The Yanghui Triangle

What is that? Well, it's used to find the coefficients of a binomial expansion. Just like Pascal's triangle. Actually, it is the same thing. Yanghui was a Chinese mathematician from the thirteenth century who did work with binomial expansions and created a triangular array to contain them. One interesting thing about the document we have of the ancient Chinese version of the triangle is that it appears to contain an error. Here is a high resolution image where you can try to find it:


An ancient scribe must have been having an off day. Notice also that for some reason, the symbol for "ten" looks a lot like the Arabic numerals version, just turned on its side. There is more history of the triangle, and the answer to where the mistake is, here.

Historical problem from Medieval China

The book The Nine Chapters on the Mathematical Art is an ancient, anonymous Chinese work that gives techniques for solving various practical problems. In 1261, a mathematician named Yang Hui wrote a commentary called "Detailed analysis of the mathematical rules in the Nine Chapters and their reclassifications." This work supposedly contains the "Problem of the Broken Bamboo" which deals with properties of right triangles. If the height of the stalk and of the break are known, the distance from the base of the stalk to the place where the tip touches the ground can be found with the Pythagorean theorem, which was known to them. The angles involved can be found using trigonometry.