History of Mathematics Project | Vedic Fire Altar Rectangle Diagonal

History of
Mathematics

Project

around 800 to 500 BCE

Vedic Fire Altar Rectangle Diagonal

Altar construction using the Pythagorean theorem

The Śulbasūtras were ritual texts written by successive generations of scribes who not only copied but also commented on and extended the work of their predecessors. Nothing is known about the authors of the Śulbasūtras other than their names and a rough idea of when they lived. They are the only sources of knowledge about Indian mathematics from the Vedic period.

The Śulbasūtras, no original texts of which are known to have survived, were Indian ritual texts containing geometrical content related to fire altar construction. In the earlier Baudhāyana Śulbasūtra, a special case of the Pythagorean theorem was given for the case of an isosceles right triangle. The general version involving a rectangle was subsequently stated in the Kātyāyana Śulbasūtra.

Artifact format

Unknown (lost)

Artifact origin

India

Timeline

Interactive Content

Computational Explanation

The Pythagorean theorem is a fundamental theorem in Euclidean geometry that states that for any right triangle with leg lengths a and b and hypotenuse c,



. A visualization of the Pythagorean theorem in terms of the areas of the squares based on each side of a right triangle can be obtained using computational geometry functionality built into the Wolfram Language:

In[]:=

RandomInstance[GeometricScene[{a,b,c,ab,ba,bc,cb,ca,ac},{triStyle[Triangle[{a,b,c}],Opacity[.8],Orange],PolygonAngle[tri]90°,GeometricAssertion[tri,"Counterclockwise"],GeometricAssertion[{Polygon[{a,b,ba,ab}],Polygon[{b,c,cb,bc}],Polygon[{c,a,ac,ca}]},"Regular","Clockwise"]}]]

Out[]=

The Śulbasūtras were Indian ritual texts containing geometrical content related to fire altar construction rules. In actuality, the Śulbasūtras were construction manuals for geometric shapes such as squares, circles and rectangles. As a result, the term sūtra came to mean a rope for measuring an altar. Unfortunately, no original Śulbasūtra texts are known to have survived, so later copies are the only sources of knowledge about Indian mathematics from the Vedic period.

In the Baudhāyana Śulbasūtra, a special case of the Pythagorean theorem was given for the case of an isosceles right triangle. The general version involving a rectangle was subsequently stated in the Kātyāyana Śulbasūtra, which gave a general statement of the Pythagorean theorem as follows:

"The rope which is stretched along the length of the diagonal of a rectangle produces an area which the vertical and horizontal sides make together."

A geometric diagram can be constructed to show this relationship, where the rope stretched across the diagonal BD gives the square BDFE and the problem states that the area of this square equals the sum of the areas of the squares constructed on the sides AD and AB:

Out[]=

The Śulbasūtras contained many examples of Pythagorean triples (triples of integers satisfying the Pythagorean theorem



), as well as rational versions of triples:

In[]:=

Function[{a,b,c},



]@@@{5,12,13},{12,16,20},{8,15,17},{15,20,25},{12,35,37},{15,36,39},

,6,

,

,10,



Out[]=

{True,True,True,True,True,True,True,True}

Numerical evidence for the version of the Pythagorean theorem appearing in the Kātyāyana Śulbasūtra can be obtained using computational geometry functionality built into the Wolfram Language:

In[]:=

ri=RandomInstance[GeometricScene[{"A","B","C","D","E","F","P","Q","X","Y"},{GeometricAssertion[Polygon[{"A","B","C","D"}],"Equiangular","Counterclockwise"],GeometricAssertion[{Polygon[{"A","B","Q","P"}],Polygon[{"D","A","X","Y"}],Polygon[{"B","D","F","E"}]},"Regular","Clockwise"]}]]

Out[]=

Plugging in the values corresponding to the preceding random scene shows the stated equality is indeed satisfied:

In[]:=

{Area[Polygon[{"A","B","Q","P"}]]+Area[Polygon[{"D","A","X","Y"}]],Area[Polygon[{"B","D","F","E"}]]}/.ri["Points"]

Out[]=

{8.72914,8.72914}

Other Resources

History of Mathematics
Vedic Fire Altar Square Diagonal

Wolfram Demonstration
Pythagorean Theorem

Additional Reading

Plofker, K. "The Śulba-Sūtras." §2.2 in Mathematics in India. Princeton, NJ: Princeton University Press, pp. 16–18, 2007.
Plofker, K. "The Śulbasūtras." Ch. 4, § II.b in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 387–393, 2007.
Sen, S. N. and Bag, A. K. The Sulbasutras. New Delhi: Indian National Science Academy, 1983.