** Ancient Indian Mathematicians** have made a significant contributions to Mathematics including

*Place-Value Arithmetical Notation and the Concept of Zero*.

*Vedic Mathematicians*

• *Baudhayana
• Katyayana
• Panini, ca. 5th c. BC, Algebraic Grammarian
• Yajnavalkya, credited with Authorship of the Shatapatha Brahmana, which contains Calculations related to altar Construction.*

*Classical Mathematicians*

*Post-Vedic Sanskrit to Pala period Mathematicians (5th c. BC to 11th c. AD)
• Aryabhata – Astronomer who gave Accurate Calculations for Astronomical constants, 476-520
• Aryabhata II
• Bhaskara I
• Bhaskara II
• Brahmagupta – Helped bring the Concept of Zero into Arithmetic
• Mahavira
• Varahamihira
• Bhavik
• Shridhara (between 650-850) – Gave a good rule for finding the Volume of a Sphere.*

*ANCIENT INDIAN SCIENCE & MATHEMATICS*

Ancient Indian Mathematician were the First to make the Expression of Mathematical understanding appears in the form of Counting Systems. Numbers in very early Societies were typically represented by Groups of Lines, though later different Numbers came to be assigned specific *Numeral Names and Symbols (as in India)* or were designated by Alphabetic Letters (such as in Rome). Although today, we take our Decimal System for granted, not all ancient Civilizations based their Numbers on a *Ten-Base System.* In **Ancient Babylon, a Sexagesimal (base 60) System** was in use.

*The Decimal System in Harappa *

Ancient Indian Mathematicians has made a remarkable contribution in Decimal System and Harappan Period as made the Proof of it by using it, which is an evidence by an analysis of Harappan Weights and Measures. Weights corresponding to *Ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500* have been identified, as have scales with Decimal Divisions. A particularly notable Characteristic of Harappan Weights and Measures is their remarkable Accuracy. A Bronze Rod Marked in Units of 0.367 Inches Points to the Degree of Precision demanded in those times. Such scales were particularly important in ensuring proper implementation of *Town Planning Rules *that required roads of fixed widths to run at right angles to each other, for Drains to be Constructed of precise Measurements, and for homes to be constructed according to specified guidelines. The existence of a gradated System of accurately marked Weights Points to the ** Development of Trade and Commerce** in Harappan Society.

*Mathematical Activity in the Vedic Period *

In the **Vedic Period**, *Records of Mathematical* activity are mostly to be found in* Vedic Texts Associated with Ritual Activities.* However, as in many other *Early Agricultural Civilizations, the Study of Arithmetic and Geometry* was also impelled by secular considerations. Thus, to some extent *early Mathematical Developments in India mirrored the Developments in Egypt, Babylon and China .* ** The System of Land Grants and Agricultural Tax Assessments required accurate Measurement of Cultivated Areas. **As Land was redistributed or consolidated, Problems of Mensuration came up that required solutions. In order to ensure that all Cultivators had equivalent amounts of Irrigated and Non-Irrigated Lands and Tracts of equivalent Fertility – individual farmers in a village often had their holdings broken up in several parcels to ensure fairness. Since Plots could not all be of the same

*Shape – Local Administrators were required to Convert Rectangular Plots or Triangular Plots to Squares of Equivalent Sizes*and so on.

*Tax Assessments were based on fixed proportions of Annual or Seasonal Crop Incomes,*but could be adjusted upwards or downwards based on a variety of factors. This Meant that an understanding of

*Geometry and Arithmetic was virtually essential for Revenue Administrators.*

*Mathematics was thus brought into the service of both the Secular and the Ritual Domains.*** Arithmetic Operations (Ganit) such as Addition, Subtraction, Multiplication, Fractions, Squares, Cubes and Roots** are enumerated in the

**attributed to**

*Narad Vishnu Purana**Ved Vyas (Pre-1000 BC).*Examples of

*Geometric Knowledge (Rekha-Ganit)*are to be found in the

**Sulva-Sutras of Baudhayana (800 BC) and Apasthmaba (600 BC)**which describe Techniques for the Construction of Ritual Altars in use during the Vedic Era. It is likely that these Texts tapped Geometric Knowledge that may have been acquired much earlier, possibly in the Harappan Period.

**Baudhayana’s Sutra**displays an understanding of Basic Geometric Shapes and Techniques of Converting one Geometric Shape (such as a Rectangle) to another of Equivalent (or Multiple, or Fractional) Area (such as a Square).*While some of the Formulations are Approximations, others are accurate and reveal a certain Degree of Practical ingenuity as well as some Theoretical understanding of Basic Geometric Principles. Modern Methods of Multiplication and Addition probably emerged from the Techniques described in the Sulva-Sutras.*

**Pythagoras – the Greek Mathematician and Philosopher** who lived in the *6th C B.C *was familiar with the *Upanishads and learnt his Basic Geometry from the Sulva Sutras.* An early statement of what is Commonly known as the *Pythagoras Theorem is to be found in Baudhayana’s Sutra:* The chord which is stretched across the Diagonal of a Square produces an Area of Double the Size. A similar observation pertaining to oblongs is also noted. His Sutra also contains *Geometric Solutions of a Linear Equation in a Single Unknown.* Examples of *Quadratic Equations *also appear. ** Apasthamba’s Sutra (An Expansion of Baudhayana’s with several original contributions) **provides a

*Value for the Square Root of 2 that is accurate to the Fifth Decimal Place.*

**Apasthamba**also looked at the

*Problems of Squaring a Circle, Dividing a Segment into Seven Equal Parts, and a Solution to the general Linear Equation.*

**Jain Texts**from the

*6th C BC*such as the

**.**

*Surya Pragyapti describe Ellipses*Modern-Day Commentators are Divided on how some of the results were generated. Some believe that these results came about through *Hit and Trial* – as Rules of Thumb, or as generalizations of observed examples. Others believe that once the Scientific Method came to be formalized in the ** Nyaya-Sutras** – Proofs for such results must have been provided, but these have either been lost or destroyed, or else were transmitted orally through the

*Gurukul System*, and only the final results were tabulated in the texts. In any case, the

*Study of Ganith*i.e Mathematics was given considerable importance in the Vedic period. The

**includes the statement:**

*Vedang Jyotish (1000 BC)**“Just as the Feathers of a Peacock and the Jewel-Stone of a Snake are placed at the Highest Point of the Body (at the Forehead), similarly, the Position of Ganith is the Highest amongst all*

**Branches of the Vedas and the Shastras**.”*(Many Centuries later, Jain Mathematician from Mysore, Mahaviracharya further emphasized the Importance of Mathematics: “Whatever object exists in this moving and Non-Moving World, cannot be understood without the Base of Ganith (i.e. Mathematics)”.)*

*Panini and Formal Scientific Notation*

A particularly Important Development in the History of Indian Science that was to have a profound impact on all Mathematical treatises that followed was the pioneering work by *Panini (6th C BC)* in the** Field of Sanskrit Grammar and Linguistics**. Besides expounding a

*Comprehensive and Scientific Theory of Phonetics, Phonology and Morphology,*

**Panini**provided Formal Production Rules and Definitions describing

**Basic elements such as Vowels and Consonants, Parts of Speech such as Nouns and Verbs were placed in Classes.**

*Sanskrit Grammar in his treatise called Asthadhyayi*.*The Construction of Compound Words and Sentences was elaborated through ordered Rules Operating on underlying Structures in a manner similar to*

**Formal Language Theory**.Today,** Panini’s Constructions** can also be seen as

*comparable to Modern Definitions of a Mathematical Function. G G Joseph,*in The crest of the Peacock argues that the Algebraic Nature of Indian Mathematics arises as a Consequence of the Structure of the Sanskrit Language. Ingerman in his paper titled Panini-Backus form finds Panini’s Notation to be equivalent in its power to that of Backus – Inventor of the Backus Normal Form used to describe the Syntax of Modern Computer Languages. Thus

*Panini’s Work provided an example of a Scientific Notational Model that could have Propelled later Mathematicians to use Abstract Notations in Characterizing Algebraic Equations and Presenting Algebraic Theorems and results in a Scientific Format.*

*Philosophy and Mathematics*

* Philosophical Doctrines also had a profound influence on the Development of Mathematical Concepts and Formulations. *Like the

**Upanishadic**World View, Space and Time were considered Limitless in Jain Cosmology. This led to a deep interest in Very Large Numbers and Definitions of Infinite Numbers. Infinite Numbers were created through Recursive Formulae, as in the

**.**

*Anuyoga Dwara Sutra***Jain Mathematicians**recognized

*Five Different Types of Infinities: Infinite in One Direction, in Two Directions, in Area, Infinite Everywhere and Perpetually Infinite.*

**Permutations and Combinations are listed in the Bhagvati Sutras (3rd C BC) and Sathananga Sutra (2nd C BC).**

*Jain Set Theory probably arose in parallel with the Syadvada System of Jain Epistemology* in which reality was described in terms of Pairs of Truth Conditions and State Changes. The **Anuyoga Dwara Sutra **demonstrates an understanding of the *Law of Indices* and uses it to develop the *Notion of Logarithms*. Terms like * Ardh Aached , Trik Aached, and Chatur Aached are used to Denote Log Base 2, Log Base 3 and Log Base 4 respectively.* In

*Satkhandagama*various sets are operated upon by

*Logarithmic Functions to Base Two, by Squaring and Extracting Square Roots*, and by raising to

*Finite or Infinite Powers.*The Operations are repeated to produce new sets. In other works the relation of the Number of Combinations to the Coefficients occurring in the

*Binomial Expansion*is noted.

Since *Jain Epistemology* allowed for a degree of indeterminacy in describing reality, it probably helped in grappling with *Indeterminate Equations and finding Numerical Approximations to Irrational Numbers. *

*Buddhist Literature ***also demonstrates an awareness of Indeterminate and Infinite Numbers. Buddhist Mathematics was classified either as Garna (Simple Mathematics) or Sankhyan (Higher Mathematics). Numbers were deemed to be of Three Types: Sankheya (Countable), Asankheya (Uncountable) and Anant (Infinite). **

*Philosophical Formulations* concerning** Shunya** – i.e. emptiness or the void may have facilitated in the* Introduction of the Concept of Zero*. While the** Zero (Bindu) **as an Empty Place Holder in the Place-Value Numeral System appears much earlier, Algebraic Definitions of the Zero and it’s relationship to Mathematical Functions appear in the Mathematical treatises of Brahmagupta in the 7th C AD. Although Scholars are Divided about How early the Symbol for Zero came to be used in Numeric Notation in India,* (Ifrah arguing that the use of Zero is already implied in Aryabhatta) *Tangible Evidence for the use of the Zero begins to proliferate towards the End of the Gupta Period. Between the *7th C and the 11th C, Indian Numerals developed into their Modern Form, and along with the Symbols Denoting various Mathematical Functions (such as Plus, Minus, Square Root, Etc) eventually became the Foundation Stones of Modern Mathematical Notation. *

*The Indian Numeral System*

Although the *Chinese were also using a Decimal Based Counting System, *the Chinese lacked a Formal Notational System that had the abstraction and elegance of the Indian Notational System, and it was the Indian Notational System that reached the Western world through the Arabs and has now been accepted as Universal. Several Factors contributed to this development whose significance is perhaps best stated by *French Mathematician, Laplace: “The ingenious method of expressing every possible Number using a set of ten Symbols (each Symbol having a Place Value and an Absolute Value) emerged in India. The idea seems so simple nowadays that its Significance and Profound Importance is no longer appreciated. It’s simplicity lies in the way it facilitated Calculation and placed Arithmetic foremost amongst useful inventions.” *

Brilliant as it was, this Invention was no accident. In the Western world, the Cumbersome Roman Numeral System posed as a major obstacle, and in China the Pictorial Script posed as a hindrance. But in India, almost everything was in place to favor such a development. There was already a long and established history in the use of *Decimal Numbers, and Philosophical and Cosmological Constructs *encouraged a creative and expansive approach to *Number Theory*. **Panini’s** studies in *Linguistic Theory and Formal Language and the Powerful Role of Symbolism and Representational Abstraction in Art and Architecture may have also provided an impetus, as might have the Rationalist Doctrines and the exacting Epistemology of the Nyaya Sutras, and the innovative abstractions of the Syadavada and Buddhist Schools of Learning. *

*Influence of Trade and Commerce, Importance of Astronomy *

The growth of *Trade and Commerce*, particularly lending and borrowing demanded an understanding of both simple and compound interest which probably stimulated the interest in Arithmetic and Geometric Series. **Brahmagupta’s **description of* Negative Numbers as Debts and Positive Numbers as Fortunes Points* to a link between Trade and Mathematical Study. Knowledge of Astronomy – particularly knowledge of the tides and the Stars was of great import to Trading Communities who crossed Oceans or Deserts at Night. This is borne out by Numerous references in the *Jataka Tales *and several other *Folk-Tales*. The young person who wished to embark on a Commercial venture was inevitably required to first gain some grounding in Astronomy. This led to a proliferation of *Teachers of Astronomy,* who in turn received training at universities such as at *Kusumpura (Bihar) or Ujjain (Central India) or at Smaller Local Colleges or Gurukuls.* This also led to the exchange of texts on Astronomy and Mathematics amongst Scholars and the Transmission of Knowledge from one part of India to another. Virtually every Indian state produced great Mathematicians who wrote Commentaries on the works of other Mathematicians (who may have lived and worked in a different part of India many Centuries earlier). *Sanskrit served as the Common Medium of Scientific Communication. *

The Science of Astronomy was also spurred by the need to have accurate Calendars and a better understanding of Climate and Rainfall patterns for timely sowing and choice of crops. At the same *Time, Religion and Astrology* also played a role in creating an interest in Astronomy and a Negative fallout of this irrational influence was the rejection of Scientific Theories that were far ahead of their time. One of the Greatest Scientists of the Gupta Period – ** Aryabhatta (Born in 476 AD, Kusumpura, Bihar)** provided a Systematic Treatment of the Position of the Planets in Space. He correctly posited the axial rotation of the Earth, and inferred correctly that the Orbits of the Planets were Ellipses. He also correctly deduced that the Moon and the Planets shined by Reflected Sunlight and provided a Valid explanation for the Solar and Lunar Eclipses rejecting the Superstitions and Mythical Belief Systems surrounding the phenomenon. Although

*Bhaskar I (Born Saurashtra, 6th C, and follower of the Asmaka School of Science, Nizamabad, Andhra )*recognized his genius and the tremendous value of his Scientific contributions, some later Astronomers continued to believe in a Static Earth and Rejected his Rational Explanations of the Eclipses. But in spite of such setbacks,

*Aryabhatta*had a profound influence on the Astronomers and Mathematicians who followed him, particularly on those from the

**Asmaka School**.

*Mathematics played a vital role in Aryabhatta’s Revolutionary understanding of the Solar System. *His

**the Circumference of the**

*Calculations on pi,***Earth (62832 Miles)**and the Length of the Solar Year (within about 13 Minutes of the Modern Calculation) were remarkably close Approximations. In making such Calculations, Aryabhatta had to solve several Mathematical problems that had not been addressed before including

**Problems in Algebra (Beej-Ganith) and Trigonometry (Trikonmiti).****Bhaskar I** continued where *Aryabhatta* left off, and discussed in further *Detail Topics such as the Longitudes of the Planets; Conjunctions of the Planets with each other and with Bright Stars; Risings and Settings of the Planets; and the Lunar Crescent.* Again, these studies required still more *Advanced Mathematics and Bhaskar I expanded on the Trigonometric Equations provided by Aryabhatta, and like Aryabhatta Correctly assessed pi to be an Irrational Number. *Amongst his most important contributions was his formula for

*Calculating the Sine Function which was 99% accurate. He also did Pioneering Work on Indeterminate Equations and considered for the First Time Quadrilaterals with all the Four Sides Unequal and None of the Opposite Sides Parallel.*

*Another important Astronomer/Mathematician was Varahamira (6th C, Ujjain) *who Compiled Previously Written Texts on Astronomy and made important additions to Aryabhatta’s Trigonometric Formulas. His works on Permutations and Combinations complemented what had been previously achieved by Jain Mathematicians and provided a Method of Calculation of nCr

*that closely resembles the much more recent*

*Pascal’s Triangle.*In the

*7th Century, Brahmagupta*did important work in enumerating the

**. In addition to Listing the**

*Basic Principles of Algebra**Algebraic Properties of Zero*, he also listed the

*Algebraic Properties of Negative Numbers*. His work on solutions to Quadratic Indeterminate Equations anticipated the work of Euler and Lagrange.

*Emergence of Calculus *

*In the Course of Developing a Precise Mapping of the Lunar Eclipse, Aryabhatta was obliged to Introduce the Concept of Infinitesimals – i.e. Tatkalika Gati to designate the Infinitesimal, or near Instantaneous Motion of the Moon, and express it in the form of a Basic Differential Equation.*

**Aryabhatta’s Equations**were elaborated on by

**who Derived the Differential of the Sine Function. Later Mathematicians used their intuitive understanding of integration in deriving the areas of curved surfaces and the volumes enclosed by them.**

*Manjula (10th C) and Bhaskaracharya (12th C)**Applied Mathematics, Solutions to Practical Problems *

Developments also took place in Applied Mathematics such as in creation of Trigonometric Tables and Measurement Units. **Yativrsabha’s work Tiloyapannatti (6th C)** gives various units for *Measuring Distances and Time and also describes the System of Infinite Time Measures. *

In the * 9th C, Mahaviracharya ( Mysore) *wrote

**Ganit Saar Sangraha**where he described the currently used

*Method of Calculating the Least Common Multiple (LCM)*of given Numbers. He also

*Derived Formulae to Calculate the Area of an Ellipse and a Quadrilateral inscribed within a Circle (something that had also been looked at by Brahmagupta)*The Solution of Indeterminate Equations also drew considerable interest in the 9th century, and several Mathematicians contributed Approximations and Solutions to Different Types of Indeterminate Equations.

In the late **9th C, Sridhara (Probably Bengal) **provided Mathematical Formulae for a variety of *Practical Problems involving Ratios, Barter, Simple Interest, Mixtures, Purchase and Sale, Rates of Travel, Wages, and Filling of Cisterns. *Some of these examples involved fairly complicated solutions and his *Patiganita *is considered an advanced Mathematical work. Sections of the book were also devoted to* Arithmetic and Geometric Progressions, including Progressions with Fractional Numbers or Terms, and Formulas for the sum of certain Finite Series* are provided. Mathematical investigation continued into the *10th C. Vijayanandi (of Benares, whose Karanatilaka was translated by Al-Beruni into Arabic) and Sripati of Maharashtra are amongst the prominent Mathematicians of the century. *

The leading light of *12th C Indian Mathematics was Bhaskaracharya* who came from a long-line of Mathematicians and was head of the Astronomical observatory at Ujjain. He left several important Mathematical texts including the

**Lilavati and Bijaganita**and the

**Siddhanta Shiromani,**an Astronomical text. He was the first to recognize that certain Types of Quadratic Equations could have two solutions. His

**of Solving Indeterminate Solutions preceded European Solutions by several Centuries, and in his**

*Chakrawaat Method***he postulated that the**

*Siddhanta Shiromani**Earth had a Gravitational Force, and broached the Fields of Infinitesimal Calculation and Integration.*In the Second Part of this treatise, there are several chapters relating to the study of the

*Sphere and it’s Properties and Applications to Geography, Planetary Mean Motion, Eccentric Epicyclical Model of the Planets, First Visibilities of the Planets, the Seasons, the Lunar Crescent etc.*He also discussed

*Astronomical Instruments and Spherical Trigonometry. Of particular interest are his Trigonometric Equations:*

**sin(a + b) = sin a cos b + cos a sin b; sin(a – b) = sin a cos b – cos a sin b;***The Spread of Indian Mathematics*

The study of Mathematics appears to slow down after the onslaught of the *Islamic *invasions and the conversion of Colleges and Universities to Madrasahs. But this was also the time when Indian Mathematical texts were increasingly being translated into Arabic and Persian. Although Arab scholars relied on a variety of sources including *Babylonian, Syriac, Greek and some Chinese texts, Indian Mathematical texts played a particularly important role.* Scholars such as I*bn Tariq and Al-Fazari (8th C, Baghdad), Al-Kindi (9th C, Basra), Al-Khwarizmi (9th C. Khiva), Al-Qayarawani (9th C, Maghreb, author of Kitab fi al-hisab al-hindi), Al-Uqlidisi (10th C, Damascus, author of The book of Chapters in Indian Arithmetic), Ibn-Sina (Avicenna), Ibn al-Samh (Granada, 11th C, Spain), Al-Nasawi (Khurasan, 11th C, Persia), Al-Beruni (11th C, born Khiva, died Afghanistan), Al-Razi (Teheran), and Ibn-Al-Saffar (11th C, Cordoba) *were amongst the many who based their own Scientific texts on translations of Indian treatises. Records of the Indian origin of many proofs, Concepts and formulations were obscured in the later centuries, but the enormous contributions of Indian Mathematics was generously acknowledged by several important Arabic and Persian Scholars, especially in Spain. Abbasid scholar Al-Gaheth wrote: *“India is the source of knowledge, thought and insight.” Al-Maoudi (956 AD) who travelled in Western India also wrote about the greatness of Indian science. Said Al-Andalusi, an 11th C Spanish scholar and court historian was amongst the most enthusiastic in his praise of Indian civilization, and specially remarked on Indian achievements in the sciences and in Mathematics. Of course, eventually, Indian Indian Algebra and Trigonometry reached Europe through a Cycle of Translations, Travelling from the Arab world to Spain and Sicily, and eventually penetrating all of Europe. At the same time, Arabic and Persian translations of Greek and Egyptian scientific texts become more readily available in India. *

*The Kerala School*

Although it appears that original work in Mathematics ceased in much of Northern India after the Islamic Conquests, Benaras survived as a center for Mathematical Study, and an Important School of Mathematics blossomed in Kerala. **Madhava (14th C, Kochi) **made important Mathematical discoveries that would not be identified by European Mathematicians till at Least Two Centuries later. His *Series Expansion of the Cos and Sine Functions anticipated Newton by Almost Three Centuries. Historians of Mathematics, Rajagopal, Rangachari and Joseph considered his contributions Instrumental in taking Mathematics to the next stage, that of Modern Classical Analysis.* **Nilkantha (15th C, Tirur, Kerala)** extended and elaborated upon the results of Madhava while **Jyesthadeva (16th C, Kerala) **provided detailed Proofs of the Theorems and derivations of the rules contained in the works of Madhava and Nilkantha.* It is also notable that Jyesthadeva’s Yuktibhasa which contained Commentaries on Nilkantha’s Tantrasamgraha included elaborations on planetary theory later adopted by Tycho Brahe, and Mathematics that anticipated work by later Europeans. Chitrabhanu (16th C, Kerala) gave Integer Solutions to Twenty-One Types of Systems of Two Algebraic Equations, using both Algebraic and Geometric Methods in developing his results. *Important Discoveries by the Kerala Mathematicians included the Newton-Gauss Interpolation Formula, *The Formula for the Sum of an Infinite Series, And a Series Notation for Pi. *Charles Whish (1835, published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland) was one of the first Westerners to recognize that the Kerala School had anticipated by almost 300 Years many European developments in the field.

Yet, few Modern Compendiums on the history of Mathematics have paid adequate attention to the often* Pioneering and Revolutionary Contributions of Indian Mathematicians.* A Significant Body of Mathematical works were produced in the Indian subcontinent. *The Science of Mathematics played a pivotal role not only in The Industrial Revolution but in the Scientific Developments that have occurred since. No other Branch of Science is complete without Mathematics. Not only did India provide the Financial Capital for the Industrial Revolution India also provided vital elements of the scientific foundation without which Humanity could not have entered this Modern Age of Science and High Technology. *

*Facts:*

** Mathematics and Music: Pingala (3rd C AD),** Author of

**Chandasutra**explored the relationship between

*Combinatorics and Musical Theory*anticipating

**Mersenne (1588-1648) Author of a Classic on Musical Theory.**** Mathematics and Architecture: **Interest in

*Arithmetic and Geometric Series*may have also been stimulated by (and influenced)

*Of course, the*

**Indian Architectural Designs**– (as in Temple Shikaras, Gopurams and Corbelled Temple Ceilings).*Relationship between Geometry and Architectural Decoration*was developed to it’s Greatest Heights by

*Central Asian, Persian, Turkish, Arab and Indian Architects in a variety of monuments Commissioned by the Islamic rulers.*

** Transmission of the Indian Numeral System:** Evidence for the transmission of the

*Indian Numeral System*to the West is provided by Joseph (Crest of the Peacock):-

• **Quotes Severus Sebokht (662) in a Syriac **text describing the **“Subtle Discoveries” **of Indian Astronomers as being *“more Ingenious than those of the Greeks and the Babylonians” and “their valuable Methods Of Computation which surpass description”* and then goes on to mention the use of Nine Numerals.

• **Quotes from Liber abaci (Book of the Abacus) by Fibonacci (1170-1250): **The nine Indian Numerals are …with these nine and with the sign 0 which in Arabic is sifr, any desired Number can be written. (Fibonaci learnt about Indian Numerals from his Arab teachers in North Africa)

** Influence of the Kerala School:** Joseph (Crest of the Peacock) suggests that Indian Mathematical Manuscripts may have been brought to Europe by Jesuit priests such as Matteo Ricci

*who spent two years in Kochi (Cochin) after being ordained in Goa in 1580. Kochi is only 70km from Thrissur (Trichur) which was then the largest repository of Astronomical Documents.*

*Whish and Hyne – Two European Mathematicians obtained their copies of works by the Kerala Mathematicians from Thrissur,*and it is not inconceivable that Jesuit monks may have also taken copies to

*Pisa (where Galileo, Cavalieri and Wallis spent time), or Padau (where James Gregory studied) or Paris (where Mersenne who was in touch with Fermat and Pascal, acted as an agent for the transmission of Mathematical ideas).*

*Upanishadic Philosophy: Preparing the Ground for Rationalism *

Although the** Upanishadic Texts** (like some of the earlier Vedic Texts) are Primarily concerned with *Acquiring Knowledge of the “Soul”, “Spirit” And “God”* – there are aspects of* Vedic and Upanishadic Literature* that also Point to an intuitive understanding of Nature And Natural Processes. In addition, many of the ideas are presented in a Philosophical and exploratory manner – rather than as strict definitions of inviolable truth.

Although the Upanishadic Texts goaded the Upanishadic Student to concentrate on comprehending The Inner Spirit, Rational Investigation of the world by other Scholars was not entirely squelched, and eventually, the Upanishadic Period gave way to an Era which was not inimical to the Development of Rational Ideas, even encouraging *Scientific Observation and Advanced Study* in the *Fields of Logic, Mathematics and the Physical Sciences*.

Following an Era When Rituals and Superstitions had begun to proliferate, in some ways the Upanishadic Texts helped to clear the ground for greater rationalism in Society. Brahmin Orthodoxy and ideas of ritual purity were superseded by a spiritual perspective that eschewed sectarianism and could be practised universally, unfettered by an individual’s Social standing. Much of the emphasis was on discovering ** “spiritual truths”** for oneself as opposed to mechanically accepting the testimony of established religious leaders. Although there is a thematic Commonality to the Upanishadic Discourses, Different Commentators offered subtly varying perspectives and insights.

The Concept of God in Upanishadic (and even earlier Vedic) thinking was quite different from the more Common definition of God as Creator and Dispenser of Reward and Punishment. The Upanishadic Concept of God was more abstract and Philosophical. Different Texts postulated the Doctrine of a Universal Soul That Embraced All Physical Beings. All Life Emanated from This Universal Soul and Death Simply Caused Individual Manifestations of the Soul to Merge or mingle back with the Universal Soul. The Concept of a Universal Soul was illustrated through Analogies from Natural phenomenon.

*“As the Bees Make Honey by Collecting the Juices of Distant Trees, And Reduce the Juice into One form. And as these Juices have no Discrimination, so that they Might Say, I Am the Juice of this Tree or that, In the Same Manner, all these Creatures, when they have become merged in the True, know not that they are merged in the True. . . .”*

*“These Rivers run, the Eastern (like the Ganges) towards the East, the Western (like the Indus) towards the West. They Go From Sea to Sea (i.e., the Clouds Lift up the Water from the Sea to the Sky and Send It Back as Rain to the Sea). They become indeed Sea. And as those Rivers, when they are in the Sea, do not know, I am this or that River, in the same manner, all these Creatures, proceeding from the True, knows not that they have proceeded from the True. . . .”*

In another story, the** “ Wise” **father, expounder of the Upanishadic Concept of God, asks his son to Dissolve Salt in Water, and asked him to taste it from the Surface, from the middle and from the bottom. In each case, the son finds the taste to be salty. To this his father replies that the ‘universal being’ though invisible resides in all of us, just as the salt, though invisible is completely dissolved in the water. (Chanddogya, VI)

As a corollary to this theory emerged the notion that even as individual beings might refer to this universal soul – i.e. God in varied ways – By Using Different Names and Different Methods of Worship – all living beings were nevertheless related to each other and to the Universal God, and capable of merging with the universal god. This approach thus laid the foundation for egalitarian and *Non-Discriminatory Philosophies such as Buddhism and Jainism* (as well As Non-Sectarian Streams of Hinduism) that followed the Upanishadic Period. As is evident, such an approach was not incompatible with secular Society, and permitted different Faiths and Sub-Faiths to coexist in relative peace and harmony.

In the course of defining their Philosophy, the Scholars of the Upanishad Period raised several questions that challenged mechanical theism (as was also done in some * Hymns from the Rig Veda and Atharva Veda*). If God existed as the unique creator of the world, they wondered who created this unique creator. The logical pursuit of such a line of questioning could either lead to an infinite series of creators, or to the rejection or abandonment of this line of questioning. The Common theist solution to this Philosophical dilemma was to simply reject logic and demand unquestioning faith on the part of the believer. A few theists attempted to use this contradiction to their own advantage by positing that God existed precisely because “He” was indescribable by mere mortals. But, by and large, this contradiction was taken very seriously by the

**Philosophers of the Upanishadic Period.**The Upanishadic Philosophers attempted to resolve this contradiction by defining God as an entity that extended infinitely in all dimensions covering both space and time. This was a Philosophical Advance in that it attempted to come to terms with at least the most obvious challenges to the notion of God as a Human-Like Creator and did not require the complete rejection of logic.

Another Philosophical Advance of the Upanishadic Period was that religion was transformed from the realm of bookish parroting of Scriptures to the realm of Advanced Intellectual Debate and Polemics. The Upanishadic Philosophers did not lay down their conclusions as rigid Doctrines or inviolable Laws but as seductive parables – sometimes displaying remarkable worldly insight and analytical skill. By attempting to win over their followers through analogies from nature, and by employing the methods of abstract reasoning and debate, they created an

*Environment Where Dialectical Thinking and Intellectual Exchanges could later flourish*.

In the very process of their questioning, (and albeit speculative reasoning about God), they had opened the door for rationalists and even outright atheists who took their tentative questioning about the role and the character of God as “*Creator*” to conclusions that rejected theism entirely. But in either case, many rationalist and/or naturalist Philosophical streams emerged from this initial foundation. Some were nominally theistic (but in the abstract Upanishadic vein), others were agnostic (as the early Jains), while the early Buddhists and the Lokayatas were atheists. Thus even though the Upanishads contained much that should rightly be dismissed as Abstruse Intellectual Jugglery and Philosophical Mumbo-Jumbo, the Upanishadic Philosophers had levelled the ground for the seeds of rationalism to flourish in Indian soil.

*The Vaisheshika School*

The ** Vaisheshika School **(considered to be founded by Kanada, Author of the Vaisesika Sutra) was an early realistic School whose main achievement lay in it’s attempt at classifying nature into like and unlike groups. It also posited that all matter was made up of tiny and indestructible particles – i.e. Atoms that aggregated in different ways to form new compounds that formed the variety of matter that existed on the Earth.

Their Philosophy was described through the enumeration of the following Concepts: **Dravya **(Substance), **Guna **(Quality), **Karma **(Action), **Samanya **(Generality), **Visesa **(Particularity), **Samavaya **(Inherence) and **Abhava **(non-existence).* *

**Dravya (or Substance) **was understood as the specific result of a particular aggregate effect – i.e. the combination of atoms in a unique way. Substances were repositories for qualities and actions. Guna or quality was that which resided in a dravya. Qualities did not however contain qualities themselves. 24 qualities were enumerated, such as – *Color, Form, Smell, Touch, Sound, Number, Magnitude, Distinctions, Conjunction, Disjunction, Nearness, Remoteness, Heaviness, Fluidity and Viscosity. *(As was typical of the times, Psychological attributes such as *Pleasure, Pain, Desire, Aversion, Effort, Tendency, Cognition, Impression, and Ethical* attributes such as merit and demerit were also included in the list, i.e. – qualities that were inapplicable to inanimate objects were not treated separately)

** Action or Karma** represented physical movement. Unlike quality which was Passive, Karma was Dynamic. Action was the determinant of conjuction and disjunction. Five Types of action were noted:

*Throwing Upwards or Downwards, Contraction, Expansion and Locomotion.*

** Satta or Physical **existence was viewed as being the Common attribute of substance, quality and action – i.e. only existing (as opposed to imaginary) entities could have substance, qualities and be capable of action.

** Samanyata or ‘Generality’ **was seen as a mental construct to create Common classes of substances, qualities or actions while Visesata (particularity) was used to identify and separate individual items from their general classes. Samavaya or inherence was a relation that existed in those things that could not be separated without destroying them.

** Four Categories of Abhava as Negation or Non-Existance were listed**:

*Pragabhava or Prior Non-Existance,*referring to the absence of an object before it’s creation;

*Dhvamsabhava or Posterior Negation*, as the absence of an object after it had been destroyed;

*Anyonyabhava or Mutual Non-Existance*, refering to an object being distinct and different from the other;

*Atyantabhava or Absolute Non-Existence*, indicating non-existence in the past, present and future, citing the example of air as permanently lacking in smell – (which was presumably true in a Period where air pollution must have been uncommon!).

An important contribution of the * Vaisheshika School *was a careful study of the

*Time-Relation*in a chain of causes and effects. In a very rudimentary way, the School (along with other such Schools) anticipated the theory of

*Time Calculus*which could also be extended to

*Space Calculus*.

The *Vaisheshika School* **thus served as an important step in the study of Science by enumerating Concepts that could further the Study of Physics and Chemistry. In addition, the Study of Medical Science (including Veterinary Science) received considerable impetus from such attempts at methodical observation and classification. **

*The Nyaya and Related Schools*

The **Nyaya Schools** complemented and built on the* Vaisheshika School* by elaborating on the process of accumulating valid scientific knowledge through accurate perception and generating valid inferences.

**The School articulated four Means of acquiring valid knowledge:** *Pratyaksha Or Perception through One of the Senses; Anumana Or Inference; Upamana or Comparison with a Well-Known Object; Or Shabda – Verbal Testimony. *

*The conditions of perception, and it’s range and limits were carefully studied. **Trasarenu – The Minima Sensible (I.E. The Minimum Visible), Anubhuta-Rupa – The Infra-Sensible, Abhibhuta – The Obscured Perception , And Anubhuta-Vriti – Potential Perception, were recognized as different types of perception.*

A general methodology of ascertaining the *Truth (Tattva) was described which consisted of Describing a Proposition (Uddesa), The Ascertainment of Essential Facts Obtained through Perception, Inference or Induction (Laksan Or Uppa-Laksana), And Finally Examination And Verification (Pariksa And Nirnaya). *This process could involve Examples *(Drishtanta), Logical Arguments (Avayava), Reasoning (Tarka) And Discussion (Vada) – , Intellectual Exchange, or Interplay of Two Opposing Sides in The Process of Arriving at a Decisive Conclusion.* A Successful Application of this *Method Could Result in a Siddhanta *– I.E. Established Principle – (or in the *Case of Mathematics – A Theorem or Theory*) *Elucidated Through Proofs (Pramana)*. Alternatively, it could lead to a rejection of the initial proposition.

The Nyaya School identified various types of arguments that hindered or obstructed the path of Genuine Scientific pursuit, suggesting perhaps, that there may have been considerable practical resistance to their unstinting devotion to truth-seeking and scientific accuracy. They list the term Jalpa – An Argument not for the sake of arriving at the truth but for the sake of seeking victory *(this term was coined perhaps to distinguish Exaggerated and Rhetorical Arguments, or Hyperbole from Genuine Arguments); Vitanda (or Cavil) *to identify arguments that were specious or frivolous, or intended to Divert attention from the substance of the debate, that were put-downs intended to lower the dignity or credibility of the opponent; and Chal – Equivocation or Ruse to Confuse the Argument. ** Three Types of Chal are Listed: Vakchala – or Verbal Equivocation where the words of the Opponent are Deliberately Misused to Mean or Suggest Something Different than What was Intended; Samanyachala or False Generalization,** where the Opponents Arguments are deliberately and Incorrectly Generalized in a way to suggest that the Original Arguments were ridiculous or absurd; Uparachala – Misinterpreting a word which is used figuratively by taking it literally. Also mentioned is

*Jati, a type of fallacious argument*where an inapplicable similarity is cited to reject an argument, or conversely an irrelevant dissimilarity is cited to reject an argument.

The Nyaya School also recognized that Intelligent and Meaningful Debates were not possible if certain Fundamental Principles and Basic Definitions and Concepts were not mutually accepted. Nigrahasthana was the term used to Identify Disagreements based on absence of Mutually Acceptable First Principles. An example might be a debate between a Theist who Rejected Logic, and a Non-Theist who Rejected Faith.

The **Nyaya School** also listed *Five Classes of Logical Fallacies (Hetvabhasa) : Savyabhichara or the inconclusive type which employed reasoning from which more than one conclusion could be drawn but was used to insist on a single specific conclusion; Viruddha or Contradictory, where the reasoning used actually contradicted the proposition to be established; Kalatita – where the elapse of Time had made the Argument Invalid; Sadhyasama, the unproven type, where the reasoning employed rested on arguments or principles that had not been proven and require proofs themselves – i.e. this was the type of fallacy where one unproven result was merely converted into another unproven result.; and finally Prakaranasama – where the reasoning employed provoked the very question it was Designed to answer – i.e. a recursive fallacy.*

In this manner, the Nyaya School defined a very sophisticated School of Rational Philosophy where the Process of Scientific Epistemology was analyzed threadbare and all the dangers of unscientific reasoning and propaganda ploys were skilfully exposed.

*Causality*

**Buddhist and Jain scholars**, as well as later Hindu scholars offered their own approaches to scientific reasoning. Virtually all the rational schools were concerned with describing causality and causal relationships, and recognized that effects may not have single causes but may require a group or conjunction of causes to occur. Buddhist scholars emphasized that cause and effect need not have a linear effect but that desired effects may also require the right conditions for their fruition. (That is to say that for a plant to grow successfully, it would not only need the right seed, but that it would also need the *right type of soil, fertilization, sunlight and water*.)

Both the **Jains and the Buddhists** correctly speculated that a potential for the desired effect must also be present in the cause or causal agent. (For instance, only a mango seed could produce a mango tree because only the mango seed incorporated the potential of developing into a mango tree.) As another example, one could note that something with brittle properties such as glass might break upon impact whereas something strong such as steel would survive. Thus a physical impact on substances of different properties would have different results.

The Nyaya school also recognized co-effects – i.e a series of antecedants could cause a series of effects – either successive and staggered in time, or near simultaneous. Nyaya texts on causality indicate that there was an awareness that light travelled at a very high speed but the transmission of light was not instantaneous.

*Buddhist and Jain Atomic Theories*

The **Buddhist and Jain Philosophers** also proposed their own variations of the *Atomic Theory*. Like the *Vaisheshikas*, atoms were perceived as infinitely small by the Jainas. But the Jainas went a step further by positing that the union of atoms required opposite qualities in the *combining atoms – as is true in the case of electrovalent bonding*. However, they erred in thinking that covalent bonding (which does not require opposite polarities in the combining atoms) could not occur. But their intuition that opposite polarities created mutual attraction and facilitated chemical reactions was correct. In the Buddhist view, matter was in fact an aggregate of rapidly recurring forces or energy waves. Their theory was illustrated with examples drawn from natural phenomenon involved with light emission. An atom was perceived as a momentary flash of light combining and separating from other atoms according to strict and definite laws of causality. Physical matter was thus seen as a denser and more concentrated form of light. Although at odds with other atomic theories of the time, their approach fit in with their general view that all things in nature were temporal, that there was constant change in nature – that degradation and renewal were continuous processes.

*The Syadvada System of Jain Logic*

Jain Philosophers also made certain important contributions to the *Science of Epistemology *by proposing that the truth of a Concept or observation could not only be true or false but indeterminate – and combinations of the above – such as true under some conditions (or true at a particular time or place – or true based on the validity of certain inferences) and false under other conditions, or true under some conditions but indeterminate under others, and so on. This led to a matrix of seven possible states of the truth (t*rue, false, true or false, indeterminate, true or indeterminate, false or indeterminate, true or false or indeterminate*).

Jain rationalists also studied the relationship between the universal and the particular and made important Points concerning generalities and inDividual peculiarities. They also noted that objects in the real world exist in a network of relationships with each other – and have specific attributes that mark them temporally and spatially: “Every real is thus hedged round by a network of relations and attributes, which we propose to call its System or context or universe of discourse, which demarcates it from others.** ” Jaina philosophers also successfully synthesized earlier debates on change and permanence by positing that all objects (or parts of objects) passed through phases of “existence, persistence, and cessation” **and that reality was therefore a complex combination of things relatively permanent yet also relatively changing.

These ideas thus formed the foundations of Indian science and contributed to the gradual elaboration of Mathematics and Astronomy, as well as agricultural and meteorological sciences. Developments in metallurgy and civil engineering also followed. Medicine and surgery perhaps received the greatest and the earliest impetus from these developments. Developments in philosophy also led to concomitant developments in the realm of art and culture.

Yet to a considerable extent, knowledge about the progress of science and reason in Indian history is often scarce. These (and other such) historical contributions were either denied or deMeaned during the process of colonization, and are only now beginning to be re-acknowledged within India and abroad. But in A. D 1068, Indian contributions to the mainstream of science were held in great esteem and readily acknowledged in some parts of the world:

Here is what **Said Al-Andalusi, an 11th C Spanish scholar, court historian and chronicler wrote then**: *“Among the nations, during the course of centuries and throughout the passage of time, India was known as the mine of wisdom and the fountainhead of justice and good government and the Indians were credited with excellent intellects, exalted ideas, universal maxims, rare inventions and wonderful talents … They have studied Arithmetic and geometry. They have also acquired copious and abundant knowledge of the movements of the stars, the secrets of the celestial sphere and all other kinds of Mathematical sciences. Moreover, of all the peoples they are the most learned in the science of medicine and thoroughly informed about the properties of drugs, the nature of composite elements and peculiarities of the existing things.”*

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