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 Sciences mathématiques: sources anciennes

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MessageSujet: Sciences mathématiques: sources anciennes   Dim 27 Nov - 18:41

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BRAHMAGUPTA (ou Brahamagupta), indien, 598-660 (?)

Ce mathématicien et astronome indien s'intéressa à l'algèbre et aux équations diophantiennes à la suite des travaux d'Aryabhata. Brahmagupta est sans doute le premier, dans des calculs commerciaux, à user des nombres négatifs pour signifier les pertes et les profits et à les utiliser en algèbre en énonçant la règle des signes.

Il emploie dans ses calculs, les chiffres décimaux (graphisme très proche de nos chiffres actuels dits arabes, fruit d'une évolution sur plusieurs siècles commençant au 3ème siècle avant J.-C.) et principalement le zéro (notation o) que les Arabes adopteront au 9è siècle avec, principalement, les travaux de Al-Khwarizmi. Le célèbre "chiffre" manqua cruellement aux grandes civilisations babyloniennes, égyptiennes et grecques. Son apparition en Inde, tout particulièrement dans l'oeuvre de Brahmagupta, est un pas de géant en algèbre.

Citation :
ARYABHATA (Arjabahr), indien, 476-550

Astronome. Il affirme la rotation de la Terre, contrairement à la doctrine géocentrique de Ptolémée alors répandue et héritée d'Aristote, selon laquelle la Terre est immobile au centre de l'univers.

Aryabhata est le premier grand mathématicien indien. Il nous fut connu par un important traité, traduit en Europe au 19è siècle, appelé l'Aryabhatiya, écrit en sanscrit (la langue sacrée des brahmanes) en 499 et relatif à l'astronomie et aux mathématiques.

Aryabhata y décrit les algorithmes de d'extraction des racines carrée et cubique, résout de difficiles équations diophantiennes par l'usage de fractions continues et fait usage d'un système décimal positionnel dont le graphisme est proche du notre et où l'usage du zéro apparaît implicitement. Le grand artisan de l'introduction du célèbre symbole sera Brahmagupta.

Citation :
BHASKARA (Bhaskaracharya), indien, 1114-1185

On ne confondra pas ce mathématicien indien avec un autre Bhaskara ayant vécu à la charnière entre 6è et 7è siècle, astronome ayant établi des formules d'approximation pour le sinus d'un angle et dont Brahmagupta se serait inspiré. On parle parfois de Bhaskara I pour évoquer ce mathématicien homonyme, Bhaskara II désignant celui qui nous intéresse ici.

De père astronome, astronome lui-même, il dirigea l'observatoire d'Ujjayani (Ujjain, une des sept villes sacrées de l'Inde), là même où vécut Brahamagupta. Son oeuvre, essentiellement algébrique marque l'apogée des mathématiques indiennes et inspirera nombre de mathématiciens arabes et occidentaux.

Trois oeuvres principales de Bhaskara nous sont parvenues :

Le Siddhanta-çiromani (couronnement du système), traité d'astronomie
Le Lilavati, traité d'arithmétique (portant le nom de sa fille), d'extractions de racines, de problèmes du 1er degré, fausse position en particulier, où il reprend et complète des résultats de Brahamagupta.
Le Bija Ganita (calcul des inconnues) important traité d'algèbre traitant des polynômes, du second degré et des racines carrées. On y trouve aussi une "preuve" du théorème de Pythagore.
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MessageSujet: Re: Sciences mathématiques: sources anciennes   Dim 27 Nov - 19:07

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Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. He was the head of the astronomical observatory at Ujjain, and during his tenure there wrote two texts on mathematics and astronomy: the Brahmasphutasiddhanta in 628, and the Khandakhadyaka in 665.

The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right. It goes well beyond that, however, stating rules for arithmetic on negative numbers and zero which are quite close to the modern understanding. The major divergence is that Brahmagupta attempted to define division by zero, which is left undefined in modern mathematics. His definition is not terribly useful; for instance, he states that 0/0 = 0.

Brahmasphutasiddhanta has four and a half chapters devoted to pure math while the twelfth chapter, the Ganita, deals with arithmetic progressions and a bit of geometry. The eighteenth chapter of Brahmagupta's work is called the Kuttaka. This is usually associated with the Aryabhata's method for solving the indeterminate equation ax - by = c. But here Kuttaka means algebra. Brahmagupta was the inventor of the method of solving indeterminate equations of the second degree (equations of the form Nx^2 + 1 = y^2). He was also the first to use algebra to solve astronomical problems. It was through Brahmagupta's Brahmasphutasiddhanta that the Arabs came to know of Indian astronomy. The Famous King Khalif Abbasid Al Mansoor (712-775) founded Baghdad, which is situated on the banks of the Tigris, and made it a center of learning. The King invited a scholar of Ujjain by the name of Kanka in 770 A.D. Kanka used the Brahmasphutasiddhanta to explain the Hindu system of arithmetic astronomy. Al Fazaii translated Brahmugupta's work into Arabic upon the request of the King. Some of the important contributions made by Brahmagupta in astronomy are: methods for calculations of the motions and places of various planets, their rising and setting, conjunctions, and the calculation of eclipses of the sun and the moon. Brahmagupta criticized the Puranic view that the earth was flat or hollow like a bowl. Instead, he observed that the earth and heaven were round. However he wrongfully believed that the earth did not move.

Citation :
Aryabhata (आर्यभट) Āryabhada) is the first of the great astronomers of the classical age of India. He was born in 476 AD in Ashmaka but later lived in Kusumapura, which his commentator Bhāskara I (629 AD) identifies with Patilputra (modern Patna).

His book, the Āryabhatīya, presented astronomical and mathematical theories in which the Earth was taken to be spinning on its axis and the periods of the planets were given with respect to the sun (in other words, it was heliocentric).He believes that the Moon and planets shine by reflected sunlight and he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses of the Sun and the Moon. The Indian belief up to that time was that eclipses were caused by a demon called Rahu. His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours. This book is divided into four chapters: (i) the astronomical constants and the sine table (ii) mathematics required for computations (iii) division of time and rules for computing the longitudes of planets using eccentrics and epicycles (iv) the armillary sphere, rules relating to problems of trigonometry and the computation of eclipses. In this book, the day was reckoned from one sunrise to the next, whereas in his Āryabhata-siddhānta he took the day from one midnight to another. There was also difference in some astronomical parameters.

He discovered how the Lunar Eclipse and the Solar Eclipse happen for the first time.

Aryabhata also gave close approximation for Pi. In the Aryabhatiya, he wrote: "Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words, π ≈ 62832/20000 = 3.1416, correct to four rounded-off decimal places.

He was the first to note that Earth is round. He says "Bhumukha sarvato golah" (Earth is round).
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MessageSujet: Re: Sciences mathématiques: sources anciennes   Dim 27 Nov - 19:10

Citation :
Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician of the Kerala school who is considered the father of mathematical analysis for having taken the decisive step from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis.

Citation :
Bhāskara (1114-1185), also called Bhāskara II and BhāskarāAchārya ("Bhaskara the teacher") was an Indian mathematician. He was born near Bijjada Bida in Bijapur district, Karnataka and became head of the astronomical observatory at Ujjain, continuing the mathematical tradition of Varahamihira and Brahmagupta.

In many ways, Bhaskaracharya represents the peak of mathematical knowledge in the 12th century. He reached an understanding of the number systems and solving equations, which was not to be achieved anywhere else in the world for several centuries. His main works are the Lilavati (dealing with arithmetic), Bijaganita (algebra) and Siddhantasiromani which consists of two parts: Goladhyaya (sphere) and Grahaganita (mathematics of the planets).

Citation :
Yajnavalkya, 1800 BC, the author of the altar mathematics of the Shatapatha Brahmana.
Lagadha - Author of a 1350 BC text on Vedic astronomy
Baudhayana, 800 BC
Manava, 750 BC
Apastamba, 700 BC
Aksapada Gautama, 550 BC, Logician
Katyayana, 400 BC
Pingala, 5th century BC
Bharata Muni, 450 BC

P.C

Aryabhata - Astronomer who gave very accurate calculations for astronomical constants, 500
Varahamihira
Bhaskara I, 620
Brahmagupta - Helped bring the concept of zero into arithmetic,
Virahanka (8th century) - Gave explicit rules for the Fibonacci series.
Shridhara (between 650-850) - Gave a good rule for finding the volume of a sphere.
Lalla, 720-790
Govindsvamin (9th century)
Virasena
Mahavira (9th century)
Jayadeva (9th century)
Prithudaka, 9th century
Halayudha, 10th century
Aryabhata II, 920-1000
Vateshvara (10th century)
Brahmadeva, 1060-1130
Sripati, 1019-1066
Gopala - Studied Fibonacci numbers before Fibonacci
Hemachandra - Also studied Fibonacci numbers before Fibonacci
Bhaskara - Conceived of Differential Calculus
Gangesha Upadhyaya, 13th century, Logician, Mithila school
Pakshadhara, sone of Gangehsa, Logician, Mithila school
Shankara Mishra, Logician, Mithila school
Narayana Pandit
Madhava - Considered the father of mathematical analysis, Founded some concepts of Calculus
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MessageSujet: Re: Sciences mathématiques: sources anciennes   Mar 15 Juin - 21:26

Interessant.
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MessageSujet: Re: Sciences mathématiques: sources anciennes   Aujourd'hui à 22:20

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