and also supplies newspapers with weekly Vedic problems. And a few organizations such as Mathvedics, which has franchises in California and Colorado, are pioneering the approach in America.

"School is just studies. This is fun," says Janhavi, who sits at the end of a couch, sandwiched by peers, in Astunkar's small apartment, four floors up in one of New Bombay's mid-rise buildings. It's Sunday morning and the students are happy, even giggling, as they speed through multiplication problems that would make the average adult sweat.

Astunkar, an aeronautical engineer who left his software firm five years ago to teach Vedic math, excels at making it fun. Using a small whiteboard, he writes out problems for the kids to race through, afterwards explaining the Vedic techniques in a flash of scribbling.

He says the methods, derived from devotional Sanskrit verses called sutras, should not replace classic math practices. But, he adds, they offer students fresh, and ultimately faster, ways to calculate.

A Hindu holy man, steeped deeply in Sanskrit and math scholarship, marshaled Vedic math into the modern era. Followers of Bharati Krishna Tirthaji believe he discovered 16 sutras in a long-lost appendix to the Vedas, Hinduism's most sacred religious texts. Using these verses, Tirthaji spent eight years (1911-18) reconstructing formulas that span math's various realms, from basic arithmetic to advanced calculus.

Indeed, Tirthaji enthusiasts claim the 16 sutras encompass all possible mathematical knowledge. But they're left with precious little guidance from their guru. He supposedly wrote one book for each sutra and entrusted them to a disciple. Near the end of his life, however, the works mysteriously disappeared. Tirthaji obliged frantic devotees and rewrote the works from memory into one compressed volume, published posthumously in 1965 under the title Vedic Mathematics.

Average readers will find the book virtually impenetrable. The sutras, simple phrases loaded with meaning, like "All from nine and last from 10," and "By one more than the one before," are spelled out but hardly illuminated. And the resulting math procedures are recorded but left largely unexplained, notes Astunkar.

"I have read a little last night and not understood a word," says his student Janhavi. (Her mother, a college professor of commerce, bought a copy out of curiosity.)

But, for most Indians, the appeal of Vedic math has less to do with its spiritual and cultural roots and more to do with the intense competition facing students on college entrance exams, says Atul Gupta, author of The Power of Vedic Maths.

Gupta's book skims over the underlying sutras, instead featuring more than 1,000 practice problems with detailed explanations of the math techniques themselves. In a workshop Gupta offered in November, all but one of the students were preparing for technical examinations in the spring.

Still, Gupta waxes poetically about the spiritual nature of Vedic math. "The simplicity and the brilliance of the techniques will make you feel humble," he says.

Of course, Vedic math is not without controversy. Many scholars say it's neither "Vedic" nor "mathematics."

Joining 15 math and Sanskrit scholars, Madhav Deshpande, a Sanskrit professor at the University of Michigan, argued as much in a signed letter sent to India's national council on school curriculum. Advocates hope to add Vedic math to the national curriculum, which these critics characterize as arithmetic "tricks."

"I believe as a Sanskrit scholar that whatever its intrinsic merits or lack thereof, the contents of 'Vedic Mathematics' have no historical connection with the Vedas," says Deshpande.

Regardless what Sanskrit scholars make of the Vedic connection, it is full steam ahead for Astunkar, Gupta and other Vedic math teachers who hope to spread the practice. The goal is not to get students mired in Sanskrit, but to get them excited about math, says Astunkar.

Recalling a student who found a way to expand a method for squaring numbers that end in 5 (see sidebar), Astunkar notes, "With Vedic math, their hidden creativity just pops up."

Thanks to Vedic math, Chicago-based writer John Myers can instantly convert any price in rupees to dollars, astounding both chaiwallas and fellow travelers throughout India.

VEDIC MATHS - THE WAY FORWARD

MATHPHOBIA! I have always felt that mathphobia is probably the most common phobia in the world of students surpassing arachnophobia, claustrophobia etc. It is universal, and I feel to a certain extent 'taught'. Though I did not hate math in school, I definitely did not like the way it was taught. I found it too be complex, and boring!

Fast track 20 years; another generation, but math has not changed. It is still being taught the same way as it was during my school days. Therefore I feel there is a need for change and Vedic Mathematics does just that. Its changes the perspective, you can see it from another angle, it makes it more interesting and, I believe, its makes your concepts clearer.

I have now given about a hundred lectures in vedic maths to a wide spectrum of people; from students in schools to the 'ordinary Jane' wanting to know a bit more about this interesting subject to mathematics lecturers. Everytime I have felt that the response to the introductory lecture has been overwhelming. The interest it generates is far from ordinary and I still remember a three day workshop that I took in Kolkata, India, where there was no room for the participants to sit! The organizers expected about a hundred people and double that amount turned up. It was mayhem, chaotic, disorganized, but at the same time, for me, it was fulfilling.

Another incident which I recall was in an up-market school in Mumbai where I was asked to demonstrate the use of vedic maths. I had asked all the teachers to be present, most importantly the maths teachers. I asked for a final examination question paper of class VIII (2 years prior to O level). It was a 2 hours question paper and I asked one of the maths teachers as to how long would she take to complete this paper excluding the geometry construction part. She told me about an hour. I completed the entire paper (arithmetic and algebra) in 20 minutes without writing down a single sum!!!!

What distinguishes vedic mathematics from pure mathematics? Its amazing subtlety, its clarity of concept, its tremendous flexibility and innovative and brilliant structure. You often wonder what unparallel insight Tirthaji, the rediscover of vedic math had, to come up with such a unique vision. As the Hindu scriptures mention,

'whatever is consistent with right reasoning should be accepted, even though it comes from a boy or even from a a parrot; and whatever is inconsistent therewith ought to be rejected, although emanating from an old man or even from the great sage Shree Shuka himself.'

All said and done, 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.

Why Vedic Mathematics, A winner’s
choice?
Vedic Mathematics is emerging as a useful tool for students appearing in
competitive examinations like SAT, iSAT, ACT, CAT, GRE, Engineering
Entrance examinations…. where “Time Factor” plays a crucial role. Vedic
mathematics born as a result of eight years intensive research done by
His Holiness, Jagad guru Swami Sri Bharati Krsna Tirthaji Maharaja.
Today this area of Mathematics functions as a meeting point of ancient
wisdom and modern requirement.
Vedic Mathematics simplifies the four basic mathematical operations like
addition, subtraction, multiplication and division. This will reduce the time
to solve a mathematical problem, especially in examination halls.
For example, if we have to multiply 86 and 98, the conventional method is
86 ×
98
688
774
8428
But by the method of Vedic Mathematics we can do it in a simple way. The
two numbers are set down (Here numbers are 86 and 98) and their
difference from a suitable base are written (Here we can take the base
100) down to the right (that is 100-86 = 14 and 100-98 = 02).
86 - 14
98 – 02
84/28 Ans: 8428
The answer comes out in two parts. The sign ‘/’ is used here to separate
these two parts. To get the first part, cross subtract, either 86 – 2 = 84
or 98 – 14 = 84.To get the second part multiply the difference of the
numbers from the base chosen.
i.e. 14 ´ 2 = 28. Now 28 is the second part of the answer.
Hence the answer is 8428.
It is a useful tool in finding squares and cubes of numbers. In Vedic
Mathematics there is a fantastic method to square numbers ending in 5.
For example Find 652
Conventional method
65 ×
65
325
390
4225
=====
By using the method in Vedic mathematics the answer comes in two parts.
To get the first part find the product of first digit (Here it is 6) and one
more than the first digit (Here it is 7) of the given number (6 x 7 = 42).
To find the second part just put down the square of 5, that is 25.By
combining the first part and second part we will get the answer as 4225When the final digits of two numbers add up to 10, then there is a
shortcut to multiply it. For example 46 x 44.Here sum of the last digits, 6
+ 4 = 10.In this case also answer comes in two parts. The left hand part
can be calculated by multiplying the penultimate digit by one more than
itself, that is 4 x 5 = 20.The right hand part can be calculated by
multiplying the two final digits i.e.6 x 4 = 24.By unifying the two parts we
will get the answer 2024.
By applying Vedic Mathematics in these ways one can reduce the time for
mathematical calculations. This will help to score high marks in
competitive examinations. In today’s scenario all the competitive
examination contains Mathematical aptitude sessions. If the candidate is
going to solve problems in a conventional manner he will take lot of time.
If one moves with Vedic mathematics in a systematic way, then he can
save the time in examination halls. Vedic Mathematics can play a decisive
role not only in Arithmetical problems but also in theory of equations
geometry etc.
For solving quadratic equations we can use some methods of Vedic mathematics. With the help of these methods one can solve the equations
in a lightning speed. For example: Solve
x + 4
4
+
x +1
1
=
x + 3
3
+
x + 2
2
In conventional method we have to solve it by taking L.C.M on both sides.
Then convert it into the form of a quadratic equation. This is a time
taking process. Especially in an examination hall. But in Vedic Mathematics
there is a shortcut. Consider the equation
x + d
d
+
x + c
c
=
x + b
b
+
x + a
a , suchwhat a+b = c+d. Then the roots of the equation are x = 0, x =
2
- (a + b) .
With the help of this method we can arrive at the answer without taking
the pen.
To solve various problems in Analytic geometry Vedic Mathematicsfunctions as a useful tool. General form of the equation of a straight lineis ax+ by + c =0. If two points in a straight line are given, we can find itsequation. Usually we are using two-point form or slope-intercept form tofind the equation of the straight line when two points are given. But thereis a comparatively simple method in Vedic Mathematics to find theequation of a straight line. This can be explained with the help of the following example. Find the equation of a straight line passing through the points (3,4) and (1,2). Standard equation of a straight line is ax + by + c =0. To get the x-coefficient, take the difference of the y co –ordinates ofthe given points. Hence coefficient of x, a = 4-2 = 2. To get the ycoefficienttake the difference of the x –coordinates of the given points. Hence y–coefficient, b = 3-1=2.To find the Constant term c = (x –coordinate of the first point) × (ycoordinate of the second Point)-(x-coordinate of the second point) × (ycoordinate of the first point).Therefore C = 3 ´ 2 -4 ´ 1 = 2.Then equation is 2x+2y+2 = 0

In integral calculus usually one will met with problems related to partial fractions. current systems have a very lengthy procedure to shape the complicated expressions to partial fractions .Suppose we have to express the expression
( )( )( )
2
x l x m x n
px qx r
- - -
+ + in the form of partial fractions.
Let
( )( )( )
2
x l x m x n
px qx r
- - -
+ + =
x n
C
x m
B
x l
A
-
+
-
+
( - ) ( )
Then we can find A, B and C by the following general relations.
A =
( - )( - )
2
l m l n
px + qx + r B =
( - )( - )
2
m l m n
px + qx + r C =
( - )( - )
2
n m n l
px + qx + r
These types of approach will quicken the process to form partial
fractions.
Vedic Mathematics simplifies not only the four basic Mathematical operations, it also simplifies problems from calculus, theory of equations,geometry etc. So it is a necessary element for candidates applying for competitive examinations. In the final analysis we can see that the real losers in the competitive examination are persons without any systematic time management. On the other hand the performers overcome “Time factor” through the systematic tackling of “time traps”. To hear the news of victory, to conquer tomorrows, to keep the presence in the current and upcoming cutthroat competitions, we can find a good companion in Vedic
Mathematics.

References:
Vedic Mathematics by Swami Bharati Krsna Thirthaji
Vedic Mathematics for schools vol-1, vol-2, vol-3 by G.T.Glover

THE FAMOUS SCIENTIST SHRI SATYEN BOSE AND JAGAT GURU SHRI SHANKRACHARYA

Indira Gandhi National Open University (IGNOU), where I happen to be a member on the Board of Management, has entrusted me the job of making a film on India's first Noble laureate and the great Scientist, Shri c. v. Raman. I was doing my studies for the same. When I saw the reference of Bose Einstein theory I was lost in the trip down memory lane of my Calcutta University postgraduate education days. Shri Satyen Bose was the top most Indian Scientist of his time. I was fortunate to be his student in the postgraduate classes of mathematics. If my memory is not failing me, we were hardly four students. Shri Satyen Bose looked like an Indian Rishi in his white Dhoti-Kurta, snow-white hair and beautiful fair complexion, round face and a strong body. It was 1947 and India had just become independent.

I was trying to ride two boats. Astronomy was my hobby and mathematics was helpful in that. I stood first in Mathematics graduation exam of Calcutta University and was a Gold medallist also. Obviously, I was keen to pursue my studies.

It was 1956, two years after the establishment of Sansad. The Puri Shankracharya Jagat Guru Swami, Shri Bharti Krishan Tirthji Maharaj (1884-1960) happened to visit Calcutta. I had heard that he was a great mathematician. I went to offer my pranaams to him. When I told him that I was also a student of mathematics under the famous Scientist, Shri Sat yen Bose, he took keen interest in me. He acquainted me with ancient Indian science of mathematics, which was much more developed than western mathematics, which is only a few hundred years old. He recited a number of Sanskrit hymns on mathematics. A number of large multiplications and divisions were done by him within a few minutes. I was stunned. I posed some questions on differential and integral calculus also and he answered the same in no time.

I got extremely excited and on the next day went to meet Shri Sat yen Bose. I was meeting the Professor after many years. I told him all, about the Jagat Guru. His first reaction was that I am superstitious and have been swayed by a saffron saint. But when I explained to him Jagat Guru's exploits, he gave me a patient hearing. Ultimately, I was able to persuade him to attend a programme being organised by the Sansad where he and Jagat Guru Shankracharya would share a platform and both would give a lecture on mathematics.

I and my other colleagues worked very hard to make this event a success. The notice was sent to various schools and colleges. I and Sh. Dhandharia met some principals personally so that we can ensure the participation of a respectable and knowledgeable audience. The Sansad hall whose capacity was for 200 people was full to its capacity. We placed a huge blackboard on the stage. I introduced Jagat Guru Shri Shankracharya and Shri Sat yen Bose and requested Jagat Guru to open the Conference. He invited people from the audience to write down complicated mathematics questions on the blackboard. Somebody from the audience wrote a long multiplication with about 8 -9 digits. Swamiji wrote the answer within a minute. A number of other people from the audience also wrote various questions, long divisions etc., and Jagat Guru was able to solve them in no time. It looked as if the whole thing was a magic. The programme created a lot of excitement. Shri Sat yen Bose expressed great satisfaction that I insisted and brought him here. After the programme was over, I delved deeper into the subject. The Vedic Rishis got this knowledge by intuition. Swami Vivekananda used to see his Ishtha Mantra shining with a golden hue. Our Rishis were therefore called "Mantra- drishta" and that was their method of acquiring knowledge.

Jagat Guru Bharti Krishanji had worked very hard for eight years to get all this knowledge. The ancient Indian knowledge comprises of four Vedas -Rig Veda, Saam Veda, Yajur Veda and Atharv Veda. Each Veda has four Upvedas and sixteen Vedangs. Sthapatya Veda is the Upveda of Atharv Veda. The science of mathematics comes under that. Jagat Guru obtained this knowledge from this Sthapatya Veda.

Large mathematical questions can be answered in no time. But it does need some practice. jagat Guru illustrated us by giving examples of eight digit multiplications. But I am giving here illustration of simple multiplications so that the reader can understand easily. This is explained by a Sanskrit hymn "Urdhva Tiryagbhyam (vertically-crosswise)".

12
13
156

1) The methodology is that one has to multiply the left digit of the multiplication namely 1 x 1 = 1. This is the left digit of the answer.-
2) Thereafter both the right digits are multiplied, i.e. 2 x 3 = 6. This is the right digit of the answer.
3) For third digit, we have to multiply crosswise namely 1 x3 = 3 and 1 x2 = 2. Thereafter we make a total of the two -three & two i.e. five. This becomes the middle digit of the answer.
The Shloka explains the multiplication quite clearly - Urdhva means vertical and Tiryag means crosswise: accordingly we multiply vertically and crosswise.
Where there are more digits, the multiplication becomes slightly complicated and one has to do certain practice but the answer comes quickly. I am giving one instance of a multiplication of 6471 x 6212.

First step consists by multiplication of 6x6 = 36

Second step consists of Multiplication and addition 6x2 + 6x4 = 36

Third step consists of multiplication of first & third digit and vertical multiplication of the second digit. The three are added The total becomes the next step. 6xl + 6x7 + 4x2 = 6 + 42 + 8 = 56

The fourth step consists of crosswise multiplication of 1st & 4th digit followed by the crosswise multiplication of 2nd & 3rd digits. The four results are added. 6x2 + 6x1 + 4x1 + 7x2 = 12 + 6 + 4 + 14 = 36

The fifth step consists of crosswise multiplication of 2nd & 4th digit followed by vertical multiplication of the 3rd digit.
4x2 + 1x2 + 7xl = 8+2+7 = 17

The sixth step consists of multiplication of 3rd & 4th digits and then their addition 7x2 + 1x1 = 14+1 = 15

The last step consists of multiplication of the 4th digits. lx2 = 2

These seven are added for the final answer: 40197852.

The meeting ended with great excitement. It was an interaction between a great scientist and a great saint. Both were the highest men of their times.

Three years later in 1956, Jagat Guru went to America and left first volume of the book for publication. His plan was to write 16 volumes but unfortunately his health deteriorated and he passed away in 1960. We are deprived of the huge knowledge, which would have been contained in the next 15 volumes.
The Banaras Hindu University took interest in it and with the help of Sh Arvind Mafatlal's donation, this book was published in 1965 under the heading "Vedic Mathematics". 16 hymns are contained in the book and they have been explained in great details. Those who are interested should read this book.

"As science went further and further into the external world, they ended up inside the atom where to their surprise they saw consciousness staring them in the face!"

The ongoing interface between Western science and Eastern mysticism is perhaps the strongest statement in modern times as to the relevance of India's ancient spiritual wisdom. That the Upanishads are influencing the reigning paradigm of modern science is good reason to look more deeply within their pages for insight in today's world.

A conference sponsered by the Bhaktivedanta Institute in San Francisco centered on the study of of consciousness within science. The Institutes international secretary, Ravi Gomatam, shared with us what he calls the third wave of the ongoing interface between science and mysticism.

Can you tell me something about the Bhaktivedanta Institute?

The word Bhaktivedanta itself connotes the synthesis of science and consciousness. Vedanta represents the rational, intellectual side, and bhakti represents the holistic, subjective inner side. The institute promotes studies and discussions on the need for and development of consciousness-based paradigms to outstanding problems in science. The Institute consists of fifteen well-trained professionals, mostly scientists and a few engineers. Our main branch is in Bombay, and we have only recently begun to hold programs in the West.
Our in-house research is based on specific paradigms for consciousness that are available within the Bhagavat tradition of Vedanta, or theistic Vedanta. We also offer research fellowships through which academic people can interact with us, and we hold broad-based conferences and workshops.

When we do conferences we recognize that the topic of consciousness is a very difficult one to deal with. Consciousness has occupied the attention of mankind for thousands of years. As conscious beings we have wondered about our essential nature, our place and our relationship to the universe in which we find ourselves, our rights, and even what are our duties—especially as we see today so many problems caused directly and indirectly by the application of science. No one can claim at this point that he has a final answer to these questions. Consequently our conferences are very broad-based. We bring together a wide variety of thoughts from different disciplines of science, and we provide a forum for discussion so that some kind of a scientific consensual understanding of consciousness can emerge on its own. Although we have our roots in India's spirituality, our work itself is very contemporary and highly objective.

How do you view the evolution of the ongoing interface between modern science and Eastern mysticism?

Capra on one hand should definitely be credited for putting the subject into the center of the stage. His work was the first wave. His essential point was that the scientific tradition and the mystical traditions are two different approaches to understanding the same reality. He managed to draw some parallels between the emerging concerns of science and existing world views of Eastern mysticism. Despite the importance of his work that started this trend, his drawing of parallels was very superficial. For example, his conjecture that the tracks that sub-atomic particles leave on a photographic plate are the dance of Shiva is really pseudo-science. He had a fair understanding of physics and, for those times, a reasonable introduction to Eastern mysticism. His ideas were commercially successful, revealing that there was a large audience for this topic, and they pointed the direction in which further exploration could be made.

The second wave, the work of Ken Wilber and others, recognized the shortcomings of Capra, Zukav, and the like. They showed that the issues of spirituality, whether Christian mysticism, Sufism, or the Vedic tradition, are dealing with a different ontology than that of modern science. Thus Ken Wilber strongly argued that we should not think that science is going to lead directly to the same understanding of reality as that afforded by mysticism. At best science could point towards the need for cultivating mysticism, for which we would then have to shift gears. This was the second wave.But the problem with this approach, although true in the ultimate sense, is that it does not chart specific pathways by which science can come closer to consciousness. Indeed, it even precludes the possiblity of an expanded science that can on day legitimately study consciousness directly. In cleaving the two in this way, in a sense, Wilber reintroduced a kind of Cartesian dualism. Instead of the mind/body problem, it became the spirituality versus science problem. This dilemma then formed the motivation for our recent conference—the third wave.

This third wave, as I see it, will begin due to the willingness on the part of scientists themselves to expand the domain of science in very new ways. The motivation for this is already coming from results in established fields, such as artificial intelligence, molecular biology, theoretical physics, as well as new emerging fields like engineering anomolies. Through these fields the causal role of consciousness in the physical world at deeper levels of matter is becoming established. What is required is to sustain this investigation so that a logical framework for discussion of consciousness results naturally within science. In the process science will doubtless discover a new middle ground between what it now thinks of as matter and what the mystics describe as consciousness. It will involve discovering levels of subtle matter presently unknown to science. This new science will become the empiric evidence for, and system by which we can better explain the causal role of consciousness. No doubt, this will require new tools of theory and experiment. Our own contribution is to facilitate this process of discovery.

That's quite a challenge for science.

Well if we survey the history of modern science we will see that major advancements came when scientists succeeded in integrating seemingly disparate phenomen. Newton, Maxwell, and Einstein are good examples.

Newton's success was that he integrated stellar motions with movements of ordinary bodies on Earth. It was a grand synthesis that launched Newtonian physics. Newtonian physics had an ontology, or mode of existence of things. In it the fabric of the universe was particles: small particles that constantly acted, reacted, and collided with one another according to very precise laws. The first synthesis was that of motions, small motions and big motions. That was considered a big success. Imagine the euphoria they experienced when they realized that an object falling from the Leaning Tower of Pisa followed the same laws that the sun follows! It was soon shown that these laws of motion could be used to understand not only the behavior of solids, but also liquids, and then gases. In this way the behavior of the entire macrocosm and microcosm was thought to be within our grasp. The second major synthesis came when Maxwell unified the concepts of electromagnetic phenomena and light.

People may be surprised to know that toward the end of the 19th century scientists thought there were no more fundamental laws to be discovered; just do more and more mathematics and everything would be explained. It was the famous physicist Lord Kelvin who said that there were only two small clouds on the horizon: "black body radiation" and "ether drift." But these turned out to be bigger than scientists thought.

In this century the two great leaps science has taken concern these two phenomena. One was Einstein's integration of space and time into one space-time continuum, which explained the absence of ether drift. The second great leap was quantum mechanics. It brought us a connection between two seemingly separate realms—physical measuring devices and human observers. The point I am making is that science has made great steps when apparently disparate phenomena were brought together under one roof. Now the time is ripe to bring together yet another pair—mind and matter. But this too requires a new conceptualization. This is now what we are attempting—to bring together science and consciousness, and take another giant step. With the development of quantum mechanics it became clear that the theory had a fundamental problem. The quantum theory has no ontology. It does not concern itself with what the world is made up of. It doesn't start with an assumption about the world's makeup and then build a theory. Rather, it talks about probabilistic connections between successive observations not the events themselves.

As Heisenberg pointed out, "Quantum theory no longer speaks of the state of the universe, but our knowledge of the state of the universe." For the first time scientists had a theory that ultimately had no objective foundation. That this may be because quantum theory does not satisfactorily account for consciousness has been pointed out by the founding fathers of quantum theory, Eugen Wigner and John von Neumann, but this line of reasoning has not been adequately pursued.

There are also other areas within science besides quantum mechanics where consideration of consciousness has become central. Artificial intelligence is an example, where the initial mood was very similar to Newtonian hubris. Newtonian physicists thought everything in the world could be explained in terms of laws governing basic motions. Similarly, artificial intelligence researchers thought that all aspects of human cognition could be explained simply in terms of rules governing our behavior. But soon AI researchers found that even the simplest aspects of human cognition could not be reproduced. Now they understand that to suceed in AI we need a basic understanding of human consciousness. In psychology too, behaviorism has proven to be insufficient, and what was called introspective psychology is coming back into fashion.

So our institute is promoting the examination of overtly consciousness-based approaches to these problems within science today. Consciousness has been talked about within science in the past, but always with a view to explain it away rather than explain it. Accepting that consciousness has a causal role in the world is a very bitter medicine for scientists to swallow, but they are beginning to do it. And metaphysicists are also beginning to see that while there is undeniable reality to the subjective dimension, any system claiming to explain it must bear relevance to the objective concerns of empiric science. This is the challenge: to answer the pressing questions arising in science that call for consideration of consciousness with genuine consciousness-based paradigms.

How did you choose your speakers for the panel?

The first thing I did was contact Sir John Eccles. Eccles is very much known for his open stand that mind is different from the brain. Eccles was described by Libet as one of the five top neuroscientists of the century. When he says that brain is different from the mind, in the very least you cannot tell him that he does not know about the brain. He was the first to accept, which he did immediately. Once he agreed, everything else fell into place. We had to choose both theorists and experimenters. Data in this field is very, very rare. We chose two people to present data that were from opposite camps. Benjamin Libet from UCSF had data which seems to show that in some cases our apparent actions of free will, such as when our hand moves spontaneously to set the clock, may well be merely action triggered by the brain a full half second before we desired to lift our hand. According to this data, our free will may well be an after thought! There are other ways to interpret his data, and Libet is the first to admit that his data deals at best with local intentionalities, not global free will. Robert Jahn and Brenda Dunn presented data that shows the opposite, that consciousness has intentionality. These were the experimenters. Although Pribram and Eccles might consider themselves experimenters as well, they presented no data. The rest of the panel consisted of theorists of different fields: neuroscience, psychology, physics, artificial intelligence, mathematics, and philosophy.

You mentioned that there is not much data in this field to draw from. What about the data in neurscience?

Yes. This point was also raised during the panel discussion. It was Pribram who complained that not enough of the existing data was sufficiently discussed at the conference. But John Searle came up with the best rejoinder when he said that the problem of discussing data collected thus far is that all this data was gathered specifically to demonstrate that consciousness does not exist. Therefore how can we speak of consciousness and use this data? First we need to do new research.

The difficulty is that science always goes by an operational definition. In order to make any concept scientific, you must have an operational definition, because then it becomes falsifiable and hence becomes scientific. An operational definition is in itself an interesting concept. What it really means is that you can propose any phenomena, like Newton proposed gravitation, but it must be eventually corelated to some adhoc physical measurements. Consciousness, however, is by definition the one that measures, the one that does the observation. So how are you going to give an operational definition of it?

I think the answer lies in seeing that the interaction between consciousness and gross matter involves subtle levels or realms of matter where other kinds of measurement than the ones that we are presently aware of can be made. The work of Robert Jahn and others are the kind of experiments in which more precise operational definitions of phenomena that are closer to consciousness than gross matter, namely mind, can be talked about. If we learn to see other orders of existence between consciousness and gross matter, such as mind and intelligence, then scientists might be better able to conceptualize the ultimate phenomena.

Why have scientists been so reluctant to discuss consciousness in the past?

Did you know that before Rutherford split the atom in 1911 scientists considered the question of what an atom is a religious question?! For them it was enough that the hypothesis of the atom was useful to explain certain physical processes. Kekule, who discovered the structure of benzene said, "The question of whether or not atoms exist has little signifigance from a chemical point of view; its discussion belongs rather to metaphysics." But today the study of what's inside the atom is physics!

Similarly, scientists in this century have regarded the issue of what consciousness is as a religious or metaphysical question. After all, Western science started out as a protest against religion. Since religion went inward, science saw its own task as going outward. But as science went further and further into the external world, they ended up inside the atom where to their surprise they saw consciousness once again staring them in the face!

Even then scientists thought a hypothesis about consciousness was all that was needed. However, just as the study of the atom has become what we call physics today, the study of what consciousness is, I feel, may very soon become the science. William James said

"When science comes to eventually understand consciousness it will be an achievement in the face of which every other achievement of science will pale into insignifigance."

Many scientists equate mind and consciousness. Yet in your personal presentation at the conference you described mind as subtle matter, different from consciousness. What is your conception of mind, matter, and consciousness?

In my talk, I approached the issue of consciousness from the perspective of AI. The first step here is to show the need for a new paradigm. That artificial intelligence needs a new paradigm has become apparent from the variety of intractable problems in cognition we face in areas such as perception, natural language processing, knowledge representation, and automatic reasoning. We have no general theory of computation yet that can produce human cognition in machines. A task that comes naturally to a one year old child—recognising the face of his or her mother—is hopelessly beyond the capacity of supercomputers. What's required is not just some new hardware/software schemes, but a fundamentally new technology.

To understand what I mean let's compare electronic computers with mechanical calculators. Both are symbol processing systems. In principle, a mechanical system of gears and levers can be constructed to reproduce the workings of any electronic computer. In practice, however, this will not be possible. A mechanical system equivilent to even the simple desktop computer would be so enormous as to fill the entire planet and consume power that all the coal mines on earth cannot supply! This advantage of speed, power, and size is present in electronic computers because IC chips involve operation of matter at a much subtler level, obeying laws of a different kind from mechanical systems. You cant hope to make smaller and smaller mechanical parts and reach IC technology.

Similarly, AI researchers today think that by making IC chips smaller and smaller we will eventually come to mind. But I argue that you can't do that. You have to go to another level to talk about mind. I am postulating different levels of matter. I am suggesting that we have to think of mind as a subtler level of matter that operates much faster and under different laws than IC chips. You cannot reach that level through nanotechnology.

Professor Bremmerman at UC Berkeley has shown that there are absolute limits to infromation processing in physical systems regardless of the details of their internal construction. For example, given a computer of total mass m, the maximum information it can ever process is mc2/h bits/second, where h is the plank's constant. He has gone on to show that even if we consider a computer that has been in operation for the duration of the entire universe, assuming that it has been in operation for the duration of the present age of the universe, its total information capacity will not be enough to solve a travelling salesman's problem involving no more than 100 cities! The conclusion is that the human brain, being a physical device, is subject to the same absolute limitations, irrespective of its internal construction. If the brain alone was involved in human cognition, we should not be able to carry out the kind of complicated cognitive operations that we do! Therefore, I have argued that what is involved in human cognition is information processing involving levels much faster and hence subtler than the brain.

If you accept this idea, that there is more to human cognition than the brain function, then there is already a model of consciousness, intelligence, mind, and brain in the Vedantic texts that closely follows these requirements. This Vedantic model describes mind as a level of matter subtler than the brain. According to this model, thought is to mind what motionis to objects, or beavior is to the body. That is, thoughts have no intrinsic semantic content. An example of this is when a driver drives a car. The idea of the journey is not intrinsic to the car's motion, but a superimposition on the part of the driver. Similarly, meaning is not intrinsic top thoughts of the material mind, but is a superimposition of subjective consciousness.

This idea, that thought is a mechanical output of matter at the subtle level of mind without intrinsic meaning is a novel idea within Western tradition. If this idea can be shown to be of practical relevance to AI, then I feel we can go one step nearer to the paradigm of consciousness, otherwise, to ask current science to jump directly to consciousness is too much. This is a necessary step in what I have mentioned about the third wave—finding the middle ground between consciousness and matter, and thus expanding the domain of current science.

What is the difference between Cartesian dualism and the Vedantic dualism you are discussing?

Descartes said, "I am that, that thinks, the soul, or the reason, or the understanding." He used all of these terms equivalantly. Thinking, reasoning, and soul were all the same for him. This is the problem with Cartesian dualism—that it lumped into one concept called mind all hierarchic cognitive traits. That is why Cartesian dualism has no relevance for science, whereas the Vedantic pluralism—in terms of consciousness, mind, and body—seems to give ideas about the presence of various levels of hierarchy in matter.

If you see a car moving on the street and you want to know why it's turning left or right, one might say, "All you need to do is study the mechanics of the car. The car is a complete system; there is nothing inside." But I come and say no, there is a driver in there. Now that is correct, but it's not sufficient. Still you have to accept that there are several levels of mechanisms within the car, and there is a specific point at which the driver is coming in contact with the car, the steering wheel and control panel. Descartes was correct in thinking that there is an irreducible subjective residio that is essentially the self. That is exactly the same as the Vedic idea tat tvam asi, thou art that. But Descartes was not able to distinguish that there is a subtle material substance called mind that is the point at which consciousness meets matter. There is a hand and there is a glove. The glove is exactly like the hand but it is a cover. So the mind is very close to consciousness but it is matter.

The Vedanta also has a monistic interpretation, monistic idealism if you will. In Shankara's view there is no objective reality to matter. It is all illusion. You hold a very different viewpoint on Vedanta.

Yes. There is a very established tradition of Vedantic thought, monism, that is close to idealism. We are proposing something different,a multidimensional, pluralistic approach to the whole issue of reality. We are talking about individual consciousness and a supreme consciousness or God. We are also talking about matter as an objective reality, the shadow of consciousness, rather than an illusion or something that really does not exist. This is theistic Vedanta.

The question is which Vedantic paradigm can import concepts that can be shown to be empirically and analytically accountable. I do not think that monism can explain any of the problems of consciousness in science in a way relevant to science simply because, according to the monistic viewpoint, in the ultimate analysis matter doesn't exist. Therefore the highesr realizations of monisim by definition can not have any bearing on modern science, which studies the domain of matter.

It seems that in attempting to bring consciousness into science, rather than keep the two separate, you are attempting to bring value into a somewhat valueless technological world view.

I certainly hope so. Today science is totaly without a framework for values. Any highschool boy or girl knows how to calculate the force with which a stone he or she throws will hit someone in the face, but nothing in those equations they use will tell them whether or not to throw it. Given the fact that science is perturbing our universe in greater and greater proportions, it is essential that we address the absence of values within science. We must note that the changes wrought by science and technology to our environment are always irreversible. That is to say we cannot go on polluting our environment for years and then one day suddenly say "Oops, that was a mistake, let's take it back." It is easy to destroy something, but much more difficult to put it back together again.

To solve the problem of values we must know what is valuable. Consciousness is the most valuable commodity. Without consciousness our own bodies as dear as they are to us, are suddenly without value. This of course is a philosophical argument, but nonetheless an pragmatic one. If we accept it, then, to bring values into science,we need to connect science with what is valuable—consciousness.

Cairns Smith is well known for his work in the field of chemical evolution. I was quite surprised to hear some of his remarks about consciousness. What is the Vedic view on evolution?

Darwinian evolution is biological. It talks about the needs of the biological system by which evolution proceeds. But it is inadequate to explain the appearance of the first biological system. Therefore we have theories of chemical evolution which precede biological evolution. Cairns Smith, as a chemical evolutionist, was pointing out that consciousness is fundamentally different from all other physical phenomena because it acts back on the system that creates it. Consciousness has a two-way interplay that Smith called interactionism. His realization was that this interactionism must be present at the most fundamental level of matter. It cannot evolve suddenly in matter.

He went to the extent of assreting that "To say that consciousness evolved from matter is to say that a TV evolved from a refrigerator. Such things do not happen." He therefore postulated what he calls protoconscious units, which are not themselves conscious, but have the potential for consciousness that molecules and atoms don't have. However, in doing so he himself is dodging the issue. If protoconscious units are not conscious, then they have the same defect as matter in that they can't give rise to consciousness. If they are conscious, then why not call them consciousness rather than protoconsciousness? This is the same thing that Minsky tried to do in his book Society of Minds. He tried to show that there are certain things called minds that are not really minds, but when they all get together, then you get mind. This degenerates ultimately into philosophical emergence, where something comes out at the top of a structure that is not at the bottom of the structure. So you can see that even materialists invoke some fundamental conscious-like units different from known matter in an attempt o explain consciousness.

We can congratulate Cairns Smith for boldly recognizing the conceptual limitations of chemical evolution, but he has not yet taken the next step, which is to postulate consciousness as a separate ontological category coexisting along with matter. This is what I feel scientists in every field should do to solve the problem of consciousness in there respective fields. It won't suffice for scientists to assume that once we posit something as non-material that we cannot study it. We simply have to develop new scientific tools.

As far as the Vedic viewpoint on the different levels of consciousness within different species, I once explained this to Wigner. According to the Vedas, just as matter has fundamental particles called atoms, so consciousness is full of fundamental particles called cit kana. While every material atom is unconscious and therefore devoid of individuality, every spiritual particle is conscious, and therefore it has to be individual. Individuality is a fundamental axiomatic property of consciousness. Material atoms are governed by the laws of physics, and spiritual atoms are governed by love because they are units of free will.

I explained to Wigner that each unit of consciousness interacts with matter, and we see its capabilities manifest in accordance with whichever material machine or body it interacts with. If you drive a motorcycle and I drive a bicycle, you may go faster than me only because of the vehicle. It has nothing to do with you or I but the vehicles we are using. He asked me if I thought an amoeba had consciousness. I told him that the Vedas do not say that an amoeba has consciousness, but rather that consciousness has an amoeba body! Just as in each vehicle you see on the road there is a different driver, similarly in each body there is an individual conscious entity. According to the Vedas, all species exist at all times. Material bodies do not evolve. But each individual conscious entity evolves, thus acquiring different bodies which correspond with the individual's particular state of conscious evolution..

This paradigm is not contra-intuitive, and different Western schools of thought can be accomodated within it. Take for example reductionism, which claims that our behavior is essentially controlled by the physical laws operating on our bodies. The Vedantic viewpoint accepts that even though I am a conscious individual transcendent to the body, because I am using this particular body, I am constrained by its operation according to material laws. Thus reductionism can be accomodated within this framework.

You can talk also of emergence. The more sophisticated my physical structure is, the more I can show my skills. Higher order structures will show higher order properties, not intrinsically but extrinsically because consciousness can manifest more of its qualities. Dualism is also accommodated because the Vedic paradigm admits that consciousness and matter are different. Phenomenology, which says that beingness is an essential aspect of every structure that has consciousness, can be accommodated.

In short this Vedic model is the proverbial elephant of which different portions are being touched by so many blind men. One blind man says that it is rationalistic, another dualistic, another idealistic monism, another realism, but no one is seeing the entire elephant of this Vedic paradigm. The elephant is that there are two ontological categories, consciousness and matter, and the two interact to form our world.

Can't you also say that matter is a vitiated form of consciousness, that everything is ultimately consciousness?

This involves a higher philosophical discussion. I can see that at some level of God consciousness we can think of consciouness and matter in these terms—as you put it, seeing matter as a vitiated form of consciousness. But presently that vitiated form of consciousness acts differently as matter, and therefore it can be considered as a separate ontological category.

As the discussion of the conscious self enters the scientific arena it seems that we are at a critical juncture. What is the future of science?

I don't think that I can do better than to quote scientists who are greater than myself, who at the ends of their careers have given some reflections. I have some favorite quotes. W. Penfield, one of the top neuroscientists of the century, said in an article called Science, the Ox, and the Spirit:

"The physical basis of the mind is the brain action in each individual. It accompanies the activity of the spirit, but the spirit is free. It is capable of some degree of initiative. The spirit is the man one knows. He must have continuity through periods of coma and sleep. I assume then that the spirit must live on somehow after death. I cannot doubt that many make contact with God and have guidance from a greater spirit. If he had only a brain and not a mind, this difficult decision would not be his."

The tendency to see the human mind in terms of the latest technology of the times is an old one. In earlier times mind was thought of as a steam engine, as a clock, and before that as a catapult. Today the attempt is to equate mind with the brain. But here is something from Ludwig Wittgenstein from his Last Writings on the Philosophy of Psychology: "Nothing seems more possible to me than that people some day will come to the definite opinion that there is no copy in the nervous system which corresponds to a particular thought or to a particular idea of memory."

Szent-Giorgi, the Nobel laureate biologist, said,

"I went through my entire scientific career searching for life, but now I see that life has somehow slipped through my fingers and all I have is electrons, protons, and particles, which have no life at all. So in my old age I am forced to retrace my steps."
So I think the great advantage of discussing the notion of the conscious self within our scientific paradigms is that we can actually enlarge our framework. In order to do that we need help, and I don't think that anyone can deny that the Vedic literatures are the single most vast body of literature that seriously deals with this topic. From page one to the end it is conscious all the way.

Science, as long as it remains bound to emperical reductionism, can say nothing about the conscious self. Many in the contemporary world have tried to define perception such that it fits into their existing paradigms, but this has only made our problems more accute. Time has come to redefine scientific procedures such that they explain the conscious self. We need as many new ideas as we can get. If we are so foolhardy as to reject the entire wisdom preceeding us, such as the Vedic paradigm I have presented, then what assurance do we have that our present-day knowledge will not similarly be rejected by future generations?

Science is rooted in observations, and our conscious self is the very tool by which we observe. Even the strongest giant can not lift the platform on which he stands. As great as scientific knowledge is, it cannot explain the conscious self within its present observational framework. To experience it is to observe it.

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This journal provides a forum for research on the forefront of mankind's expanding knowledge of the universe. It is devoted to exploration of the unified field of all the laws of nature through the combined approaches of modern science and ancient Vedic science, as brought to light by Maharishi Mahesh Yogi. The identification of the unified field by modern physics is only the first glimpse of a new area of investigation that underlies all disciplines of knowledge, and which can be explored not only through objective science but through a new technology of consciousness developed by Maharishi.
The unified field is now beginning to be understood through modern physics as the unified source of the entire universe, as a unified state of all laws of nature from which all force and matter fields sequentially emerge according to exact dynamical principles. As each science and each academic discipline progresses to uncover its own most basic laws and foundational principles, each is beginning to discover that the roots of these laws and principles can be traced to the unified field.

This journal recognizes a new method of gaining knowledge of the unified field that combines the approach of the modern sciences with that of the most ancient of sciences, the ancient tradition of Vedic science. Many thousands of years ago, the seers of the Himalayas discovered, through exploration of the silent levels of awareness, a unified field where all the laws of nature are found together in a state of wholeness. This unity of nature was directly experienced to be a self-referral state of consciousness which is unbounded, all-pervading, unchanging, and the self-sufficient source of all existing things. They experienced and gave expression to the self-interacting dynamics through which this unified field sequentially gives rise to the diversity of all laws of nature. That experience is expressed in the ancient Vedic literature.

In our own time, Maharishi has brought to light the knowledge of this ancient science and integrated it with the modern sciences in such a way that Vedic science and modern science are now seen as complementary methods of gaining knowledge of the same reality-the unified field of all the laws of nature. The knowledge of this ancient science that Maharishi has brought to light is known as Maharishi's Vedic Science.

Maharishi's Vedic Science is to be understood, first of all, as a reliable method of gaining knowledge, as a science in the most complete sense of the term. It relies upon experience as the sole basis of knowledge, not experience gained through the senses only, but experience gained when the mind, becoming completely quiet, is identified with the unified field. This method, examined in relation to the modern sciences, proves to be an effective means of exploring the unified field of all laws of nature. On the basis of this method, complete knowledge of the unified field becomes possible. It is possible to know the unified field both subjectively on the level of direct experience through exploration of consciousness and objectively through the investigative methods of modern science. Vedic Science gives complete knowledge of consciousness, or the knower, complete knowledge of the object known, and complete knowledge of the process of knowing. In knowing the unified field, all three-knower, known, and process of knowing?re united in a single unified state of knowledge in which the three are one and the same.

Maharishi has developed and made available a technology for the systematic exploration of the unified field. This technology is a means by which anyone can gain access to the unified field and explore it through experience of the simplest and most unified state of consciousness. As this domain of experience becomes universally accessible, the unified field becomes available as a direct experience that is a basis for universal knowledge. The technology for gaining access to the unified field is called the Maharishi Technology of the Unified Field, and the science based on this experience, which links modern science and Vedic Science in a single unified body of knowledge, is called the Science of Creative Intelligence.

Maharishi is deeply committed to applying the knowledge and technology of the unified field for the practical benefit of life. He has developed programs to apply this knowledge to every major area of human concern, including the fields of health, education, rehabilitation, and world peace. These applications of the Maharishi Technology of the Unified Field have laid it open to empirical verification and demonstrated its practical benefit to mankind. Hundreds of scientific studies have already established its usefulness. From these results it is clear that the Maharishi Technology of the Unified Field is far more beneficial than technologies based on present day empirical science; it promises to reduce and even eliminate war, terrorism, crime, ill health, and all forms of human suffering.

The Maharishi Technology of the Unified Field, the applied value of Vedic Science, represents a great advance in methods of gaining knowledge. Past science was based on a limited range of knowledge gained through the senses. This new technology opens to mankind a domain of experience of a deeper and more far reaching import. It places within our grasp a new source of discovery of laws of nature that far exceeds the methods of modern science yet remains complementary to these methods.

Modern science and Vedic Science, explored together, constitute a radically new frontier of knowledge in the contemporary world, opening out vistas of what it is possible for mankind to know and to achieve, which extend far beyond present conceptions, and which demand a revaluation of current paradigms of reality and a reassessment of old conceptions of the sources and limits of human knowledge.

This introductory essay will provide a preliminary understanding of what the unified field is, what Vedic Science is, and how Vedic Science and modern science are related. It also defines fundamental concepts and terminology that will be frequently used in this journal and surveys the practical applications of this new technology. We begin with a description of the unified field as understood in modern science.

The Unified Field of Modern Science
Within the last few years, modern theoretical physics has identified and mathematically described a unified field at the basis of all observable states of physical nature. Einstein's hope of finding a unified field theory to unite the electromagnetic, gravitational, and other known force fields has now been virtually realized in the form of unified quantum field theories. Instead of having several ir reducible and distinct force fields, physics can now mathematically derive all four known force fields from a single supersymmetric field located at the Planck scale (10(-33cm) or 10-(43sec.), the most fundamental time-distance scale in nature. This field constitutes an unbounded continuum of non-changing unity pervading the entire universe. All matter and energy in the universe are now understood to be just excitations of this one, all-pervading field.

Physics now has the capacity to accurately describe the sequence by which the unified field of natural law systematically gives rise, through its own self-interacting dynamics, to the diverse force and matter fields that constitute the universe. With a precision almost undreamed of a few years ago, the modern science of cosmology can now account for the exact sequence of dynamical symmetry breaking by which the unified field, the singularity at the moment of cosmogenesis, sequentially gave rise to the diverse force fields and matter fields. It is now possible to determine the time and sequence in which each force and matter field decoupled from the unified field, often to within a precision of minute fractions of a second. This gives us a clear understanding of how all aspects of the physical universe emerge from the unified field of natural law.

Mathematics, physiology, and other sciences have also located a unified source and basis of all the laws of nature in their respective disciplines. In mathematics, the foundational area of set theory provides an account of the sequential emergence of all of mathematics out of the single concept of a set and the relationship of set membership. The iterative mechanics of set formation at the foundation of set theory directly present the mechanics of an underlying unified field of intelligence that is self-sufficient, self-referral, and infinitely dynamic in its nature. Investigations into the foundations of set theory are ultimately investigations of this unified field of intelligence from which all diversity of the discipline emerges in a rigorous and sequential fashion. In physiology, it is the DNA molecule that contains, either explicitly or implicitly, the information specifying all structures and functions of the individual physiology. In this sense, therefore, it is DNA that unifies the discipline by serving as a unified source to which the diversity of physiological functioning can be traced.

Each of the modern sciences may indeed be said to have glimpsed a unified state of complete knowledge in which all laws of nature are contained in seed form. Each has gained some knowledge of how the unified field of natural law sequentially unfolds into the diverse expressions of natural law constituting its field of study. Modern science is now discovering and exploring the fundamental unity of all laws of nature.

Vedic Science
Maharishi's Vedic Science is based upon the ancient Vedic tradition of gaining knowledge through exploration of consciousness, developed by the great masters in the Himalayas who first expressed this knowledge and passed it on over many thousands of years in what is now the oldest continuous tradition of knowledge in existence. Maharishi's work in founding Vedic Science is very much steeped in that ancient tradition, but his work is also very much imbued with the spirit of modern science and shares its commitment to direct experience and empirical testing as the foundation and criteria of all knowledge. For this reason, and other reasons to be considered below, it is also appropriately called a science. The name "Vedic Science" thus indicates both the ancient traditional origins of this body of knowledge and the modern commitment to experience, system, testability, and the demand that knowledge be useful in improving the quality of human life.

The founders of the ancient Vedic tradition discovered the capability of the human mind to settle into a state of deep silence while remaining awake, and therein to experience a completely unified, simple, and unbounded state of awareness, called pure consciousness, which is quite distinct from our ordinary waking, sleeping, or dreaming states of consciousness. In that deep silence, they discovered the capability of the mind to become identified with a boundless, all-pervading, unified field that is experienced as an eternal continuum underlying all existence. They gave expression to the self-sufficient, infinitely dynamic, self-interacting qualities of this unified state of awareness; and they articulated the dynamics by which it sequentially gives rise, through its own self-interacting dynamics, to the field of space-time geometry, and subsequently to all the distinct forms and phenomena that constitute the universe. They perceived the fine fabric of activity, as Maharishi explains it, through which this unity of pure consciousness, in the process of knowing itself, gives rise sequentially to the diversity of natural law and ultimately to the whole of nature.

This experience was not, Maharishi asserts, on the level of thinking, or theoretical conjecture, or imagination, but on the level of direct experience, which is more vivid, distinct, clear, and orderly than sensory experience„perhaps much in the same way that Newton or Einstein, when they discovered the laws of universal gravitation or special relativity, enjoyed a vivid experience of sudden understanding or a kind of direct "insight" into these laws. The experience of the unified field of all the laws of nature appears to be a direct experience of this sort, except that it includes all laws of nature at one time as a unified totality at the basis of all existence?n experience obviously far outside the range of average waking state experience.

The ancient Vedic literature, as Maharishi interprets it, expresses in the sequence of its flow and the structure of its organization, the sequence of the unfoldment of the diversity of all laws of nature out of the unified field of natural law. The Veda is thus to be understood as the sequential flow of this process of the oneness of pure consciousness giving rise to diversity; and Vedic Science is to be understood as a body of knowledge based on the direct experience of the sequential unfoldment of the unified field into the diversity of nature. It is an account, according to Maharishi, of the origin of the universe from the unified field of natural law, an account that is open to verification through direct experience, and is thus to be understood as a systematic science.

These ancient seers of the Vedic tradition developed techniques to refine the human physiology so that it can produce this level of experience, techniques that were passed on over many generations, but were eventually lost. Maharishi's revival and reinterpretation of ancient Vedic science is based on his revival of these techniques which have now been made widely accessible through the training of thousands of teachers of the Maharishi Technology of the Unified Field. He has thus provided a reliable method of access to this field of direct experience where the oneness of pure consciousness gives rise to the diversity of the laws of nature; and he has also developed applications of this technology that render it open to experimental testing. These applications will be considered below.

Maharishi describes the experience of this unified field of consciousness as an experience of a completely unchanging, unbounded unity of consciousness, silently awake within itself. Gaining intimate familiarity with the silence of pure consciousness, Maharishi holds, one gains the ability to experience within that silence an eternal "fabric" or "blueprint" of all laws of nature that govern the universe, existing at the unmanifest basis of all existence. This unmanifest basis of life, where all laws of nature eternally reside in a collected unity, is experienced as the fabric of the silent field of consciousness itself, which is not in space and time, but lies at the unmanifest basis of all manifest activity in space and time. Through Maharishi's work, this experience comes to be understood (as we see below) as a normal state of consciousness that arises in the natural course of human development.

Glimpses of this universal domain of experience where all possibilities reside together in an eternally unified state have been reported in almost every culture and historical epoch, from Plato to Plotinus and Augustine, and from Leibniz to Hegel and Whitehead. Scientists like Kepler, Descartes, Cantor, and Einstein also appear to have written of it and seemingly drew their insights into the laws of nature from this experience. Descartes writes, for example, of an experience that he had as a young man of "penetrating to the very heart of the kingdom of knowledge" and there comprehending all the sciences, not in sequence, but "all at once." Scientists and writers from many traditions have described this experience of unity, which confirms that it is completely universal, and not a product of a particular cultural tradition. Just as the Vedic tradition has been misunderstood, however, so have those descriptions of consciousness found in these different cultural traditions; for without a technique that makes the experience systematically accessible to everyone, the understanding that this is a universal experience of the most fundamental level of nature's activity has been obscured, and has not before now emerged into the light of universal science.

According to Maharishi's Vedic Science, it is not only possible to gain direct experience of the unity of natural law at the basis of the manifest universe, but one can also directly experience the unity of nature sequentially giving rise to the diversity of natural law through its own self-interacting dynamics. Maharishi's most recent research has centered on delving deeply into the analysis of these selfinteracting dynamics of consciousness.

The Self-lnteracting Dynamics of Consciousness

When one gains the capability, through the practice of the Maharishi Technology of the Unified Field, of remaining awake while becoming perfectly settled and still, one gains the ability to experience a completely simple, unified, undifferentiated, self-referral state of pure consciousness, which is called samhita in the Vedic literature, in which knower, known, and process of knowing are one and the same. Consciousness is simply awake to itself, knowing its own nature as simple, unified pure consciousness. Yet in knowing itself, the state of pure consciousness creates an intellectually conceived distinction between itself as knower, itself as known, and itself as process of knowing. In Vedic literature, this is reflected in the distinction between rishi (knower), devata (process of knowing), and chhandas (object of knowledge). According to Maharishi, from the various interactions and transformations of these three intellectually conceived values in the unified state of pure consciousness, all diverse forms of knowledge, all diverse laws of nature, and ultimately all diversity in material nature itself sequentially emerge.

The conscious mind, awake at this totally settled and still level of awareness, can witness the mechanics by which this diversification of the many out of the unity of pure consciousness takes place. The mechanics of rishi, devata, and chhandas transforming themselves into samhita , samhita transforming itself into rishi , devata , and chhandas , and rishi , devata and chhandas transforming themselves into each other are the mechanics by which the unity of pure consciousness gives rise to the diversity of natural law. These mechanics are expressed in the sequential unfoldment of Vedic literature. These are the self-interacting dynamics of consciousness knowing itself, which, Maharishi asserts, sequentially give rise to all diversity in nature.

Maharishi (1986) describes this self-referral state of consciousness as the basis of all creative processes in nature:

This self-referral state of consciousness is that one element in nature on the ground of which the infinite variety of creation is continuously emerging, growing, and dissolving. The whole field of change emerges from this field of non-change, from this selfreferral, immortal state of consciousness. The interaction of the different intellectually conceived components of this unified self-referral state of consciousness is that allpowerful activity at the most elementary level of nature. That activity is responsible for the innumerable varieties of life in the world, the innumerable streams of intelligence in creation. (pp.25-26)
The Structure of Maharishi's Vedic Science

One of Maharishi's most important contributions to Vedic scholarship has been his discovery of the Apaurusheya Bhashya , the "uncreated commentary" of the Rig Veda , which brings to light the dynamics by which the Veda emerges sequentially from the self-interacting dynamics of consciousness. According to Maharishi's analysis, the Veda unfolds through its own commentary on itself, through the sequential unfoldment, in different sized packets of knowledge, of its own knowledge of itself. All knowledge of the Veda is contained implicitly even in the first syllable "Ak" of the Rig Veda , and each subsequent expression of knowledge elaborates the meaning inherent in that packet of knowledge through an expanded commentary. The phonology of that syllable, as analyzed by Maharishi, expresses the self-interacting dynamics of consciousness knowing itself. As pure consciousness interacts with itself, at every stage of creation a new level of wholeness emerges to express the same self-interacting dynamics of rishi , devata , and chhandas .

Thus the body of Vedic literature reflects, in its very organization and structure, the sequential emergence of all structures of natural law from the unity of pure consciousness. Each unit of Vedic literature -Rig Veda, Sama Veda, Yajur Veda, Atharva Veda, Upanishads, Aranyakas, Brahmanas, Vedangas, Upangas, Itihasa, Puranas, Smritis , and Upaveda -expresses one aspect or level of the process. As Maharishi (1986) describes it:

The whole of Vedic literature is beautifully organized in its sequential development to present complete knowledge of the reality at the unmanifest basis of creation and complete knowledge of all of its manifest values. (p.28)
Veda, Maharishi asserts, is the self-interaction of consciousness that ultimately gives rise to the diversity of nature. The diversity of creation sequentially unfolding from the unity of consciousness is the result of distinctions being created within the wholeness of consciousness, as consciousness knows itself. Thus from the perspective of Vedic Science, the entire universe is just an expression of consciousness moving within itself: all activity in nature is just activity within the unchanging continuum of the wholeness of consciousness.

Through the texts of ancient Vedic science, as interpreted by Maharishi, we possess a rich account of the emergence of diversity out of the unity of natural law. On the basis of this account, it becomes feasible to compare the Vedic description of the origin of the universe with that of the modern sciences.

The reconstructions of our earliest science are based not only on the Vedas but also on their appendices called the Vedangas. Briefly, the Vedic texts present a tripartite and recursive world view. The universe is viewed as three regions of earth, space, and sky with the corresponding entities of Agni, Indra, and Vishve Devah (all gods).
In Vedic ritual the three regions are assigned different fire altars. Furthermore, the five categories are represented in terms of altars of five layers. The great altars were built of a thousand bricks to a variety of dimensions which coded astronomical knowledge.

In the Vedic world view, the processes in the sky, on earth, and within the mind are taken to be connected. The Vedic rishis were aware that all descriptions of the universe lead to logical paradox. The one category transcending all oppositions was termed brahman. Understanding the nature of consciousness was of paramount importance in this view but this did not mean that other sciences were ignored. Vedic ritual was a symbolic retelling of this world view.

Chronology

To place Vedic science in context it is necessary to have a proper understanding of the chronology of the Vedic literature. There are astronomical references in the Vedas which recall events in the third or the fourth millennium BCE and earlier. The recent discovery that Sarasvati, the preeminent river of the Rigvedic times, went dry around 1900 BCE due to tectonic upheavals implies that the Rigveda is to be dated prior to this epoch. Traditionally, Rigveda is taken to be prior to 3100 BCE.

Vedic cognitive science

The Rigveda speaks of cosmic order. It is assumed that there exist equivalences of various kinds between the outer and the inner worlds. It is these connections that make it possible for our minds to comprehend the universe. It is noteworthy that the analytical methods are used both in the examination of the outer world as well as the inner world. This allowed the Vedic rishis to place in sharp focus paradoxical aspects of analytical knowledge. Such paradoxes have become only too familiar to the contemporary scientist in all branches of inquiry.

In the Vedic view, the complementary nature of the mind and the outer world, is of fundamental significance. Knowledge is classified in two ways: the lower or dual; and the higher or unified. What this means is that knowledge is superficially dual and paradoxical but at a deeper level it has a unity. The Vedic view claims that the material and the conscious are aspects of the same transcendental reality.

The idea of complementarity was at the basis of the systematization of Indian philosophic traditions as well, so that complementary approaches were paired together. We have the groups of: logic (nyaya) and physics (vaisheshika), cosmology (sankhya) and psychology (yoga), and language (mimamsa) and reality (vedanta). Although these philosophical schools were formalized in the post-Vedic age, we find an echo of these ideas in the Vedic texts.

In the Rigveda there is reference to the yoking of the horses to the chariot of Indra, Ashvins, or Agni; and we are told elsewhere that these gods represent the essential mind. The same metaphor of the chariot for a person is encountered in Katha Upanishad and the Bhagavad Gita; this chariot is pulled in different directions by the horses, representing senses, which are yoked to it. The mind is the driver who holds the reins to these horses; but next to the mind sits the true observer, the self, who represents a universal unity. Without this self no coherent behavior is possible. In the Taittiriya Upanishad, the individual is represented in terms of five different sheaths or levels that enclose the individual's self.

The Sankhya and the yoga systems take the mind as consisting of five components: manas, ahankara, chitta, buddhi, and atman. Manas is the lower mind which collects sense impressions. Its perceptions shift from moment to moment. This sensory-motor mind obtains its inputs from the senses of hearing, touch, sight, taste, and smell. Each of these senses may be taken to be governed by a separate agent. Ahankara is the sense of I-ness that associates some perceptions to a subjective and personal experience. Once sensory impressions have been related to I-ness by ahankara, their evaluation and resulting decisions are arrived at by buddhi, the intellect. Manas, ahankara, and buddhi are collectively called the internal instruments of the mind.

Chitta is the memory bank of the mind. These memories constitute the foundation on which the rest of the mind operates. But chitta is not merely a passive instrument. The organization of the new impressions throws up instinctual or primitive urges which creates different emotional states.

This mental complex surrounds the innermost aspect of consciousness which is called atman, the self, brahman, or jiva. Atman is considered to be beyond a finite enumeration of categories.

The reconstructions of our earliest science are based not only on the Vedas but also on their appendices called the Vedangas. Briefly, the Vedic texts present a tripartite and recursive world view. The universe is viewed as three regions of earth, space, and sky with the corresponding entities of Agni, Indra, and Vishve Devah (all gods).
In Vedic ritual the three regions are assigned different fire altars. Furthermore, the five categories are represented in terms of altars of five layers. The great altars were built of a thousand bricks to a variety of dimensions which coded astronomical knowledge.

In the Vedic world view, the processes in the sky, on earth, and within the mind are taken to be connected. The Vedic rishis were aware that all descriptions of the universe lead to logical paradox. The one category transcending all oppositions was termed brahman. Understanding the nature of consciousness was of paramount importance in this view but this did not mean that other sciences were ignored. Vedic ritual was a symbolic retelling of this world view.

Chronology

To place Vedic science in context it is necessary to have a proper understanding of the chronology of the Vedic literature. There are astronomical references in the Vedas which recall events in the third or the fourth millennium BCE and earlier. The recent discovery that Sarasvati, the preeminent river of the Rigvedic times, went dry around 1900 BCE due to tectonic upheavals implies that the Rigveda is to be dated prior to this epoch. Traditionally, Rigveda is taken to be prior to 3100 BCE.

Vedic cognitive science

The Rigveda speaks of cosmic order. It is assumed that there exist equivalences of various kinds between the outer and the inner worlds. It is these connections that make it possible for our minds to comprehend the universe. It is noteworthy that the analytical methods are used both in the examination of the outer world as well as the inner world. This allowed the Vedic rishis to place in sharp focus paradoxical aspects of analytical knowledge. Such paradoxes have become only too familiar to the contemporary scientist in all branches of inquiry.

In the Vedic view, the complementary nature of the mind and the outer world, is of fundamental significance. Knowledge is classified in two ways: the lower or dual; and the higher or unified. What this means is that knowledge is superficially dual and paradoxical but at a deeper level it has a unity. The Vedic view claims that the material and the conscious are aspects of the same transcendental reality.

The idea of complementarity was at the basis of the systematization of Indian philosophic traditions as well, so that complementary approaches were paired together. We have the groups of: logic (nyaya) and physics (vaisheshika), cosmology (sankhya) and psychology (yoga), and language (mimamsa) and reality (vedanta). Although these philosophical schools were formalized in the post-Vedic age, we find an echo of these ideas in the Vedic texts.

In the Rigveda there is reference to the yoking of the horses to the chariot of Indra, Ashvins, or Agni; and we are told elsewhere that these gods represent the essential mind. The same metaphor of the chariot for a person is encountered in Katha Upanishad and the Bhagavad Gita; this chariot is pulled in different directions by the horses, representing senses, which are yoked to it. The mind is the driver who holds the reins to these horses; but next to the mind sits the true observer, the self, who represents a universal unity. Without this self no coherent behavior is possible. In the Taittiriya Upanishad, the individual is represented in terms of five different sheaths or levels that enclose the individual's self.

The Sankhya and the yoga systems take the mind as consisting of five components: manas, ahankara, chitta, buddhi, and atman. Manas is the lower mind which collects sense impressions. Its perceptions shift from moment to moment. This sensory-motor mind obtains its inputs from the senses of hearing, touch, sight, taste, and smell. Each of these senses may be taken to be governed by a separate agent. Ahankara is the sense of I-ness that associates some perceptions to a subjective and personal experience. Once sensory impressions have been related to I-ness by ahankara, their evaluation and resulting decisions are arrived at by buddhi, the intellect. Manas, ahankara, and buddhi are collectively called the internal instruments of the mind.

Chitta is the memory bank of the mind. These memories constitute the foundation on which the rest of the mind operates. But chitta is not merely a passive instrument. The organization of the new impressions throws up instinctual or primitive urges which creates different emotional states.

This mental complex surrounds the innermost aspect of consciousness which is called atman, the self, brahman, or jiva. Atman is considered to be beyond a finite enumeration of categories.

Astrology is the primary Vedanga or limb of the Vedas, as through it the right timing of actions can be determined. It is called the science of reason or causes "Hetu Shastra" as through it we can discern the karmic patterns behind events. Another name for it is "Jyotish", the science of light, as it deals with the subtle astral light patterns which inform and sustain our physical being and determine our destiny in life.

Vedic astrology is not just another system of astrological interpretation. Its purpose is not just to tell us what our destiny is via the stars. It does not leave us helpless before fate but shows how we can use the planetary energies operative in our lives in the best possible way. It also has a practical side, its Yoga. This is its series of remedial measures aimed at purifying our subtle or psychic environment, balancing our planetary influences and maximizing our karma. Its methods include the use of gems, colors, mantras, deities, rituals, herbs and foods. Hence a competent Vedic astrologer can provide us with tools to harmonize our entire being with the stars and align us thereby with the beneficent forces of the entire cosmos. Astrology is the basis for a total examination of our life on all levels, inner and outer.

We all clean our bodies and our houses on a regular basis. Yet few of us know how to clear our psychic or mental space. Things as the use of rituals, mantras and the science of astrology help us to do this. The foremost of these psychic influences we have to deal with is that of the planets. Vedic astrology teaches us to see these influences and gives us the means of promoting those which are beneficial and warding off those which are harmful.

Modern science deals with the sensory perceivable and measurable world. Astrology deals more with the astral world than the physical, with the energy behind form rather than the forms themselves and with the energy of the mind more so than the energy of matter.

Classical Vedic astrology uses the seven visible planets; the Sun, Moon, Mars, Mercury, Jupiter, Venus and Saturn, along with the two lunar nodes, the north and south nodes, Rahu and Ketu.

The effects of the distant planets is not as crucial. Many of their effects can be discerned through the functioning of the lunar nodes. Yet though Vedic astrology uses fewer planets it requires more calculations than a regular western astrological chart as it goes into much more detail in regard to the location and strength of the planets. Vedic astrology uses the twelve signs and twelve houses and planetary aspects, much like western astrology but with some differences, particularly in regard to the aspects.

In the earliest period of Indian science, it is exceptional when we know the authorship of a text or an idea. For example, although Yajnavalkya and Lagadha describe considerable astronomy, we do not know if this was developed by them or they merely summarized what was then well known. Likewise we are not sure of the individual contributions in the Shulba Sutras--- of Baudhayana, Apastamba, and other authors--- which describe geometry, or in Pingala's Chhandahsutra which shows how to count in a binary manner. The major exception to the anonymous nature of early Indian science is the grammatical tradition starting with Panini. This tradition is an application of the scientific method where the infinite variety of linguistic data is generated by means of a limited number of rules.
With Aryabhata, we enter a new phase in which it becomes easier to trace the authorship of specific ideas. But even here there remain other aspects which are not so well understood. For example, the evolution of Indian medicine is not as well documented as that of Indian mathematics. Neither do we understand well the manner in which the philosophical basis underlying Indian science evolved.

Thus many texts speak of the relativity of time and space---abstract concepts that developed in the scientific context just a hundred years ago. The Puranas speak of countless universes, time flowing at different rates for different observers and so on.

The Mahabharata speaks of an embryo being divided into one hundred parts each becoming, after maturation in a separate pot, a healthy baby; this is how the Kaurava brothers are born. There is also mention of an embryo, conceived in one womb, being transferred to the womb of another woman from where it is born; the transferred embryo is Balarama and this is how he is a brother to Krishna although he was born to Rohini and not to Devaki. There is an ancient mention of space travellers wearing airtight suits in the epic Mahabharata which may be classified as an early form of science fiction.

Universes defined recursively are described in the famous episode of Indra and the ants in Brahmavaivarta Purana. Here Vishnu, in the guise of a boy, explains to Indra that the ants he sees walking on the ground have all been Indras in their own solar systems in different times! These flights of imagination are to be traced to more than a straightforward generalization of the motions of the planets into a cyclic universe. They must be viewed in the background of an amazingly sophisticated tradition of cognitive and analytical thought.

The context of modern science fiction books is clear: it is the liberation of the earlier modes of thought by the revolutionary developments of the 20th century science and technology. But how was science fiction integrated into the mainstream of Indian literary tradition two thousand years ago? What was the intellectual ferment in which such sophisticated ideas arose?

Of the eighteen early siddhantas the summaries of only five are available now. In addition to these siddhantas, practical manuals, astronomical tables, description of instruments, and other miscellaneous writings have also come down to us. The Puranas also have some material on astronomy. Here we just list some of the main names in astronomy after 450 CE.

Aryabhata (born 476) is the author of the first of the later siddhantas called Aryabhatiyam which sketches his mathematical, planetary, and cosmic theories. This book is divided into four chapters: (i) the astronomical constants and the sine table, (ii) mathematics required for computations, (iii) division of time and rules for computing the longitudes of planets using eccentrics and epicycles, (iv) the armillary sphere, rules relating to problems of trigonometry and the computation of eclipses.

The parameters of Aryabhatiyam have, as their origin, the commencement of Kaliyuga on Friday, 18th February, 3102 BCE. He wrote another book where the epoch is a bit different.

Aryabhata took the earth to spin on its axis; this idea appears to have been his innovation. He also considered the heavenly motions to go through a cycle of 4.32 billion years; here he went with an older tradition, but he introduced a new scheme of subdivisions within this great cycle.

That Aryabhata was aware of the relativity of motion is clear from this passage in his book,``Just as a man in a boat sees the trees on the bank move in the opposite direction, so an observer on the equator sees the stationary stars as moving precisely toward the west.''

Varahamihira

Varahamihira (died 587) lived in Ujjain and he wrote three important books: Panchasiddhantika, Brihat Samhita, and Brihat Jataka. The first is a summary of five early astronomical systems including the Surya Siddhanta. (Incidently, the modern Surya Siddhanta is different in many details from this ancient one.) Another system described by him, the Paitamaha Siddhanta, appears to have many similarities with the ancient Vedanga Jyotisha of Lagadha.

Brihat Samhita is a compilataion of an assortment of topics that provides interesting details of the beliefs of those times. Brihat Jataka is a book on astrology which appears to be considerably influenced by Greek astrology.

Brahmagupta

Brahmagupta of Bhilamala in Rajasthan, who was born in 598, wrote his masterpiece, Brahmasphuta Siddhanta, in 628. His school, which was a rival to that of Aryabhata, has been very influential in western and northern India. Brahmagupta's work was translated into Arabic in 771 or 773 at Baghdad and it became famous in the Arabic world as Sindhind.

One of Brahmagupta's chief contributions is the solution of a certain second order indeterminate equation which is of great significance in number theory.

Another of his books, the Khandakhadyaka, remained a popular handbook for astronomical computations for centuries.

Bhaskara

Bhaskara (born 1114), who was from the Karnataka region, was an outstanding mathematician and astronomer. Amongst his mathematical contributions is the concept of differentials. He was the author of Siddhanta Shiromani, a book in four parts: (i) Lilavati on arithmetic, (ii) Bijaganita on algebra, (iii) Ganitadhyaya, (iv) Goladhyaya on astronomy. He epicyclic-eccentric theories of planetary motions are more developed than in the earlier siddhantas.

Subsequent to Bhaskara we see a flourishing tradition of mathematics and astronomy in Kerala which saw itself as a successor to the school of Aryabhata. We know of the contributions of very many scholars in this tradition, of whom we will speak only of two below.

Madhava

Madhava (c. 1340-1425) developed a procedure to determine the positions of the moon every 36 minutes. He also provided methods to estimate the motions of the planets. He gave power series expansions for trigonometric functions, and for pi correct to eleven decimal places.

Nilakantha Somayaji

Nilakantha (c. 1444-1545) was a very prolific scholar who wrote several works on astronomy. It appears that Nilakantha found the correct formulation for the equation of the center of the planets and his model must be considered a true heliocentric model of the solar system. He also improved upon the power series techniques of Madhava.

The methods developed by the Kerala mathematicians were far ahead of the European mathematics of the day.

Source: T.R.N. Rao and S. Kak, Computing Science in Ancient India. USL Press, Lafayette, 1998.
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