THE SUTRAS OF VEDIC MATHEMATICS
For the purposes of this paper, by ‘Vedic Mathematics’ is meant the system outlined by Sri Bharati Krsna Tirthaji in his book of that title, initially published by Banaras Hindu University in 1965 and then by Motilal Banarsidass (Reference 1). By ‘The Sutras of Vedic Mathematics’ is meant the Sutras given in that work.
Sri Bharati Krsna Tirthaji (1884-1960) was a brilliant scholar. He spent eight years between 1911 and 1918 in forest solitude in his attempt to interpret Sanskrit texts that other scholars had dismissed. This led, he says, to his discovery of a mathematical system based on sixteen Sutras. He later became Shankaracarya of Puri. He led a remarkable and exemplary life and however extraordinary his assertions about Vedic Mathematics may seem his work is worthy of serious consideration.
Many questions and criticisms have been put forward relating to the Sutras and also the mathematical system based on them. Regarding the system, any open-minded inquirer making a careful study of the book and subsequent research would come to the conclusion that the Vedic system has many merits and is more unified and coherent (at least at an elementary level) than the current, modern system. But we will not discuss this further here as this paper is about issues relating to the Sutras.
The Vedic Mathematics system uses a collection of sixteen sutras and some sub-sutras. These are given in Sanskrit in the book “Vedic Mathematics”. We have Sri Bharati Krsna Tirthaji’s English translation for some but not those marked with an asterisk in the list below. [please note the Sanskrit and transliteration are not included here although they were in the original article - most books on Vedic Mathematics will have these.]
1. By One More than the One Before
A list of sixteen Sutras and thirteen sub-Sutras appears in the book “Vedic Mathematics” and is followed by a note in brackets that “This list has been compiled from stray references in the text – Editor”. There are a number of discrepancies between this list and the text however. For example the Sutra Lopanasthapanabhyam is always quoted in the text as being a Sutra but is given as a sub-Sutra in the list. So the question arises as to the validity of this list. In 1981 Andrew Nicholas, a UK academic, met Mr Somanath Mahapatra, a disciple of Sri Bharati Krsna Tirthaji and was shown a list of Sutras and sub-Sutras written in Sri Bharati Krsna Tirthaji’s own hand. That list was the same as the one given in the book: both Sutras and sub-Sutras were listed in the same order, except that there was an extra sub-Sutra, Chakravat.
Studying the book we also find that Sri Bharati Krsna Tirthaji frequently refers to a sub-Sutra as a Sutra. It appears he is just being brief here, just as he refers to the Sutra Nikhilam Navatascaramam Dasatah as the Nikhilam Sutra. This explains why Lopanasthapanabhyam is referred to as a Sutra but listed as a sub-Sutra. For these reasons we will assume that the list given in “Vedic Mathematics” is Sri Bharati Krsna Tirthaji’s list.
ORIGIN IN THE VEDAS
The main criticism in India of Vedic Mathematics is that the Sutras have not been located in the extant Vedic literature and that therefore it is not right to give the book the name Vedic Mathematics. Three responses are worth making to this:
1) Since the Vedic literature is very extensive, with many documents not having been recorded, it is still quite possible that the Sutras could be located in some texts. Some Vedic texts have also been lost in floods etc.
2) The word ‘Vedic’ refers not only to actual texts but also has a literal meaning of ‘knowledge’. Sri Bharati Krsna Tirthaji in the Prolegomena to his book wrote:
“Immemorial tradition has it and historical research confirms the orthodox belief that the Sages, Seers and Saints of ancient India (who are accredited with having observed, studied and meditated in the Aranya (i.e. forest-solicitude) – on physical Nature around them and deduced their grand Vedantic Philosophy therefrom as the result not only of their theoretical reasonings but also of what may be more fittingly described as True Realisation by means of Actual Visualisation, seem to have similarly observed, studied and meditated on the mysterious workings of numbers, figures etc. of the mathematical world (to wit, Nature) around them and deduced their Mathematical Philosophy therefrom by a similar process of what one may, equally correctly, describe as processes of True Realisation by means of Actual Visualisation.”
It seems reasonable that just as a mathematical technique (say vertically and crosswise multiplication) could have been rediscovered many times, so too when the processes of nature, physical or mental, are studied the same conclusions or structures could be arrived at.
This could explain why, when questioned about the occurrence of the Sutras in the Artharvaveda, Sri Bharati Krsna Tirthaji said they were in his own Parisista. He may have believed he had rediscovered the same universal laws that were known in Vedic times.
3) It does not make any difference to the potency or application of Vedic Mathematics whether the Sutras are located in ancient texts or not. In the West the validity of the name is not an important issue.
Nor can the question about the authenticity of the Sutras be answered, at least from a western point of view, by finding some reference to them in the extant literature. Because they appear in a book or other text does not make them valid, it just means somebody once thought they were. But there are references in the Vedic literature to sixteen phases of the Purusha (in the Prasno Upanishad for example).
According to Sri Bharati Krsna Tirthaji (Page xvi):
(i) The Sutras (aphorisms) apply to and cover each and every part of each and every chapter of each and every branch of mathematics (including arithmetic, algebra, geometry – plane and solid, trigonometry – plane and spherical, conics- geometrical and analytical, astronomy, calculus – differential and integral etc., etc. In fact, there is no part of mathematics, pure or applied, which is beyond their jurisdiction;
(ii) The Sutras are easy to understand, easy to apply and easy to remember; and the whole work can be truthfully summarised in one word “mental”!
Sri Bharati Krsna Tirthaji tells us (Reference 1, Page xx) he wrote sixteen volumes expounding the Sutras; but it seems these were lost (see Reference 2).
If the Vedic system is a valid system of mathematics it is imperative to establish the authenticity of the Sutras on which it is based. So the most important question, and one that does not seem to get asked, is: “how do these sixteen Sutras form a basis for mathematics?”
Even if applications of the Sutras were demonstrated in all the main areas of modern mathematics we would still probably not understand why and how the Sutras form a basis for mathematics in general. The only option is to show that the Sutras themselves have some sort of universality and can thereby form a set of principles that inevitably cover all of mathematics. In fact we can go further and say that if sixteen Sutras cover all of mathematics they must express universal principles and they must in some way form a complete set. So it could be useful to examine the Sutras from this point of view: as general or universal principles.
And if the Sutras express universal principles these would find manifestation in other areas apart from mathematics. They could for example describe the way the mind works. In fact it might be easier to see the mental expressions of these principles than to see more cosmic expressions of them. And if the Sutras do describe laws of the mind this could be immensely useful in all sorts of ways.
NATURE OF THE SUTRAS
To understand the Sutras, and for them to be useful to us, it is necessary to see them as something we can relate to in a natural way. What follows explores the possibility of the Sutras describing natural functions of mind.
Our mind is extremely subtle and fluid. There seems to be no limit to what we can think and imagine. But if we carefully watch our thoughts we find that we all use certain specific techniques: of reversal, comparison, extension, generalisation for example. And it is interesting that we all seem to develop the same mental techniques, even though we are not specifically taught how to think. It is also perhaps surprising that on examination there do not appear to be very many distinct techniques. The sixteen Sutras can be related to these naturally evolved mental processes. It is our mind that constructs the mathematics and this is done according to the natural functions of mind, which may be expressible by the Vedic Sutras.
We are not normally aware of our thought processes. We have built up layers of mental techniques since we were very young and rely totally on them. We never question them unless we are forced to do so: when for example we are confronted with a contradiction.
To appreciate the types of mental action that follow we need to see them in action in daily life, then we begin to realise just how frequently we use them. In my experience people vary a great deal in their ability to observe their thoughts and thought processes so please have an open mind and be prepared to see things in a different way.
Some of the sutras seem to fit neatly with a well-known mental function. For example, Transpose and Apply would describe the reversing that we often use in our thinking. We look at something from the other person’s point of view; we feel compassion; we ‘change our mind’ and do the opposite of what we were going to do. It makes sense as a cosmic principle as we see plants grow and then die away, the sun rises and sets. In one of Aesop’s fables a bird was trying to drink from a jug but could not reach down to the level of the water. After a while instead of struggling to get his head down far enough the bird went and got some stones which he dropped into the jug until the level had risen enough for him to drink the water.
By One More than the One Before is also easy to see as a principle: everywhere we see things that follow on from, or are created out of, other things, due to the lawful nature of the universe. The Sutra expresses how one idea naturally leads to another: just by resting the mind on an idea another idea arises. This can happen sub-consciously when our mind wanders from one thought to another (sub-Sutra) or it can be a deliberate conscious choice to develop an idea. So, given a square you know that it has four right angles, and seeing someone walking towards you, you expect to pass them shortly.
The sutra By Addition and by Subtraction refers to the mental process of comparison. In comparing two things we look for similarities (addition) and differences (subtraction) between them. Addition involves bringing things together into the same group, and observing similarities involves noting those qualities which the two things have in common (i.e. can be grouped together). In subtraction we observe those qualities which differ in the two things. Shown two photographs of the same person taken ten years apart we notice similarities (we recognise common features which we mentally group together) and we notice differences (the person is taller, has different hair colour etc.). This is a very common mental technique. As a natural phenomenon we see that the forces of nature continually throwing things together and separating them.
The sutra Vertically and Crosswise can be related to decision making and evaluation. In assessing something we weigh up all its aspects, perhaps giving strengths to each of them and then finally coming to a decision by summing all these results. For example, in deciding at what time to leave home for a meeting we may consider traffic, weather, how long it takes to prepare, implications of being late and so on. We assess the importance of each one and sum them, rather like multiplying pairs of numbers and adding the results, as we do in Vertically and Crosswise multiplication. This is also a very common mental process. As a principle of nature it seems to describe how everything is interconnected.
All from Nine and the Last from Ten. Here we first need to interpret nine and ten in more general terms in order to understand them as more general entities. In the generation of whole numbers from number one each new digit is one more than the previous one and is more complex than the previous one. In a sense nine is the most complex digit as the next number, ten, is one with zero (i.e. 10). Ten represents a new unity, a new order of unity and it contains the previous nine numbers. Nine is the stage just before unity is reached. So the Sutra describes repeated non-unity, followed by unity. In following this sentence, for example, all the parts of it are absorbed before the full comprehension of the meaning of the sentence and we know at each step that we are moving towards a unity. All the parts are like nines, they are all short of the unity sought, but together they lead to a new unity, ten.
In fact if we look we see this principle operating everywhere. A house is built: the foundations are laid, the walls go up, the water pipes are put in etc. and the final result, ten, is the house. A plant grows until it reaches its fulfilment in its flower and its seed. We listen to someone and each point that is made is understood after hearing the words that make up the point being expressed. And each point may itself be leading to a further idea so that the same principle may apply on different levels. Whenever a new unity is realised there were steps that led to it. The same mental activity will be recognised in the way we sometimes continually think about something in order to try to understand it or to come to terms with it.
The Sutra refers to the reaching of a new state. So understanding, comprehension, realisation, enlightenment etc. all express this. A ball rolls along a table top, but when it reaches the edge it goes into a different state. Then it falls until it reaches an obstacle when it again changes its state of motion. We may see parallels in our own life.
If the Samuccaya is the same it is Zero. The
word ‘samuccaya’ has several meanings we are told, but usually it can
be interpreted as total’ or ‘combination’. So we may say ‘If the
Total is the Same it is Zero’. Sometimes it is the sameness of something
that is significant. We do not realise that the clock was ticking until
it stops. A word constantly repeated becomes meaningless. As long as
things are as normal (the same) we do not register them (it is zero).
In a Sherlock Holmes story Sherlock Holmes points out to Dr Watson that
the fact that the dog did not bark is significant (it meant that
the dog knew the intruder). If walking to a place by a different route
to the normal one has no advantage we are not inclined to take that
route. We ignore what is irrelevant or what we prefer not to see. We
‘put it out of our mind’.
If One is in Ratio the other One is Zero. We can put our mind onto anything we like. We choose our opinions and beliefs and so ignore contrary views. We choose a meal from a menu, we select certain beliefs that we choose to hold, discarding all others. A country chooses a leader, he speaks and acts for the people (he is in ratio) the others need do nothing. We make an assumption or we take a stand. We frequently decide to take something as true/given (in ratio) and proceed on this assumption, leaving other possibilities aside (equal to zero) for the moment. When looking for a blue sock among a drawer full of socks we can mentally tell our mind to see only blue. (The word ‘ratio’ comes from a Latin word that means ‘reason’).
By the Completion or Non-Completion. Sometimes we can see a whole even though the whole is not there. Three-quarters of a circle immediately gives the impression of a full circle. We see symmetry in the beauty of a flower even though perfect symmetry is not there. When we ‘jump to a conclusion’ we see, or think we see, some idea. We complete the whole. So, someone is speaking and we know what they are saying before they finish saying it. The detective pieces together what has happened by a series of clues. We say “I get the picture”.
Completion implies non-completion, both are there together. We may deliberately leave something incomplete, we may wish to indicate something without actually saying it: it can be implied.
Differential Calculus. Calculus is all about change and motion, which are fundamental to creation. Mentally we may imagine the effect of changing something (while keeping other things constant); we can conduct ‘thought experiments’. We may consider the effect if a tree continues to grow, if interest rates continue to rise, if the rain doesn’t stop, what it might be like to jump on the moon and so on.
By the Deficiency. We often see how things fall short of an ideal. Differences from the norm stand out. We examine something we have made and see its shortcomings for example. These shortcomings are then the focus of our attention. (Looking for faults may be valuable in a car mechanic but not in a personal relationship).
It is the deficiencies or excesses that stand out and get noticed. We are interested in how much we are driving above or below the speed limit or whether we have thanked someone enough, or overdone it. A wrong note in a piece of music stands out.
Specific and General. This relates to the mental process of generalisation from the specific, which we do all the time. In fact this is just what we have been doing with the Vedic Sutras in this paper: we take the idea of, say, addition and subtraction in the mathematical sense and look for a more abstract meaning of those words. Then we can look for mental processes that use this more abstract meaning: which is going from general to specific. Metaphors use this Sutra: when we say “leave no stone unturned” or “food for thought” it is the general quality that has been described by means of something concrete that has that quality.
We can use this technique for interpreting dreams because a dream may be a specific formulation of a general worry or concern: we may examine the essence of the dream and look for examples in the life of the dreamer of how that essential quality may manifest.
The Remainders by the Last Digit. Everything in the world is being continually modified by the things around it. We have ideas and sets of ideas about things and one observation or thought can lead to the whole structure being modified.
We can regard the ideas we have already as the remainders and the latest idea is the last digit that operates on those ideas and modifies them. For example, we learn something about someone and that modifies our opinion of them. Driving a different car to the usual one or treating someone differently because they are in an emotional state, we have to adapt our habits to cope with the change.
We also set up more permanent structures in the mind that apply for a lifetime. As we get older we can get more ‘set in our ways’, it is more difficult to adapt, to change our way of doing things. We say ‘You can’t teach an old dog new tricks’.
The Ultimate and Twice the Penultimate. In nature we often see that one thing triggers another: warm, wet weather causes seeds to germinate for example. The mind may act habitually in a certain way in response to a certain stimulus. One idea (the ultimate) may trigger a reaction, a stronger set of ideas already fixed in the mind (the penultimate). I want to say something to a certain person when I see him, the sight of that person triggers the memory. At the time I decided to speak to that person I conditioned my mind to react when that person is sighted. One idea takes over from another. It is because the fixed ideas are stronger and take over that they count twice. We can use this to our advantage by alerting the mind to be selective in some way. Mother hears baby cry in spite of all the other sounds; she has conditioned her mind to be alerted to that sound. When we say ‘it rings a bell’ an internal idea has been awakened by something outside. The saying ‘Once bitten, twice shy’ also involves this idea.
By One Less than the One Before. Sometimes we stand back and take an objective look at a situation. We may be struggling with some problem and decide that it is time to look at the bigger picture or perhaps we are feeling the pressure of too many commitments and think it is time to step back, relax and take a fresh look. Here you are in this room, in this country, on this planet etc.
This is the reverse of the Sutra By One More than the One Before (which describes deduction) and it also relates to the mental process of induction where a conclusion is reached based on a number of instances. For example the child sees objects fall to the ground when they are released and so naturally thinks that the next object released will fall. This has the same quality of stepping back.
The Product of the Sum. Sometimes we have to assimilate several ideas and draw a result, or product, from them. ‘Sum’ suggests bringing some ideas together, and ‘product’ is a result that follows from this – it is the product of the sum. It is the overall impression. Bring a lot of men together and you have a crowd or an army: the whole is more than the sum of the parts.
All the Multipliers. Sometimes we need to summarise a number of things. The final speaker at a conference brings together all the ideas that have been aired and discussed. You are saying goodbye to someone after a meeting and bring to your mind the important points from the meeting that are relevant in order to say the appropriate things.
These interpretations of the Sutras may or may not be on the right lines; and even if they are on the right lines considerable revision may be necessary. This is because the mind is very subtle and the Sutras seem to merge together at some points. Hopefully others will be able to extend and modify these ideas.
Together with the sixteen Sutras Sri Bharati Krsna Tirthaji lists thirteen sub-Sutras. For example Proportionately, By Alternate Elimination and Retention and By Mere Observation are three of them. Two of the sixteen Sutras (By One More than the One Before and By Addition and By Subtraction) are indicated to be sub-Sutras also, so the total is fifteen sub-Sutras. However we are given the impression that although there are exactly sixteen Sutras, the sub-Sutras are not fixed in number. According to Sri Somanath Mahapatra (mentioned earlier) Sri Bharati Krsna Tirthaji would ‘pluck sub-Sutras out of the air’. The sub-Sutra Chakravat has already been mentioned and is not in the list given in “Vedic Mathematics”.
Suppose you are asked how many triangles there are in this figure.
There are sixteen of these but did you notice how you could mentally make the various shapes stand out, seeing some lines and ignoring others, and seeing different triangles, at will? The Vedic sub-Sutra By Alternate Elimination and Retention describes this attribute of our mind and has many more obviously mathematical applications (in eliminating first x and then y when solving a pair of simultaneous equations, for example). In fact you may have used Alternate Elimination and Retention in another way in the problem above as you may have decided to consider only triangles of a certain type first and then of another type and so on.
It is not clear just how the sub-Sutras relate to the Sutras, but if we think of the physical body we know it has evolved special specific functions (walking, communicating, holding etc.) and also functions which are less obvious (blood circulation, digestion etc.). Similarly the mind also has a more manifest and sophisticated level of operation as well as more subconscious operating levels. Perhaps there are sixteen clear, conscious types of mental activity we can perform, corresponding to the sixteen Vedic Sutras and a number of other more subconscious activities.
Many of the sub-Sutras are clearly related to one of the Sutras. This is what we would expect since the sixteen Sutras are said to cover all of mathematics: each sub-Sutra must relate to at least one of the Sutras. The sub-Sutra described above (By Alternate Elimination and Retention) is similar to If One is in Ratio the Other One is Zero.
One argument used against the Sutras is that they don’t tell you exactly what to do to solve a problem. But they are general principles and are not meant to do that. The first step we face when confronted with a problem is the crucial one. We know we have a variety of ways of using our mind but we may be unsure which to use: we don’t know how to start. The Sutras help to make that initial step: not by trying out each Sutra to see which one works but by being familiar with, and practised in, the Sutras, the mind knows the best way forward. A fuller exposition of the Sutras as functions of mind is under preparation.
Since our mind constructs mathematics it would follow that a mathematical system based on a complete set of natural mental functions would be hugely beneficial personally, and would also be more efficient and seem easier. This explains why students enjoy the Vedic system. Just as you would not get enough physical exercise by twiddling your thumbs, so your mind needs to have all its faculties exercised to fully develop in a mental way, and to enjoy the activity. As things are at present many people do not enjoy mathematics.
In a talk in 1958 at the Institute of Technology, Pasadena, California, USA (see Reference 3, Page 170) Sri Bharati Krsna Tirthaji said: “People who have practical knowledge of the application of the sutras, need not go in for the theory side of it at all”. This fascinating comment makes a clear distinction between those who learn to do mathematics by just practising the Sutras and those who can also learn the theory of it as well. The non-mathematician and the applied mathematician just want to use mathematics. There are those of us who just want to get on and do the job and those who want to understand the details.
Anyone familiar with Vedic system will know that it is more integrated, more efficient and more fun than conventional mathematics. It leads to greater flexibility of mind, increased mental agility and stimulates the creative faculty that is in all students. Further research is needed to establish the nature of the Sutras and their full range of application, possibly along the lines described here and for this we would probably need the participation of psychologists and cognitive scientists.