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Abstract,
Introduction,
The sutras of Vedic Mathematics in Geometry :
1. Invariance,
2. Balance,
3. Proportion,
4. Self-similarity,
5. Transposition,
6. Deficiency,
7. Observation,
8. Progression,
9. Completion/non-completion,
10. Addition/subtraction,
11. Elimination/Retention,
1. Bisecting an angle,
2. Intersecting chord theorem,
3. Apollonius’ theorem,
4. Polygon problem,
5. Drawing square roots,
6. The problem of the greater perimeter,
7. Golden Rectangle,
8. Construct a square equal in area to a given rectangle,
1.
The sutras of Vedic Mathematics in Geometry
James Glover
During the last fifty years or so geometry in education has diminished both in quantity and in
quality and yet the pedagogic use and applications remain as important now as they were then. A
new and holistic set of principles provides a simple approach to the fundamentals as well as to the
vast array of applications. Since the sutras of Vedic Maths apply to all areas of mathematics it
seems reasonable to suppose that they also apply to geometry. One of the hallmarks of the sutras is
that they include the human experience. One example is Vilokanam, By Observation. Accepting
both the mathematical and psychological nature of the sutras leads to a new orientation within
geometry. This paper looks at eleven sutras which provide general principles. Each sutra expresses
a simple concept that has a broad range of applications.
In its widest sense geometry deals with the relationship of form within space. Both form and space
are innate properties of the known universe and the human psyche. Geometry informs and enables
us to understand many aspects of the world around us. It develops our spatial awareness and is
integral to mathematics. So much of the world in which we live depends upon this developed
awareness, such as in design, architecture, physics, chemistry, crystallography, moelcular
modelling, engineering, art, and even in sport such as cricket, that it renders the subject a vital part
of modern education. Not only is geometry functional in providing engineers, and the like, with the
concepts and faculties required for their work but also it contains an aesthetic appeal because it
reveals beauty, harmony, symmetry and balance. Furthermore, it is a useful pedagogical tool
because it gives visual representation to mathematical relationships some of which are far easier to
understand in spacial terms than in digital format.
In education geometry has become analytical in the sense that it is invariably related to algebra. For
instance, lines and circles are frequently associated with equations. There is nothing wrong with
algebraic geometry but there are many geometric relationships that are more easily understood
visually. For example, the incentre and circumcentre of triangles, in terms of the concurrence of
angle bisectors and perpendicular bisectors, are far more easily understood visually than
2.
The angle bisectors all meet at the same point that is the incentre. All points along the bisectors are
equidistant from their adjacent sides. Hence the perpendiculars from the incentre to the sides of the
triangle are all the same length. Therefore a circle can be drawn that touches all three sides. This
can be proved algebraically but the “visual proof” is far easier.
In the past geometry has been a pillar of western education. For nearly a thousand years the
geometry of Euclid was taught at schools and universities throughout Europe. It was regarded as a
powerful tool primarily because it provided a visual means to learn how to think logically, with
rigour and clarity, and how to establish the validity of propositions through deductive reasoning.
The discovery and invention of other geometries during the 19th C and a less formal approach in
education loosened the grip of Euclid in education. The decline of geometry in UK schools
proliforated to the extent that today formal proofs are absent from all mathematics curricular for 11
– 18 year-olds. Geometry exists as a small fragment of primary education with the result that little
progress is accomplished at secondary level. One of the beautiful aspects of geometry is its
practicality and in an obvious way. The practicality and usefulness of teaching some aspects of
algebra, such as the difference of two squares, is not so readily seen. It is not uncommon for
disgruntled teenagers, who do not have a thirst for mathematical knowledge, to ask, Why am I
having to learn this? In pure geometry, such questions are sometimes easier to answer because
problems frequently admit to simple visual relationships. Diagrams can be drawn and so it appeals
greatly to those who are inclined towards kinesthetic or visual learning. Yet, virtually no child
enters a secondary school being able to draw a circle with a pair of compasses!
One of the great assets of geometry is that the spatial relationships involved are visual and, in the
past, this has assisted education in deductive reasoning. Unfortunately, Euclidean propositions and
proofs are very formal and this is not entirely suited to modern appraoches to education.
Euclid laid out a system of propositions based on three fundamental types of principles, axioms,
postulates and definitions. His axioms are not specific to geometry. They are quite general
statements of self-evident truths such as, if equals be added to equals the totals are equal. The five
postulates were entirely practical, such as, a line can be drawn between two points, and intended to
be specifically related to geometry and which had to be accepted in order to develop the
propositions. The definitions were highly important as they sought to clarify anything that might not
be clear. His aim was to demonstrate the validity of various propositions based only on the
principles he set out. Euclid’s Elements is a highly systematic exposition but, for the modern reader,
may seem rather pedantic.
A more modern approach to geometry, one that takes in all available methods of proof, can be much
more user-friendly than pure Euclid. The sutras of Vedic Mathematics can be applied to all types of
geometry, practical and theoretical and offer a new perspective. They do not go against established
norms but emphasise the underlying structure of mathematical thought.
The sutras of Vedic Mathematics in Geometry
Inasmuch as the sutras express thought processes and mathematical principles, many of them are
applicable to geometrical problems. Of course, a sutra such as “All form nine and the last from 10”
seems only to apply to number but others, such as Particular and General, have a closer relationship
to geometry.
3.
By using the key idea within a sutra it is possible to formulate geometric principles or processes.
Wherever possible I have tried to use a single idea or word to encapsulate how the sutras apply in
1. Invariance
This is based on the enigmatic sutra Sishyate Seshasanja, which Sri Tirtha translates as The
remainder remains constant. A more literal translation can be The remainder is what remains and,
at first glance, appears to be tautalogical. In mathematics, however, it largely relates to
invarianceFor example, in geometric transformations an enlargement with scale factor 1 is
invariant. What remains following a process is the same as before the process. Another example
will be referred to later when sectioning off squares from a golden rectangle; the remaining
rectangle’s proportions are invariant.
2. Balance
Balance expresses equality and symmetry. When two things are equal they are in balance. This is an
important principle not only in geometry but in all aspects of life. This concept is expressed in the
sutra, Sunyam Samyasamuccaye, When the total is the same it is zero. The idea is that when two
objects are the same then the difference is nothing. Seemingly tautalogical, this idea frequently
occurs in mathematics whenever equality is seen for example with two sides of an equation. A
simple example can be seen with congruence, that is, when two figures have the same size and same
shape. Congruence is frequently used in geometrical proofs.
3. Proportion
The sutra involved here is Anurupyena, Proportionately. The Sanskrit word for form or shape is
rupa and the literal meaning of the sutra is “by the same form”. The classical meaning of proportion
is equality of ratios. For example, two rectangles are similar when their sides are in the same ratio
and so there is a proportion.
4. Self-similarity
This is part of the Vyashti Samashti sutra, Particular and general. It gives expression to the
perception of a particular shape or form reflected in the form or shape of the whole. There are many
examples of this in geometry, particularly in chaos theory.
5. Transposition
Transformations and transpositions are common throughout geometry. The sutra is Paravartya
Yojayet meaning Transpose and Apply or Transpose and Adjust (Sri Tirtha uses both translations).
It is a very far-reaching process. In fact, in his book, this is the most commonly used sutra. This
sutra expresses such processes as applying a formula to a new situation, reversing processes,
geometrical and algebraic transformations, and even when transposing a word problem into
mathematical language.
4.
6. Deficiency
The single-word sutra, Yavadunam, is applicable wherever a problem is solved by using or referring
to a defficiency from some whole.
7. Observation
Vilokanam translates as By mere observation and comes into play wherever the answer to a problem
appears just by inspection. The fact that this is included in the list of sutras indicates that the mental
process of mathematical working is just as important as what is conventionally taken as
mathematical working.
8. Progression
The Ekadhikena Purvena sutra, By one more than the one before, relates to sequencing and
The next set of principles are based on the couplet sutras, Lopanasthapanabhyam – By elimination
and retention, Puranapuranabhyam – By completion and non-completion and Sankalana
Vyavakalanabhyam- By addition and subtraction.
The “bhyam” ending is dual case and so indicates both. On first reading it appears that both
processes are involved. For example, when something is eliminated it usually implies that
something else is retained. Certainly, in the physical sense when something is added it must be
subtracted from somewhere, for example, if money is added to an overdraft it must come from
somewhere and so there is a subtraction from the place where it comes. Again, if I give £9 to a
friend my pocket will be £9 shorter. The question then arises, if I set a problem for a child and
create a sum such as 56 plus 9, where does the 9 come from? Is there a subtraction? Clearly there is
no end to the potential of creating problems with plus 9 and so we may assume that the source of
the 9 is unlimited. 9 can be subtracted, as it were, from an infinite source.
In practice we may focus on one or other part of these couplets. For example, we may solve a
problem By Addition, and not refer to the subtraction. This implies that the dual case ending in the
sutras can be taken as either meaning “and” or “or”.
9. Completion/non-completion
10. Addition/subtraction
11. Elimination/Retention
The following eight examples are used to provide a brief illustratration of how these principles
5.
1. Bisecting an angle
This uses the principle of Balance. An arc is drawn, cutting the two lines at A and B. With A and B
as centres, two equal arcs are drawn intersecting at C. The angle bisector is the locus of points
equidistant from the two lines.
The same principle is involved with bisecting a line into two equal parts.
2. Intersecting chord theorem
This theorem uses the fact that angles in the same segment are equal.
Proof Principles
Angles in the same segment are equal and so DÂP = BĈP and Balance
AD̂P = CB̂P .
It follows that the angles of triangles DAP and BCP are the same and so Proportion
the triangles are similar.
Since they are similar, corresponding sides are in ratio with,
AP DP
=
CP BP
From this it follows that AP ´ BP = CP ´ DP . Transposition
This is the intersecting chord theorem.
6.
3. Apollonius’ theorem
The triangle has sides a, b and c and a median is drawn that divides the side, a, into two equal parts,
2 2 2
(
Apollonius’ theorem states that b + c = 2 m + d
2
)
Proof Principles
Using the Cosine rule,
b2 = m2 + d 2 - 2md cosq Transposition
and c2 = m2 + d 2 - 2md cosq '
But cosq ' = -cosq Transposition
And so c2 = m2 + d 2 + 2md cosq
Adding the two equations, Addition
(
b2 + c2 = 2 m2 + d 2 ) Elimination and
retention
4. Polygon problem
A common polygon problem found in school exams involves two adjacent regular polygons with
different number of sides. The problem is to find the missng angle, x.
7.
In the figures the polygons are incomplete (Non-completion) but it is easy to imagine them as
being whole (Completion). It is possible to calculate the interior angles of each polygon and then
subtract these from 360˚ to find x. But an easier method is found by extending (Transposition) the
adjacent side to form two exterior angles, a and b. The figure is now set up (Completion) for using
the exterior angles. By dividing 360˚ by the number of sides in each case the exterior angles are
found to be 40˚ and 24˚. x is therefore 64˚.
5. Drawing square roots
In the diagram each right-angled triangle has a unit side perpendicular to its base. The principle of
Progression, demonstrated here is a formulation of the sutra, By one more than the one before.
6. The problem of greater perimeter
A square has side 8 units. A circle is drawn through two corners of the square so that the opposite
side of the square is a tangent. Which has the greater perimeter?
8.
The clue to an easy solution is in the side of 8 units. By Observation 8 can be split into 3 and 5 and
4 is half the side length. This menas that a 3, 4, 5 triangle can be drawn revealing the radius of the
circle to be 5 units. The circumference is therefore 10π which is about 31.4 units whilst the square
has perimeter of 32 units.
Another simple solution comes from extending the centre line of the square to form a diameter, by
the principle of Completion.
Using the intersecting chord theorem, 8(D - 8) = 42 , from which D = 10 . Again, this gives the
circumference as 31.4 units.
A point to note is that in both cases previsouly known mathematics have been used. In the first
instance the Pythagorean triple and in the second, the intersecting chord theorem. This is an
example of a very common procedure and comes under the principle of Transposition. For
example, the intersecting chord theorem is formulaic and is applied to the situation in hand. A
transposition occurs whenever a formula is used or applied in mathematics.
7. Golden Rectangle
9.
Ö5 +1
This rectangle has sides in the ratio f :1 , where f = = 1.61803398...
2
By cutting off a square the remaining rectangle has the same proportions. This can be repeated with
successive squares being cut off, each time leaving a golden rectangle as the remainder. Three
principles are involved, Proportion, Invariance and Self-Similarity. The proportionality of the
rectangle is unique since no other rectangle has this property. Invariance ocurrs in the sense that the
remainder remains constant (in shape). Self-similarity is where the shapes of the part is reflected in
the shape of the whole and vice versa.
8. Construct a square equal in area to a given rectangle
Invariably, constructions in geometry require a number of steps each one of which can involve one
or more of the principles.
The rectangle has sides a and b. The side b is rotated
to the horizontal to form a line of length a + b.
(Transposition)
The midpoint of this line is found by contructing the
perpendicular bisector. (Balance)
A semicircle with radius 0.5(a + b) is then drawn
and the side b is extended upwards to meet the
circle. This line has length c.
By drawing lines from the ends of the diameter three
similar triangles are formed (Completion).
a c
Then = (Proportion)
c b
This leads to ab = c 2 (Transposition)
10.
In fact c is the geometric mean of a and b.
A square can now be drawn equal in area to the
rectangle whose area is ab. (Completion)
Concluding Remarks
It should be noted that, in none of the cases above, is the mathematics unconventional. All of these
are well-known solutions and problems often have alternative solutions. But it is the way they are
looked at that is different. The Vedic Maths sutras provide a simple set of principles that can be
applied singly or in combination in a very wide variety of ways. In some of the above problems
there is a multi-step approach, such as with Apolloniius’ theorem, intersecting chord theorem and
creating a square equal to a rectangle, and in these it is possible to see the sutras at work within
individual steps of solving the problem. So in one sense the sutras provide alternative names for
steps or working that are entirely conventional. Additionally, there are cases, such as bisecting an
angle, where the sutra simply express an idea or aspect of knowledge that lies behind the steps of
Uniquely, this approach reveals that behind mathematical thought processes and ideas there are
very few principles, each one of which has huge potential in many apparantly diverse areas of
mathematics. Not only do the sutras apply to arithmetic and algebra but also in pure geometry. The
very fact that so few aphorisms find expression within a multitude of problems has a unifying
Above all the sutras provide us with simplicity and this can be a great asset in teaching and
learning. As the famous Renaissance artist and scientist Leonado Da Vinci said, “Simplicity is the
ultimate sophistication”.
11.
In 2001 the Royal Society and the Joint Mathematical Council produced the report, Teaching and
learning geometry 11 – 19. The report layed out key principles for the learning of geometry,
together with sixteen recommendations for change. The curriculum authority within the department
for Education has implemented none of them.
Also look at
Jones, K. (2000), Critical Issues in the Design of the Geometry Curriculum. In: Bill Barton (Ed), Readings in
Mathematics Education. Auckland, New Zealand: University of Auckland. pp 75-90.
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