 #### The sulva-sutras, a collection of Hindu religious documents, were written between 600 and 300 B.C. Although this is quite a bit later than when the Vedas originate, the ideas set forth in them must have come from the earlier Vedic society. And also like the Vedas, the sulva-sutras were not mathematics textbooks! These contained the instructions for making special altars to the gods. These altars, made out of mud-brick, could be very complicated in shape and size, and often required the use of mathematical formulas. A common example is wanting to build a square-shaped altar (or section of one) which had the same area as a circular altar (a math problem which has baffled mathematicians for years). For this to work, they needed an approximation of pi, calculating procedures, and accurate construction methods6. Sulva means "cord", and the sulva-sutras are so named because the Hindu priests used a simple cord or string for all (or at least most) of their constructions. This cord could work as a straight edge (stretched tight), a compass (drawn around a point), and more (such as getting proportional lengths, by doubling the string over itself the desired number of times). A pole was also often used, for functions such as the making of circles and also sometimes in marking corners of rectangles, so that the cord could be stretched around the poles to make various measurements7. The sulva-sutras begin by giving the names and measures of the units that will be used. To give you an idea of the kind of units that were used: "The following is the measure of an Angula: Fourteen grains of the anu-plant."8 Then, they go on to give the rules for various "constructions" which will be used later on to build the actual altars. These constructions are what give us most of our information about the Hindus' mathematical knowledge. One striking discovery is that there were many examples included of the wrongly named "Pythagorean theorem", or the rule for the lengths of sides of right triangles (now written as a2+b2=c2). It was expressed as "the cord stretched in the diagonal of a rectangle produces both areas which the cords forming the longer and the shorter side of a rectangle produce separately."9 In other words, the sum of the squares of the two different sides of the rectangle equals the square of its diagonal. ("Squares" probably meant the area of actual squares drawn on the three sides of the triangle formed.) "Pythagorean triplets", sets of side lengths that can be expressed as integers (for example 3,4,5 and 5,12,13), were used as a sort of proof of that proposition, and were also used to make right angles. One problem which the priests must have run into was that they could not find any squares with the sides and diagonal of integral lengths.10 In the end, they came up with an incredibly close approximation (6 digits of accuracy) to the square root of 2 (the length of the diagonal of a square whose sides are one unit). #### Along with the "simple" mathematics involved in altar-building, there were very complicated problems to decide what math to use where. One of these problems can be understood just by the fact that the most basic altar was in the form of a falcon! However, all of the bricks retained a square, rectangular, triangular, or circular shape. So the precise dimensions and area of every brick had to be calculated, after the specific design was decided upon. Usually, the total area was 7-1/2 units, because that was convenient for the falcon shape. But to make matters exponentially "worse", when two or more altars were made for the same purpose, subsequent ones usually had to be one unit bigger than the last; and the areas of each brick had to stay proportional.12 As you can infer, this was one heck of a math problem! And it had to be done twice for each altar, because a different design was needed for alternating layers, so that spaces between bricks weren't directly over each other.13  