 Good afternoon. First I would like to thank Professor Peter Fluegel for having given me the opportunity to speak today. So I will speak about the treatment of series in the Ganita Sar Sangraha of Mahaviracharya and its connections to Jain cosmology. The Ganita Sar Sangraha is a very significant treatise on mathematics composed by Mahaviracharya, a Digambur Jain Acharya, living in South India in the 9th century. We know this date with fair accuracy as the first chapter contains verses praising the king Amogha Varsha. Le texte était découvert par Prof. Rangacharya, qui a changé sur des manuscripts dans le gouvernement oriental manuscript d'Abrélie, à Madras, où il était curateur. Il a édité, translété et a publié le texte en 1912 et cette édition est encore autoritative aujourd'hui. Le texte a un size imprécif, car nous avons plus de 1000 stanzas dans les 9 chapters. Il y a une introduction et 8 sujets de treatment, ou des pratiques, via Vahar. Le premier via Vahar a les opérations par E. Kallmann. Il donne des procédures ou des algorithmes et des problèmes simples. Normalement, dans la tradition mathématique de la Sanskrit, il y a huit opérations, qui sont ces deux. Mais dans le Ghanita Sarasangraha, les premières deux, addition et subtraction, sont mises et sont répliquées par l'addition et la subtraction des séries. Qu'est-ce qu'il veut dire par une série ? Ici, dans le Ghanita Sarasangraha, il y a deux types de séries, qui sont des séquences de numéros suivant une certaine loi. Ici, il y a une progression arithmétique. Il faut toujours ajouter deux, par exemple, ici. Et une progression géométrique, où vous multipliez constamment par le même numéro pour arriver à la prochaine étape. Le traitement de la séries arithmétique n'est pas un nouveau sujet. Beaucoup d'Aryabhata et Brahmagupta ont donné quelques règles dans leur texte. Mais dans le GSS, Mahavira Charya a arrêté et amplifié les opérations sur la série dans une manière nouvelle. Comme vous pouvez le voir sur cette table, Mahavira Charya est la seule autorité à commencer avec l'opération de multiplication, parce que les additions et subtractions sont à la fin. Il n'y a pas d'explanation apparente pour cette option. En fait, dans la synthèse du master des mathématiques en Indie, Kim Plovker évoque quelques questions qui sont encore en réponse. Et elle dit, par exemple, pourquoi se considère-t-elle possible dans le GANITA, pour dépasser les additions et subtractions des numéros comme des opérations canoniques ? Pour cette question, je pourrais ajouter une autre question. Pourquoi Mahavira Charya n'a pas seulement commencé avec la multiplication, mais aussi pourquoi il a donné tant d'importance à ces deux opérations sur la série. Comme vous pouvez le voir, la première étape est occupée par les premières six opérations et les dernières deux sont occupées à la fin de la première étape. Donc, si nous devons travailler sur la série, nous avons besoin de quelques termes techniques. Et je vais essayer d'être le plus clair et simple, parce qu'il y aura des mathématiques dans le monde. Donc, la terminologie est comme ça. Pour la première étape de la série, nous prenons quelque chose qui signifie le début, l'origine, la source, etc. Pour la différence commune, nous avons Charya, Prachaya, Uttara, Vridhi, quelque chose qui signifie « augmentation » ou « accession » et le nombre de termes est généralement Pada ou Gaccha. Et nous avons le total summe, Salvador. Donc, avec ces quatre quantités, généralement, nous connaissons trois de ces quantités et il y a quelques formules qui nous permettent de trouver le fort. Un peu plus technique. Non. Nous allons voir la calculation du total summe. Donc, si nous le faisons aujourd'hui, nous donnerons une formule. Mais à la fois de la GSS, il n'y avait pas de notation algébrique. Donc, généralement, c'est une présentation algoretmique. Il s'agit comme ça. On prend une, on enlève une, on divise par deux, on multiplie par l'air, on ajoute le premier terme et on multiplie par le nombre de termes. Vous allez avoir le summe. Si vous faites ces calculations dans un ordre propre, ou une autre option, c'est de faire comme ça, ça donne exactement le même résultat. Des termes plus techniques, qui sont très spécifiques, et à mon avis, ce n'est pas utilisé dans d'autres textes mathématiques. Mais je suis peut-être méstricé, parce que je n'ai pas fait une surveille complète pour ça. Donc, ces termes sont adidana et utradeana. Adidana n'est pas la valeur du premier terme, c'est la valeur de l'accumulation du premier terme. Donc, c'est u1 x n. Et le utradeana est la valeur de l'accumulation de la différence commune. Mahaviracharya est aussi la seule autorité à proposer que si l'un change l'ordre des termes, le total sera la même. Les dernières termes techniques, qui sont très communes et utilisés par tous les autorités, sont les dernières termes et les principales termes. Donc, maintenant, let's see how the sample problems are presented in the manuscript. So, in this manuscript, there is an exercise where you have to calculate ten sums, because you have ten series. As you can see, the first line, there is always go for gotcha, then u for utrade and r for adi. And you have the numbers, which are there. It seems that this presentation is always the same in all manuscripts, whatever is the region or the paper manuscript or palm leaf. Here is the palm leaf manuscript from Madras, where you have the same exercise. And in Canada script, you have again the same formulation with the three values here. Rangacharya, sorry, in his edition, he didn't know that the presentation is always the same. And in this exercise, this is the preprint copy, which is available in Madras. You can see that he crossed the u from utara to put pra, because in the text itself, it's pracharya and not utara, which is used. So let's do quickly an example. So we have the case here of a Shravak who offers two gems in the beginning and increase by three gems. And we have to calculate how many we get after five like that. So it goes like that, you calculate the beginning multiplied by the number of terms is ten and this is the adidana. Then the number of terms minus one divided by two is two. You multiply by the common increase, you get six. And then you multiply by the number of terms, you get 30. This is exactly the utara dana. And if you add both of them, you get 40. This is the total sum. And now we can also, we get simple problems where we have to find the number of terms. So the formula we would use today would be this one, but again it would be an algorithmic presentation. How is it shown in the manuscript? You can see that when one of these three values is unknown, there is a zero which is put there because we don't know the value of the gacha as this is the one we have to find. So, again if we know the total sum, like 90 here, you follow the algorithm and then you multiply by eight, again by eight, etc. And you get the answer finally. Ok, now what about geometric progressions? In fact, there is hardly any sign showing that the type of series studied has changed. Here in the spam leaf manuscript you have a smaller pattern, decorative pattern which indicates that the topic has changed. In fact, it is the terminology which indicates the change because the first term is the same but now it's a common ratio and it's Guna or Guna Torah. Again, you have Guna Dana and Guna Sanculate for the sum of the series. How do we do to calculate the sum in the case of a geometric series? We have two formulas which are these two on top and in this exercise you have to know that 3 is the beginning, 5 is the common ratio and 15 is the number of term. So, it means that you have to calculate 5 into the power of 15 which is an absolutely huge number. And to calculate this you can obviously multiply 5 by 5 by 5 by 5 the number of time necessary but there is a sutra which gives a procedure very similar to the one used in prosody in the Pingala text to calculate 2 to the power of n. And in the same spam leaf manuscript in the commentary you have that algorithm which is explained. So, the number of term is 15 if it is even you divide by 2 if it is uneven you remove 1 and then you put 0 you put 0 otherwise you put 1 and after you go from down to upward and you multiply by the common ratio if you have 1 you multiply by 5 if you have 0 you square and like that in only 6 operations you have the answer instead of 14 multiplications if you do the other one. Now, this process is very known and in a very famous text like the patiganita of Shridhara or the Lila Vati of Baskaracharya you have the same process but instead of writing 1 or 0 1 is supposed to write multiply or square. So, now turn to the connections between this treatment of series and the gene cosmology. For that I will look at the teloyopanati which is a work written in prakrit a very big work and which has plenty of numerical data and results but also which has general procedures stated. So, let's look at the most common illustrations of the locus that we have and we will first deal with the lower region the hills. So, the lower region is a 7 radios high so the radios is an an uncountable unit of land and there are 7 earths which have 1 radios each height and usually the illustration is this one on this kind but this illustration is very deceptive because it looks like the intervals between the different earths are very small or in fact it should be the opposite but it's impossible to represent. So, this illustration is deceptive as I have indicated the thickness of every earth so 1,80,000 eogenas is absolutely negligible in front of 1 radio So, what is important is that every earth has layers and you can see that this is arithmetical regression because you minus 2 every time starting from 13. Now, in these layers the hellish souls live in huge holes and the first earth has 30 lakhs of them and the total of holes is 84 lakhs in the hells so again we get the 20 lakhs So, each layer has one central hole in Draca and some ordered or aligned role there are 49 in the main directions and 48 in the sub directions for the first layer and then it's minus 1 every time so the second layer it's 48 and 47 then you have some scattered holes also So, the aim obviously is to calculate how many holes you will have in any layer so as it is an arithmetical progression we can use the formulas so here if you start from the top the first layer if you add the central and aligned holes you have 389 holes and then you minus 8 every time you change the layer so you can calculate and the last one is 5 now, in the teloyopanaty again it is explained that you can also start from the bottom and say that the first term is 5 instead of being 389 if you do like that the increase will be 8 instead of the decrease being minus 8 and that remains remarque by Maviracharya that if you change the order of terms you you get to the same formula total so now there are also arithmetical series for each earth and then you have to calculate the beginnings so the beginnings are these ones the number of holes in these layers and here there is a formula which is stated in a general form you can calculate the number of holes in any desired earth of level D so D is from 1 to 7 and it has alpha layers again it's a formula for calculating the sum of arithmetical progressions but much more interesting and very specific to the Thilloyapanati is the Sutra 265 the Gata 265 because it gives a general formula even if you don't know the beginning you can still calculate the sum of the holes in the different layers of one earth not only the number of holes follow arithmetical progressions but also the width of the Indracas and the depth of the holes also are in arithmetical progression for the width you can notice that the first one which is on top is 45 lakhs which is exactly the same as the size of the Udaidwip the two and a half continents and the last one which is the lowest one is one lakh which is exactly the size of the Jambudwip so it's not mere coincidences now let's examine the cosmological structure of the middle world it is divided into concentric rings with the Sumeru mountain at the center and the last diameter is one Raju the number of rings is an Asankyat number and the sizes of the rings double each time so like that the first one is the Jambudwip it's one lakh then the Lavana C is two lakhs the Taki land is four lakhs and eight lakhs then 16 lakhs Leogina since the sizes of the rings go into geometric progression then we can use the formulas for the geometric progression as the common Raju is two if we want the width of any ring the formula for this is w for the width is equal to 2 to the power of n d is the width of the Jambudwip so the problem which comes because the number of rings is uncountable so if we want to know the width of the last C how do we do because we can't calculate 2 to the power of an incountable number because we don't know that number so the how is it solved in the there is a lot of gathas a lot of many sutras which have been stated for example you can show it requires a little bit of calculation but it's not very difficult that the inner diameter of a ring follows this formula this is the middle diameter and this is the outer diameter so knowing that the outer diameter of the middle world is one Raju if you know this value then you can calculate the width of the last C so you get 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1