AI 中文总结
本文论证《几何原本》第七卷数论源自毕达哥拉斯音乐算术化，明确Hippasus是关键人物，其发现的音乐互反减法及传递的算术互反减法为相关数学发展奠定基础。
AI 中文摘要
《几何原本》第七卷代表了一项卓越的数学成就，标志着数论的诞生。尽管它未明确陈述算术基本定理（即每个自然数都能唯一表示为素数的乘积），但包含了证明该定理所需的全部工具：最小原理（等价于数学归纳法）和用于求最大公约数的算术互反减法（anthyphairesis）。本文提出新的论证，表明第七卷直接源自早期毕达哥拉斯学派算术化的音乐。 Vincenzo Galilei 证明了将这种算术化归因于毕达哥拉斯声学实验的说法是虚构的；相反，替代实验以及 Hippasus 的四弦（高度为6、8、9、12的青铜圆柱体）通过欧拉管定律被证实为物理上正确。这一实验基础促成了音程的算术化以及音乐（乘法）互反减法的发现。应用亚里士多德《论题篇》158b24-29 中的原理，我们重建了音乐互反减法如何通过 Smyrna 的 Theon 所述的从倍数和超特定比开始的比率归纳阶梯，被转化为其算术（加法）对应形式。此外，我们表明第七卷定义和证明中的数学特性仅能通过其音乐起源得到令人信服的解释。最终，Hippasus 成为关键人物：他发现了音乐互反减法（演变为 Philolaus 的残篇6），并将算术互反减法传递给几何学，这对毕达哥拉斯学派的不可公度性以及无限与有限的原理至关重要。
英文摘要
Book VII of Euclid's Elements represents a remarkable mathematical achievement, marking the birth of number theory. Although it falls short of explicitly stating the Fundamental Theorem of Arithmetic (that every natural number is uniquely a product of primes), it contains all the tools necessary for its proof: the Principle of the Least (equivalent to Mathematical Induction) and arithmetical anthyphairesis for finding the greatest common divisor. Our work presents novel arguments that Book VII evolved directly from early Pythagorean arithmetized music. The accounts attributing this arithmetization to Pythagoras' acoustical experiments were shown to be fictitious by Vincenzo Galilei. In contrast, the alternative experiments and Hippasus' 4-chord (bronze cylinders of heights 6, 8, 9, 12) are validated as physically correct by Euler's Law for pipes. This experimental foundation led to the arithmetization of musical intervals and to the discovery of musical (multiplicative) anthyphairesis. Applying Aristotle's Topics 158b24-29 Principle, we reconstruct how musical anthyphairesis was transferred - through an inductive ladder of ratios starting with multiple and epimoric ratios recounted by Theon of Smyrna - to its arithmetical (additive) counterpart. Furthermore, we show that the mathematical peculiarities in the definitions and proofs of Book VII find a convincing explanation only through their musical origin. Ultimately, Hippasus emerges as the pivotal figure who discovered musical anthyphairesis (which evolved into Philolaus' Fragment 6) and transferred arithmetical anthyphairesis to geometry, crucial for Pythagorean incommensurability and the principles of the Infinite and Finite.
Comments88 pages, 12 figures, Preliminary version of a chapter to appear in a collective volume
