Indian mathematics
Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student.
[11][12] A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) by mathematicians of the Kerala school in the 15th century CE.
They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry.
[19][20] Hollow cylindrical objects made of shell and found at Lothal (2200 BCE) and Dholavira are demonstrated to have the ability to measure angles in a plane, as well as to determine the position of stars for navigation.
[2]The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta (RV 10.90.4): With three-fourths Puruṣa went up: one-fourth of him again was here.The Satapatha Brahmana (c. 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.
[22] The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. 700–400 BCE) list rules for the construction of sacrificial fire altars.
[23] Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement,"[24] that of constructing fire altars which have different shapes but occupy the same area.
"[25] Since the statement is a sūtra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.
"Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India.
The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time.
He wrote the Katyayana Sulba Sutra, which presented much geometry, including the general Pythagorean theorem and a computation of the square root of 2 correct to five decimal places.
"[42] Memorisation of "what is heard" (śruti in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India.
Modern scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia.
[45] Similar methods were used for memorising mathematical texts, whose transmission remained exclusively oral until the end of the Vedic period (c. 500 BCE).
The rest of the instruction must have been transmitted by the so-called Guru-shishya parampara, 'uninterrupted succession from teacher (guru) to the student (śisya),' and it was not open to the general public" and perhaps even kept secret.
[57] Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.
"[58] Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept.
"[58] A third decimal representation was employed in a verse composition technique, later labelled Bhuta-sankhya (literally, "object numbers") used by early Sanskrit authors of technical books.
[61] Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India.
It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE.
[62] According to Plofker,[60] These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion.
[65]The prose commentary accompanying the example solves the problem by converting it to three (under-determined) equations in four unknowns and assuming that the prices are all integers.
[74] Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers[73] and is considered the first systematic treatment of the subject.
Bhāskara II (1114–1185) was a mathematician-astronomer who wrote a number of important treatises, namely the Siddhanta Shiromani, Lilavati, Bijaganita, Gola Addhaya, Griha Ganitam and Karan Kautoohal.
His contributions include: Arithmetic: Algebra: Geometry: Calculus: Trigonometry: The Navya-Nyāya or Neo-Logical darśana (school) of Indian philosophy was founded in the 13th century by the philosopher Gangesha Upadhyaya of Mithila.
Gangeśa's book Tattvacintāmaṇi ("Thought-Jewel of Reality") was written partly in response to Śrīharśa's Khandanakhandakhādya, a defence of Advaita Vedānta, which had offered a set of thorough criticisms of Nyāya theories of thought and language.
[83] Navya-Nyāya developed a sophisticated language and conceptual scheme that allowed it to raise, analyze, and solve problems in logic and epistemology.
Jyesthadeva presented proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other Kerala School mathematicians.
Some scholars have recently suggested that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries.
[71] However, they did not (as Newton and Leibniz did) "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".
