The Sulvasutras are ancient Hindu texts from 600-300 BC that describe geometric constructions and formulas to help build Vedic altars. They contain some of the earliest concepts resembling the Pythagorean theorem and approximations of irrational numbers like the square root of 2. The Sulvasutras were created by Vedic priests and mathematicians to precisely lay out fire altars and ritual spaces according to religious ceremonies and principles of symmetry.

1. Sulvasutras By Sarah Berman http://abish.byui.edu/library/libguides/francisl/Hinduism.jpg

2. What are the Sulvasutras? The Sulvasutras, or Sulbasutras, are a collection of sacred Hindu documents dating around 600-300 B.C. that were used to show members of the ancient Vedic culture how to construct Hindu altars. The Sulvasutras are made up of 9 or so texts, 8 of which can still be found in print and manuscript today. http://www.harekrsna.com/sun//features/11-07/geometry1a.jpg

3. More about the Sulvasutras The Sulvasutras are believed to be mathematical appendices to the Vedas, a large group of texts that discuss Hindu traditions, rules, and hymns, much like any religious text today. The name Sulvasutras literally translates to “rules of the cord,” or rope, and the mathematics within them are based on instructions as to how to use ropes to construct the desired architectural shape.

4. Where They’re From The Sulvasutras were created within the ancient Vedic civilization. The people of the Vedic period were Indo-Europeans who travelled from the Northwest of modern-day Middle East in what is now Iran, and eventually settled down around the Ganges river in India This map shows a rough outline of where the Vedic peoples occupied http://mstreitwieser.com/civilization/images/c_indus_map.jpg

5. Vedic Civilization The Vedic period is the era preceding the development of Sanskrit literature(up to about 600 B.C. or so). The word Vedic comes from the Greek word for knowledge, vid, which is very symbolic of Vedic culture and society based on all that was discovered and created during the era. The Vedic people created the sacred books of India that are considered the original building blocks for Hinduism and even delved into other Hindu creations such as yoga.

6. Images of Vedic Culture & Society Left: a scene depicting a Vedic village Right: An image of one of the many Hindu gods & goddesses http://www.vedicworld.org/images/entryleft.jpg http://img444.imageshack.us/i/sutra1oq8.jpg/ Here depicts an ancient Vedic sacrificing ritual still in use today A typical woman in Vedic society http://images.blogs.hindustantimes.com/she-baba/post/usha.jpg http://beacononline.files.wordpress.com/2008/07/puja-2.gif

7. Who Wrote The Sulvasutras? The Sulvasutras were written by scribes, not scribes functioning as copiers of other texts, but scribes who most likely developed the math within the texts themselves. All we know about the authorship of these documents is the name of the writers of each of the texts: The Baudhayana: by Baudhayana, mathematician and priest The Apustamba: by Apustamba as well as a group of Brahmins The Manava: by Manava, his work discusses the religious rituals of Vedic religion The Katyayana: by Katyayana, who wrote about the eternal history behind the origin of words The other 4 in print are also named for their authors, the Maitrayaniya, Varaha, Vadhula, and Hiranyakeshin and are of lesser importance as they largely resemble the primary 4 texts.

8. What The Sulvasutras Teach The Sulvasutras tackled all problems that these mathematicians needed to solve in order to build sturdy, beautiful altars no matter what shape. There are even references to altars in the shapes of falcons and turtles. They were used practically as religious texts and as the true way to build an altar worthy of sacrificial rituals.

9. Pythagorean Theorem Anyone? Although owed to many other mathematicians like Pythagoras, the Babylonians, or even the Egyptians, many consider the Sulvasutras to have the first known use of mathematics resembling the Pythagorean Theorem. Today, we understand the Pythagorean Theorem as a²+b²=c² but the writers of the Sulvasutras had a different way of finding lengths and areas of triangles.

10. Pythagorean Theorem Continued… The study of what we call the Pythagorean theorem begins with these words in the Sulvasutras: “The diagonal of a square produces double the area [of the square].”(wikipedia) We see this as simply 2 times the area of an isosceles triangle, or A=bh

11. More Pythagoras… Next we have to break down what the Pythagorean Theorem really means: The Pythagorean Theorem is actually just the calculations of the areas of squares. When you use the formula, you are really finding the areas of all the squares off of the sides of the triangle, because the area of a triangle is A=b² http://students.umf.maine.edu/~michaur/MATHWEBQUEST/Pythagoras%20diagram.jpg

12. What Vedic Scribes Used It For In the Sulvasutras the process is next described as: “The areas [of the squares] produced separately by the lengths of the breadth of a rectangle together equal the area [of the square] produced by the diagonal.”(wikipedia) This means that they discovered the areas of the squares in the diagram ABPQ and DAXY added together equal the area of the larger square. From here, they went on to discover Pythagorean triples and right triangles in order to create their altars. http://archanaraghuram.files.wordpress.com/2007/10/sulbasutras_1.gif

13. Pythagorean Triples Once they discovered that their version of the Pythagorean Theorem was only true with right triangles, they also discovered that there were rational constants for which the idea could always be applied. We know these as Pythagorean triples, the numbers that always came out nicely when you plugged them into the formula (like 3,4,5). The mathematicians, however, used the rectangle in the last picture as their reference point instead of a triangle, as we do today.

14. √2 The way that the scribes of the Vedic era found irrational numbers like √2 was very different from how we do it today, and yet was still accurate to 5 decimal places. The formula they used is thought to have been something along the lines of √(a²+r)≈a+(r/2a) – [(r/2a)²/2(a+(r/2a))] So, for √2, when a=4/3 and r=2/9 their answer was 1.4142 (The real answer is 1.4142, pretty incredible...) Their formula could have possibly come from Mesopotamian tablets, but ours today comes from a little theory called Taylor’s Polynomials…

15. Our √2 Like we learned in class, to approximate a square root using Taylor’s Polynomials, we have to understand local linearity, as well as the equation we came up with. Local Linearity means that the tangent line at a point touches the line at that point, so therefore a very close approximation can be made for numbers right around it. Now, if we want to find √2, we have to figure out what else is close to √2. The closest is probably √1.

16. √2 Continued… So, what we want to do is create a polynomial centered around x=1 that approximates f(x)= √x We need a constant approximation which in this case will be y=1 We need to find the tangent line using the derivative and point-slope form, so, since we have a point, it shouldn’t be too hard f(x)= √x, so f(1)=1 f’(x)=1/(2 √x), so f’(1)=½ Now we have the point (1,1) as well as the slope m=½ so we can now put it in point-slope form y – 1= ½(x – 1), or y = ½(x – 1)+1 With this equation, when x=2, we get 1.5, not too far off from 1.4142. If we had decided to keep on going into a quadratic approximation, the number would be even closer to the actual value of √2

17. Like the Rhind Papyrus… Both deal with triangles and trigonometry. While the Rhind Papyrus teaches scribes to do basic math, the Sulvasutras dictate how exactly to create triangles using certain lengths and positioning of ropes. Both texts also delve into a definition of Π, and require an accurate approximation to carry out both the problems on the Rhind Papyrus, and the actual construction of altars in Vedic society.

18. Bibliography “Geometry: Finding the Right Angle.” Encyclopaedia Britannica Online School Edition. Britannica Digital Learning, n.d. Web. 21 Oct. 2009. <http://school.eb.com/‌all/‌eb/‌article-217473>. O’Connor, J. J., and E. F. Robertson. “The Indian Sulbasutras.” MacTutor History of Mathematics. University of St. Andrews, n.d. Web. 21 Oct. 2009. <http://www-history.mcs.st-andrews.ac.uk/‌HistTopics/‌Indian_sulbasutras.html >. Scott, J. A. 86.55, An Eight-Point Circle. 325-326. JSTOR. The Mathematical Gazette, 2002. Web. 21 Oct. 2009. <http://www.jstor.org/‌stable/‌3621878?&Search=yes&term=sulvasutras&list=hide&searchUri=%2Faction%2FdoBasicSearch%3FQuery%3Dsulvasutras%26wc%3Don%26dc%3DAll%2BDisciplines&item=4&ttl=18&returnArticleService=showArticle>. “Shulba Sutras.” Wikipedia: The Free Encyclopaedia. Wikimedia Foundation, 7 June 2009. Web. 21 Oct. 2009. <http://en.wikipedia.org/‌wiki/‌Shulba_Sutras>. Stewart, Robin. “Sulva-Sutras.” robinstewart.com. Robin Stewart, 30 Sept. 2008. Web. 21 Oct. 2009. <http://www.robinstewart.com/‌personal/‌learn/‌indiamath/‌sutras.html>. “Vedic Literature.” Encyclopaedia Americana. Vol. 27. N.p.: Grolier, 1998. 925-928. Print. Encyclopaedia Americana International Edition.