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LA " 301 Mathematics in Manasollasa Mathematics is considered to be the most important and essential science. Mathematics helps the growth of other sciences. Mathematics did not develop in ancient India as a separate branch of knowledge. It was one of the important accessories to a body of knowledge which was helpful to the practical interest in the life of ancient Indian people. Like other Sastras and Vidyas, the study of the science of Mathematics was also connected with their religious life. In Vedanga-Jyotisa it is said, "As the crests on the heads of peacocks, as the gems on the hoods of serpents, so is Mathematics (to be reckoned) at the crown of the sciences known as 'Vedanga'". So, 31 Mathematics is very important among all the sciences. Rgveda, Yajurveda, and Atharvaveda mantras make several references to arithmetic principles. The Yajurveda appears to mention additions of two, and additions of four. 31. vedangajyotisa sastramukham - 40 yatha sikha mayuranam naganam manayo yatha | " tadvadbhedangasastranam ganita murdhani sthitam || 32. yajurveda . 18.24-25. eka ca me tistrasca me tistrasca me panca ca me panca ca me sapta ca me sapta ca me nava ca me 1+2=3 3+2 = 5 5+2=7 7+2=9 .. || 24 | ' 32

302 i In the 19 th Mandala of the Atharvaveda, there seems to be a reference to multiplication. "The ninety-nine supervisors (sentinel stars), O night, who look upon mankind, eighty eight in number or seven and seventy are they, sixty and six, 0 opulent, fifty and five. O happy one, forty and four and thirty three are they, O though enriched with spoil, twenty and two hast thou O might, eleven, yes and fewer still" 33 (Griffith). In the Atharvaveda, there is the consecutivity of of numbers from one to ten, and additions of numbers with catastrasca me astau ca me astau ca me dvadasa ca me dvadasa ca me sodasa ca me sodasa ca me vimsatisca me 33. atharvaveda - 19.47. 4+4 = 8 8 + 4 = 12 12 + 4 = 16 16+4 = 20 .. || 25 ye te ratri nrcaksaso drastaro navatirnava | asitih santyasta uto te saptasaptati || sastisca sat ca revati pancasata panca sumnaya | catvarascatvarimsatisca trayastrimsacca vajini | dvau ca te vimsatisca te ratryekadasavamah ||

303 34 multiples of ten. The Yajurveda mentions, the decimal numerical system. "鈼� Agni, may bricks be mine own milchkine, one and ten, ten and a hundred, a hundred and a thousand, a thousand and ten thousands, myrical and hundred thousand, a million and a hundred millions, an ocean, middle and end, and hundred thousand millions, and billion. May these bricks be mine own milchkine in 35 yonder world and in this world. (Griffith). 34. tadeva . 13.4. 16-18. ya etam devamekavrtam veda na dvitiyo na trtiyascaturthi namyucyate | na pancamo na sastah saptami napyucyate | nastamo na navami dasamo napyucyate || tadeva . 5.15.15. eka ca me dasa ca me dve ca me vimsati ca me tistrasca me trimsasca me 1 + 10 = || 2 + 20 = 22 3+ 30 = 33 catastrasca me catvarimsacca me 4 + 40 = 44 panca ca me pancasacca me 5 + 50 = 55... 35. yajurveda . 17.2. ima me agna istaka navah | santveka ca dasa ca dasa ca satam ca satam ca sahastram ca sahastram cayutam ca ayutam ca niyutam ca arbudam ca nyarbudam ca niyutam ca prayutam ca samudrasca madhyam cantasca parardhascaita me agna istaka dhenavah santvamutramusmimlloke ||

304 About this numerical system, D.D.Mehta says that 'The extensiveness of this numerical system is unique in the world. ,36 Dr. Shiva Shekhara Misra tells - "Hindu Science, indeed, especially in the sphere of mathematics, reaches a high standard, and the tendency to employ figures even in the other branches of learning which 37 this people cultivated is unmistakable. Somesvara's Manasollasa gives us a very vast numerical system. He says, 'there are eighteen place values of numbers. And further he mentions all the names of eighteen place values of numbers. If there is a zero after one, the place value of number one is called Dasa. Similarly, if there are two zeros after the number one, the place value of number one is called Sata. Further details of the names for the place value of one, corresponding to the number of zeroes after it, are listed in 36. D.D.Mehta, 'Positive Sciences in the Vedas'. (Arnold Heinemann Publishers, Delhi 1974), p.114. 37. Dr.Shiva Shekhar Misra. Fine Arts and Technical Sciences in Ancient India. (Krishnadas Academy, Varanasi, 1982), p.149.

the following table. Number 38 305 No. of Zeros after the number One Name of the Place Value of One 1,000 10,000 1,00,000 10,00,000 1,00,00,000 10,00,00,000 1,00,00,00,000 3 Sahasra " 4 Ayuta 5 Laksa 6 Prayuta " 10,00,00,00,000` 1,00,00,00,00,000 10,00,00,00,00,000 1,00,00,00,00,00,000 10,00,00,00,00,00,000 14 1,00,00,00,00,00,00,000 10,00,00,00,00,00,00,000 16 1,00,00,00,00,00,00,00,000 8 6000 9 10 11 27231995 Koti Arbuda Padma Kharva Nikharva Mahapadma sahkha Samudra Antya Madhyama Parardha 38. bindureko dasasthane te bindudvayam bhavet | bindutrayam sahastre syadayute taccatustayam || bindavah panca laksa syuh prayute bindavastu sat | bindavah sapta kotau syurarbude casta bindavah || bindavo nava padume syuh kharve syurdasa bindavah | ekadasa nikharve tu dvadasa syurmahambuje || sankha trayodasa proktah samudre manubindavah | antyasamjne patamakhyata bindavastithisamjnaya || dvirasta - bindavo madhye parardhe dasa sapta ca | evamastadasasthanam ganitam vyavaharikam || 2.2.98-102.

306 According to D.D.Mehta, "It is admitted by scholars that the modern decimal value notation was known in India in the 4 th CenturyB.C... paying a tribute to Indian genius, Laplace, the great scientist says, 'How grateful we should be, to the Hindus who discovered this great decimal system that did not occur in the minds of such mighty mathematicians as Archimedes and Apollonius'."39 In Manasollasa, it is stated, "By writing a zero after the number one (1), the value of the number one will be ten (10). But by writing the same zero after the number two (2), the value of the number two will be twenty (20). Similarly by writing a zero after the numbers three (3), four (4), and five (5), the values of 40 them will be 30, 40, 50 etc., respectively. . CE . tarkasamgrahah haridasa samskrta granthamala . 160 pr . 16. ekam dasa satam caiva sahastramayutam tatha | laksam ca niyutam caiva kotirarbudameva ca || vrndam kharvo nikharvasca prakukhah padmasca sagarah | antyam madhyam parardham ca dasavrdvayo yathakramam || 39. D.D.Mehta. Op.cit. p.116. 40. ekaike bindurekasced dasakam tat prakirtitam | dvitiyake purobindo sankhya vimsatirisyate || evam trtiyakesa binduh syat purato yadi | trimsadadha tada sankhya navatyanta prakirtita || 2.2.103-104.

307 Similarly, the value of the number two (2), by placing two zeros, three zeros... seventeen zeros after it, will be two hundred, two thousand ... two parardha. about Further, describing the fractions, Somesvara tells (1) Rupa (Integers or whole numbers); (2) Amsa (Numerator) and (3) Chheda (Denominator) 41 Rupa means a whole thing. Amsa means the part taken after dividing the thing. And Cheda means the total number of divisions made in the thing. Further, the multiplication of fractions is dealt with in Manasollasa. According to this, "Amsa of the fraction is to be multiplied by the Amsa of the other. And Cheda of the one fraction is to be multiplied by the Cheda of the other. The product of Amsa is to be divided by the product of the 42 Cheda. This concept is made clear in the following example. " 41. sampurna kathyate rupamamsa uddharito bhavet | tasyamsasya vibhago yah sa cchedah parikirtitah || 2.2.119. 42. gunayaidamsamamsana cchedam chedena buddhiman | phlamsam vibhajet tajjnah phlena cchedajanmana || 2.2.120.

308 3/4 x 1/5) The product of Amsa = 3 * 1 = 3 3 x 1 1 The product of Chedas = 4 x 5 = 20 4 * 5 3/20 The product of Amsa The product of Cheda Proceeding further, we find in Manasollasa, the description of (1) The conversion of mixed fraction into improper fraction and (2) the Division of fractions. 1. Conversion of mixed fraction into improper fraction According to Manasollasa, the Cheda is to be multiThe product plied with the Rupa of the mixed fraction. thus obtained is to be added to the Amsa of the mixed fraction. The sum thus obtained becomes the Amsa of the improper fraction. The denominator (Cheda) of the mixed fraction becomes the denominator of the improper fraction also. This concept is made clear in the following example. 3 is mixed fraction. The product of Rupa and Cheda 3 x 6 = 18. Product of Rupa and Cheda + Amsa = 18 + 5 = 23. Cheda in both the fractions The result is 23/6. 6.

1 309 ` 2. Division of Fractions 43 To divide a fraction by the other, Manasollasa says, "The dividend is to be multiplied with the divisor as same in sloka 120, after writing the Aisa and Cheda of the divisor as the Cheda and Amsa. Example: 3/4 + 1/5 = 3/4 x 5/1 = 15/4. Thus we find that Somesvara has referred to important fundamental Zoperations in Mathematics.