Abstract Amid the debate over infinitesimals, Bolzano introduced the alternative notion of variable quantities ω in his 1816 work on the binomial theorem. It has often been assumed that his 1817 definition of continuity using them is practically the modern one. This paper explores Bolzano's early mathematical works and diaries to gain insight into the subtleties in his definition of ω and some of his mathematical procedures. We show that those quantities are not clearly ‘proto-Weierstrassian’ and argue that Bolzano was in the process of refinement and increasing abstraction of the idea of quantity that eventually led to the development of a theory of real numbers.

中文翻译：

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博尔扎诺早期作品中变量 ω 的概念

摘要 在关于无穷小的争论中，博尔扎诺在他 1816 年关于二项式定理的工作中引入了变量 ω 的替代概念。人们经常假设他在 1817 年使用它们对连续性的定义实际上是现代定义。本文探讨了 Bolzano 的早期数学著作和日记，以深入了解他对 ω 的定义和他的一些数学程序中的微妙之处。我们表明这些量显然不是“原始魏尔斯特拉斯”，并认为博尔扎诺正处于对数量概念的提炼和日益抽象的过程中，最终导致了实数理论的发展。