这个 小弟不才 如果有错还有请板上各位强人校正 因为小弟本身不是很厉害 文中有很多地方都还是到处找资料才看懂 为了让内文看起来顺畅(其实是让我自己这个外行人看懂@@) 有些部分有多加了一些原文没有的部分、或是修改 还请见谅 m(_ _)m ------------------------------------------------------------------------------ 原文连结: http://ppt.cc/i_-4 A great many descriptions of ancient Maya mathematical notation read something like this: 有关古马雅数学标示的描述大部分如下: The Maya made use of a base-20 (vigesimal) system with the units of 1, 20, 400, 8,000, 160,000, etc.. To write a number, a scribe would show multiples of these units in a set columnar order, moving down from highest to lowest, and add them accordingly. “32” for example would be written as single dot for 1, representing one unit of 20, above the two bars and two dots for 12, corresponding to the “ones” unit (1×20 + 12×1 = 32). A larger number such as 823 would be written in three places as two dots followed by one dot followed in turn by three dots, standing for the necessary multiples of 400, 20, and 1 respectively (2×400 + 1×20 + 3×1 = 823). 马雅人使用20进位法(vigesimal)系统，单位有1、20、400、8000、160000等。书记人员 要记录一个数字的时候会利用这些「单位列」，以横列方式表示乘数关系、再以栏柱从上 到下加总而成。举例来说，「32」会是：在20的单位列有一个点代表20；下列会有两横杠 (横杠代表5)还有两点代表12。以上的组合就对应到我们想要的数字「单位」(1X20 + 12X1 =32)。较大的数字如823会分三列，最上列会有两点，第二列一点，然後最下列会有 三点；以乘数关系表示就会是2个400加上1个20+3个1(2*400+1*20+3*1= 823)。 Similar descriptions of Maya math pervade the literature, textbooks and the internet. For example Michael Coe writes in the latest edition of The Maya (p. 232): Unlike our system adopted from the Hindus, which is decimal and increasing in value from right to left, the Maya was vigesimal and increased from bottom to top in vertical columns. Thus, the first and lowest place has the value of one; the next above it the value of twenty; then 400; and so on. It is immediately apparent that “twenty” would be written with a nought in the lowest place and a dot in the second. 相同的描述主宰了大部份的文献、教科书还有网路。例如Michael Coe在最新版的 《马雅》(The Maya)，第232页: 有别於我们从印度人学来的十进位法是以每十单位进位且从右到左增值，马雅人是以每二 十单位进位并且由下到上增值。也就是说，最下面且最底部的地方(列)就是一的值；在上 去就是二十的值；再上去就是四百的单位，以此类推。显而易见得我们可以说20的写法会 是在最下列的地方画上零，然後在第二列上画上一个点。 The illustration accompanying this text provides many examples of this purely vigesimal system: 下列图片为一些二十进位法的一些表现方式: http://ppt.cc/cWXU Maya mathematical notation is described the same way in a number of other influential books widely read in classrooms and seminars, such as The Ancient Maya: New Perspectives (McKillop 2004:277) or the venerable The Ancient Maya(Sharer and Traxler 2006:101). In the latter work, two types of counts are represented (see below) – the purely vigesimal or base-20 count (with units of 1, 20, 400, and 8,000) alongside what’s called the “chronological count” (with units of 1, 20, 360, 7,200). The second is of course the basis for the familiar Long Count system. 在大部分课堂还有研讨会上有影响力的着作也都是如此论述马雅标记方式 (书名下略) 在 Sharer 跟Traxler写的《The Ancient Maya》中有提到两种计数方式 http://ppt.cc/9ftV 一个是纯二十进位或是以二十为底的计算法 (以1、20、400还有8000为单位) 另一个则是所谓的「年序计」 (以1、20、360、7200为单位) 後者就是我们熟悉的长纪系统的基础。 A big problem exists with all of these seemingly straightforward descriptions of Maya mathematical notation. As far as I am aware no purely vigesemal place-notation system was ever written this way. It’s true that in Mayan languages numbers are base-20 in their overall structure, just as in most Mesoamerican languages. In Colonial Yukatek, for example, we have familiar terms for these units: k’al (20), bak’ (400), pik (8,000), and so on. However, ancient scribes never represented these units in a columnar place notation system, as is so commonly described in the textbooks. That format was instead always reserved for a for the count of time, in what we know as the Long Count. That system is mostly vigesimal, but it is skewed in one of its units (the Tun, of 360 days) in order to conform as much as possible to the number of days in the solar year (365). To reiterate: the columns of numbers we find in the pages of the Dresden Codex or painted on the walls of Xultun (stay tuned, folks…) are all day counts; the positional notation system was never used for reckoning anything else. 这麽一来，就跟我们所熟悉的马雅计算描述有很大的不同。就个人的认知里并没有 纯二十进位系统可以这麽写的。整体结构而言在马雅语里面数字都是以二十为底， 符合大部分中美洲语系的特性。举例来说在犹加敦玛雅语里有下列我们熟悉的词汇 做为单位: K'al (20), Bak' (400), pik (8000)...以此类推。然而古代书记没有 像教科书中普遍描写的那样将这些单位以成列方式书写；这样的非成列格式总是用 来计时用，也就是我们说的长纪历法。长纪历法大致上是20进位，但是在其中一个 单位列却有些许不一样 (Tun单位，360天)，为了可以契合一般太阳年中的365天。 重申一遍:我们在Dresden Codex德勒斯登刻本页上还有Xultun夏敦的墙上看到的 「数柱」全是用来计日的；定位计数系统从来没有用来测算其他东西。 In the ancient inscriptions non-calendrical counts using large numbers are quite rare, mostly found in connection to tribute tallies, such as the counting of bundled cacao beans. But in those settings the scribes always seem to show nice rounded numbers (as in ho’ pik kakaw, “5×8,000 [40,000] cacao beans,” shown in the murals of Bonampak) without all the place units we know from the Long Count. In the Dresden and Madrid codices, counts of food offerings are given as groupings of WINIK (20) signs with accompanying bars and dots for 1-19. In this way a cluster of four such elements (4×20) with 19 writes 96 (See Love 1994:58-59; Stuart, in press). 在古碑文中鲜少有用在非历法上的大数字；大半都是进贡的纪载用来记录可可豆束的 数量。即便如此，书记通常都会故意四舍五入成很漂亮的整数(例如在Bonampak波南 帕克的墙上有纪载着 ho' pik kakaw 五个八千束[四万]的可可豆)，并没有在用到 长纪法中的列单位。在德勒斯登还有马德里刻本里进贡的食物纪录都是以WINIK(20) 的符号分组还有横杠还有点代表1到19的数字，这麽一来四个WINIK跟19遍组成了96 这段我真的看了很久...最後我放弃去理解他了) There is a good deal we still don’t know about the ways the Maya wrote quantities, especially of non-calendrical things. The pattern nonetheless seems clear that the place notation system of the Long Count was restricted to time reckoning, and never applied to the purely vigesimal counting structure we see reflected in Mayan languages. The descriptions of written numbers found in the many texts about the ancient Maya therefore need to be corrected. 我们对於马雅人如何纪录数量规则并不是很了解，特别是非历法上的纪录更是毫无头 绪。从种种迹象来说，长纪历法中的成列书写只限於计算时间，并没有像马雅语中所 隐含的那样用来做纯计算。因此许多文献中对於马雅计算方式需要更正。 -------------------------------------------------------------------------- 说真的我翻完我也看不太懂啊 (汗颜) -- But I, being poor, have only my dreams. I have spread my dreams under your feet. Tread softly, because you tread on my dreams. --

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