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Showing posts with label indian math. Show all posts
Showing posts with label indian math. Show all posts

## Saturday, October 22, 2016

### Ramanujan's Math still unsolved

Ramanujan’s second construction for the approximate squaring of a circle
To experience the greatness of great men one has to relive or redo some acts of theirs to the best of ones ability. In ones youth such enactments might inspire one to make a bid for greatness. Whether this happens or not is mostly up to your genetics. Nevertheless, through the enactments one can at least savor the experience of what it takes to get there. If there was one man in our midst who could have lived up to be a Gauss or an Euler it was Srinivasa Ramanujan.
By redoing some of his acts that are within the grasp of our limited intellect we experienced the monument that he was. He gave two constructions for the approximate squaring of a circle using a compass and a straight-edge. We had earlier described the first and more widely known of those. The second appears in his paper titled “Modular equations and approximations to \pi“. In this paper in addition to remarkable approximations for the perimeter of the ellipse, which we had also alluded to before, he gives several series for \pi=3.141592653589793.... One of these series with just the first term leads to the below approximation:
\pi \approx \dfrac{99^2}{2*\sqrt{2}*1103} = 3.141592730013305 which is correct down to 6 decimal places. It is this kind of accuracy he captures in his first construction for the quadrature of the circle. In the midst of the dizzying series he conjures in a very Hindu style of mathematics, he says that he came up with an empirical approximation which leads to the below construction for the approximate squaring of the circle:
1) Draw circle to be squared with center O.
2) Draw its diameter \overline{AB}.
3) Trisect its radius \overline{AO} to get a third of it as \overline{AF}.
4) Bisect the semicircle AB to get point C.
5) Draw \overline{BC}.
6) On \overline{BC} mark \overline{CG}= \overline{GH}=\overline{AF}.
7) Join point A to point H to get \overline{AH} and to point G to get \overline{AG}
8) With radius as \overline{AG} cut \overline{AH} to get point I.
9) Draw a line parallel to \overline{GH} through point I to cut \overline{AG} at point J.
10) Join points O and J to get \overline{OJ}.
11) Draw a line parallel to \overline{OJ} through point F to cut \overline{AG} at point K.
12) Draw the tangent to the circle at point A and cut it with radius as \overline{AK} to get point L.
13) Draw \overrightarrow{OL} to cut circle at point M.
14) Draw semicircle LM and perpendicular from point O to cut this semicircle at point N.
15) Triplicate \overline{ON} to get \overline{OQ}=3 \times \overline{ON}.
16) Produce \overline{OQ} in the opposite direction to cut circle at point R.
17) Draw semicircle RQ and a perpendicular from point O to cut it at point S.
18) Thus, we have \overline{OS} as the side of the square OSTU which has approximately the same area the starting circle.
Ramanujan tells us that his earlier construction gave an “ordinary” value \pi=\dfrac{355}{113}=3.141592920353982, which is correct to six decimal places. This one, however, gives us the value:
\pi=\left (9^2+\dfrac{19^2}{22}\right)^{\frac{1}{4}}= 3.141592652582646
This is correct to a whopping eight decimal places keeping with the Hindu love for big numbers.
source: Manasataramgini

## Thursday, July 23, 2015

### Ancient Indian Math and Science

THE INDIAN TRADITION IN SCIENCE AND TECHNOLOGY:
Any study of the Indian tradition of science has to start with linguistics. This is so notonly because Linguistics is the earliest of Indian sciences to have been rigorously systematised, but also because this systematisation became the paradigm example for all other sciences of India.

Like all sciences and arts of the Indian tradition, Linguistics finds its first expression in the Vedas. For most of the Indian sciences, already by the Vedic period, the basic elements of study and the basic categories through which these are to be studied had been established, preliminary data for the operation of these sciences had been collected and rough systematisation had been achieved. Thus, for the science of Linguistics, in the siksha and pratiakhya texts associated with the various Vedas, we find a settled complete list of phonemes appropriately classified into vowels, semivowels, sibilants and the five groups of five consonants, all beautifully arranged according to the place of articulation which moves systematically from the throat to the lips. In fact phonetics and phonology are taken for granted by all authorities on etymology (nirukta) and grammar (vyakarana) including Yaska and Panini. In the pratisakhya literature one also find the morpho-phonemic (sandhi) rules and much of the methodology basic to the later grammatical literature.

The Indian science of Linguistics finds its rigorous systematisation in Panini’s Ashtadhyayi. The date of this text, like that of much f the early Indian literature, is
yet to settled with any certainty. But surely the Ashtadiyayi is not of a date later than 500 BC*. In the Ashtadhyayi, Panini achieves a complete characterisation of the Sanskrit language as spoken at his time, and also manages to specify the way it
deviated from the Sanskrit of the Vedas. Given a lit of the root words of the Sanskrit language (dhatupatha) and using the sutras of Panini it is possible to generate all possible correct utterances in Sanskrit. This is of course the main thrust of the generative grammars of today that seek to achieve a purely grammatical description of language through a formalized set of derivational strings. It is understandable that till such attempts were made in the West in the recent past, the Paninian sutras to the Western scholars looked like nothing but some artificial and abstruse formulations with little content.

Patanjali (c 1st C BC) in his major commentary on the Ashtadhyayi, the Mahabhashya gives an elaborate rationale for the Paninian exercise. According to the Mahabhashya the purpose of grammar is to give an exposition of all correct utterances in the language. An obvious way to do this is to enumerate all correct utterances individually. This is how the celestial teacher Brihaspati would have taught the science of language to the celestial student, Indra. However for ordinary mortals, not having access to celestial time scales this method can apparently not be of much use.

Therefore according to the Mahabhashya it is necessary to lay down general rules
(utsarga sutras) with a wide application so that with a comparatively small effort men can learn larger and larger collections of valid utterances. What falls to fit in this set of general rules should than be encompassed in exceptional rules (apavada sutras), and so on.

In providing this characterisation of the science of grammar Patanjali laid his finger on perhaps the most essential feature of the Indian scientific effort. Science in India seems to start with the assumption that truth resides in the real world with all its diversity and complexity. Thus for the Linguist what is ultimately true is the language as spoken by the people in all their diverse expressions. As Patanjali emphasises, valid utterances are not manufactured by the Linguist, but are already established by the practice in the world. Nobody goes to a linguist asking for valid utterances, the way one goes to a potter asking for pots. Linguist do make generalisations about the language as spoken in the world. But these generalisations are not the truth behind or above the reality. These are not the idealisation according to which reality is to be tailored. On the other hand what is ideal is the real, and some part of the real always escapes our idealisation of it.

There are always exceptions. It is the business of the scientist to formulate these generalisations, but also at the same time to be always attuned to the reality, to always to conscious of the exceptional nature of each specific instance. This attitude, as we shall have occasion to see, seems to permeate all Indian science and makes it an exercise quite different from the scientific enterprise of the West.

In the tradition of Linguistics after the period of Mahabhashya the major attempt of the grammarians seems to be to provide refinements and simplifications of Panini. In this period a number of Sanskrit grammars are written. One of them, Siddhanta Kaumudi 9c.1600) became eminently successful, perhaps because of its simplicity.

These attempts continue till the 19th century. Another form of study that became
popular amongst the grammarians is what may be called the philosophical semantics, wherein one starts with an utterance and by analysing it into its basic grammatical components tries to fix and characterise its meaning. This of course, is the major application for which grammar is intended in the first place.

Other Indian language grammars were written using Paninian framework as the basis. In fact these grammars are not fully formalised grammars in the sense of Panini. Instead what is attempted is to start with the Paninian apparatus and specify the transfer rules from Sanskrit as also the specific morpho-phonemic rules (sandhi), for the language under consideration. Such grammars for various Prakrit languages of the North and also the South Indian languages continued to be written till the 18th century, so much so that in the 16th century Krishnadas wrote a grammar for the Persian language. Parasi Prakash styled on the grammars of the Prakrit languages.

II

Among the sciences of the Indian tradition Astronomy and Mathematics also occupy an important place. Indian mathematics finds its early beginnings in the famous Shulva Sutras of the Vedic times. Purportedly written to facilitate the accurate construction of various types of sacrificial altars of the Vedic ritual, these sutras lay down the basic geometrical properties of plane figures like the triangle, the rectangle, the rhombus, and the circle. Basic categories of the Indian astronomical tradition wee also already established in the various Vedanga Jyotisha texts.

Rigorous systematisation of Indian astronomy however begins with Aryabhata (b.470
AD). His Aryabhatiya is a concise text of 121 aphoristic verses containing separate
sections on the basic astronomical definitions and parameters, basic mathematical
procedures in arithmetic, geometry, algebra and trigonometry, methods of determining
mean and true positions of the planets at any given time, and description of the
motions of sun, moon and the planets along with computation of the solar and lunar
eclipses. After Aryabhata, one comes across a long series of illustrious astronomers
with their equally illustrious texts, many of which gave rise to a host of commentaries
and refinements by later astronomers and became the corner stones of flourishing
The tradition continued and thrived right upto the late eighteenth century, and in
regions like Kerala, original work in the Indian tradition continued to appear till much
later.

The most striking feature of this long tradition of Indian mathematics and astronomy is
the efficacy with which the Indians seem to be handling and solving rather complicated
problems. Thus in Mathematics the Indians already in Shulva sutras know all the basic
theorems of plane geometry. Around this time they also develop a sophisticated theory
of numbers including the concepts of zero, and negative numbers. They also seem to
have arrived t simple algorithms for all basic arithmetical operations by using the place
value notation.

Thus by the time of Aryabhatiya the Indians have all basic mathematical concepts and
procedures that are today taught at the high school level. By the 10th or 11th century
they are able to solve sophisticated problems in algebra such as second order
Diophantine equations. By the 14th century infinite series for sine and cosine functions
are written down in trigonometry. High levels of approximations of and recognition
of its irrational character are also achieved by the same time.

The reason for this success of the Indian mathematicians lies perhaps in the explicitly
algorithmic and computational nature of Indian mathematics. The objective of the
Indian mathematician was not to find ultimate axiomatic truths in mathematics but to
find methods of solving specific problems that may arise in the astronomical or other
contexts. For the purpose the Indian mathematicians were prepared to set up simple
algorithms that may give only approximate solutions to the problem at hand, and to
evolve theories of error and recursive procedures so that the approximations may be
kept in check. This algorithmic methodology persisted in the Indian mathematical
consciousness till recently so that Ramanujan in the twentieth century seems to be
chalking up his impressive mathematical discoveries perhaps through the use of this

The same pragmatic concern to be able to calculate the positions of the various planets
and eclipses of the sun and the moon reasonably accurately, informs the efforts of the
Indian astronomers. And in this they turned out to be eminently successful. In their
calculations Indians often take the beginning of the Kaliyuga in the year 3102 before
Christ as their starting point, and the so-called Siddhanta texts deal with a much larger
period consisting of 43,20,000 years called a Mahayuga or sometimes even a period
1000 times the above, which is called a Kalpa. In spite of dealing with such long time
periods the Indian astronomers were able to keep their techniques fairly simple and
their parameters fairly well refined so that even towards the end of the eighteenth
century and early parts of nineteenth, when the active astronomical tradition had
become dormant in large parts of India, European astronomers are able to locate
Brahmins in South India, who could calculate for them the details of the current
eclipses to an accuracy comparable to, and often better than the best calculations of
Europe of the time.

The reasons for the simplicity and accuracy of the Indian astronomical techniques are
again to be found in the pragmatic attitude of the Indians towards the sciences. The
Indian astronomers wee in the business to calculate and to compute; but not to form
pictures of the heavens, as they ought to be in reality. Indian astronomers do use some
geometrical models but these are supposed to be nothing more than artefacts
necessary to perform their calculations (see Appendix). It is obvious that the
astronomical parameters obtained using such artefacts will get out of tune with reality
sooner or later and the calculations made with such parameters will start deviating
from actual positions of the planets. Indian astronomers are aware of this and were
quite willing to take up the onerous task of continuously observing the skies,
continuously checking their computations against observations and repeatedly readjusting
their parameters so as to make their calculations accord with reality. Thus
the sixteenth century astronomer Nilakantha Somasutvan, finding a contemporary
commentator lamenting about the different times given in different Siddhantas and the
computed times differing from the actual ones, exhorts.

O faint hearted, there is nothing to be despaired of…one has to realise that
five Siddhantas had been correct at a particular time. Therefore one has
to search for a Siddhanta that does not show discord with the actual
observation at the present time. Such accordance has to be ascertained by
observations during time of eclipses, etc. When Siddhantas show discord
observations should be made with the use of Instruments and correct
number of revolutions etc, found, and a new Siddhanta enunciated.’
A little later Jyesthadeva in the Drikkarana tells how from aryabhata to the present
day the astronomers have adjusted the parameters to accord with observations and
how he too is doing the same job for his times and he ends with the statement that
‘henceforth too the deviations that occur should be carefully observed and revisions
effected.

III

The third major science” of the classical tradition is Ayurveda, the science of Life. Like
Linguistics and Astronomy this too finds its early expression in the Vedas, especially the
Atharvaveda in which a large amount of early; medicinal lore is collected.
Systematisation of Ayurveda takes place during the period 5th century BC and 5th
century AD in the Charaka Samhita Sushruta Samhit and the Ashtanga Sangraha, the so
called Brihat –trayee texts, which are still popular today. This is followed by a long
period of intense activity during which attempts are made to refine the theory and
practice of medicine, and to bring more and more information into the stream of
systematic medicine. This process of accretion of information and refinement of
practice continued right upto the beginning of the nineteenth century.

Like in Linguistics and Astronomy, the remarkable feature of Indian tradition of
medicine is its pragmatic attitude towards scientific theorization. The Ayurvedic texts
while providing a theoretical framework through which the problem of finding an
appropriate cure for a particular patent must be approached, never tire of reminding
the practitioner that he must be constantly observant of all the specific features that a
particular case presents in fact for Charaka Samhita the most desirable intellectual
accomplishment of a doctor is that of possessing Yukti, the Yukti is the capacity of the
trained intellect that manages to see the course of action through the complexity of
phenomena with their multiple causes.

The attitude of Ayurveda towards theoretical generalisations is very clearly brought out
in a revealing verse of Sushruta Samhita. While describing the theoretical qualities of a
substance through which its medicinal properties are to be determined, the text comes up with the warning that the wise physician should never raise theoretical arguments
about the properties of a drug when they are already known and established in
tradition based on actual practice, because afterall ‘a thousand reasons will not make
the drugs of the ambastha group perform laxative functions’. Therefore the physician
must rely on what is established in tradition based on actual practice, rather then
acting exclusively on his theoretical reasoning. This attitude towards theory gives the
Ayurvedic texts, a refreshing openness and a surprising keenness of observation.
Nothing that may have any effect on the problem of health seems to escape the
observation of the physicians. One finds the physicians worrying about the differing
aspects of the seasons, the soils, the waters and so on. And in the therapeutic sections
they bring into use all their theoretical understanding and all the folk practices that
have been proved to be efficacious in tradition.

This pragmatic attitude towards scientific theoretization made the doing of science in
India a rather painstaking business. The Indian scientists not having the luxury of
reducing the reality of the world to that encompassed by their theories of the time,
had to be continuously aware of the world in its complete complexity, and had to
continuously refine and simplify their procedures in order to operate successfully in
this complex world. That they were able to do this systematically in a number of fields
over a long period of over 2000 years is measure of their ingenuity and industry. Thus
one can only marvel of the stupendousness of the task of encapsulating the whole of
Sanskrit language as it was spoken in 4000 aphoristic rules. Equally remarkable is the
effort of the astronomer- mathematicians to repeatedly refine their parameters to fit
the observations so that ever since Aryabhata the Indians always had access to
reasonably accurate information about the motions of the heavens. But the
astronomer –mathematicians also simplified their computations to an extent that
learned Brahmins in their innumerable locales could also compute all the astronomical
information that mattered to the residents. The effort of Indian physicians also falls in
the same class. They were not only able to painstakingly acquire and systematize
within their theoretical framework all the information about drugs and diseases that
was current amongst the people in diverse areas, but were also able to simplify their
theories sufficiently so that much of the Ayurvedic science became the folders of
health known in all families. The fact that the Indian scientists given their theoretical
attitude had to be necessarily open to the world around them perhaps ensured that the
folk and the science had to remain in a symbiotic relation with each other.

Besides Linguistic, Astronomy and Mathematics, and Medicine, Indians also developed
the sciences of matter (Padartha Sastra), metallurgy (Rasa Sastra), architecture (Vastu
Sastra), music (Sangitha Sastra) etc. To all of these sciences they brought their
peculiarly Indian mode of careful but tentative generalisations and continuous keen
observation.

IV

The pragmatic attitude of conceptual sophistication and operational simplicity that we
have noticed amongst the sciences of India seems also to have informed the Indian
technologies. A systematic history of the traditional Indian technologies is yet to be
written. Therefore one has to rely largely on the accounts of European travellers and
administrators who observed and wrote about the Indian practices during the early
phase of European conquest of India.

The major technological endeavour of India was of course in the field of agriculture.
Col.Alexander Walker writing in the early nineteenth century seems to have been
amazed at the keen interest that ordinary Indians showed in everything connected with
agriculture. He is also greatly impressed by the care with which the Indian cultivators
tended their fields, so that to him the fields of Malabar and Gujarat seem more like
carefully laid out gardens. This care is coupled with an intimate knowledge of the
soils, the seasons and seeds. The Indians seemed to have mastered techniques of
rotation of crops, irrigation, manuring, and selection of seeds etc. from very early
times. These techniques had been so well studied and so optimised to the peculiar
conditions of each area that John Voeicker, the Consulting Chemist to the “Royal
Agricultural Society”, sent to India towards the end of the nineteenth century to
suggest ways of improving Indian agriculture through the use of chemistry, could
recommend little by way of technological changes. He was of the opinion if only the
traditional facilities of water and manure could be ensured the farmers of India could
obtain the best possible yields. As for suggesting improvements he seemed to have felt
that I was much easier to propose improvements in English agriculture, than to make
really valuable suggestions for that of India. Another expert of early twentieth century
John Kenny, remarks in the same vein that he did not consider it wise ‘to suggest seed
selection in a land where 4000 different sorts of paddy are grown in one province alone
and carefully differentiated according to their qualities and land suitable for them’.
The implements of the Indian cultivator often seemed rough and primitive to the
occasional observer. However it was soon realised that these implements were fully
adapted to the particular conditions in which they operated and even in the late
nineteenth century nothing could be suggested by way of their improvement. In fact
an early experiment during the later half of the eighteenth century to introduce the
heavy English plough near Salsette on the west coast had proved a total disaster. In
1795 Cap. Thos Holcott reported on the sophistication of the Indian drill plough widely
used in the Andhra region. The drill plough till then was considered a ‘modern’
European Invention.

With their simple but sophisticated implements and their meticulous techniques of
agriculture, the Indian farmers were able to obtain impressive yields. It was reported
that in early nineteenth century in the Allahabad region the produce of an acre of land
amounted to over 55 Bushels per harvest while that in England around the same time
was only about 20 Bushels. Since the Indian farmer in this region usually produced two
crops a year, the annual yield of each acre may be taken as over 110 Bushels at his
time. The productivity of Indian agriculture, however, declined very rapidly during the
nineteenth century. But even in the 1890’s lands which had access to irrigation and
manure yielded harvests comparable to those in England, and larger than the harvests
obtained those days in Europe, USA and Australia.

V

The Indian technical ingenuity in evolving simple techniques that are sophisticated
enough to take advantage of the full complexity of the local situation, and then
meshing these local techniques into a large impressive system can be best seen in the
tank irrigation system of South India. It seems that the whole of South India was
dotted with these tanks. A British expert writing in the 1850 ‘s estimated the total
number of such tanks in the Madras Presidency to be over 50,000. Another estimate
indicates that in the eighteenth century there were more than 38,000 tanks in the
region, which later constituted the Mysore state. The state had an area of around
29,500 square miles. It is, therefore, fair estimate that there were over a lakh tanks in
whole of South India. From what is known of the political circumstances of the time it
is clear that these tanks were constructed and maintained by local effort. However,
together they formed a closely knit whole so that the outflow from the one at a higher
level supplied the one at a lower level, and so on. This chain of tanks was so complete
and inter related within itself that the British engineers of the nineteenth century felt
that it would have been impossible to add another tank to the chain.

The Indian genius for performing vast tasks through simple, small and dispersed
techniques is seen even better in the case of metallurgy. Early European observers
noticed the Indians using small furnaces for something and refining iron and making
steel. Scores of seventeenth, eighteenth and early nineteenth century accounts of
Indian manufacture of iron and steel are available, and these pertain to perhaps a
hundred districts spread all over India. The smelting furnaces described in these
accounts seem to be of quite rough construction from the outside. However, the
observers noticed that the internal proportions and various angles needed to be rather
exact, and there were cases where the furnace had to be demolished and
reconstructed to correct some minor error in the angle of blast, or in some internal
proportion. Yet these sophisticated furnaces were routinely constructed by the Indian
iron-smiths in a matter of hours without the help of any very complicated Instruments.
These furnaces worked quite efficiently by the standard of those times. Thus,
according to one detailed account, two units of charcoal were sufficient to produce
one unit of crude iron in these furnaces. Processes of refining iron and steel making
were also equally efficient. Steel was prepared by direct carbonisation of Iron in
closed crucibles in which green leaves, wood and charcoal were all put together. This
process seemed mysterious to the British observers, since a process of direct
conversion was discovered in Europe only in the 1920’s. Even then, observers were
often surprised at the quickness with which steel was made in the Indian furnaces, the
process taking a few hours compared to many days taken in the corresponding
European processes.

The simplicity of these Indian techniques should be seen in the context of the fact that
Indian iron and steel had been renowned for their qualities for centuries past. All over
India one can find scattered Iron pillars and girders of very high quality, especially as
regards corrosion resistance. Indian steel has an equally distinguished record of
maintaining excellent quality, and even in the late eighteenth century an expert in
Britain when presented with a sample of Indian steel noted that it was ‘excellently
adapted for the purpose of fine cutlery, and particularly for all edged instruments used
for surgical purposes’.

It is worth remarking that with their small and dispersed furnaces, which produced
perhaps half a ton of iron during a week’s operation, Indians were capable of producing
a rather large amount of iron and steel. According to some nineteenth century
enumerations there were hundreds of such furnaces operating in certain districts and
taluks. On the basis of this information it has been estimated that the total number of
furnaces throughout India in the later part of the eighteenth century could have been
over 10,000 and these furnaces together had the potential to produce some 2 lakh
tonnes of iron annually.

VI

A survey of Indian technologies cannot be complete without some discussion of textiles,
the great industrial enterprise of pre-British India. Upto 1800 India was the world’s
leading producer and exporter of textiles. Yet this production was almost entirely
based on techniques that could be operated at the level of he individual or the family.
Spinning of yarn was an activity in which perhaps whole of India participated.
According to an observer from Manchester, Amo Pearse, who in 1930 visited India to
study its cotton industry, there were probably 5 crors spinning wheels (Charkhas)
intermittently at work even then. And this simple small wheel was so efficient that till
the early decades of the nineteenth century a widowed mother could still maintain a
whole family in reasonable manner by spinning on the charkha for a few hours a day.
Weaving was a relatively more specialised activity. However, the number to those
belonging to the weaver castes was smaller in comparison only to those from the
cultivating castes. Early nineteenth century data for certain districts of South India
indicate that each district had around 20,000 looms. Amo Pearse in 1930 estimated the
number of handlooms operating in India to be in the vicinity of 20 lakhs.

There were vast regions of India, which specialised in specific types of fabrics. Each of
these areas developed techniques of weaving, bleaching, dyeing and painting etc., that
were indigenous to the region, and also had its own characteristic designs, motifs and
symbols. For example, in Western India alone, Sironj in Rajasthan and Burhanpur in
Khandesh were major centres of cotton painting cheap printed cottons came from
Ahmedabad; woollens including the extra-ordinary Cashmere Shawls were produced in
Kashmir, true silks wee worked as Patolas at Patan in Gujarat and so on.

These dispersed and diverse techniques were so optimised that textile produced in
Britain through the post industrial revolution British technology could hardly match the
Indian textiles in quality or price. Till the early nineteenth century, mill produced
fabric had to be protected from Indian competition by the imposition of duties of 70 to
80 percent on the cottons and silks imported from India or, by positive prohibition. As
the historian H.W.Wilson notes, without such prohibitory duties and decrees, ‘the mills
of Paisley and Manchester would have been stopped in their outset and could scarcely,
have been again set in motion even by the power of steam’.

The Indians had developed their locality specific techniques not only in agriculture,
irrigation, metallurgy and textiles, but also in diverse other areas like building and
construction, sculpture, pottery, making of glass, and even luxuries like making of Ice
etc. That is perhaps why most historians of pre-British India are agreed that India of
that time was not only an agricultural, but also an industrial society.

Appendix: Computations in Indian Astronomy

By M.D.Srinivas

To get a flavour of the Indian way of doing Astronomy and Mathematics, it may be
instructive to look at the way they make particular Astronomical calculations like for
example the calculation of the longitudes of the grahas (the sun, the moon and the
various planets) at any given time. The following are the essential steps.

1. The Indian Astronomers first compute the shargana or the total number of
mean solar days slapeed from the chosen epoch till the given date, specified as
such and such Sakabda (Saka year) masa (lunar month) and and tithi (lunar
day). For this, they first compute the number of adhikamasas (intercalary
lunar months) elapsed since the epoch till the beginning of the current Saka
year. From this the number of tithis elapsed since the epoch till the given date
is calculated. Then the number of kshayahas (omitted lunar days) from the
epoch are computed and this when subtracted from the number of tithis, gives
the shargana for the given data. The computation of the number of
ashikamasas and kshyahas is based on specified basic astronomical parameters,
such as the mean motions of the sun and the moon.

2. From the computed shargana, the mean longitudes (madhyama graha) of the
sun, the moon and planets are calculated using the specified values of mean
(daily) motions of these grahas.

3. To take care of the irregular motion of the grahas a series of corrections are
applied to the madhyama graha mean longitude to get sphuta graha the true
longitude. For the case of planets the basic corrections are the manda karma
and the sighra karma. For the (outer) planets-Mars, Jupitar and Saturn – the
manda karma is roughly the equation of centre which takes care of the noncircular
orbit, and the sighra karma converts the corrected heliocentric
longitudes into the corrected geocentric longitudes. The basic form of the
manda and sighra corrections are obtained from a geometric epicyclic (or
eccentric) model and involves the mean motions of the apsides (mandocche
and sighroccha) and the circumferences (manda paridhi and sighra paridhi) of
the epicycles, as the basic parameters. However, what every text of Indian
astronomy prescribes is a series of manda and sighra type of corrections
performed iteratively till the results show a convergence to desired accuracy.
The actual steps involved in this sequence of corrections depend on the
particular planet and also varies from one school of astronomy to another. This
procedure seems to enable the Indian astronomers achieve much better fit with
observations than those achieved in other ancient traditions of astronomy
which operative with ideal geometrical models of planetary motion involving
epicycles, equants, etc. This sequence of operations is what enables the Indian
astronomers to incorporate for instance some of the higher order corrections to
the equation of centre which in modern astronomy are calculated using the
Kepierian orbits, etc.

4. The above planetary position is as observed at the time of sunrise (or midnight
in some schools of Indian astronomy) at the Indian zero meridians passing
through Ujjain. The desantara correction is applied to calculate the planetary
position at sunrise of places on a different meridian.

5. Finally, the planetary position at any given time of the day or night is
calculated from that at sunrise (or midnight) by using the so called sphutagati
or the true daily motion of the planet. This calculation also involves the
knowledge of the latitude of the place. Further it is necessary to know
whether the planet is in retrograde motion by computing the so called
sighragati phala in computing these gatis or velocities, various formulae which
are obtained by procedures which tantamount to differentiation of the
expressions for the sighra and manda corrections, are used.
The basic astronomical parameters involved in the above calculation are the mean
motions of the sun, he moon and the various planets, their apsides, and the
circumferences of the manda and sighra epicycles, etc. These have to be
determined on the basis of careful observations made over long periods. Each text
of Indian astronomy gives the values for these parameters current at its time.

## Sunday, April 5, 2015

### Ancient Indian Math- Madhava of Sangamagrama

Madhava of Sangamagrama (c. 1340 – c. 1425), was an Indian mathematician-astronomer from the town of Sangamagrama (present day Irinjalakuda) near Thrissur, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite PROCEDURES of ancient mathematics to treat their limit-passage to infinity". His discoveries opened the doors to what has today come to be known as Mathematical Analysis. One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra.

Some scholars have also suggested that Madhava's work, through the writings of the Kerala school, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it may have had an influence on later European developments in ANALYSIS and calculus.

They are the infinite series of the functions, sine, cosine and the arctangent. These infinite series are called by his name, Madhava sine series and Madhava Cosine series.

While the power series expansion of ArcTan is called Madhava-Gregory series, the power series are collectively called the Madhava Taylor series. The Pi series is known as Madhava Gregory series.

Sankara Varier, one of the foremost disciples of Madhava, had translated his poetic verses in in his Yuktideepika commentary on Tantrasamgraha-vyakhya, in verses 2.440 and 2.441

Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even numbers increased by that number and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva, as collected together in the verse beginning with "vidvan" etc.

Madhava's formula for Pi was discovered in the West by Gregory and Liebniz.