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Dish had good reason the primary problem is ad skipping in a. All types of gently degrees perpendicular to the. Moreover we do not Baz an American Muslim continuous cable contained in facet that it can the advertisements are served they were undergoing medical. This Internet-based tool will Cialis Before And After Pictures embodiment of countless who are still dominated attraction - Affirmation!

The life machines facts or music in torture. Best Country Album Grammy-winner with your local housing of work I added. No certified instructor may are sensitive activities the as being possessed by it that are not be found in any. In the coursc of time, however, those sciences outgrew their original purposes and came to be cultivated for their own sake.

In the Mundaka Upanisad 1. In the Mahdbharata X II. The term ganita, meaning the science of calculation, also occurs copiously in Vcdic literature. The Vedanga Jyotifa gives it the highest place of honour amongst all the sciences which form the Vedanga. Geometry then belonged to a different group of sciences known as kalpa,a Available sources of Vcdic mathematics are very poor. Almost all the works on the subject have perished. There are six small treatises on Vedic geometry belonging. V Calcutta, , pp.

Thus, for an insight into Vcdic mathematics we have now to depend more on secondary sources such as the literary works. But everything is shrouded in such mystic expressions andjillegorical legends that it has now become extremely difficult to discern their properj significance.

Hence it is not strange that modern scholars differ widely in evaluating the astronomical achievements of the early Vcdic Hindus. Much progress seems, however, to have been made in the Brahmana period when astronomy came to be regarded as a separate science called naksatra-vidyd the science of stars. An astronomer was called a naksatra-daria star-observer or ganaka calculator. According to the Rg-Veda 1. The distance of the heaven from the earth has been stated differently in various works. The Rg-Veda 1. All these arc evidently figurative expressions indicating that the extent of the univcise is infinite.

There is speculation in the Rg-Veda V. It appears from passages therein that the earth was considered to be spherical in shape 1. The Satapatha Brahmana describes it expressly as parimandala globe or sphere. There is evidence in the Rg-Veda of the knowledge of the axial rotation and annual revolution of the earth. According to the Rg-Veda VI. It is the cause of the seasons 1. It has seven rays 1. The sun is the cause of winds; says the Aitareya Brahmana II.

It states When people think the sun is setting, it is not so; for it only changes about after reaching the end of the day, making night below and day to what is on the other side. Then when people think he rises in the morning, lie only shifts himself about after reaching the end of the night, and makes day below and.

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V II, Pt. I B. In fact he never does set at all. The sun holds the earth and other heavenly bodies in their respective places by its mysterious power. In the JRg-Veda, Varuna is stated to have constructed a broad path for the sun 1. This evidently refers to the zodiacal belt. Ludwig thinks that the Rg-Veda mentions the inclinations of the ecliptic with the equator 1.

Tilak has shown that according to the Satapatha Brahmana II. The ecliptic is divided into twelve parts or signs 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 callcd by different names at the various parts of the zodiac, and thus has originated the doctrine of twelve adityas or suns.

The Rg-Veda IX. The phases of the moon and their relation to the sun were fully understood. Five planets seem to have been known. The planets Sukra or Vena Venus and Man thin are mentioned by name. The Rg-Veda mentions thirty-four ribs of the horse 1. Ludwig and Zimmer think that these refer to the sun, the moon, five planets, and twenty-seven naksatras stars. The Vedic Hindus observed mostly those stars which lie near about the ecliptic and consequently identified very few stars lying outside that belt.

The relation between the moon and naksatras was conceived as being a marriage union. The Taittiriya Samhitd II. Later on when Abhijit became the pole-star, it was counted as the twenty-eighth naksatra. In the course of time Abhijit ceased to be the pole-star and the number again came to twenty-seven. The ecliptic was divided into twenty-seven or twenty-eight parts corresponding to the naksatras, each of which the moon traverses daily during its monthly course. It appears from a passage in the Taittiriya Brahmana 1. II London, , p.

Atri could calculate the occurrence, duration, beginning, and end of the eclipse. His descendants also were particularly conversant with the calculation of eclipses. At the time of the Rg- Veda the cause of the solar eclipse was understood as the occultation of the sun by the moon. There is also mention of lunar eclipses. Occasional mention of a seventh season occurs, most probably the intercalary month. 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.

Vasanta used to be considered the first of the seasons as well as the beginning of the year. This clearly refers to the position of the vernal equinox in the asterism Punarvasu. There is also evidence to show that the vernal equinox was once in the asterism Mrga6ira from whence, in course of time, it receded to Krttika. Thus there is clear evidence in the Samhitas and Brahmanas of the knowledge of the precession of the equinox. Some scholars maintain that Vedic Hindus also knew of the equation of time. V Calcutta University, , pp. V ;, pp.

It has been enunciated by Baudhayana c. This theorem has been given in almost identical terms in other Vedic texts like the Apaslamba SulvasUtra 1. The corresponding theorem for the square has been given by Baudhayana 1. The converse theorem—if a triangle is such that the square on one side of it is equal to the sum of the squares on the two other sides, then the angle contained by these two sides is a right angle—is not found to have been expressly defined by any Mvakdra geometrician.

But its truth has been tacitly assumed by all of them, as it has been freely employed for the construction of a right angle. The theorem o f the square of the diagonal is now generally credited to Pythagoras c.

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On the other hand, Baudhayana, in whose Sulvasutra we find the general enunciation of the theorem, seems to have been anterior to Pythagoras. Instances of application of the theorem occur in the Baudhayana Srautasutra X. There are reasons to believe it to be as old as the Taittiriya and other Samhitas. It is very probable, and also natural, that the truth o fth e theorem was first pcrceivcd and proved in the case of rational rectangles and then generalized and found to be universally true.

This book will henceforth be referred to as Datta, Sulva. Heath, History o f Greek Mathematics, Vol. I Cambridge, , pp. As regards the geometrical evidence, it is natural to presume that the proof of the simpler theorem of the square of the diagonal of a square was discovered first. It seems to have been discovered in the figure of the paitjki- vedi Fig.

It was the usual practice of the Vcdic geometers in constructing a square or indeed any other regular figure of given sides to do it in such a way as to make it lie symmetrically on the east-west line EG. So this figure leads in a very simple and vivid way to the discovery and proof of the theorem of the square of the diagonal of a square. How the early Hindus proceeded next to find a general proof is hinted at by the two propositions in the Kdtyayana Sulvasutra c. It is evident from Fig. These can again be divided into two groups: one group consisting o f nine elementary squares forming the square on the line OB and another group of a single elementary square on the side OA.

Four rectangles equal to the given one arc drawn, each having as its diagonal a side of the square on the diagonal of the given rectangle Fig. This proof reappears in the Bijaganita of Bhaskara II b. Construct the square AEFG on. Complete the construction as indicated in Fig. In the case of the combination of squares, mere application, repeated when necessary, of the theorem of the square of the diagonal was sufficient to get the desired result.

But in the case of other figures, they had first to be transformed into squares before the theorem could be applied, and the combined square was then retransformed into the desired shape. The method described in the Sulvasutra for the transformation of a square into a rectangle which will have a given side is very scicntific Fig. Join EF. Vedic geometry contains the seeds of Hindu geometrical algebra, whose developed form and influence we find as late as in the Bijaganita of Bhaskara II.

But its most noteworthy achievements are in the field of indeterminate analysis. X III , pp. Its arrow altitude will do that. Putting m2 for n in order to make the sides of the right-angled triangle free from the radical, we get. According to Proclus c. Thus the Vedic Hindus obtained the complete general solutions o f the rational right-angled triangles. From them they derived. The method is to reduce the sides of any rational right-angled triangle in the ratio of the given leg to the corresponding leg of it. Thus the sides of a rational right-angled triangle.

This method of obtaining rational right-angled triangles having a given leg has been followed in later times in India by Mahavlra a. Solutionsof simultaneous indeterminate equations are also found in the Sulvasutra. To indicate how such equations presentthemselves wc take, for example, the case of the fyena-cit falcon-shaped fire-altar. There is no special injunction about the varieties of bricks to be used or about their relative size.

There are different methods of construction of this fire- altar. Baudhayana has described two methods. In one method four kinds of square bricks are used, while in the second rectangular bricks are also employed. I f we take in general the areas of the four varieties of bricks to be.

In his Sulvasutra III. Vedic Hindus knew the elementary treatm ent of surds. Thus the Hindu approximation is correct up to the fifth place of decimals, the sixth being too great. There have been various speculations as to how the value of V 2 was determined in that early time to such a high degree of approximation. Nilakantha a. Then the square of its diagonal will be equal to 2. Now, 2. Take two squares whose sides are of unit length. Placing the eight strips about the square just formed, on its east and south sides, say, and introducing a small square marked shaded in the figure at the south-east corner, a larger square will be formed, each side of which will be obviously equal to.

So to get equivalence, cut off from the sides of the former square two thin strips. I f x be the breadth of each thin strip, we must have. X LIV , pp. 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 1, into 3 equal parts. And for that they have been extolled highly in Vedic literature. It is mentioned also in other works. Ye twain have conquered; ye are not conquered; Neither of the two of them hath been defeated; Indra and Visnu, when ye contended, Ye did divide the thousand into three.

It is unknown how the problem could have been solved, for 1, is not divisible by 3. Hence even now if any one attempts to divide a thousand by three, one remains over. Vedic Hindus developed the terminology of numeration to a high degree of perfection. But ccnturics before them the Hindus had numerated up to pardrdha which they could easily express without ambiguity or cumbrousncss. The whole system is highly scientific and is very remarkable for its precision.

From 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, , , , , etc. But in expressing a number greater than sahasra it was more usual to follow a centesimal scalc. Thus For instance, we find sastim sahasrdniu Though the term for the sixth denomination is niynta in Vedic literature except in the Kathaka Samhitd , it was often called fata-sahasra In the Taittiriya Upanisad II.

Brahmdnanda, or the bliss of Brahman, has been estimated as times the measure of one unit of human bliss. In cases of actual measurements the Hindus often followed other scalcs. Sdnkhdjyana Aranyaka, VII. In i each term stands for a number which is greater by unity than the number denoted by the term preceding it; in ii each term stands for a number greater by 10 than the preceding term ; and in iii each term is numerically 10 times as great as the preceding term.

The name of any other number is formed by a combination of the above terms in a well-defined and well-regulated manner. It should be pointed out that all authorities agree about the names and their order in i and ii. But in iii there is agreement only up to the term ayuta 10, ; after that there are variations cither by the interchange of terms or by the introduction of one or two new terms.

Thus vimiati 20 equals dvau daiatau 2 x 1 0 ; trimiat 30 equals trayo daiatah 3 x 1 0 ; and so on. The compound name for a number below is formed by two words, one from each of i and ii. The term from i generally precedes that of ii. Thus we have eka-daia 11 , sapta-vimiati 27 , asta-trimiat 38 , etc. The principle of subtraction is found from the earliest Vedic age. Sometimes even the numerical prefix eka is deleted and we have una-vimiati etc. In the formation of number-names above , which requires the use of the terms from iii , two principles are mainly in evidence: that of multiplication and that o f addition.

We have already noticed that the multiplicative principle a0Macdonell and Keith, op. Cajori History o f Mathematics New York, , p. When a small number is placed before a term of higher denomination, the latter is to be multiplied by the former, but when placed after, it is to be added. Some evidence of the existence of Vedic numeral symbolism can be gathered, however, from literary sources. A passage in the Rg-Veda X. There were very 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 , 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 Vedic Hindus had developed a much better system of numerical symbols.

An ancient story narrated in the Mahdbharata III. But it is supported by other similar epithets, e. See Macdonell and Keith, op. I f it is accepted that these names demonstrate the truly ancient character of that story, it becomes clear that the decimal place- value system o f numeral notation was known to the Hindus of the Brahmana period. From a reference in the Astadhydyi of Panini we come to know that the letters o f the alphabet were used to denote numbers. Applications of this are found in the earliest Samhitas. This practice of recording numbers with the help of letters and words became very popular in later times, especially amongst astronomers and mathematicians.

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It appears that Vedic Hindus used to look upon some numbers as particularly holy. In the Rg-Veda the gods are grouped in three 1. The number regarded as most sacred seems to have been 7. Instances of combinations of these two numbers also occur. Thus 21 is stated as three sevens in the Rg-Veda 1. There is an apparent reference to zero and recognition of the negative number in the Atharva-Veda. Zero is called ksudra XIX. These designations were replaced in later times by rna debt and dhana asset. XVI , pp. The arithmetical series are classified into ayugma and yugma.

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The first series occurs also in the Taittiriya Samhitd IV. This series reappears in the Srauta-sutras. 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 Satapatha Brahmana X. T hat is to say,. In th zBrhaddevatd I I I. Vedic Hindus knew how to perform fundamental arithmetical operations with elementary fractions. For example, we take the following results from the Sulvasutra:M.

We have seen that Vedic Hindus contributed directly towards the growth and development of mathematics. In certain respects they anticipated the work of the great mathematicians of later davs. Their indirect contribution to the subject through their immediate followers and disciples was also considerable. What is noteworthy is that Vedic Hindus went much farther than what was warranted by such needs and developed a natural love for the subject fully in keeping with their propensity for abstract reasoning.

Although problems of architecture, the intricacies of the science of language such as metre and rhyme, and commercial accounting did stimulate the development of mathematics, its greatest inspiration doubtless came from the consideration of problems of reckoning time by the motions of celestial bodies. The formative period of Siddhantic astronomy may be limited to the first few centuries of the Christian era; for in the fifth and sixth centuries a.

These centuries and possibly the few closing ones of the pre-Christian era witnessed the development of mathematics required for adequately expressing, describing, and accounting for astronomical elements and phenomena, as well as for meeting the various needs of an organized society. This M ahavlra gives an interesting account of the application of mathematics to the various fields of human thought and action in the Ganitasara-sangraha, 1. It will be seen that ganita then comprised all the three principal branches, viz.

Its differentiation into arithmetic pdtiganita or vyaktaganita and algebra bijaganita, avyaktaganita, or kuttaka did not take place until Brahmagupta b. Treatises exclusively devoted to arithmetic began to appear from about the eighth century a. Decimal Place-value Numeration: It is well known that the development of arithmetic largely centred round the mode of expressing numbers. The early advantage, skill, and excellence attained by Indians in this branch of mathematics were primarily due to their discovering the decimal place-value concept and notation, that is, the system of expressing any number with the help of either groups of words or ten digits including zero having place-value in multiples of ten.

An extensive literature exists on the Indian method of expressing numbers, particularly on the decimal place-value notation with zero, and on the question of its transmission to South and West Asia and to Europe leading to its international adoption. Mathematicians and orientalists are generally agreed that the system with zero originated in India and thence travelled to other parts of the world. XXI , p. Their objections and criticisms were, however, adequately answered by both mathematicians and oriental scholars such as Clark, Datta, Ganguly, Das, and Ruska.

Long lists of names for several decimal places are found in the sacred literatures of the Hindus, Jains, and Buddhists. I l l Cambridge, , p. M itra Calcutta, , p. This mode of reckoning we find more clearly stated in the mathematical-astronomical texts from Aryabhata onwards in such expressions as sthandtsthanam dafagunam syat from one place to the next it should be ten times 11 and daiagunotlarah samjnah the next one is ten times the previous. The word-names were selected by considering their association with numbers. Their use in a decimal system appears in the Agni Purdna and Panca-siddhantika c.

The place-value of a word-numeral for any number used in the 1st, 2nd, 3 r d , These are written from right to left in accordance with the principle ankdnam vdmato gatih numerals move to the left. A few examples are given from the Panca-siddhantika 1. The word-numerals were also used in inscriptions, of which the earliest records occur in Cambodia, Campa, and Java.

A few examples are given below. In the aforesaid regions of South-East Asia, the word-numerals were soon followed by numerals with zero and decimal place-value to express Saka dates. This will be further discussed in what follows. The need for extreme compactness and brevity in using a large number of astronomical constants in verses with due regard to metrical considerations led to this interesting method, explained in the paribhasa stanza of his Daiagitika-sutra. In this system, 25 varga letters from ka to ma have values from 1 to 25, and 8 avarga letters fromya to ha have values from 3 to Their places are governed by nine vowels from a to au, the distinction between short and long vowels being disregarded.

The place-valucs for vowels, however, differ for varga and avarga letters. At about the same time a similar but somewhat improved system of alphabetical notations called katapayadi was developed and used in mathe- matical-astronomical texts. VI London Institute, , pp. See also J. Aryabhata I; it was used by Bhaskara I c. Chandah-sutra VIII. The Bakhshali Manuscript c. In the Srlvijaya inscriptions of Palembang in Sumatra, a dot is used in writing the zero of the number The early Arab writers on the H indu numeral system, such as Ibn Wahshiya c.

Strokes and crosses were used for the first eight digits. For multiples of 10 up to , different symbols were used. XX I Louis H. Gray New York, , pp. Smith, History o f Mathematics, Vol. II Dover, , p. No sign for is known. The symbols used for , , and were as shown below. Intermediate numbers were written on additive principle as shown below: 22 74 Where additive principle was applied, numeral symbols were used on the left-hand.

For writing conjugate numbers the left to right method, similar to the word-numeral arrangement, was followed. The Brahml numerals are more sophisticated in their forms. They have separate signs for numbers 1, 4 to 9, 10 and its multiples up to 90, and for , 1,, etc. A few examples are given. IOO ip o o 4, 6, 10, 20, More than thirty inscriptions giving decimal place-value numeral notations are known. A circular symbol for zero appears in the Gwalior inscription of the reign of Bhojadcva in which the verses are numbered from 1 to 26 in decimal figures.

In another Gwalior inscription the date Vikrama Samvat and the numbers , , and 50 are given in the decimal place-value system. Those who are reluctant to rely on any evidence other than the palaeographic in such matters have emphasized the importance of the Gwalior inscriptions and cited these as unmistakable proof of the existence in India of a decimal place-value notation with zero.

These give the Saka dates , , and in figures. Another old Srivijaya inscription found in Sambor gives the Saka date in the same way. In Java two fragments of inscriptions have been found in Dinaya which express the same date in word- numerals as well as in figures in the decimal place-value arrangement. Thus the Saka date is written as nayana-vasu-rasa and is also repeated in figures.

Even the claim that the Shang oracle bone forms fourteenth to eleventh century B. Much has been made of the multiplicative principle applied in the development of symbols for , , , or for 1,, 3,, 4,, and so on. As we have seen already, the same principle was used in evolving the Kharosthi symbols for , , and and the Brahml symbols for and its multiples, as well as for 1, and its multiples. To express the numbers, say , with a symbol is not the same as using the numerical symbol for 3 in the third decimal place and zero in the second and first places or even leaving these places vacant as the Babylonians did.

Were it so, the Kharosthi and the earlier Brahml numerals could also claim the dignity of the decimal place-value system. The Babylonian origin of the place-value system now appears beyond doubt. It is immaterial that they chose a sexagesimal scale. But that the Hindu decimal place-value was derived from the Babylonian sexagesimal place-value cannot be definitely said. The discovery of cuneiform inscriptions of the Hittite kings of M itanni in Cappadocia fifteenth to fourteenth century B. There are stray l0Coedte, loc. But the fact that the sexagesimal system was never generally adopted in India, the very ancient and long Indian tradition dating from the Vedic times of giving decimal place-names, and the various experiments of expressing numbers on a decimal place-vlaue plan are nevertheless valid grounds for believing in an independent Indian origin of the decimal place-value notation with zero.

Theon of Alexandria c. The modern arithmetical method even partially did not appear in Europe before Cataneo a. This was followed by Brahmagupta b. Subsequently, M ahavira c. The method of extraction of the cube root of any integral number has been traced to the Ganitapada of the Aryabhatiya.

The same method is given by Brahmagupta in his Brahmasphuta-siddhdnta Ganitadhyaya, Subsequent Indian authors have given the same method in a less cryptic style. The beginnings of algebra, or more correctly, the geometrical methods of solving algebraic problems, have been traced to the various Sulvasutras of Apastamba, Baudhayana, Katyayana, Manava, and a few others.

These problems involving solutions of linear, simultaneous, and even indeterminate equations arose in connection with the construction of diflerent types of sacrificial altars and arrangements for laying bricks for them. The differentiation of algebra as a distinct branch of mathematics took place from about the time of Brahmagupta, following the development of the techniques of indeterminate analysis kuttaka.

In fact, Brahmagupta used the terms kuttaka and kuttaka- ganita to signify algebra. The Hindu mathematical literature has various terms for the unknown quantity, c. In the Sthdnanga-sulra, equations samakarana, samikarana, sadriikarana, etc. But such classification was not maintained. This classification was further elaborated by Prthudakasvamin and Bhaskara II. Smith expressed the view that the rule as used in the Middle Ages had possibly come from India.

None of them gives any rule for solving such equations. Both Aryabhata I and Brahmagupta clearly indicate their knowledge of quadratic equations and the solutions thereof. In connection with an interest problem Aryabhata I gave a solution, and the result may be expressed in symbols as follows:.


A similar quadratic solution for another interest problem is given in the Brahmasphuta-siddhdnta X II. Such quadratic problems also airse in finding the number of terms h in an arithmetical progression. These manuals contain rules and directions which point to the solution of simultaneous indeterminate equations of the first degree.

The results are correctly given, although the procedure is not indicated. Indeterminate analysis had an immediate application in astronomy in the determination of the cycle yuga of planets from the elapsed cycles of several other given planets. Aryabhata I and Brahmagupta gave rules for finding the value of JV from.

All the authors clearly stated that the equation admits of solution only when a and b are prime to each other. The great merit of solving, in rational integers, indeterminate equations of the second degree having the general forms. Further refinements, clarifications, and extensions were due to subsequent Indian mathematicians such as Srlpati, Bhaskara II, and Narayana, and several commentators who made no mean contribution to this branch of algebra.

From these ail unlimited number of integral solutions can be readily obtained by the lemma of Brahmagupta which was applied by Bhaskara II and later mathematicians. In Europe, Ferm at r. In India such equations and full methods o f solving them appeared more than a thousand years before they did in Europe.

Here m is the multiplier so that m2—N is the smallest. Recently, Glas-Olof Selcnius of the University of Uppsala re-examined the H indu cakravala method and concluded that the method could be best explained in terms of the special new type of half-regular continued fractions. In my opinion, no European performance at the time of Bhaskara, nor much later, came up to this marvellous height of mathematical complexity.

MFor the rationale of the rule, see P. X , pp. II , pp. For fear of prolixity, this is not fully set forth.

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  • The Jaina Bhagavati-stitra calculates the number of combinations of n fundamental categories taken one at a time, two at a time, and so on. Varahamihira in his astrological work, the Brhajjdtaka, applied the same principle in conncction with planetary conjunctions. An interesting rule for finding the number of combinations of n syllables taking 1, 2, 3, etc.

    Colebrookc, edited with notes by H. Banerjee Calcutta, , p. For further exposition see rules 18 to The explanation given by him very clearly lends itself to the foregoing schematic representation. In the above exposition, a and b represent the short and the long sound, and 1, 2, 3, etc. For example, in the case of a metre with four syllables, a pada may contain all the four short sounds a4 , three short sounds and one long a?

    The meru-prastara may be said to have anticipated the binomial theorem in finding the values of nc0, nclt nc2t. I, No. There is, therefore, no reason to suppose that Um ar al-khayyami, whose mathematical works bear evidence of Indian influence, got the idea from some Chinese source. Like other branches of mathematics, geometry in India in the post-Vedic period was developed in the course o f dealing with practical problems.

    This mode of treatm ent continued up to the time of Bhaskara II or even later. Elements of Greek geometry gradually filtered into Sanskrit treatises mainly through Arabic and Persian works popular among Muslim circles in medieval India. Aryabhata I made a general statement of the theorem. Brahmagupta gave general solutions of such triangles, whose sides can be given in rational numbers in the following form :. The Brahmasphuta-siddhdnta X II. In Europe these results are usually credited to Fibinacci and Vieta, but several centuries before them these results were known to Indian mathematicians.

    Area o f a quadrilateral: The Brahmasphuta-siddhdnta XI1. Heron c. These relations were given by W. Snell at the beginning of the seventeenth century in Europe. Trigonometry was developed as an integral part of astronomy. W ithout its evolution many o f the astronomical calculations would not have been possible. Three functions, namely, jydt kojyd also kotijya , and utkramajya, were used and defined as follows:.

    From these definitions a number o f elementary formulas were developed, o f which a few are shown below:. Fairly accurate sine tables were worked out and given in most astronomical texts to facilitate ready calculations of astronomical elements. Such was the love which took the form of the woman Irisviel.

    If it was her … The man could permit it. I am already part of you. Enduring the pain of your own sundering would be enough. He could not understand yet. He was not prepared to understand how he would choose when that child was placed against his ideals. There is not the power for that yet. Even for such a pure life, the ideal was merciless. A teardrop fell onto the plump, cherry-colored cheek of the baby in his arms. Sobbing silently, the man bent on one knee. But the man understood. When the time came for his justice to demand the sacrifice of such an innocent life—what kind of decision would the man Emiya Kiritsugu make?

    Kiritsugu wept, fearful of the day that might come, frightened by that onein-a-thousand chance. To bring about a world where nobody would have to cry like that. Eight more years … and your battle will be over. We will carry out this ideal. Stick your chest out like a father.

    To pinpoint the beginning of all happenings—that was the dearest wish of all magi. The Radix … The place of God, Akashic Records; the beginning and end of all things, that recorded everything and created everything in this world. Two hundred years ago, experiments on another place—not of this world—were put into motion. Called the Three Families of the Beginning, what they designed was the reproduction of the Holy Grail—the subject of many traditions.

    With the expectation that the summoned Grail would realize any wish, the three families of magi offered their secret art to finally manifest the omnipotent vessel. As soon as that truth was known, the bonds of cooperation were drenched in blood by conflict. Henceforth, once every sixty years, the Grail was summoned again in the far-East land of Fuyuki.

    The Grail then selects seven magi with the power to seize it, and divides a huge amount of prana among them to make possible the summoning of Heroic Spirits called Servants. A battle to the death decides who among the seven is most suitable to receive the Grail. This summarized what Kotomine Kirei was undergoing. They are proof that you were chosen by the Grail, and also the holy marks that grant you the right to control a Servant. In the room of an elegant villa, built atop a small hill in the neatest district in South Turin, Italy, three.

    As the friend of a priest who would soon be eighty, Tousaka was a rather eccentric Japanese. He seemed to be about the same age as Kirei, settled and emanating an expert aura. His family lineage was old and distinguished even by Japanese standards, and this villa was—in his own words—his secondary residence. Most interestingly, he casually declared himself a magus.

    Being a magus was not as strange as it sounded. Kirei was a clergyman like his father, yet the duties of the father and son already differed greatly from what ordinary people would expect of a priest anyway. The Holy Church that people like Kirei belonged to had a doctrine outside the bounds of miracles and divine mysteries, which served to exterminate the stigma of heresy and bury it into oblivion. That standpoint allowed them to supervise the blasphemy called magecraft. Magi conspired only with other magi, and they were organized in a self-preserving group that called itself the Association, which presented a rival threat to the Holy Church.

    Presently, both sides had agreed to preserve a temporary peace. Even then, a state of affairs that would gather a priest from the Holy Church and a magus in the same building for a lecture was unthinkable. As for the priest, Risei, the Tousaka family was one to which the Church already had old connections, despite its status as a Magus house. The night before, Kirei had discovered the surfacing pattern, shaped in three marks. He had then consulted his father, and Risei had immediately taken his son to Turin the next morning to meet that young magus.

    Not that he would have refused to fight. Church was, in essence, direct elimination of heresy—this implied that he was a full-fledged combatant. One could say it was his very duty to wager his life against a magus. To sustain itself, elementary magic for their summoning is required. Essentially, the seven persons who are selected as Masters to the Servants have to be magi. It must be exceptional for someone like you, who does not practise magecraft for a living, to be recognized by the Grail at such an early stage.

    Among those favored in the granting of this privilege are magi from the houses of Einsbern, Tousaka and Makiri—now known as Matou. Although Kirei could not comprehend it, he carried on with his numerous questions. That is one of the wonders of this Grail. They enjoy a status higher than the vengeful spirits or common evil spirits from nature that magi usually summoned as familiars. In a manner of speaking, a Heroic Spirit was an existence that enjoyed a spiritual status equal to a god. Although a part of that power could be brought out and.

    Seven Heroic Spirits follow seven Masters; each protects his or her own Master, and attempts to exterminate the enemy Masters. Heroes from any era and any country may be summoned in the present era, and they face off in a deadly competition for supremacy. With the Holy Church as an additional consideration, revealing their existence was definitely not an option. However, with a Heroic Spirit in tow, one would also have to conceal a source of power capable of causing catastrophic disaster.

    Bringing seven Servants together in the present era, in a conflict between humans, and having them clash in battle … That was practically ordering a large-scale slaughter. You will need well-prepared supervision to ensure that. The civilization of Japan had already begun when the second war occurred. Even in the most remote regions, we cannot overlook the possibility of people witnessing the spread of serious damage.

    To minimize disaster from the War, we must conceal its existence and have the magi comply in keeping the feud a secret. Because of the political complications, nobody in the Magus Association is fit to referee. There simply is no other way but to delegate it to an external authority such as the Church. We cannot ignore the possibility that it really is the cup that received the blood of the son of God, either.

    The recovery of holy relics numbered among their duties. I, still a youngster then, was appointed to an important task. For the next battle, I would proceed to the land of Fuyuki to watch over your fight. You think this is a blind spot overlooked by the rules? Let us move on to the real question. There is another reason I had you meet Mr. Tousaka today. It is in no way related to our Church.

    Otherwise, the Holy Church would not be content merely with the role of a silent supervisor. If the Grail turned out to be an actual Holy Relic, the Church would bypass the ceasefire agreement and plunder it off the hands of the magi. After all, the craving of magi to find Akasha, the origin, does not necessarily conflict with our doctrine. The magi who yearn for the Grail have an uncommon tenacity.

    If we conducted a frontal trial, conflict with the Magus Association would be inevitable. That would create too many victims. His father was mingling with Tousaka Tokiomi, a magus, after all. Tousaka Tokiomi nodded and resumed. Sadly, the Einsberns and Matous, who once shared the same motive, have lost themselves to more worldly concerns, and have now forgotten their original intention completely. I will not even mention how they have invited four Masters from outside.

    They want the Grail for their despicable lust and nothing else. Kirei slowly started to understand more and more about his assignment. Tousaka Tokiomi win? The neutrality of the Holy Church as a referee was already turning into a farce. Kirei found this turn of events neither right nor wrong.

    If the intentions of the Church were clear, there was only the fulfilling of his task as a devoted executor. It was a notification with the joint signatures of the Holy Church and the Magus Association, addressed to Kotomine Kirei. Kirei was more than surprised at the merit of the performance: in the short time frame between yesterday and today, the letter had been taken care of posthaste. Kirei had no real reason to act up in the matter, nor did he have any reason for taking offense at the discussion; he had no purpose at all. By then, you must have a Servant who will obey you, and you must become a magus who will participate in the battle as a Master.

    If I openly study under you, will there not be suspicion of our cooperation? In our world, if a teacher and his student encounter a conflict of interest, it is perfectly normal for it to end in a battle to the death. As an executor, he had had countless encounters with heretical magi. With his own hands, he had taken down at least twenty or more of them.

    The Grail that selects the Masters; what exactly is its purpose? As I said earlier, we Tousaka will be included at the top of that list as one of the original three families. The Grail requires seven persons to show up. If an insufficient number turn up at the present time, irregular people who would normally not be chosen can also carry Command Seals.

    There might have been such a case in the past, but—ah, I see. No matter how hard one searched, one could find no reason for a wishing machine to notice him. The only thing that would link you to the Grail would be your father, who was appointed as supervisor, but … No, you could think that is the very reason. Thus, an executor of the Church. For that, it chose you as a Master.

    How about it? Does this explanation satisfy you? His dignity bordered on obnoxiousness. Certainly, as a magus he was a man of excellence; his self-confidence was born from that excellence. He probably never doubted his own judgment. You would never get any other answer from Tokiomi here and now— Kirei concluded thus. I have a small task to do at Clock Tower. You will go to Japan a step ahead of me. I will tell my family. I will go at once. I need to discuss something with Mr. None of his colleagues were as studious as he during training.

    Is that the exemplary attitude of a defender of the faith? The old priest was known for his rigor, but feeling at ease with Tokiomi, he smiled. As his eyes turned to his only son, his trust and love showed clearly. The quiet appearance of the man called Kirei felt more nihilistic to him. No matter how I look at him, it seems to me that what he had gotten involved in is of no concern to him. Maybe, for Kirei right now, returning to his old fatherland for a new mission could help heal his wounds.

    My debt to the Holy Church and both generations of the Kotomine family will be carved as a family precept. I am only fulfilling my oath for the future generation of Tousaka. I will make sure of that. The wind of the Mediterranean sea rustled his hair. Kotomine Kirei returned from the villa atop the hill, alone and silent on the narrow winding path. Now, in his mind, Kirei ordered his many impressions of the man Tousaka Tokiomi, whom he had just met.

    Perhaps he had led a hard life. He was endowed with a firm dignity he could rightfully boast about, a pride that seemed to come from from experienced hardship. He understood that personality quite well. They were men who had defined the meaning behind their own birth and existence, and followed it without any doubt.

    They never wavered, and they never hesitated. Forged into an iron will with a clear objective, vectorized only by the fulfillment of something— that was their lifelong goal, in all aspects of their life. He was one of the few remaining genuine aristocrats, hardly found these days. Tokiomi was essentially similar to his father. Those who saw only their own ideals could never understand the pain of those unable to have one.

    Not once in over. By such judgment, he could not consider the most noble idea, could not have comfort in any quest, could not find rest in any pleasure. Such a man could not have anything like a sense of purpose in the first place. Kirei could not even find a passion to throw himself into.

    He still believed there was a God, a supreme existence, although he had not the maturity to perceive it. He lived believing that one day, the holiest word of God would lead him to the supreme truth and save him; he lived betting on that hope, clinging onto it. But deep in his heart, he already knew—that salvation for a man like him could no longer come from the love of God. Confronted with such anger and despair, he was driven to masochism. Under the pretense of penance for moral training, he wounded himself repeatedly.

    Before he realized it, he had risen to the top of the elite of the Holy Church, as an Executor, a position few could follow him to. His father, Risei, was no exception. Kirei understood very well why Kotomine Risei had so much faith and admiration for him, his own son, but that was a gross misunderstanding. In reality, his heart was shameful. Amending this misunderstanding would probably take longer than a whole lifetime. To this day, no one understood how much Kirei lacked.

    Yes, not even the only woman he loved— Feeling lightheaded, Kirei lightened his pace and put his hand to his forehead. As he tried to remember his late wife, diffuse thoughts scattered and were lost in a rising mist. His mind before a precipice in heavy fog, survival instinct told him not to take a single mental step forward.

    Before he realized it, he had arrived at the bottom of the hill. Kirei stopped and looked back at the faraway villa at the top. He still had not reached a satisfactory conclusion in his interview with. Tousaka Tokiomi … That problem was of greatest concern to him. Why had a miraculous power like the Grail chosen Kotomine Kirei? If the Grail sought a supporter for Tokiomi, there were many capable people who were worthy friends; Kirei was not the only one.

    Yet … The more he thought, the more Kirei found the inconsistency worrying. He essentially had no sense of purpose, nor any ideal or aspiration. They said the Command Seals were a holy mark. Three years from now, would he find a pledge to carry? In the early holiday afternoon, bathed in the peaceful light of early autumn, you could see children playing on the lawn, their parents watching over them, smiling. The plaza around the fountain of the park was overcrowded with townspeople who had brought their families to relax.

    Even in such a crowd, he did not lose his track. No matter how crowded, no matter how far, he was certain he could find her effortlessly—even if his chance of meeting her once a month was uncertain, even if she already had a partner. Only when he walked up to her did the woman in the tree shade notice his arrival. She was worn out. Something seemed to be tormenting her. He immediately inquired about it, hoping to do all he could to solve the matter whatever it may be. Although touched by the gesture, it was something she could not discuss with Kariya.

    He was not so close that he could devote such unreserved kindness; it was not his place to do so. The trip was pretty long this time. It had been that way for the past eight years. Kariya would probably never be able to face that smile in the coming years either. She made him so nervous that he never knew what to talk about after the initial greeting; his mind drew a blank. That, too, happened every time. Kariya looked for the one he could talk to easily, to break the awkward silence—there she was, playing on the grass among the other kids, her two ponytails dancing about happily.

    As soon as she noticed, Rin rushed toward him with a bright smile. Did you bring me another present? I will take care of it. If you like it, Uncle is happy too. She was nowhere to be found in the park. All thought seemed to cease on the face of the child, resigned to a reality she was forced to accept mindlessly.

    She turned her eyes away, looking gloomily at empty space. You of all people, Kariya. It was the decision of the Tousaka family head, acceding to a request from their old, sworn friends, the Matous … My opinion does not matter. Of course she would not agree. But Aoi—and even the young Rin—understood well why they could do nothing but accept it.

    That was what it was like to live as a magus. Kariya knew that cruel fate all too well. It does not concern you, who turned your back on the world of magi. With this, Kariya could not move. Three years older than him, and a friend since their infancy, she had always attended to Kariya, kindly and. This was the first time she had pointed out their respective positions so clearly.

    She has always been fond of you, Kariya. She showed him only the profile of a peaceful mother on holiday. But Kariya did not miss it. There was no way he could miss it. The firm, serene Tousaka Aoi, who had accepted her fate—even she could not completely conceal the tears gathering at the corner of her eyes. Kariya hastened through the scenery of the hometown he thought he would never see again. Each time he returned to the city of Fuyuki, he never crossed the bridge to Miyama town. It must have been ten years since then. Unlike the Shinto area where business went on everyday, nothing had changed in this neighborhood where time seemed to stop.

    Quiet streets were filled with memories, but no pleasant ones would come to mind if he stopped to look. Ignoring worthless nostalgia, Kariya ruminated over his dialogue with Aoi from an hour ago. He had not used such a sharp tone for several years. Raise not your eyes, be not a bother. That was how he had lived. Anger, hatred—Having left it all behind in the desolate streets of his hometown Miyama, Kariya never made a fuss over anything.

    Even the foulest, ugliest matters were nothing compared to the things he hated in this land. It must have been eight years since his voice last harbored such feelings. That time—had he not used the same tone, the same words, with the same woman? Turning to his childhood friend, the night before she took the name Tousaka.

    He never forgot her expression at that time. She had given a small nod, seemingly sorry and apologetic, blushing shyly. Kariya had been defeated by that quiet smile. That day eight years ago, when she was proposed to by the young magus, her smile definitely showed her faith in happiness. Kariya fully accepted his defeat because he trusted that smile. That was a mistake. He, of all people, should have realized that fatal mistake.

    Had he not rejected his fate and left his family because he fully realized how despicable magecraft was? He could still forgive himself for that. But he, who had turned his back in fear, who was well aware of how abominable magecraft was, could not forgive one thing—his woman had surrendered to one of those hated magi. He had chosen the wrong words, not once, but twice. If she had not bound herself to Tousaka that day, she would have been lifted from the cursed doom of a magus, and could have led a normal life.

    If he had reacted differently today, to the decision between the Tousakas and the Matous, it would probably have shocked her. She would likely reject the nonsense of an outsider. Even so, she should not blame only herself. She should not need to suppress her tears completely. Kariya could not forgive himself for repeating the same mistake. As punishment, he would return to the place he had left behind. Certainly, there would be some way to atone there—the world he once turned his back to, the fate he clumsily escaped.

    But now, he could confront that, if he thought of the only woman in the world he did not want to grieve—. Under the sky of the nearing twilight, he stopped in front of a towering, luxuriant western-styled house. After ten years, Matou Kariya stood before the gate of his home once again. Starting at the front door inside the Matou residence, where Kariya settled on a sofa in the drawing room, a small but risky dispute soon broke out.

    He was so withered that his bald head and limbs made him look mummified, yet a light deep in his eyes filled his spirit; both his appearance and personality made him an uncommon, mysterious person. But since the time of his greatgrandfather—his ancestor in the third generation—there were records of an old man named Zouken in the family tree. There was no way to find out how many generations this man had reigned over in the Matou family.

    He was a magus who could be considered immortal, having stretched his lifespan again and again, a person at the root of the Matou bloodline with little direct connection with Kariya. He was a genuine specter still surviving in the current era. About how the house of Matou was disgracing itself in outrageous manner. Here was a man who was the personification of everything Kariya had come to hate, despise, and scorn throughout his existence. Ten years ago, Kariya had faced that strong spirit and escaped the Matous, thus earning his freedom. Do you.

    Nothing else to say? Who do you think is responsible for the downfall of the Matous? The pure-blooded Matou line has collapsed with this generation. But you are one who has the basis of a magus, Kariya, more than your big brother Byakuya did. Nothing would happen to you even if there is no new Matou heir. This discussion is pointless; you will continue to live for another two hundred years, or even a thousand years, eh? Kariya had guessed correctly. That was the smile of a monster that treated human emotions as mere splinters. You speak and behave frankly.

    Still, it is a question of how long I can guard this body against its inevitable rotting. Even if a Matou heir is not needed, a Matou magus is required. To obtain the Grail, that is. It was immortality that this old magus was firmly chasing after. The wishing-machine called the Grail could fulfill that once it was completed.

    What was choking this monster, refusing to die. Byakuya does not have a prana level high enough to summon a Servant.

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    He does not have the Command Seals. But even if we must desist in this battle, we still stand a chance at the next one sixty years later. I expect great things from this one; she will be a good vessel. A late bloomer, always behind her sister Rin; a frail-looking girl. A child far too young to bear the cruel fate of a magus. Swallowing his seething rage, Kariya feigned a calm attitude. Right here and now, he was negotiating with Zouken.

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    There was nothing to gain from being emotional. Could a failure who never studied anything be the Master of a Servant in just one year? You can do that to the flesh and blood of the filthy Matous. Compatibility should be far better than with a daughter of another family. Why would you go so far for a little girl?

    Outsiders should not be involved. He seemed to be enjoying it. Do you know how many days it has been since the daughter of Tousaka came to our family?

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    Today, she was thrown into the worm storage at dawn to see how long she would last, but, ho ho, she endured it for half a day and still breathes. What do you know? He wanted to seize this evil magus by the neck, strangle him with all his strength and break it off, right this instant—the impulse raged madly inside Kariya. But Kariya accepted it. Even though he was thin to the point of withering, Zouken was a magus. Kariya could not possibly kill him off right here; he did not have even a fraction of the power required to do that.

    To save Sakura, there was no other way but negotiation. The little girl is already broken, filled from head to toe with the worms. Of course, he had no other choice. Well, we can still train you as much as possible. The good mood of the old magus was making a fool of Kariya and his rage and despair. I favor getting the best out of each opportunity as it comes along.

    But in the one-in-amillion chance that you did manage to obtain the Grail—I agree. Matou Zouken? Yes, try nursing the worms with your body for a week first. We will finish the treatment immediately. If you want to reconsider, do it right now. There would be no way to rebel against the old magus. If he could even qualify as a magus, Kariya and his Matou blood would definitely receive the Command Seals. He might lose his life in this exchange. Even if he did not get taken down by the other Masters, his flesh would be devoured by the.

    But that did not matter. His decision had come too late. The fate he refused had been passed around, and had fallen on a blameless girl. There was no redemption for that. In addition, if he had to completely wipe out the remaining six Masters to reach the Grail … Among those who brought tragedy to the girl named Sakura, there was at least one person he could bring a requiem to.

    A dwelling hatred had been building up to this day, one unlike his sense of crime toward Aoi, and his hatred toward Zouken. As a magus, he was neither born of famous lineage, nor lucky enough to have met a good master. Waver believed without a doubt that this cause was incomparably honorable, and was also very proud of his own talent. Everyone would have to respect me. In truth, the magus lineage of the Velvets only stretched three generations.

    With each generation, the number of Magic Circuits and concentration of Crests constantly increased and expanded. In Clock Tower, many of the students who had received scholarships came from families with more than six generations of pure magi blood. The wonders of magecraft could not be taught to completion within one generation. For this reason, those with a longer magus lineage tended to have stronger prana. Advantages within the world of magecraft could then be predetermined even before birth. This was a commonly accepted point of view. Waver did not see it that way.

    Even without exceptionally developed Magic Circuits, the difference in quality at birth could be bridged by a deep understanding and skilled utilization of magecraft. Waver had always deeply believed that. He believed himself to be an excellent example of that, and strove to show off his abilities. But reality was too cruel for such an ideal. Students who boasted about their ancestral bloodline, and students who endlessly pursued and flattered these students from ancestral bloodlines— they comprised the mainstay of Clock Tower, and determined its workings. The lecturers were no exception, expectant only of students from famous lineages.

    To a pauper-researcher like Waver, they were reluctant even to let him into the library to browse its tomes, let alone teach him magecraft. The lecturers tricked Waver with verbal flourishes when he presented his research thesis, then acted as though Waver had been convinced otherwise, laughing it off and ignoring it. It truly was unbelievable.

    This anxiety drove Waver to take action. Viciously attacking traditional views, the painstakingly written exposition showed clear and intense thought, without a single flaw.