This essay discusses the instructional value of mathematical proofs using different interpretations of the analysis cum synthesis method in Apollonius’ Conics as a case study. My argument is informed by Descartes’ complaint about ancient geometers and William Thurston’s discussion on how mathematical understanding is communicated. Three historical frameworks of the analysis/synthesis distinction are used to understand the instructive function of the analysis cum synthesis method: the directional interpretation, the structuralist interpretation, and the phenomenological interpretation. I apply these interpretations to the analysis cum synthesis method in order reveal how the same underlying mathematical activity occurs at different levels of scale: at the level of an individual proof, at the level of a collection of proofs, and at the level of a single line within a proof. On the basis of this investigation, I argue that the instructive value of mathematical proof lies in engendering in the reader the same mathematical activity experienced by the author themselves.

本文讨论了使用阿波罗尼乌斯圆锥曲线分析和综合方法的不同解释对数学证明的指导价值作为案例研究。我的论点来自笛卡尔对古代几何学的抱怨以及威廉·瑟斯顿关于如何传达数学理解的讨论。使用分析/综合区分的三个历史框架来理解分析和综合方法的指导功能：方向性解释，结构主义解释和现象学解释。我将这些解释应用于分析暨综合方法，以揭示相同的基础数学活动如何在不同的规模级别上发生：在单个证明级别，在证明集合级别以及在单个证明级别证明内的线。根据这项调查，