Ancient Indian Mathematician Baudhayana’s Theorems and Practical Usage
The Shulba Sutras are part of the larger group of texts called the Shrauta Sutras (A knowledge gained by listening in the superconscious state). They are the only sources of knowledge of Indian mathematics from the Vedic period.
Before we start exploring more on ancient Indian mathematician work, it is important to note that what could be the reason to trigger for such inventions.
There are unique fire-altar shapes that were associated with unique gifts from the Gods. For example,
- “One who desires heaven is to construct a fire-altar in the form of a falcon”.
- “A fire-altar in the form of a tortoise is to be constructed by one desiring to win the world of Brahman”
- “Those who wish to destroy existing and future enemies should construct a fire-altar in the form of a rhombus”
There is a view that Indian mathematics originated in the service of religion. Astronomy was developed to help determine the auspicious day and hour for performing rituals.
In the same way, complex formulas were required to draw complex structures which were the necessity during the Vedic prayers. Like the Mahavedi (Great Altar) was laid out as an isosceles trapezium with the bases 24 prakrama and 30 prakrama and width 36 prakrama. A prakrama is about 0.8 meters.
These alters were present in the Satapatha Brahmana and the Taittiriya Samhita, whose contents date to the late second millennium or early first millennium BC. One of the triangles appears in the above falcon listed in the Baudhayana Shulba Sutra.
Three of the more mathematically important Sulbasutras were the ones recorded by Baudhayana, Apastamba, and Katyayana.
It is difficult to assign firm dates to these three texts. All we can say is that the earliest of them, the one composed by Baudhayana, was probably first recorded between 800 and 500 B.C and that the other two were recorded one or two centuries later.
Baudhayana’s Sulbasutra is offering instructions to those conducting the Vedic rituals as to how to construct altars of various shapes using stakes and marked cords. We can discern from these instructions a general statement of the:
- Pythagorean theorem
- An approximation procedure for obtaining the square root of 2 correct to five decimal places
- Number of area-preserving transformations for “squaring the circle”
- Constructing rectilinear shapes whose area was equal to the sum or difference of areas of other shapes.
Pythagoras theorem is such a phenomenon in math which is associated with humans since long back in history. Egyptian was the oldest one using this but didn’t have any connection with mathematics.
The second record we find on Mesopotamian clay tablets where we see the mathematic relationship between numbers. Indian version we can see that Budhayana proves this theorem with the help of area and finally Pythagoras who proved this theorem geometrically.
The Baudhayana actual statement of the Pythagorean theorem, expressed in terms of the sides and diagonals of squares and rectangles, is found in both the Baudhayana and Apastamba Sulbasutras. Baudhayana states:
- The rope which is stretched across the diagonal of a square produces an
area double the size of the original square.
- The rope [stretched along the length] of the diagonal of a rectangle makes a [area] which the vertical and horizontal sides make together
Practical Usage of Theorem
The above figure illustrates how this proposition was applied in the construction of altars. It shows a drawing of the base of the Mahavedi (Great Altar) for the Soma ritual.
Its base had to be constructed to precise dimensions to get the desired results. It had to be an isosceles trapezium like ABCD, with AD and BC being 24 and 30 padas (literally feet).
The altitude of the trapezium (i.e., the distance between the midpoints X and Y of AD and BC) had to be precisely 36 padas.
The instructions are given for the construction of this altar in Apastamba’s Sulbasutra are, in modern notation, as follows:
- With the help of a rope mark out XY, which is precisely 36 pads.
- Along this line, locate points P, R, and Q such that XP, XR, and XQ equal 5, 28, and 35 padas respectively.
- Construct perpendiculars at X and Y.
- Use the fact that the triangles APX, DPX, BRY, and CRY are right-angled
triangles with integral-valued sides to locate points A, B, C, and D. In other words, make AXD 24 padas and BYC 30 padas. Join AB, BC, CD, and DA.
Implied in these directions for construction are the following right-angled triangles with integral sides:
- DAPX and DDPX with sides 5, 12, 13
- DAOX and DDOX with sides 12, 16, 20
- DBRY and DCRY with sides 8, 15, 17
- DBOY and DCOY with sides 15, 20, 25
- DAQX and DDQX with sides 12, 35, 37
- DBXY and DCXY with sides 15, 36, 39
Another example is to merge two equal or unequal squares to obtain a third square. In modern notation,
- Let ABCD and PQRS be the two squares to be combined and let DX be equal to SR.
- Draw a line to join A and X.
- The square on AX is equal to the sum of the squares ABCD and PQRS.
- The original explanation then points out that DX2 + AD2 = AX2 = SR2 + AD2, which shows the use of the Pythagorean theorem.
Construct a Square of N Times
However, it is another practical usage of the Pythagoras theorem. Katyayana says, as much one less of a given times of given square side forms the base. Half of one plus of given times forms a side of the triangle. The arrow of that [traingle] makes the required number. The height of this triangle will give the required length of the new square.
Current square Area = 〖a〗^2; new square will be n times of given square. As given in the above illustration, ABD is making a right angle triangle. So,
Let’s take one example. The 10 times area of the square which side is 10 will be 1000.
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The Value of √ 2
A remarkable achievement of Vedic mathematics is the discovery of a procedure for evaluating square roots to a high degree of approximation.
The problem may have originally arisen from an attempt to construct a square
twice the area of a given square altar. The problem, which the reader may wish to try, is one of constructing a square twice the area of a given square, A, of side 1 unit.
The procedure given in the three Sulbasutras discussed earlier may be restated as “Increase the measure by its third and this third by its own fourth less the thirty-fourth part of that fourth”. This is the value with a special quantity in excess. Sulabhasutras have no clue as to how this extraordinary approximation took place.
Construction of Square
- Take a rope of length you want to create Square.
- Construct a circle of radius equal to the length of half rope.
- Attach both the points of the rope on the circle drawn above.
- Draw circle from both the points of the radius equal to the rope length.
- Draw four circles in four directions of radius equal to the half rope.
- Join the intersection point. Your square is created
By looking into the above formulas we can believe that Indian mathematics was way ahead of their time and extraordinary theorems and new inventions were happening in those days.
A few inventions happened due to the need for Vedic rituals but certainly, a few inventions must have happened due to the architectural needs. There were many grand universities and libraries in ancient India like “Taxila” and “Nalanda” where mathematics must have been taught as a separate branch.
It also seems true when we see the grand temples and their architecture of ancient India which is not possible without knowing the extraordinary math.
Above mathematical explanation was taken as it is from the book “The Crest of the Peacock Princeton University Press | Non-European Roots of Mathematics by George Gheverghese Joseph”
From the lectures of K. Ramasubramaniam: http://vedicheritage.gov.in/mathematics/
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