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I don’t know how to have a complete answer for this. But to a certain extent, the research I did on Vedic Mathematics impels me to say that they make sense. And here is what I came across in my research.Vedic MathematicsIn many ancient cultures and civilizations, the development of mathematics was necessitated on account of religious practices and observances. These required an accurate calculation of the times of certain festivals and of the times auspicious for the performances of certain sacrifices and rituals. They also required a correct knowledge of the times of rising and the setting of the sun and the moon, and of the occurrences of the solar and lunar eclipses. All these meant a good knowledge of arithmetic, plane and spherical geometry and trigonometry, and possibly also the know-how of the construction of simple astronomical instruments.[1]The impression that science started only in Europe was deeply embedded in the minds of educated people all over the world until recently. The alchemists of Arab countries were occasionally mentioned, but there was very little reference to India and China. Thanks to the work of the Indian National Science Academy and other learned bodies, the development of science in India during both the ancient and medieval periods has recently been studied. It is becoming clearer from these studies that India has consistently been a scientific country right from Vedic to modern times with the usual fluctuations that can been expected of any country. In fact, a research will not find another civilization except that of ancient Greece, which accorded an exalted place to knowledge and science as in India.[2]It has to be universally acknowledged that much of mathematical knowledge in the world originated in India and moved from East to West. The high degree of sophistication in the use of mathematical symbols and developments in arithmetic, algebra, trigonometry, especially the work attributed to Aryabhatta, is indeed remarkable and should be a source of inspiration to all of us in India. The articles which describe Indian contributions to science from the ancient times to the very modern period bring out quite clearly the continuity of scientific thought as part of our cultural heritage. It is however, unfortunate that the period of decline in India coincided with that of ascendancy of Europe. It is perhaps the contrast during this period that made Europeans believe that all modern science was European.[3]An impending danger which I recognized as part of this study is the fact that Vedic Mathematics is now understood and taught in some institutions merely as an alternate method in facilitating easy and quick mental calculations. So, my study is twofold in nature. The first goal is to have an integral understanding about the domain of Vedic mathematics and second is to have a better understanding about the misconceived notion about Vedic Mathematics, which is becoming prevalent as part of some political or religious agenda. This is a very subjective and personal inquiry towards the disciplines dealt with in Vedic Mathematics which had substantial impact in the Mathematical pursuits of probably all the Indian Mathematicians and Schools of Mathematics.History and Development of Vedic MathematicsThe mathematical tradition in India goes back at least to the Vedas. For compositions with a broad scope covering all aspects of live, spiritual as well as secular, the Vedas show a great fascination for large numbers. As the transmission of the knowledge was oral the numbers were not written, but they were expressed as combinations of powers of 10, and it would be reasonable to believe that when the decimal place value system for written numbers came into being it owed a great deal to the way numbers were discussed in the older compositions.[4] (See the second section of this chapter)It is well known that Geometry was pursued in India in the context of construction of vedis[5] for the yaj̀nas of the Vedic period. The Śulvasūtras[6]contain elaborate descriptions of constructions of vedis and also enunciate various geometric principles. These were composed in the first millennium BC, the earliest Baudhāyana Śulvasūtra dating back to about 800 BC. The Śulvasūtra geometry did not go very far in comparison to the Euclidean geometry developed by the Greeks, who appeared on the scene a little later, in the seventh century BC. It was however an important stage of development in India too. The Śulvasūtra geometers were aware, among other things, of what is now called the Pythagorean theorem, over two hundred years before Pythagoras (all the four major Śulvasūtras contain explicit statement of the theorem), addressed (within the framework of their geometry) issues such as finding a circle with the same area as a square and vice versa and worked out a very good approximation to the square root of 2, in the course of their studies.[7]Though it is generally not recognized, the Śulvasūtras geometry was itself evolving. This is seen in particular from the differences in the contents of the four major extant Śulvasūtras. Certain revisions are especially striking. For instance, in the early Śulvasūtras period the ratio of the circumference to the diameter was, like in other ancient cultures, thought to be 3, as seen in the Sūtra of Baudhāyana, but in the Manava Śulvasūtra a new value was proposed, as 3 and one-fifth; interestingly the Sūtra describing it ends with an exultation “not a hair breadth remains”, and though we see that it is still substantially off the mark, it is a gratifying instance of an advance made. In the Manava Śulvasūtra one also finds an improvement over the method described by Baudhāyana for finding the circle with the same are as that of a given square.[8]Vedic Hindus evinced special interest in two particular branches of Mathematics, namely geometry (Śulva) and astronomy (Jyotiṣa). Sacrifice (Yaj̀na) was their prime religious avocation. Each sacrifice has to be performed on an altar of prescribed size and shape. They were very strict regarding this and thought that even a slight irregularity in the form and size of the altar would nullify the object of the whole ritual and might even lead to adverse effect. So, the greatest care was taken to have the right shape and size of the sacrificial altar. Thus the problems of geometry and consequently the science of geometry originated.[9]As it is evident, available sources of Vedic Mathematics are very poor. Almost all works on the subject have perished. At present we find only a very short treatise on Vedic astronomy in three rescensions, namely, in Ārca Jyotiṣa, Yājuṣa Jyotiṣa and Atharva Jyotiṣa. There are six small treatises on Vedic Geometry belonging to the six schools of the Vedas. Thus for an insight into Vedic Mathematics, we have to now depend more on secondary sources such as the literary works.[10]The study of astronomy began and developed chiefly out of the necessity for fixing the proper time for the sacrifice. This origin of the sciences as an aid to religion is not at all unnatural, for it’s generally found that the interest of a people in a particular branch of knowledge, in all climes and times, has been aroused and guided by specific reasons. In the case of the Vedic Hindu that specific reason was religious. In the course of time, however, those sciences outgrew their original purposes and came to be cultivated for their own sake.[11]The Chāndogya Upaniśad (VII.1.2, 4) mentions among other sciences the science of numbers (rāśi). In the Muņdaka Upaniṣad {I.2. 4-5} knowledge is classified as superior (parā) and inferior (aparā). In the second category is included the study of astronomy (jyotiṣa). In the Mahābhārata (XII.201) we come across a reference to the science of stellar motion (nakṣatragati). The term gaņita, meaning the science of calculation, also occurs copiously in Vedic literature. The Vedāńga jyotiṣa gives it the highest place of honour amongst all the sciences which form the Vedāńga. Thus it was said: ‘As are the crests on the heads of peacocks, as are the gems on the hoods of snakes, so is the gaņita at the top of the sciences known as the Vedāńga’. At that remote period gaņita included astronomy, arithmetic, and algebra, but not geometry. Geometry then belonged to a different group of sciences known as kalpa.[12]Vedic Astronomy – JyotiṣaVedāńga Jyotiṣa is one of the six ancillary branches of knowledge called the Ṣad- Vedāńgas, essentially dealing with astronomy. It must be remarked that although the word Jyotiṣa, in the modern common parlance, is used to mean predictive astrology, in the traditional literature, the word always meant the science of astronomy. Of course, mathematics was considered as a part of this subject. Vedāńga Jyotiṣa is the earliest Indian astronomical work available.[13]The purpose of Vedāńga Jyotiṣa was primarily to fix suitable times for performing the different sacrifices. It is found in two rescensions: the Ŗgveda Jyotiṣa and the Yajurveda Jyotiṣa. Though the contents of both the rescensions are the same, they differ in the number of verses contained in them. While the Ŗgvedic version contains only 36 verses, the Yajurvedic contains 44 verses. This difference in the number of verses is perhaps due to the addition of explanatory verses by the adhvaryu priests with whom it was in use.[14]Authorship and DateIn one of the verses of the Vedāńga Jyotiṣa, it is said, “I shall write on the lore of Time, as enunciated by sage Lagadha”. Therefore, the Vedāńga Jyotiṣa is attributed to Lagadha.[15]According to the text, at the time of Lagadha, the winter solstice was at the beginning of the constellation Śrāviṣṭhā (Delphini) and the summer solistice was at the midpoint of Āśleṣā. Since Vārāhamihira (505 – 587AD) stated that in his own time the summer solstice was at the end of the first quarter of Punarvasu and the winter solstice at the end of the first quarter of Uttarāṣāḍhā, There had been a precession of one and three quarters of nakṣatra or 230 20`. Since the rate of precession is about a degree in 72 years, the time interval for a precession of 230 20` is 72 * 231/3 = 1680 years prior to Vārāhamihira’s time i.e. around 1150 BC. According to the famous astronomer Prof. T. S. Kuppanna Sastri, if instead of the segment of nakṣatra, the group itself it smeant, which is about 30 within it, Lagadha’s time would be around 1370 BC. Therefore, the composition of Lagadha’s Vedāńga Jyotiṣa can be assigned to the period of 12th to 14th century BC.[16]The Vedāńga Jyotiṣa belongs to the last part of the Vedic age. The text proper can be considered as the records of the essentials of astronomical knowledge needed for the day to day life of the people of those times. The Vedāńga Jyotiṣa is the culmination of the knowledge developed and accumulated over thousands of years of the Vedic period prior to 1400 BC.[17]Furthermore, there is considerable material on astronomy in the Vedic Samhitas. But everything is shrouded in such mystic expressions and allegorical legends that it has now become extremely difficult to discern their proper significance. Hence it is not strange that modern scholars differ widely in evaluating the astronomical achievements of the early Vedic Hindus. Much progress seems, however, to have been made in the Brāhmana period when astronomy came to be regarded as a separate science called nakṣatra-vidyā (the science of stars). An astronomer was called a nakṣatra-darśa (star observer) or ganaka (calculator).[18]Numerous Discoveries AnticipatedNumerous amounts of mental calculations have been done in the Vedic era. The distance of the heaven from the earth have been calculated differently in various works. All of them are figurative expressions indicating that the extent of the universe is infinite. There is speculation in the Ŗg-Veda (V.85.5, VIII.42.1) about the extent of the earth. It appears from passages therein that the earth was considered to be spherical in shape (I.33.8) and suspended freely in the air. (IV.53.3) The Śatapatha Brāhmaņa describes it clearly as parimaņdala (globe or sphere). There is evidence in the Ŗg-Veda of the knowledge of the axial rotation and annual revolution of the earth. It was known that these motions are caused by the sun.[19] In fact, all the postulations and conclusions had been arrived at couple of millenniums prior to the discovery of the same by the westerners.There are also evident mentions about the zodiacal belt, the inclinations of the ecliptic with the equator and the axis of the earth. We see that the apparent annual course of the sun is divided into two halves. We also see that the ecliptic is divided into twelve parts or sings of the zodiac corresponding to the twelve months of the year, the sun moving through the consecutive signs during the successive months. The sun is called by different names at various parts of the zodiac, and thus has originated the idea of twelve ādityas or suns.[20]The Ŗg-Veda (IX.71.9 etc.) says that the moon shines by the borrowed light of the sun. The phases of the moon and their relations to the sun were fully understood. Five planets seem to have been known. The planets Sukra or Vena (Venus) and Manthin are mentioned by name.[21]Knowledge Through ObservationsIt appears from a passage in the Taittrīya Brāhmaņa (I.5.2.1) that Vedic astronomers ascertained the motion of the sun by observing with the naked eye the nearest visible stars rising and setting with the sun from day to day. This passage is considered very important ‘as it describes the method of making celestial observations in old times’. Observations of several solar eclipses are mentioned in the Ŗg-Veda, a passage of which states that the priests of the Atri family observed a total eclipse of the sun caused by its being covered by Svarbhānu, the darkening demon (V.40.5-9).[22] The Atri priests could calculate the occurrence, duration, beginning, and the end of the eclipse. Their descendants were particularly conversant with the calculation of eclipses. At the time of the Ŗg-Veda, the cause of the solar eclipse was understood as the occultation of the sun by the moon. There are also mentions of lunar eclipses.[23]Concept of Time and SeasonsThe Vedic people had considerable knowledge about the seasons of for sowing, reaping etc. Apart from that, they had acquired knowledge required for their religious activities, like the times and periodicity of the full and new moons, the last disappearance of the moon and its first appearance etc. This type of information was necessary for their monthly rites like the darśapūrņamāsa and seasonal rites cāturmāsya.[24]Vedic Hindus counted the beginning of a season on the sun’s entering a particular asterism. After a long interval of time, it was observed that the same season began with the sun entering a different asterism. Thus they discovered the falling back of the seasons with the position of the sun among the asterisms. There are also clear references to the vernal equinox in the asterism Punarvasu. There is also evidence to show that the vernal equinox was once in the asterism Mŗgaśirā from whence, in course of time, it receded to Kŗttikā. Thus there is clear evidence in the Samhitās and Brāhmaņas of the knowledge of the precession of the equinox. Some scholars maintain that Vedic Hindus also knew of the equation of time.[25] The practical way of measuring time is mentioned as the time taken by a specific quantity of water to flow through the orifice of a specified clepsydra (water-clock), as one nāḍikā i.e., 1/60 part of a day. The day is divided into 124 bhāgas (or parts) so that the ending moments of parvas and tithis can be given in whole units. The day is also divided into 603 units called kalās so that the duration of the lunar nakṣatra is given in whole units as 610 kalās. The nakṣatra is divided into 124 amśas so that the nakṣatraspassed at the ends of the parvas may be expressed in whole amśas. [26]During the Yajurveda period, it was known that the solar year had 365 days and a fraction more. In the Kŗṣna Yajurveda (Taittrīya Samhita VII. 2.6) it is mentioned that the extra 11 days over 12 lunar months, Caitra, Vaiśākha etc. (totaling to 354 days), complete the ŗtus by the performance of the ekādaśarātra or eleven day sacrifice. Again, the Taittrīya Samhita says that 5 days more were required over the sāvana year of 360 days to complete the season adding specifically that 4 days are too short and 6 days too long.[27]In the Yajurveda period, they had recognized the six ŗtus (seasons) in a period of 12 tropical months of the year and named them as follows:SeasonsMonthsi.Vasanta ŗtuMadhu and Mādhavaii.Grīṣma ŗtuŚukra and Śuciiii.Varṣa ŗtuNalcha and Nabhasyaiv.Śarad ŗtuIṣa and Ūrjav.Hemanta ŗtuSaha and Sahasyavi.Śiśira ŗtuTapa and TapasyaThe sacrificial year commenced with Vasanta ŗtu. Thy had alos noted that the shortest day was at the winter solstice when the seasonal year Śiśira began with Uttarāyana and rose to a maximum at the summer solstice.[28]Vedic ArithmeticIndia’s recognized contribution to mathematics was chiefly in the fields of arithmetic and algebra. In fact, Indian arithmetic is what is now used world over. The topics discussed in the Hindu Mathematics of early renaissance included the following:-[29]a.Parikarma (The four fundamental operations)b.Vyavahāra (determination)c.Rajju (meaning rope referring to geometry)d.Trairāśi (the rule of three)e.Yāvat tāvat (simple equations)f.Kalasavarņa (operations with fractions)g.Varga and Vargamūla (square and square root)h.Ghana and ghana-mūla (cube and cube root)i.Prastāra and Vikalpa (Permutations and Combinations)Sources of information on Vedic arithmetic being very meagre, it is difficult to define the topics for discussion and their scope of treatment. One problem that appears to have attracted the attention and interest of Vedic Hindus was to divide 1000 into 3 equal parts. It is unknown how the problem could be solved, for 1000 is not divisible by 3. So, an attempt has been made to explain the whole thing as a metaphorical statement. But a passage in the Śatapatha Brāhmaņa (III.3.1.13) seems clearly to belie all such speculations, saying: ‘When Indra and Viṣṇu divided a thousand into three parts, one remained in excess, and that they caused to be reproduced into three parts. Hence even now if any one attempts to divide a thousand by three, one remains over.’ In any case, it was a mathematical exercise.[30]Vedic Hindus developed the terminology of numeration to a high degree of perfection. The highest terminology that ancient Greeks knew was ‘myriad’ which denoted 104 and which came into use only about the fourth century BC. The Romans had to remain content with a ‘mille’ (103). But centuries before them, the Hindus had numerated up to parārdha (1014) which they could easily express without ambiguity or cumbrousness. The whole system is highly scientific and is very remarkable for its precision.[31]Scales of NumerationFrom the time of the Vedas, the Hindus adopted the decimal scale of numeration. They coined separate names for the notational places corresponding to 1, 10, 102, 103, 104, 105, etc. and any number, however big, used to be expressed in terms of them. But in expressing a number greater than 103 (sahasra) it was more usual to follow a centesimal scale. Thus, 50.103 was a more common form than 5.104. In Taittrīya Upaniṣad (II.8) the centesimal scale has been adopted in describing the different orders of bliss. Brahmānanda, or the bliss of Brahman, has been estimated as 10010 times the measure of one unit of human bliss.[32]In cases of actual measurements, the Hindus often followed other scales. For instance, we have in the Śatapatha Brāhmaņa (XII.3.2.5 et seq.) the minute subdivision of time on the scale of 15. The smallest unit prāṇa is given by 1/155 of a day. In the Vedāńga jyotiṣa (verse 31), a certain number is indicated as eka-dvi-saptika. If it really means ‘two-sevenths and one’ as it seems to do, then it will have to be admitted that there was once a septismal scale.[33]Representation of NumbersThe whole vocabulary of the number-names of the Vedic Hindus consisted mainly of thirty fundamental terms which can be divided into three groups, the ones, tens and the powers of ten. Furthermore, we can understand that Vedic Hindus had a unique and powerful method of their own in representing numbers. Because from the seals and inscriptions of Mohenjo-daro we can see that in the third millennium before the Christian era, numbers were represented in the Indus valley by means of vertical strokes arranged side by side or one group upon another. There were probably other signs for bigger numbers. Those rudimentary and cumbrous devices of rod-numerals were, however, quite useless for the representation of large numbers mentioned in the Vedas. In making calculations with such large numbers, as large as 1012, Vedic Hindus must have found the need for some shorter and more rapid method of representing numbers. This and other considerations give sufficient grounds for concluding that the Vedic Hindus had developed a much better system of numerical symbols.[34]From a reference in the Aṣṭādhyāyī of Pāṇini, we come to know that the letters of the alphabet were used to denote numbers. Another favourite device of Vedic Hindus to indicate a particular number was to employ the names of things permanently connected with that number by tradition or other associations, and sometimes vice versa. Applications of this are found in the earliest Saṁhitās. This practice of recording numbers with the help of letters and words became very popular in later times, especially amongst astronomers and mathematicians.[35]Holiness Attributed to NumbersIt appears that Vedic Hindus used to look upon some numbers as particularly holy. One such number is 3. In the Ṙg-Veda, the gods are grouped in three (I.105.5) and the mystical ‘three dawns’ are mentioned (VIII.41.3, X.67.4). Cases of magic where 3 is employed in a mysterious occult manner occur in the Ṙg-Veda and the Atharva-Veda. Even the number 180 is mentioned in the Ṙg-Veda as three sixties (VIII.96.8) and 210 as three seventies (VIII.19.37). The number regarded as most sacred seems to have been 7. Thus in the Ṙg-Veda, we get ‘seven seas’ (VIII.40.5), ‘seven rays of the sun’ (I.105.9), and ‘seven sages’ (IV.42.8, IX.92.2, etc.); and the number 49 is stated as seven sevens. Instances of combinations of these two numbers also occur. Thus 21 is stated as three sevens in the Ṙg-Veda (I.133.6, 191.12) and the Atharva-Veda (I.1.1), and 1,470 as three seven seventies in the Ṙg-Veda (VIII.46.26).[36]Classification of NumbersNumbers were divided into even (yugma, literally pair) and odd (ayugma, literally non pair), but there is no further subdivisions of numbers. There is an apparent reference to Zero and recognition of the negative number in the Atharvaveda. Zero is called kṣudra (XIX.22.6) meaning trifling’; the negative number is indicated by the epithet anṛca (XIX.23.22), meaning ‘without a hymn’; and the positive number by ṛca (XIX.23.1), meaning ‘a sacred verse’. These designations were replaced in later times by ṛṇa (debt) and dhana (asset).[37] The fractions like half, quarter, one-eighth and one-sixteenth are referred to for the first time, in the history of mathematics, in the Ŗgveda. These fractions are respectively called ardha, pāda, śapha and kalā.[38]Number SeriesVedic Hindus became interested in numbers forming series or progressions. The TaittrīyaSaṁhitā (VII.2.12-17) mentions the following arithmetical series:(i)1, 3, 5,...19,.......99;(ii)2, 4, 6, ...............100;(iii)4, 8, 12, .............100;(iv)5, 10, 15, ............100; and(v)10, 20, 30, ..........100.The arithmetical series are classified into ayugma and yugma. The Vajasaneyi Saṁhitā (XVIII.24.25) has given the following two instances:(i)1, 3, 5, .............31 and(ii)4, 8,12,...............48.The first series occurs also in the Taittrīya Saṁhitā (IV.3.10). The Paňcavimśa Brāhmana (XVIII.3) describes a list of sacrificial gifts forming a geometrical series of some interest and particular nature.12, 24, 48, 96, 192, ..............49152, 98304, 196608, 393216.This series reappears in the Śrauta-sūtras. Some method for the summation of series was also known. Thrice the sum of an arithmetical progression whose first term is 24, the common difference 4 and number of terms 7 is stated correctly in the ŚatapathaBrāhmaņa as 756.[39] . Moreover, from the texts available to us, it is obvious that Vedic Hindus knew how to perform fundamental arithmetical operations even with elementary fractions.[40]Regarding solutions of quadratic equations, we see that problems for solving quadratic equations of the form ax2 = c and ax2 + bx = c are stated and proved. We also find some indeterminate equations of the first degree (wrongly called as Diophantine equations by modern mathematicians). Furthermore, an important feature, peculiar to the Vedic mathematics, was to use geometrical diagrams and methods to solve algebraic problems and identities. Such geometrical techniques are beautifully adopted in the Śulvasūtras to solve algebraic equations.[41]The ŚulvasūtrasThe most ancient mathematical texts of the Vedic lore are the Śulvasūtras which form a part of the Śrauta section of the Kalpa Vedāńga. In the Śulvasūtras are seen very remarkable and rich principles of mathematics, particularly geometry. The word Śulva (or Śulba) is derived from the verb-root, Śulv or Śulb which means to measure. Since for measure length and breadth rope (rajju) was used, the word Śulva, in course of time came to mean a rope. In fact, Geometry (now referred to as Rekhāgaņita) was called Śulva Sāstra or Rajju Sāstra in ancient http://times.It is believed that these Śulvasūtras were composed around eight or nine centuries before Christ. [42]In the Vedic religion, every household man (barring the Sanyasis who would concentrate on meditation for years uninterruptedly) had to do certain acts of worship every day. It would be sinful if he neglected them. For purposes of worship, he would constantly maintain in his house three types of Agnis or fires sheltering them in certain altars of special designs. The required altars had to be constructed with great care so as to conform to certain specific shapes and areas.[43] While the three Agnis were to be used for the daily or routine Pujas or acts of worship, there were more elaborate sacrifices or Pujas for attaining cherished objects or wants. They were called Kamyagnis. The sacrificial altars for these Kamyagnis required more complicated constructions involving combinations of rectangles, triangles and trapeziums. It is clear that these processes require a clear knowledge of the properties of triangles, rectangles and squares, properties of similar figures, and a solution of the problem of ‘squaring the circle’ and its converse, ‘circling the square’ (i.e., to construct a square equal in area to a given circle, and vice versa).[44]Source and OriginOnly seven of the Śulvasūtras are known at present. They are known by the names Baudhāyana, Āpasthamba, Kātyāyana, Manava, Maitrayana, Varaha and Vadhūla after the names of the Rishis or sages who wrote them. The Kātyāyana Sūtra belongs to the section of the Vedas called Sukla Yajurveda while all the rest belong to Krishna Yajurveda. The Bodhayana, Āpasthamba and Kātyāyana Śulvas are of importance from the mathematical point of view.[45]The dates of these Śulvasūtras have been estimated to be between 800 BC and 500 BC. There is no knowledge about the existence of any Śulvas prior to these seven Sūtras. It must be emphasized that the writers of the Śulvasūtras only wrote down and codified the rules for the constructions of the altars, which were in vogue from ancient times. They were not the persons who specified and directed the rules for the constructions of the altars.[46]Simple Theorems in ŚulvasūtrasThe Śulvas explain a large number of simple geometrical constructions- constructions of squares, rectangles, parallelograms and trapeziums. The following geometrical theorems are either explicitly mentioned or clearly implied in the constructions of the altars of the prescribed shapes and sizes.[47]a.The diagonal of a rectangle divides it into two equal parts.b.The diagonal of a rectangle bisect each other and the opposite areas are equal.c.The perpendicular through the vertex of an isosceles triangle on the base divides the triangle into equal halves.d.A rectangle and a parallelogram on the same base and between the same parallels are equal in area.e.The diagonals of a rhombus bisect each other at right angles.f.The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. (The famous theorem known after the name of Pythagoras)g.Properties of similar rectilinear figures.h.If the sum of the squares on two sides of a triangle be equal to the square on the third side, then the triangle is right-angled. (This is the converse of the Pythagoras theorem, nay the Śulva theorem!)How the Mathematical Theorems were derived?These above mentioned theorems of Śulvasūtras cover roughly the first two books and the sixth book of Euclid’s ‘Elements’. How these theorems were actually obtained is a matter for which no definite answer is available. We all know that Euclid’s geometry is based upon certain axioms and postulates as I pointed out in the first chapter[48] and the proofs involve strict logical application of these. The logical methods of Greek Geometry are certainly not discernible in Hindu geometry. No book on Hindu mathematics explains the system of axioms and postulates assumed, and this itself should go some way in refuting the concocted claim that Hindu mathematics is borrowed from the Greeks. At the same time, it may not be correct to conclude that the above theorems were asserted as a matter of experience and measurement. The people who could make out and solve complicated problems of arithmetic, algebra and spherical trigonometry should be credited with some amount of logic in their work. The Śulvas are not formal mathematical treatises. They are only adjuncts to certain religious works. The question has to end with these remarks.[49] And probably from those very remarks does spring up, valid support for the idea, hypothesis and ultimate destiny of my attempt in this research paper.To make the view clearer, it could be stated that what Pythagoras’ theorem states was known, proved and applied in cases by the Vedic Hindus even before Pythagoras was born in the 6th century BC. Of course, the Vedic scholars did not prove the theorem though they stated and used it. But then there is no evidence that Pythagoras proved it either! The well known proof given by Euclid in this regard may be his own. But the theorem itself was known and widely used from very early times as we mentioned some 5 or 6 centuries prior to the birth of Euclid. It is interesting to note that A. Burk even argues that the much travelled Pythagoras borrowed the result from India![50]Geometrical Constructions contained in ŚulvasThe Vedis discussed in the Śulvasūtras are of various forms. Their constructions require a good knowledge of the properties of the square, the rectangle, the rhombus, the trapezium, the triangle, and of course, the circle. The following are among the important geometrical constructions used in the Śulvasūtras.[51](i)To divide a line-segment into any number of equal parts(ii)To divide a circle into any number of equal areas by drawing diameters (Baudhāyana, II. 73-74, Āpasthamba VII. 13-14)(iii)To divide a triangle into a number of equal and similar areas (Baudhāyana, III. 256)(iv)To draw a straight line at right-angles to a given line (Kātyāyana I. 3)(v)To draw a straight line at right-angles to a given line from a given point on it. (Kātyāyana I. 3)(vi)To construct a square on a given side(vii)To construct a rectangular of given length and breadth. (Baudhāyana I. 36-40)(viii)To construct an isosceles trapezium of given altitude, face and base. (Baudhāyana I. 41, Āpasthamba V. 2-5)(ix)To construct a parallelogram having the given sides at a given inclination. (Āpasthamba XIX. 5)(x)To construct a square equal to the sum of two different squares. (Baudhāyana I. 51-52, Āpasthamba II. 4-6, Kātyāyana II. 22)(xi)To construct a square equivalent to two given triangles.(xii)To construct a square equivalent to two given pentagons. (Baudhāyana III. 68, 288, Kātyāyana IV. 8)(xiii)To construct a square equal to a given rectangle in area. (Baudhāyana I. 58, Āpasthamba II. 7, Kātyāyana III. 2)(xiv)To construct a rectangle having a given side and equivalent to a given square.(xv)To construct an isosceles trapezium having a given face and equivalent to a given square or rectangle. (Baudhāyana I. 55)(xvi)To construct a triangle equivalent to a given square. (Baudhāyana I. 56)(xvii)To construct a square equivalent to a given isosceles triangle. (Kātyāyana IV. 5)(xviii)To construct a rhombus equivalent to a given square or rectangle. (Baudhāyana I. 57, Āpasthamba XII. 9)(xix)To construct a square equivalent to a given rhombus. (Kātyāyana IV. 6)Squaring a CircleOne of the greatest problems that had remained unsolved for centuries in the history of mathematics, till recently, was what is popularly known as “squaring the circle”, i.e., to construct – using only ruler and compasses – a square whose area is equal to that of a given circle. IT is really remarkable that this very problem and vice versa, was tackled by the authors of the Śulvasūtras. They gave practical methods for constructing a square whose area is equal to that of a given circle and vice versa. Of course, their constructions involved approximating the value of the well-known constant number Pi to 3.088 (Pi is the ratio of the circumference of any circle to its diameter). The approximation made by the Vedic Ŗṣis is quite justifiable and admirable in the light of crude mathematical tools and methods available to them thousands of years ago. IT is now, however, in modern mathematics established that an exact construction to “Square a circle” or “to circle a square” (in the sense of the areas) is impossible. Incidentally, it must be pointed out that the approximate value of Pi used in the Śulvasūtras is certainly far better than the Biblical value, 3 (see Kings VII. 23 and Chronicles IV. 2) given many centuries later! In fact, it is only as late as in 1761 AD that Lambert proved that Pi is “irrational” and only in 1882 AD that Lindemann established further that Pi is transcendental.[52]Vedic Knowledge of Irrational NumbersThe essentially arithmetical background of the Śulva mathematics must be contrasted with the essentially geometrical background characteristic of Greek mathematics. Simple fractions and operations on them are available in the Śulvas. Moreover, we find the use of Irrational numbers in them. Surds (Irrational numbers) of the form √2, √3 etc. are called Karanis, thus √2 is dwi-karani, √3 is tri-karani, √1/3 is triteeya karani, √1/7 is saptama karani, so on.[53]The shape of Ashwamedhiki Vedika is an isosceles trapezium whose head, foot and altitude are respectively 24√2, 30√2 , 36√2 prakramas. Its area is stated to be 1944 prakramas (square is to be understood).Area = 24√2 * 1/2(30√2 * 36√2) = 1944.This indicates knowledge of the method of finding the area of a trapezium and simple operations on surds.[54]A remarkable approximation of √2 occurs in each of the three Śulvas Bodhayana, Āpasthamba and Kātyāyana.√2 = 1 + 1/3 + 1/ (3.4) – 1/(3.4 34)....This gives √2 = 1.4142156........., whereas the true value is 1.414213............ The approximation is thus correct to five decimal places, and is expressed by means of simple unit fractions. The problem evidently arises in the construction of a square double a given square in area. The Śulvas contain no clue at all as to the manner in which this remarkable approximation was arrived at. Many theories or plausible explanations have been proposed.[55]This point to the fact that the India, thanks to Vedic mathematics, was the first nation to use irrational numbers. It has to be believed that the Vedic Hindus knew its irrationality. The Greeks also used irrational numbers. If AB is a given segment, Pythagoras and others described the methods of constructing the segments of length√2AB, √3AB, √5AB, etc. But no rational approximation to √2, √3 etc. are found in Greek mathematics, nor are there any problems of the arithmetical operations on irrational numbers. This is easily explained, because the requisite knowledge of arithmetic was not available to the Greeks. It will also be borne in mind that according to unprejudiced estimates, the Śulvasūtras are about two or three centuries prior to Pythagoras.[56][1] Iyengar, The History of Ancient Indian Mathematics, 6.[2]Raja, “The Cultural Heritage of India”,1.[3] Raja, “The Cultural Heritage of India”, 2.[4] Dani, “Ancient Indian Mathematics - A conspectus”, 236.[5] The Vedic Culture had the unique feature of performances of five rituals, known as yajnas. These involved well laid out altars, the vedis, and fire platforms, known as citi or agni, elaborately constructed in the form of birds, tortoise, wheel, etc.[6] The SulvaSūtras are compositions, in the form of manuals for construction of vedis and citis, but they also discuss the geometric principles involved.[7] Dani, “Ancient Indian Mathematics - A conspectus”, 238.[8] Dani, “Ancient Indian Mathematics - A conspectus”, 239.[9] Datta, “Vedic Mathematics”, 18.[10] Datta, “Vedic Mathematics”, 19.[11] Datta, “Vedic Mathematics”, 18.[12] Datta, “The Scope and Development of Hindu Ganita”, 480.[13] Rao, Indian Mathematics and Astronomy, 23.[14] Rao, Indian Mathematics and Astronomy, 23.[15] Rao, Indian Mathematics and Astronomy, 23.[16] Rao, Indian Mathematics and Astronomy, 24.[17] Rao, Indian Mathematics and Astronomy, 24.[18] Datta, “Vedic Mathematics”, 19.[19] Ghosh, “Studies on Rig-Vedic Deities”, 11.[20] Tilak, “The Orion or Researches into the Antiquity of the Vedas, 25.[21] Datta, “Vedic Mathematics”, 20.[22] Datta, “Vedic Mathematics”, 21.[23] Tilak, “The Orion or Researches into the Antiquity of the Vedas, 26.[24] Rao, Indian Mathematics and Astronomy, 25.[25] Mookerjee, “Notes on Indian Astronomy”, 137.[26] Rao, Indian Mathematics and Astronomy, 25.[27] Rao, Indian Mathematics and Astronomy, 25.[28] Rao, Indian Mathematics and Astronomy, 26.[29] Rao, Indian Mathematics and Astronomy, 3.[30] Datta, “Vedic Mathematics”, 30.[31] Rao, Indian Mathematics and Astronomy, 3..[32] Datta, “Vedic Mathematics”, 31.[33] Datta, “Vedic Mathematics”, 31.[34] Datta, “Vedic Mathematics”, 33.[35] Datta, “Vedic Mathematics”, 34.[36] Hopkins, “Numerical Formulae in the Veda and their Bearing on Vedic Criticism”, 279.[37] Shukla, “The Deceptive Title of Swamiji’s Book”, 36.[38] Rao, Indian Mathematics and Astronomy, 14.[39] Shukla, “The Deceptive Title of Swamiji’s Book”, 37.[40] Mehta, Positive Sciences in the Vedas, 114.[41] Rao, Indian Mathematics and Astronomy, 12.[42] Rao, Indian Mathematics and Astronomy, 12.[43] Mehta, Positive Sciences in the Vedas, 117.[44] Iyengar, The History of Ancient Indian Mathematics, 6.[45] Iyengar, The History of Ancient Indian Mathematics, 7.[46] Iyengar, The History of Ancient Indian Mathematics, 8.[47] Rao, Indian Mathematics and Astronomy, 14.[48] Chapter I (1.2.1) Geometry, 16-21.[49] Iyengar, The History of Ancient Indian Mathematics, 9.[50] Rao, Indian Mathematics and Astronomy, 15.[51] Rao, Indian Mathematics and Astronomy, 15-17.[52] Rao, Indian Mathematics and Astronomy, 18.[53] Iyengar, The History of Ancient Indian Mathematics, 13.[54] Iyengar, The History of Ancient Indian Mathematics, 13.[55] Iyengar, The History of Ancient Indian Mathematics, 14.[56] Iyengar, The History of Ancient Indian Mathematics, 15.

Here is the result of what I got to know about Vedic Mathematics. And I have presented these in my research paper.Vedic MathematicsIn many ancient cultures and civilizations, the development of mathematics was necessitated on account of religious practices and observances. These required an accurate calculation of the times of certain festivals and of the times auspicious for the performances of certain sacrifices and rituals. They also required a correct knowledge of the times of rising and the setting of the sun and the moon, and of the occurrences of the solar and lunar eclipses. All these meant a good knowledge of arithmetic, plane and spherical geometry and trigonometry, and possibly also the know-how of the construction of simple astronomical instruments.[1]The impression that science started only in Europe was deeply embedded in the minds of educated people all over the world until recently. The alchemists of Arab countries were occasionally mentioned, but there was very little reference to India and China. Thanks to the work of the Indian National Science Academy and other learned bodies, the development of science in India during both the ancient and medieval periods has recently been studied. It is becoming clearer from these studies that India has consistently been a scientific country right from Vedic to modern times with the usual fluctuations that can been expected of any country. In fact, a research will not find another civilization except that of ancient Greece, which accorded an exalted place to knowledge and science as in India.[2]It has to be universally acknowledged that much of mathematical knowledge in the world originated in India and moved from East to West. The high degree of sophistication in the use of mathematical symbols and developments in arithmetic, algebra, trigonometry, especially the work attributed to Aryabhatta, is indeed remarkable and should be a source of inspiration to all of us in India. The articles which describe Indian contributions to science from the ancient times to the very modern period bring out quite clearly the continuity of scientific thought as part of our cultural heritage. It is however, unfortunate that the period of decline in India coincided with that of ascendancy of Europe. It is perhaps the contrast during this period that made Europeans believe that all modern science was European.[3]An impending danger which I recognized as part of this study is the fact that Vedic Mathematics is now understood and taught in some institutions merely as an alternate method in facilitating easy and quick mental calculations. So, my study is twofold in nature. The first goal is to have an integral understanding about the domain of Vedic mathematics and second is to have a better understanding about the misconceived notion about Vedic Mathematics, which is becoming prevalent as part of some political or religious agenda. This is a very subjective and personal inquiry towards the disciplines dealt with in Vedic Mathematics which had substantial impact in the Mathematical pursuits of probably all the Indian Mathematicians and Schools of Mathematics.History and Development of Vedic MathematicsThe mathematical tradition in India goes back at least to the Vedas. For compositions with a broad scope covering all aspects of live, spiritual as well as secular, the Vedas show a great fascination for large numbers. As the transmission of the knowledge was oral the numbers were not written, but they were expressed as combinations of powers of 10, and it would be reasonable to believe that when the decimal place value system for written numbers came into being it owed a great deal to the way numbers were discussed in the older compositions.[4] (See the second section of this chapter)It is well known that Geometry was pursued in India in the context of construction of vedis[5] for the yaj̀nas of the Vedic period. The Śulvasūtras[6]contain elaborate descriptions of constructions of vedis and also enunciate various geometric principles. These were composed in the first millennium BC, the earliest Baudhāyana Śulvasūtra dating back to about 800 BC. The Śulvasūtra geometry did not go very far in comparison to the Euclidean geometry developed by the Greeks, who appeared on the scene a little later, in the seventh century BC. It was however an important stage of development in India too. The Śulvasūtra geometers were aware, among other things, of what is now called the Pythagorean theorem, over two hundred years before Pythagoras (all the four major Śulvasūtras contain explicit statement of the theorem), addressed (within the framework of their geometry) issues such as finding a circle with the same area as a square and vice versa and worked out a very good approximation to the square root of 2, in the course of their studies.[7]Though it is generally not recognized, the Śulvasūtras geometry was itself evolving. This is seen in particular from the differences in the contents of the four major extant Śulvasūtras. Certain revisions are especially striking. For instance, in the early Śulvasūtras period the ratio of the circumference to the diameter was, like in other ancient cultures, thought to be 3, as seen in the Sūtra of Baudhāyana, but in the Manava Śulvasūtra a new value was proposed, as 3 and one-fifth; interestingly the Sūtra describing it ends with an exultation “not a hair breadth remains”, and though we see that it is still substantially off the mark, it is a gratifying instance of an advance made. In the Manava Śulvasūtra one also finds an improvement over the method described by Baudhāyana for finding the circle with the same are as that of a given square.[8]Vedic Hindus evinced special interest in two particular branches of Mathematics, namely geometry (Śulva) and astronomy (Jyotiṣa). Sacrifice (Yaj̀na) was their prime religious avocation. Each sacrifice has to be performed on an altar of prescribed size and shape. They were very strict regarding this and thought that even a slight irregularity in the form and size of the altar would nullify the object of the whole ritual and might even lead to adverse effect. So, the greatest care was taken to have the right shape and size of the sacrificial altar. Thus the problems of geometry and consequently the science of geometry originated.[9]As it is evident, available sources of Vedic Mathematics are very poor. Almost all works on the subject have perished. At present we find only a very short treatise on Vedic astronomy in three rescensions, namely, in Ārca Jyotiṣa, Yājuṣa Jyotiṣa and Atharva Jyotiṣa. There are six small treatises on Vedic Geometry belonging to the six schools of the Vedas. Thus for an insight into Vedic Mathematics, we have to now depend more on secondary sources such as the literary works.[10]The study of astronomy began and developed chiefly out of the necessity for fixing the proper time for the sacrifice. This origin of the sciences as an aid to religion is not at all unnatural, for it’s generally found that the interest of a people in a particular branch of knowledge, in all climes and times, has been aroused and guided by specific reasons. In the case of the Vedic Hindu that specific reason was religious. In the course of time, however, those sciences outgrew their original purposes and came to be cultivated for their own sake.[11]The Chāndogya Upaniśad (VII.1.2, 4) mentions among other sciences the science of numbers (rāśi). In the Muņdaka Upaniṣad {I.2. 4-5} knowledge is classified as superior (parā) and inferior (aparā). In the second category is included the study of astronomy (jyotiṣa). In the Mahābhārata (XII.201) we come across a reference to the science of stellar motion (nakṣatragati). The term gaņita, meaning the science of calculation, also occurs copiously in Vedic literature. The Vedāńga jyotiṣa gives it the highest place of honour amongst all the sciences which form the Vedāńga. Thus it was said: ‘As are the crests on the heads of peacocks, as are the gems on the hoods of snakes, so is the gaņita at the top of the sciences known as the Vedāńga’. At that remote period gaņita included astronomy, arithmetic, and algebra, but not geometry. Geometry then belonged to a different group of sciences known as kalpa.[12]Vedic Astronomy – JyotiṣaVedāńga Jyotiṣa is one of the six ancillary branches of knowledge called the Ṣad- Vedāńgas, essentially dealing with astronomy. It must be remarked that although the word Jyotiṣa, in the modern common parlance, is used to mean predictive astrology, in the traditional literature, the word always meant the science of astronomy. Of course, mathematics was considered as a part of this subject. Vedāńga Jyotiṣa is the earliest Indian astronomical work available.[13]The purpose of Vedāńga Jyotiṣa was primarily to fix suitable times for performing the different sacrifices. It is found in two rescensions: the Ŗgveda Jyotiṣa and the Yajurveda Jyotiṣa. Though the contents of both the rescensions are the same, they differ in the number of verses contained in them. While the Ŗgvedic version contains only 36 verses, the Yajurvedic contains 44 verses. This difference in the number of verses is perhaps due to the addition of explanatory verses by the adhvaryu priests with whom it was in use.[14]Authorship and DateIn one of the verses of the Vedāńga Jyotiṣa, it is said, “I shall write on the lore of Time, as enunciated by sage Lagadha”. Therefore, the Vedāńga Jyotiṣa is attributed to Lagadha.[15]According to the text, at the time of Lagadha, the winter solstice was at the beginning of the constellation Śrāviṣṭhā (Delphini) and the summer solistice was at the midpoint of Āśleṣā. Since Vārāhamihira (505 – 587AD) stated that in his own time the summer solstice was at the end of the first quarter of Punarvasu and the winter solstice at the end of the first quarter of Uttarāṣāḍhā, There had been a precession of one and three quarters of nakṣatra or 230 20`. Since the rate of precession is about a degree in 72 years, the time interval for a precession of 230 20` is 72 * 231/3 = 1680 years prior to Vārāhamihira’s time i.e. around 1150 BC. According to the famous astronomer Prof. T. S. Kuppanna Sastri, if instead of the segment of nakṣatra, the group itself it smeant, which is about 30 within it, Lagadha’s time would be around 1370 BC. Therefore, the composition of Lagadha’s Vedāńga Jyotiṣa can be assigned to the period of 12th to 14th century BC.[16]The Vedāńga Jyotiṣa belongs to the last part of the Vedic age. The text proper can be considered as the records of the essentials of astronomical knowledge needed for the day to day life of the people of those times. The Vedāńga Jyotiṣa is the culmination of the knowledge developed and accumulated over thousands of years of the Vedic period prior to 1400 BC.[17]Furthermore, there is considerable material on astronomy in the Vedic Samhitas. But everything is shrouded in such mystic expressions and allegorical legends that it has now become extremely difficult to discern their proper significance. Hence it is not strange that modern scholars differ widely in evaluating the astronomical achievements of the early Vedic Hindus. Much progress seems, however, to have been made in the Brāhmana period when astronomy came to be regarded as a separate science called nakṣatra-vidyā (the science of stars). An astronomer was called a nakṣatra-darśa (star observer) or ganaka (calculator).[18]Numerous Discoveries AnticipatedNumerous amounts of mental calculations have been done in the Vedic era. The distance of the heaven from the earth have been calculated differently in various works. All of them are figurative expressions indicating that the extent of the universe is infinite. There is speculation in the Ŗg-Veda (V.85.5, VIII.42.1) about the extent of the earth. It appears from passages therein that the earth was considered to be spherical in shape (I.33.8) and suspended freely in the air. (IV.53.3) The Śatapatha Brāhmaņa describes it clearly as parimaņdala (globe or sphere). There is evidence in the Ŗg-Veda of the knowledge of the axial rotation and annual revolution of the earth. It was known that these motions are caused by the sun.[19] In fact, all the postulations and conclusions had been arrived at couple of millenniums prior to the discovery of the same by the westerners.There are also evident mentions about the zodiacal belt, the inclinations of the ecliptic with the equator and the axis of the earth. We see that the apparent annual course of the sun is divided into two halves. We also see that the ecliptic is divided into twelve parts or sings of the zodiac corresponding to the twelve months of the year, the sun moving through the consecutive signs during the successive months. The sun is called by different names at various parts of the zodiac, and thus has originated the idea of twelve ādityas or suns.[20]The Ŗg-Veda (IX.71.9 etc.) says that the moon shines by the borrowed light of the sun. The phases of the moon and their relations to the sun were fully understood. Five planets seem to have been known. The planets Sukra or Vena (Venus) and Manthin are mentioned by name.[21]Knowledge Through ObservationsIt appears from a passage in the Taittrīya Brāhmaņa (I.5.2.1) that Vedic astronomers ascertained the motion of the sun by observing with the naked eye the nearest visible stars rising and setting with the sun from day to day. This passage is considered very important ‘as it describes the method of making celestial observations in old times’. Observations of several solar eclipses are mentioned in the Ŗg-Veda, a passage of which states that the priests of the Atri family observed a total eclipse of the sun caused by its being covered by Svarbhānu, the darkening demon (V.40.5-9).[22] The Atri priests could calculate the occurrence, duration, beginning, and the end of the eclipse. Their descendants were particularly conversant with the calculation of eclipses. At the time of the Ŗg-Veda, the cause of the solar eclipse was understood as the occultation of the sun by the moon. There are also mentions of lunar eclipses.[23]Concept of Time and SeasonsThe Vedic people had considerable knowledge about the seasons of for sowing, reaping etc. Apart from that, they had acquired knowledge required for their religious activities, like the times and periodicity of the full and new moons, the last disappearance of the moon and its first appearance etc. This type of information was necessary for their monthly rites like the darśapūrņamāsa and seasonal rites cāturmāsya.[24]Vedic Hindus counted the beginning of a season on the sun’s entering a particular asterism. After a long interval of time, it was observed that the same season began with the sun entering a different asterism. Thus they discovered the falling back of the seasons with the position of the sun among the asterisms. There are also clear references to the vernal equinox in the asterism Punarvasu. There is also evidence to show that the vernal equinox was once in the asterism Mŗgaśirā from whence, in course of time, it receded to Kŗttikā. Thus there is clear evidence in the Samhitās and Brāhmaņas of the knowledge of the precession of the equinox. Some scholars maintain that Vedic Hindus also knew of the equation of time.[25] The practical way of measuring time is mentioned as the time taken by a specific quantity of water to flow through the orifice of a specified clepsydra (water-clock), as one nāḍikā i.e., 1/60 part of a day. The day is divided into 124 bhāgas (or parts) so that the ending moments of parvas and tithis can be given in whole units. The day is also divided into 603 units called kalās so that the duration of the lunar nakṣatra is given in whole units as 610 kalās. The nakṣatra is divided into 124 amśas so that the nakṣatraspassed at the ends of the parvas may be expressed in whole amśas. [26]During the Yajurveda period, it was known that the solar year had 365 days and a fraction more. In the Kŗṣna Yajurveda (Taittrīya Samhita VII. 2.6) it is mentioned that the extra 11 days over 12 lunar months, Caitra, Vaiśākha etc. (totaling to 354 days), complete the ŗtus by the performance of the ekādaśarātra or eleven day sacrifice. Again, the Taittrīya Samhita says that 5 days more were required over the sāvana year of 360 days to complete the season adding specifically that 4 days are too short and 6 days too long.[27]In the Yajurveda period, they had recognized the six ŗtus (seasons) in a period of 12 tropical months of the year and named them as follows:SeasonsMonthsi.Vasanta ŗtuMadhu and Mādhavaii.Grīṣma ŗtuŚukra and Śuciiii.Varṣa ŗtuNalcha and Nabhasyaiv.Śarad ŗtuIṣa and Ūrjav.Hemanta ŗtuSaha and Sahasyavi.Śiśira ŗtuTapa and TapasyaThe sacrificial year commenced with Vasanta ŗtu. Thy had alos noted that the shortest day was at the winter solstice when the seasonal year Śiśira began with Uttarāyana and rose to a maximum at the summer solstice.[28]Vedic ArithmeticIndia’s recognized contribution to mathematics was chiefly in the fields of arithmetic and algebra. In fact, Indian arithmetic is what is now used world over. The topics discussed in the Hindu Mathematics of early renaissance included the following:-[29]a.Parikarma (The four fundamental operations)b.Vyavahāra (determination)c.Rajju (meaning rope referring to geometry)d.Trairāśi (the rule of three)e.Yāvat tāvat (simple equations)f.Kalasavarņa (operations with fractions)g.Varga and Vargamūla (square and square root)h.Ghana and ghana-mūla (cube and cube root)i.Prastāra and Vikalpa (Permutations and Combinations)Sources of information on Vedic arithmetic being very meagre, it is difficult to define the topics for discussion and their scope of treatment. One problem that appears to have attracted the attention and interest of Vedic Hindus was to divide 1000 into 3 equal parts. It is unknown how the problem could be solved, for 1000 is not divisible by 3. So, an attempt has been made to explain the whole thing as a metaphorical statement. But a passage in the Śatapatha Brāhmaņa (III.3.1.13) seems clearly to belie all such speculations, saying: ‘When Indra and Viṣṇu divided a thousand into three parts, one remained in excess, and that they caused to be reproduced into three parts. Hence even now if any one attempts to divide a thousand by three, one remains over.’ In any case, it was a mathematical exercise.[30]Vedic Hindus developed the terminology of numeration to a high degree of perfection. The highest terminology that ancient Greeks knew was ‘myriad’ which denoted 104 and which came into use only about the fourth century BC. The Romans had to remain content with a ‘mille’ (103). But centuries before them, the Hindus had numerated up to parārdha (1014) which they could easily express without ambiguity or cumbrousness. The whole system is highly scientific and is very remarkable for its precision.[31]Scales of NumerationFrom the time of the Vedas, the Hindus adopted the decimal scale of numeration. They coined separate names for the notational places corresponding to 1, 10, 102, 103, 104, 105, etc. and any number, however big, used to be expressed in terms of them. But in expressing a number greater than 103 (sahasra) it was more usual to follow a centesimal scale. Thus, 50.103 was a more common form than 5.104. In Taittrīya Upaniṣad (II.8) the centesimal scale has been adopted in describing the different orders of bliss. Brahmānanda, or the bliss of Brahman, has been estimated as 10010 times the measure of one unit of human bliss.[32]In cases of actual measurements, the Hindus often followed other scales. For instance, we have in the Śatapatha Brāhmaņa (XII.3.2.5 et seq.) the minute subdivision of time on the scale of 15. The smallest unit prāṇa is given by 1/155 of a day. In the Vedāńga jyotiṣa (verse 31), a certain number is indicated as eka-dvi-saptika. If it really means ‘two-sevenths and one’ as it seems to do, then it will have to be admitted that there was once a septismal scale.[33]Representation of NumbersThe whole vocabulary of the number-names of the Vedic Hindus consisted mainly of thirty fundamental terms which can be divided into three groups, the ones, tens and the powers of ten. Furthermore, we can understand that Vedic Hindus had a unique and powerful method of their own in representing numbers. Because from the seals and inscriptions of Mohenjo-daro we can see that in the third millennium before the Christian era, numbers were represented in the Indus valley by means of vertical strokes arranged side by side or one group upon another. There were probably other signs for bigger numbers. Those rudimentary and cumbrous devices of rod-numerals were, however, quite useless for the representation of large numbers mentioned in the Vedas. In making calculations with such large numbers, as large as 1012, Vedic Hindus must have found the need for some shorter and more rapid method of representing numbers. This and other considerations give sufficient grounds for concluding that the Vedic Hindus had developed a much better system of numerical symbols.[34]From a reference in the Aṣṭādhyāyī of Pāṇini, we come to know that the letters of the alphabet were used to denote numbers. Another favourite device of Vedic Hindus to indicate a particular number was to employ the names of things permanently connected with that number by tradition or other associations, and sometimes vice versa. Applications of this are found in the earliest Saṁhitās. This practice of recording numbers with the help of letters and words became very popular in later times, especially amongst astronomers and mathematicians.[35]Holiness Attributed to NumbersIt appears that Vedic Hindus used to look upon some numbers as particularly holy. One such number is 3. In the Ṙg-Veda, the gods are grouped in three (I.105.5) and the mystical ‘three dawns’ are mentioned (VIII.41.3, X.67.4). Cases of magic where 3 is employed in a mysterious occult manner occur in the Ṙg-Veda and the Atharva-Veda. Even the number 180 is mentioned in the Ṙg-Veda as three sixties (VIII.96.8) and 210 as three seventies (VIII.19.37). The number regarded as most sacred seems to have been 7. Thus in the Ṙg-Veda, we get ‘seven seas’ (VIII.40.5), ‘seven rays of the sun’ (I.105.9), and ‘seven sages’ (IV.42.8, IX.92.2, etc.); and the number 49 is stated as seven sevens. Instances of combinations of these two numbers also occur. Thus 21 is stated as three sevens in the Ṙg-Veda (I.133.6, 191.12) and the Atharva-Veda (I.1.1), and 1,470 as three seven seventies in the Ṙg-Veda (VIII.46.26).[36]Classification of NumbersNumbers were divided into even (yugma, literally pair) and odd (ayugma, literally non pair), but there is no further subdivisions of numbers. There is an apparent reference to Zero and recognition of the negative number in the Atharvaveda. Zero is called kṣudra (XIX.22.6) meaning trifling’; the negative number is indicated by the epithet anṛca (XIX.23.22), meaning ‘without a hymn’; and the positive number by ṛca (XIX.23.1), meaning ‘a sacred verse’. These designations were replaced in later times by ṛṇa (debt) and dhana (asset).[37] The fractions like half, quarter, one-eighth and one-sixteenth are referred to for the first time, in the history of mathematics, in the Ŗgveda. These fractions are respectively called ardha, pāda, śapha and kalā.[38]Number SeriesVedic Hindus became interested in numbers forming series or progressions. The TaittrīyaSaṁhitā (VII.2.12-17) mentions the following arithmetical series:(i)1, 3, 5,...19,.......99;(ii)2, 4, 6, ...............100;(iii)4, 8, 12, .............100;(iv)5, 10, 15, ............100; and(v)10, 20, 30, ..........100.The arithmetical series are classified into ayugma and yugma. The Vajasaneyi Saṁhitā (XVIII.24.25) has given the following two instances:(i)1, 3, 5, .............31 and(ii)4, 8,12,...............48.The first series occurs also in the Taittrīya Saṁhitā (IV.3.10). The Paňcavimśa Brāhmana (XVIII.3) describes a list of sacrificial gifts forming a geometrical series of some interest and particular nature.12, 24, 48, 96, 192, ..............49152, 98304, 196608, 393216.This series reappears in the Śrauta-sūtras. Some method for the summation of series was also known. Thrice the sum of an arithmetical progression whose first term is 24, the common difference 4 and number of terms 7 is stated correctly in the ŚatapathaBrāhmaņa as 756.[39] . Moreover, from the texts available to us, it is obvious that Vedic Hindus knew how to perform fundamental arithmetical operations even with elementary fractions.[40]Regarding solutions of quadratic equations, we see that problems for solving quadratic equations of the form ax2 = c and ax2 + bx = c are stated and proved. We also find some indeterminate equations of the first degree (wrongly called as Diophantine equations by modern mathematicians). Furthermore, an important feature, peculiar to the Vedic mathematics, was to use geometrical diagrams and methods to solve algebraic problems and identities. Such geometrical techniques are beautifully adopted in the Śulvasūtras to solve algebraic equations.[41]The ŚulvasūtrasThe most ancient mathematical texts of the Vedic lore are the Śulvasūtras which form a part of the Śrauta section of the Kalpa Vedāńga. In the Śulvasūtras are seen very remarkable and rich principles of mathematics, particularly geometry. The word Śulva (or Śulba) is derived from the verb-root, Śulv or Śulb which means to measure. Since for measure length and breadth rope (rajju) was used, the word Śulva, in course of time came to mean a rope. In fact, Geometry (now referred to as Rekhāgaņita) was called Śulva Sāstra or Rajju Sāstra in ancient http://times.It is believed that these Śulvasūtras were composed around eight or nine centuries before Christ. [42]In the Vedic religion, every household man (barring the Sanyasis who would concentrate on meditation for years uninterruptedly) had to do certain acts of worship every day. It would be sinful if he neglected them. For purposes of worship, he would constantly maintain in his house three types of Agnis or fires sheltering them in certain altars of special designs. The required altars had to be constructed with great care so as to conform to certain specific shapes and areas.[43] While the three Agnis were to be used for the daily or routine Pujas or acts of worship, there were more elaborate sacrifices or Pujas for attaining cherished objects or wants. They were called Kamyagnis. The sacrificial altars for these Kamyagnis required more complicated constructions involving combinations of rectangles, triangles and trapeziums. It is clear that these processes require a clear knowledge of the properties of triangles, rectangles and squares, properties of similar figures, and a solution of the problem of ‘squaring the circle’ and its converse, ‘circling the square’ (i.e., to construct a square equal in area to a given circle, and vice versa).[44]Source and OriginOnly seven of the Śulvasūtras are known at present. They are known by the names Baudhāyana, Āpasthamba, Kātyāyana, Manava, Maitrayana, Varaha and Vadhūla after the names of the Rishis or sages who wrote them. The Kātyāyana Sūtra belongs to the section of the Vedas called Sukla Yajurveda while all the rest belong to Krishna Yajurveda. The Bodhayana, Āpasthamba and Kātyāyana Śulvas are of importance from the mathematical point of view.[45]The dates of these Śulvasūtras have been estimated to be between 800 BC and 500 BC. There is no knowledge about the existence of any Śulvas prior to these seven Sūtras. It must be emphasized that the writers of the Śulvasūtras only wrote down and codified the rules for the constructions of the altars, which were in vogue from ancient times. They were not the persons who specified and directed the rules for the constructions of the altars.[46]Simple Theorems in ŚulvasūtrasThe Śulvas explain a large number of simple geometrical constructions- constructions of squares, rectangles, parallelograms and trapeziums. The following geometrical theorems are either explicitly mentioned or clearly implied in the constructions of the altars of the prescribed shapes and sizes.[47]a.The diagonal of a rectangle divides it into two equal parts.b.The diagonal of a rectangle bisect each other and the opposite areas are equal.c.The perpendicular through the vertex of an isosceles triangle on the base divides the triangle into equal halves.d.A rectangle and a parallelogram on the same base and between the same parallels are equal in area.e.The diagonals of a rhombus bisect each other at right angles.f.The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. (The famous theorem known after the name of Pythagoras)g.Properties of similar rectilinear figures.h.If the sum of the squares on two sides of a triangle be equal to the square on the third side, then the triangle is right-angled. (This is the converse of the Pythagoras theorem, nay the Śulva theorem!)How the Mathematical Theorems were derived?These above mentioned theorems of Śulvasūtras cover roughly the first two books and the sixth book of Euclid’s ‘Elements’. How these theorems were actually obtained is a matter for which no definite answer is available. We all know that Euclid’s geometry is based upon certain axioms and postulates as I pointed out in the first chapter[48] and the proofs involve strict logical application of these. The logical methods of Greek Geometry are certainly not discernible in Hindu geometry. No book on Hindu mathematics explains the system of axioms and postulates assumed, and this itself should go some way in refuting the concocted claim that Hindu mathematics is borrowed from the Greeks. At the same time, it may not be correct to conclude that the above theorems were asserted as a matter of experience and measurement. The people who could make out and solve complicated problems of arithmetic, algebra and spherical trigonometry should be credited with some amount of logic in their work. The Śulvas are not formal mathematical treatises. They are only adjuncts to certain religious works. The question has to end with these remarks.[49] And probably from those very remarks does spring up, valid support for the idea, hypothesis and ultimate destiny of my attempt in this research paper.To make the view clearer, it could be stated that what Pythagoras’ theorem states was known, proved and applied in cases by the Vedic Hindus even before Pythagoras was born in the 6th century BC. Of course, the Vedic scholars did not prove the theorem though they stated and used it. But then there is no evidence that Pythagoras proved it either! The well known proof given by Euclid in this regard may be his own. But the theorem itself was known and widely used from very early times as we mentioned some 5 or 6 centuries prior to the birth of Euclid. It is interesting to note that A. Burk even argues that the much travelled Pythagoras borrowed the result from India![50]Geometrical Constructions contained in ŚulvasThe Vedis discussed in the Śulvasūtras are of various forms. Their constructions require a good knowledge of the properties of the square, the rectangle, the rhombus, the trapezium, the triangle, and of course, the circle. The following are among the important geometrical constructions used in the Śulvasūtras.[51](i)To divide a line-segment into any number of equal parts(ii)To divide a circle into any number of equal areas by drawing diameters (Baudhāyana, II. 73-74, Āpasthamba VII. 13-14)(iii)To divide a triangle into a number of equal and similar areas (Baudhāyana, III. 256)(iv)To draw a straight line at right-angles to a given line (Kātyāyana I. 3)(v)To draw a straight line at right-angles to a given line from a given point on it. (Kātyāyana I. 3)(vi)To construct a square on a given side(vii)To construct a rectangular of given length and breadth. (Baudhāyana I. 36-40)(viii)To construct an isosceles trapezium of given altitude, face and base. (Baudhāyana I. 41, Āpasthamba V. 2-5)(ix)To construct a parallelogram having the given sides at a given inclination. (Āpasthamba XIX. 5)(x)To construct a square equal to the sum of two different squares. (Baudhāyana I. 51-52, Āpasthamba II. 4-6, Kātyāyana II. 22)(xi)To construct a square equivalent to two given triangles.(xii)To construct a square equivalent to two given pentagons. (Baudhāyana III. 68, 288, Kātyāyana IV. 8)(xiii)To construct a square equal to a given rectangle in area. (Baudhāyana I. 58, Āpasthamba II. 7, Kātyāyana III. 2)(xiv)To construct a rectangle having a given side and equivalent to a given square.(xv)To construct an isosceles trapezium having a given face and equivalent to a given square or rectangle. (Baudhāyana I. 55)(xvi)To construct a triangle equivalent to a given square. (Baudhāyana I. 56)(xvii)To construct a square equivalent to a given isosceles triangle. (Kātyāyana IV. 5)(xviii)To construct a rhombus equivalent to a given square or rectangle. (Baudhāyana I. 57, Āpasthamba XII. 9)(xix)To construct a square equivalent to a given rhombus. (Kātyāyana IV. 6)Squaring a CircleOne of the greatest problems that had remained unsolved for centuries in the history of mathematics, till recently, was what is popularly known as “squaring the circle”, i.e., to construct – using only ruler and compasses – a square whose area is equal to that of a given circle. IT is really remarkable that this very problem and vice versa, was tackled by the authors of the Śulvasūtras. They gave practical methods for constructing a square whose area is equal to that of a given circle and vice versa. Of course, their constructions involved approximating the value of the well-known constant number Pi to 3.088 (Pi is the ratio of the circumference of any circle to its diameter). The approximation made by the Vedic Ŗṣis is quite justifiable and admirable in the light of crude mathematical tools and methods available to them thousands of years ago. IT is now, however, in modern mathematics established that an exact construction to “Square a circle” or “to circle a square” (in the sense of the areas) is impossible. Incidentally, it must be pointed out that the approximate value of Pi used in the Śulvasūtras is certainly far better than the Biblical value, 3 (see Kings VII. 23 and Chronicles IV. 2) given many centuries later! In fact, it is only as late as in 1761 AD that Lambert proved that Pi is “irrational” and only in 1882 AD that Lindemann established further that Pi is transcendental.[52]Vedic Knowledge of Irrational NumbersThe essentially arithmetical background of the Śulva mathematics must be contrasted with the essentially geometrical background characteristic of Greek mathematics. Simple fractions and operations on them are available in the Śulvas. Moreover, we find the use of Irrational numbers in them. Surds (Irrational numbers) of the form √2, √3 etc. are called Karanis, thus √2 is dwi-karani, √3 is tri-karani, √1/3 is triteeya karani, √1/7 is saptama karani, so on.[53]The shape of Ashwamedhiki Vedika is an isosceles trapezium whose head, foot and altitude are respectively 24√2, 30√2 , 36√2 prakramas. Its area is stated to be 1944 prakramas (square is to be understood).Area = 24√2 * 1/2(30√2 * 36√2) = 1944.This indicates knowledge of the method of finding the area of a trapezium and simple operations on surds.[54]A remarkable approximation of √2 occurs in each of the three Śulvas Bodhayana, Āpasthamba and Kātyāyana.√2 = 1 + 1/3 + 1/ (3.4) – 1/(3.4 34)....This gives √2 = 1.4142156........., whereas the true value is 1.414213............ The approximation is thus correct to five decimal places, and is expressed by means of simple unit fractions. The problem evidently arises in the construction of a square double a given square in area. The Śulvas contain no clue at all as to the manner in which this remarkable approximation was arrived at. Many theories or plausible explanations have been proposed.[55]This point to the fact that the India, thanks to Vedic mathematics, was the first nation to use irrational numbers. It has to be believed that the Vedic Hindus knew its irrationality. The Greeks also used irrational numbers. If AB is a given segment, Pythagoras and others described the methods of constructing the segments of length√2AB, √3AB, √5AB, etc. But no rational approximation to √2, √3 etc. are found in Greek mathematics, nor are there any problems of the arithmetical operations on irrational numbers. This is easily explained, because the requisite knowledge of arithmetic was not available to the Greeks. It will also be borne in mind that according to unprejudiced estimates, the Śulvasūtras are about two or three centuries prior to Pythagoras.[56][1] Iyengar, The History of Ancient Indian Mathematics, 6.[2]Raja, “The Cultural Heritage of India”,1.[3] Raja, “The Cultural Heritage of India”, 2.[4] Dani, “Ancient Indian Mathematics - A conspectus”, 236.[5] The Vedic Culture had the unique feature of performances of five rituals, known as yajnas. These involved well laid out altars, the vedis, and fire platforms, known as citi or agni, elaborately constructed in the form of birds, tortoise, wheel, etc.[6] The SulvaSūtras are compositions, in the form of manuals for construction of vedis and citis, but they also discuss the geometric principles involved.[7] Dani, “Ancient Indian Mathematics - A conspectus”, 238.[8] Dani, “Ancient Indian Mathematics - A conspectus”, 239.[9] Datta, “Vedic Mathematics”, 18.[10] Datta, “Vedic Mathematics”, 19.[11] Datta, “Vedic Mathematics”, 18.[12] Datta, “The Scope and Development of Hindu Ganita”, 480.[13] Rao, Indian Mathematics and Astronomy, 23.[14] Rao, Indian Mathematics and Astronomy, 23.[15] Rao, Indian Mathematics and Astronomy, 23.[16] Rao, Indian Mathematics and Astronomy, 24.[17] Rao, Indian Mathematics and Astronomy, 24.[18] Datta, “Vedic Mathematics”, 19.[19] Ghosh, “Studies on Rig-Vedic Deities”, 11.[20] Tilak, “The Orion or Researches into the Antiquity of the Vedas, 25.[21] Datta, “Vedic Mathematics”, 20.[22] Datta, “Vedic Mathematics”, 21.[23] Tilak, “The Orion or Researches into the Antiquity of the Vedas, 26.[24] Rao, Indian Mathematics and Astronomy, 25.[25] Mookerjee, “Notes on Indian Astronomy”, 137.[26] Rao, Indian Mathematics and Astronomy, 25.[27] Rao, Indian Mathematics and Astronomy, 25.[28] Rao, Indian Mathematics and Astronomy, 26.[29] Rao, Indian Mathematics and Astronomy, 3.[30] Datta, “Vedic Mathematics”, 30.[31] Rao, Indian Mathematics and Astronomy, 3..[32] Datta, “Vedic Mathematics”, 31.[33] Datta, “Vedic Mathematics”, 31.[34] Datta, “Vedic Mathematics”, 33.[35] Datta, “Vedic Mathematics”, 34.[36] Hopkins, “Numerical Formulae in the Veda and their Bearing on Vedic Criticism”, 279.[37] Shukla, “The Deceptive Title of Swamiji’s Book”, 36.[38] Rao, Indian Mathematics and Astronomy, 14.[39] Shukla, “The Deceptive Title of Swamiji’s Book”, 37.[40] Mehta, Positive Sciences in the Vedas, 114.[41] Rao, Indian Mathematics and Astronomy, 12.[42] Rao, Indian Mathematics and Astronomy, 12.[43] Mehta, Positive Sciences in the Vedas, 117.[44] Iyengar, The History of Ancient Indian Mathematics, 6.[45] Iyengar, The History of Ancient Indian Mathematics, 7.[46] Iyengar, The History of Ancient Indian Mathematics, 8.[47] Rao, Indian Mathematics and Astronomy, 14.[48] Chapter I (1.2.1) Geometry, 16-21.[49] Iyengar, The History of Ancient Indian Mathematics, 9.[50] Rao, Indian Mathematics and Astronomy, 15.[51] Rao, Indian Mathematics and Astronomy, 15-17.[52] Rao, Indian Mathematics and Astronomy, 18.[53] Iyengar, The History of Ancient Indian Mathematics, 13.[54] Iyengar, The History of Ancient Indian Mathematics, 13.[55] Iyengar, The History of Ancient Indian Mathematics, 14.[56] Iyengar, The History of Ancient Indian Mathematics, 15.Popular misconceived notions about Vedic MathematicsVedic Mathematics has been extensively dealt with in the earlier sections of this chapter. But it has to be acknowledged that the idea of Vedic Mathematics prevalent in many of the common people around us is highly stereotyped. And the so called teachers or students of Vedic Mathematics, along with those who have got some vague idea of their training might conceive it as a collection of shortcut methods used for mental calculation. Their interest may be of propagating and validating the supremacy of Vedas and the sciences in it. But, I feel that what they argue as supposed to be Vedic is not exactly so. So, in this section of the chapter, I would like to deal with the misconceived notions about Vedic Mathematics and present the reality and originality of the Mathematics in the Vedas. Obviously, this is only a very faint attempt with the help of some resource books, but I do feel that distinguishing the reality from the popular fallacy is important so that I can state my eventual arguments better.So, first of all, I am presenting the concepts of the so called ‘popular’ idea of Vedic Mathematics, and then I would proceed to talking about how far this idea could be considered Vedic in nature.Conventional to Unconventionally OriginalFrom a popular perspective, Vedic Mathematics deals mainly with various Vedic mathematical formulae and their applications for carrying out tedious and cumbersome arithmetical operations and to a very large extent, executing them mentally. In this field of mental arithmetical operations the works of the famous Mathematicians Trachtenberg and Lester Meyers (High Speed Maths) are elementary compared to that of Jagadguru SwāmīŚrī Bhāratī Kṛṣna Tīrthajī Mahārāja.[1]They sort of argue that some people may find it difficult, at first reading, to understand the arithmetical operations although they have been very lucidly explained by Jagadguruji. It is not because the methods are totally unconventional. Some people are so deeply rooted in the conventional methods that they probably, subconsciously reject to see the logic in unconventional methods.[2] In fact, I agree that using unconventional methods for mathematical computations is a very creative and innovative idea. But I am totally against their claim that the unconventional methods are Vedic in nature.What they are considering now as Vedic mathematics is a work claimed to be forming a class by itself not pragmatically conceived and worked out as in the case of other scientific works, but is the result of the intuitional visualization of fundamental mathematical truths and principles during the course of eight years of highly concentrated mental endeavour on the part of the author and therefore appropriately has been the given the title of “mental” mathematics appearing more as a miracle than the usual approach of hard-baked science, as the author himself states.[3]Rediscovery of a “So Called” Vedic System of MathematicsAs mentioned earlier, Vedic Mathematics hence is now considered as a system, which was rediscovered from Vedic texts earlier last century by Bhāratī Kṛṣna Tīrthajī (1884-1960). Bhāratī Kṛṣna is said to have studied the ancient Indian texts between 1911 and 1918 and reconstructed a mathematical system based on Sixteen Sūtras (formulas) and some sub-Sūtras. He subsequently wrote sixteen volumes, one on each Sūtra, but unfortunately these were all lost. Bhāratī Kṛṣna intended to rewrite the books, but has left us only one introductory volume, written in 1957. This is the book “Vedic Mathematics” published in 1965 by Banaras Hindu University and by Motilal Banarsidass.[4]The Vedic system presents a new approach to mathematics, offering simple, direct, one-line, mental solutions to mathematical problems. The Sūtras on which it is based are given in word form, which renders them applicable in a wide variety of situations. They are easy to remember, easy to understand and a delight to use. The contrast between the Vedic system and conventional mathematics is striking. Modern methods have just one way of doing, say, division and this is so cumbrous and tedious that the students are now encouraged to use a calculating device. This sort of constraint is just one of the factors responsible for the low esteem in which mathematics is held by many people nowadays.[5]The Vedic system on the other hand, does not have just one way of solving a particular problem; there are often many methods to choose from. This element of choice in the Vedic system, and even of innovation, together with the mental approach, brings a new dimension to the study and practice of mathematics. The variety and simplicity of the methods brings fun and amusement, the mental practice leads to a more agile, alert and intelligent mind and innovation naturally follows.[6]It may seem strange to some people that mathematics could be based on sixteen word formulae; but mathematics, more patently than other systems of thought, is constructed by internal laws, natural principles inherent in our consciousness and by whose action more complex edifices are constructed. From the very beginning of life, there must be some structure in consciousness enabling the young child to organise its perceptions, learn and evolve. If these principles could be formulated and used they would give us the easiest and more efficient system possible for all our mental enquiries. This system of Vedic Mathematics given to us by Bhāratī Kṛṣna Tīrthajī points towards a new basis for mathematics, and a unifying principle by which we can simultaneously extend our understanding of the world and of our self.[7]Contents of Vedic Mathematics of Bhāratī Kṛṣna Tīrthajī·The first chapter deals with the conversion of vulgar fractions into decimal or recurring decimal fractions.·The second and third chapters deal with the methods of multiplication·Chapters 4 to 6 and 27 deal with methods of division.·Chapters 7 to 9 deal with factorization of algebraic expressions·The tenth chapter deals with the HCF of algebraic expressions·Chapter 11 to 14 and 16 deal with the various kinds of simple equations. These are similar to those occurring in modern works of algebra.·Chapters20 and 21 deal with various types of simultaneous algebraic equations. These are also similar to those taught to intermediate students.·Chapter 17 deals with quadratic equations·Chapter 18 deals with cubic equations·Chapter 19 deals with biquadratic equations.·Chapter 22 deals with successive differentiation, covering the theorems of Leibniz, Maclaurin and Taylor, among others.·Chapter 23 deals with partial fractions·Chapter 24 deals with integration of partial fractions.·Chapter 25 is talking about the Kaṭapayādi system of expressing numbers by means of letters in the Sanskrit alphabet.·The 26th chapter deals with the recurring decimals.·Chapter 28 deals with auxiliary fractions·Chapters 29 and 30 deal with divisibility and the so called osculators.·Chapter 31 deals with the sum and difference of squares·Chapters 32 to 36 deal with squaring and cubing, square-root and cube-root.·Chapter 37 deals with Pythagoras theorem·Chapter 38 deals with Apollonius theorem.·Chapter 39 deals with analytical conics·The final chapter consists of some miscellaneous methods.[8]Propeties of this system of Vedic MathematicsThe essence of the Vedic system is that it is simple, direct, one-line and mental. The Vedic Sūtras are applicable to a wide variety of problems. They are very easy to understand and a delight to use. Some of their properties are:a.The property of numbers is very extensively exploited in Vedic mathematics, particularly in the field of computations. The Vedic approach has many special methods – indeed many more than conventional methods. This large flexibility of methods finds itself reflected in the mind, when approaching a problem from the Vedic viewpoint. This indeed is the great benefit of the approach. One wonders at the supreme simplicity and ease of the Vedic method, which is lacking in most of our usual methods.b.This Vedic mathematics is perfectly adapted to oral teaching and mental calculation. The Sūtras, of course, demand regular practice of simple problems. They are perfectly in tune with the oral Vedic tradition supplemented by terse statements of important points called Sūtras, which act as an aid to the memory.c.Even a little exposure to the Vedic mathematical approach, coupled with some practice clearly shows us that we are dealing with an entirely new and direct way of thinking.d.Many persons are using the Vedic addition process with slight variation. And certain other historical events point that there are different facts of Vedic Mathematics, most of which were scattered due to mass destruction and killings at the ancient Vedic seats of learning. There is need to do further extensive research both in applications and cross-linkage. But whatever is available today is sufficient to make an excellent start both at the teaching as well as at the research level.e.Vedic mathematics, owing to its multi-choice facility, provides wonderful opportunities for the development of the innovative and research faculty of young students. A large number of Vedic maths participants have been coming forward with very interesting and useful links. The age and qualifications are no barrier in this process. The extensions of the Nikhilam method for multiplying three or more numbers and the use of different bases and sub bases have all been of great application value in terms of quick and accurate calculations.f.The Vedic mathematics is primarily for mental calculation and has the potential of developing the human computer to a wonderful level where the results of the problem flow very naturally, with the least amount of effort, in the typical Vedic way. This approach provides a corrective methodology to the problem of mental slavery to calculators.g.Further, Vedic algorithms have considerable potential for automatic computations and small computers can handle much bigger problems owing to the ease and simplicity of the processes and lesser memory requirements.[9]Deception of the Title “Vedic Mathematics”The title of the book, Vedic Mathematics or Sixteen Simple Mathematical Formulae from the Vedas, written by Jagadguru Swāmī Śrī Bhāratī Kṛṣna Tīrthajī Mahārāja of Govardhana Matha, Puri, bears the impression that it deals with the mathematics contained in the Vedas – ŖgVeda, Sāmaveda, Yajurveda and Atharvaveda. This is indeed not the case, as the book deals not with the Vedic Mathematics but with modern elementary mathematics up to the intermediate standard.[10]Tīrthajī’s explanation of Vedic OriginThe question naturally arises as to whether the Sūtras which form the basis of this treatise exist anywhere in the Vedic literature as known to us. But this criticism was met by Tīrthajī through his renewed understanding and concept of Vedas and Vedic knowledge.[11] To quote Tīrthajī:“The very word “Veda” has this derivational meaning, i.e. the fountain-head and illimitable storehouse of all knowledge. This derivation, in effect, means, connotes, and implies that the Vedas should contain within themselves all the knowledge needed by mankind relating not only to the so called “spiritual” (or other worldly) matters but also to those described as purely “secular”, “temporal”, or “worldly” and also to the means required by humanity as such for the achievement of all-round, complete and perfect success in all conceivable directions and that there can be no adjectival or restrictive epithet calculated or tending to limit that knowledge down in any sphere, any direction or any respect what so ever.“In other words, it connotes and implies that our ancient Indian Vedic lore should be all round, complete and perfect and able to throw the fullest necessary light on all matters which any aspiring seeker after knowledge cna possibly seeks to be enlightened on.”[12]It is the whole essence of his assessment of Vedic tradition that it is not to be approached from a factual stand point but from an ideal stand point, namely, as the Vedas as traditionally accepted in India as the repository of all knowledge should be and not what they are in human possession. That approach entirely turns the tables on all critics, for the authorship of Vedic Mathematics then need not be laboriously searched in the texts as preserved from antiquity.[13] But this in a way is a proof that the so called ‘Vedic Mathematics’ is only a name given to a modern work and it has no connection with the so called Vedas in whom we are tracing the origin of Indian Mathematics. But it is unfortunate that the public understanding of the work of Tīrthajī is that he is presenting the whole mathematics that is contained in the Vedas. In fact, if we look carefully in his book, we can see instances where he states that the Sūtras having a Vedic origin, which is absolutely false.In his preface to his Vedic Mathematics, Swamiji has stated that the sixteen Sūtras dealt with by him in that book were contained in the Pariśiṣṭa (the appendix) of the Atharvaveda. But this is also not a fact; for they are untraceable in the known Pariśiṣṭas of the Atharvaveda edited by G. M. Bolling and J. Von Negelein (Leipzig, 1909-10). Sometime in the 1950, when Swamiji visited Lucknow to give a black board demonstration of the sixteen Sūtras of his ‘Vedic Mathematics’, he was asked to point out the places where the sixteen Sūtras demonstrated by him occurred in the Pariśiṣṭas. He replied off hand, without even touching the book, that the sixteen Sūtras demonstrated by him were not in those Pariśiṣṭas, rather they occurred in his own Pariśiṣṭa and not in any other.[14]The argument of Tīrthajī that his work be regarded as a new Pariśiṣṭa by itself is fallacious. The question is whether any book written in modern times on a modern subject can be regarded as a Pariśiṣṭa of a Veda. The answer is definitely in the negative. So, from all that is said, it is evident that the sixteen Sūtras of Swamiji’s Vedic Mathematics are his own compositions and have nothing to do with the mathematics of the Vedic period.[15]Deceiving even the Learned ScholarsAlthough there is nothing Vedic in his book, Swamiji designates his preface to the book as ‘A Descriptive Prefactory Note on the Astounding Wonders of Ancient Indian Mathematics’ and at places calls his mathematical processes as Vedic processes. The deceptive title of Swamiji’s book and the attribution of the sixteen Sūtras to the Pariśiṣṭas of the Atharvaveda, etc. have confused and baffled the readers who have failed to recognize the real nature of the book whether it is Vedic or non-Vedic. Some scholars, in their letters addressed to the author of this article (K. S. Shukla, well versed and learned in Vedic Scriptures) are said to have sought to know whether these sixteen Sūtras stated by Swamiji occurred anywhere in the Vedas or the Vedic Literature.[16]Even the Rashtriya Veda Vidya Pratishthan, who conducts workshops on Vedic Mathematics, in their circular issued through the Ministry of Human Resource Development, were under the impression that the sixteen Sūtras were actually reconstructed from materials in the various parts of the Vedas and the sixteen formulae contained in them were based on an Appendix (Pariśiṣṭa) of the Atharvaveda, which appendix was not known to exist before the publication of Swamiji’s book.[17]This move might have been for another reason also. Vedic Mathematics was not so much interesting or worth pursuing for the most part of our history. Very few did realize the pearls it carried and fewer dared to carry out extensive researches to find out the depth and immense contributions of it. But, in the 20th century, it has to be acknowledged that it was the book of Tīrthajī that gave a renewed interest for Vedic Mathematics. And obviously, it is something good for Hinduism as a religion also, to have their traditional religious books considered as having enormous knowledge of application value. But this has given a completely wrong notion about the Vedic Mathematics in general. Now, many people even learned philosophers of renowned institutes are stereotyped with the idea that Vedic Mathematics is just another method of computation which gives some shortcut for mental calculations. That is the misconception that I want to clarify through this section of my research work.Paradoxes Seen in the Vedic Mathematics of TīrthajīObviously there are all the possibilities that many more paradoxes can be found in the books of Tīrthajī. All these are valid arguments against the popular idea that the book is the expounding of Vedic Sūtras. These are some of the most simple and trivial paradoxes seen in the work:[18]i.Nobody in the Vedic period could think of decimal or recurring decimal fractions. The decimal fractions were first introduced by the Belgian Mathematician Simon Stevin only in 1585.ii.The multiplication and division methods used in the book are entirely different from the traditional Hindu methods seen in Vedas.iii.Factorization and H.C.F of algebraic expressions were never included in any Hindu work on Algebra.iv.The simple equations dealt in the book show more affinity towards the modern methods of computation rather than the older ones of Vedic era.v.Simultaneous algebraic equations do not occur in any Hindu work on algebra; rather they correspond to the syllabus of intermediate students.vi.The successive differentiation, partial fractions and integration by partial fractions are all modern topics to which Vedic Hindus had no access to.vii.The so called Vedic numerical code, Kaṭapayādi system of expressing numbers by means of the Sanskrit alphabet, has not been used anywhere in the Vedic literature.viii.Recurring decimals, Auxiliary fractions, Divisibility and Osculators are all topics never found in any Hindu work on algebra.Conclusion over the MisconceptionFrom the various evidences given and the paradoxes shown in the contents of the Book, it is evident that the mathematics dealt with in the book Vedic Mathematics is far removed from that of the Vedic period. Instead, they are mathematical aids which could be useful for student of High School and Intermediate classes so that they may find the computations interesting. The results mentioned in the book are indeed the result of Swamiji’s own experience as a teacher of mathematics in his early life. Not a single method described is Vedic, even though Swamiji has declared all the methods and processes explained by him as Vedic and ancient for some unknown reason.[19]Even though one has to admit that the term Vedic Mathematics became popular, interesting and amusing for many due to the advent of the book of Tīrthajī, I would say that it has done equal harm too because, now Vedic mathematics is taught in the schools and understood by teachers and students alike as some very easy shortcut methods for fast calculations. But this doesn’t do justice to what it actually is. The great contributions of Vedic Mathematics must not be belittled by considering it only as an aid for some mental workouts. Rather, we have to convey the idea that Vedic Mathematics is that great source of knowledge which empowered and triggered a great leap for the scientific expeditions of this country even before many other countries could even think about it. Unless, the misconceived notion of Vedic Mathematics is duly corrected, it would be difficult for all those stereotyped minds to grasp the reality concerning it and the enormous impact it has had in all the circles of scientific research and study over centuries.[1] Tiwari, Vedic Mathematics, xv.[2] Tiwari, Vedic Mathematics, xv.[3] Agrawala, Vedic Mathematics, v.[4] Williams, Discover Vedic Mathematics, ix.[5] Williams, Discover Vedic Mathematics, ix.[6] Williams, Discover Vedic Mathematics, ix.[7] Williams, Discover Vedic Mathematics, x.[8] Shukla, “The Deceptive Title of Swamiji’s Book”, 34.[9] Shukla, “The Deceptive Title of Swamiji’s Book”, 58.[10] Shukla, “The Deceptive Title of Swamiji’s Book”, 31.[11] Agrawala, Vedic Mathematics, v.[12] Agrawala, Vedic Mathematics, vi.[13]Agrawala, Vedic Mathematics, v.[14] Shukla, “The Deceptive Title of Swamiji’s Book”, 32.[15] Shukla, “The Deceptive Title of Swamiji’s Book”, 33.[16] Shukla, “The Deceptive Title of Swamiji’s Book”, 33.[17] Shukla, “The Deceptive Title of Swamiji’s Book”, 33.[18] Shukla, “The Deceptive Title of Swamiji’s Book”, 34.[19] Shukla, “The Deceptive Title of Swamiji’s Book”, 35.

Vedic MathematicsIn many ancient cultures and civilizations, the development of mathematics was necessitated on account of religious practices and observances. These required an accurate calculation of the times of certain festivals and of the times auspicious for the performances of certain sacrifices and rituals. They also required a correct knowledge of the times of rising and the setting of the sun and the moon, and of the occurrences of the solar and lunar eclipses. All these meant a good knowledge of arithmetic, plane and spherical geometry and trigonometry, and possibly also the know-how of the construction of simple astronomical instruments.[1]The impression that science started only in Europe was deeply embedded in the minds of educated people all over the world until recently. The alchemists of Arab countries were occasionally mentioned, but there was very little reference to India and China. Thanks to the work of the Indian National Science Academy and other learned bodies, the development of science in India during both the ancient and medieval periods has recently been studied. It is becoming clearer from these studies that India has consistently been a scientific country right from Vedic to modern times with the usual fluctuations that can been expected of any country. In fact, a research will not find another civilization except that of ancient Greece, which accorded an exalted place to knowledge and science as in India.[2]It has to be universally acknowledged that much of mathematical knowledge in the world originated in India and moved from East to West. The high degree of sophistication in the use of mathematical symbols and developments in arithmetic, algebra, trigonometry, especially the work attributed to Aryabhatta, is indeed remarkable and should be a source of inspiration to all of us in India. The articles which describe Indian contributions to science from the ancient times to the very modern period bring out quite clearly the continuity of scientific thought as part of our cultural heritage. It is however, unfortunate that the period of decline in India coincided with that of ascendancy of Europe. It is perhaps the contrast during this period that made Europeans believe that all modern science was European.[3]An impending danger which I recognized as part of this study is the fact that Vedic Mathematics is now understood and taught in some institutions merely as an alternate method in facilitating easy and quick mental calculations. So, my study is twofold in nature. The first goal is to have an integral understanding about the domain of Vedic mathematics and second is to have a better understanding about the misconceived notion about Vedic Mathematics, which is becoming prevalent as part of some political or religious agenda. This is a very subjective and personal inquiry towards the disciplines dealt with in Vedic Mathematics which had substantial impact in the Mathematical pursuits of probably all the Indian Mathematicians and Schools of Mathematics.History and Development of Vedic MathematicsThe mathematical tradition in India goes back at least to the Vedas. For compositions with a broad scope covering all aspects of live, spiritual as well as secular, the Vedas show a great fascination for large numbers. As the transmission of the knowledge was oral the numbers were not written, but they were expressed as combinations of powers of 10, and it would be reasonable to believe that when the decimal place value system for written numbers came into being it owed a great deal to the way numbers were discussed in the older compositions.[4] (See the second section of this chapter)It is well known that Geometry was pursued in India in the context of construction of vedis[5] for the yaj̀nas of the Vedic period. The Śulvasūtras[6]contain elaborate descriptions of constructions of vedis and also enunciate various geometric principles. These were composed in the first millennium BC, the earliest Baudhāyana Śulvasūtra dating back to about 800 BC. The Śulvasūtra geometry did not go very far in comparison to the Euclidean geometry developed by the Greeks, who appeared on the scene a little later, in the seventh century BC. It was however an important stage of development in India too. The Śulvasūtra geometers were aware, among other things, of what is now called the Pythagorean theorem, over two hundred years before Pythagoras (all the four major Śulvasūtras contain explicit statement of the theorem), addressed (within the framework of their geometry) issues such as finding a circle with the same area as a square and vice versa and worked out a very good approximation to the square root of 2, in the course of their studies.[7]Though it is generally not recognized, the Śulvasūtras geometry was itself evolving. This is seen in particular from the differences in the contents of the four major extant Śulvasūtras. Certain revisions are especially striking. For instance, in the early Śulvasūtras period the ratio of the circumference to the diameter was, like in other ancient cultures, thought to be 3, as seen in the Sūtra of Baudhāyana, but in the Manava Śulvasūtra a new value was proposed, as 3 and one-fifth; interestingly the Sūtra describing it ends with an exultation “not a hair breadth remains”, and though we see that it is still substantially off the mark, it is a gratifying instance of an advance made. In the Manava Śulvasūtra one also finds an improvement over the method described by Baudhāyana for finding the circle with the same are as that of a given square.[8]Vedic Hindus evinced special interest in two particular branches of Mathematics, namely geometry (Śulva) and astronomy (Jyotiṣa). Sacrifice (Yaj̀na) was their prime religious avocation. Each sacrifice has to be performed on an altar of prescribed size and shape. They were very strict regarding this and thought that even a slight irregularity in the form and size of the altar would nullify the object of the whole ritual and might even lead to adverse effect. So, the greatest care was taken to have the right shape and size of the sacrificial altar. Thus the problems of geometry and consequently the science of geometry originated.[9]As it is evident, available sources of Vedic Mathematics are very poor. Almost all works on the subject have perished. At present we find only a very short treatise on Vedic astronomy in three rescensions, namely, in Ārca Jyotiṣa, Yājuṣa Jyotiṣa and Atharva Jyotiṣa. There are six small treatises on Vedic Geometry belonging to the six schools of the Vedas. Thus for an insight into Vedic Mathematics, we have to now depend more on secondary sources such as the literary works.[10]The study of astronomy began and developed chiefly out of the necessity for fixing the proper time for the sacrifice. This origin of the sciences as an aid to religion is not at all unnatural, for it’s generally found that the interest of a people in a particular branch of knowledge, in all climes and times, has been aroused and guided by specific reasons. In the case of the Vedic Hindu that specific reason was religious. In the course of time, however, those sciences outgrew their original purposes and came to be cultivated for their own sake.[11]The Chāndogya Upaniśad (VII.1.2, 4) mentions among other sciences the science of numbers (rāśi). In the Muņdaka Upaniṣad {I.2. 4-5} knowledge is classified as superior (parā) and inferior (aparā). In the second category is included the study of astronomy (jyotiṣa). In the Mahābhārata (XII.201) we come across a reference to the science of stellar motion (nakṣatragati). The term gaņita, meaning the science of calculation, also occurs copiously in Vedic literature. The Vedāńga jyotiṣa gives it the highest place of honour amongst all the sciences which form the Vedāńga. Thus it was said: ‘As are the crests on the heads of peacocks, as are the gems on the hoods of snakes, so is the gaņita at the top of the sciences known as the Vedāńga’. At that remote period gaņita included astronomy, arithmetic, and algebra, but not geometry. Geometry then belonged to a different group of sciences known as kalpa.[12]Vedic Astronomy – JyotiṣaVedāńga Jyotiṣa is one of the six ancillary branches of knowledge called the Ṣad- Vedāńgas, essentially dealing with astronomy. It must be remarked that although the word Jyotiṣa, in the modern common parlance, is used to mean predictive astrology, in the traditional literature, the word always meant the science of astronomy. Of course, mathematics was considered as a part of this subject. Vedāńga Jyotiṣa is the earliest Indian astronomical work available.[13]The purpose of Vedāńga Jyotiṣa was primarily to fix suitable times for performing the different sacrifices. It is found in two rescensions: the Ŗgveda Jyotiṣa and the Yajurveda Jyotiṣa. Though the contents of both the rescensions are the same, they differ in the number of verses contained in them. While the Ŗgvedic version contains only 36 verses, the Yajurvedic contains 44 verses. This difference in the number of verses is perhaps due to the addition of explanatory verses by the adhvaryu priests with whom it was in use.[14]Authorship and DateIn one of the verses of the Vedāńga Jyotiṣa, it is said, “I shall write on the lore of Time, as enunciated by sage Lagadha”. Therefore, the Vedāńga Jyotiṣa is attributed to Lagadha.[15]According to the text, at the time of Lagadha, the winter solstice was at the beginning of the constellation Śrāviṣṭhā (Delphini) and the summer solistice was at the midpoint of Āśleṣā. Since Vārāhamihira (505 – 587AD) stated that in his own time the summer solstice was at the end of the first quarter of Punarvasu and the winter solstice at the end of the first quarter of Uttarāṣāḍhā, There had been a precession of one and three quarters of nakṣatra or 230 20`. Since the rate of precession is about a degree in 72 years, the time interval for a precession of 230 20` is 72 * 231/3 = 1680 years prior to Vārāhamihira’s time i.e. around 1150 BC. According to the famous astronomer Prof. T. S. Kuppanna Sastri, if instead of the segment of nakṣatra, the group itself it smeant, which is about 30 within it, Lagadha’s time would be around 1370 BC. Therefore, the composition of Lagadha’s Vedāńga Jyotiṣa can be assigned to the period of 12th to 14th century BC.[16]The Vedāńga Jyotiṣa belongs to the last part of the Vedic age. The text proper can be considered as the records of the essentials of astronomical knowledge needed for the day to day life of the people of those times. The Vedāńga Jyotiṣa is the culmination of the knowledge developed and accumulated over thousands of years of the Vedic period prior to 1400 BC.[17]Furthermore, there is considerable material on astronomy in the Vedic Samhitas. But everything is shrouded in such mystic expressions and allegorical legends that it has now become extremely difficult to discern their proper significance. Hence it is not strange that modern scholars differ widely in evaluating the astronomical achievements of the early Vedic Hindus. Much progress seems, however, to have been made in the Brāhmana period when astronomy came to be regarded as a separate science called nakṣatra-vidyā (the science of stars). An astronomer was called a nakṣatra-darśa (star observer) or ganaka (calculator).[18]Numerous Discoveries AnticipatedNumerous amounts of mental calculations have been done in the Vedic era. The distance of the heaven from the earth have been calculated differently in various works. All of them are figurative expressions indicating that the extent of the universe is infinite. There is speculation in the Ŗg-Veda (V.85.5, VIII.42.1) about the extent of the earth. It appears from passages therein that the earth was considered to be spherical in shape (I.33.8) and suspended freely in the air. (IV.53.3) The Śatapatha Brāhmaņa describes it clearly as parimaņdala (globe or sphere). There is evidence in the Ŗg-Veda of the knowledge of the axial rotation and annual revolution of the earth. It was known that these motions are caused by the sun.[19] In fact, all the postulations and conclusions had been arrived at couple of millenniums prior to the discovery of the same by the westerners.There are also evident mentions about the zodiacal belt, the inclinations of the ecliptic with the equator and the axis of the earth. We see that the apparent annual course of the sun is divided into two halves. We also see that the ecliptic is divided into twelve parts or sings of the zodiac corresponding to the twelve months of the year, the sun moving through the consecutive signs during the successive months. The sun is called by different names at various parts of the zodiac, and thus has originated the idea of twelve ādityas or suns.[20]The Ŗg-Veda (IX.71.9 etc.) says that the moon shines by the borrowed light of the sun. The phases of the moon and their relations to the sun were fully understood. Five planets seem to have been known. The planets Sukra or Vena (Venus) and Manthin are mentioned by name.[21]Knowledge Through ObservationsIt appears from a passage in the Taittrīya Brāhmaņa (I.5.2.1) that Vedic astronomers ascertained the motion of the sun by observing with the naked eye the nearest visible stars rising and setting with the sun from day to day. This passage is considered very important ‘as it describes the method of making celestial observations in old times’. Observations of several solar eclipses are mentioned in the Ŗg-Veda, a passage of which states that the priests of the Atri family observed a total eclipse of the sun caused by its being covered by Svarbhānu, the darkening demon (V.40.5-9).[22] The Atri priests could calculate the occurrence, duration, beginning, and the end of the eclipse. Their descendants were particularly conversant with the calculation of eclipses. At the time of the Ŗg-Veda, the cause of the solar eclipse was understood as the occultation of the sun by the moon. There are also mentions of lunar eclipses.[23]Concept of Time and SeasonsThe Vedic people had considerable knowledge about the seasons of for sowing, reaping etc. Apart from that, they had acquired knowledge required for their religious activities, like the times and periodicity of the full and new moons, the last disappearance of the moon and its first appearance etc. This type of information was necessary for their monthly rites like the darśapūrņamāsa and seasonal rites cāturmāsya.[24]Vedic Hindus counted the beginning of a season on the sun’s entering a particular asterism. After a long interval of time, it was observed that the same season began with the sun entering a different asterism. Thus they discovered the falling back of the seasons with the position of the sun among the asterisms. There are also clear references to the vernal equinox in the asterism Punarvasu. There is also evidence to show that the vernal equinox was once in the asterism Mŗgaśirā from whence, in course of time, it receded to Kŗttikā. Thus there is clear evidence in the Samhitās and Brāhmaņas of the knowledge of the precession of the equinox. Some scholars maintain that Vedic Hindus also knew of the equation of time.[25] The practical way of measuring time is mentioned as the time taken by a specific quantity of water to flow through the orifice of a specified clepsydra (water-clock), as one nāḍikā i.e., 1/60 part of a day. The day is divided into 124 bhāgas (or parts) so that the ending moments of parvas and tithis can be given in whole units. The day is also divided into 603 units called kalās so that the duration of the lunar nakṣatra is given in whole units as 610 kalās. The nakṣatra is divided into 124 amśas so that the nakṣatras passed at the ends of the parvas may be expressed in whole amśas. [26]During the Yajurveda period, it was known that the solar year had 365 days and a fraction more. In the Kŗṣna Yajurveda (Taittrīya Samhita VII. 2.6) it is mentioned that the extra 11 days over 12 lunar months, Caitra, Vaiśākha etc. (totaling to 354 days), complete the ŗtus by the performance of the ekādaśarātra or eleven day sacrifice. Again, the Taittrīya Samhita says that 5 days more were required over the sāvana year of 360 days to complete the season adding specifically that 4 days are too short and 6 days too long.[27]In the Yajurveda period, they had recognized the six ŗtus (seasons) in a period of 12 tropical months of the year and named them as follows:SeasonsMonthsi. Vasanta ŗtuMadhu and Mādhavaii. Grīṣma ŗtuŚukra and Śuciiii. Varṣa ŗtuNalcha and Nabhasyaiv. Śarad ŗtuIṣa and Ūrjav. Hemanta ŗtuSaha and Sahasyavi. Śiśira ŗtuTapa and TapasyaThe sacrificial year commenced with Vasanta ŗtu. Thy had alos noted that the shortest day was at the winter solstice when the seasonal year Śiśira began with Uttarāyana and rose to a maximum at the summer solstice.[28]Vedic ArithmeticIndia’s recognized contribution to mathematics was chiefly in the fields of arithmetic and algebra. In fact, Indian arithmetic is what is now used world over. The topics discussed in the Hindu Mathematics of early renaissance included the following:-[29]a. Parikarma (The four fundamental operations)b. Vyavahāra (determination)c. Rajju (meaning rope referring to geometry)d. Trairāśi (the rule of three)e. Yāvat tāvat (simple equations)f. Kalasavarņa (operations with fractions)g. Varga and Vargamūla (square and square root)h. Ghana and ghana-mūla (cube and cube root)i. Prastāra and Vikalpa (Permutations and Combinations)Sources of information on Vedic arithmetic being very meagre, it is difficult to define the topics for discussion and their scope of treatment. One problem that appears to have attracted the attention and interest of Vedic Hindus was to divide 1000 into 3 equal parts. It is unknown how the problem could be solved, for 1000 is not divisible by 3. So, an attempt has been made to explain the whole thing as a metaphorical statement. But a passage in the Śatapatha Brāhmaņa (III.3.1.13) seems clearly to belie all such speculations, saying: ‘When Indra and Viṣṇu divided a thousand into three parts, one remained in excess, and that they caused to be reproduced into three parts. Hence even now if any one attempts to divide a thousand by three, one remains over.’ In any case, it was a mathematical exercise.[30]Vedic Hindus developed the terminology of numeration to a high degree of perfection. The highest terminology that ancient Greeks knew was ‘myriad’ which denoted 104 and which came into use only about the fourth century BC. The Romans had to remain content with a ‘mille’ (103). But centuries before them, the Hindus had numerated up to parārdha (1014) which they could easily express without ambiguity or cumbrousness. The whole system is highly scientific and is very remarkable for its precision.[31]Scales of NumerationFrom the time of the Vedas, the Hindus adopted the decimal scale of numeration. They coined separate names for the notational places corresponding to 1, 10, 102, 103, 104, 105, etc. and any number, however big, used to be expressed in terms of them. But in expressing a number greater than 103 (sahasra) it was more usual to follow a centesimal scale. Thus, 50.103 was a more common form than 5.104. In Taittrīya Upaniṣad (II.8) the centesimal scale has been adopted in describing the different orders of bliss. Brahmānanda, or the bliss of Brahman, has been estimated as 10010 times the measure of one unit of human bliss.[32]In cases of actual measurements, the Hindus often followed other scales. For instance, we have in the Śatapatha Brāhmaņa (XII.3.2.5 et seq.) the minute subdivision of time on the scale of 15. The smallest unit prāṇa is given by 1/155 of a day. In the Vedāńga jyotiṣa (verse 31), a certain number is indicated as eka-dvi-saptika. If it really means ‘two-sevenths and one’ as it seems to do, then it will have to be admitted that there was once a septismal scale.[33]Representation of NumbersThe whole vocabulary of the number-names of the Vedic Hindus consisted mainly of thirty fundamental terms which can be divided into three groups, the ones, tens and the powers of ten. Furthermore, we can understand that Vedic Hindus had a unique and powerful method of their own in representing numbers. Because from the seals and inscriptions of Mohenjo-daro we can see that in the third millennium before the Christian era, numbers were represented in the Indus valley by means of vertical strokes arranged side by side or one group upon another. There were probably other signs for bigger numbers. Those rudimentary and cumbrous devices of rod-numerals were, however, quite useless for the representation of large numbers mentioned in the Vedas. In making calculations with such large numbers, as large as 1012, Vedic Hindus must have found the need for some shorter and more rapid method of representing numbers. This and other considerations give sufficient grounds for concluding that the Vedic Hindus had developed a much better system of numerical symbols.[34]From a reference in the Aṣṭādhyāyī of Pāṇini, we come to know that the letters of the alphabet were used to denote numbers. Another favourite device of Vedic Hindus to indicate a particular number was to employ the names of things permanently connected with that number by tradition or other associations, and sometimes vice versa. Applications of this are found in the earliest Saṁhitās. This practice of recording numbers with the help of letters and words became very popular in later times, especially amongst astronomers and mathematicians.[35]Holiness Attributed to NumbersIt appears that Vedic Hindus used to look upon some numbers as particularly holy. One such number is 3. In the Ṙg-Veda, the gods are grouped in three (I.105.5) and the mystical ‘three dawns’ are mentioned (VIII.41.3, X.67.4). Cases of magic where 3 is employed in a mysterious occult manner occur in the Ṙg-Veda and the Atharva-Veda. Even the number 180 is mentioned in the Ṙg-Veda as three sixties (VIII.96.8) and 210 as three seventies (VIII.19.37). The number regarded as most sacred seems to have been 7. Thus in the Ṙg-Veda, we get ‘seven seas’ (VIII.40.5), ‘seven rays of the sun’ (I.105.9), and ‘seven sages’ (IV.42.8, IX.92.2, etc.); and the number 49 is stated as seven sevens. Instances of combinations of these two numbers also occur. Thus 21 is stated as three sevens in the Ṙg-Veda (I.133.6, 191.12) and the Atharva-Veda (I.1.1), and 1,470 as three seven seventies in the Ṙg-Veda (VIII.46.26).[36]Classification of NumbersNumbers were divided into even (yugma, literally pair) and odd (ayugma, literally non pair), but there is no further subdivisions of numbers. There is an apparent reference to Zero and recognition of the negative number in the Atharvaveda. Zero is called kṣudra (XIX.22.6) meaning trifling’; the negative number is indicated by the epithet anṛca (XIX.23.22), meaning ‘without a hymn’; and the positive number by ṛca (XIX.23.1), meaning ‘a sacred verse’. These designations were replaced in later times by ṛṇa (debt) and dhana (asset).[37] The fractions like half, quarter, one-eighth and one-sixteenth are referred to for the first time, in the history of mathematics, in the Ŗgveda. These fractions are respectively called ardha, pāda, śapha and kalā.[38]Number SeriesVedic Hindus became interested in numbers forming series or progressions. The Taittrīya Saṁhitā (VII.2.12-17) mentions the following arithmetical series:(i) 1, 3, 5,...19,.......99;(ii) 2, 4, 6, ...............100;(iii) 4, 8, 12, .............100;(iv) 5, 10, 15, ............100; and(v) 10, 20, 30, ..........100.The arithmetical series are classified into ayugma and yugma. The Vajasaneyi Saṁhitā (XVIII.24.25) has given the following two instances:(i) 1, 3, 5, .............31 and(ii) 4, 8,12,...............48.The first series occurs also in the Taittrīya Saṁhitā (IV.3.10). The Paňcavimśa Brāhmana (XVIII.3) describes a list of sacrificial gifts forming a geometrical series of some interest and particular nature.12, 24, 48, 96, 192, ..............49152, 98304, 196608, 393216.This series reappears in the Śrauta-sūtras. Some method for the summation of series was also known. Thrice the sum of an arithmetical progression whose first term is 24, the common difference 4 and number of terms 7 is stated correctly in the Śatapatha Brāhmaņa as 756.[39] . Moreover, from the texts available to us, it is obvious that Vedic Hindus knew how to perform fundamental arithmetical operations even with elementary fractions.[40]Regarding solutions of quadratic equations, we see that problems for solving quadratic equations of the form ax2 = c and ax2 + bx = c are stated and proved. We also find some indeterminate equations of the first degree (wrongly called as Diophantine equations by modern mathematicians). Furthermore, an important feature, peculiar to the Vedic mathematics, was to use geometrical diagrams and methods to solve algebraic problems and identities. Such geometrical techniques are beautifully adopted in the Śulvasūtras to solve algebraic equations.[41]The ŚulvasūtrasThe most ancient mathematical texts of the Vedic lore are the Śulvasūtras which form a part of the Śrauta section of the Kalpa Vedāńga. In the Śulvasūtras are seen very remarkable and rich principles of mathematics, particularly geometry. The word Śulva (or Śulba) is derived from the verb-root, Śulv or Śulb which means to measure. Since for measure length and breadth rope (rajju) was used, the word Śulva, in course of time came to mean a rope. In fact, Geometry (now referred to as Rekhāgaņita) was called Śulva Sāstra or Rajju Sāstra in ancient http://times.It is believed that these Śulvasūtras were composed around eight or nine centuries before Christ. [42]In the Vedic religion, every household man (barring the Sanyasis who would concentrate on meditation for years uninterruptedly) had to do certain acts of worship every day. It would be sinful if he neglected them. For purposes of worship, he would constantly maintain in his house three types of Agnis or fires sheltering them in certain altars of special designs. The required altars had to be constructed with great care so as to conform to certain specific shapes and areas.[43] While the three Agnis were to be used for the daily or routine Pujas or acts of worship, there were more elaborate sacrifices or Pujas for attaining cherished objects or wants. They were called Kamyagnis. The sacrificial altars for these Kamyagnis required more complicated constructions involving combinations of rectangles, triangles and trapeziums. It is clear that these processes require a clear knowledge of the properties of triangles, rectangles and squares, properties of similar figures, and a solution of the problem of ‘squaring the circle’ and its converse, ‘circling the square’ (i.e., to construct a square equal in area to a given circle, and vice versa).[44]Source and OriginOnly seven of the Śulvasūtras are known at present. They are known by the names Baudhāyana, Āpasthamba, Kātyāyana, Manava, Maitrayana, Varaha and Vadhūla after the names of the Rishis or sages who wrote them. The Kātyāyana Sūtra belongs to the section of the Vedas called Sukla Yajurveda while all the rest belong to Krishna Yajurveda. The Bodhayana, Āpasthamba and Kātyāyana Śulvas are of importance from the mathematical point of view.[45]The dates of these Śulvasūtras have been estimated to be between 800 BC and 500 BC. There is no knowledge about the existence of any Śulvas prior to these seven Sūtras. It must be emphasized that the writers of the Śulvasūtras only wrote down and codified the rules for the constructions of the altars, which were in vogue from ancient times. They were not the persons who specified and directed the rules for the constructions of the altars.[46]Simple Theorems in ŚulvasūtrasThe Śulvas explain a large number of simple geometrical constructions- constructions of squares, rectangles, parallelograms and trapeziums. The following geometrical theorems are either explicitly mentioned or clearly implied in the constructions of the altars of the prescribed shapes and sizes.[47]a. The diagonal of a rectangle divides it into two equal parts.b. The diagonal of a rectangle bisect each other and the opposite areas are equal.c. The perpendicular through the vertex of an isosceles triangle on the base divides the triangle into equal halves.d. A rectangle and a parallelogram on the same base and between the same parallels are equal in area.e.The diagonals of a rhombus bisect each other at right angles.f. The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. (The famous theorem known after the name of Pythagoras)g. Properties of similar rectilinear figures.h. If the sum of the squares on two sides of a triangle be equal to the square on the third side, then the triangle is right-angled. (This is the converse of the Pythagoras theorem, nay the Śulva theorem!)How the Mathematical Theorems were derived?These above mentioned theorems of Śulvasūtras cover roughly the first two books and the sixth book of Euclid’s ‘Elements’. How these theorems were actually obtained is a matter for which no definite answer is available. We all know that Euclid’s geometry is based upon certain axioms and postulates as I pointed out in the first chapter[48] and the proofs involve strict logical application of these. The logical methods of Greek Geometry are certainly not discernible in Hindu geometry. No book on Hindu mathematics explains the system of axioms and postulates assumed, and this itself should go some way in refuting the concocted claim that Hindu mathematics is borrowed from the Greeks. At the same time, it may not be correct to conclude that the above theorems were asserted as a matter of experience and measurement. The people who could make out and solve complicated problems of arithmetic, algebra and spherical trigonometry should be credited with some amount of logic in their work. The Śulvas are not formal mathematical treatises. They are only adjuncts to certain religious works. The question has to end with these remarks.[49] And probably from those very remarks does spring up, valid support for the idea, hypothesis and ultimate destiny of my attempt in this research paper.To make the view clearer, it could be stated that what Pythagoras’ theorem states was known, proved and applied in cases by the Vedic Hindus even before Pythagoras was born in the 6th century BC. Of course, the Vedic scholars did not prove the theorem though they stated and used it. But then there is no evidence that Pythagoras proved it either! The well known proof given by Euclid in this regard may be his own. But the theorem itself was known and widely used from very early times as we mentioned some 5 or 6 centuries prior to the birth of Euclid. It is interesting to note that A. Burk even argues that the much travelled Pythagoras borrowed the result from India![50]Geometrical Constructions contained in ŚulvasThe Vedis discussed in the Śulvasūtras are of various forms. Their constructions require a good knowledge of the properties of the square, the rectangle, the rhombus, the trapezium, the triangle, and of course, the circle. The following are among the important geometrical constructions used in the Śulvasūtras.[51](i) To divide a line-segment into any number of equal parts(ii) To divide a circle into any number of equal areas by drawing diameters (Baudhāyana, II. 73-74, Āpasthamba VII. 13-14)(iii) To divide a triangle into a number of equal and similar areas (Baudhāyana, III. 256)(iv) To draw a straight line at right-angles to a given line (Kātyāyana I. 3)(v) To draw a straight line at right-angles to a given line from a given point on it. (Kātyāyana I. 3)(vi) To construct a square on a given side(vii) To construct a rectangular of given length and breadth. (Baudhāyana I. 36-40)(viii) To construct an isosceles trapezium of given altitude, face and base. (Baudhāyana I. 41, Āpasthamba V. 2-5)(ix) To construct a parallelogram having the given sides at a given inclination. (Āpasthamba XIX. 5)(x) To construct a square equal to the sum of two different squares. (Baudhāyana I. 51-52, Āpasthamba II. 4-6, Kātyāyana II. 22)(xi) To construct a square equivalent to two given triangles.(xii) To construct a square equivalent to two given pentagons. (Baudhāyana III. 68, 288, Kātyāyana IV. 8)(xiii) To construct a square equal to a given rectangle in area. (Baudhāyana I. 58, Āpasthamba II. 7, Kātyāyana III. 2)(xiv) To construct a rectangle having a given side and equivalent to a given square.(xv) To construct an isosceles trapezium having a given face and equivalent to a given square or rectangle. (Baudhāyana I. 55)(xvi) To construct a triangle equivalent to a given square. (Baudhāyana I. 56)(xvii) To construct a square equivalent to a given isosceles triangle. (Kātyāyana IV. 5)(xviii) To construct a rhombus equivalent to a given square or rectangle. (Baudhāyana I. 57, Āpasthamba XII. 9)(xix) To construct a square equivalent to a given rhombus. (Kātyāyana IV. 6)Squaring a CircleOne of the greatest problems that had remained unsolved for centuries in the history of mathematics, till recently, was what is popularly known as “squaring the circle”, i.e., to construct – using only ruler and compasses – a square whose area is equal to that of a given circle. IT is really remarkable that this very problem and vice versa, was tackled by the authors of the Śulvasūtras. They gave practical methods for constructing a square whose area is equal to that of a given circle and vice versa. Of course, their constructions involved approximating the value of the well-known constant number Pi to 3.088 (Pi is the ratio of the circumference of any circle to its diameter). The approximation made by the Vedic Ŗṣis is quite justifiable and admirable in the light of crude mathematical tools and methods available to them thousands of years ago. IT is now, however, in modern mathematics established that an exact construction to “Square a circle” or “to circle a square” (in the sense of the areas) is impossible. Incidentally, it must be pointed out that the approximate value of Pi used in the Śulvasūtras is certainly far better than the Biblical value, 3 (see Kings VII. 23 and Chronicles IV. 2) given many centuries later! In fact, it is only as late as in 1761 AD that Lambert proved that Pi is “irrational” and only in 1882 AD that Lindemann established further that Pi is transcendental.[52]Vedic Knowledge of Irrational NumbersThe essentially arithmetical background of the Śulva mathematics must be contrasted with the essentially geometrical background characteristic of Greek mathematics. Simple fractions and operations on them are available in the Śulvas. Moreover, we find the use of Irrational numbers in them. Surds (Irrational numbers) of the form √2, √3 etc. are called Karanis, thus √2 is dwi-karani, √3 is tri-karani, √1/3 is triteeya karani, √1/7 is saptama karani, so on.[53]The shape of Ashwamedhiki Vedika is an isosceles trapezium whose head, foot and altitude are respectively 24√2, 30√2 , 36√2 prakramas. Its area is stated to be 1944 prakramas (square is to be understood).Area = 24√2 * 1/2(30√2 * 36√2) = 1944.This indicates knowledge of the method of finding the area of a trapezium and simple operations on surds.[54]A remarkable approximation of √2 occurs in each of the three Śulvas Bodhayana, Āpasthamba and Kātyāyana.√2 = 1 + 1/3 + 1/ (3.4) – 1/(3.4 34)....This gives √2 = 1.4142156........., whereas the true value is 1.414213............ The approximation is thus correct to five decimal places, and is expressed by means of simple unit fractions. The problem evidently arises in the construction of a square double a given square in area. The Śulvas contain no clue at all as to the manner in which this remarkable approximation was arrived at. Many theories or plausible explanations have been proposed.[55]This point to the fact that the India, thanks to Vedic mathematics, was the first nation to use irrational numbers. It has to be believed that the Vedic Hindus knew its irrationality. The Greeks also used irrational numbers. If AB is a given segment, Pythagoras and others described the methods of constructing the segments of length√2AB, √3AB, √5AB, etc. But no rational approximation to √2, √3 etc. are found in Greek mathematics, nor are there any problems of the arithmetical operations on irrational numbers. This is easily explained, because the requisite knowledge of arithmetic was not available to the Greeks. It will also be borne in mind that according to unprejudiced estimates, the Śulvasūtras are about two or three centuries prior to Pythagoras.[56][1] Iyengar, The History of Ancient Indian Mathematics, 6.[2]Raja, “The Cultural Heritage of India”,1.[3] Raja, “The Cultural Heritage of India”, 2.[4] Dani, “Ancient Indian Mathematics - A conspectus”, 236.[5] The Vedic Culture had the unique feature of performances of five rituals, known as yajnas. These involved well laid out altars, the vedis, and fire platforms, known as citi or agni, elaborately constructed in the form of birds, tortoise, wheel, etc.[6] The SulvaSūtras are compositions, in the form of manuals for construction of vedis and citis, but they also discuss the geometric principles involved.[7] Dani, “Ancient Indian Mathematics - A conspectus”, 238.[8] Dani, “Ancient Indian Mathematics - A conspectus”, 239.[9] Datta, “Vedic Mathematics”, 18.[10] Datta, “Vedic Mathematics”, 19.[11] Datta, “Vedic Mathematics”, 18.[12] Datta, “The Scope and Development of Hindu Ganita”, 480.[13] Rao, Indian Mathematics and Astronomy, 23.[14] Rao, Indian Mathematics and Astronomy, 23.[15] Rao, Indian Mathematics and Astronomy, 23.[16] Rao, Indian Mathematics and Astronomy, 24.[17] Rao, Indian Mathematics and Astronomy, 24.[18] Datta, “Vedic Mathematics”, 19.[19] Ghosh, “Studies on Rig-Vedic Deities”, 11.[20] Tilak, “The Orion or Researches into the Antiquity of the Vedas, 25.[21] Datta, “Vedic Mathematics”, 20.[22] Datta, “Vedic Mathematics”, 21.[23] Tilak, “The Orion or Researches into the Antiquity of the Vedas, 26.[24] Rao, Indian Mathematics and Astronomy, 25.[25] Mookerjee, “Notes on Indian Astronomy”, 137.[26] Rao, Indian Mathematics and Astronomy, 25.[27] Rao, Indian Mathematics and Astronomy, 25.[28] Rao, Indian Mathematics and Astronomy, 26.[29] Rao, Indian Mathematics and Astronomy, 3.[30] Datta, “Vedic Mathematics”, 30.[31] Rao, Indian Mathematics and Astronomy, 3..[32] Datta, “Vedic Mathematics”, 31.[33] Datta, “Vedic Mathematics”, 31.[34] Datta, “Vedic Mathematics”, 33.[35] Datta, “Vedic Mathematics”, 34.[36] Hopkins, “Numerical Formulae in the Veda and their Bearing on Vedic Criticism”, 279.[37] Shukla, “The Deceptive Title of Swamiji’s Book”, 36.[38] Rao, Indian Mathematics and Astronomy, 14.[39] Shukla, “The Deceptive Title of Swamiji’s Book”, 37.[40] Mehta, Positive Sciences in the Vedas, 114.[41] Rao, Indian Mathematics and Astronomy, 12.[42] Rao, Indian Mathematics and Astronomy, 12.[43] Mehta, Positive Sciences in the Vedas, 117.[44] Iyengar, The History of Ancient Indian Mathematics, 6.[45] Iyengar, The History of Ancient Indian Mathematics, 7.[46] Iyengar, The History of Ancient Indian Mathematics, 8.[47] Rao, Indian Mathematics and Astronomy, 14.[48] Chapter I (1.2.1) Geometry, 16-21.[49] Iyengar, The History of Ancient Indian Mathematics, 9.[50] Rao, Indian Mathematics and Astronomy, 15.[51] Rao, Indian Mathematics and Astronomy, 15-17.[52] Rao, Indian Mathematics and Astronomy, 18.[53] Iyengar, The History of Ancient Indian Mathematics, 13.[54] Iyengar, The History of Ancient Indian Mathematics, 13.[55] Iyengar, The History of Ancient Indian Mathematics, 14.[56] Iyengar, The History of Ancient Indian Mathematics, 15.