> The set 的 numbers that form 帕斯卡’s triangle were known before 帕斯卡. However, 帕斯卡 developed many uses 的 it 和 was the first one to organize all the 在formation together 在 his treatise, Traitédu trianglearithmétique （1653）。这些数字最初来自印度教对组合数和二项式数以及希腊人的研究’研究比喻数字。
> In 13th century, 杨辉 [杨辉]（1238–1298）提出了与Pascal相同的算术三角形’s triangle. 帕斯卡’三角形叫杨辉’在中国的三角形。的“Yang Hui’s triangle”是中国数学家于11世纪初在中国认识的 贾宪 [贾宪 ]（1010-1070）。
> To appreciate the nature 和 significance 的 the coming “singularity,” it is important to ponder the nature 的 指数增长. Toward this end, I am fond 的 telling the tale 的 the 在ventor 的 棋 和 his patron, the emperor 的 China. In response to the emperor’s 的fer 的 a reward for his new beloved game, the 在ventor asked for a single grain 的 rice on the first square, two on the second square, four on the third, 和 so on. The Emperor quickly granted this seemingly benign 和 humble request. One version 的 the story has the emperor going bankrupt as the 63 doublings ultimately totaled 18 million trillion grains 的 rice. 在 ten grains 的 rice per square 在ch, this requires rice fields covering twice the surface area 的 the Earth, oceans 在cluded. Another version 的 the story has the 在ventor losing his head.
> It should be pointed out that as the emperor 和 the 在ventor went through the first half 的 the 棋 board, things were fairly uneventful. The 在ventor was given spoonfuls 的 rice, then bowls 的 rice, then barrels. By the end 的 the first half 的 the 棋 board, the 在ventor had accumulated one large field’s worth (4 billion grains), 和 the emperor did start to take notice. It was as they progressed through the second half 的 the 棋board that the situation quickly deteriorated. Incidentally, with regard to the doublings 的 computation, that’s about where we stand now–there have been slightly more than 32 doublings 的 performance since the first programmable computers were 在vented during World War II.
> This is the nature 的 指数增长. Although technology grows 在 the exponential domain, we humans live 在 a linear world. So technological trends are not noticed as small levels 的 technological power are doubled. Then seemingly out 的 nowhere, a technology explodes 在to view. For example, when the Internet went from 20,000 to 80,000 nodes over a two year period during the 1980s, this progress remained hidden from the general public. A decade later, when it went from 20 million to 80 million nodes 在 the same amount 的 time, the impact was rather conspicuous.
> Joseph Needham posits that “image-chess,” a recreational game associated with divination, was developed 在 China 和 transmitted to India, where it evolved 在to the form 的 modern military 棋. Needham notes that dice were transmitted to China from India, 和 were used 在 the game 的 “image-chess.”
> Another alternative theory contends that 棋 arose from 象棋 或其前身，自公元前2世纪起就存在于中国。李鸿章（David H. Li）是一位退休会计师，会计学教授和古代汉语译者，他假设汉新将军借鉴了刘博的早期游戏，以在公元前204-203年冬季开发出早期的中国象棋形式。但是，德国象棋历史学家彼得·巴纳沙克（Peter Banaschak）指出，李’s main hypothesis “毫无根据”. He notes that the “Xuanguai lu,”由唐朝大臣牛僧如（779–847）撰写，仍然是中国象棋变体的第一个真实来源 象棋.