 The Lila Vati, beautiful of Bhaskara, is one of the most comprehensive sources of medieval Indian mathematics. According to the internet of the 15th century, that would be a bunch of scribes sitting around recording things, Bhaskara calculated the exact time his daughter should be married to insurer happiness, and built a water-clock to make sure the event would happen on time. Unfortunately a pearl disrupted the flow of water, and the wedding took place at the wrong time. Lila Vati's husband died a year after the wedding, and to console her, Bhaskara taught her mathematics and named the book after her. Lila Vati, the book that is, is a comprehensive treatise on many areas of mathematics. Like the Ryan Papyrus, it gives us insight into how the ancient Indians performed computations, so in the Lila Vati, Bhaskara presented many different ways to multiply. And he illustrated these with a product of 135 and 12. Now for those of us who can't read Sanskrit, like me, there are English translations available, so we can see how Bhaskara wrote. His first method is vaguely described, and we might infer from that that the method is familiar enough that he doesn't need to give a full description. We'll get back to this method in a moment. As an alternative, he broke 12 into 8 and 4 and computed the product 135 times 8, to which he added 135 times 4. As a third method, he factored 12 as 4 times 3, then he computed 135 times 4 times 3. And he also presented other possibilities. Bhaskara's first method appears to be what we now call the grating method. Here the factors are set along the sides of a rectangle, which is divided into split cells. The products of the digits are found, and if the product is a two-digit number, the digits are put into the separate compartments, then the diagonals are added. To see how that might work, let's multiply 135 by 17. So we'll first put down our two factors along the sides of a rectangle, then divide the rectangle into cells, where our cells are big enough to hold one digit at a time, and split the cells. Now we'll multiply each of the digits of the multiplicand by the digits of the multiplicator. So we'd multiply one by one, then record the product in the cell. Now since the product is a one-digit number, that product will be written below the diagonal. First we'll multiply one by three and write down our product, add finally one times five. Then we'll repeat the process with seven times one. When we find seven times three we get a two-digit number. So we'll split the digits along that diagonal, and similarly for seven times five. Now we'll add the terms along the diagonals beginning with the rightmost diagonal, which just has the number five in it. The next diagonal has one, three, and five, and adding them together gives us nine. The next diagonal, seven, two, and three, adds 12, so we'll write down the two and hold on to the one. And for now we could just write it in this last diagonal, which we'll add. And that carried one was never really in this diagonal, so we'll get rid of it. Reading the digits along the outside corner we get the product 2,295.