Shulba Sutras
शुल्बसूत्राणि
Geometric altar construction with Pythagorean theorem.
Shlokas (9)
+ Add ShlokaThe Three Sacred Fire Altars
Baudhayana Sulba Sutra · Chapter 1 · Verse 1
गार्हपत्यो वृत्तः — पूर्वः। आहवनीयः समचतुरश्रः — पश्चिमः। दक्षिणाग्निः अर्धवृत्तः — दक्षिणः। एते त्रयोऽग्नयः यज्ञमण्डले प्रतिष्ठाप्यन्ते॥
gārhapatyo vṛttaḥ — pūrvaḥ. āhavanīyaḥ samacaturaśraḥ — paścimaḥ | dakṣiṇāgniḥ ardhavṛttaḥ — dakṣiṇaḥ | ete trayo'gnayaḥ yajñamaṇḍale pratiṣṭhāpyante ||
The three sacred fire-altars of the Vedic yajna: (1) Gārhapatya — circular, in the east — the householder's fire; (2) Āhavanīya — square, in the west — where oblations are offered; (3) Dakṣiṇāgni — semicircular, in the south — the ancestors' fire. Their precise dimensions and geometry are specified in the Śulba Sūtras.
Apastamba's Theorem — Pythagorean Triplets
Apastamba Sulba Sutra · Chapter 1 · Verse 4
त्रिकं चतुष्कं च पञ्चकं च। द्वादशकं पञ्चदशकं नवदशकं च। पञ्चदशकं षट्त्रिंशकं नवत्रिंशकं च। एते सर्वे समकोणत्रिभुजस्य भुजाः॥
trikaṃ catuṣkaṃ ca pañcakaṃ ca | dvādaśakaṃ pañcadaśakaṃ navadaśakaṃ ca | pañcadaśakaṃ ṣaṭtriṃśakaṃ navatriṃśakaṃ ca | ete sarve samakoṇatribhujaṣya bhujāḥ ||
Āpastamba's Śulba Sūtra lists explicit Pythagorean triplets: (3,4,5), (12,15,19... wait — actually 12,16,20 = 3:4:5), (5,12,13), (8,15,17), (7,24,25), and others. These are integer solutions where a²+b²=c². Āpastamba knew these by ~600 BCE, confirming systematic knowledge of right-triangle geometry in ancient India.
Baudhayana Theorem (Pythagorean Theorem)
Baudhayana Shulba Sutra · Chapter 1 · Verse 12
दीर्घचतुरश्रस्याक्ष्णया रज्जुः पार्श्वमानी तिर्यङ्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति
dīrgha-caturasrasyākṣṇayā rajjuḥ pārśvamānī tiryaṅmānī ca yat pṛthag bhūte kurutas tad ubhayaṃ karoti
The diagonal of a rectangle produces by itself both the areas which the two sides of the rectangle produce separately.
Baudhayana's Pythagorean Theorem — Sulba Sutra
Baudhayana Sulba Sutra · Chapter 1 · Verse 48
दीर्घचतुरश्रस्याक्ष्णयारज्जुः पार्श्वमानी तिर्यग्मानी च यत् पृथग्भूते कुरुतस्तदुभयं करोति।
dīrghacaturaśrasyākṣṇayārajjuḥ pārśvamānī tiryagmānī ca yat pṛthagbhūte kurūtas tad ubhayaṃ karoti |
The diagonal of a rectangle produces (an area) equal to what both the length and the breadth produce separately (i.e., when used as sides of squares). This is the Śulba Sūtra statement of what the West calls the Pythagorean theorem — dating to ~800 BCE, approximately 250 years before Pythagoras.
Square Construction — Rope and Peg Method
Baudhayana Sulba Sutra · Chapter 1 · Verse 50
समचतुरश्रं चिकीर्षन् — यावत् कर्तुमिच्छति तावती रज्जुं चतुर्धा विभजेत्। तस्य त्रिभागेन पार्श्वं, तद्वर्गाद् अपरं वर्गं मुञ्चेत्। अक्ष्णया रज्जुः समचतुरश्रं करोति॥
samacaturaśraṃ cikīrṣan — yāvat kartum icchati tāvatī rajjuṃ caturdhā vibhajet | tasya tribhāgena pārśvaṃ, tadvargād aparaṃ vargaṃ muñcet | akṣṇayā rajjuḥ samacaturaśraṃ karoti ||
To construct a square: divide the rope into four equal parts. Use three parts as the side; construct squares on side and hypotenuse separately — the diagonal gives you the perfect square. The Śulba Sūtras provide rope-and-peg constructions equivalent to the ruler-and-compass methods of classical Greek geometry.
Combining Two Squares Into One
Baudhayana Sulba Sutra · Chapter 1 · Verse 51
द्वे समचतुरश्रे समासेतुमिच्छन् — प्रथमस्य करणीम् आनेत्य। द्वितीयस्य पार्श्वेन छिन्द्यात् — यच्छिष्टे तदपच्छेदे। अपच्छेदे कर्ण्या — तद् वर्गद्वयसमम् चतुरश्रम्॥
dve samacaturaśre samāsetum icchan — prathamasya karaṇīm ānetya | dvitīyasya pārśvena chindyāt — yac chiṣṭe tad apacchede | apacchede karṇyā — tad vargadvayasamam caturaśram ||
To combine two squares into one: take the diagonal of the first square; from it, cut off the side of the second square; with the remainder as the new diagonal — you get a square equal to both original squares combined. This is the geometric proof of a² + b² = c² using actual square construction.
Squaring the Circle — Approximate Method
Baudhayana Sulba Sutra · Chapter 2 · Verse 9
वृत्तं चतुरश्रं चिकीर्षन् — विष्कम्भमष्टौ भागान् कृत्वा। एकं भागमपोह्याष्टाविंशत्याऽपि तद् भागस्य। चतुर्थोऽष्टभागः शिष्यते — समचतुरश्रं भवति॥
vṛttaṃ caturaśraṃ cikīrṣan — viṣkambham aṣṭau bhāgān kṛtvā | ekaṃ bhāgam apohyāṣṭāviṃśatyā'pi tad bhāgasya | caturtha'ṣṭabhāgaḥ śiṣyate — samacaturaśraṃ bhavati ||
To square a circle: divide the diameter into 8 parts, subtract 1 part, then subtract 1/29 of that part, and subtract 1/4 of that result. The remaining value gives the side of the equivalent square. This is Baudhayana's approximation — it gives π ≈ 3.088, a remarkable early approximation.
Approximate Value of √2 — Baudhayana
Baudhayana Sulba Sutra · Chapter 2 · Verse 12
समस्य द्विकरणी — प्रमाणं तृतीयेन वर्धयेत्। तच्चतुर्थेनात्मचतुस्त्रिंशोनेन — समासेत्। साविशेषः॥
samasya dvikaraṇī — pramāṇaṃ tṛtīyena vardhayet | tac caturthenātmacatustriṃśonena — samāset | sāviśeṣaḥ ||
The diagonal of a square = side × (1 + 1/3 + 1/(3×4) − 1/(3×4×34)). This gives √2 ≈ 1 + 1/3 + 1/12 − 1/408 = 1.4142156... — correct to 5 decimal places. A remarkable rational approximation obtained by Baudhayana centuries before any Western mathematician.
The Falcon-Shaped Altar — Syenachit
Apastamba Sulba Sutra / Shatapatha Brahmana · Chapter 16 · Verse 1
श्येनचिद् — गरुडपक्षाकारा वेदिका। सप्तशतानि च द्विपञ्चाशदिष्टकाः। पुनर्जन्मार्थं चीयते — इति श्रुतिः॥
śyenacid — garuḍapakṣākārā vedikā | saptaśatāni ca dvipañcāśad iṣṭakāḥ | punarjanmārthaṃ cīyate — iti śrutiḥ ||
The Syenacit (falcon-shaped altar) is constructed in the form of a bird with spread wings — resembling Garuḍa. It is built of 752 bricks, arranged in 5 layers, and covers approximately 7.5 square puruṣas of area. It is built for the one who wishes to overcome repeated death (punar-mṛtyu) and attain liberation. The geometry of this altar is described in detail in the Śatapatha Brāhmaṇa and Āpastamba Śulba Sūtra.