 I have to start with an apology. I don't know quite as much about gynomethematics as some of the other speakers at this conference. As a matter of fact, I'm not even going to speak about any particular gynomethematical texts at all. I'm going to talk about a bramanical mathematical text but as I will argue later, this particular text engages with another mathematician who is a gyn. And I will try to analyze the approach of the author I'm going to present to you. Now, this author is called Bhaskara. He is the first to write a commentary on the famous text by Arya Bhatta that has been mentioned earlier today. Arya Bhatta wrote his work probably in 510. There's some disagreement about that. Others think it was 499, but it was around the year 500. And Bhaskara, his commentator, he finished his work, his Bhashya, his commentary, the Arya Bhattiyya Bhashya in 629 of the Common Era. So that is some 120, perhaps 130 years later. Now, both Arya Bhatta's text and Bhaskara's commentary on it deal in part with geometry. So unlike some of the other speakers, I will not talk about the knowledge of numbers and the manipulation of numbers. I will talk about the parts in these texts that deal with geometry. And both these texts, they present a number of geometrical theorems. Now, Bhaskara's Bhashya, he also provides diagrams and exercises. In one sense, it's the earliest really surviving, really mathematical treatise, and it addresses a public of people who have to learn mathematics. There are exercises, there are diagrams, it's quite amusing to read. However, neither of these two texts of Arya Bhatta and of Bhaskara, they ever provide proofs for their theorems. This has given rise to quite a discussion among modern scholars. Why are there geometrical theorems which without any proofs being given, you cannot imagine in Euclid's geometry any theorem to be given without a proof. So people have often supposed that the proofs were there but they were simply not provided because they were orally transmitted. This assumption is demonstrably incorrect. At least for this case, I don't talk about other texts that I don't know half as well as this particular one. Both the texts by Arya Bhatta and the one by Bhaskara, they contain some theorems that are false, they are not correct. They were false in the form they were given by Arya Bhatta around the year 500, as we saw, and they were still false in Bhaskara's text in 629. They had apparently been handed down in this erroneous form for some 130 years and no one during these 130 years had ever noticed that they were mistaken. This would have been impossible if generations of students had bent over the proofs of these theorems. Discovering the mistakes would have been a matter of routine and Bhaskara's Baasje would not have repeated incorrect theorems. If you are interested, the two theorems, one of them is about the volume of a pyramid. Bhaskara says it is the surface of the bottom times half the height. And of course you know from school geometry it is a third of that. Bhaskara didn't know, Arya Bhatta didn't know and no one in between found out the mistake. And there is another one, I won't enter into details. It is perhaps not surprising that Bhaskara was not critical with regard to the text he had received from Arya Bhatta. In his comments on the first chapter he states, for example, that all knowledge derives from Brahma, Arya Bhatta pleased Brahma on account of his great aesthetic practices and could then compose for the wellbeing of the world his texts. That means he had not reasoned it out, he had received it from Brahma and Bhaskara, 130 years later, would not disagree with Arya Bhatta because the origin of this knowledge was from higher up, so to say. Elsewhere Bhaskara attributes superhuman qualities to Arya Bhatta, he calls him attindriyarte darshin, seeing things that are beyond the reach of the senses. Somewhere else, he quotes verse from earlier non-identified astronomical text and calls it smrti, that means sacred tradition. All this confirms that Bhaskara handed down a tradition that he looked upon as authoritative and more or less sacred and not a body of doctrines that could be improved upon by critical reflection. Clearly, independent critical reasoning with the possible result that a respected authority is shown to be wrong could have no place in his work. Now, we can contrast this with what we know about the history of Indian philosophy. Indian philosophers were not indifferent to proofs. They used proofs in their treatises and theorized about it. Clear ideas about what constitutes valid proof were well in place at a time when Bhaskara wrote his Arya Bhatiya Bhashya. Don't forget that this is in the seventh century. Dignaga, Dharmakirti, all the Nayayikas who had thought about the nature of proofs they had lived before him. The question is therefore, why was Bhaskara not interested? Why does he not give proofs? To find an answer, we must stay a little longer with the philosophers. Their interest in proofs and in logic in general has to be understood in the specific circumstances in which Indian philosophy developed. The history of Indian philosophy can at least to some extent be described as the history of different schools of thought that confront each other. The opposition between Buddhists and Brahmins is particularly important here, but it is not the only one. Philosophers try to defend their own positions and show that the positions of their opponents were incoherent or worse. In these confrontations, which may sometimes have taken the shape of debates, sometimes public debates, logic, and proofs were vital. Indian mathematicians at the time of Bhaskara did not normally find themselves in a situation of confrontation. They did not normally have to prove that others were wrong and only they themselves right. Debates about mathematics may have been rare or even nonexistent. Recall in this context that Buddhists did not practice mathematics. It's one field of knowledge where Buddhists had withdrawn from mathematics, astronomy, and astrology. And they had left this task essentially to Brahmins. I've explained that in a book that has come out a few years ago. Mathematicians as a result did not feel threatened and they did not have to prove that their theorems were correct. The result of this situation we have seen. Some of Bhaskara's theorems were in actual fact incorrect, but no one over a period of one and a half centuries and perhaps longer ever noticed. There is one important exception to what I have just said. In one passage, Bhaskara discusses an opinion with which he disagrees. A central point of disagreement is the value of pi and Mr. Shah earlier has pointed out that most giants and also, in fact, also many others believed that the value of pi is the square root of 10. Bhaskara did not agree. So Bhaskara's opponent, whom he quotes, holds that it is the square root of 10. Pi is the square root of 10. Bhaskara disagrees and he criticizes that but not without quoting his opponent and he does that in the line in Prakrit. Now, since Brahmins would never use Prakrit to my knowledge for in their writings, the opponent was not a Brahmin. He was not a Buddhist either and I conclude that the opponent of Bhaskara in this case was a giant and the fact that it was about the value of pi being the square root of 10, that makes it all the more plausible because as Mr. Shah pointed out earlier, the square root of 10 is the giant value of pi, not only giant, to be sure. There are, in fact, many examples of texts that use this value and I repeat, not only giant text but also others but I will not enter into these details. Now, Bhaskara doesn't accept it because he thinks that this value has not been demonstrated. He himself has a wonderful value for pi. I don't remember by heart but it is exactly but it is very close to what in modern mathematics we consider the value of pi. The interesting part follows next. So Bhaskara opposes the view that the value of pi is the square root of 10. And then in order to criticize this, he quotes a theorem of his opponent and this theorem concerns the length of an arch. Now I'll try to show the picture. Is it a PC that I should press, press? Ah, there it is. There it is. As you know, it concerns this part of the circle and it is like a bowl, this is the cord and this part is the arrow, the sagitta. And this opponent of Bhaskara has a formula to calculate the length of the arch. Once you know the value of the sagitta, of the arrow and of the cord. Now Bhaskara rightly points out that in a particular case where the arrow has a certain value, the cord has a certain value, if you apply that formula, you get the length of the arch which is shorter than the cord, which is totally bizarre, the arch than the cord. And so the formula which leads to a result in which the arch is shorter than the cord, of course, is totally incorrect. But interestingly, Bhaskara formulates that in a way as if that is the result of the incorrect value of pi. In fact, it isn't. You can take another value of i, pi, take the same formula and you still get a wrong result. You can even take the value of pi that Bhaskara himself presents, put it in the formula and still be shorter than all that in a criticism of an opponent and why is this opponent criticized because he thinks pi is the square root of 10. So here for once, we have a situation where Bhaskara does disagree with an opponent. But and he attacks the words of this opponent, but he also shows that he has no idea of logic. He mixes up the thing. He criticizes, he says, the wrong result is a result of pi being a square root of 10, which is wrong. But in fact, it is the result of a formula which would give wrong results wherever and whenever you would apply it. So I will not give you all the details and the exact formulation he chooses. You have to take that for the time being on trust. And of course, it's easy to verify what I say because I want to draw some conclusions from what I've said so far. What does it teach us? I think it shows that the same Bhaskara who could not discover mistakes made in his own tradition has no difficulty finding mistakes in a tradition different from his own. And this tradition different from his own in this case appears to be a giant tradition of geometry. However, it looks as if Bhaskara's criticism of the giant tradition contains a logical error. The sutra, that means the theorem he criticizes is no doubt incorrect. But this fact has nothing to do with the value that the giant attributed to Pi. Whatever value one attributes to Pi, the sutra will always lead to an absurdity. It is unlikely that the contemporary philosopher of his time, trained in logic, could have made such a mistake, really an elementary mistake. Now having made a comparison with the history of Indian philosophy, it is almost a pity that there were apparently so few occasions for confrontation between Bhaskara's and other traditions of geometry in classical India, at least at that period. If there had been, each of these traditions might have developed a deeper critical sense first with regard to others, ultimately also with regard to themselves. The outcome might have been positive for all. For one thing, more self-criticism might have induced Bhaskara and no doubt others to critically evaluate the theorems that had been handed down to them and ask themselves why they should accept them. Incorrect theorems might in this way have been weeded out. In reality, no such thing happened, not at least at the time of Bhaskara. We do not know whether mathematicians in the giant tradition cared about Bhaskara's criticism. We have seen that they continued to assign the value square root of 10 to pi, they did it for a thousand years to come after Bhaskara until many centuries after him. The problem that Bhaskara found with this value, as we have seen, was linked to a rule about the length of the arc of a circle. Interestingly, many giant mathematicians in works that have survived, they use an altogether different rule to calculate the length. I cannot project it, but in fact, Mr. Shah predicted one of these rules right here an hour ago. And that is a rule in which, as Mr. Shah pointed out, the outcome is very close to the correct value of the arc. But that is not the formula that Bhaskara is criticizing. He criticizes the formula that is quite different. And this other form that we find in so many giant texts is not open to Bhaskara's criticism. For here, if you apply it, the arc turns out to be certainly longer than the chord. And well, this better formula had already been known by Umaswati. And of course, it was known later to mathematicians like Mahavira, Arya Bhattattu, and there was a second one of the same name, Shri Patti, and a number of others. I don't need to go into the details. I conclude, it seems clear that geometry at a time of Bhaskara, and that is Bhaskara one, the one I've been talking about, was practiced in different schools. To some extent, these schools were independent of each other and concentrated on their own traditional teachings. If these traditional teachings were incorrect, they were preserved without anyone being aware of it. Criticism only came into play where representatives of different schools confronted each other. Only in such cases was there place for criticism. Unfortunately, such confrontations were, at a time of Bhaskara, too infrequent to allow a more general atmosphere of criticism to become part of the traditions. Thank you for your attention.