 What we know about Jordanus Numerarius, who lived in the 13th century? Well, nothing really. Jordanus wrote several mathematical treatises, including Demonstratio de Algarismo, an algorithm for whole number operations, and Demonstratio de Minuitis, an algorithm for fractions. He also wrote treatises on statics, which is very important when you're constructing a large building that you'd rather not collapse right away, like a gothic cathedral, and Astronomy. His most important work was probably De Numerus Datis. The title refers to the fact that many of its problems are about finding some numbers with given properties. For example, a given number might be separated into two parts where the difference is known, the product is known, or the quotient is known. For example, the very first problem in De Numerus Datis is if a given number is separated into two parts with a given difference, then each part can be found. Notice that Jordanus is claiming a theoretical result. It is possible to solve this problem. The proof is the algorithm for finding the two parts. So Jordanus's argument is that the lesser plus the difference is the greater. The lesser and the greater make the whole. So the lesser plus the lesser plus the difference is the whole amount. Now this verbal algebra of Jordanus may be a little difficult to follow. So let's anachronistically introduce some modern notation. So note that we've separated a given number into two parts. So we actually know the sum of the two parts that's a given, and we're given the difference. Then the larger of the two numbers is the smaller, plus whatever the difference is. Now since we know the sum of the two numbers, we can replace the larger of the numbers with smaller plus the difference, and that gives us an algebraic equation we can solve. Then the rest of the problem is finding one of these numbers. So Jordanus gives the algorithm, if you subtract the difference from the whole, you get twice the smaller number. Take half to find the lesser number, and add the difference to find the greater. Jordanus gave an example, separate 10 into two numbers with difference 2. So subtract 2 to get 8, half of 8 is 4, the lesser number, and 2 more is 6, the greater number. And again, we can anachronistically introduce some modern notation. We know that the sum is 10, and the difference is 2. So if we subtract the one from the other we get, divide by 2 gives us, and that's our smaller number, and we can compute the larger number.