History of Mathematics Project | Vedic Fire Altar Square Diagonal

History of
Mathematics

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around 800 to 500 BCE

Vedic Fire Altar Square Diagonal

Altar construction via a special case of 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. The Baudhāyana-śulba-sūtra gives a special case of the Pythagorean theorem.

The Śulbasūtras, no original texts of which are known to have survived, were Indian ritual texts containing geometrical content related to fire altar construction. The Baudhāyana-śulba-sūtra, transmitted orally as early as 800 BCE (but possibly as late as 500 BCE), stated, "The rope which is stretched across the diagonal of a square produces an area double the size of the original square" (which is a special case of the Pythagorean theorem corresponding to an isosceles right triangle) and prescribed a series of circles laid out via ropes prior to building one (of many possible shapes) of a Vedic fire altar.

Artifact format

Unknown (lost)

Artifact origin

India

Timeline

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Computational Explanation

It is straightforward to construct a square using methods similar to those used by the ancient Indians to build fire altars. First, take a length of string, fold it in half and mark the midpoint. Then lay the string horizontally, draw a line along it and make a mark at the center. The line segment so marked is the west-east line:

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Remove the string, place a pin at the red point marked in the center of the west-east line, loop the string around it so that two ends of the string coincide, pull it taut and use it as a compass to draw a red circle centered at the pin:

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Now place pins at the west and east points, fix one end of the string at each point in turn and draw green circles around them using the entire length of the string as the diameter:

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Draw the line connecting the two points of intersection of the green circles and mark its intersections with the original red circle as north and south:

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nsline=Line[RegionIntersection[c1,c2][[1]]]

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Line0,-

,0,



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{south,north}=RegionIntersection[nsline,c0][[1]]

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{{0,-1},{0,1}}

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At each of the points N, S, E and W, place a pin and, looping the string symmetrically around each point, draw a blue circle:

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The pairwise intersections of these four circles (excluding their common intersection at the origin) then determine the vertices of a square:

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squarecorners=DeleteCases[Join@@(RegionIntersection/@Partition[circles,2,1,1])[[All,1]],{0,0}]

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{{-1,1},{1,1},{1,-1},{-1,-1}}

Connecting the corners then gives the desired square:

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The construction can also be done directly using the Wolfram Language's built-in geometric scene language. Use of a string creates the west-east line by fixing "m" at the origin and "e" at (1, 0) and designating "m" as the midpoint of the line between "e" and "w":

RandomInstance[GeometricScene[{"m"{0,0},"e"{1,0},"w"},{"m"Midpoint[Style[Line[{"e","w"}],Hue[0,1,.8]]]}]]

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However, when constructed mathematically, this step is redundant since the condition that the red circle to be constructed in step 2 is centered at "m" and goes through "e" and "w" automatically implies that "m" is the midpoint of the line between "e" and "w":

RandomInstance[GeometricScene[{"m"{0,0},"e"{1,0},"w"},{Style[{Line[{"e","m","w"}],CircleThrough[{"e","w"},"m"]},Hue[0,1,.8]]}]]

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For step 3, draw two green circles: one centered at "e" and going through "w" and the other centered at "w" and going through "e":

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RandomInstance[GeometricScene[{"m"{0,0},"e"{1,0},"w"},{Style[{Line[{"e","m","w"}],CircleThrough[{"e","w"},"m"]},Hue[0,1,.8]],Style[{CircleThrough[{"w"},"e"],CircleThrough[{"e"},"w"]},ColorData[97][3]]}]]

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For step 4, draw the line connecting the two points of intersection of the green circles, "n1" and "s1", and mark its intersections with the original red circle as "n" and "s":

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RandomInstance[GeometricScene[{"m"{0,0},"e"{1,0},"w","n","s","n1","s1"},{Style[{Line[{"e","m","w"}],CircleThrough[{"e","w","n","s"},"m"]},Hue[0,1,.8]],GeometricAssertion[{"e","n","w","s"},"Counterclockwise"],Style[{CircleThrough[{"w","n1","s1"},"e"],CircleThrough[{"e","n1","s1"},"w"],Line[{"n1","n","m","s","s1"}]},ColorData[97][3]]}]]

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For the penultimate step 5, draw four blue circles centered at "e", "n", "w" and "s", with each going through "m":

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To complete the geometric construction in the sixth step, place "ne", "nw", "sw" and "se" at the pairwise intersections of the blue circles (excluding their mutual intersection point "m") and draw the square determined by these vertices:

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RandomInstance[GeometricScene[{"m"{0,0},"e"{1,0},"w"{-1,0},"n","s","n1","s1","ne","nw","sw","se"},{Style[{Line[{"e","m","w"}],CircleThrough[{"e","w","n","s"},"m"]},Hue[0,1,.8]],GeometricAssertion[{"e","n","w","s"},"Counterclockwise"],Style[{CircleThrough[{"w","n1","s1"},"e"],CircleThrough[{"e","n1","s1"},"w"],Line[{"n1","n","m","s","s1"}]},ColorData[97][3]],Style[{CircleThrough[{"m","se","ne"},"e"],CircleThrough[{"m","ne","nw"},"n"],CircleThrough[{"m","nw","sw"},"w"],CircleThrough[{"m","sw","se"},"s"]},ColorData[97][1]],GeometricAssertion[{"m","ne","nw","sw","se"},"Distinct"],Polygon[{"ne","nw","sw","se"}]}]]

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Other Resources

History of Mathematics
Vedic Fire Altar Rectangle 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.