The Śulba Sūtras (c. 800–500 BCE) are the earliest known texts in Indian mathematics related to geometry, written during the Vedic period. These texts were practical manuals for constructing altars (yajña-vedis) used in Vedic rituals and contained geometric rules for measuring and shaping sacrificial fire altars.
The first Śulba Sūtra was written by Baudhāyana around 800 BCE.
It is one of the four surviving Śulba Sūtras, the others being written by Āpastamba, Mānava, and Kātyāyana.
The name “Śulba” means “cord” or “rope”, indicating that geometric constructions were done using ropes (similar to ancient Egyptian and Mesopotamian methods).
- Pythagorean Theorem (Before Pythagoras)
Baudhāyana’s Śulba Sūtra contains the earliest known statement of the Pythagorean theorem:
“The diagonal of a rectangle produces the same area as the sum of the squares on the two sides.”
He provided practical applications of this rule for constructing perpendiculars and doubling altar areas.
- Geometric Constructions
The texts explain how to construct squares, rectangles, rhombuses, and circles using ropes.
Methods for transforming one geometric shape into another while preserving the same area were described.
- Irrational Numbers
Baudhāyana gave an approximation of 2\sqrt{2} as:
2≈1+13+13×4−13×4×34≈1.414215\sqrt{2} \approx 1 + \frac{1}{3} + \frac{1}{3 \times 4} – \frac{1}{3 \times 4 \times 34} \approx 1.414215
This value is accurate up to five decimal places, making it one of the earliest recorded approximations of an irrational number.
- Right-Angle Constructions
The 3-4-5 triangle method was used to construct right angles, which is still used in surveying and carpentry today.
- Doubling the Square (Geometric Algebra)
The Śulba Sūtras provided an algorithmic method to double the area of a square using diagonal constructions.
| Civilization | Geometric Contributions | Approximate Date |
| India (Baudhāyana’s Śulba Sūtra) | Pythagorean theorem, right-angle construction, irrational numbers | 800 BCE |
| Babylonians | Approximation of 2\sqrt{2}, area calculations | 1800 BCE |
| Egyptians | Rope-stretching for right angles, basic area formulas | 2000 BCE |
| Greeks (Euclid, Pythagoras) | Formal proofs, deductive geometry | 500–300 BCE |
The Śulba Sūtras provided the foundation for later Indian mathematics and astronomy.
Their practical approach to geometry influenced temple architecture, town planning, and land measurement.
Later Indian mathematicians like Brahmagupta and Bhaskara II built upon these ideas in algebra and trigonometry.
