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a² + b² = c². The most famous equation in mathematics, drilled into every school student's head as 'Pythagoras's theorem.' But the earliest known statement of this result appears not in Greece, but in India — in the Baudhayana Sulba Sutra, a text on Vedic fire altar construction written around 800 BCE. Pythagoras was born around 570 BCE — nearly 300 years later.
The Sulba Sutras are appendices to the Vedas, specifically dealing with the geometry required to construct fire altars (Agni) of precise shapes and areas. The word 'Sulba' literally means cord or rope — these were rope-and-peg geometry manuals, used to measure and lay out sacred altar spaces with mathematical exactness. The requirement was strict: altars had to be specific shapes (falcon, tortoise, wheel) with exact areas, because Vedic ritual demanded mathematical precision. Any deviation was considered ritually invalid.
Four principal Sulba Sutras survive: Baudhayana (~800 BCE), Apastamba (~600 BCE), Katyayana (~300 BCE), and Manava (~750 BCE). Of these, Baudhayana is the oldest — and the one that contains the explicit general theorem. Beyond the Pythagorean theorem, the Sulba Sutras also contain area-preserving transformations (converting a rectangle to a square of equal area, and vice versa), methods for circling the square (approximating a circle with the same area as a given square), and constructing altars of complex shapes while preserving exact areas — problems that anticipate integral geometry by two millennia.
Principal Sulba Sutras
Baudhayana Sulba Sutra
Oldest — contains the general theorem
Manava Sulba Sutra
Geometric transformations
Apastamba Sulba Sutra
Refined √2, additional constructions
Katyayana Sulba Sutra
Generalised geometric transformations
Baudhayana Sulba Sutra 1.48 states the theorem in Sanskrit. The verse reads: 'दीर्घचतुरश्रस्याक्ष्णयारज्जुः पार्श्वमानी तिर्यङ्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति' — which translates as: 'The diagonal of a rectangle produces both [areas] which its length and breadth produce separately.' This IS the Pythagorean theorem: diagonal² = length² + breadth². Note that it is stated as a GENERAL rule, applicable to all rectangles — not just specific cases.
दीर्घचतुरश्रस्याक्ष्णयारज्जुः पार्श्वमानी तिर्यङ्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति
The diagonal of a rectangle produces both [areas] which its length and breadth produce separately.
— Baudhayana Sulba Sutra 1.48, ~800 BCE
Baudhayana's meaning
"The diagonal of a rectangle produces" the area that "its length and breadth produce separately." In modern notation: c² = a² + b². A general rule for ALL rectangles.
Significance
This is not a special case — it is a general theorem. Baudhayana states it as a universal rule applying to all rectangles.
Baudhayana did not stop at the theorem. He also gave an extraordinarily accurate approximation of √2, needed to compute the diagonal of a unit square. His formula: √2 ≈ 1 + 1/3 + 1/(3×4) − 1/(3×4×34) = 1.4142156... The modern value is 1.4142135... That is correct to 5 decimal places — achieved with no calculators, no decimal notation, no computer algebra. No other civilization came close to this accuracy for centuries.
√2 ≈ 1 + 1/3 + 1/(3×4) − 1/(3×4×34)
= 1.4142156... (modern: 1.4142135...)
| Source | Value | Delta |
|---|---|---|
| Baudhayana (~800 BCE) | 1.4142156 | +0.0000021 |
| Apastamba (~600 BCE) | 1.4142135 | ~0.0000000 |
| Modern (IEEE 754) | 1.4142136 | reference |
Baudhayana's value differs from the modern value by only 0.0000021 — correct to 5 decimal places.
Baudhayana lists specific right triangles that satisfy a² + b² = c²: the triples (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). These are what we call 'Pythagorean triples' — though they might more accurately be called 'Baudhayana triples.' The Sulba Sutras use them for altar construction requiring precise right angles: if you stretch a rope of length 13 between pegs at 5 and 12, you get a perfect right angle. For comparison, the Babylonian Plimpton 322 tablet (~1800 BCE) lists some triples, but gives no general theorem. Baudhayana gives both the triples AND the theorem.
(3, 4, 5)
9 + 16 = 25
✓
(5, 12, 13)
25 + 144 = 169
✓
(8, 15, 17)
64 + 225 = 289
✓
(7, 24, 25)
49 + 576 = 625
✓
All used in altar construction to create precise right angles using rope-and-peg geometry.
Here is a concrete example of the theorem in ritual use: the problem of doubling a square altar. If the original altar has side s, the new (double-area) altar must have side s√2 — which is precisely the diagonal of the original square. So to construct a square with double the area, you take the diagonal of the original. This is a direct application of the theorem: diagonal² = s² + s² = 2s², so diagonal = s√2. The famous falcon-shaped altar (Syena-chiti) required even more complex geometric transformations — all dependent on this theorem.
The formula
If original square has side s, the doubled-area square has side = diagonal of original = s√2. Because: diagonal² = s² + s² = 2s².
Pythagoras (~570–495 BCE) almost certainly learned geometry during his extensive travels — to Egypt, Babylon, and possibly further east. The Greek tradition credits him with the first formal proof of the theorem, not merely its discovery. But here is the problem: no written work by Pythagoras himself survives. Everything attributed to him comes from his followers (the Pythagoreans) or from later ancient writers. The earliest surviving Greek proof appears in Euclid's Elements (~300 BCE), Book I, Proposition 47 — written over two centuries after Pythagoras lived. The fair assessment: Indians discovered and systematically applied the theorem 300 years before Pythagoras. Greeks may have provided the first formal deductive proof — though this too is debated, since it relies entirely on later accounts.
~800 BCE
Baudhayana stated it
General theorem + triples + √2
~570 BCE
Pythagoras born
230 years after Baudhayana
~300 BCE
Euclid's formal proof
Earliest surviving Greek proof
The chronology of the theorem across civilisations, from the earliest known statement to modern mathematics.
Plimpton 322 (Babylon)
Lists Pythagorean triples — no general theorem
Baudhayana Sulba Sutra
General theorem stated + √2 to 5 decimal places + triples (3,4,5), (5,12,13), (8,15,17), (7,24,25)
Apastamba Sulba Sutra
Refined √2, additional geometric constructions
Pythagoras born
Born in Samos, Greece — 230 years after Baudhayana
Euclid's Elements, Book I, Prop. 47
Earliest surviving formal Greek proof
Aryabhatiya
Uses the theorem for astronomical calculations — planetary distances, eclipse geometry