Pingala
Aryabhata: 5th century
Aryabhata is the author of several treatises on mathematics and astronomy.
His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.
Place value system and zero
The place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah explains that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients[13]
Approximation of π
Aryabhata worked on the approximation for pi (), and may have come to the conclusion that is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:
caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached." [15]
This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.
It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert.[16]
After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.[12
Varāhamihira: 6th century
Bhaskara I: 7th century
Bhaskara wrote three astronomical contributions. In 629 he commented the Aryabhatiya, written in verses, about mathematical astronomy. The comments referred exactly to the 33 verses dealing with mathematics. There he considered variable equations and trigonometric formulae.
His work Mahabhaskariya divides into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for sin x, that is
which he assigns to Aryabhata. It reveals a relative error of less than 1.9% (the greatest deviation at ). Moreover, relations between sine and cosine, as well as between the sine of an angle >90° >180° or >270° to the sine of an angle <90° are given. Parts of Mahabhaskariya were later translated into Arabic.
Bhaskara already dealt with the assertion that if p is a prime number, then 1 + (p–1)! is divisible by p.[dubious – discuss] It was proved later by Al-Haitham, also mentioned by Fibonacci, and is now known as Wilson's theorem.
Moreover, Bhaskara stated theorems about the solutions of today so called Pell equations. For instance, he posed the problem: "Tell me, O mathematician, what is that square which multiplied by 8 becomes - together with unity - a square?" In modern notation, he asked for the solutions of the Pell equation . It has the simple solution x = 1, y = 3, or shortly (x,y) = (1,3), from which further solutions can be constructed, e.g., (x,y) = (6,17).
Brahmagupta: 7th century
Brahmagupta was the first to use zero as a number. He gave rules to compute with zero. Brahmagupta used negative numbers and zero for computing. The modern rule that two negative numbers multiplied together equals a positive number first appears in Brahmasputa siddhanta. It is composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta's mathematics was derived.[1]
Mahavira: 9th century
Mahavira was a 9th-century south Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. He gave the sum of a series whose terms are squares of an arithmetical progression and empirical rules for area and perimeter of an ellipse. He was patronised by the great Rashtrakuta king Amoghavarsha.[1]
Mahavira was the author of Ganit Saar Sangraha. He separated Astrology from Mathematics. He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. He is highly respected among Indian Mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle. Mahavira's eminence spread in all South India and his books proved inspirational to other Mathematicians in Southern India.[2] It was translated into Telugu language by Pavuluri Mallana as Saar Sangraha Ganitam.
Bhaskara II: 12th century
Bhāskara (1114–1185), was a south Indian mathematician and astronomer. He was born near Vijjadavida (Bijāpur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of ancient India. He lived in the Sahyadri region (Patnadevi, Jalgaon, Maharashtra).[1]
Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India.[2] His main work Siddhānta Shiromani, (Sanskrit for "Crown of treatises,"[3]) is divided into four parts called Lilāvati, Bijaganita, Grahaganita and Golādhyāya.[4] These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karna Kautoohala.
Bhāskara's work on calculus predates Newton and Leibniz by over half a millennium.[5][6] He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.[7]
Acharya Hemachandra: 12th century
Mahendra Sūri: 14th century
Madhava of Sangamagrama: 14th century
Madhava of Sangamagrama (1340 – c. 1425), was a south Indian mathematician-astronomer from the town of Sangamagrama (present day Irinjalakuda) near Cochin, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. He was the first in the world to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".[1] His discoveries opened the doors to what has today come to be known as Mathematical Analysis.[4] One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra.
Some scholars have also suggested that Madhava's work, through the writings of the Kerala school, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it may have had an influence on later European developments in analysis and calculus.[5]
Parameshvara: 15th century
He was a major south Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer. Parameshvara was a proponent of observational astronomy in medieval India and he himself had made a series of eclipse observations to verify the accuracy of the computational methods then in use. Based on his eclipse observations, Parameshvara proposed several corrections to the astronomical parameters which had been in use since the times of Aryabhata. The computational scheme based on the revised set of parameters has come to be known as the Drgganita system. Parameshvara was also a prolific writer on matters relating to astronomy. At least 25 manuscripts have been identified as being authored by Parameshvara.[1]
Parameshvara wrote commentaries on many mathematical and astronomical works such as those by Bhaskara I and Aryabhata. He made a series of eclipse observations over a 55 year period, and constantly attempted to compare these with the theoretically computed positions of the planets. He revised planetary parameters based on his observations.
Parameshvara's most significant contribution is his mean value type formula for the inverse interpolation of the sine. He was the first mathematician to give the radius of circle with an inscribed quadrilateral, an expression that is normally attributed to Lhuilier (1782), 350 years later. With the sides of the cyclic quadrilateral being a, b, c, and d, the radius R of the circumscribed circle is:
Nilakantha Somayaji: 15th century
Kelallur Nilakantha Somayaji (Malayalam: നീലകണ്ഠ സോമയാജി, Nīlakaṇṭa Sōmayāji ?, Sanskrit: नीलकण्ठ सोमयाजि
(1444–1544) (also referred to as Kelallur Comatiri[1]) was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501. He had also composed an elaborate commentary on Aryabhatiya called the Aryabhatiya Bhasya. In this Bhasya, Nilakantha had discussed infinite series expansions of trigonometric functions and problems of algebra and spherical geometry. Grahapareeksakrama is a manual on making observations in astronomy based on instruments of the time.
Aryabhata: 5th century
Aryabhata is the author of several treatises on mathematics and astronomy.
His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.
Place value system and zero
The place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah explains that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients[13]
Approximation of π
Aryabhata worked on the approximation for pi (), and may have come to the conclusion that is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:
caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached." [15]
This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.
It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert.[16]
After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.[12
Varāhamihira: 6th century
Bhaskara I: 7th century
Bhaskara wrote three astronomical contributions. In 629 he commented the Aryabhatiya, written in verses, about mathematical astronomy. The comments referred exactly to the 33 verses dealing with mathematics. There he considered variable equations and trigonometric formulae.
His work Mahabhaskariya divides into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for sin x, that is
which he assigns to Aryabhata. It reveals a relative error of less than 1.9% (the greatest deviation at ). Moreover, relations between sine and cosine, as well as between the sine of an angle >90° >180° or >270° to the sine of an angle <90° are given. Parts of Mahabhaskariya were later translated into Arabic.
Bhaskara already dealt with the assertion that if p is a prime number, then 1 + (p–1)! is divisible by p.[dubious – discuss] It was proved later by Al-Haitham, also mentioned by Fibonacci, and is now known as Wilson's theorem.
Moreover, Bhaskara stated theorems about the solutions of today so called Pell equations. For instance, he posed the problem: "Tell me, O mathematician, what is that square which multiplied by 8 becomes - together with unity - a square?" In modern notation, he asked for the solutions of the Pell equation . It has the simple solution x = 1, y = 3, or shortly (x,y) = (1,3), from which further solutions can be constructed, e.g., (x,y) = (6,17).
Brahmagupta: 7th century
Brahmagupta was the first to use zero as a number. He gave rules to compute with zero. Brahmagupta used negative numbers and zero for computing. The modern rule that two negative numbers multiplied together equals a positive number first appears in Brahmasputa siddhanta. It is composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta's mathematics was derived.[1]
Mahavira: 9th century
Mahavira was a 9th-century south Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. He gave the sum of a series whose terms are squares of an arithmetical progression and empirical rules for area and perimeter of an ellipse. He was patronised by the great Rashtrakuta king Amoghavarsha.[1]
Mahavira was the author of Ganit Saar Sangraha. He separated Astrology from Mathematics. He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. He is highly respected among Indian Mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle. Mahavira's eminence spread in all South India and his books proved inspirational to other Mathematicians in Southern India.[2] It was translated into Telugu language by Pavuluri Mallana as Saar Sangraha Ganitam.
Bhaskara II: 12th century
Bhāskara (1114–1185), was a south Indian mathematician and astronomer. He was born near Vijjadavida (Bijāpur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of ancient India. He lived in the Sahyadri region (Patnadevi, Jalgaon, Maharashtra).[1]
Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India.[2] His main work Siddhānta Shiromani, (Sanskrit for "Crown of treatises,"[3]) is divided into four parts called Lilāvati, Bijaganita, Grahaganita and Golādhyāya.[4] These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karna Kautoohala.
Bhāskara's work on calculus predates Newton and Leibniz by over half a millennium.[5][6] He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.[7]
Acharya Hemachandra: 12th century
Mahendra Sūri: 14th century
Madhava of Sangamagrama: 14th century
Madhava of Sangamagrama (1340 – c. 1425), was a south Indian mathematician-astronomer from the town of Sangamagrama (present day Irinjalakuda) near Cochin, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. He was the first in the world to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".[1] His discoveries opened the doors to what has today come to be known as Mathematical Analysis.[4] One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra.
Some scholars have also suggested that Madhava's work, through the writings of the Kerala school, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it may have had an influence on later European developments in analysis and calculus.[5]
Parameshvara: 15th century
He was a major south Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer. Parameshvara was a proponent of observational astronomy in medieval India and he himself had made a series of eclipse observations to verify the accuracy of the computational methods then in use. Based on his eclipse observations, Parameshvara proposed several corrections to the astronomical parameters which had been in use since the times of Aryabhata. The computational scheme based on the revised set of parameters has come to be known as the Drgganita system. Parameshvara was also a prolific writer on matters relating to astronomy. At least 25 manuscripts have been identified as being authored by Parameshvara.[1]
Parameshvara wrote commentaries on many mathematical and astronomical works such as those by Bhaskara I and Aryabhata. He made a series of eclipse observations over a 55 year period, and constantly attempted to compare these with the theoretically computed positions of the planets. He revised planetary parameters based on his observations.
Parameshvara's most significant contribution is his mean value type formula for the inverse interpolation of the sine. He was the first mathematician to give the radius of circle with an inscribed quadrilateral, an expression that is normally attributed to Lhuilier (1782), 350 years later. With the sides of the cyclic quadrilateral being a, b, c, and d, the radius R of the circumscribed circle is:
Nilakantha Somayaji: 15th century
Kelallur Nilakantha Somayaji (Malayalam: നീലകണ്ഠ സോമയാജി, Nīlakaṇṭa Sōmayāji ?, Sanskrit: नीलकण्ठ सोमयाजि
(1444–1544) (also referred to as Kelallur Comatiri[1]) was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501. He had also composed an elaborate commentary on Aryabhatiya called the Aryabhatiya Bhasya. In this Bhasya, Nilakantha had discussed infinite series expansions of trigonometric functions and problems of algebra and spherical geometry. Grahapareeksakrama is a manual on making observations in astronomy based on instruments of the time.
