Bharatiya Ganit Ka Sankshipta Itihas

Here is a comprehensive summary in English of the provided Jain text, "Bharatiya Ganit ka Sankshipta Itihas" by R S Pandey:

The book "A Concise History of Indian Mathematics" by R S Pandey emphasizes the paramount importance of mathematics in India from ancient times. Quoting from "Vedanga Jyotisha" (1000 BCE), it states that just as the crest on a peacock's head and the jewel on a serpent's head are the highest parts of their bodies, mathematics holds the supreme position among all the branches of Vedic knowledge and scriptures.

The history of Indian mathematics begins with the Rigveda.

Vedic Period (Up to 1000 BCE): The Vedas clearly mention numbers and a decimal system. A Rigvedic verse uses numbers like twelve, three, three hundred, and sixty, demonstrating an understanding of the decimal system. The invention of "zero" and the "decimal place-value system" are India's unprecedented contributions to mathematics. While the exact time and inventor of zero are unknown, its usage dates back to the Vedic period. This system is now globally prevalent. The "Narada Vishnu Purana" by Maharishi Vedvyas describes mathematical concepts within the context of astronomical calculations, listing large numbers like ten, hundred, thousand, ten thousand (Ayuta), lakh (Laksha), crore (Koti), ten crore (Arbuda), arab (Abja), ten arab (Kharba), and so on, up to Shankhu (Neel), Jaladhi (Dashneel), Antya (Padma), Madhya (Dashpadma), and Parardha (Shankha, equivalent to 10^17). It details arithmetic operations like addition, subtraction, multiplication, division, fractions, squares, square roots, cubes, cube roots, and rule of three. The decimal place-value system of writing numbers originated in India, traveled to Arabia, and then to the Western countries. Arabs refer to digits 1-9 as "Ilm Hindsa," while Westerners call the digits (0-9) Hindu-Arabic numerals.

Late Vedic Period (1000 BCE to 500 BCE): Shulva and Vedanga Jyotisha Period: This era saw the development of geometry, particularly through the "Shulva Sutras," which contain rules for constructing altars for sacrifices with precise measurements. "Shulva" refers to the rope used for measurement in these constructions. The Shulva Sutras are also known as "Raju Ganita" (rope mathematics), which evolved into geometry. Besides measuring distances for altars, they were used for land measurement, leading to the development of "kshetramiti" (area measurement), "jyaamiti" (geometry), and "bhumiti" (land measurement). The term "geometry" in Greek is "geometria," and it's believed the English term is a derivative. An ancient Jain text highlights the importance of geometry by stating, "Geometry is the lotus of mathematics; all else is insignificant." A major achievement of the Shulva period is the Pythagorean theorem: the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. This theorem was widely known in India centuries before Pythagoras. The Taittiriya Samhita (3000 BCE) already contains the fact 392 = 362 + 152. Baudhayana, a great mathematician of the Shulva period and author of the Shulva Sutras, stated this theorem, predating Pythagoras by about 450 years. Therefore, it would be more appropriate to call it the "Baudhayana Theorem." Baudhayana also provided methods for constructing squares equal to the sum or difference of two squares and a formula for calculating the value of surds up to five decimal places. Ancient Indian geometers typically provided only concise formulas without detailed proofs, reflecting the inherent nature of the Indian people. This period also saw the development of astronomy for timekeeping and calculating the positions and movements of stars. "Vedanga Jyotisha" (1000 BCE) indicates that astronomers of that time possessed knowledge of addition, multiplication, and division.

Surya Prajnapti Period: Jain literature provides extensive details about mathematics of this period. The "Suryaprajnapti" and "Chandraprajnapti" (500 BCE), prominent Jain religious texts, extensively discuss mathematical principles, demonstrating Jainism's tendency to make knowledge accessible to the common people. "Suryaprajnapti" clearly mentions the ellipse, referred to as "parimandala," which is the circumcircle of a rectangle. This indicates that Indians knew about ellipses about 150 years before Minimax (350 BCE). The Bhagavati Sutra (300 BCE) also uses the term "parimandala" for ellipse and describes two types: "pratarparimandala" and "ghanparimandala." Jain scholars made commendable contributions to the development of mathematics and astronomy. They meticulously analyzed mathematical concepts, including number notation, fractions, rule of three, compound ratios, algebraic equations and their applications, progressions, permutations, combinations, rules of exponents and logarithms, and set theory. They discussed various types of infinite sets and provided examples of finite, infinite, and singleton sets. For logarithms, they used terms like "ardhacheda" (half-division), "tricched" (three divisions), and "chaturched" (four divisions), corresponding to log₂, log₃, and log₄ respectively. Logarithms were invented and widely applied in India long before John Napier (1550-1617 CE).

Early Middle Period (500 BCE to 400 CE): While most works from this period are lost, surviving fragments of "Bakhshali Ganita," "Surya Siddhanta," and "Ganitanuyoga," along with the literature of medieval mathematicians like Aryabhata and Brahmagupta, indicate significant mathematical development. Key texts include "Sthananga Sutra," "Bhagavati Sutra," and "Anuyogadvara Sutra." Other notable Jain works are "Tattvarthadhigama Sutra Bhashya" by Jain philosopher Umasvati (135 BCE) and "Tiloyapannatti" by Acharya Yatishabh (around 176 CE). "Bakhshali Ganita" details basic arithmetic operations, numbers in the decimal system, operations with fractions, squares, cubes, rule of three, interest calculations, retail sales problems, and mixture problems, including methods for verifying answers. It proves the widespread use of current arithmetic methods in India by 300 BCE. "Sthananga Sutra" discusses five types of infinity and four types of measures. It also describes permutations and combinations, known as "bhang" and "vikalp." "Bhagavati Sutra" explains combinations (taking 1-1 or 2-2 items from 'n' types) as "ekaka," "dvika," etc., with values like 'n' and 'n(n-1)/2', which are still in use. "Surya Siddhanta" provides a detailed exposition of modern trigonometry, including sine, verse-sine, and cosine. The term "jya" (sine) transformed into "jaib" in Arabic and then "sinus" in Latin. The word "trigonometry" itself is of Indian origin. Indians used trigonometry for astronomical calculations.

Algebra saw revolutionary advancements during this period. The "Bakhshali" manuscript uses an unknown quantity in place of 1 or 100 in the "Rule of False Position," which is considered the origin of algebraic expansion. Indians developed rules for addition, subtraction, multiplication, and division of positive and negative numbers. Brahmagupta (628 CE) stated that the product of a positive and negative number is negative, the product of two negative numbers is positive, and the product of two positive numbers is positive. Indians used symbols for powers like squares and cubes, which are still in use. "Anuyogadvara Sutra" contains algebraic exponent rules. Algebra, like arithmetic, was transmitted from India to Arabia. AL-KHOWARIZMI's books "Aljabr" and "Almuqabala" are based on Indian algebra, and the subject's name is derived from his works. Greek mathematics, despite its golden age, lacked algebra in the modern sense. While Greeks could solve complex problems, their solutions were geometric. Algebraic solutions first appeared in the works of Diophantus (around 275 CE). Indian mathematicians were far ahead of other nations in algebra during this time.

Middle Ages or Golden Age (400 CE to 1200 CE): This period is considered the golden age of Indian mathematics, producing brilliant mathematicians like Aryabhata, Brahmagupta, Mahaviracharya, and Bhaskaracharya, who advanced all branches of mathematics.

• Aryabhata (499 CE): From Patna, Aryabhata's "Aryabhatiya Ganitapada" summarizes fundamental mathematical principles in 332 verses. The first two parts deal with mathematics, and the last two with astronomy. The "Dasham Geetika" section introduces a method for representing large numbers using alphabets. The "Ganita Pada" covers arithmetic, geometry, algebra, and trigonometry. He provided detailed solutions for linear and quadratic equations and indeterminate equations ("Kutaka"). Aryabhata was the first to use the verse sine, later known as "verse sine" in the West. He calculated Pi to four decimal places as 3.1416, which remains accurate today. He stated the circumference of a circle with a diameter of 20,000 units is 62,832 units (Pi = 62832/20000 = 3.1416). He also described methods for finding square and cube roots and the rule of three. Aryabhata was the first to clearly propose that the Earth is mobile and the Sun is stationary, a concept acknowledged by Copernicus 1100 years later in the 16th century.
• Varahamihira (550 CE): A great mathematician and philosopher, his works include "Panch Siddhantika," "Brihat Samhita," and "Brihajataka." His "Varaha Shulvasutra" sheds light on all aspects of mathematics, including many algebraic formulas related to altar construction.
• Bhaskara I (600 CE): In his works "Maha Bhaskariya," "Aryabhatiya Bhashya," and "Lagu Bhaskariya," he further developed and expanded upon Aryabhata's principles, making significant contributions to indeterminate equations ("Kutaka").
• Brahmagupta (628 CE): In 25 chapters of "Brahmasphutasiddhanta," he elaborated on mathematical principles and methods, detailing 20 operations and 8 procedures. He outlined rules for solving equations and indeterminate quadratic equations, which were later rediscovered by Euler and Lagrange. Brahmagupta found formulas for the volumes of prisms and cones and the sum of geometric progressions. He also provided a detailed description of the Shulva Sutra for right-angled triangles. Notably, Brahmagupta conceptualized infinity, stating that any positive or negative number divided by zero results in infinity.
• Mahaviracharya (850 CE): A Jain scholar and courtier of King Amoghavarsha of the Rashtrakuta dynasty, he authored the extensive arithmetic treatise "Ganita Sara Sangraha." He discovered the modern rule for the least common multiple (LCM), first used in Europe in 1500 CE. He also derived formulas for the areas of cyclic quadrilaterals and ellipses.
• Sridharacharya (850 CE): He wrote on arithmetic and algebra, including "Navashatika," "Trishatika," and "Pati Ganita." His quadratic equation solution method, "Sridharacharya Vidhi," is still widely used. His "Pati Ganita" was translated into Arabic as "Hisab-ul-Tarabut."
• Aryabhata II (950 CE): From Maharashtra, he wrote "Mahasiddhanta," with chapters on arithmetic and first-degree indeterminate equations ("Kutaka"). This work likely contains the accurate value of Pi (22/7).
• Shripati Mishra (1039 CE): Also from Maharashtra, he authored "Siddhanta Shekhar" and "Ganita Tilak." He made significant contributions to permutations and combinations.
• Nemichandra Siddhanta Chakravarti (11th Century): His famous work is "Gommatasara," divided into "Karma Kanda" and "Jiva Kanda," discussing souls, karma, and set theory. The method of one-to-one correspondence was used centuries later by Galileo and Georg Cantor.
• Bhaskara II (1114 CE): The last and most outstanding mathematician of the medieval period, his works "Siddhanta Shiromani," "Lilavati," "Bijaganita," "Goladhyaya," "Griha Ganitam," and "Karana Kutuhala" represent the culmination of Indian mathematics. The principles stated in the Vedas in sutra form found their complete expression in Bhaskara II's writings. He elaborated on Brahmagupta's 20 operations and 8 procedures. "Lilavati" provides a foundational and creative exposition of the number system, forming the backbone of modern arithmetic and algebra. The poetic and simple problems in "Lilavati" are examples of the confluence of art and science. Bhaskara II's "Chakravart" method for solving indeterminate equations in algebra was highly praised by scholar Hankel as the "foremost achievement in number theory before Lagrange." Fermat used this rule in 1657. In "Siddhanta Shiromani," Bhaskara II discussed trigonometry extensively, including relationships between sine, cosine, and verse-sine, and introduced calculus (differential and integral). He also understood Earth's gravitational force, stating that the Earth possesses an attractive power that pulls objects towards its surface.

Late Middle Period (1200 CE to 1800 CE): Following Bhaskara II, there was less original work in mathematics; instead, commentaries on ancient texts were prevalent. Kerala mathematicians like Nilakantha (1500 CE) derived the sine series expansion: sin x = x - x³/3! + x⁵/5! ..., a formula known today as the Gregory series.

• Narayana Pandit (1356 CE): Authored the extensive arithmetic treatise "Ganita Kaumudi," notable for its discussion of Magic Squares.
• Nilakantha (1587): Wrote "Tajika Nilakanthi" on astronomical mathematics.
• Kamalakar (1608 CE): Wrote "Siddhanta Tatva Viveka."
• Samrat Jagannath (1731 CE): Authored "Samrat Siddhanta" and "Rekha Ganita" (Geometry), the latter forming the basis for much of modern geometrical terminology. Notable Kerala mathematicians of this era include Madhava (1350-1410), author of "Yukti Bhasha," Jyesthadeva (1500-1610), and Shankara Parashara (1500-1560), commentator of "Lilavati."

Sudhakar Dwivedi: Wrote several books on subjects like ellipses, spherical geometry, equation analysis, and calculus. He also made ancient texts by Brahmagupta and Bhaskara accessible by writing commentaries.

Ramanujan (1889 CE): A modern mathematical genius in the Vedic tradition of presenting and proving mathematical principles in sutras. His brilliance is evident from the fact that proving even one or two of his 50 theorems took mathematicians years of dedicated effort, and some remain unproven. His work was published as "Ramanujan's Diary" in his centenary year.

Swami Bharati Krishna Tirtha (1884-1960): A great mathematician and philosopher, Jagadguru Shankaracharya, he is a principal commentator on Vedic mathematics in modern times. In his book "Vedic Mathematics," he re-presented Vedic sutras and explained their principles and methods in such simple and clear language that even an ordinary student can master them and solve complex problems quickly. His book is an authoritative text on Vedic mathematics, awakening scholars and students alike to its hidden potential.

Current Era (After 1800 CE): Brief introductions to some prominent mathematicians and their works from the current era are provided.

• Nrising Bapu Dev Shastri (1831 CE): Wrote books on Indian and Western mathematics, including geometry, trigonometry, and astronomy.

The book highlights the profound and continuous development of mathematics in India, emphasizing its originality and its contribution to global mathematical knowledge.