 Chairman sir thank you very much but I have a complaint against Peter for putting me in this lot after lunch and talking about mathematics I was reminded of one of my sessions in Tanzania I was addressing a class of 35 trainees and after lunch they will have very heavy lunch mainly non-vegetarian and then after the lunch the session will be almost washed out people will start dosing so once out of desperation I told a fellow was sleeping dosing and I told his neighbor please wake him up immediately he retorted sir why should I wake him up you made him sleep better you wake him up I was actually taken aback and you are some out of truth is statement also anyway I will not go into very high technicalities of mathematics because everybody I don't know why they have some fear about the mathematics I will just try to highlight the contribution of Jinnabhadra Gani Kshama Sharman a great Shwetambar Acharya of 56 century to the Jaina mathematics Indian mathematics and world mathematics actually lived for 105 years and during which he produced excellent works like Vishisha Vashya Bhashya Burhat Sangra Hani and mainly we are concerned with Burhat Kshetra Samas which is nothing but geography of human world human world as per the Jaina tradition isn't it? I am taking you back to about 1500 to 1600 years when the golden era of Greek classical mathematics came to an end with Papas Proclus etc etc and it was only revived about a thousand years in Western Europe and the Western mathematicians owe their allegiance to Greek mathematicians whatever it may be but what about the period it is also called dark period of European mathematics Western mathematics but what about mathematics in that period yes in India a school emerged a very big school emerged of mathematics spearheaded by Vedic mathematicians like Aryabhatta, Brahma Gupta, Varahamiheer etc etc and in India in Jaina mathematicians just as Sarvanandhi then Jaina Bhadrakani Kshama Sharman then so many other mathematicians of Jaina school like they were talking about the Yati Vreshabha whose contribution is very great he also talks about the entire universe and he gives a lot of formulas various formulas about the universe its area its volume etc etc whatever it may be but what is the contribution of India during this period the greatest contribution is the system of numerals symbols of numerals from 0 1 2 3 9 this 10 symbols that is the biggest contribution of India which is recognized by all western scholars also and to give 0 its place value that was a much more important function we will see what is this place value etc is there so anyway the first clear evidence of this I am not taking credit that Jaina mathematicians did this but it is sure that the first clear evidence is of Jaina Bhadrakani who employed these symbols very easily he made big very big calculations with the help of this etc so that's why I'm just highlighting his very important contribution he was so adept so skillful with matters as a student of mathematics today today he could add subtract multiply divide deal with fractions calculate square roots of very large numbers etc etc isn't as you know I'm not going much detail the scale all of us in India know that zero ten hundred thousand this was a decimal scale increasing by multiplication multiplication of ten then ten thousand then what we call one lakh or hundred thousand then million that is ten lakh within kuri kuri is a very important that it is ten raised to seven that means one followed by seven zeros and Jaina Bhadrakani in many of his calculations goes up to Koda Koda Kodi that means ten raised to seven multiplied by ten raised to seven is equal to ten raised to fourteen that is one followed by fourteen zeros you I'm giving you some examples immediately for example volume of Lavanasi which is surrounding Jambudvipa of two lakh Eugen Witz he calculates by this sixteen digit number now even for me it will be very difficult to immediately spell out but he very easily says it is ten Koda Kodi ninety three Kodi say as a Hassa thirty nine nine nine hundred Kodi say a fifteen fifty say as a Hassa this is place value each place he's assigning some value that is like a hundred trillion then trillion then billion and hundred thousand etc etc this is nothing even he tackles a very large number that is the number of stars in the human world now this also I cannot immediately if you ask me to read out in the modern term I'm not able to but he says it is forty eight say a Sahasa twenty two Sahasa two sayam so much Koda Kodi it is a 23 digit number it's a very big number actually we cannot imagine also the even as per the modern astronomy the number of stars in the universe is ten raised to 23 this comes very near to that actually sometimes he drops the space values that is don't say say a Sahasa etc etc simply reads like as we read even today one thousand five hundred eighty one here he says Panneras Ekasei which means fifteen eighty one etc now I'm coming to his contribution for symbol zero very clear evidence is here now you will say where he has mentioned those symbols because his work was poetic in nature he could not use symbols but he indicates very clearly that he was aware of the symbol of zero one to nine etc etc how see the square of called of north Bharata Bharat means our India not Bharata called means east to west span east to west span is the square of that is seventy five six hundred thousand thousand Kala Kala means one nineteenth of Eugen or nineteen Kala is equal to one Eugen this time I will explain you little later why he uses Kala that there is also very very good reason and he expresses it that seventy five six and eight zeros so it very clear evidence isn't it then square of Mount Himalaya Himalaya span he and his square he expresses as one zero five and followed by ten zeros and writes like that in his poetry in his verse form that it is hundred five with ten zeros ten zeros square of called of Nishada a mountain Nishada is which is identified by so many people with Alta E etc etc I'm not going into those details of geographical details but it is thirty two two zero four and then followed by eight zeros and he writes like that that he reads it as thirty two two zeros four and eight zeros this I'm only illustrating that he was very much aware of the symbols of this one nine and zero etc this which fractions also in a very skillful manner nowadays we write two upon five by a separating a line between two and five in those days probably I'm say why I'm saying probably it was it was said two out of five parts and written as two and below five that separating line was not there at that time why I'm saying this probably because the Bakshali manuscript which was found out and it belonged to the period just nearby to Jain Bhadrakini it it was mentioned like this so probably I say it was written like this but he was very adept with the calculation with probably LCM and GCF etc because Arya Bhatta employs all these LCM and GCF etc of two numbers three numbers four numbers etc and he was aware of this technique so because Arya Bhatta was contemporary to Jain Bhadrakini it means definitely it was evolved even earlier friends Arya Bhatta was a great mathematician and astronomer but there are very strong evidence that he was influenced very much by Jain school of mathematics he himself accepts that he was he studied at Kusumpura this Patliputra and Patliputra was the center of Jain studies at those times he uses the terms which are never found in any other Vedic literature like Utsarpini, Ausarpini, Sushama, Dushama etc etc so there is a clear evidence that he was greatly influenced by he gave a very correct value of Pi 3.1416 but he always used the Jain value of Pi which is root of 10 which is slightly incorrect but he used always that Brahma Gupta also uses the value of root 10 and that way it is rightly called Jain value of Pi Pi means the ratio of the circumference of the circle to the diameter which is a constant now see some of the fractions in the last line I have shown even such big fractions he was able to simplify by dividing the numerator and denominator by 12 the first fraction isn't it and it was not divisible any further that was also here ensure these are prime to each other these numbers similarly the second fraction also you can see which he divides by 35 both numerator and denominator and simplifies it so this also he exhibits very high skill in tackling this chord of South Bharata he shows his square root now I am coming to square root such square roots he calculates even today we shiver while calculating the square roots of such numbers etc 34 30 80 7500 he calculates there in the fraction also he calculate 18 5 2 2 4 and the fraction is numerator this much and denominator this much and he writes it colors and now just after sometime I will explain you why he converts eugens into colors by multiplying eugens by 19 because it helps him in calculating but such big numbers it tackles area of human he gives us so much and there you will see instead of dividing by 19 he divorced by 361 that is 19 square because you know areas length multiplied by width so eugens and eugens square so if it is color square we'll have to divide it by 361 to convert it into so he calculates the area of mount human as this much another contribution which is recognized by all the scholars that is rule of three it is of Indian origin Indian origin that sort of proportions ratios etc on that basis for example I have given one example that the Mount Meru of Jane mythology is a 90,000 eugens standing above the ground with the on the base it is 10,000 eugens and at the top it is 1,000 eugens and then there are various gardens on that at 500 feet 500 eugens height at 33,500 eugens height etc etc etc and the circumference areas of those gardens are calculated by all Acharya not only in Madhugani it was calculated even earlier in Agamas etc and there they have used the rule of proportion for example suppose at the height is 1,000 so what will be the height sorry diameter is 1,000 at the top so what will be at the bottom so by rule of three applying that in 99,000 height the how much it has decreased so at any height they could calculate the diameter of the Mount Meru and dimensions of its circumference its area etc etc because majority of the mountains in Jaina mythology are either trapezoidal or they are like frustum of cone a cone is slashed no see there is one big charge against all Indian mathematicians western scholars generally blame that you are all mathematics was empirical like Greeks you have not gone into the depth you have not talked about the proof you have not talked about the generalization but first time genobahs organic shaman shaman or to some extent even earlier scriptures like anubhav darsutra etc but I will say by stating the formulas clearly it is genobahs organic who first time came to the generalization of various things for example pi is equal to root 10 c is equal to that of a pi into d or for calculation purpose he says root of 10 d square isn't it called ab called what is the formula he has given specifically stated the formula in his work similarly this arc arc ab which is called dhanusha that also he has given the formula c square plus 6 h square gives very accurate measurements sometimes the error is just 2 percent or 3 percent or maximum 5 percent but it gives in my article which may be circulated to you afterwards I have given the various explanations how this formula could have been errored etc but I am not dealing with it just now but I only want to point out that the attempt of generalization started with genobahs organic area is equal to c upon 4 into d and from purpose of calculation he converts into 10 d fourth upon 16 and square root of that that I am giving you some example jambudweep the we say that the diameter is 1 lakh so its circumference will be 1 followed by 11 zeros square root of that and he calculates his l 3 1 6 2 2 7 and his fraction now it is for if the diameter is 1 this pi will work out to 3.16 2 2 7 while the correct value is 3.1416 but he gives a very near a value and he calculates that this was calculation was given in the earlier scriptures also Bhagwati all this Arnabdwar jiva jiva everywhere but they have not explained how it was calculated etc but here jambudweep explains that how it is calculated and he the most important thing he gives that fraction also and that fraction he converts into then kosh yoshan kosh dhanu etc etc and he gives it that it is 3 1 6 2 2 7 yoshan 3 kosh 128 dhanushya 13 and a half a little above 13 and half you want to that accuracy he goes angula angula is a very small measure width of this finger isn't it. Similarly for card of Bharata he makes the calculation and ultimately he arrives at root of such a big number 18 5 2 2 4 color he which he converts into yoshan by dividing by 19 and says it is 9 7 48 yoshan and 12 color means 12 by 19 12 by 19 that fraction. Similarly arc of south Bharata this south Bharata I have shown that a to b along the sarcophagus of the circle he now I am explaining you why he uses color he multiples by 19 squares it so it becomes a big whole number whole number because he if he wants to take the diameter then he will come into trouble so he converts into color and calculates it this thing and see he arrives at the value of this arc of the circle as 9764 yoshan 1 color his error is only of 0.02 percent among 100 yoshan the error is only 0.02 percent and you will see that area between two parallel cores this is his unique formula nowhere in any mathematics gen mathematics or Greek mathematics this formula is shown and there he says that a b and cdr do two parallel cores why why he is interested in this because all our mountains are of this shape in the Bharata in the jambudvipa all the mountains is to waste of our this this shape so he calculates the area as root of c1 square plus c2 square upon 2 into h2 minus h1 see friends earlier he has given a incorrect formula c1 plus c2 upon 2 into height height between the two cores isn't it but he himself rejects it he says no no this is not approximate formula it doesn't give correct result and he uses this in statistics there is a very important principle that is called root mean square isn't it and he shows that he probably knew that root mean square is more than the arithmetical mean and that is one clear evidence that he himself rejects no no I will take this and he calculates the value I have given this some examples that if we calculate as per modern trigonometry and if we calculate as per this formula you will see the error is just very minor again 0.02 percent 0.0 come to this now this why I'm giving you this example he tries to calculate the volume of lamana samudra which is surrounding our jambudvipa and it is said that it is 1000 deep it is trapezoidal and then in the high tide it rises up to 16000 I don't want to deal with this because then the immediately our scriptures also say the heretic asked that sir if it is so tall if the whole water will come and it will wash out jambudvipa on one side and the dhatki khanda on the other side he said no no the millions of gods billions of gods are there they will come and they will stop the such havoc whatever it may be I'm not going into that but he calculates the surface area of the lamana samudra by techniques which are very near to modern calculus I'm not saying modern calculus the holy and entirely as per modern calculus but technique very similar to modern calculus calculus we say is originated with libniz and newton but even earlier in western countries in 15th century 16th century there were robberwald torricelli etc so they also calculate about that is firma they actually are the originator of calculus it is almost recognized because they adopted techniques and that helped libniz and newton to carry forward this their work isn't so surface area is also calculates so much see there is one boat above the another boat they are fishing each other like that and that he takes the mean diameter of that multiplies it by this width average width and then he arrives at this and then to that he multiplies by 17000 and I was at the volume of 11 samudra which is such a big thing which I have given to you in earlier so now see square root also I was also flabbergasted we are also calculating square root by this method only but what we do above we write the divisors he doesn't write divisors then how could he arrive at answer the last divisor 38 93 52 he house it divides by two and that becomes the answer and then it is also not perfectly accurate so there is a fraction and the fraction is the last divide divided by last dividend dividend to that extent he gives the correct answer of the volume of 117 and sorry mount white ad here now we have talked about in the pre-lunch session about the arithmetic progression where ganit sar sanghara was cited but yet he was also new ap gp that in geometric progression arithmetic progression he gives 19 formulas yet he was a very complicated formulas of volumes weights areas etc etc but even jinnabhadra can he has given this has tested the formula in his work that is about ap last term is equal to first term a plus number of terms into difference minus difference there is same formula as what we are doing today or the sum of that is n by two the number of term divided by two into first term plus last term etc and why why does this that as in the morning the example was given of nark that means lower world he gives the example of upper world heavens there are 12 heavens or 16 heavens whatever it may be and in the first heaven there are 13 layers in which the vimanas the abode or airships of the gods heavenly gods are there and how many are there he wanted to calculate in each layer similarly 13 year in the first layer there are 201 in the second layer there are 205 in the third layer there are 209 increasing by 4 and he calculates in the 13th heaven there are 249 so 201 plus 202 he doesn't calculate all but he just does this and by using this formula he says 13th layer there are 249 airships vimanas and how much total this is in one direction not in all direction what is the total 13 by 2 201 plus 249 is equal to 29 25 this is the total number of vimanas in all the direction so he was perfectly aware with the arithmetic series one very important thing a very important and powerful technique of modern mathematics number theory is what is called modular arithmetic which goes as per if I ask you that one followed by some 20 numbers huh what is how it is divisible can we find out what are its divisors etc so for that purpose this modular arithmetic is used isn't it the similarly he says that out of three abodes first is triangular second is square shape and third is circular now where he uses triangular means he says first is it is written as one more three means if it is divided by three it leaves a remainder of one two more three means if he's divided by three it leaves a remainder of two and if it divided by three it leaves a remainder of three means again it is divisible by three so remainder is zero only so this is called I'm not going into detail but this modern technique which is attributed to Gauss of 17th century who he has employed in calculation of this how he writes like this I'm not going into detail one example only 62 is three into 20 plus two into one that means if 62 is divided by three it will leave a remainder of two two 61 it will remainder of one last term last term isn't it so it doesn't matter I'm not going into much detail but see please finish now what he's done he sums all this not just common sum but he says there are 20 attempts three attempts of multiply by 20 that is 60 huh 19 into 3 18 into 3 17 into 3 16 you can make it in the exercise at your own leisure time plus five into two five numbers are such which leave a remainder of two four are such which is a remainder of one total 728 in one direction in four direction 2912 and there are 13 layer 13 central airships so if they are added again the same number which was seen earlier 2925 but where he uses this model modular arithmetic powerfully is this how many are triangular how many are circular square and how many are circular so for that purpose he uses this technique that earlier we have seen 728 is equal to 3 into 238 plus 5 into 2 into 4 into 1 that means 238 are circular they are perfectly divisible by 3 then five are such which are not circular they are leaving a remainder of two so they are square shape so there are five more that means 238 plus 5 243 and four more are triangular that is 247 triangular so he's finished the exact number of how many are circular how many are triangular how many are square by using this technique of yes please finish please finish time is over okay just five minutes not five minutes within two minutes okay okay i'm finishing i'm not dealing with this figure eight number some the modern technique in number theory that somewhere and some numbers can be represented by figures and jinn bhajagani displays a lot of skill in this also but i'm not going to deal with this because it is not subject matter of my talk today not about the computational skills similarly friends one more thing which is a very good area for research in gynaology in gyna mathematics is infinitesimal now today we say we have gone in the earlier lecture from zero to asankhar ananta etc ananta also ananta ananta but what about the infinitesimal smaller so jinn bhajagani goes up to 26 stages we are reminded of modern physics where molecule then atom then electron proton neutron etc then they are formed of baryons leptons etc etc and now we have gone up to god particle that is boson and even lesser than that is the quantum length plank length that is 10 there is 2 minus 44 isn't it but jinn bhajagani's calculations are unimaginable i i'm not dealing with that he goes 26 stages down each stage is multiplied by divided by ananta and this way he goes avdhariksharir molecule atoms ultimate particles then vikariksharir then ahariksharir then so many other karman karman particles are very fine and this way he goes up to 26 i'm not dealing with this but this is the completion we can say he's the pioneer in indian mathematics actually i had discussion with dr nupamji where i said sarvanandhi who wrote the he also displays all these skills and he is of same era about 450 ed which was composed in 458 ed but anupamji pointed out to me that the original is not available it is available with a translation in Sanskrit of that prakrit work by simha suri so unless we get the original you cannot claim this so i accepted this so we can say then as per the scriptures available jinn bhajagani can be treated as a pioneer of modern computational mathematics he does it in the most modern manner and exhibits the all the skills very high skills of competition that's why i have titled it as thank you it is hard