Mathematics in the Time of the Pharaohs
Mathematics in the Time of the Pharaohs
复制标题
法老时代的数学
作者：
R. J. Gillings
在写第一本书的长度研究古埃及数学，理查德·吉林斯提出的证据表明，埃及人在这一领域的成就比以前认为的要多得多。他这样做的方式不仅会引起埃及和数学历史学家的兴趣，而且会引起那些喜欢以新颖方式操纵数字的人的兴趣。他检查了所有现存的来源，特别注意其中最广泛的-莱因德数学纸草，一个收集的训练练习抄写员。这张纸莎草纸除了处理埃及人发展数学所需的实际商业计算外，还包括一系列以更一般的方式陈述的抽象数值问题，所用的数学运算数量极其有限，但适用于许多应用。埃及的数字系统是十进制的，数字按顺序排列（很像我们的数字，但阅读从右到左），使他们能够轻松地进行加减。他们可以将任何数字乘以2，并使用二进制过程来完成更多的扩展乘法，依次将结果乘以2并将导致正确结果的部分乘积相加。分裂也是以类似的方式进行的。他们可以完全操纵分数，即使所有分数（有一个例外）都是以单位分数的求和的笨拙形式表示的-那些以“1”作为分子的分数。(The例外是2/3。抄写员认识到这是一个非常特殊的数量，并采取2/3的整数或分数的变化时，提出自己在计算过程中。在将一个有理数表示为一系列单位分数时，抄写员通常能够从许多-有时是数千-可能的解决方案中选择一个简单而直接的解决方案。如果没有现代计算机，就像没有现代机械就能建造金字塔一样，用这些有限的操作手段解决的数学问题的范围比许多数学史学家所承认的要广得多。Gillings举例说明埃及人能够解决正比例和反比例问题;评估某些平方根;引入两个数字之间的“调和平均”概念;解决一阶线性方程组和两个联立方程组，其中一个是二阶方程组;找到算术和几何级数项的总和;计算一个圆和圆柱形（甚至可能是球形）表面的面积;计算截头金字塔和圆柱形粮仓的体积;并利用基本的三角函数来描述金字塔的斜率。历史学家们总是不加批判地一个接一个地重复埃及人的成就，但吉林斯找不到任何证据来支持这一点：埃及人知道毕达哥拉斯定理，至少在3-4-5直角三角形的特殊情况下是这样。
In writing the first book-length study of ancient Egyptian mathematics, Richard Gillings presents evidence that Egyptian achievements in this area are much more substantial than has been previously thought. He does so in a way that will interest not only historians of Egypt and of mathematics, but also people who simply like to manipulate numbers in novel ways. He examines all the extant sources, with particular attention to the most extensive of these--the Rhind Mathematical Papyrus, a collection of training exercises for scribes. This papyrus, besides dealing with the practical, commercial computations for which the Egyptians developed their mathematics, also includes a series of abstract numerical problems stated in a more general fashion.The mathematical operations used were extremely limited in number but were adaptable to a great many applications. The Egyptian number system was decimal, with digits sequentially arranged (much like our own, but reading right to left), allowing them to add and subtract with ease. They could multiply any number by two, and to accomplish more extended multiplications made use of a binary process, successively multiplying results by two and adding those partial products that led to the correct result. Division was done in a similar way. They could fully manipulate fractions, even though all of them (with one exception) were expressed in the unwieldy form of sumes of unit fractions--those having "1" as their numerator. (The exception was 2/3. The scribes recognized this as a very special quantity and took 2/3 of integral or fractional numbers whenever the change presented itself in the course of computation.) In expressing a rational quantity as a series of unit fractions, the scribes were generally able to choose a simple and direct solution from among the many--sometimes thousands--that are possible. Doing this without modern computers would seem quite as remarkable as building pyramids without modern machinery.The range of mathematical problems that were solved using these limited operational means is far wider than many historians of mathematics acknowledge. Gillings gives examples showing that the Egyptians were able, for example, to solve problems in direct and inverse proportion; to evaluate certain square roots; to introduce the concept of a "harmonic mean" between two numbers; to solve linear equations of the first degree, and two simultaneous equations, one of the second degree; to find the sum of terms of arithmetic and geometric progressions; to calculate the area of a circle and of cylindrical (possibly even spherical) surfaces; to calculate the volumes of truncated pyramids and cylindrical granaries; and to make use of rudimentary trigonometric functions in describing the slopes of pyramids. The Egyptian accomplishment that historians have tended to repeat uncritically, one after another, is one that Gillings can find no evidence to support: that the Egyptians knew the Pythagorean theorem, at least in the special case of the 3-4-5 right triangle.