section_path: History/Origins/Āryabhaṭa, Brahmagupta, Bhāskara
section_name: Āryabhaṭa, Brahmagupta, Bhāskara
paragraph_text: Āryabhaṭa, Brahmagupta, Bhāskara
  
  While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry, it seems to be the case that Indian mathematics is otherwise an indigenous tradition; in particular, there is no evidence that Euclid's Elements reached India before the 18th century.
  
  Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences n ≡ a 1 mod m 1 {\displaystyle n\equiv a_{1}{\bmod {m}}_{1}}, n ≡ a 2 mod m 2 {\displaystyle n\equiv a_{2}{\bmod {m}}_{2}} could be solved by a method he called kuṭṭaka, or pulveriser; this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations.
  
  Brahmagupta (628 AD) started the systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala, or "cyclic method") for solving Pell's equation was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in Bhāskara II 's Bīja-gaṇita (twelfth century).
  
  Indian mathematics remained largely unknown in Europe until the late eighteenth century; Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke.