Showing posts with label MATHMATICS. Show all posts
Showing posts with label MATHMATICS. Show all posts

## Friday, October 28, 2016

### Indian mathematics -Still Amazing

INDIAN MATHEMATICS, STILL AMAZING TO THIS PRESENT DAY.
Despite developing quite independently of Chinese (and probably also of Babylonian mathematics), some very advanced mathematical discoveries were made at a very early time in India.
Mantras from the early Vedic period (before 1000 BCE) invoke powers of ten from a hundred all the way up to a trillion, and provide evidence of the use of arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes and roots. A 4th Century CE Sanskrit text reports Buddha enumerating numbers up to 1053, as well as describing six more numbering systems over and above these, leading to a number equivalent to 10421. Given that there are an estimated 1080 atoms in the whole universe, this is as close to infinity as any in the ancient world came. It also describes a series of iterations in decreasing size, in order to demonstrate the size of an atom, which comes remarkably close to the actual size of a carbon atom (about 70 trillionths of a metre).
As early as the 8th Century BCE, long before Pythagoras, a text known as the “Sulba Sutras” (or "Sulva Sutras") listed several simple Pythagorean triples, as well as a statement of the simplified Pythagorean theorem for the sides of a square and for a rectangle (indeed, it seems quite likely that Pythagoras learned his basic geometry from the "Sulba Sutras"). The Sutras also contain geometric solutions of linear and quadratic equations in a single unknown, and give a remarkably accurate figure for the square root of 2, obtained by adding 1 + 1⁄3 + 1⁄(3 x 4) - 1⁄(3 x 4 x 34), which yields a value of 1.4142156, correct to 5 decimal places.
As early as the 3rd or 2nd Century BCE, Jain mathematicians recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite. Ancient Buddhist literature also demonstrates a prescient awareness of indeterminate and infinite numbers, with numbers deemed to be of three types: countable, uncountable and infinite.
Like the Chinese, the Indians early discovered the benefits of a decimal place value number system, and were certainly using it before about the 3rd Century CE. They refined and perfected the system, particularly the written representation of the numerals, creating the ancestors of the nine numerals that (thanks to its dissemination by medieval Arabic mathematicians) we use across the world today, sometimes considered one of the greatest intellectual innovations of all time.
The Indians were also responsible for another hugely important development in mathematics. The earliest recorded usage of a circle character for the number zero is usually attributed to a 9th Century engraving in a temple in Gwalior in central India. But the brilliant conceptual leap to include zero as a number in its own right (rather than merely as a placeholder, a blank or empty space within a number, as it had been treated until that time) is usually credited to the 7th Century Indian mathematicians Brahmagupta - or possibly another Indian, Bhaskara I - even though it may well have been in practical use for centuries before that. The use of zero as a number which could be used in calculations and mathematical investigations, would revolutionize mathematics.
Brahmagupta established the basic mathematical rules for dealing with zero: 1 + 0 = 1; 1 - 0 = 1; and 1 x 0 = 0 (the breakthrough which would make sense of the apparently non-sencical operation 1 ÷ 0 would also fall to an Indian, the 12th Century mathematician Bhaskara II). Brahmagupta also established rules for dealing with negative numbers, and pointed out that quadratic equations could in theory have two possible solutions, one of which could be negative. He even attempted to write down these rather abstract concepts, using the initials of the names of colours to represent unknowns in his equations, one of the earliest intimations of what we now know as algebra.
The so-called Golden Age of Indian mathematics can be said to extend from the 5th to 12th Centuries, and many of its mathematical discoveries predated similar discoveries in the West by several centuries, which has led to some claims of plagiarism by later European mathematicians, at least some of whom were probably aware of the earlier Indian work. Certainly, it seems that Indian contributions to mathematics have not been given due acknowledgement until very recently in modern history.
Golden Age Indian mathematicians made fundamental advances in the theory of trigonometry, a method of linking geometry and numbers first developed by the Greeks. They used ideas like the sine, cosine and tangent functions (which relate the angles of a triangle to the relative lengths of its sides) to survey the land around them, navigate the seas and even chart the heavens. For instance, Indian astronomers used trigonometry to calculated the relative distances between the Earth and the Moon and the Earth and the Sun. They realized that, when the Moon is half full and directly opposite the Sun, then the Sun, Moon and Earth form a right angled triangle, and were able to accurately measure the angle as 1⁄7°. Their sine tables gave a ratio for the sides of such a triangle as 400:1, indicating that the Sun is 400 times further away from the Earth than the Moon.
Although the Greeks had been able to calculate the sine function of some angles, the Indian astronomers wanted to be able to calculate the sine function of any given angle. A text called the “Surya Siddhanta”, by unknown authors and dating from around 400 CE, contains the roots of modern trigonometry, including the first real use of sines, cosines, inverse sines, tangents and secants.
As early as the 6th Century CE, the great Indian mathematician and astronomer Aryabhata produced categorical definitions of sine, cosine, versine and inverse sine, and specified complete sine and versine and inverse sine, and specified complete sine and versine tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. Aryabhata also demonstrated solutions to simultaneous quadratic equations, and produced an approximation for the value of π equivalent to 3.1416, correct to four decimal places. He used this to estimate the circumference of the Earth, arriving at a figure of 24,835 miles, only 70 miles off its true value. But, perhaps even more astonishing, he seems to have been aware that π is an irrational number, and that any calculation can only ever be an approximation, something not proved in Europe until 1761.
Bhaskara II, who lived in the 12th Century, was one of the most accomplished of all India’s great mathematicians. He is credited with explaining the previously misunderstood operation of division by zero. He noticed that dividing one into two pieces yields a half, so 1 ÷ 1⁄2 = 2. Similarly, 1 ÷ 1⁄3 = 3. So, dividing 1 by smaller and smaller factions yields a larger and larger number of pieces. Ultimately, therefore, dividing one into pieces of zero size would yield infinitely many pieces, indicating that 1 ÷ 0 = ∞ (the symbol for infinity).
However, Bhaskara II also made important contributions to many different areas of mathematics from solutions of quadratic, cubic and quartic equations (including negative and irrational solutions) to solutions of Diophantine equations of the second order to preliminary concepts of infinitesimal calculus and mathematical analysis to spherical trigonometry and other aspects of trigonometry. Some of his findings predate similar discoveries in Europe by several centuries, and he made important contributions in terms of the systemization of (then) current knowledge and improved methods for known solutions.
The Kerala School of Astronomy and Mathematics was founded in the late 14th Century by Madhava of Sangamagrama, sometimes called the greatest mathematician-astronomer of medieval India. He developed infinite series approximations for a range of trigonometric functions, including π, sine, etc. Some of his contributions to geometry and algebra and his early forms of differentiation and integration for simple functions may have been transmitted to Europe via Jesuit missionaries, and it is possible that the later European development of calculus was influenced by his work to some extent.
story of mathematics.com

## Saturday, July 25, 2015

### Bakhshali Manuscript – Ancient Indian mathematical manuscript on math

The Bakhshali Manuscript is an Ancient Indian mathematical manuscript written on "birch bark" which was found near the village of Bakhshali in 1881 in what was then the North-West Frontier Province of British India (now Khyber Pakhtunkhwa province, in Pakistan).
Bakhshali Manuscript is written in Śāradā script and in Gatha dialect (which is a combination of the ancient Indian languages of Sanskrit and Prakrit). The manuscript is incomplete, with only seventy leaves of birch bark, many of which are mere scraps. Many remain undiscovered. The Bakhshali manuscript, which is currently too fragile to be examined by scholars, is currently housed in the Bodleian Library at the University of Oxford and is too fragile to be examined by scholars.
It does not appear to belong to any specific period. Although that said, G Joseph classes it as a work of the early ‘classical period’, while E Robertson and J O’Connor suggest it may be a work of Jaina mathematics, and while this is chronologically plausible there is no proof it was composed by Jain scholars. L Gurjar discusses its date in detail, and concludes it can be dated no more accurately than ‘between 2nd century BC and 2nd century AD’. He offers compelling evidence by way of detailed analysis of the contents of the manuscript (originally carried by R Hoernle). His evidence includes the language in which it was written (‘died out’ around 300 AD), discussion of currency found in several problems, and the absence of techniques known to have been developed by the 5th century.
Historians who have placed the date at pre 450 AD and identified the ‘current’ version as a copy.
Avoiding further debate, L Gurjar states that the Bakshali manuscript is the:
Capstone of the advance of mathematics from the Vedic age up to that period…
Although, as much work was lost between ‘periods’, we cannot fully gauge continuity of progress and it is possible the composer(s) of the Bakhshali manuscript were not fully aware of earlier works and had to start from ‘scratch’. This would make the work an even more remarkable achievement.
The arithmetic contained within the work is of such a high quality that it has been suggested:
…In fact [the] Greeks [are] indebted to India for much of the developments in Arithmetic…
Some of the contents of Manuscript are elaborated here.
Examples of the rule of three (and profit and loss and interest).Solution of linear equations with as many as five unknowns.The solution of the quadratic equation (development of remarkable quality). Arithmetic (and geometric) progressions.Compound Series (some evidence that work begun by Jainas continued).Quadratic indeterminate equations (origin of type ax/c = y). Simultaneous equations.Fractions and other advances in notation including use of zero and negative sign.Improved method for calculating square root (and hence approximations for irrational numbers). The improved method (shown below) allowed extremely accurate approximations to be calculated:A = (a2 + r) = a + r/2a – {(r/2a)2 / 2(a + r/2a)}
Example 6.1: Application of square root formula.
Again we can calculate 10, where a = 3 and r = 1.10 = (32 + 1) = 3 + 1/6 – {(1/36)/2(3 + 1/6)}= 3 + 1/6 – {(1/36)/(19/3)}= 3 + 1/6 – 1/228= 3.16228… in decimal form root(10) = 3.16228 when calculated on a calculator and rounded to five decimal places.
Example 6.2: Quadratic equation as found in B. Ms.
If the equation given is dn2 + (2a – d)n -2s = 0Then the solution is found using the equation:n = (- (2a – d) (2a – d)2 +8ds))/2dWhich is the quadratic equation with a = d, b = 2a – d, and c = 2s.
Example 6.3: Linear equation with 5 variables.
The following problem is stated : ”Five merchants together buy a jewel. Its price is equal to half the money possessed by the first together with the money possessed by the others, or one-third the money possessed by the second together with the moneys of the others, or one-fourth the money possessed by the third together with the moneys of the others…etc. Find the price of the jewel and the money possessed by each merchant.
Solution :We have the following systems of equations:
x1/2 + x2 + x3 + x4 + x5
= px1 + x2/3 + x3 + x4 + x5
= px1 + x2 + x3/4 + x4 + x5
= px1 + x2 + x3 + x4/5 +x5
= px1 + x2 + x3 + x4 + x5/6 = p
Then if x1/2 + x2/3 + x3/4 + x4/5 + x5/6 = q ,
the equations become (377/60 )q = p.
A number of possible answers can be obtained. This is the origin of the indeterminate equation of the type ax/c = y, the theory of which was greatly developed, and later perfected by Bhaskara II, four hundred years before it was discovered in Europe. If q = 60 then p = 377 and x1 = 120, x2 = 90, x3 = 80, x4 = 75 and x5 = 72
Ms. Historians of mathematics debate whether true algebra ‘began’ in Greece or Arabia, and little mention is ever made of Indian algebra. In light of my own research I feel that early Arabic algebra (c. 800 AD) in no way surpasses the level of understanding of 6th century Indian scholars.
The Bakhshali manuscript is a unique piece of work and while it not only contains mathematics of a remarkably high standard for the time period, also, in contrast to almost all other Indian works composed before and after, the method of the commentary follows a highly systematic order of:
i. Statement of the rule (sutra)
ii. Statement of the examples (udaharana)
iii. Demonstration of the operation (karana) of the rule.
By the end of the 2nd century AD mathematics in India had attained a considerable stature, and had become divorced from purely practical and religious requirements, (although it is worth noting that over the next 1000 years the majority of mathematical developments occurred within works on astronomy).
The topics of algebra, arithmetic and geometry had developed significantly and it is widely thought that the decimal place value system of notation had been (generally) perfected by 200 AD, the consequence of which was far reaching.
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## Monday, February 3, 2014

### भारतीय गणितज्ञ “रामानुजम” (1887-1920)

गणित के क्षेत्र में अनेक विद्वान हुए हैं, जिनमे रामानुजम भी एक उच्चकोटि के गणितज्ञ रहे हैं, इनका जन्म ‘तमिलनाडु’ प्रांत के ‘इरोद’ नामक ग्राम मे एक निर्धन ब्राह्मण परिवार मे 22 दिसम्बर, 1887 मे हुआ इनके पिता ‘कुम्भ कोनम्’ ग्राम मे रहते थे और वही पर एक कपङे वाले के यहां मुनीमी करते थे ।
रामानुज के जन्म के बारे मे एक किंवदंती प्रचलित है, कहा जाता है कि विवाह होने के कई वर्ष बाद तक उनकी माता को कोई संतान नही हुई । इससे वह हमेशा चिंतित रहती थी । अपनी पुत्री को चिंतकुल देख कर रामानुजम् के नाना नामकल गाँव मे जाकर वहाँ नामगिरी देवी की आराधना की, इसी के फलस्वरूप ‘श्रीनिवास रामानुजम्’ का जन्म हुआ ।
5 वर्ष की अवस्था मे रामानुजम् को स्कूल भेजा गया । वहाँ 2 वर्ष के बाद इनको कुम्भ-कोनम् हाईस्कूल मे पढ़ने भेजा गया, इन्हे गणितशास्त्र मे विशेष रुचि थी । वे अपने सहपाठियो और गुरुओ से यह कभी नक्षत्रों के बारे मे तो कभी परिधि के बारे मे प्रश्न पूछते थे, वह जब कक्षा 3 मे पढ़ते थे, तो एक दिन जब अध्यापक समझा रहे थे कि किसी संख्या को उसी से भाग देने पर भागफल 1 होता है, तो रामानुजम् ने उसी समय पूछा, “क्या यह नियम शून्य के लिए भी लागू होता है ?” उन्होने इसी कक्षा मे बीजगणित की तीनों श्रेणियो – समान्तर श्रेणी, गुणोत्तर श्रेणी और हरात्मक श्रेणी को पढ़ लिया था, जो कि आजकल इंटर मीडिएट कक्षा मे पढ़ाई जाती है । कक्षा 4 मे त्रिकोणमिति तथा कक्षा 5 मे Sin (ज्या) और Cos (कोज्या) का विस्तार समाप्त कर लिया था ।
17 वर्ष की आयु मे इनहोने हाईस्कूल की परीक्षा योग्यता सहित उत्तीर्ण की, और इन्हे सरकारी छात्रवृत्ति प्रदान की गई । परंतु कालेज के प्रथम वर्ष तक पहुंचते- पहुंचते यह गणितशास्त्र मे इतने तल्लीन हो गए कि गणित के सिवाय और किसी के नही रहे और उसका परिणाम यह हुआ कि यह फेल हो गये, इससे इनकी छात्रवृत्ति रोक दी गई । अतः आर्थिक स्थिति खराब होने के कारण इनको विश्बविधालय कि शिक्षा समाप्त करनी पङी ।
उन दिनो रामानुजम् को आर्थिक कठिनाइयो ने परेशान कर दिया इसी समय इनका विवाह भी कर दिया गया । विवाह हो जाने के कारण कठिनाइयां दुगनी हो गई, और वह शीघ्र नौकरी ढूँढने के लिए मजबूर हो गए । बङी कठिनाइयो के बाद इनको मद्रास ट्रस्ट मे 30 रुपये मासिक की नौकरी मिल गई, इसी बीच ड़ॉ॰ वाकर गणित मे उनकी दिलचस्पी से बहुत प्रभावित हुये उनके प्रयत्न से रामानुजम् को मद्रास विश्बविधालय से 2 वर्ष के 75 रूपये मासिक छात्रवृत्ति मिल गई तथा इनको क्लर्कीसे छुटकारा मिल गया । आर्थिक चिंताओ से मुक्त होकर इन्हे अपना सारा समय गणित के अध्ययन मे लगाने का सुअवसर प्राप्त हो गया ।
फिर इनहोने कुछ लेख लिखकर ट्रिनिटी कालेज के गणित के फैलो ड़ॉ॰ हार्ड़ी के पास भेजे, इन लेखो को देखकर ड़ॉ॰ हार्ड़ी तथा दूसरे अंग्रेज़ गणितज्ञ बहुत प्रभावित हुए । अतः वे लोग रामानुजम् को कैम्ब्रिज बुलाने का प्रयत्न करने लगे ।
सन् 1914 मे जब ट्रिनिटी कालेज के ड़ॉ॰नोविल भारत आये तो हार्डी ने उसे रामानुजम् से मिलने तथा उनको कैम्ब्रिज लाने का अनुरोध कर दिया था । भारत आने पर ड़ॉ नोविल ने रामानुजम् से भेट की । चरित नायक ने नोविल महोदय की प्रार्थना को स्वीकार कर लिया । इस पर नोविल महोदय (साहव) ने इनको 250 पौंड की छात्रवृत्ति देने के अतिरिक्त प्रारम्भिक व्यय तथा यात्रा व्यय देना भी स्वीकार कर लिया । इसमे से 60 रूपये प्रति मास अपनी मटा को देने का प्रबंध करके 17 मार्च 1917 ई० को मि० नोबिल के साथ आप विलायत के लिए रवाना हो गए ।
28 फरवरी 1918 को आप रायल सोसाइटी के फैलो वन गए, इस सम्मान को प्राप्त करने वाले आप प्रथम भारतीय थे । 27 फरवरी 1919 को आप लंदन से भारत के लिए रवाना हुए और 27 मार्च को आप मुंबई पहुंचे । विदेश मे जलवायु अनुकूल न होने से आपका स्वस्थ्य गिर गया था । स्वस्थ्य खराब होने से इनको कावेरी कोदू मंडी ले जाया गया । वहाँ से उनको कुम्भ कोणम ले जाया गया । इनका स्वस्थ्य दिन पर दिन गिरता गया । फिर भी मस्तिष्क का प्रकाश अंत तक मंद नहीं हुआ, अंतिम समय तक वह कार्य मे लगे रहे। ‘Mock Theta Function’ पर उनका सब कार्य रोग शैय्या पर ही हुआ । हालात ज्यादा खराब होती देखकर मद्रास ले जाए गये । 26 अप्रैल 1920 ई० को मद्रास के पास चेतपुर ग्राम मे इस विश्व विख्यात गणितज्ञ का शरीरांत हो गया

## Tuesday, January 28, 2014

### Aryabhata-GREAT INDIAN MATHMATICIAN,ASTROLOGER

Aryabhata is also known as Aryabhata I to distinguish him from the later mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed t...o believe that there were two different mathematicians called Aryabhata living at the same time. He therefore created a confusion of two different Aryabhatas which was not clarified until 1926 when B Datta showed that al-Biruni's two Aryabhatas were one and the same person.

We know the year of Aryabhata's birth since he tells us that he was twenty-three years of age when he wrote Aryabhatiya which he finished in 499. We have given Kusumapura, thought to be close to Pataliputra (which was refounded as Patna in Bihar in 1541), as the place of Aryabhata's birth but this is far from certain, as is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final verdict can be given regarding the locations of Asmakajanapada and Kusumapura.

We do know that Aryabhata wrote Aryabhatiya in Kusumapura at the time when Pataliputra was the capital of the Gupta empire and a major centre of learning, but there have been numerous other places proposed by historians as his birthplace. Some conjecture that he was born in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that he was born in the north-east of India, perhaps in Bengal. In [8] it is claimed that Aryabhata was born in the Asmaka region of the Vakataka dynasty in South India although the author accepted that he lived most of his life in Kusumapura in the Gupta empire of the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th century. It is now thought by most historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on the Aryabhatiya.
Aryabhata

### Madhava--GREAT INDIAN MATHMATICIAN

Madhava of Sangamagramma was born near Cochin on the coast in the Kerala state in southwestern India. It is only due to research into Keralese mathematics over the last twenty-five years that the remarkable cont...ributions of Madhava have come to light. In Rajagopal and Rangachari put his achievement into context when they write:-

[Madhava] took the decisive step onwards from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis.

All the mathematical writings of Madhava have been lost, although some of his texts on astronomy have survived. However his brilliant work in mathematics has been largely discovered by the reports of other Keralese mathematicians such as Nilakantha who lived about 100 years later.

Madhava discovered the series equivalent to the Maclaurin expansions of sin x, cos x, and arctan x around 1400, which is over two hundred years before they were rediscovered in Europe. Details appear in a number of works written by his followers such as Mahajyanayana prakara which means Method of computing the great sines. In fact this work had been claimed by some historians such as Sarma (see for example [2]) to be by Madhava himself but this seems highly unlikely and it is now accepted by most historians to be a 16th century work by a follower of Madhava. This is discussed in detail in [4].

Jyesthadeva wrote Yukti-Bhasa in Malayalam, the regional language of Kerala, around 1550. In [9] Gupta gives a translation of the text and this is also given in [2] and a number of other sources. Jyesthadeva describes Madhava's series as follows:-

The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.

This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion. Perhaps we should write down in modern symbols exactly what the series is that Madhava has found. The first thing to note is that the Indian meaning for sine of θ would be written in our notation as r sin θ and the Indian cosine of would be r cos θ in our notation, where r is the radius. Thus the series is

r θ = r(r sin θ)/1(r cos θ) - r(r sin θ)3/3r(r cos θ)3 + r(r sin θ)5/5r(r cos θ)5- r(r sin θ)7/7r(r cos θ)7 + ...

putting tan = sin/cos and cancelling r gives

θ = tan θ - (tan3θ)/3 + (tan5θ)/5 - ...

which is equivalent to Gregory's series

tan-1θ = θ - θ3/3 + θ5/5 - ...

Now Madhava put q = π/4 into his series to obtain

π/4 = 1 - 1/3 + 1/5 - ...

and he also put θ = π/6 into his series to obtain

π = √12(1 - 1/(3×3) + 1/(5×32) - 1/(7×33) + ...

We know that Madhava obtained an approximation for π correct to 11 decimal places when he gave

π = 3.14159265359

which can be obtained from the last of Madhava's series above by taking 21 terms. In [5] Gupta gives a translation of the Sanskrit text giving Madhava's approximation of π correct to 11 places.

Perhaps even more impressive is the fact that Madhava gave a remainder term for his series which improved the approximation. He improved the approximation of the series for π/4 by adding a correction term Rn to obtain

π/4 = 1 - 1/3 + 1/5 - ... 1/(2n-1) ± Rn

Madhava gave three forms of Rn which improved the approximation, namely

Rn = 1/(4n) or
Rn = n/(4n2 + 1) or
Rn = (n2 + 1)/(4n3 + 5n).

There has been a lot of work done in trying to reconstruct how Madhava might have found his correction terms. The most convincing is that they come as the first three convergents of a continued fraction which can itself be derived from the standard Indian approximation to π namely 62832/20000.

Madhava also gave a table of almost accurate values of half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle. It is thought that the way that he found these highly accurate tables was to use the equivalent of the series expansions

sin θ = θ - θ3/3! + θ5/5! - ...

cos θ = 1 - θ2/2! + θ4/4! - ...

Jyesthadeva in Yukti-Bhasa gave an explanation of how Madhava found his series expansions around 1400 which are equivalent to these modern versions rediscovered by Newton around 1676. Historians have claimed that the method used by Madhava amounts to term by term integration.

Rajagopal's claim that Madhava took the decisive step towards modern classical analysis seems very fair given his remarkable achievements. In the same vein Joseph writes in :-

We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition, making him almost the equal of the more recent intuitive genius Srinivasa Ramanujan, who spent his childhood and youth at Kumbakonam, not far from Madhava's birthplace.

### JYESTHADEVA- GREAT INDIAN MATHMATICIAN

Jyesthadeva lived on the southwest coast of India in the district of Kerala. He belonged to the Kerala school of mathematics built on the work of Madhava, Nilakantha Somayaji, Paramesvara and others.

Jyesthadeva... wrote a famous text Yuktibhasa which he wrote in Malayalam, the regional language of Kerala. The work is a survey of Kerala mathematics and, very unusually for an Indian mathematical text, it contains proofs of the theorems and gives derivations of the rules it contains. It is one of the main astronomical and mathematical texts produced by the Kerala school. The work was based mainly on the Tantrasamgraha of Nilakantha.

The Yuktibhasa is a major treatise, half on astronomy and half on mathematics, written in 1501. The Tantrasamgraha on which it is based consists of 432 Sanskrit verses divided into 8 chapters, and it covers various aspects of Indian astronomy. It is based on the epicyclic and eccentric models of planetary motion. The first two chapters deal with the motions and longitudes of the planets. The third chapter Treatise on shadow deals with various problems related with the sun's position on the celestial sphere, including the relationships of its expressions in the three systems of coordinates, namely ecliptic, equatorial and horizontal coordinates.

The fourth and fifth chapters are Treatise on the lunar eclipse and On the solar eclipse and these two chapters treat various aspects of the eclipses of the sun and the moon. The sixth chapter is On vyatipata and deals with the complete deviation of the longitudes of the sun and the moon. The seventh chapter On visibility computation discusses the rising and setting of the moon and planets. The final chapter On elevation of the lunar cusps examines the size of the part of the moon which is illuminated by the sun and gives a graphical representation of it.

The Yuktibhasa is very important in terms of the mathematics Jyesthadeva presents. In particular he presents results discovered by Madhava and the treatise is an important source of the remarkable mathematical theorems which Madhava discovered. Written in about 1550, Jyesthadeva's commentary contained proofs of the earlier results by Madhava and Nilakantha which these earlier authors did not give. In [4] Gupta gives a translation of the text and this is also given in [2] and a number of other sources. Jyesthadeva describes Madhava's series as follows:-

The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.

This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion. To see how this description of the series fits with Gregory's series for arctan(x) see the biography of Madhava. Other mathematical results presented by Jyesthadeva include topics studied by earlier Indian mathematicians such as integer solutions of systems of first degree equation solved by the kuttaka method, and rules of finding the sines and the cosines of the sum and difference of two angles.

Not only does the mathematics anticipate work by European mathematicians a century later, but the planetary theory presented by Jyesthadeva is similar to that adopted by Tycho Brahe.

### Bhaskaracharya- The mathmatician , astrologer.

Bhaskaracharya – The Crown Jewel of Mathematics and Astronomy

A Glance at the Astronomical Achievements of Bhaskaracharya...

1. The Earth is not flat, has no support and has a power of attraction.

2. The north and south poles of the Earth experience six months of day and six months of night.

3. One day of Moon is equivalent to 15 earth-days and one night is also equivalent to 15 earth-days.

4. Bhaskaracharya had accurately calculated apparent orbital periods of the Sun and orbital periods of Mercury, Venus, and Mars. There is slight difference between the orbital periods he calculated for Jupiter and Saturn and the corresponding modern values.

5. Earth’s atmosphere extends to 96 kilometers and has seven parts.

6. There is a vacuum beyond the Earth’s atmosphere.

There have been several unscrupulous attempts to argue that there are traces of Diophantine influence in Bhaskara's work, but this once again seems like an attempt by European scholars to claim European influence on (all) the great works of mathematics. These claims should be ignored. Particularly in the field of algebra, Diophantus only looked at specific cases and did not achieve the general methods of the Indians.
The period between 500 and 1200 AD was the golden age of Indian Astronomy. In this long span of time Indian Astronomy flourished mainly due to emi...nent astronomers like Aryabhat, Lallacharya, Varahamihir, Brahmagupta, Bhaskaracharya and others. Bhaskaracharya’s Siddhanta Shiromani is considered as the pinnacle of all the astronomical works of those 700 hundred years. It can be aptly called the “essence” of ancient Indian Astronomy and mathematics. In the ninth century Brahmagupta’s Brahmasphutasiddhanta was translated in Arabic. The title of the translation was ‘Sind Hind’. This translation proved to be a watershed event in the history of numbers. The Arabs quickly grasped the importance of the Indian decimal system of numbers. They played a key role in transmitting this system of numbers to Europeans. For a long time Europeans were using Roman Numerals, which were very tedious to handle. After accepting the decimal system of numbers, European mathematicians made a remarkable progress in mathematics, but that was about 500 years after Bhaskaracharya.
From 750 AD Onwards India was engulfed in waves of foreign attacks. In 1205 AD Bakhtiyar Khilji destroyed the magnificent Nalanda University, which was a renowned center of knowledge for about 800 years. India was in utter chaotic state till the country was colonized by British. All universities and learning centers in India were destroyed, knowledge was lost and hardly any progress was made in mathematics and astronomy. A few scholars like Keshav Daivadnya, Ganesh Daivadnya Madhav, Sawai Jai Singh and others tried to keep the flame of knowledge burning in that dark period.

Birth and Education of Bhaskaracharya :

Ganesh Daivadnya has bestowed a very apt title on Bhaskaracharya. He has called him ‘Ganakchakrachudamani’, which means, ‘a gem among all the calculators of astronomical phenomena.’ Bhaskaracharya himself has written about his birth, his place of residence, his teacher and his education, in Siddhantashiromani as follows,
‘ A place called ‘Vijjadveed’, which is surrounded by Sahyadri ranges, where there are scholars of three Vedas, where all branches of knowledge are studied, and where all kinds of noble people reside, a brahmin called Maheshwar was staying, who was born in Shandilya Gotra (in Hindu religion, Gotra is similar to lineage from a particular person, in this case sage Shandilya), well versed in Shroud (originated from ‘Shut’ or ‘Vedas’) and ‘Smart’ (originated from ‘Smut’) Dharma, respected by all and who was authority in all the branches of knowledge. I acquired knowledge at his feet’.
From this verse it is clear that Bhaskaracharya was a resident of Vijjadveed and his father Maheshwar taught him mathematics and astronomy. Unfortunately today we have no idea where Vijjadveed was located. It is necessary to ardently search this place which was surrounded by the hills of Sahyadri and which was the center of learning at the time of Bhaskaracharya. He writes about his year of birth as follows,
‘I was born in Shake 1036 (1114 AD) and I wrote Siddhanta Shiromani when I was 36 years old.’
Bhaskaracharya has also written about his education. Looking at the knowledge, which he acquired in a span of 36 years, it seems impossible for any modern student to achieve that feat in his entire life. See what Bhaskaracharya writes about his education,
‘I have studied eight books of grammar, six texts of medicine, six books on logic, five books of mathematics, four Vedas, five books on Bharat Shastras, and two Mimansas’.
Bhaskaracharya calls himself a poet and most probably he was Vedanti, since he has mentioned ‘Parambrahman’ in that verse.

SIDDHANTASHIROMANI
...
Bhaskaracharya wrote Siddhanta Shiromani in 1150 AD when he was 36 years old. This is a mammoth work containing about 1450 verses. It is divided into four parts, Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact each part can be considered as separate book. The numbers of verses in each part are as follows,
Lilawati has 278, Beejaganit has 213, Ganitadhyaya has 451 and Goladhyaya has 501 verses.
One of the most important characteristic of Siddhanta Shiromani is, it consists of simple methods of calculations from Arithmetic to Astronomy. Essential knowledge of ancient Indian Astronomy can be acquired by reading only this book. Siddhanta Shiromani has surpassed all the ancient books on astronomy in India. After Bhaskaracharya nobody could write excellent books on mathematics and astronomy in lucid language in India. In India, Siddhanta works used to give no proofs of any theorem. Bhaskaracharya has also followed the same tradition.
Lilawati is an excellent example of how a difficult subject like mathematics can be written in poetic language. Lilawati has been translated in many languages throughout the world. When British Empire became paramount in India, they established three universities in 1857, at Bombay, Calcutta and Madras. Till then, for about 700 years, mathematics was taught in India from Bhaskaracharya’s Lilawati and Beejaganit. No other textbook has enjoyed such long lifespan.

BHASKAR’S MATHEMATICS

Lilawati and Beejaganit together consist of about 500 verses. A few important highlights of Bhaskar’s mathematics are as follows,
Terms for numbers
In English, cardinal numbers are only in multiples of 1000. They have terms such as thousand, million, billion, trillion, quadrillion etc. Most of these have been named recently. However, Bhaskaracharya has given the terms for numbers in multiples of ten and he says that these terms were coined by ancients for the sake
of positional values.

Bhaskar’s terms for numbers are as follows:
eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), arbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017).

Kuttak

Kuttak is nothing but the modern indeterminate equation of first order. The method of solution of such equations was called as ‘pulverizer’ in the western world. Kuttak means to crush to fine particles or to pulverize. There are many kinds of Kuttaks. Let us consider one example.
In the equation, ax + b = cy, a and b are known positive integers. We want to also find out the values of x and y in integers. A particular example is,
100x +90 = 63y
Bhaskaracharya gives the solution of this example as, x = 18, 81, 144, 207… And y=30, 130, 230, 330…
Indian Astronomers used such kinds of equations to solve astronomical problems. It is not easy to find solutions of these equations but Bhaskara has given a generalized solution to get multiple answers.

Chakrawaal

Chakrawaal is the “indeterminate equation of second order” in western mathematics. This type of equation is also called Pell’s equation. Though the equation is recognized by his name Pell had never solved the equation. Much before Pell, the equation was solved by an ancient and eminent Indian mathematician, Brahmagupta (628 AD). The solution is given in his Brahmasphutasiddhanta. Bhaskara modified the method and gave a general solution of this equation. For example, consider the equation 61x2 + 1 = y2. Bhaskara gives the values of x = 22615398 and y = 1766319049
There is an interesting history behind this very equation. The Famous French mathematician Pierre de Fermat (1601-1664) asked his friend Bessy to solve this very equation. Bessy used to solve the problems in his head like present day Shakuntaladevi. Bessy failed to solve the problem. After about 100 years another famous French mathematician solved this problem. But his method is lengthy and could find a particular solution only, while Bhaskara gave the solution for five cases. In his book ‘History of mathematics’, see what Carl Boyer says about this equation,
‘In connection with the Pell’s equation ax2 + 1 = y2, Bhaskara gave particular solutions for five cases, a = 8, 11, 32, 61, and 67, for 61x2 + 1 = y2, for example he gave the solutions, x = 226153980 and y = 1766319049, this is an impressive feat in calculations and its verifications alone will tax the efforts of the reader’
Henceforth the so-called Pell’s equation should be recognized as ‘Brahmagupta-Bhaskaracharya equation’.

Simple mathematical methods

Bhaskara has given simple methods to find the squares, square roots, cube, and cube roots of big numbers. He has proved the Pythagoras theorem in only two lines. The famous Pascal Triangle was Bhaskara’s ‘Khandameru’. Bhaskara has given problems on that number triangle. Pascal was born 500 years after Bhaskara. Several problems on permutations and combinations are given in Lilawati. Bhaskar. He has called the method ‘ankapaash’. Bhaskara has given an approximate value of PI as 22/7 and more accurate value as 3.1416. He knew the concept of infinity and called it as ‘khahar rashi’, which means ‘anant’. It seems that Bhaskara had not notions about calculus, One of his equations in modern notation can be written as, d(sin (w)) = cos (w) dw.