Brahmasphutasiddhanta by Brahmagupta (Introduction)

by Acharyavara Ram Swarup Sharma | 1966 | 93,779 words

Hinduism Ganitashastra Jyotisha

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Summary: This is the English introduction the Brahmasphutasiddhanta, a Sanskrit text dealing with ancient Indian mathematical astronomy and algebra written by Brahmagupta in the 7th century. The Brahma-sphuta-siddhanta highlights a thorough understanding of zero, rules for handling negative and positive numbers, and solutions for linear and quadratic equations. It further presents the earliest description of the quadratic formula and notable rules and theorems. The work consists of 24 chapters with varying verse counts, addressing topics from lunar and solar eclipses to astronomical instruments.

This is the English introduction portion of a four-volume publication of the Sanskrit text, which includes the commentaries: Vasana, Vijnana and various Hindi commentaries. The introduction discusses Brahmagupta's controversy and eventual acceptance of Aryabhata's astronomical systems and examines how the Brahma-sphuta-siddhanta formed the basis for Indian astronomy for centuries. Moreover, it showcases Brahmagupta's originality, such as his second-difference interpolation and the introduction of the sine rule in Indian trigonometry, underscoring his significant advancement beyond his predecessors, particularly Aryabhata.

Source 1: exoticindiaart.com
Source 2: archive.org

Contents of this online book ( + / - )

The full text of the Brahmasphutasiddhanta by Brahmagupta (Introduction) in English is available here and publically accesible (free to read online). Of course, I would always recommend buying the book so you get the latest edition. You can see all this book’s content by visiting the pages in the below index:

Warning! This document has not been proofread. However, for each page you can (and should) verify the text with the original source PDF file.

+ Chapter 1 - Astronomy in Ancient Nations

. Part 1 - Introduction—Astronomy in Ancient Nations

. Part 2 - Origin of the Zodiac

. Part 3 - First notions of the Constellations

. Part 4 - Indo-Greek Contacts

. Part 5 - Earth as a Sphere (in ancient Indian astronomy)

. Part 6 - Earth rotation by a current of aerial fluid

. Part 7 - Contacts with Arabs

. Part 8 - Arabs and Greeks

. Part 9 - Western Countries under Islam

. Part 10 - Figure of Earth

. Part 11 - Moon and its orbit

. Part 12 - Abu al-Wafa' and his Almagest

. Part 13 - Abu al-Wafa' and Ptolemy

. Part 14 - Al Farghani and others on Planets

. Part 15 - Ali Qushji and diameters of planets

. Part 16 - History of Al Qazwini, Abu al-Faraj and al-Jaghmini

. Part 17 - History of Ibn Bajja, Ibn Tufayl and Abu 'al-Walid

. Part 18 - History of Nasir al-Din al-Tusi

. Part 19 - Arabian astronomers and Ptolemaic system

. Part 20 - Arabs on motions of fixed stars

+ Chapter 2 - Personal References of Brahmagupta

. Part 1 - Introduction—Personal References of Brahmagupta

. Part 2 - A Note on Bhillamala

. Part 3 - Brahmagupta's own References

. Part 4 - Reference to Aryabhata

. Part 5 - Brahmagupta opposed to Shrisena

. Part 6 - Reference to Romaka and Paulisa systems

Chapter 3 - Manuscripts of the Brahmasphuta Siddhanta

+ Chapter 4 - Subject-matter classified in the Brahmasphutasiddhanta and Khandakhadyaka

. Part 1 - Divisions of Aryabhatiyam

. Part 3 - Major Landmarks in Indian astronomy

. Part 4 - Division of Mahabhaskariya and Laghu-bhaskariya

. Part 5 - Contents of the Brahmasphutasiddhanta

. Part 6 - Aryabhata and Brahmagupta Controversy

. Part 7 - Al-biruni and Brahmagupta

+ Chapter 5 - Brahmagupta's Originality in the Khandakhadyaka

. Part 1 - Introduction Brahmagupta's Khandakhadyaka

. Part 2 - Brahmagupta corrects Mars's Aphelion Point

. Part 3 - Brahmagupta First to Use Second Differences

. Part 4 - Brahmagupta First to Introduce Sine Rule in Indian Plane Trigonometry

. Part 5 - Brahmagupta corrects Dimensions of the Epicycle of Apsis

+ Chapter 6 - Indian Luni-Solar Astronomy

. Part 1 - Introduction to Solar Astronomy

. Part 2 - Introduction to Lunar Astronomy

. Part 3 - Description of Moon's Mean Motion

. Part 4 - Moon's Equations: The First Equation

. Part 5 - Moon's Second Inequality or Equation

. Part 6 - Manjula's Second Equation of the Moon (932 A.D.)

. Part 7 - Shripati's Second Inequality of the Moon

. Part 8 - Bhaskara II on Moon's Inequalities

. Part 9 - Candrasekhara of Orissa on the Moon's Inequalities

+ Chapter 7 - Greek and Hindu Methods in Spherical Astronomy

. Part 1 - Introduction—Greek and Hindu Methods in Spherical Astronomy

. Part 2 - Ancient Indian Methods in Spherical Astronomy

. Part 3 - Problem 3: To find the “Time-altitude” Equation

. Part 4 - Problem 4: The Altazimuth Equation

. Part 5 - Problem 5: To find the Angle between the Ecliptic and the Meridian

. Part 6 - Problem 6: To find the Angle between the Ecliptic and the Horizon

. Part 7 - Problem 7: To find the Angle made by the Vertical ...

. Part 8 - Problem 8: To convert the Celestial Longitude ...

+ Chapter 8 - Epicyclic Theory of Ancient Indians

. Part 1 - Apparent Motions of the Sun and Moon

. Part 2 - Superior Planets

. Part 3 - Inferior Planets

+ Chapter 9 - Brahmagupta and Arithmetic

. Part 1 - Scope of Ganita

. Part 2 - Aryabhata, Bhaskara and Brahmagupta use Place Value Notations

. Part 3 - Operations and Determinations in Patiganita

. Part 4 - The concept of Multiplication (in Indian mathematics)

. Part 5 - The concept of Square (in Indian mathematics)

. Part 6 - The concept of Cube (in Indian mathematics)

. Part 7 - The concept of Square-Root (in Indian mathematics)

. Part 8 - The concept of Cube Root (in Indian mathematics)

. Part 9 - The concept of Fractions (in Indian mathematics)

. Part 10 - The Rule of Three (in Indian mathematics)

. Part 11 - The Rule of Compound Proportion (in Indian mathematics)

. Part 12 - Problem Containing Quadratic Equation

+ Chapter 10 - Brahmagupta as an Algebraist

. Part 1 - Algebra goes to Europe from India

. Part 2 - Technical Terms (in ancient Indian Algebra)

. Part 3 - Laws of Signs (in ancient Indian Algebra)

. Part 4 - Algebraic Operations (in ancient Indian Mathematics)

. Part 5 - The concept of Equation (in ancient Indian Algebra)

. Part 6 - Shodhana (Clearance of an Equation)

. Part 7 - Classification of Equations (in ancient Indian Algebra)

. Part 8 - Linear Equations with One Unknown and Their Solutions

. Part 9 - Samkramana (the rule of Concurrence)

. Part 10 - Linear Equations with Several Unknowns

. Part 11 - Quadratic Equations (in ancient Indian Algebra)

. Part 12 - Two Roots of a Quadratic Equation and Brahmagupta

. Part 13 - Simultaneous Quadratic Equations (in ancient Indian Algebra)

. Part 14 - Rule of Dissimilar Operations (in ancient Indian Algebra)

. Part 15 - Indeterminate Equations of the First Degree

. Part 16 - Preliminary Operations in Kuttaka-Karma

. Part 17 - Bhaskara I and Kuttaka Operation

. Part 18 - Brahmagupta's Rules Concerning Indeterminate Analysis of the First Degree

. Part 19 - One Linear Equation in More Than Two Unknowns

. Part 20 - General Problem of Remainders

. Part 21 - The concept of Square-Nature (Varga-prakriti or Kriti-prakriti)

. Part 22 - Lemmas of Brahmagupta

. Part 23 - Cakravala or Cyclic Method

. Part 24 - Solution of Indeterminate Quadratic Equation

. Part 25 - Rational Geometrical Figures

. Part 26 - Isosceles Triangles with a Given Altitude

. Part 27 - Rational Isosceles Trapeziums

. Part 28 - Rational Trapeziums With Three Equal Sides

. Part 29 - Rational Inscribed Quadrilaterals

. Part 30 - Double Equations of the First Degree

Chapter 11 - Arabic and Indian Divisions of the Zodiac

+ Chapter 12 - Brahmagupta's Astronomy: Its Highlights

. Part 1 - The Beginning or Starting Point (for the reckoning of Ahargana)

. Part 2 - The method of calculating Ahargana

. Part 3 - Mean Longitude of a Planet

. Part 4 - Calculating without using Ahargana

. Part 5 - Concordance of Working Rules

. Part 6 - Tables of Constants

. Part 7 - Brahmagupta: A great critic

. Part 8 - Brahmagupta and Srisena

+ Chapter 13 - Brahmagupta and Astronomical Instruments

. Part 1 - Introduction to Brahmagupta and Astronomical Instruments

. Part 2 - Setting up of the Gnomon (Shanku-yantra)

. Part 3 - Gnomon Used for Finding the Directions

. Part 4 - Golayantra or Armillary Sphere

. Part 5 - How to Observe Places of Stars

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Last Updated: 30 May, 2025

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