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<p>We interpret Newton’s theory of <i>first and last ratios</i> as developed in Book I of the <i>Principia</i>. We understand the basic concept of the theory – the <i>ratio of equality</i> – as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{A:B=1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">A</mi> <mo mathvariant="bold">:</mo> <mi mathvariant="bold-italic">B</mi> <mo mathvariant="bold">=</mo> <mn mathvariant="bold">1</mn> </mrow> </math></EquationSource> </InlineEquation>. However, since Newton’s arguments approximate ratios rather than establish exact equalities, we show that his lemmas lead to the conclusion that these ratios are infinitely close to <b>1</b>. To this end, we combine Euclidean proportions with the modern concept of infinitesimals. Accordingly, we model the Newtonian plane as the Cartesian product of hyperreal numbers, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb {R}}^*\times {\mathbb {R}}^*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∗</mo> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Reading Newton’s Principia through Euclidean Proportion and Nonstandard Analysis

  • 期刊论文

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  • 发表日期：

    2025年7月21日

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  • Piotr Błaszczyk

摘要

We interpret Newton’s theory of first and last ratios as developed in Book I of the Principia. We understand the basic concept of the theory – the ratio of equality – as \(\varvec{A:B=1}\) A : B = 1 . However, since Newton’s arguments approximate ratios rather than establish exact equalities, we show that his lemmas lead to the conclusion that these ratios are infinitely close to 1. To this end, we combine Euclidean proportions with the modern concept of infinitesimals. Accordingly, we model the Newtonian plane as the Cartesian product of hyperreal numbers, \({\mathbb {R}}^*\times {\mathbb {R}}^*\) R ∗ × R ∗ .