800 BC Baudhayana 750 BC Manava

600 BC Apastamba 520 BC Panini

200 BC Katyayana 120 AD Yavanesvara

476 Aryabhata I 500 Yativrsabha

505 Varahamihira 598 Brahmagupta

600 Bhaskara I 720 Lalla

800 Govindasvami 800 Mahavira

830 Prthudakasvami 840 Sankara

870 Sridhara 920 Aryabhata II

940 Vijayanandi 1019 Sripati

1060 Brahmadeva 1114 Bhaskara II

1340 Mahendra Suri 1340 Narayana

1350 Madhava 1370 Paramesvara

1444 Nilakantha 1500 Jyesthadeva

1616 Kamalakara 1690 Jagannatha

*BAUDHAYANA* (Born: about 800 BC in India Died: about 800 BC in India)

To write a Biography of **Baudhayana **is essentially impossible since nothing is known of him except that he was the author of one of the earliest Sulbasutras. We do not know his dates accurately enough to even guess at a life span for him, which is why we have given the same approximate Birth Year as Death Year.

He was neither a Mathematician in the sense that we would understand it today, nor a scribe who simply copied manuscripts like *Ahmes*. He would certainly have been a man of very considerable learning but probably not interested in Mathematics for its own sake, merely interested in using it for Religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for Religious rites and it would appear an almost certainty that Baudhayana himself would be a Vedic Priest.

The Mathematics given in the Sulbasutras is there to enable the accurate Construction of altars needed for sacrifices. It is clear from the writing that Baudhayana, as well as being a Priest, must have been a skilled craftsman. He must have been himself skilled in the Practical use of the Mathematics he described as a Craftsman who himself constructed sacrificial altars of the highest quality.

The Sulbasutras are discussed in detail in the article Indian Sulbasutras. Below we give one or two details of Baudhayana’s Sulbasutra, which contained three chapters, which is the oldest which we possess and, it would be fair to say, one of the two most important.

The Sulbasutra of Baudhayana contains Geometric Solutions (but not Algebraic ones) of a Linear Equation in a single unknown. Quadratic Equations of the forms ax2 = c and ax2 + bx = c appear.

Several values of π occur in Baudhayana’s Sulbasutra since when giving different Constructions Baudhayana uses different Approximations for constructing circular shapes. Constructions are given which are equivalent to taking π equal to 676/225 (where 676/225 = 3.004), 900/289 (where 900/289 = 3.114) and to 1156/361 (where 1156/361 = 3.202). None of these is particularly accurate but, in the context of constructing altars they would not lead to noticeable errors.

An interesting, and quite accurate, Approximate value for √2 is given in Chapter 1 verse 61 of Baudhayana’s Sulbasutra. The Sanskrit text gives in words what we would write in symbols as

√2 = 1 + 1/3 + 1/(3×4) – 1/(3×4×34)= 577/408

which is, to nine Places, 1.414215686. This gives √2 correct to Five Decimal Places. This is surprising since, as we mentioned above, great Mathematical accuracy did not seem necessary for the building work described. If the Approximation was given as

√2 = 1 + 1/3 + 1/(3×4)

then the error is of the order of 0.002 which is still more accurate than any of the values of π. Why then did Baudhayana feel that he had to go for a better Approximation?

*MANAVA (Born: about 750 BC in India Died: about 750 BC in India)*

Manava was the Author of one of the Sulbasutras. The Manava Sulbasutra is not the oldest (the one by Baudhayana is older) nor is it one of the most important, there being at least three Sulbasutras which are considered more important. We do not know Manava’s Dates accurately enough to even guess at a life span for him, which is why we have given the same Approximate Birth Year as Death year. Historians disagree on 750 BC, and some would put this Sulbasutra later by one hundred or more years.

Manava would have not have been a Mathematician in the sense that we would understand it today. Nor was he a scribe who simply copied Manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in Mathematics for its own sake, merely interested in using it for Religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for Religious rites and it would appear an almost certainty that Manava himself would be a Vedic Priest.

The Mathematics given in the Sulbasutras is there to enable accurate Construction of altars needed for sacrifices. It is clear from the writing that Manava, as well as being a Priest, must have been a skilled craftsman.

Manava’s Sulbasutra, like all the Sulbasutras, contained Approximate Constructions of Circles from Rectangles, and Squares from Circles, which can be thought of as giving Approximate values of π. There appear therefore different values of π throughout the Sulbasutra, essentially every Construction involving Circles leads to a different such Approximation. The paper contradict is concerned with an interpretation of verses 11.14 and 11.15 of Manava’s work which give π = 25/8 = 3.125.

*APASTAMBA (Born: about 600 BC in India Died: about 600 BC in India)*

To write a biography of Apastamba is essentially impossible since nothing is known of him except that he was the Author of a Sulbasutra which is certainly later than the Sulbasutra of Baudhayana. It would also be fair to say that Apastamba’s Sulbasutra is the most interesting from a Mathematical point of view. We do not know Apastamba’s Dates accurately enough to even guess at a life span for him, which is why we have given the same Approximate Birth Year as Death year.

Apastamba was neither a Mathematician in the sense that we would understand it today, nor a scribe who simply copied Manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in Mathematics for its own sake, merely interested in using it for Religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for Religious rites and to improve and expand on the rules which had been given by his predecessors. Apastamba would have been a Vedic Priest instructing the people in the ways of conducting the Religious rites he describes.

The Mathematics given in the Sulbasutras is there to enable the accurate Construction of altars needed for sacrifices. It is clear from the writing that Apastamba, as well as being a Priest and a Teacher of Religious practices, would have been a skilled craftsman. He must have been himself skilled in the Practical use of the Mathematics he described as a Craftsman who himself constructed sacrificial altars of the highest quality.

The Sulbasutras are discussed in detail in the article Indian Sulbasutras. Below we give one or two details of Apastamba’s Sulbasutra. This work is an expanded version of that of Baudhayana. Apastamba’s work consisted of six chapters while the earlier work by Baudhayana contained only three.

The general Linear Equation was solved in the Apastamba’s Sulbasutra. He also gives a remarkably accurate value for √2 namely

1 + 1/3 + 1/(3×4) – 1/(3×4×34).

which gives an answer correct to Five Decimal Places. A possible way that Apastamba might have reached this remarkable result is described in the article Indian Sulbasutras.

As well as the problem of squaring the circle, Apastamba considers the problem of dividing a Segment into 7 equal parts. The article contradict looks in detail at a Reconstruction of Apastamba’s version of these two problems.

*PANINI (Born: about 520 BC in Shalatula (near Attock) Died: about 460 BC in India)*

Panini was born in Shalatula, a town near to Attock on the Indus river in present day Pakistan. The Dates given for Panini are pure guesses. Experts give Dates in the 4th, 5th, 6th and 7th century BC and there is also no agreement among historians about the extent of the work which he undertook. What is in little doubt is that, given the period in which he worked, he is one of the most innovative people in the whole development of knowledge. We will say a little more below about how historians have gone about trying to pinpoint the date when Panini lived.

Panini was a Sanskrit Grammarian who gave a comprehensive and Scientific Theory of Phonetics, Phonology, and Morphology. Sanskrit was the Classical Literary Language of the Indian Hindus and Panini is considered the founder of the Language and Literature. It is interesting to note that the word “Sanskrit” means “Complete” or “Perfect” and it was thought of as the divine language, or Language of the gods.

A treatise called Astadhyayi (or Astaka ) is Panini’s major work. It consists of eight chapters, each subdivided into quarter chapters. In this work Panini distinguishes between the Language of sacred texts and the usual Language of communication. Panini gives formal production rules and definitions to describe Sanskrit Grammar. Starting with about 1700 basic elements like nouns, verbs, vowels, consonants he put them into classes. The Construction of sentences, compound nouns etc. is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways Panini’s Constructions are similar to the way that a Mathematical function is defined today. Joseph writes in

contradict:-

*[Sanskrit's] potential for Scientific use was greatly enhanced as a result of the thorough systemisation of its Grammar by Panini. … On the basis of just under 4000 sutras [rules expressed as aphorisms], he built virtually the whole structure of the Sanskrit language, whose general ‘shape’ hardly changed for the next two thousand years. … An indirect consequence of Panini’s efforts to increase the linguistic facility of Sanskrit soon became apparent in the character of Scientific and Mathematical Literature. This may be brought out by comparing the Grammar of Sanskrit with the geometry of Euclid – a particularly apposite comparison since, whereas Mathematics grew out of philosophy in ancient Greece, it was … partly an outcome of linguistic developments in India.*

Joseph goes on to make a convincing argument for the Algebraic nature of Indian Mathematics arising as a consequence of the structure of the Sanskrit language. In particular he suggests that Algebraic reasoning, the Indian way of representing numbers by words, and ultimately the development of modern number systems in India, are linked through the structure of language.

Panini should be thought of as the forerunner of the modern formal Language Theory used to specify computer languages. The Backus Normal Form was discovered independently by John Backus in 1959, but Panini’s notation is equivalent in its power to that of Backus and has many similar properties. It is remarkable to think that concepts which are fundamental to today’s theoretical computer science should have their origin with an Indian genius around 2500 years ago.

At the beginning of this article we mentioned that certain concepts had been attributed to Panini by certain historians which others dispute. One such Theory was put forward by B Indraji in 1876. He claimed that the Brahmi numerals developed out of using letters or syllables as numerals. Then he put the finishing touches to the Theory by suggesting that Panini in the eighth century BC (earlier than most historians place Panini) was the first to come up with the idea of using letters of the alphabet to represent numbers.

There are a number of pieces of evidence to support Indraji’s Theory that the Brahmi numerals developed from letters or syllables. However it is not totally convincing since, to quote one example, the symbols for 1, 2 and 3 clearly do not come from letters but from one, two and three lines respectively. Even if one accepts the link between the numerals and the letters, making Panini the originator of this idea would seem to have no more behind it than knowing that Panini was one of the most innovative geniuses that world has known so it is not unreasonable to believe that he might have made this step too.

There are other works which are closely associated with the Astadhyayi which some historians attribute to Panini, others attribute to authors before Panini, others attribute to authors after Panini. This is an area where there are many theories but few, if any, hard facts.

We also promised to return to a discussion of Panini’s dates. There has been no lack of work on this topic so the fact that there are theories which span several hundreds of years is not the result of lack of effort, rather an indication of the difficulty of the topic. The usual way to date such texts would be to examine which authors are referred to and which authors refer to the work. One can use this technique and see who Panini mentions.

There are ten scholars mentioned by Panini and we must assume from the context that these ten have all contributed to the study of Sanskrit Grammar. This in itself, of course, indicates that Panini was not a solitary genius but, like Newton, had “stood on the shoulders of giants”. Panini must have lived later than these ten but this is absolutely no help in providing Dates since we have absolutely no knowledge of when any of these ten lived.

What other internal evidence is there to use? Well of course Panini uses many phrases to illustrate his Grammar any these have been examined meticulously to see if anything is contained there to indicate a date. To give an example of what we mean: if we were to pick up a text which contained as an example “I take the train to work every day” we would know that it had to have been written after railways became common. Let us illustrate with two actual examples from the Astadhyayi which have been the subject of much study. The first is an attempt to see whether there is evidence of Greek influence. Would it be possible to find evidence which would mean that the text had to have been written after the conquests of Alexander the Great? There is a little evidence of Greek influence, but there was Greek influence on this north east part of the Indian subcontinent before the time of Alexander. Nothing conclusive has been identified.

Another angle is to examine a reference Panini makes to nuns. Some argue that these must be Buddhist nuns and therefore the work must have been written after Buddha. A nice argument but there is a counter argument which says that there were Jaina nuns before the time of Buddha and Panini’s reference could equally well be to them. Again the evidence is inconclusive.

There are references by others to Panini. However it would appear that the Panini to whom most refer is a poet and although some argue that these are the same person, most historians agree that the linguist and the poet are two different people. Again this is inconclusive evidence.

Let us end with an evaluation of Panini’s contribution by Cardona in contradict:-

*Panini’s Grammar has been evaluated from various points of view. After all these different evaluations, I think that the Grammar merits asserting … that it is one of the greatest monuments of human intelligence. *

*KATYAYANA (Born: about 200 BC in India Died: about 200 BC in India)*

We cannot attempt to write a biography of Katyayana since essentially nothing is known of him except that he was the Author of a Sulbasutra which is much later than the Sulbasutras of Baudhayana and Apastamba. It would also be fair to say that Katyayana’s Sulbasutra is the least interesting from a Mathematical point of view of the three best known Sulbasutras. It adds very little to that of Apastamba written several hundreds of years earlier. We do not know Katyayana’s Dates accurately enough to even guess at a life span for him, which is why we have given the same Approximate Birth Year as Death year.

Katyayana was neither a Mathematician in the sense that we would understand it today, nor a scribe who simply copied Manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in Mathematics for its own sake, merely interested in using it for Religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for Religious rites and to improve and expand on the rules which had been given by his predecessors. Katyayana would have been a Priest instructing the people in the ways of conducting the Religious rites he describes.

Katyayana lived in a period when the Religious rites that the Sulbasutras were written to support were becoming less influential. People were turning to other religions and perhaps this lack of vigour in the religion at this time partly explains why several hundreds of years after Apastamba Katyayana adds little of importance to the Sulbasutra which he wrote.

*YAVANESVARA (Born: about 120 in Western India Died: about 180 in India)*

Indian astrology was originally known as Jyotisha, which means “science of the stars”. Until around the first century AD no real distinction was made between astrology and astronomy and in fact most astronomical theories were propounded to support the Theory that the positions of the heavenly bodies directly influenced human events.

The Indian methods of computing horoscopes all date back to the translation of a Greek astrology text into Sanskrit prose by Yavanesvara in 149 AD. Yavanesvara (or Yavanaraja) literally means “Lord of the Greeks” and it was a name given to many officials in western India during the period 130 AD – 390 AD. During this period the Ksatrapas ruled Gujarat (or Madhya Pradesh) and these “Lord of the Greeks” officials acted for the Greek merchants living in the area.

The particular “Lord of the Greeks” official Yavanesvara who we are interested in here worked under Rudradaman. Rudradaman became ruler of the Ksatrapas in around 130 AD and it was during the period of his rule that Yavanesvara worked as an official and made his translation. We know of Rudradaman because information is recorded in a lengthy Sanskrit inscription at Junagadh written around 150 AD.

The Greek astrology text in question was written in Alexandria some time round about 120 BC. Yavanesvara did far more than just translate the Greek text for such a translation would have had little relevance to the Indians. He therefore not only translated the Language but he translated the context too. Instead of the Greek gods who appear in the original, Yavanesvara used Hindu images. Again he worked the Indian caste system into the work and made the work one which would fit well with the Indian thought.

The work was written with the aim of letting Indians became astrologers so it had to present astronomy in a form in which it could be used for astrology. In order to do this Yavanesvara put into his work an explanation of the Greek version of the Babylonian Theory of the motions of the planets. All this he wrote in Sanskrit prose but sadly the original has not survived. We do have, however, a version written in Sanskrit verse 120 years after Yavanesvara’s work appeared.

Yavanesvara had an important influence on the whole of astrology in India for centuries after he made his popular translation. Although the influence was more than on astrology, as the science of astronomy split from astrology, the influence of Yavanesvara’s work reached into astronomy too.

*ARYABHATA I (Born: 476 in Kusumapura (now Patna), India Died: 550 in India)*

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 to 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:-

*… 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 contradict 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.

We should note that Kusumapura became one of the two major Mathematical centres of India, the other being Ujjain. Both are in the north but Kusumapura (assuming it to be close to Pataliputra) is on the Ganges and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a communications network which allowed learning from other parts of the world to reach it easily, and also allowed the Mathematical and astronomical advances made by Aryabhata and his school to reach across India and also eventually into the Islamic world.

As to the texts written by Aryabhata only one has survived. However Jha claims in contradict that:-

*… Aryabhata was an Author of at least three astronomical texts and wrote some free stanzas as well.*

The surviving text is Aryabhata’s masterpiece the Aryabhatiya which is a small astronomical treatise written in 118 verses giving a summary of Hindu Mathematics up to that time. Its Mathematical section contains 33 verses giving 66 Mathematical rules without proof. The Aryabhatiya contains an introduction of 10 verses, followed by a section on Mathematics with, as we just mentioned, 33 verses, then a section of 25 verses on the reckoning of time and planetary models, with the final section of 50 verses being on the sphere and eclipses.

There is a difficulty with this layout which is discussed in detail by van der Waerden in. Van der Waerden suggests that in fact the 10 verse Introduction was written later than the other three sections. One reason for believing that the two parts were not intended as a whole is that the first section has a different meter to the remaining three sections. However, the problems do not stop there. We said that the first section had ten verses and indeed Aryabhata titles the section Set of ten giti stanzas. But it in fact contains eleven giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have been added and he identifies a small number of verses in the remaining sections which he argues have also been added by a member of Aryabhata’s school at Kusumapura.

The Mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines. Let us examine some of these in a little more detail.

First we look at the system for representing numbers which Aryabhata invented and used in the Aryabhatiya. It consists of giving numerical values to the 33 consonants of the Indian alphabet to represent 1, 2, 3, … , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000, …. In fact the system allows numbers up to 1018to be represented with an alphabetical notation. Ifrah in contradict argues that

Aryabhata was also familiar with numeral symbols and the place-value system. He writes in contradict:-

*… it is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.*

Next we look briefly at some algebra contained in the Aryabhatiya. This work is the first we are aware of which examines integer Solutions to Equations of the form by = ax + c and by = ax – c, where a, b, c are integers. The problem arose from studying the problem in astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to solve problems of this type. The word kuttaka means “to pulverise” and the method consisted of breaking the problem down into new problems where the coefficients became smaller and smaller with each step. The method here is essentially the use of the Euclidean algorithm to find the highest common factor of a and b but is also related to continued fractions.

Aryabhata gave an accurate Approximation for π. He wrote in the Aryabhatiya the following:-

*Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.*

This gives π = 62832/20000 = 3.1416 which is a surprisingly accurate value. In fact π = 3.14159265 correct to 8 Places. If obtaining a value this accurate is surprising, it is perhaps even more surprising that Aryabhata does not use his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how he found this accurate value but, for example, Ahmad contradict considers this value as an Approximation to half the perimeter of a regular polygon of 256 sides inscribed in the unit circle. However, in contradict Bruins shows that this result cannot be obtained from the doubling of the number of sides. Another interesting paper discussing this accurate value of π by Aryabhata is where Jha writes:-

*Aryabhata I’s value of π is a very close Approximation to the modern value and the most accurate among those of the ancients. There are reasons to believe that Aryabhata devised a particular method for finding this value. It is shown with sufficient grounds that Aryabhata himself used it, and several later Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata’s value of π is of Greek origin is critically examined and is found to be without foundation. Aryabhata discovered this value independently and also realised that π is an irrational number. He had the Indian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit of discovering this exact value of π may be ascribed to the celebrated mathematician, Aryabhata I.*

We now look at the trigonometry contained in Aryabhata’s treatise. He gave a table of sines calculating the Approximate values at intervals of 90°/24 = 3° 45′. In order to do this he used a formula for sin(n+1)x – sin nx in terms of sin nx and sin (n-1)x. He also introduced the versine (versin = 1 – cosine) into trigonometry.

Other rules given by Aryabhata include that for summing the first n integers, the Squares of these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of a circle which are correct, but the formulae for the volumes of a sphere and of a pyramid are claimed to be wrong by most historians. For example Ganitanand in describes as “Mathematical lapses” the fact that Aryabhata gives the incorrect formula V = Ah/2 for the volume of a pyramid with height h and triangular base of area A. He also appears to give an incorrect expression for the volume of a sphere. However, as is often the case, nothing is as straightforward as it appears and Elfering (see for example contradict) argues that this is not an error but rather the result of an incorrect translation.

This relates to verses 6, 7, and 10 of the second section of the Aryabhatiya and in contradict Elfering produces a translation which yields the correct answer for both the volume of a pyramid and for a sphere. However, in his translation Elfering translates two technical terms in a different way to the meaning which they usually have. Without some supporting evidence that these technical terms have been used with these different meanings in other Places it would still appear that Aryabhata did indeed give the incorrect formulae for these volumes.

We have looked at the Mathematics contained in the Aryabhatiya but this is an astronomy text so we should say a little regarding the astronomy which it contains. Aryabhata gives a systematic treatment of the position of the planets in space. He gave the circumference of the earth as 4 967 yojanas and its diameter as 1 5811/24 yojanas. Since 1 yojana = 5 miles this gives the circumference as 24 835 miles, which is an excellent Approximation to the currently accepted value of 24 902 miles. He believed that the apparent rotation of the heavens was due to the axial rotation of the Earth. This is a quite remarkable view of the nature of the solar system which later commentators could not bring themselves to follow and most changed the text to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the radius of the planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sun. He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses of the Sun and the Moon. The Indian belief up to that time was that eclipses were caused by a demon called Rahu. His value for the length of the Year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours.

Bhaskara I who wrote a commentary on the Aryabhatiya about 100 years later wrote of Aryabhata:-

*Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world. *

*YATIVRSABHA (Born: about 500 in India Died: about 570 in India)*

Yativrsabha (or Jadivasaha) was a Jaina Mathematician who studied under Arya Manksu and Nagahastin. We know nothing of Yativrsabha’s Dates except for a reference which he makes to the end of the Gupta dynasty which he says was after 231 years of ruling. This ended in 551 so we must assume that 551 AD is a date which occured during Yativrsabha’s lifetime. This fits with the only other information regarding his Dates which are that his work is referenced by Jinabhadra Ksamasramana in 609 and that Yativrsabha himself refers to a work written by Sarvanandin in 458.

Yativrsabha’s work Tiloyapannatti gives various units for measuring distances and time and also describes the system of infinite time measures. It is a work which describes Jaina cosmology and gives a description of the universe which is of historical importance in understanding Jaina science and mathematics. The Jaina belief was in an infinite world, both infinite in space and in time. This led the Jainas to devise ways of measuring larger and larger distances and longer and longer intervals of time. It led them to consider different measures of infinity, and in this respect the Jaina mathematicians would appear to be the only ones before the time when Cantor developed the Theory of infinite cardinals to envisage different magnitudes of infinity.

*VARAHAMIHIRA (Born: 505 in Kapitthaka, India Died: 587 in India)*

Our knowledge of Varahamihira is very limited indeed. According to one of his works, he was educated in Kapitthaka. However, far from settling the question this only gives rise to discussions of possible interpretations of where this place was. Dhavale in contradict discusses this problem. We do not know whether he was born in Kapitthaka, wherever that may be, although we have given this as the most likely guess. We do know, however, that he worked at Ujjain which had been an important centre for Mathematics since around 400 AD. The school of Mathematics at Ujjain was increased in importance due to Varahamihira working there and it continued for a long period to be one of the two leading Mathematical centres in India, in particular having Brahmagupta as its next major figure.

The most famous work by Varahamihira is the Pancasiddhantika (The Five Astronomical Canons) dated 575 AD. This work is important in itself and also in giving us information about older Indian texts which are now lost. The work is a treatise on Mathematical astronomy and it summarises Five earlier astronomical treatises, namely the Surya, Romaka, Paulisa, Vasistha and Paitamaha siddhantas. Shukla states in contradict:-

*The Pancasiddhantika of Varahamihira is one of the most important sources for the history of Hindu astronomy before the time of Aryabhata I I. *

One treatise which Varahamihira summarises was the Romaka-Siddhanta which itself was based on the epicycle Theory of the motions of the Sun and the Moon given by the Greeks in the 1st century AD. The Romaka-Siddhanta was based on the tropical Year of Hipparchus and on the Metonic cycle of 19 years. Other works which Varahamihira summarises are also based on the Greek epicycle Theory of the motions of the heavenly bodies. He revised the calendar by updating these earlier works to take into account precession since they were written. The Pancasiddhantika also contains many examples of the use of a place-value number system.

There is, however, quite a debate about interpreting data from Varahamihira’s astronomical texts and from other similar works. Some believe that the astronomical theories are Babylonian in origin, while others argue that the Indians refined the Babylonian models by making observations of their own. Much needs to be done in this area to clarify some of these interesting theories.

In contradict Ifrah notes that Varahamihira was one of the most famous astrologers in Indian history. His work Brihatsamhita (The Great Compilation) discusses topics such as contradict:-

*… descriptions of heavenly bodies, their movements and conjunctions, meteorological phenomena, indications of the omens these movements, conjunctions and phenomena represent, what action to take and operations to accomplish, sign to look for in humans, animals, precious stones, etc. *

Varahamihira made some important Mathematical discoveries. Among these are certain trigonometric formulae which translated into our present day notation correspond to

sin x = cos(π/2 – x),

sin2x + cos2x = 1, and

(1 – cos 2x)/2 = sin2x.

Another important contribution to trigonometry was his sine tables where he improved those of Aryabhata I giving more accurate values. It should be emphasised that accuracy was very important for these Indian mathematicians since they were computing sine tables for applications to astronomy and astrology. This motivated much of the improved accuracy they achieved by developing new interpolation methods.

The Jaina school of Mathematics investigated rules for computing the number of ways in which r objects can be selected from n objects over the course of many hundreds of years. They gave rules to compute the binomial coefficients nCr which amount to

nCr = n(n-1)(n-2)…(n-r+1)/r!

However, Varahamihira attacked the problem of computing nCr in a rather different way. He wrote the numbers n in a column with n = 1 at the bottom. He then put the numbers r in rows with r = 1 at the left-hand side. Starting at the bottom left side of the array which corresponds to the values n = 1, r = 1, the values of nCr are found by summing two entries, namely the one directly below the (n, r) position and the one immediately to the left of it. Of course this table is none other than Pascal’s triangle for finding the binomial coefficients despite being viewed from a different angle from the way we build it up today. Full details of this work by Varahamihira is given in contradict.

Hayashi, in contradict, examines Varahamihira’s work on magic squares. In particular he examines a pandiagonal magic square of order four which occurs in Varahamihira’s work.

*BRAHMAGUPTA (Born: 598 in (possibly) Ujjain, India Died: 670 in India)*

Brahmagupta, whose father was Jisnugupta, wrote important works on Mathematics and astronomy. In particular he wrote Brahmasphutasiddhanta (The Opening of the Universe), in 628. The work was written in 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamala which today is the city of Bhinmal. This was the capital of the lands ruled by the Gurjara dynasty.

Brahmagupta became the head of the astronomical observatory at Ujjain which was the foremost Mathematical centre of ancient India at this time. Outstanding mathematicians such as Varahamihira had worked there and built up a strong school of Mathematical astronomy.

In addition to the Brahmasphutasiddhanta Brahmagupta wrote a second work on Mathematics and astronomy which is the Khandakhadyaka written in 665 when he was 67 years old. We look below at some of the remarkable ideas which Brahmagupta’s two treatises contain. First let us give an overview of their contents.

The Brahmasphutasiddhanta contains twenty-Five chapters but the first ten of these chapters seem to form what many historians believe was a first version of Brahmagupta’s work and some Manuscripts exist which contain only these chapters. These ten chapters are arranged in topics which are typical of Indian Mathematical astronomy texts of the period. The topics covered are: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon’s crescent; the moon’s shadow; conjunctions of the planets with each other; and conjunctions of the planets with the fixed stars.

The remaining fifteen chapters seem to form a second work which is major addendum to the original treatise. The chapters are: examination of previous treatises on astronomy; on mathematics; additions to chapter 1; additions to chapter 2; additions to chapter 3; additions to chapter 4 and 5; additions to chapter 7; on algebra; on the gnomon; on meters; on the sphere; on instruments; summary of contents; versified tables.

Brahmagupta’s understanding of the number systems went far beyond that of others of the period. In the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-

*When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.*

He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-

*A debt minus zero is a debt.*

A fortune minus zero is a fortune.

Zero minus zero is a zero.

A debt subtracted from zero is a fortune.

A fortune subtracted from zero is a debt.

The product of zero multiplied by a debt or fortune is zero.

The product of zero multipliedby zero is zero.

The product or quotient of two fortunes is one fortune.

The product or quotient of two debts is one fortune.

The product or quotient of a debt and a fortune is a debt.

*The product or quotient of a fortune and a debt is a debt. *

Brahmagupta then tried to extend arithmetic to include division by zero:-

*Positive or negative numbers when divided by zero is a fraction the zero as denominator.
Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. *

*Zero divided by zero is zero. *

Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt to extend arithmetic to negative numbers and zero.

We can also describe his methods of multiplication which use the place-value system to its full advantage in almost the same way as it is used today. We give three examples of the methods he presents in the Brahmasphuta siddhanta and in doing so we follow Ifrah in contradict. The first method we describe is called “gomutrika” by Brahmagupta. Ifrah translates “gomutrika” to “like the trajectory of a cow’s urine”. Consider the product of 235 multiplied by 264. We begin by setting out the sum as follows:

2 235

6 235

4 235

Now multiply the 235 of the top row by the 2 in the top position of the left hand column. Begin by 2 × 5 = 10, putting 0 below the 5 of the top row, carrying 1 in the usual way to get

2 235

6 235

4 235

470

Now multiply the 235 of the second row by the 6 in the left hand column writing the number in the line below the 470 but moved one place to the right

2 235

6 235

4 235

470

1410

Now multiply the 235 of the third row by the 4 in the left hand column writing the number in the line below the 1410 but moved one place to the right

2 235

6 235

4 235

470

1410

940

Now add the three numbers below the line

2 235

6 235

4 235

470

1410

940

62040

The variants are first writing the second number on the right but with the order of the digits reversed as follows

235 4

235 6

235 2

940

1410

470

62040

The third variant just writes each number once but otherwise follows the second method

235

940 4

1410 6

470 2

62040

Another arithmetical result presented by Brahmagupta is his algorithm for computing square roots. This algorithm is discussed in contradict where it is shown to be equivalent to the Newton-Raphson iterative formula.

Brahmagupta developed some Algebraic notation and presents methods to solve quardatic equations. He presents methods to solve indeterminate Equations of the form ax + c = by. Majumdar in contradict writes:-

*Brahmagupta perhaps used the method of continued fractions to find the integral solution of an indeterminate Equation of the type ax + c = by.*

In contradict Majumdar gives the original Sanskrit verses from Brahmagupta’s Brahmasphuta siddhanta and their English translation with modern interpretation.

Brahmagupta also solves quadratic indeterminate Equations of the type ax2 + c = y2 and ax2 – c = y2. For example he solves 8×2 + 1 = y2 obtaining the Solutions (x, y) = (1, 3), (6, 17), (35, 99), (204, 577), (1189, 3363), … For the Equation 11×2 + 1 = y2 Brahmagupta obtained the Solutions (x, y) = (3, 10), (161/5, 534/5), … He also solves 61×2 + 1 = y2 which is particularly elegant having x = 226153980, y = 1766319049 as its smallest solution.

A example of the type of problems Brahmagupta poses and solves in the Brahmasphutasiddhanta is the following:-

Five hundred drammas were loaned at an unknown rate of interest, The interest on the money for four months was loaned to another at the same rate of interest and amounted in ten mounths to 78 drammas. Give the rate of interest.

Rules for summing series are also given. Brahmagupta gives the sum of the Squares of the first n natural numbers as n(n+1)(2n+1)/6 and the sum of the cubes of the first n natural numbers as (n(n+1)/2)2. No proofs are given so we do not know how Brahmagupta discovered these formulae.

In the Brahmasphutasiddhanta Brahmagupta gave remarkable formulae for the area of a cyclic quadrilateral and for the lengths of the diagonals in terms of the sides. The only debatable point here is that Brahmagupta does not state that the formulae are only true for cyclic quadrilaterals so some historians claim it to be an error while others claim that he clearly meant the rules to apply only to cyclic quadrilaterals.

Much material in the Brahmasphutasiddhanta deals with solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a static Earth and he gave the length of the Year as 365 days 6 hours 5 minutes 19 seconds in the first work, changing the value to 365 days 6 hours 12 minutes 36 seconds in the second book the Khandakhadyaka. This second values is not, of course, an improvement on the first since the true length of the years if less than 365 days 6 hours. One has to wonder whether Brahmagupta’s second value for the length of the Year is taken from Aryabhata I since the two agree to within 6 seconds, yet are about 24 minutes out.

The Khandakhadyaka is in eight chapters again covering topics such as: the longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon’s crescent; and conjunctions of the planets. It contains an appendix which is some versions has only one chapter, in other versions has three.

Of particular interest to Mathematics in this second work by Brahmagupta is the interpolation formula he uses to compute values of sines. This is studied in detail in contradict where it is shown to be a particular case up to second order of the more general Newton-Stirling interpolation formula.

*BHASKARA I (Born: about 600 in Saurastra India Died: about 680 in Asmaka, India)*

We have very little information about Bhaskara I’s life except what can be deduced from his writings. Shukla deduces from the fact that Bhaskara I often refers to the Asmakatantra instead of the Aryabhatiya that he must have been working in a school of mathematicians in Asmaka which was probably in the Nizamabad District of Andhra Pradesh. If this is correct, and it does seem quite likely, then the school in Asmaka would have been a collection of scholars who were followers of Aryabhata I and of course this fits in well with the fact that Bhaskara I himself was certainly a follower of Aryabhata I.

There are other references to Places in India in Bhaskara’s writings. For example he mentions Valabhi (today Vala), the capital of the Maitraka dynasty in the 7th century, and Sivarajapura, which were both in Saurastra which today is the Gujarat state of India on the west coast of the continent. Also mentioned are Bharuch (or Broach) in southern Gujarat and Thanesar in the eastern Punjab which was ruled by Harsa for 41 years from 606. Harsa was the preeminent ruler in north India through the first half of Bhaskara I’s life. A reasonable guess would be that Bhaskara was born in Saurastra and later moved to Asmaka.

Bhaskara I was an Author of two treatises and commentaries to the work of Aryabhata I. His works are the Mahabhaskariya, the Laghubhaskariya and the Aryabhatiyabhasya. The Mahabhaskariya is an eight chapter work on Indian Mathematical astronomy and includes topics which were fairly standard for such works at this time. It discusses topics such as: the longitudes of the planets; conjunctions of the planets with each other and with bright stars; eclipses of the sun and the moon; risings and settings; and the lunar crescent.

Bhaskara I included in his treatise the Mahabhaskariya three verses which give an Approximation to the trigonometric sine function by means of a rational fraction. These occur in Chapter 7 of the work. The formula which Bhaskara gives is amazingly accurate and use of the formula leads to a maximum error of less than one percent. The formula is

sin x = 16x (π – x)/[5π2 - 4x (π - x)]

and Bhaskara attributes the work as that of Aryabhata I. We have computed the values given by the formula and compared it with the correct value for sin x for x from 0 to π/2 in steps of π/20.

x = 0 formula = 0.00000 sin x = 0.00000 error = 0.00000

x = π/20 formula = 0.15800 sin x = 0.15643 error = 0.00157

x = π/10 formula = 0.31034 sin x = 0.30903 error = 0.00131

x = 3π/20 formula = 0.45434 sin x = 0.45399 error = 0.00035

x = π/5 formula = 0.58716 sin x = 0.58778 error = -0.00062

x = π/4 formula = 0.70588 sin x = 0.70710 error = -0.00122

x = π/10 formula = 0.80769 sin x = 0.80903 error = -0.00134

x = 7π/20 formula = 0.88998 sin x = 0.89103 error = -0.00105

x = 2π/5 formula = 0.95050 sin x = 0.95105 error = -0.00055

x = 9π/20 formula = 0.98753 sin x = 0.98769 error = -0.00016

x = π/2 formula = 1.00000 sin x = 1.00000 error = 0.00000

In 629 Bhaskara I wrote a commentary, the Aryabhatiyabhasya, on the Aryabhatiya by Aryabhata I. The Aryabhatiya contains 33 verses dealing with mathematics, the remainder of the work being concerned with Mathematical astronomy. The commentary by Bhaskara I is only on the 33 verses of mathematics. He considers problems of indeterminate Equations of the first degree and trigonometric formulae. In the course of discussions of the Aryabhatiya, Bhaskara I expressed his idea on how one particular rectangle can be treated as a cyclic quadrilateral. He was the first to open discussion on quadrilaterals with all the four sides unequal and none of the opposite sides parallel.

One of the Approximations used for π for many centuries was √10. Bhaskara I criticised this Approximation. He regretted that an exact measure of the circumference of a circle in terms of diameter was not available and he clearly believed that π was not rational.

In contradict, contradict, contradict and contradict Shukla discusses some features of Bhaskara’s Mathematics such as: numbers and symbolism, the classification of mathematics, the names and solution methods of Equations of the first degree, quadratic equations, cubic Equations and Equations with more than one unknown, symbolic algebra, unusual and special terms in Bhaskara’s work, weights and measures, the Euclidean algorithm method of solving Linear indeterminate equations, examples given by Bhaskara I illustrating Aryabhata I’s rules, certain tables for solving an Equation occurring in astronomy, and reference made by Bhaskara I to the works of earlier Indian mathematicians.

*LALLA (Born: about 720 in India Died: about 790 in India)*

Lalla’s father was Trivikrama Bhatta and Trivikrama’s father, Lalla’s paternal grandfather, was named Samba. Lalla was an Indian astronomer and Mathematician who followed the tradition of Aryabhata I. Lalla’s most famous work was entitled Shishyadhividdhidatantra. This major treatise was in two volumes. The first volume, On the computation of the positions of the planets, was in thirteen chapters and covered topics such as: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; syzygies; risings and settings; the shadow of the moon; the moon’s crescent; conjunctions of the planets with each other; conjunctions of the planets with the fixed stars; the patas of the moon and sun, and a final chapter in the first volume which forms a conclusion.

The second volume was On the sphere. In this volume Lalla examined topics such as: graphical representation; the celestial sphere; the principle of mean motion; the terrestrial sphere; motions and stations of the planets; geography; erroneous knowledge; instruments; and finally selected problems.

In Shishyadhividdhidatantra Lalla uses Sanskrit numerical symbols. Ifrah writes in contradict:-

*… over the centuries, Sanskrit has lent itself admirably to the rules of prosody and versification. This explains why Indian astronomers [like Lalla] favoured the use of Sanskrit numerical symbols, based on a complex symbolism which was extraordinarily fertile and sophisticated, possessing as it did an almost limitless choice of synonyms.*

Despite writing the most famous treatise giving the views of Aryabhata I, Lalla did not accept his Theory given in the Aryabhatiya that the earth rotated. Lalla argues in his commentary, like many other Indian astronomers before him such as Varahamihira and Brahmagupta, that if the earth rotated then the speed would have to be such that one would have to ask how do the bees or birds flying in the sky come back to their nests? In fact Lalla misinterpreted some of Aryabhata I’s statements about the rotating earth. One has to assume that the idea appeared so impossible to him that he just could not appreciate Aryabhata I’s arguments. As Chatterjee writes in contradict, Lalla in his commentary:-

*… did not interpret the relevant verses in the way meant by Aryabhata I. *

Astrology at this time was based on astronomical tables and often the horoscopes allow one to identify the tables used. Some Arabic horoscopes were based on astronomical tables calculated in India. The most frequently used tables were by Aryabhata I. Lalla improved on these tables and he produced a set of corrections for the Moon’s longitude. One aspect of Aryabhata I’s work which Lalla did follow was his value of π. Lalla uses π = 62832/20000, i.e. π = 3.1416 which is a value correct to the fourth Decimal place.

Lalla also wrote a commentary on Khandakhadyaka, a work of Brahmagupta. Lalla’s commentary has not survived but there is another work on astrology by Lalla which has survived, namely the Jyotisaratnakosa. This was a very popular work which was the main one on the subject in India for around 300 years.

*GOVINDASVAMI (Born: about 800 in India Died: about 860 in India)*

Govindasvami (or Govindasvamin) was an Indian Mathematical astronomer whose most famous treatise was a commentary on the Mahabhaskariya of Bhaskara I.

Bhaskara I wrote the Mahabhaskariya in about 600 A. D. It is an eight chapter work on Indian Mathematical astronomy and includes topics which were fairly standard for such works at this time. It discussed topics such as the longitudes of the planets, conjunctions of the planets with each other and with bright stars, eclipses of the sun and the moon, risings and settings, and the lunar crescent.

Govindasvami wrote the Bhasya in about 830 which was a commentary on the Mahabhaskariya. In Govindasvami’s commentary there appear many examples of using a place-value Sanskrit system of numerals. One of the most interesting aspects of the commentary, however, is Govindasvami’s Construction of a sine table.

Indian mathematicians and astronomers constructed sine table with great precision. They were used to calculate the positions of the planets as accurately as possible so had to be computed with high degrees of accuracy. Govindasvami considered the sexagesimal fractional parts of the twenty-four tabular sine differences from the Aryabhatiya. These lead to more correct sine values at intervals of 90 °/24 = 3 °45 ‘. In the commentary Govindasvami found certain other empirical rules relating to computations of sine differences in the argumental range of 60 to 90 degrees. Both of the references contradict and contradict are concerned with the sine tables in Govindasvami’s work.

*MAHAVIRA (Born: about 800 in possibly Mysore, India Died: about 870 in India)*

Mahavira (or Mahaviracharya meaning Mahavira the Teacher) was of the Jaina religion and was familiar with Jaina mathematics. He worked in Mysore in southern Indian where he was a member of a school of mathematics. If he was not born in Mysore then it is very likely that he was born close to this town in the same region of India. We have essentially no other biographical details although we can gain just a little of his personality from the acknowledgement he gives in the introduction to his only known work, see below. However Jain in contradict mentions six other works which he credits to Mahavira and he emphasises the need for further research into identifying the complete list of his works.

The only known book by Mahavira is Ganita Sara Samgraha, dated 850 AD, which was designed as an updating of Brahmagupta’s book. Filliozat writes contradict:-

*This book deals with the teaching of Brahmagupta but contains both simplifications and additional information. … Although like all Indian versified texts, it is extremely condensed, this work, from a pedagogical point of view, has a significant advantage over earlier texts. *

It consisted of nine chapters and included all Mathematical knowledge of mid-ninth century India. It provides us with the bulk of knowledge which we have of Jaina Mathematics and it can be seen as in some sense providing an account of the work of those who developed this mathematics. There were many Indian mathematicians before the time of Mahavira but, perhaps surprisingly, their work on Mathematics is always contained in texts which discuss other topics such as astronomy. The Ganita Sara Samgraha by Mahavira is the earliest Indian text which we possess which is devoted entirely to mathematics.

In the introduction to the work Mahavira paid tribute to the mathematicians whose work formed the basis of his book. These mathematicians included Aryabhata I, Bhaskara I, and Brahmagupta. Mahavira writes:-

*With the help of the accomplished holy sages, who are worthy to be worshipped by the lords of the world … I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are picked from the sea, gold from the stony rock and the pearl from the oyster shell; and I give out according to the power of my intelligence, the Sara Samgraha, a small work on arithmetic, which is however not small in importance. *

The nine chapters of the Ganita Sara Samgraha are:

1. Terminology

2. Arithmetical operations

3. Operations involving fractions

4. Miscellaneous operations

5. Operations involving the rule of three

6. Mixed operations

7. Operations relating to the calculations of areas

8. Operations relating to excavations

9. Operations relating to shadows

Throughout the work a place-value system with nine numerals is used or sometimes Sanskrit numeral symbols are used. Of interest in Chapter 1 regarding the development of a place-value number system is Mahavira’s description of the number 12345654321 which he obtains after a calculation. He describes the number as:-

*… beginning with one which then grows until it reaches six, then decreases in reverse order. *

Notice that this wording makes sense to us using a place-value system but would not make sense in other systems. It is a clear indication that Mahavira is at home with the place-value number system.

Among topics Mahavira discussed in his treatise was operations with fractions including methods to decompose integers and fractions into unit fractions. For example

2/17 = 1/12 + 1/51 + 1/68.

He examined methods of squaring numbers which, although a special case of multiplying two numbers, can be computed using special methods. He also discussed integer Solutions of first degree indeterminate Equation by a method called kuttaka. The kuttaka (or the “pulveriser”) method is based on the use of the Euclidean algorithm but the method of solution also resembles the continued fraction process of Euler given in 1764. The work kuttaka, which occurs in many of the treatises of Indian mathematicians of the classical period, has taken on the more general meaning of “algebra”.

An example of a problem given in the Ganita Sara Samgraha which leads to indeterminate Linear Equations is the following:

*Three merchants find a purse lying in the road. One merchant says “If I keep the purse, I shall have twice as much money as the two of you together”. “Give me the purse and I shall have three times as much” said the second merchant. The third merchant said “I shall be much better off than either of you if I keep the purse, I shall have Five times as much as the two of you together”. How much money is in the purse? How much money does each merchant have?*

If the first merchant has x, the second y, the third z and p is the amount in the purse then

p + x = 2(y + z), p + y = 3(x + z), p + z = 5(x + y).

There is no unique solution but the smallest solution in positive integers is p = 15, x = 1, y = 3, z = 5. Any solution in positive integers is a multiple of this solution as Mahavira claims.

Mahavira gave special rules for the use of permutations and combinations which was a topic of special interest in Jaina mathematics. He also described a process for calculating the volume of a sphere and one for calculating the cube root of a number. He looked at some geometrical results including right-angled triangles with rational sides, see for example contradict.

Mahavira also attempts to solve certain Mathematical problems which had not been studied by other Indian mathematicians. For example, he gave an Approximate formula for the area and the perimeter of an ellipse. In contradict

Hayashi writes:-

*The formulas for a conch-like figure have so far been found only in the works of Mahavira and Narayana. *

It is reasonable to ask what a “conch-like figure” is. It is two unequal semiCircles (with diameters AB and BC) stuck together along their diameters. Although it might be reasonable to suppose that the perimeter might be obtained by considering the semiCircles, Hayashi claims that the formulae obtained:-

*… were most probably obtained not from the two semiCircles AB and BC. *

*PRTHUDAKASVAMI (Born: about 830 in India Died: about 890 in India)*

Prthudakasvami is best known for his work on solving equations.

The solution of a first-degree indeterminate Equation by a method called kuttaka (or “pulveriser”) was given by Aryabhata I. This method of finding integer Solutions resembles the continued fraction process and can also be seen as a use of the Euclidean algorithm.

Brahmagupta seems to have used a method involving continued fractions to find integer Solutions of an indeterminate Equation of the type ax + c = by. Prthudakasvami’s commentary on Brahmagupta’s work is helpful in showing how “algebra”, that is the method of calculating with the unknown, was developing in India. Prthudakasvami discussed the kuttaka method which he renamed as “bijagnita” which means the method of calculating with unknown elements.

To see just how this new idea of algebra was developing in India, we look at the notation which was being used by Prthudakasvami in his commentary on Brahmagupta’s Brahma Sputa Siddhanta. In this commentary Prthudakasvami writes the Equation 10x + 8 = x2 + 1 as:

yava 0 ya 10 ru 8

yava 1 ya 0 ru 1

Here yava is an abbreviation for yavat avad varga which means the “square of the unknown quantity”, ya is an abbreviation for yavat havat which means the “unknown quantity”, and ru is an abbreviation for rupa which means “constant term”. Hence the top row reads

0×2 + 10x + 8

while the second row reads

x2 + 0x + 1

The whole Equation is therefore

0×2 + 10x + 8 = x2 + 0x + 1

or

10x + 8 = x2 + 1.

*SANKARA NARAYANA (Born: about 840 in India Died: about 900 in India)*

Sankara Narayana (or Shankaranarayana) was an Indian astronomer and mathematician. He was a disciple of the astronomer and Mathematician Govindasvami. His most famous work was the Laghubhaskariyavivarana which was a commentary on the Laghubhaskariya of Bhaskara I which in turn is based on the work of Aryabhata I.

The Laghubhaskariyavivarana was written by Sankara Narayana in 869 AD for the Author writes in the text that it is written in the Shaka Year 791 which translates to a date AD by adding 78. It is a text which covers the standard Mathematical methods of Aryabhata I such as the solution of the indeterminate Equation by = ax ± c (a, b, c integers) in integers which is then applied to astronomical problems. The standard Indian method involves using the Euclidean algorithm. It is called kuttakara (“pulveriser”) but the term eventually came to have a more general meaning like “algebra”. The paper contradict examines this method. The reader who is wondering what the determination of “mati” means in the title of the paper contradict then it refers to the optional number in a guessed solution and it is a feature which differs from the original method as presented by Bhaskara I.

Perhaps the most unusual feature of the Laghubhaskariyavivarana is the use of katapayadi numeration as well as the place-value Sanskrit numerals which Sankara Narayana frequently uses. Sankara Narayana is the first Author known to use katapayadi numeration with this name but he did not invent it for it appears to be identical to a system invented earlier which was called varnasamjna. The numeration system varnasamjna was almost certainly invented by the astronomer Haridatta, and it was explained by him in a text which many historians believe was written in 684 but this would contradict what Sankara Narayana himself writes. This point is discussed below. First we should explain ideas behind Sankara Narayana’s katapayadi numeration.

The system is based on writing numbers using the letters of the Indian alphabet. Let us quote from contradict:-

*… the numerical attribution of syllables corresponds to the following rule, according to the regular order of succession of the letters of the Indian alphabet: the first nine letters represent the numbers 1 to 9 while the tenth corresponds to zero; the following nine letters also receive the values 1 to 9 whilst the following letter has the value zero; the next Five represent the first Five units; and the last eight represent the numbers 1 to 8. *

Under this system 1 to 5 are represented by four different letters. For example 1 is represented by the letters ka, ta, pa, ya which give the system its name (ka, ta, pa, ya becomes katapaya). Then 6, 7, 8 are represented by three letters and finally nine and zero are represented by two letters.

The system was a spoken one in the sense that consonants and vowels which are not vocalised have no numerical value. The system is a place-value system with zero but one may reasonably ask why such an apparently complicated numeral system might ever come to be invented. Well the answer must be that it lead to easily remembered mnemonics. In fact many different “words” could represent the same number and this was highly useful for works written in verse as the Indian texts tended to be.

Let us return to the interesting point about the date of Haridatta. Very unusually for an Indian text, Sankara Narayana expresses his thanks to those who have gone before him and developed the ideas about which he is writing. This in itself is not so unusual but the surprise here is that Sankara Narayana claims to give the list in chronological order. His list is

Aryabhata I

Varahamihira

Bhaskara I

Govindasvami

Haridatta

**[Note that we have written Bhaskara I where Sankara Narayana simply wrote Bhaskara. The more famous Bhaskara II lived nearly 300 years after Sankara Narayana.] **

The chronological order in the list agrees with the Dates we have for the first four of these mathematicians. However, putting Haridatta after Govindasvami would seem an unlikely mistake for Sankara Narayana to make if Haridatta really did write his text in 684 since Sankara Narayana was himself a disciple of Govindasvami. If the dating given by Sankara Narayana is correct then katapayadi numeration had been invented only a few years before he wrote his text.

**SRIDHARA (Born: about 870 in possibly Bengal, India Died: about 930 in India)**

Sridhara is now believed to have lived in the ninth and tenth centuries. However, there has been much dispute over his date and in different works the Dates of the life of Sridhara have been placed from the seventh century to the eleventh century. The best present estimate is that he wrote around 900 AD, a date which is deduced from seeing which other pieces of Mathematics he was familiar with and also seeing which later mathematicians were familiar with his work. We do know that Sridhara was a Hindu but little else is known. Two theories exist concerning his birthplace which are far apart. Some historians give Bengal as the place of his birth while other historians believe that Sridhara was born in southern India.

Sridhara is known as the Author of two Mathematical treatises, namely the Trisatika (sometimes called the Patiganitasara ) and the Patiganita. However at least three other works have been attributed to him, namely the Bijaganita, Navasati, and Brhatpati. Information about these books was given the works of Bhaskara II (writing around 1150), Makkibhatta (writing in 1377), and Raghavabhatta (writing in 1493). We give details below of Sridhara’s rule for solving quadratic Equations as given by Bhaskara II.

There is another Mathematical treatise Ganitapancavimsi which some historians believe was written by Sridhara. Hayashi in contradict, however, argues that Sridhara is unlikely to have been the Author of this work in its present form.

The Patiganita is written in verse form. The book begins by giving tables of monetary and metrological units. Following this algorithms are given for carrying out the elementary arithmetical operations, squaring, cubing, and square and cube root extraction, carried out with natural numbers. Through the whole book Sridhara gives methods to solve problems in terse rules in verse form which was the typical style of Indian texts at this time. All the algorithms to carry out arithmetical operations are presented in this way and no proofs are given. Indeed there is no suggestion that Sridhara realised that proofs are in any way necessary. Often after stating a rule Sridhara gives one or more numerical examples, but he does not give Solutions to these example nor does he even give answers in this work.

After giving the rules for computing with natural numbers, Sridhara gives rules for operating with rational fractions. He gives a wide variety of applications including problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns. Some of the examples are decidedly non-trivial and one has to consider this as a really advanced work. Other topics covered by the Author include the rule for calculating the number of combinations of n things taken m at a time. There are sections of the book devoted to arithmetic and Geometric progressions, including progressions with a fractional numbers of terms, and formulae for the sum of certain finite series are given.

The book ends by giving rules, some of which are only approximate, for the areas of a some plane polygons. In fact the text breaks off at this point but it certainly was not the end of the book which is missing in the only copy of the work which has survived. We do know something of the missing part, however, for the Patiganitasara is a summary of the Patiganita including the missing portion.

In contradict Shukla examines Sridhara’s method for finding rational Solutions of Nx2 ± 1 = y2, 1 – Nx2 = y2, Nx2 ± C = y2, and C – Nx2 = y2 which Sridhara gives in the Patiganita. Shukla states that the rules given there are different from those given by other Hindu mathematicians.

Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation. Unfortunately, as we indicated above, the original is lost and we have to rely on a quotation of Sridhara’s rule from Bhaskara II:-

*Multiply both sides of the Equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root.*

To see what this means take

ax2 + bx = c.

Multiply both sides by 4a to get

4a2x2 + 4abx = 4ac

then add b2 to both sides to get

4a2x2 + 4abx + b2= 4ac + b2

and, taking the square root

2ax + b = √(4ac + b2).

There is no suggestion that Sridhara took two values when he took the square root.

*ARYABHATA II (Born: about 920 in India Died: about 1000 in India)*

Essentially nothing is known of the life of Aryabhata II. Historians have argued about his date and have come up with many different theories. In contradict Pingree gives the date for his main publications as being between 950 and 1100. This is deduced from the usual arguments such as which authors Aryabhata II refers to and which refer to him. G R Kaye argued in 1910 that Aryabhata II lived before al-Biruni but Datta contradict in 1926 showed that these Dates were too early.

The article contradict argues for a date of about 950 for Aryabhata II’s main work, the Mahasiddhanta, but R Billiard has proposed a date for Aryabhata II in the sixteenth century. Most modern historians, however, consider the most likely Dates for his main work as around 950 and we have given very Approximate Dates for his birth and Death based on this hypothesis. See contradict for a fairly recent discussion of this topic.

The most famous work by Aryabhata II is the Mahasiddhanta which consists of eighteen chapters. The treatise is written in Sanskrit verse and the first twelve chapters form a treatise on Mathematical astronomy covering the usual topics that Indian mathematicians worked on during this period. The topics included in these twelve chapters are: the longitudes of the planets, eclipses of the sun and moon, the projection of eclipses, the lunar crescent, the rising and setting of the planets, conjunctions of the planets with each other and with the stars.

The remaining six chapters of the Mahasiddhanta form a separate part entitled On the sphere. It discusses topics such as geometry, geography and algebra with applications to the longitudes of the planets.

In Mahasiddhanta Aryabhata II gives in about twenty verses detailed rules to solve the indeterminate equation: by = ax + c. The rules apply in a number of different cases such as when c is positive, when c is negative, when the number of the quotients of the mutual divisions is even, when this number of quotients is odd, etc. Details of Aryabhata II’s method are given in contradict.

Aryabhata II also gave a method to calculate the cube root of a number, but his method was not new, being based on that given many years earlier by Aryabhata I, see for example contradict.

Aryabhata II constructed a sine table correct up to Five Decimal Places when measured in Decimal parts of the radius, see contradict. Indian mathematicians were very interested in giving accurate sine tables since they were used to calculate the planetary positions as accurately as possible.

**VIJAYANANDI (Born: about 940 in Benares (now Varanasi), India Died: about 1010 in India)**

Vijayanandi (or Vijayanandin) was the son of Jayananda. He was born into the Brahman caste which meant he was from the highest ranking caste of Hindu Priests. He was an Indian Mathematician and astronomer whose most famous work was the Karanatilaka. We should note that there was another astronomer named Vijayanandi who was mentioned by Varahamihira in one of his works. Since Varahamihira wrote around 550 and the Karanatilaka was written around 966, there must be two astronomers both named “Vijayanandi”.

The Karanatilaka has not survived in its original form but we know of the text through an Arabic translation by al-Biruni. It is a work in fourteen chapters covering the standard topics of Indian astronomy. It deals with the topics of: units of time measurement; mean and true longitudes of the sun and moon; the length of daylight; mean longitudes of the Five planets; true longitudes of the Five planets; the three problems of diurnal rotation; lunar eclipses, solar eclipses; the projection of eclipses; first visibility of the planets; conjunctions of the planets with each other and with fixed stars; the moon’s crescent; and the patas of the moon and sun.

The Indians had a cosmology which was based on long periods of time with astronomical events occurring a certain whole number of times within the cycles. This system led to much work on integer Solutions of Equations and their application to astronomy. In particular there was, according to Aryabhata I, a basic period of 4320000 years called a mahayuga and it was assumed that the sun, the moon, their apsis and node, and the planets reached perfect conjunctions after this period. Hence it was assumed that the periods were rational multiples of each other.

All the planets and the node and apsis of the moon and sun had to have an integer number of revolutions in the mahayuga. Many Indian astronomers proposed different values for these integral numbers of revolutions. For the number of revolutions of the apsis and node of the moon per mahayuga, Aryabhata I proposed 488219 and 232226, respectively.

However Vijayanandi changed these numbers to 488211 and 232234. The reasons for giving the new numbers is unclear. Like other Indian astronomers, Vijayanandi made contributions to trigonometry and it appears that his calculation of the periods was computed by using tables of sines and versed sines. It is significant that accuracy was need in trigonometric tables to give accurate astronomical theories and this motivated many of the Indian mathematicians to produce more accurate methods of approximating entries in tables.

*SRIPATI (Born: 1019 in (probably) Rohinikhanda, Maharashtra, India Died: 1066 in India)*

Sripati’s father was Nagadeva (sometimes written as Namadeva) and Nagadeva’s father, Sripati’s paternal grandfather, was Kesava. Sripati was a follower of the teaching of Lalla writing on astrology, astronomy and mathematics. His Mathematical work was undertaken with applications to astronomy in mind, for example a study of spheres. His work on astronomy was undertaken to provide a basis for his astrology. Sripati was the most prominent Indian mathematicians of the 11th Century.

Among Sripati’s works are: Dhikotidakarana written in 1039, a work of twenty verses on solar and lunar eclipses; Dhruvamanasa written in 1056, a work of 105 verses on calculating planetary longitudes, eclipses and planetary transits; Siddhantasekhara a major work on astronomy in 19 chapters; and Ganitatilaka an incomplete arithmetical treatise in 125 verses based on a work by Sridhara.

The titles of Chapters 13, 14, and 15 of the Siddhantasekhara are Arithmetic, Algebra and On the Sphere. Chapter 13 consists of 55 verses on arithmetic, mensuration, and shadow reckoning. It is probable that the lost portion of the arithmetic treatise Ganitatilaka consisted essentially of verses 19-55 of this chapter. The 37 verses of Chapter 14 on algebra state various rules of algebra without proof. These are given in verbal form without Algebraic symbols. In verses 3, 4 and 5 of this chapter Sripati gave the rules of signs for addition, subtraction, multiplication, division, square, square root, cube and cube root of positive and negative quantities. His work on Equations in this chapter contains the rule for solving a quadratic Equation and, more impressively, he gives the identity:

√(x + √y) = √[(x + √(x2 - y)]/2 + √[(x - √(x2 - y)]/2)

Other Mathematics included in Sripati’s work includes, in particular, rules for the solution of simultaneous indeterminate Equations of the first degree that are similar to those given by Brahmagupta

Sripati obtained more fame in astrology than in other areas and it is fair to say that he considered this to be his most important contributions. He wrote the Jyotisaratnamala which was an astrology text in twenty chapters based on the Jyotisaratnakosa of Lalla. Sripati wrote a commentary on this work in Marathi and it is one of the oldest works to have survived that is written in that language. Marathi is the oldest of the regional languages in Indo-Aryan, dating from about 1000.

Another work on astrology written by Sripati is the Jatakapaddhati or Sripatipaddhati which is in eight chapters and is contradict:-

*… one of the fundamental textbooks for later Indian genethlialogy, contributing an impressive elaboration to the computation of the strengths of the planets and astrological Places. It was enormously popular, as the large number of manuscripts, commentaries, and imitations attests. *

Genethlialogy was the science of casting nativities and it was the earliest branch of astrology which claimed to be able to predict the course of a person’s life based on the positions of the planets and of the signs of the zodiac at the moment the person was born or conceived.

There is one other work on astrology the Daivajnavallabha which some historians claim was written by Sripati while other claim that it is the work of Varahamihira. As yet nobody has come up with a definite case to show which of these two is the author, or even whether the Author is another astrologer.

*Brahmadeva (Born: about 1060 in (possibly) Mathura, India Died: about 1130 in India)*

Brahmadeva was the son of Candrabudha. The family came from the Mathura district of Uttar Pradesh in northern India. He was born into the Brahman caste which meant he was from the highest ranking caste of Hindu Priests. The Brahman caste had a strong tradition of education so Brahmadeva would have received one of the best educations of men of his time.

We have only one work by Brahmadeva and this is Karanaprakasa which is a commentary on the Aryabhatiya by Aryabhata I. Brahmadeva’s work is in nine chapters and it follows the contents of the original Aryabhatiya. Topics covered include the longitudes of the planets, problems relating to the daily rotation of the heavens, eclipses of the sun and the moon, risings and settings, the lunar crescent, and conjunctions of the planets.

The work contains some contributions to trigonometry, motivated by its application to Mathematical astronomy. It is this aspect of the work which is mentioned by Gupta in contradict.

*Different commentaries on the Aryabhatiya achieved popularity in different parts of India. Brahmadeva’s commentary seems to have been particularly popular in Madras, Mysore and Maharastra. The more important commentaries on the Aryabhatiya became the basis for further commentaries and indeed this is what happened to the Karanaprakasa. Commentaries on Brahmadeva’s work continued to appear up to the seventeenth century. *

*Bhaskara (Born: 1114 in Vijayapura, India Died: 1185 in Ujjain, India)*

Bhaskara is also known as Bhaskara II or as Bhaskaracharya, this latter name meaning “Bhaskara the Teacher”. Since he is known in India as Bhaskaracharya we will refer to him throughout this article by that name. Bhaskaracharya’s father was a Brahman named Mahesvara. Mahesvara himself was famed as an astrologer. This happened frequently in Indian society with generations of a family being excellent mathematicians and often acting as teachers to other family members.

Bhaskaracharya became head of the astronomical observatory at Ujjain, the leading Mathematical centre in India at that time. Outstanding mathematicians such as Varahamihira and Brahmagupta had worked there and built up a strong school of Mathematical astronomy.

In many ways Bhaskaracharya represents the peak of Mathematical knowledge in the 12th century. He reached an understanding of the number systems and solving Equations which was not to be achieved in Europe for several centuries.

Six works by Bhaskaracharya are known but a seventh work, which is claimed to be by him, is thought by many historians to be a late forgery. The six works are: Lilavati (The Beautiful) which is on mathematics; Bijaganita (Seed Counting or Root Extraction) which is on algebra; the Siddhantasiromani which is in two parts, the first on Mathematical astronomy with the second part on the sphere; the Vasanabhasya of Mitaksara which is Bhaskaracharya’s own commentary on the Siddhantasiromani ; the Karanakutuhala (Calculation of Astronomical Wonders) or Brahmatulya which is a simplified version of the Siddhantasiromani ; and the Vivarana which is a commentary on the Shishyadhividdhidatantra of Lalla. It is the first three of these works which are the most interesting, certainly from the point of view of mathematics, and we will concentrate on the contents of these.

Given that he was building on the knowledge and understanding of Brahmagupta it is not surprising that Bhaskaracharya understood about zero and negative numbers. However his understanding went further even than that of Brahmagupta. To give some examples before we examine his work in a little more detail we note that he knew that x2 = 9 had two solutions. He also gave the formula

Bhaskaracharya studied Pell’s Equation px2 + 1 = y2 for p = 8, 11, 32, 61 and 67. When p = 61 he found the Solutions x = 226153980, y = 1776319049. When p = 67 he found the Solutions x = 5967, y = 48842. He studied many Diophantine problems.

Let us first examine the Lilavati. First it is worth repeating the story told by Fyzi who translated this work into Persian in 1587. We give the story as given by Joseph in contradict:-

*Lilavati was the name of Bhaskaracharya’s daughter. From casting her horoscope, he discovered that the auspicious time for her wedding would be a particular hour on a certain day. He placed a cup with a small hole at the bottom of the vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. When everything was ready and the cup was placed in the vessel, Lilavati suddenly out of curiosity bent over the vessel and a pearl from her dress fell into the cup and blocked the hole in it. The lucky hour passed without the cup sinking. Bhaskaracharya believed that the way to console his dejected daughter, who now would never get married, was to write her a manual of mathematics! *

*This is a charming story but it is hard to see that there is any evidence for it being true. It is not even certain that Lilavati was Bhaskaracharya’s daughter. There is also a Theory that Lilavati was Bhaskaracharya’s wife. The topics covered in the thirteen chapters of the book are: definitions; arithmetical terms; interest; arithmetical and geometrical progressions; plane geometry; solid geometry; the shadow of the gnomon; the kuttaka; combinations. *

In dealing with numbers Bhaskaracharya, like Brahmagupta before him, handled efficiently arithmetic involving negative numbers. He is sound in addition, subtraction and multiplication involving zero but realised that there were problems with Brahmagupta’s ideas of dividing by zero. Madhukar Mallayya in contradict argues that the zero used by Bhaskaracharya in his rule (a.0)/0 = a, given in Lilavati, is equivalent to the modern concept of a non-zero “infinitesimal”. Although this claim is not without foundation, perhaps it is seeing ideas beyond what Bhaskaracharya intended.

Bhaskaracharya gave two methods of multiplication in his Lilavati. We follow Ifrah who explains these two methods due to Bhaskaracharya in contradict. To multiply 325 by 243 Bhaskaracharya writes the numbers thus:

243 243 243

3 2 5

——————-

Now working with the rightmost of the three sums he computed 5 times 3 then 5 times 2 missing out the 5 times 4 which he did last and wrote beneath the others one place to the left. Note that this avoids making the “carry” in ones head.

243 243 243

3 2 5

——————-

1015

20

——————-

Now add the 1015 and 20 so positioned and write the answer under the second line below the sum next to the left.

243 243 243

3 2 5

——————-

1015

20

——————-

1215

Work out the middle sum as the right-hand one, again avoiding the “carry”, and add them writing the answer below the 1215 but displaced one place to the left.

243 243 243

3 2 5

——————-

4 6 1015

8 20

——————-

1215

486

Finally work out the left most sum in the same way and again place the resulting addition one place to the left under the 486.

243 243 243

3 2 5

——————-

6 9 4 6 1015

12 8 20

——————-

1215

486

729

——————-

Finally add the three numbers below the second line to obtain the answer 78975.

243 243 243

3 2 5

——————-

6 9 4 6 1015

12 8 20

——————-

1215

486

729

——————-

78975

Despite avoiding the “carry” in the first stages, of course one is still faced with the “carry” in this final addition.

The second of Bhaskaracharya’s methods proceeds as follows:

325

243

——–

Multiply the bottom number by the top number starting with the left-most digit and proceeding towards the right. Displace each row one place to start one place further right than the previous line. First step

325

243

——–

729

Second step

325

243

——–

729

486

Third step, then add

325

243

——–

729

486

1215

——–

78975

Bhaskaracharya, like many of the Indian mathematicians, considered squaring of numbers as special cases of multiplication which deserved special methods. He gave four such methods of squaring in Lilavati.

*Here is an example of explanation of inverse proportion taken from Chapter 3 of the Lilavati. Bhaskaracharya writes:-
In the inverse method, the operation is reversed. That is the fruit to be multiplied by the augment and divided by the demand. When fruit increases or decreases, as the demand is augmented or diminished, the direct rule is used. Else the inverse. *

*Rule of three inverse: If the fruit diminish as the requisition increases, or augment as that decreases, they, who are skilled in accounts, consider the rule of three to be inverted. When there is a diminution of fruit, if there be increase of requisition, and increase of fruit if there be diminution of requisition, then the inverse rule of three is employed.
As well as the rule of three, Bhaskaracharya discusses examples to illustrate rules of compound proportions, such as the rule of Five (Pancarasika), the rule of seven (Saptarasika), the rule of nine (Navarasika), etc. Bhaskaracharya’s examples of using these rules are discussed in contradict. *

An example from Chapter 5 on arithmetical and geometrical progressions is the following:-

**Example: On an expedition to seize his enemy’s elephants, a king marched two yojanas the first day. Say, intelligent calculator, with what increasing rate of daily march did he proceed, since he reached his foe’s city, a distance of eighty yojanas, in a week? **

Bhaskaracharya shows that each day he must travel 22/7 yojanas further than the previous day to reach his foe’s city in 7 days.

An example from Chapter 12 on the kuttaka method of solving indeterminate Equations is the following:-

*Example: Say quickly, mathematician, what is that multiplier, by which two hundred and twenty-one being multiplied, and sixty-Five added to the product, the sum divided by a hundred and ninety-Five becomes exhausted. *

Bhaskaracharya is finding integer solution to 195x = 221y + 65. He obtains the Solutions (x, y) = (6, 5) or (23, 20) or (40, 35) and so on.

In the final chapter on combinations Bhaskaracharya considers the following problem. Let an n-digit number be represented in the usual Decimal form as

d1d2… dn (*)

where each digit satisfies 1 ≤ dj ≤ 9, j = 1, 2, … , n. Then Bhaskaracharya’s problem is to find the total number of numbers of the form (*) that satisfy

d1 + d2 + … + dn = S.

In his conclusion to Lilavati Bhaskaracharya writes:-

**Joy and happiness is indeed ever increasing in this world for those who have Lilavati clasped to their throats, decorated as the members are with neat reduction of fractions, multiplication and involution, pure and perfect as are the solutions, and tasteful as is the speech which is exemplified. **

The Bijaganita is a work in twelve chapters. The topics are: positive and negative numbers; zero; the unknown; surds; the kuttaka; indeterminate quadratic equations; simple equations; quadratic equations; Equations with more than one unknown; quadratic Equations with more than one unknown; operations with products of several unknowns; and the Author and his work.

Having explained how to do arithmetic with negative numbers, Bhaskaracharya gives problems to test the abilities of the reader on calculating with negative and affirmative quantities:-

*Example: Tell quickly the result of the numbers three and four, negative or affirmative, taken together; that is, affirmative and negative, or both negative or both affirmative, as separate instances; if thou know the addition of affirmative and negative quantities. *

Negative numbers are denoted by placing a dot above them:-

**The characters, denoting the quantities known and unknown, should be first written to indicate them generally; and those, which become negative should be then marked with a dot over them.
**

*Example: Subtracting two from three, affirmative from affirmative, and negative from negative, or the contrary, tell me quickly the result … *

In Bijaganita Bhaskaracharya attempted to improve on Brahmagupta’s attempt to divide by zero (and his own description in Lilavati ) when he wrote:-

**A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth. **

So Bhaskaracharya tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskaracharya has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero.

Equations leading to more than one solution are given by Bhaskaracharya:-

*Example: Inside a forest, a number of apes equal to the square of one-eighth of the total apes in the pack are playing noisy games. The remaining twelve apes, who are of a more serious disposition, are on a nearby hill and irritated by the shrieks coming from the forest. What is the total number of apes in the pack? *

The problem leads to a quadratic Equation and Bhaskaracharya says that the two solutions, namely 16 and 48, are equally admissible.

The kuttaka method to solve indeterminate Equations is applied to Equations with three unknowns. The problem is to find integer Solutions to an Equation of the form ax + by + cz = d. An example he gives is:-

**Example: The horses belonging to four men are 5, 3, 6 and 8. The camels belonging to the same men are 2, 7, 4 and 1. The mules belonging to them are 8, 2, 1 and 3 and the oxen are 7, 1, 2 and 1. all four men have equal fortunes. Tell me quickly the price of each horse, camel, mule and ox. **

Of course such problems do not have a unique solution as Bhaskaracharya is fully aware. He finds one solution, which is the minimum, namely horses 85, camels 76, mules 31 and oxen 4.

Bhaskaracharya’s conclusion to the Bijaganita is fascinating for the insight it gives us into the mind of this great mathematician:-

*A morsel of tuition conveys knowledge to a comprehensive mind; and having reached it, expands of its own impulse, as oil poured upon water, as a secret entrusted to the vile, as alms bestowed upon the worthy, however little, so does knowledge infused into a wise mind spread by intrinsic force. *

*It is apparent to men of clear understanding, that the rule of three terms constitutes arithmetic and sagacity constitutes algebra. Accordingly I have said … The rule of three terms is arithmetic; spotless understanding is algebra. What is there unknown to the intelligent? Therefore for the dull alone it is set forth. *

The Siddhantasiromani is a Mathematical astronomy text similar in layout to many other Indian astronomy texts of this and earlier periods. The twelve chapters of the first part cover topics such as: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; syzygies; lunar eclipses; solar eclipses; latitudes of the planets; risings and settings; the moon’s crescent; conjunctions of the planets with each other; conjunctions of the planets with the fixed stars; and the patas of the sun and moon.

The second part contains thirteen chapters on the sphere. It covers topics such as: praise of study of the sphere; nature of the sphere; cosmography and geography; planetary mean motion; eccentric epicyclic model of the planets; the armillary sphere; spherical trigonometry; ellipse calculations; first visibilities of the planets; calculating the lunar crescent; astronomical instruments; the seasons; and problems of astronomical calculations.

There are interesting results on trigonometry in this work. In particular Bhaskaracharya seems more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskaracharya are:

sin(a + b) = sin a cos b + cos a sin b

and

sin(a – b) = sin a cos b – cos a sin b.

Bhaskaracharya rightly achieved an outstanding reputation for his remarkable contribution. In 1207 an educational institution was set up to study Bhaskaracharya’s works. A medieval inscription in an Indian temple reads:-

*Triumphant is the illustrious Bhaskaracharya whose feats are revered by both the wise and the learned. A poet endowed with fame and Religious merit, he is like the crest on a peacock. *

*Mahendra Suri (Born: about 1340 in Western India Died: about 1410 in India)*

Mahendra Suri was a Jain. Jainism began around the sixth century BC and the religion had a strong influence on Mathematics particularly in the last couple of centuries BC. By the time of Mahendra Suri, however, Jainism had lost support as a national religion and was much less vigorous. It had been influenced by Islam and in particular Islamic astronomy came to form a part of the background. However, Pingree in contradict writes that this filtering of Islamic astronomy into Indian culture was:-

*… not allowed to affect in any way the structure of the traditional science. *

Mahendra Suri was a pupil of Madana Suri. He is famed as the first person to write a Sanskrit treatise on the astrolabe. Ohashi writes in contradict of the early history of the astrolabe in the Delhi Sultanate in India:-

**The astrolabe was introduced into India at the time of Firuz Shah Tughluq (reign AD 1351 – 88), and Mahendra Suri wrote the first Sanskrit treatise on the astrolabe entitled Yantraraja (AD 1370). **

The Delhi Sultanate was established around 1200 and from that time on Muslim culture flourished in India. The ideas of Islamic astronomy began to appear in works in the Sanskrit Language and it is the Islamic ideas on the astrolabe which Mahendra Suri wrote on in his famous text. It is clear from the various references in the text and also from the particular values that Mahendra Suri uses for the angle of the ecliptic etc. that his work is based on Islamic rather than traditional Indian astronomy works.

**Narayana Pandit (Born: about 1340 in India Died: about 1400 in India)**

Narayana was the son of Nrsimha (sometimes written Narasimha). We know that he wrote his most famous work Ganita Kaumudi on arithmetic in 1356 but little else is known of him. His Mathematical writings show that he was strongly influenced by Bhaskara II and he wrote a commentary on the Lilavati of Bhaskara II called Karmapradipika. Some historians dispute that Narayana is the Author of this commentary which they attribute to Madhava.

In the Ganita Kaumudi Narayana considers the Mathematical operation on numbers. Like many other Indian writers of arithmetics before him he considered an algorithm for multiplying numbers and he then looked at the special case of squaring numbers. One of the unusual features of Narayana’s work Karmapradipika is that he gave seven methods of squaring numbers which are not found in the work of other Indian mathematicians.

He discussed another standard topic for Indian mathematicians namely that of finding triangles whose sides had integral values. In particular he gave a rule of finding integral triangles whose sides differ by one unit of length and which contain a pair of right-angled triangles having integral sides with a common integral height. In terms of geometry Narayana gave a rule for a Segment of a circle. Narayana contradict:-

*… derived his rule for a Segment of a circle from Mahavira’s rule for an ‘elongated circle’ or an ellipse-like figure. *

Narayana also gave a rule to calculate Approximate values of a square root. He did this by using an indeterminate Equation of the second order, Nx2 + 1 = y2, where N is the number whose square root is to be calculated. If x and y are a pair of roots of this Equation with x < y then √N is approximately equal to y/x. To illustrate this method Narayana takes N = 10. He then finds the Solutions x = 6, y = 19 which give the Approximation 19/6 = 3.1666666666666666667, which is correct to 2 Decimal Places. Narayana then gives the Solutions x = 228, y = 721 which give the Approximation 721/228 = 3.1622807017543859649, correct to four Places. Finally Narayana gives the pair of Solutions x = 8658, y = 227379 which give the Approximation 227379/8658 = 3.1622776622776622777, correct to eight Decimal Places. Note for comparison that √10 is, correct to 20 Places, 3.1622776601683793320. See contradict for more information.

The thirteenth chapter of Ganita Kaumudi was called Net of Numbers and was devoted to number sequences. For example, he discussed some problems concerning arithmetic progressions.

The fourteenth chapter (which is the last one) of Naryana's Ganita Kaumudi contains a detailed discussion of magic Squares and similar figures. Narayana gave the rules for the formation of doubly even, even and odd perfect magic Squares along with magic triangles, Rectangles and Circles. He used formulae and rules for the relations between magic Squares and arithmetic series. He gave methods for finding "the horizontal difference" and the first term of a magic square whose square's constant and the number of terms are given and he also gave rules for finding "the vertical difference" in the case where this information is given.

*Madhava of SangamaGramma (Born: 1350 in SangamaGramma (near Cochin), Kerala, India Died: 1425 in India)*

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 contributions of Madhava have come to light. In contradict 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 contradict) 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 contradict.

Jyesthadeva wrote Yukti-Bhasa in Malayalam, the regional Language of Kerala, around 1550. In contradict Gupta gives a translation of the text and this is also given in contradict 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 contradict 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 contradict:-

*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. *

*Paramesvara (Born: about 1370 in Alattur, Kerala, India Died: about 1460 in India)*

Paramesvara was an Indian astronomer and Mathematician who wrote many commentaries on earlier works as well as making many observations. Although his father has not been identified, we know that Paramesvara was born into a Namputiri Brahmana family who were astrologers and astronomers. The family home was Vatasseri (sometimes called Vatasreni) in the village of Alattur. This village was in Kerala and Paramesvara himself gives its coordinates with respect to Ujjain. This puts it at latitude 10° 51′ north. It is on the north bank of the river Nila at its mouth.

From Paramesvara’s writing we know that Rudra was his teacher, and Nilakantha, who knew Paramesvara personally, tells us that Paramesvara’s teachers included Madhava and Narayana. We can be fairly confident that the Dates we have given for Paramesvara are roughly correct since he made eclipe observations over a period of 55 years. We will say a little more about these observations below. He played an important part in the remarkable developments in Mathematics which took place in Kerala in the late 14th and early part of the 15th century.

The commentaries by Paramesvara on a number of works have been published. For example the Karmadipika is a commentary on the Mahabhaskariyam, an astronomical and Mathematical work by Bhaskara I, and its text is given in contradict. In contradict the text of Paramesvara’s commentary on the Laghubhaskariyam of Bhaskara I is given. Munjala wrote the astronomical work Laghumanasam in the Year 932 and Paramesvara wrote a commentary (see contradict). It is a work containing typical topics for Indian Mathematical astronomy works of this period: the mean motions of the heavenly bodies; the true motions of the heavenly bodies; miscellaneous Mathematical rules; the systems of coordinates, direction, place and time; eclipses of the sun and the moon; and the operation for apparent longitude.

Aryabhata gave a rule for determining the height of a pole from the lengths of its shadows in the Aryabhatiya. Paramesvara gave several illustrative examples of the method in his commentary on the Aryabhatiya.

Like many mathematicians from Kerala, Madhava clearly had a very strong influence on Paramesvara. One can see throughout his work that it is teachings by Madhava which direct much of Paramesvara’s Mathematical ideas. One of Paramesvara’s most remarkable Mathematical discoveries, no doubt influenced by Madhava, was a version of the mean value theorem. He states the theorem in his commentary Lilavati Bhasya on Bhaskara II’s Lilavati. There are other examples of versions of the mean value theorem in Paramesvara’s work which we now consider.

The Siddhantadipika by Paramesvara is a commentary on the commentary of Govindasvami on Bhaskara I’s Mahabhaskariya. Paramesvara gives some of his eclipse observations in this work including one made at Navaksetra in 1422 and two made at Gokarna in 1425 and 1430. This work also contains a mean value type formula for inverse interpolation of the sine. It presents a one-point iterative technique for calculating the sine of a given angle. In the Siddhantadipika Paramesvara also gives a more efficient Approximation that works using a two-point iterative algorithm which turns out to be essentially the same as the modern secant method. See contradict and contradict for further details.

The expression for the radius of the circle in which a cyclic quadrilateral is inscribed, given in terms of the sides of the quadrilateral, is usually attributed to Lhuilier in 1782. However Paramesvara described the rule 350 years earlier. If the sides of the cyclic quadrilateral are a, b, c and d then the radius r of the circumscribed circle was given by Paramesvara as:

r2 = x/y where

x = (ab + cd) (ac + bd) (ad + bc)

and y = (a + b + c – d) (b + c + d – a) (c + d + a – b) (d + a + b – c).

The original text by Paramesvara describing the rule is given in contradict.

Paramesvara made a series of eclipse observations between 1393 and 1432 which we have referred to above. The last observation which we know he made was in 1445 but Nilakantha quotes a verse by Paramesvara in which he claims to have made observations spanning 55 years. The known observatons by Paramesvara do not quite square with this statement, there being a discrepancy of three years. Although we do not know when Paramesvara died we do know, again from Nilakantha, that the two knew each other personally. Since we have a definite date for Nilakantha’s birth of 1444 it is hard to believe that Paramesvara died before 1460.

Using his observations, Paramesvara made revisions of the planetary parameters and, like many other Indian astronomers, he constantly attempted to compare the theoretically computed positions of the planets with those which he actually observed. He was keen to improve the theoretical model to bring it into as close an agreement with observations as possible.

**Nilakantha Somayaji (Born: 14 June 1444 in Trkkantiyur (near Tirur), Kerala, India Died: 1544 in India)**

Nilakantha was born into a Namputiri Brahmin family which came from South Malabar in Kerala. The Nambudiri is the main caste of Kerala. It is an orthodox caste whose members consider themselves descendants of the ancient Vedic religion.

He was born in a house called Kelallur which it is claimed coincides with the present Etamana in the village of Trkkantiyur near Tirur in south India. His father was Jatavedas and the family belonged to the Gargya gotra, which was a Indian caste that prohibits marriage to anyone outside the caste. The family followed the Ashvalayana sutra which was a manual of sacrificial ceremonies in the Rigveda, a collection of Vedic hymns. He worshipped the personified deity Soma who was the “master of plants” and the healer of disease. This explains the name Somayaji which means he was from a family qualified to conduct the Soma ritual.

Nilakantha studied astronomy and Vedanta, one of the six orthodox systems of Indian Hindu philosophy, under the Teacher Ravi. He was also taught by Damodra who was the son of Paramesvara. Paramesvara was a famous Indian astronomer and Damodra followed his father’s teachings. This led Nilakantha also to become a follower of Paramesvara. A number of texts on Mathematical astronomy written by Nilakantha have survived. In all he wrote about ten treatises on astronomy.

The Tantrasamgraha is his major astronomy treatise written in 1501. It 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 Tantrasamgraha is very important in terms of the Mathematics Nilakantha uses. In particular he uses results discovered by Madhava and it is an important source of the remarkable Mathematical results which he discovered. However, Nilakantha does not just use Madhava’s results, he extends them and improves them. An anonymous commentary entitled Tantrasangraha-vakhya appeared and, somewhat later in about 1550, Jyesthadeva published a commentary entitled Yuktibhasa that contained proofs of the earlier results by Madhava and Nilakantha. This is quite unusual for an Indian text in giving Mathematical proofs.

The series π/4 = 1 – 1/3 + 1/5 – 1/7 + … is a special case of the series representation for arctan, namely

tan-1x = x – x3/3 + x5/5 – x7/7 + …

It is well known that one simply puts x = 1 to obtain the series for π/4. The Author of contradict reports on the appearance of these series in the work of Leibniz and James Gregory from the 1670s. The contributions of the two European mathematicians to this series are well known but in contradict the results on this series in the work of Madhava nearly three hundred years earlier as presented by Nilakantha in the Tantrasamgraha is also discussed.

Nilakantha derived the series expansion

tan-1x = x – x3/3 + x5/5 – x7/7 + …

by obtaining an Approximate expression for an arc of the circumference of a circle and then considering the limit. An interesting feature of his work was his introduction of several additional series for π/4 that converged more rapidly than

π/4 = 1 – 1/3 + 1/5 – 1/7+ … .

The Author of contradict provides a Reconstruction of how he may have arrived at these results based on the assumption that he possessed a certain continued fraction representation for the tail series

1/(n+2) – 1/(n+4) + 1/(n+6) – 1/(n+8) + …. .

The Tantrasamgraha is not the only work of Nilakantha of which we have the text. He also wrote Golasara which is written in fifty-six Sanskrit verses and shows how Mathematical computations are used to calculate astronomical data. The Siddhanta Darpana is written in thirty-two Sanskrit verses and describes a planetary model. The Candracchayaganita is written in thirty-one Sanskrit verses and explains the computational methods used to calculate the moon’s zenith distance.

The head of the Nambudiri caste in Nilakantha’s time was Netranarayana and he became Nilakantha’s patron for another of his major works, namely the Aryabhatiyabhasya which is a commentary on the Aryabhatiya of Aryabhata I. In this work Nilakantha refers to two eclipses which he observed, the first on 6 March 1467 and the second on 28 July 1501 at Anantaksetra. Nilakantha also refers in the Aryabhatiyabhasya to other works which he wrote such as the Grahanirnaya on eclipses which have not survived.

*Jyesthadeva (Born: about 1500 in Kerala, India Died: about 1575 in Kerala, India)*

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 contradict Gupta gives a translation of the text and this is also given in contradict 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.

*Kamalakara (Born: about 1616 in Benares (now Varanasi), India Died: about 1700 in India)*

Kamalakara was an Indian astronomer and Mathematician who came from a family of famous astronomers. Kamalakara’s father was Nrsimha who was born in 1586. Two of Kamalakara’s three brothers were also famous astronomer/ mathematicians, these being Divakara, who was the eldest of the brothers born in 1606, and Ranganatha who was younger than Kamalakara.

As was common throughout the classical period of Indian mathematics, members of the family acted as teachers to other family members. In particular Kamalakara was taught by his elder brother Divakara while Divakara himself had been taught by their uncle Siva. Pingree writes in contradict:-

*[Kamalakara] combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists (especially Ulugh Beg). Following his family’s tradition he wrote a commentary, Manorama, on Ganesa’s Grahalaghava and, like his father, Nrsimha, another commentary on the Suryasiddhanta, called the Vasanabhasya … *

Kamalakara’s most famous work, the Siddhanta-tattva-viveka, was commented on by Kamalakara himself. The work was completed in 1658. It is a work of fifteen chapters covering standard topics for Indian astronomy texts at this time. It deals with the topics of: units of time measurement; mean motions of the planets; true longitudes of the planets; the three problems of diurnal rotation; diameters and distances of the planets; the earth’s shadow; the moon’s crescent; risings and settings; syzygies; lunar eclipses, solar eclipses; planetary transits across the sun’s disk; the patas of the moon and sun; the “great problems”; and a final chapter which forms a conclusion.

The third chapter of the Siddhanta-tattva-viveka contains some of the most interesting Mathematical results. In that chapter Kamalakara used the addition and subtraction theorems for the sine and the cosine to give trigonometric formulae for the sines and cosines of double, triple, quadruple and quintuple angles. In particular he gives formulae for sin(A/2) and sin(A/4) in terms of sin(A) and iterative formulae for sin(A/3) and sin(A/5). See for example contradict and contradict for a discussion of the details of Kamalakara’s work in this area.

The Siddhanta-tattva-viveka is a Sanskrit text and in it Kamalakara makes frequent use of the place-value number system with Sanskrit numerals. This and many other aspects of the work are discussed in contradict.

*Jagannatha Samrat (Born: about 1690 in Amber (now Jaipur), India Died: about 1750 in India)*

Jagannatha had Jai Singh Sawai as his patron. Jai Singh Sawai was the ruler of Amber, now Jaipur, in eastern Rajasthan. He began his rule in 1699 and by clever use of tax rights on land that was rented by the state to an individual person he became the most important ruler in the region. His financial success let him finance the scholarly works of people such as Jagannatha. It is worth noting that Jai Singh’s importance was recognised by Amber which was then called Jaipur in his honour.

Jai Singh ruled Amber throughout the period in which Jagannatha was producing his Scientific work. He realised that the health of the country required Indian culture and science to be revitalised and returned to its position of leading importance which it had possessed. So Jai Singh employed Jagannatha to make Sanskrit translations of the important Greek Scientific works which at that time were only available in Arabic translations.

Jagannatha translated Euclid’s Elements from the Arabic translation by Nasir al-Din al-Tusi made nearly 500 years earlier. His Sanskrit version was called Rekhaganita and it was completed by 1727. We know this date since a copy was made by a scribe and he dated the start of his work as 1727.

Ptolemy’s Almagest had been one of the works which Arabic scientists had studied intently and, in 1247, al-Tusi wrote Tahrir al-Majisti (Commentary on the Almagest) in which he introduced various trigonometrical techniques to calculate tables of sines. Jagannatha translated al-Tusi’s Arabic version but he did more than this for he included in the same work, which he called Siddhantasamrat, his own comments on related work of other Arabic Mathematical astronomers such as Ulugh Beg and al-Kashi.

It is clear from Jagannatha’s work that he is working as one of a group of mathematicians and astronomers gathered by Jai Singh in his scheme to bring the best in Scientific ideas from outside India to reinvigorate the Scientific scene in India.

In contradict Gupta looks at the history of the result

sin(π/10) = (√5 – 1)/4

in Indian mathematics. The result appears for the first time in the work of Bhaskara II, but there were a number of interesting proofs of the result by later Indian mathematicians. One of the proofs presented by Gupta in contradict was by Jagannatha who gave a proof which was essentially Geometric in nature but, interestingly, contained an analytic procedure in terms of trigonometric and Algebraic steps.

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