Indian mathematics

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 (and approximately equal to the English ounce or Greek uncia). 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 (approximately 1.32 inches or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integer multiples of this unit of length. 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. ==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| (1200–900 BCE), numbers as high as were being included in the texts. Śulba Sūtras The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. 700–400 BCE) 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. 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. which are particular cases of Diophantine equations. They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square." Baudhayana (c. 8th century BCE) 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 (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together." ::\sqrt{2} \approx 1 + \frac{1}{3} + \frac{1}{3\cdot4} - \frac{1}{3\cdot 4\cdot 34} = 1.4142156 \ldots 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 (1900–1600 BCE): "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) 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 (fl. 750–650 BCE) and the Apastamba Sulba Sutra, composed by Apastamba (c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra. ;Vyakarana The Vedic period saw the work of Sanskrit grammarian (c. 520–460 BCE). His grammar includes a precursor of the Backus–Naur form (used in the description programming languages). ==Pingala (300 BCE – 200 BCE)==

Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is Pingala (') (fl. 300–200 BCE), a music theorist who authored the Chhandas Shastra (', also Chhandas Sutra ''''), a Sanskrit treatise on prosody. Pingala's work also contains the basic ideas of Fibonacci numbers (called maatraameru). 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 (literally "the staircase to Mount Meru"), has this to say: The text also indicates that Pingala was aware of the combinatorial identity: :: {n \choose 0} + {n \choose 1} + {n \choose 2} + \cdots + {n \choose n-1} + {n \choose n} = 2^n ;Kātyāyana Kātyāyana (c. 3rd century BCE) 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 (6th century BCE), 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 (and exponents) 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 (literally void in Sanskrit) to refer to zero. This word is the ultimate etymological origin of the English word "zero", as it was calqued into Arabic as and then subsequently borrowed into Medieval Latin as , finally arriving at English after passing through one or more Romance languages (cf. French , Italian ). and the Ṣaṭkhaṅḍāgama (c. 2nd century CE). Important Jain mathematicians included Bhadrabahu (d. 298 BCE), the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti; Yativrisham Acharya (c. 176 BCE), who authored a mathematical text called Tiloyapannati; and Umasvati (c. 150 BCE), who, although better known for his influential writings on Jain philosophy and metaphysics, composed a mathematical work called the Tattvārtha Sūtra. ==Oral tradition==

Mathematicians of ancient and early medieval India were almost all Sanskrit pandits (' "learned man"), who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar ('), exegesis ('') and logic (nyāya'')." Styles of memorisation Prodigious energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity. For example, memorisation of the sacred Vedas included up to eleven forms of recitation of the same text. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the '''' (literally "mesh recitation") in which every two adjacent words in the text were first recited in their original order, then repeated in the reverse order, and finally repeated in the original order. The recitation thus proceeded as: word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; ... In another form of recitation, ' The need to conserve the sound of sacred text by use of ' (phonetics) and chhandas (metrics); to conserve its meaning by use of '' (grammar) and nirukta (etymology); and to correctly perform the rites at the correct time by the use of kalpa'' (ritual) and ' (astrology), gave rise to the six disciplines of the '. The brevity achieved in a sūtra is demonstrated in the following example from the Baudhāyana Śulba Sūtra (700 BCE). The domestic fire-altar in the Vedic period was required by ritual to have a square base and be constituted of five layers of bricks with 21 bricks in each layer. One method of constructing the altar was to divide one side of the square into three equal parts using a cord or rope, to next divide the transverse (or perpendicular) side into seven equal parts, and thereby sub-divide the square into 21 congruent rectangles. The bricks were then designed to be of the shape of the constituent rectangle and the layer was created. To form the next layer, the same formula was used, but the bricks were arranged transversely. The process was then repeated three more times (with alternating directions) in order to complete the construction. In the Baudhāyana Śulba Sūtra, this procedure is described in the following words: According to Filliozat, the officiant constructing the altar has only a few tools and materials at his disposal: a cord (Sanskrit, rajju, f.), two pegs (Sanskrit, śanku, m.), and clay to make the bricks (Sanskrit, '', f.). Concision is achieved in the sūtra, by not explicitly mentioning what the adjective "transverse" qualifies; however, from the feminine form of the (Sanskrit) adjective used, it is easily inferred to qualify "cord." Similarly, in the second stanza, "bricks" are not explicitly mentioned, but inferred again by the feminine plural form of "North-pointing." Finally, the first stanza, never explicitly says that the first layer of bricks are oriented in the east–west direction, but that too is implied by the explicit mention of "North-pointing" in the second'' stanza; for, if the orientation was meant to be the same in the two layers, it would either not be mentioned at all or be only mentioned in the first stanza. All these inferences are made by the officiant as he recalls the formula from his memory. ==The written tradition: prose commentary==

With the increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation. The earliest mathematical prose commentary was that on the work, Aryabhatiya| (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the '' was composed of 33 sūtras (in verse form) consisting of mathematical statements or rules, but without any proofs. However, according to Hayashi, "this does not necessarily mean that their authors did not prove them. It was probably a matter of style of exposition." From the time of Bhaskara I (600 CE onwards), prose commentaries increasingly began to include some derivations (upapatti''). Bhaskara I's commentary on the '''', had the following structure: • Rule ('sūtra') in verse by Aryabhata| • Commentary by Bhāskara I, consisting of: • Elucidation of rule (derivations were still rare then, but became more common later) • Example (uddeśaka) usually in verse. • Setting (nyāsa/sthāpanā) of the numerical data. • Working (karana) of the solution. • Verification ('''', literally "to make conviction") of the answer. These became rare by the 13th century, derivations or proofs being favoured by then. Typically, for any mathematical topic, students in ancient India first memorised the sūtras, which, as explained earlier, were "deliberately inadequate" in explanatory details (in order to pithily convey the bare-bone mathematical rules). The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards (i.e. boards covered with dust). The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, Brahmagupta (fl. 7th century CE), to characterise astronomical computations as "dust work" (Sanskrit: dhulikarman). ==Numerals and the decimal number system==

It is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe. The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers. The earliest extant script used in India was the Kharoṣṭhī| script used in the Gandhara culture of the north-west. It is thought to be of Aramaic origin and it was in use from the 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the Brāhmī script, appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially not based on a place-value system. The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE. A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate. Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE. Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier. the number 4, for example, could be represented by the word "Veda" (since there were four of these religious texts), the number 32 by the word "teeth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon). 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. According to Plofker, 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." ==Bakhshali Manuscript==

The oldest extant mathematical manuscript in India is the Bakhshali Manuscript, a birch bark manuscript written in "Buddhist hybrid Sanskrit" The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near Peshawar (then in British India and now in Pakistan). Of unknown authorship and now preserved in the Bodleian Library in the University of Oxford, the manuscript has been dated recently as 224–383 CE. The surviving manuscript has seventy leaves, some of which are in fragments. Its mathematical content consists of rules and examples, written in verse, together with prose commentaries, which include solutions to the examples. ==Classical period (400–1300)==

This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji give broader and clearer shape to many branches of mathematics. Their contributions would spread to Asia, the Middle East, and eventually to Europe. Unlike Vedic mathematics, their works included both astronomical and mathematical contributions. In fact, mathematics of that period was included in the 'astral science' (jyotiḥśāstra) and consisted of three sub-disciplines: mathematical sciences (gaṇita or tantra), horoscope astrology (horā or jātaka) and divination (saṃhitā). This tripartite division is seen in Varāhamihira's 6th century compilation—Pancasiddhantika (literally panca, "five", siddhānta, "conclusion of deliberation", dated 575 CE)—of five earlier works, Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhanta, which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy. As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries. This ancient text uses the following as trigonometric functions for the first time: • Sine (Jya). • Cosine (Kojya). • Inverse sine (Otkram jya). Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East. ;Chhedi calendar This Chhedi calendar (594) contains an early use of the modern place-value Hindu–Arabic numeral system now used universally. ;Aryabhata I Aryabhata (476–550) wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained: • Quadratic equations • Trigonometry • The value of π, correct to 4 decimal places. Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include: Trigonometry: (See also : Aryabhata's sine table) • Introduced the trigonometric functions. • Defined the sine (jya) as the modern relationship between half an angle and half a chord. • Defined the cosine (kojya). • Defined the versine (utkrama-jya). • Defined the inverse sine (otkram jya). • Gave methods of calculating their approximate numerical values. • Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy. • Contains the trigonometric formula sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − (1/225)sin nx. • Spherical trigonometry. Arithmetic: • Simple continued fractions. Algebra: • Solutions of simultaneous quadratic equations. • Whole number solutions of linear equations by a method equivalent to the modern method. • General solution of the indeterminate linear equation . Mathematical astronomy: • Accurate calculations for astronomical constants, such as the: • Solar eclipse. • Lunar eclipse. • The formula for the sum of the cubes, which was an important step in the development of integral calculus. ;Varahamihira Varahamihira (505–587) produced the Pancha Siddhanta (The Five Astronomical Canons). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosine functions: • \sin^2(x) + \cos^2(x) = 1 • \sin(x)=\cos\left(\frac{\pi}{2}-x\right) • \frac{1-\cos(2x)}{2}=\sin^2(x) Seventh and eighth centuries states that AF = FD. In the 7th century, two separate fields, arithmetic (which included measurement) and algebra, began to emerge in Indian mathematics. The two fields would later be called ' (literally "mathematics of algorithms") and ' (lit. "mathematics of seeds", with "seeds"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations). Brahmagupta, in his astronomical work Brahmasphutasiddhanta| (628 CE), included two chapters (12 and 18) devoted to these fields. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral: Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers :\ x^2-Ny^2=1, where N is a nonsquare integer. He did this by discovering the following identity: Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was: but differs on other numbers, more closely resembling the 2-adic order. Virasena also gave: • The derivation of the volume of a frustum by a sort of infinite procedure. It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE. : \ \sin w' - \sin w could be approximately expressed as : \ (w' - w)\cos w This was elaborated by his later successor Bhaskara ii thereby finding the derivative of sine. • Computed π, correct to five decimal places. • Calculated the solar year to 9 decimal places. Trigonometry: • Developments of spherical trigonometry • The trigonometric formulas: • \ \sin(a+b)=\sin(a) \cos(b) + \sin(b) \cos(a) • \ \sin(a-b)=\sin(a) \cos(b) - \sin(b) \cos(a) ==Medieval and early modern mathematics (1300–1800)==

Navya-Nyaya The Navya-Nyāya or Neo-Logical darśana (school) of Indian philosophy was founded in the 13th century by the philosopher Gangesha Upadhyaya of Mithila. It was a development of the classical Nyāya darśana. Other influences on Navya-Nyāya were the work of earlier philosophers Vācaspati Miśra (900–980 CE) and Udayana (late 10th century). 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. Navya-Nyāya developed a sophisticated language and conceptual scheme that allowed it to raise, analyze, and solve problems in logic and epistemology. It involves naming each object to be analyzed, identifying a distinguishing characteristic for the named object, and verifying the appropriateness of the defining characteristic using pramanas. Kerala School c.1530 The Kerala school of astronomy and mathematics was founded by Madhava of Sangamagrama in Kerala, South India and included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school astronomers independently created a number of important mathematics concepts. The most important results, series expansion for trigonometric functions, were given in Sanskrit verse in a book by Neelakanta called Tantrasangraha and a commentary on this work called Tantrasangraha-vakhya of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhāṣā (c.1500–c.1610), written in Malayalam, by Jyesthadeva. Their discovery of these three important series expansions of calculus—several centuries before calculus was developed in Europe by Isaac Newton and Gottfried Leibniz—was an achievement. However, the Kerala School did not invent calculus, • A semi-rigorous proof (see "induction" remark below) of the result: 1^p+ 2^p + \cdots + n^p \approx \frac{n^{p+1}}{p+1} for large n. The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as: However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhāṣā given in two papers, a commentary on the Yuktibhāṣā's proof of the sine and cosine series and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary). Parameshvara (c. 1370–1460) wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His Lilavati Bhasya, a commentary on Bhaskara II's Lilavati, contains one of his important discoveries: a version of the mean value theorem. Nilakantha Somayaji (1444–1544) composed the Tantra Samgraha (which 'spawned' a later anonymous commentary Tantrasangraha-vyakhya and a further commentary by the name Yuktidipaika, written in 1501). He elaborated and extended the contributions of Madhava. Citrabhanu (c. 1530) was a 16th-century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms: : \begin{align} & x + y = a,\ x - y = b,\ xy = c, x^2 + y^2 = d, \\[8pt] & x^2 - y^2 = e,\ x^3 + y^3 = f,\ x^3 - y^3 = g \end{align} For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric. Jyesthadeva (c. 1500–1575) was another member of the Kerala School. His key work was the Yukti-bhāṣā (written in Malayalam, a regional language of Kerala). Jyesthadeva presented proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other Kerala School mathematicians. Others Narayana Pandit was a 14th century mathematician who composed two important mathematical works, an arithmetical treatise, Ganita Kaumudi, and an algebraic treatise, Bijganita Vatamsa. Ganita Kaumudi is one of the most revolutionary works in the field of combinatorics with developing a method for systematic generation of all permutations of a given sequence. In his Ganita Kaumudi Narayana proposed the following problem on a herd of cows and calves: Translated into the modern mathematical language of recurrence sequences: : for , with initial values :. The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... . The limit ratio between consecutive terms is the supergolden ratio. . Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Ganita Kaumudia(or Karma-Paddhati). ==Charges of Eurocentrism==

It has been suggested that Indian contributions to mathematics have not been given due acknowledgement in modern history and that many discoveries and inventions by Indian mathematicians are presently culturally attributed to their Western counterparts, as a result of Eurocentrism. According to G. G. Joseph's take on "Ethnomathematics": [Their work] takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness [of Indian and Arabic mathematics] is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilisations – most notably China and India, are perceived either as borrowers from Greek sources or having made only minor contributions to mainstream mathematical development. An openness to more recent research findings, especially in the case of Indian and Chinese mathematics, is sadly missing" Historian of mathematics Florian Cajori wrote that he and others "suspect that Diophantus got his first glimpse of algebraic knowledge from India". He also wrote that "it is certain that portions of Hindu mathematics are of Greek origin". More recently, as discussed in the above section, the infinite series of calculus for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described in India, by mathematicians of the Kerala school, some two centuries earlier. 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. According to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century". Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus. 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". The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate predecessors of Newton and Leibniz, "including, in particular, Fermat and Roberval, [may have] learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware." This is an area of current research, especially in the manuscript collections of Spain and Maghreb, and is being pursued, among other places, at the CNRS. ==See also==