Kalpa — ritual procedure, and the world's oldest geometry

कल्प

Also written: kalpa sutra · shrauta sutra · grihya sutra · shulba sutra · vedic geometry

Kalpa is the hands of the Veda — the science of doing. It divides into four families of sūtras, and one of the four accidentally became the foundation of Indian mathematics.

Sūtra family	Governs
Śrauta Sūtra	The great public sacrifices requiring three fires and multiple priests — agnihotra, darśapūrṇamāsa, soma rites, aśvamedha
Gṛhya Sūtra	The household rites on the single domestic fire — the sixteen saṃskāras, daily offerings, the five great yajñas
Dharma Sūtra	Conduct, duty, law, the āśramas and the king's obligations — the ancestor of the Dharma Śāstra
Śulba Sūtra	The measurement and construction of altars — geometry

The Śulba Sūtras

Fire altars had to be built to exact area and shape, and often rebuilt in a different shape of identical area. That single ritual requirement produced, in the Baudhāyana Śulba Sūtra, an explicit statement of the theorem on the diagonal of a rectangle — centuries before Pythagoras — along with methods for squaring the circle, doubling a square, and a value of √2 accurate to five decimal places.

दीर्घचतुरश्रस्याक्ष्णया रज्जुः पार्श्वमानी तिर्यग्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति।

dīrgha-caturaśrasyākṣṇayā rajjuḥ pārśvamānī tiryaṅmānī ca yat pṛthag bhūte kurutas tad ubhayaṁ karoti

The rope stretched along the diagonal of a rectangle produces an area equal to that produced by both the length and the breadth together.

— Baudhāyana Śulba Sūtra 1.48

Kalpa — ritual procedure, and the world's oldest geometry

The Śulba Sūtras

The sources of knowledge in Sanātana Dharma

Śruti — that which was heard

Smṛti — that which was remembered

Nigama and Āgama — the two revealed streams

The four Vedas

Ṛgveda