The Baudhayana Sulbasutra, About 800 BCE, States the Rope Rule for a Rectangle's Diagonal and Gives the Square Root of 2 Correct to Five Decimal Places

The Baudhayana Sulbasutra, a manual for building fire altars that historians place near 800 BCE, states in words that the rope stretched along the diagonal of a rectangle makes the two squares on its sides added together. It lists six pairs of side lengths where this can be checked with whole numbers, from 3 and 4 to 15 and 36. The same text gives the diagonal of a square as 1, plus a third, plus a quarter of that third, minus a thirty-fourth of that quarter. That works out to 577/408, or 1.4142157, which agrees with the square root of 2 through the fifth decimal place.

A manual of ropes and poles

The word Sulbasutra means “sutras of the cord,” in the translation printed in 1875 by the Sanskritist George Thibaut, then a professor at Benares College. A sutra is a short rule (Panini’s Sanskrit grammar, covered in another article, is also a book of short rules), and the Sulbasutras are the sections of the Vedic ritual manuals that explain how to measure and build the sacrificial grounds and altars. Thibaut wrote that these texts prove the earliest geometry among Indians arose from the needs of sacrifice, and he gave the first place among them to Baudhayana for a clearer and fuller treatment than the others.

The MacTutor history of mathematics at the University of St Andrews dates the Baudhayana text to about 800 BCE, the Manava to about 750 BCE, Apastamba to about 600 BCE and Katyayana to about 200 BCE. Baudhayana’s, it says, is the oldest Sulbasutra we possess, though the authors are known only by name and rough period. “We do not know his dates accurately enough to even guess at a life span for him,” its Baudhayana biography says, and a chapter by Ian Pearce on the same site dates the whole group between 800 and 200 BCE.

The rule for the rectangle, and six pairs of sides

Thibaut’s edition prints two rules by Baudhayana a few verses apart. One says the cord along the diagonal of a square makes an area double the size of the square. The other says the cord along the diagonal of an oblong, which is a rectangle, makes both the areas that its long side and its short side make separately.

A later rule, in Thibaut’s translation, says this is seen in oblongs whose sides are three and four, twelve and five, fifteen and eight, seven and twenty-four, twelve and thirty-five, and fifteen and thirty-six. The text does not state the diagonals, which are 5, 13, 17, 25, 37 and 39, as Thibaut points out. The last pair, he adds, is the second one tripled, so the list holds five different shapes. Datta, in his 1932 study for the University of Calcutta, numbers these as verses 45, 48 and 49 of the first chapter.

One detail of the record needs flagging. MacTutor’s page on the Sulbasutras quotes only the square form for Baudhayana and credits Katyayana, centuries later, with the general rectangle statement. Thibaut and Datta both print the rectangle rule under Baudhayana’s name, and the six side pairs depend on it.

What the cord did on the ground

The triples were working tools. In rules from Apastamba that Thibaut prints, the cord marks out the ground for the great soma sacrifices. That enclosure had a west side of 30 units called padas, a central east-west line of 36 and an east side of 24. A builder tied a cord of 54 units to poles at both ends of the 36-unit line, with marks on the cord at 12 and 15 units from the western end.

Taking the cord at the mark at 15 and pulling it taut to the south fixes a corner, because the cord then forms a triangle with sides of 15, 36 and 39. That triangle has a right angle where the 15 meets the 36, and 15 squared plus 36 squared is 1,521, the square of 39. According to Thibaut, the whole process rests on the knowledge that a triangle with those sides is rectangular. A marked cord and a few poles gave a priest a square corner.

Seven and a half square purushas

The altar itself, the agni, is where the geometry becomes demanding. Thibaut counts five layers of bricks, each layer of 200 bricks, so 1,000 in all, laid so that no brick lies on top of one of the same size and form. The basic altar roughly imitates a falcon in outline and is built of square bricks. The sutras give its area as seven and a half square purushas, where a purusha is the height of a man standing with his arms raised.

Four of those squares form the bird’s body. Three more make the two wings and the tail, and the wings and tail are then lengthened a little, by a fifth and a tenth of a purusha, to look more like a bird. The ritual texts list other shapes: a falcon with curved wings and a spread tail, a heron, a triangle, a chariot wheel with sixteen spokes, a tub, a tortoise. Each had to have the same area as the plain falcon, so the builders had to turn a rectangle into a square, a square into a triangle and a square into a circle.

The hardest rule came with rebuilding. According to Thibaut, the second altar built by a sacrificer added one square purusha to the area and the third added two more, and Datta writes that the overall shape was not allowed to change. Datta calls the diagonal theorem indispensable for altar geometry, along with constructions such as a square equal to the sum of two other squares. Baudhayana’s construction of a square equal to a given rectangle appears on MacTutor too, built by cutting and moving rectangles and finished with a rope turned about a point.

577/408, the diagonal of a square

The rule for the square root of 2 is short enough to give as Thibaut renders it: increase the measure by its third part, and this third by its own fourth, less the thirty-fourth part of that fourth. Thibaut says Apastamba gives the rule in the same words and that Katyayana gives it too. Nothing in the rule says it concerns a square’s diagonal. The commentaries, Thibaut explains, leave no doubt that the measure is a side and the increased measure the diagonal, and he calls the match with the true value very close. Verse 61 of Baudhayana’s first chapter holds the rule, per MacTutor’s biography.

Written out, 1 plus 1/3 plus 1/12 is 17/12, which is 1.41667 and a little too large. Taking away 1/408 leaves 577/408. The true value of the square root of 2 is 1.4142136, so the error is about two parts in a million.

How the authors reached the correction is a modern reconstruction, not something the text says. Thibaut proposed in 1875, and Datta took up in 1932, that they began with a square of side 12, whose diagonal squared is 288 small squares. A side of 17 gives 289, one too many. Shaving a strip a thirty-fourth of a unit wide from two edges of the 17-unit square removes that one square, and a tiny overlap at the corner leaves the last correction. Why Baudhayana wanted such precision, when the altars did not need it, is a question MacTutor’s biography asks and leaves open.

Circles, and the weaker values of pi

The same text shows less accuracy with circles. MacTutor gives the standard Sulbasutra method for squaring the circle as a square whose side is 13/15 of the diameter, which implies a pi of 676/225, or 3.004. Other constructions in Baudhayana imply 900/289 (3.114) and 1156/361 (3.202). None of these is particularly accurate, MacTutor says, but they would not produce noticeable errors on an altar. Roughly thirteen centuries later, in 499 CE, Aryabhata set pi at 3.1416 in the Aryabhatiya, and the Kerala school of Madhava of Sangamagrama (c. 1340 to 1425) later found infinite series for pi, as the account of Kerala mathematics explains.

Older tablets, and what the Sulbasutras do not say

Babylonian clay tablets are older. Plimpton 322, held at Columbia University, is dated by MacTutor to between 1800 and 1650 BCE and by the University of British Columbia’s mathematics department to around 1800 BCE. Its columns appear to list Pythagorean triples. A tablet in the Yale Babylonian Collection, YBC 7289, shows a square with its diagonals marked and a base-60 value of 1;24,51,10 for the square root of 2, about 1.41421296. That is closer to the truth than the Indian figure: its error is about six parts in ten million, against two in a million. New York University’s Institute for the Study of the Ancient World dates the tablet to the 19th to 17th century BCE and says it shows Babylonian scribes knew the theorem.

Scholars read this evidence in different ways. MacTutor’s Sulbasutra page says some historians argue the Indian authors copied an approximation the Babylonians already had. Abraham Seidenberg, a University of California mathematician, concluded that Babylonian, Greek and Indian geometry derive from a common origin in which altar construction and the theorem played a central part, according to the obituary reproduced on MacTutor. Datta found no mention, direct or indirect, of a proof of the theorem in the early Indian literature, and MacTutor states that the Sulbasutras contain no proofs.

Thibaut’s 1875 paper credits A. C. Burnell, in his catalogue of Sanskrit manuscripts, with first drawing attention to these texts, quoting his remark that the earliest beginnings of geometry among the Brahmans must be sought in the Sulva portions of the ritual manuals. Thibaut adds that he had begun publishing Baudhayana’s Sulbasutra with a commentary and a translation in The Pandit, the journal of Benares College.