這個 小弟不才 如果有錯還有請板上各位強人校正 因為小弟本身不是很厲害 文中有很多地方都還是到處找資料才看懂 為了讓內文看起來順暢(其實是讓我自己這個外行人看懂@@) 有些部分有多加了一些原文沒有的部分、或是修改 還請見諒 m(_ _)m ------------------------------------------------------------------------------ 原文連結: http://ppt.cc/i_-4 A great many descriptions of ancient Maya mathematical notation read something like this: 有關古馬雅數學標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). 馬雅人使用20進位法(vigesimal)系統，單位有1、20、400、8000、160000等。書記人員 要記錄一個數字的時候會利用這些「單位列」，以橫列方式表示乘數關係、再以欄柱從上 到下加總而成。舉例來說，「32」會是：在20的單位列有一個點代表20；下列會有兩橫槓 (橫槓代表5)還有兩點代表12。以上的組合就對應到我們想要的數字「單位」(1X20 + 12X1 =32)。較大的數字如823會分三列，最上列會有兩點，第二列一點，然後最下列會有 三點；以乘數關係表示就會是2個400加上1個20+3個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獻、教科書還有網路。例如Michael Coe在最新版的 《馬雅》(The Maya)，第232頁: 有別於我們從印度人學來的十進位法是以每十單位進位且從右到左增值，馬雅人是以每二 十單位進位並且由下到上增值。也就是說，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；再上去就是四百的單位，以此類推。顯而易見得我們可以說20的寫法會 是在最下列的地方畫上零，然後在第二列上畫上一個點。 The illustration accompanying this text provides many examples of this purely vigesimal system: 下列圖片為一些二十進位法的一些表現方式: 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. 在大部分課堂還有研討會上有影響力的著作也都是如此論述馬雅標記方式 (書名下略) 在 Sharer 跟Traxler寫的《The Ancient Maya》中有提到兩種計數方式 http://ppt.cc/9ftV 一個是純二十進位或是以二十為底的計算法 (以1、20、400還有8000為單位) 另一個則是所謂的「年序計」 (以1、20、360、7200為單位) 後者就是我們熟悉的長紀系統的基礎。 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. 這麼一來，就跟我們所熟悉的馬雅計算描述有很大的不同。就個人的認知裡並沒有 純二十進位系統可以這麼寫的。整體結構而言在馬雅語裡面數字都是以二十為底， 符合大部分中美洲語系的特性。舉例來說在猶加敦瑪雅語裡有下列我們熟悉的詞彙 做為單位: K'al (20), Bak' (400), pik (8000)...以此類推。然而古代書記沒有 像教科書中普遍描寫的那樣將這些單位以成列方式書寫；這樣的非成列格式總是用 來計時用，也就是我們說的長紀曆法。長紀曆法大致上是20進位，但是在其中一個 單位列卻有些許不一樣 (Tun單位，360天)，為了可以契合一般太陽年中的365天。 重申一遍:我們在Dresden Codex德勒斯登刻本頁上還有Xultun夏敦的牆上看到的 「數柱」全是用來計日的；定位計數系統從來沒有用來測算其他東西。 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). 在古碑文中鮮少有用在非曆法上的大數字；大半都是進貢的紀載用來記錄可可豆束的 數量。即便如此，書記通常都會故意四捨五入成很漂亮的整數(例如在Bonampak波南 帕克的牆上有紀載著 ho' pik kakaw 五個八千束[四萬]的可可豆)，並沒有在用到 長紀法中的列單位。在德勒斯登還有馬德里刻本裡進貢的食物紀錄都是以WINIK(20) 的符號分組還有橫槓還有點代表1到19的數字，這麼一來四個WINIK跟19遍組成了96 這段我真的看了很久...最後我放棄去理解他了) 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. 我們對於馬雅人如何紀錄數量規則並不是很了解，特別是非曆法上的紀錄更是毫無頭 緒。從種種跡象來說，長紀曆法中的成列書寫只限於計算時間，並沒有像馬雅語中所 隱含的那樣用來做純計算。因此許多文獻中對於馬雅計算方式需要更正。 -------------------------------------------------------------------------- 說真的我翻完我也看不太懂啊 (汗顏) -- 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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