Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, Varāhamihira, and Madhava. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry
was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of 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. This was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solution. In the prose section, the form was not considered so important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is the birch bark Bakhshali Manuscript, discovered in 1881 in the village of Bakhshali, near Peshawar and is likely from the 7th century CE.
A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions by mathematicians of the Kerala school in the 15th century CE. Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series. However, they did not formulate a systematic theory of differentiation and integration, nor is there any evidence of their results being transmitted outside Kerala.
Prehistory
Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley civilisation have uncovered evidence of the use of "practical mathematics". The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams. They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry.The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length.
Hollow cylindrical objects made of shell and found at Lothal 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.
Vedic period
Samhitas and Brahmanas
The texts of the Vedic Period provide evidence for the use of large numbers. By the time of the Yajurveda|, numbers as high as were being included in the texts. For example, the mantra at the end of the annahoma performed during the aśvamedha, and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:Fractions are mentioned, as in the Purusha Sukta :
The Satapatha Brahmana contains rules for ritual geometric constructions that are similar to the Sulba Sutras.
Śulba Sūtras
The Śulba Sūtras list rules for the construction of sacrificial fire altars. Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement", that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.According to Hayashi, the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."
The diagonal rope of an oblong produces both which the flank and the horizontalSince 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.produce separately."
They contain lists of Pythagorean triples, which are particular cases of Diophantine equations. They also contain statements about squaring the circle and "circling the square."
Baudhayana composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as:,,,, and, as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square." It also contains the general statement of the Pythagorean theorem : "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together." Baudhayana gives an expression for the square root of two:
The expression is accurate up to five decimal places, the true value being 1.41421356... This expression is similar in structure to the expression found on a Mesopotamian tablet from the Old Babylonian period :
which expresses in the sexagesimal system, and which is also accurate up to 5 decimal places.
According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written c. 1850 BCE "contains fifteen Pythagorean triples with quite large entries, including which is a primitive triple, indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "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." Dani goes on to say:
In all, three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by Manava and the Apastamba Sulba Sutra, composed by Apastamba, contained results similar to the Baudhayana Sulba Sutra.
;Vyakarana
The Vedic period saw the work of Sanskrit grammarian . His grammar includes a precursor of the Backus–Naur form.
Pingala (300 BCE – 200 BCE)
Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is Pingala , a music theorist who authored the Chhandas Shastra, a Sanskrit treatise on prosody. Pingala's work also contains the basic ideas of Fibonacci numbers. Although the Chandah sutra has not survived in its entirety, a 10th-century commentary on it by Halāyudha has. Halāyudha, who refers to the Pascal triangle as Meru-prastāra, has this to say:The text also indicates that Pingala was aware of the combinatorial identity:
;Kātyāyana
Kātyāyana is notable for being the last of the Vedic mathematicians. 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.
Jain mathematics (400 BCE – 200 CE)
Although Jainism as a religion and philosophy predates its most famous exponent, Mahavira, most Jain texts on mathematical topics were composed after the 6th century BCE. Jain mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "classical period."A significant historical contribution of Jain mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints. In particular, their fascination with the enumeration of very large numbers and infinities led them to classify numbers into three classes: enumerable, innumerable and infinite. Not content with a simple notion of infinity, their texts define five different types of infinity: the infinite in one direction, the infinite in two directions, the infinite in area, the infinite everywhere, and the infinite perpetually. In addition, Jain mathematicians devised notations for simple powers of numbers like squares and cubes, which enabled them to define simple algebraic equations. Jain mathematicians were apparently also the first to use the word shunya to refer to zero. This word is the ultimate etymological origin of the English word "zero", as it was calqued into Arabic as ṣifr and then subsequently borrowed into Medieval Latin as zephirum, finally arriving at English after passing through one or more Romance languages.
In addition to Surya Prajnapti, important Jain works on mathematics included the Sthānāṅga Sūtra ; the Anuyogadwara Sutra, which includes the earliest known description of factorials in Indian mathematics; and the Ṣaṭkhaṅḍāgama. Important Jain mathematicians included Bhadrabahu, the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti; Yativrisham Acharya, who authored a mathematical text called Tiloyapannati; and Umasvati, who, although better known for his influential writings on Jain philosophy and metaphysics, composed a mathematical work called the Tattvārtha Sūtra.