Before I write on what Kerala Calculus is, I feel it is necessary to remind us Indians of the fact that our continued ignorance, indifference reluctance to learn our highly scientific texts have resulted in the Myth of Aryan Invasion Theory by the Eurocentric colonialists gained currency and it has taken a lot of efforts to deny the theory which attempted to deny India its glorious past.
There are other insidious attempts too.
That the scientificancient texts are vague, they do not really talk about science and people use it, after the modern science discovers something,Indians say that the facts were mentioned in the Hindu Texts.
And another ingenious attempt to deny India its rich scientific heritage is by stating that the data referred to in the tests were not used by the ancient Indians !
The only way one can counter these arguments is for us to study the original texts and bring into light the scientific thoughts found in our Texts.
I am trying to instill a sense of pride in our heritage by posting articles on various astounding facts in out texts, our stupendous Temples Astronomy.
It is for the specialists to delve deep into the Texts and reveal what they contain.
Unfortunately those who know Sanskrit are not generally aware of advanced modern Scientific Theories and those familiar with modern concepts do not know Sanskrit.
My request is that both the groups must get familiar with what they do not know or get together to highlight the highly advanced scientific nature of Hindu Thought.
I shall continue to post articles giving general directions.
Now to Calculus.
Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves),and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science, engineering and economics and can solve many problems that algebra alone cannot.
Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called “the calculus of infinitesimals”, or “infinitesimal calculus”. The word “calculus” comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, calculus of variations, lambda calculus, and process calculus.”
The Kerala Calculus.
The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school 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 independently created a number of important mathematics concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in 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 Yuktibhasa (c. 1500 – c. 1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.
Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala ..
The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following (infinite) geometric series:
The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs. They used this to discover a semi-rigorous proof of the result:
- for large n. This result was also known to Alhazen.
They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for , , and .TheTantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:
- where, for , the series reduce to the standard power series for these trigonometric functions, for example:
- (The Kerala school did not use the “factorial” symbolism.)
The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e.computation of area under the arc of the circle), was not yet developed.) They also made use of the series expansion of to obtain an infinite series expression (later known as Gregory series) for :
Their rational approximation of the error for the finite sum of their series are of particular interest. For example, the error, , (for n odd, and i = 1, 2, 3) for the series:
- They manipulated the terms, using the partial fraction expansion of : to obtain a more rapidly converging series for :
They used the improved series to derive a rational expression, for correct up to nine decimal places, i.e. . They made use of an intuitive notion of a limit to compute these results. The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions, though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.
The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835, though there exists another work, namely Kala Sankalita by J. Warren from 1825 which briefly mentions the discovery of infinite series by Kerala astronomers. According to Whish, the Kerala mathematicians had “laid the foundation for a complete system of fluxions” and these works abounded “with fluxional forms and series to be found in no work of foreign countries.” 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 Yuktibhasa given in two papers, a commentary on the Yuktibhasa’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)..
The following notes by westerners would justify what I had written at the beginning of the Post.
- (Bressoud 2002, p. 12) Quote: “There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert  that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use.”
- Plofker 2001, p. 293 Quote: “It is not unusual to encounter in discussions of Indian mathematics such assertions as that “the concept of differentiation was understood [in India] from the time of Manjula (… in the 10th century)” [Joseph 1991, 300], or that “we may consider Madhava to have been the founder of mathematical analysis” (Joseph 1991, 293), or that Bhaskara II may claim to be “the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus” (Bag 1979, 294). … The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). … It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian “discovery of the principle of the differential calculus” somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential “principle” was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here”
- Pingree 1992, p. 562 Quote: “One example I can give you relates to the Indian Mādhava’s demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Whish, in the 1830s, it was heralded as the Indians’ discovery of the calculus. This claim and Mādhava’s achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish’s article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava’s mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.”
Reference and citation.