Unit 3: Indian Mathematics & Astronomy
1. Great Indian Mathematicians & Core Contributions
Indian mathematicians made foundational contributions across arithmetic, algebra, geometry, and infinite series long before the European Renaissance.
GREAT INDIAN MATHEMATICIANS
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BAUDHĀYANA ĀRYABHAṬA BRAHMAGUPTA BHĀSKARĀCĀRYA MĀDHAVA OF
(c. 800 BCE) (476–550 CE) (598–668 CE) (1114–1185 CE) SANGAMAGRAMA
(1340–1425 CE)
Key Mathematicians & Their Works
- • Baudhāyana (c. 800 BCE): Author of Baudhāyana Śulba Sūtra; formulated geometric principles including early statements of the Pythagorean theorem.
- • Āryabhaṭa I (476–550 CE): Author of Āryabhaṭīya; calculated the value of π, stated that Earth rotates on its axis, and laid the foundations of Indian trigonometry.
- • Brahmagupta (598–668 CE): Author of Brāhmasphuṭasiddhānta; formulated operational rules for zero, negative numbers, and solved second-order indeterminate quadratic equations (Nx2 + 1 = y2, known as Pell’s equation).
- • Bhāskarācārya II (1114–1185 CE): Author of Līlāvatī, Bījagaṇita, and Siddhānta Śiromaṇi; developed the Cakravāla method for solving quadratic indeterminate equations and anticipated differential calculus concepts.
- • Mādhava of Saṅgamagrāma (1340–1425 CE): Founder of the Kerala School of Astronomy and Mathematics; discovered infinite series expansions for trigonometric functions (sin x, cos x, and arctan x) centuries before Gregory, Leibniz, and Newton.
2. Arithmetic Operations & Geometric Foundations
GEOMETRIC & ARITHMETIC TEXTS
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ŚULBA SŪTRAS ĀRYABHAṬĪYA-BHĀṢYA
(Geometric Construction Rules) (Arithmetic & Algebraic Proofs)
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• Altar construction geometry • Commentaries by Bhāskara I &
• Transformation of shapes (square to circle) Nīlakaṇṭha Somayājī
Śulba Sūtras & Geometry
The Śulba Sūtras (Baudhāyana, Āpastamba, Kātyāyana) are manual guides for constructing Vedic altars (Vedis) using geometric rules:
- • Pythagorean Theorem Statement: Baudhāyana states: “The diagonal of a rectangle produces both areas which its sides produce separately.” (a2 + b2 = c2)
- • Square-to-Circle & Circle-to-Square Transformation: Developed geometric algorithms for transforming a square into a circle of equal area and vice versa.
3. Value of π, Trigonometry, & Algebra
Approximation of π
Āryabhaṭa’s Formula (Āryabhaṭīya 2.10):
“Add 4 to 100, multiply by 8, and then add 62,000. By this rule, the circumference of a circle with a diameter of 20,000 can be given.”
Āryabhaṭa noted that this value was an āsanna (approximate) value.
Mādhava’s Infinite Series for π (Mādhava-Leibniz Series):
TRIGONOMETRIC EVOLUTION
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JYĀ (Sine) KOṬI-JYĀ (Cosine) ŚARA (Versine)
R sin(θ) R cos(θ) R(1 − cos(θ))
Trigonometric Functions
Indian astronomy introduced modern trigonometric definitions using half-chords (Ardha-jyā), which evolved into the term Jyā (Sine), Koṭi-jyā (Cosine), and Utkrama-jyā or Śara (Versine). Āryabhaṭa produced the first Sine Table at intervals of 3.75° (225′).
Algebra (Bījagaṇita)
- • Kuttaka Method (Linear Indeterminate Equations): Developed by Āryabhaṭa and expanded by Bhāskara I to solve ax + c = by using an algorithm similar to Euclidean continued fractions.
- • Cakravāla Method (Cyclic Algorithm): Perfected by Bhāskarācārya II to solve Nx2 + 1 = y2, praised by 20th-century European mathematicians like Hermann Hankel for its efficiency.
4. Indian Astronomy & The Celestial Coordinate System
CELESTIAL COORDINATE SYSTEM
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ECLIPTIC SYSTEM RĀŚIS (Zodiac Signs) NAKṢATRAS (Lunar Mansions)
(Planetary Paths) (12 Divisions) (27 Divisions)
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• Earth's axial rotation • 360° / 12 = 30° each • 360° / 27 = 13° 20' each
• Heliocentric considerations • Aries to Pisces • Ashwini to Revati
Siddhāntic Astronomy
Indian astronomical literature is classified under Siddhāntas (treatises providing complete mathematical models of planetary motions, eclipses, and coordinate transformations).
- • Celestial Spheres: Observed movements mapped relative to the equator (Visuvadvṛtta) and the ecliptic (Kāntivṛtta).
- • Nakṣatra System: A lunar-centric division of the ecliptic into 27 segments (13° 20′ each), matching the Moon’s orbital period relative to background stars.
- • Rāśi System: 12 equal divisions (30° each) of the solar ecliptic.
5. Elements of the Indian Calendar (Pañcāṅga)
The Pañcāṅga is the traditional Indian luni-solar calendar system based on 5 primary astronomical parameters (Pañca = 5, Aṅga = limbs):
THE 5 LIMBS OF PAÑCĀṄGA
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1. TITHI 2. VĀRA 3. NAKṢATRA 4. YOGA 5. KARAṄA
(Lunar Day) (Solar Day) (Moon Segment) (Angular Sum) (Half Lunar Day)
6. Astronomical Instruments (Yantras) & Jantar Mantar
To validate mathematical calculations against observational reality, Indian astronomers designed sophisticated observational instruments (Yantras).
ASTRONOMICAL INSTRUMENTS (YANTRAS)
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HISTORICAL YANTRAS JANTAR MANTAR OBSERVATORIES KEY YANTRAS
(Wood, Brass, & String) (Raja Jai Singh Sawai II) (Masonry Architecture)
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• Śaṅku (Gnomon) • Built at 5 sites: Delhi, • Samrāṭ Yantra (Sun Dial)
• Cakra Yantra (Circular ring) Jaipur, Ujjain, Varanasi, • Jai Prakāś Yantra
• Turiya Yantra (Quadrant) Mathura • Rāma Yantra
Jantar Mantar Observatory Complex
Founded by Maharaja Sawai Jai Singh II of Jaipur in the early 18th century, these open-air masonry observatories scaled up traditional brass instruments to achieve high precision in measuring celestial coordinates.
Key Instruments (Yantras)
- • Samrāṭ Yantra (The Supreme Instrument): A massive triangular gnomon sundial capable of measuring local solar time down to an accuracy of 2 seconds.
- • Jai Prakāś Yantra: Hemispherical bowl instruments set into the ground with suspended wires, mapped with inverted coordinates to track celestial positions directly.
- • Rāma Yantra: Cylindrical open-top structures used to measure the altitude and azimuth of celestial objects.
Key Learning Summary