对Grassmann，Peano及其学派仿射外代数的修正
Reworking on affine exterior algebra of Grassmann, Peano and his school
摘要：
本文从历史的观点考察了Grassmann的仿射外代数的一个构造，并特别注意了Peano及其学派对这一问题的修正。尽管向量空间上的外代数是一个众所周知的概念，但点与向量共存的仿射空间上的外代数的构造却被忽略了。本文试图弥补这一不足。一些注意力被给予引进定义的抽象（今天称为定义的约束或等价关系），一个程序，由于和使用皮亚诺定义几何形式，基本要素的仿射外代数。皮亚诺的这种创新的定义方式，是对数学的一个相关贡献。据观察，在建设一个仿射外代数的欧几里德三维空间，格拉斯曼和皮亚诺利用度量概念：一个准确的分析表明，在某些情况下，度量方面可以消除，把证据充分的基本仿射结构的欧几里得空间。在本文的最后一部分，一些几何和机械的应用和解释的仿射外代数格拉斯曼和皮亚诺。
英文摘要：
In this paper a construction of affine exterior algebra of Grassmann, with a special attention to the revisitation of this subject operated by Peano and his School, is examined from a historical viewpoint. Even if the exterior algebra over a vector space is a well known concept, the construction of an exterior algebra over an affine space, in which points and vectors coexist, has been neglected. This paper wants to fill this lack. Some attention is given to the introduction of defining by abstraction (today called definition by quotienting or by equivalence relation), a procedure due to and used by Peano to define geometric forms, basic elements of an affine exterior algebra. This Peano's innovative way of defining, is a relevant contribution to mathematics. It is observed that in the construction of an affine exterior algebra on the Euclidean three-dimensional space, Grassmann and Peano make use of metric concepts: an accurate analysis shows that, in some cases, the metric aspects can be eliminated, putting into evidence the sufficiency of the underlying affine structure of the Euclidean space. In the final part of the paper some geometrical and mechanical applications and interpretations of the affine exterior algebra given by Grassmann and Peano are presented.
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