As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when "The Nine Chapters on the Mathematical Art" reached its final form, while the "Book on Numbers and Computation" and "Huainanzi" are roughly contemporary with classical Greek mathematics. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory. The Pythagorean theorem for example, has been attested to the time of the Duke of Zhou. Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal, such as the Song dynasty Chinese polymath Shen Kuo.