Newton, Principia (1687), Book I Section I: the two scholia on division, in Latin

Companion source

Source: Latin Wikisource, Philosophiae Naturalis Principia Mathematica, Liber I, Sect. I, a proofread transcription of the first edition (London, 1687); local verbatim excerpt .txt; rendered text retrieved 2026-09-23 through the MediaWiki parse API. Metadata: Isaac Newton, Philosophiae Naturalis Principia Mathematica (London: Joseph Streater for the Royal Society, 1687), Book I, Section I, “De Methodo rationum primarum & ultimarum”. Extraction: five passages copied verbatim from the rendered page (the second paragraph of the closing scholium added 2026-09-27, same page, same API); layout line breaks kept; superscript exponents appear flattened (AD3 for AD cubed). Rights: the 1687 text is in the public domain; Wikisource transcriptions of public-domain texts are freely reusable. Limit: a transcription of the first edition. Corollary numbering and some wording changed in 1713 and 1726; the English versions in the notes are this repository’s own renderings, and the paper must cite a critical edition (Koyré and Cohen’s variorum) and Cohen–Whitman for the English.

What it supplies

The Latin witness, in this repository, for what Newton wrote in Book I about division, in the first edition. It answers a gap found by the adversarial review of 2026-09-23: the paper quoted the Section I scholium from a sibling repository’s passage store, with no witness here.

Lemma XI, Corollary 3 (1687 numbering). The ultimate curvilinear areas between tangent and arc are “duæ tertiæ partes triangulorum rectilineorum”, two thirds of the rectilinear triangles, and the segments “partes tertiæ”, one third, “ex natura Parabolæ”. These are the thirds the Planck-gap notes use: the inertial–parabola area is two thirds of the tangent triangle, the chord’s segment one third.
The scholium after Lemma XI, on contact angles. Series of contact angles can be continued without end, and “inter duos quosvis ex his angulis potest series utrinq; in infinitum pergens angulorum intermediorum inseri”; between any two of the new series another can be inserted, “Neq; novit natura limitem”: nature knows no limit. Newton’s own statement, in Book I, that insertion has no end in the geometry.
The scholium closing Section I, first paragraph. The lemmas were put first “ut effugerem tædium deducendi perplexas demonstrationes, more veterum Geometrarum, ad absurdum”: to avoid the tedium of the ancient geometers’ reductio, the method of exhaustion that Archimedes credits to Eudoxus; the method of indivisibles is shorter, “Sed quoniam durior est indivisibilium Hypothesis”, Newton prefers limits, and where he later speaks of particles he means “evanescentia divisibilia”.
The same scholium, second paragraph. The objection that vanishing quantities have no ultimate ratio, “quippe quæ, antequam evanuerunt, non est ultima, ubi evanuerunt, nulla est”, and its extension to motion: there would be no ultimate velocity of a body reaching a place. Newton’s answer: the ultimate velocity is “illam ipsam velocitatem quacum corpus attingit locum ultimum”, the very velocity with which the body reaches its last place; “Extat limes quem velocitas in fine motus attingere potest, non autem transgredi. Hæc est velocitas ultima”, and since “hic limes sit certus & definitus”, determining it is a truly geometrical problem. This is Newton’s reply to the instant-versus-interval objection of Zeno’s arrow: a determinate velocity assigned to arrival at a place, which the fifth-postulate note reads as joint determinacy of place and motion.
The same scholium, third paragraph. The objection that given ultimate ratios give ultimate magnitudes, so that “quantitas omnis constabit ex indivisibilibus, contra quam Euclides de incommensurabilibus, in libro decimo Elementorum, demonstravit”, is answered by limits, and the scholium closes “cave intelligas quantitates magnitudine determinatas, sed cogita semper diminuendas sine limite”.

Passage anchors

Passage	Excerpt section	What it gives
Lemma XI, Corol. 3	first	Curvilinear areas two thirds, segments one third, of the rectilinear triangles
Scholium after Lemma XI	second	Contact angles; series inserted without end; “Neq; novit natura limitem”
Section I Scholium, par. 1	third	The ancients’ reductio avoided; indivisibles rejected; “evanescentia divisibilia”
Section I Scholium, par. 3	fourth	Euclid X against ultimate magnitudes; “cave intelligas quantitates magnitudine determinatas”

Coverage and limits

Reading level passage for the four anchors; the second paragraph of the Section I scholium (on the last velocity) is not excerpted. Figures and the scan are not consulted, so the flattened exponents follow the transcription. Byte identity of the excerpt is registered in SHA256SUMS.