A Categorical Approach to Euclidean Ratios and Proportions
Abstract
A categorial approach to the non-metric geometry in Books V and VI of Euclid's \textit{Elements} is presented. Specifically, we introduce a diagrammatic syntax that can be overlaid immediately on his diagrams, thus bridging intuitive presentation with fidelity to Euclid's arguments. This syntax makes complicated definitions like V.5, and indeed the arguments throughout books V and VI, including arguments about similar figures, intuitively clear. We show in an appendix that this syntax can be used to solve a puzzle regarding ancient mathematics. Finally, we offer evidence that this approach to Euclidean diagrams is rooted in the Aristotelian tradition itself, and that a similar syntax was utilized, in antiquity, for related questions of numeric and proportions. Thus the syntax is plausibly faithful to Euclid's own thought-world, and not an outside-imposition.
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BibTeX
@misc{petersen2026categorical,
title = {{A Categorical Approach to Euclidean Ratios and Proportions}},
author = {Matthew Petersen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09537},
eprint = {2609.09537},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09537},
}
RIS
TY - UNPB
TI - A Categorical Approach to Euclidean Ratios and Proportions
AU - Matthew Petersen
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09537
UR - https://arxiv.org/abs/2609.09537
AB - A categorial approach to the non-metric geometry in Books V and VI of Euclid's \textit{Elements} is presented. Specifically, we introduce a diagrammatic syntax that can be overlaid immediately on his diagrams, thus bridging intuitive presentation with fidelity to Euclid's arguments. This syntax makes complicated definitions like V.5, and indeed the arguments throughout books V and VI, including arguments about similar figures, intuitively clear. We show in an appendix that this syntax can be used to solve a puzzle regarding ancient mathematics. Finally, we offer evidence that this approach to Euclidean diagrams is rooted in the Aristotelian tradition itself, and that a similar syntax was utilized, in antiquity, for related questions of numeric and proportions. Thus the syntax is plausibly faithful to Euclid's own thought-world, and not an outside-imposition.
ER -