Okay, thank you very much, Marius, for the introduction. Thanks to Enrique for inviting

me to the colloquium. I'm happy to be with you. I'm happy you're here. It's always a

little bit strange to speak in front of a screen, but I just feel I can't give my

best. I prefer to run around in the room and write on a blackboard formulas and so

on. This is not possible right now, so I'll try to do my best and I hope that

one day we can do this in real with another topic maybe. So I have chosen one

of my favorite mathematicians of 19th century China and I have chosen to talk

about a formula or combinatorial identity that at least some people know

or they can Google it because it is one result from the history of

Chinese mathematics that has somehow entered modern collections of formulas

or manuals about combinatorics. But of course I am interested in the

mathematics behind it but not in the formula itself and you will understand

very quickly why I say that because if we move on to the next slide, please

scream when you can't see the slide. Can you also see my pointer? Just a

short question. Yes we can. Okay so when I point out something you can see that.

So this formula, this Li-Shun-Lan identity, when you check it on the

internet, I think there's even a Wikipedia page about this formula, you

would find something like this, right? It's a combinatorial identity that states

an equality that relates basically to the squares of the binomial coefficients

where you form a certain sum and you obtain again another squared binomial

coefficient. So this is the kind of formula we would see and we could of

course just stop there and say okay great this Chinese mathematician in 19th

century he obtained this formula, how wonderful. But if we of course look at

the formula like this, it doesn't reflect at all the way this Chinese

mathematician has been working in 19th century and that's what I'm of course

most interested in, to understand how Li-Shun-Lan has obtained this formula, how

eventually has justified its correctness and so on. So in what context, in what

historical context did he develop this? So as a historian we're of course always

more interested in such kind of questions. That doesn't mean that the

mathematics doesn't play a role. Li-Shun-Lan identity or sometimes also

find it in the manuals as Li-Ren-Shu identity. Ren-Shu was the style name of

Li-Shun-Lan, it's just like a, it's just another name for the same person and we

find it in a book which is called, when we translate the title literally,

Comparable Categories of Discrete Accumulations that he published in 1867.

And of course as I said in 1867 we wouldn't find formulas of that kind in a

Chinese mathematics book. So this is maybe a little bit surprising for a

Western mathematician to hear that at that time in China people did not write

mathematics with formulas. What we find rather in the book is pages like this. So

that's what the formula actually looks like in the Chinese original version.

That means mathematics in mid 19th century were still written entirely

rhetorically, at least in a context where mathematicians worked in a

traditional framework. There was also in parallel another movement but I will say

a word afterwards. So all the kind of mathematics that were based more on

traditional Chinese mathematics had no formalism. They were all written with

words, all algorithms. So it's not even that we could say it's a formula, it's

actually an algorithm. It's a list of operations that you have to perform in

order to calculate something. So this is what Li Shanlan's book looks like.

It's all algorithms and some diagrams and I will talk about this later. Diagrams

of two kinds, one kind is arithmetic triangles and the other kind are

figure-red numbers. So we'll see that later. So these are basically the three