Issue 103 - Lines of Thought

Vedic Mathematics Newsletter No. 103

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Vedic Mathematics is becoming increasingly popular as more and more people are introduced to the beautifully unified and easy Vedic methods. The purpose of this Newsletter is to provide information about developments in education and research and books, articles, courses, talks etc., and also to bring together those working with Vedic Mathematics. If you are working with Vedic Mathematics - teaching it or doing research - please contact us and let us include you and some description of your work in the Newsletter. Perhaps you would like to submit an article for inclusion in a later issue or tell us about a course or talk you will be giving or have given. If you are learning Vedic Maths, let us know how you are getting on and what you think of this system.


This issue’s article is by Kenneth Williams, editor of this newsletter. It looks at the Vedic Mathematics Sutras: in particular the first three Sutras. The title is “Lines of Thought”.



A researcher working on a Ph.D in Industrial Engineering wishes to apply Vedic Mathematics in Big Data Analysis. If you have knowledge of Data Analysis and wish to help please email at


A head of mathematics is required at a school in Ghana starting January 2016. If this may be of interest to you please see the details at:


Article on Vedic Maths in India's 'The Telegraph' (19th August)


Novan Education and Training, in collaboration with Academic City College, hosted an education conference in Accra last month.
Miracule D. Gavor and Kenneth Williams presented a paper entitled: “Improving quality of mathematics education in Ghana. A case study of Vedic mathematics Globally”. This was very well received and led to radio interviews. See:


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Lines of Thought

It is still widely believed that mathematics is based on formal logic, even though that idea was laid to rest many years ago with the failed attempt by Bertrand Russell and Alfred North Whitehead to demonstrate that.

Though he does not specifically state as much, Sri Bharati Krishna Tirthaji suggests that mathematics is based on the sixteen Sutras that he gives in his book “Vedic Mathematics”1. How it might be possible for these brief aphorisms to form a basis for mathematics is a big question, but it has an appeal and since the foundations of mathematics are not established it makes sense to look carefully into this possibility.

As described elsewhere2, the Sutras can all be related to natural mental functions that we use all the time. If we were to analyse the types of methods we use when thinking we would discover, not only that there are not very many but also that every one of us has developed the same mental methods, without being specifically taught them.

This suggests that these mental techniques are natural. And it follows that the most effective mathematics comes from those who use those techniques most efficiently. And they would most likely be the people who are taught from that basis.

If as we learn and understand mathematics we apply the sixteen mental techniques, we can say that the structure of mathematics in our mind is due to the operation of those Sutras: that the Sutras effectively construct and structure the mathematics for us. This gives a clear explanation how the Sutras can form a basis for mathematics.

That the authenticity of Vedic mathematics can be established in this way was confirmed by the current Shankaracharya of Puri, Swami Nischalananda Saraswati, when I asked him a few years ago.

Research is needed to understand the Sutras better, to fully relate them to our ways of thinking and to establish if those taught Vedic mathematics understand and can do mathematics better than others.

Mental Impressions
There are various types of impressions that come into our awareness. Sense impressions, thoughts, a general emotional feeling and so on. These can be linked in all kinds of ways. And the mind may be clear, dull, agitated etc.

Thoughts come from we know not where, but when they come they can have various effects; leading to physical reactions, further thoughts, new feelings and so on.

The more focussed mind might observe that we have the ability to do things with a thought: we can think about it in different ways.

What actually happens when you ‘think about’ something? The thought is presented by the mind to the discriminative faculty which then considers it. At this point there are many possibilities as the prior state of the mind, memory and so on will influence what happens next.

The discriminative faculty, called the Buddhi in the Vedic literature, has several ways in which it can deal with a thought. For the purpose of illustration let us take a simple mathematical thought.

Suppose you are a young child, newly introduced to the notion of prime numbers. As you think about prime numbers, what comes to mind?

1) You may think of writing out a sequence of primes one after the other.

2) Then you may then think: where is it leading, what is the last prime number?

These corellate with the first two Sutras.

1) By One More than the One Before describes an ordered sequence.

2) All from 9 and the Last from 10 describes the closure or fulfilment sought during mental activity. The number 9 represents the maximum degree of complexity (being the largest digit): after 9 we get 10 which means unity (a unity that contains all the previous digits).

Sutra 1 is like a line gradually extending from some initial point. Sutra 2 is like the completion of the line which then stands whole and fixed.

Even if we decide there is no end to the sequence of prime numbers the realisation of this fact brings fulfilment and closure.

So in Sutra 1 we start in unity with an idea/intention which ultimately finds its fulfilment in the unity of the 10 in Sutra 2.

Now you may take another ‘line of inquiry’ regarding the primes (and we may consider how Sutra 3 is formed by a combination of Sutras 1 and 2). For example, you may look at non-primes or at the differences between successive primes.

In this way you build up your knowledge of prime numbers step by step by combining several lines of inquiry. Your next thought (Sutra 1) might be to look for a pattern in the prime numbers, and when this line of inquiry is concluded (Sutra 2) you have a fuller appreciation of the subject as you have combined knowledge of all the lines of inquiry.

Understanding Evolves
So combining sutras 1 and 2 together we have a series of structured ideas which together give a set of ideas about some topic. This is described by the third sutra: Vertically and Crosswise. That is to say, each of the vertical and crosswise lines that characterise the 3rd Sutra indicate a line of thought (Sutra 1) and the combination of all these lines leads to a new understanding when brought to finalisation by Sutra 2.

We see Sutras 1 and 2 are both present in Sutra 3: Sutra 1 in the successive vertical/crosswise steps, and Sutra 2 in the evolving state of understanding.

Thus we see in this illustration how the Buddhi applies different operations to an idea, developing and expanding it; until it is found to be finished or the attention is attracted away by something else.

Our understanding generally evolves in this way, and other Sutras come in as well. For example, switching the attention to non-primes involves Transpose and Apply, and looking at the differences of consecutive primes comes under By Addition and Subtraction. Looking for patterns means using Specific and General.

These correlations between the Sutras and functions of mind are not mere, convenient, mental constructs: they are natural and observable.

1 "Vedic Mathematics", by Sri Bharati Krishna Tirthaji, Motilal Banarsidass, 1965.
2 "The Sutras of Vedic Mathematics", by Kenneth Williams, in the Journal of the Oriental Institute, Vol. L, Nos 1-4, Sept 2000 - June 2001, pages 145 to 156.

End of article.

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Editor: Kenneth Williams

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4th October 2015


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