Commensurability and symmetry have diverged from a common Greek origin. We review the history of this divergence. In mathematics, symmetry is now a kind of measure that is different from size, though analogous to it. Size being given by numbers, the concept of numbers and their equality comes into play. For Euclid, two magnitudes were symmetric when they had a common measure; also, numbers were magnitudes, commonly represented as bounded straight lines, for which equality was congruence. When Billingsley translated Euclid into English in the sixteenth century, he used the word “commensurable” for Euclid’s symmetric magnitudes; but the word had been used differently before. Symmetry has always had also a vaguer sense, as a certain quality that contributes to the beauty of an object. Today we can precisely define the symmetry of a mathematical structure as the automorphism group of the structure, or as the isomorphism class of that group. However, when we consider symmetry philosophically as a component of beauty, we can have no foolproof algorithm for it.

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关于可共通性和对称性

可通约性和对称性与希腊的共同起源背道而驰。我们回顾了这种分歧的历史。在数学中，对称是一种类似于大小的度量，尽管与之类似。由数字给定大小，数字的概念及其相等性开始起作用。对欧几里得来说，两个量度具有相同的度量是对称的。同样，数字是数量级，通常表示为有界直线，对于这些数量级，相等性是同等的。Billingsley在16世纪将Euclid译为英语时，他将“相称”一词用于Euclid的对称幅度。但是这个词之前的用法有所不同。对称始终具有模糊的感觉，因为一定的质量有助于物体的美观。今天，我们可以精确地将数学结构的对称性定义为该结构的自同构组或该组的同构类。但是，当我们从哲学上将对称视为美的组成部分时，就没有万无一失的算法。