Newton on discovery, demonstration and limiting ratios

H01 source audit, 2026-09-05; exposition revised 2026-09-06.

Findings

NATP00385 preserves Newton’s retrospective account of how he developed and presented his mathematics. Its central distinction is between analysis as a method of discovery and synthesis as a form of geometrical demonstration. His discussions of moments and fluxions explain the role of generated quantities and their first and last ratios.

Paragraphs 520 and 522 say that the ancients discovered propositions by analysis and demonstrated them by synthesis. Newton presents his own practice in that tradition. In paragraph 522 he describes the Principia propositions as discovered through fluxions and published in synthetic form. Parallel passages occur at 141, 149 and 155.

Generated quantities and ratios

Paragraph 489 treats time as uniformly flowing, fluxions as velocities of increase and moments as parts generated during a moment of time. Successive moments use powers of the symbol o. Approximation belongs to investigation; exact geometrical treatment belongs to demonstration. The contrast between first moments and ratios of nascent moments is a supralinear addition in the TEI.
Paragraph 490 locates the method in first and last ratios. Its diplomatic version records additions to the terms describing nascent and evanescent quantities.
Paragraphs 484–485 use a finite velocity for a fluxion and an infinitely small particle for o. Paragraph 334 develops the contrast with indivisibles.
Paragraph 9 describes moments as infinitely small generated parts; paragraph 796 uses infinitely short lines and narrow surfaces, distinguishing these from geometrical points and lines.

These passages show several explanatory registers within one revised dossier. Their draft layers and vocabulary are material for the historical interpretation of Newton’s limiting arguments.

Publication and controversy

Passage	Object discussed	What Newton’s account says
558, 564, 793	Composition of the Principia	Writing in 1684/1685 and publication in 1687
686	Earlier mathematical methods, including a treatise communicated in 1669	Contentions discouraged earlier publication; the syntax is heavily revised
701	Priority assertion within the Principia	Newton says he avoided controversy by omitting the assertion
1093	A 1671 fluxions/series treatise	Newton attributes its nonpublication to a preference for quiet

In paragraph 686 a long supralinear addition and an addSpan make the construction around “was soon … after … about it” unstable. In paragraph 701 the statement about subsequently declining contention is an addition. These revision details matter when dating and interpreting the claims.

The proposed link between doubts about shrinking areas and a delay of the 1687 edition remains open (H002). The passages above concern methods, presentation and controversy. A causal account of the book’s timing requires dated evidence connecting the limiting argument to a publication decision; H02/H03 carry that chronological task.

The normalized search found one delay* hit, paragraph 975, about Leibniz’s alleged suppression of an invention. The doubt* hits concern receipt of letters or documentary sincerity; the search returned zero classical hits.

Source identity

Title: Unarranged fragments, mostly relating to the dispute with Leibniz.
Catalogue author: Isaac Newton; the collection includes several hands, copied letters and quotations.
Shelfmark: Cambridge University Library, MS Add. 3968, ff. 594r–619v.
Electronic editor: Robert Ralley; work begun 29 March 2019 and completed 31 January 2020; TEI publication statement dated 2020.
Catalogue date/extent: c. 1665–1727; c. 113,053 words; “16 ff.”
Languages: English, Latin and French.
Representations: normalized, diplomatic and TEI/XML.

The catalogue extent, shelfmark span and internal labels from <1r> to <158v> need a verified concordance. The table below therefore uses displayed electronic labels with stable paragraph/line anchors. Some page breaks fall within a paragraph. Claims attributed to other people in the priority argument retain their status as Newton’s report of those people.

The dossier’s recurrent scholium reference concerns Book II, Lemma II. The six Book III Classical Scholia have a separate witness/edition inventory in H02, followed by the selected-edition reading in H03.

Exact focused reading coverage

The normalized view was searched for moment*, momentan*, fluxi*, fluent*, evanesc*, vanish*, infinitesim*, infinitely/indefinitely small, nascent*, ultim*, limit*, geometr*, demonstrat*, synthe*, composition, Principia/Principles, 1684-1687, doubt*, delay*, hesitat*, scrupl*, discourag*, desist*, and publication variants. Hits were grouped by stable paragraph and nearby context was read. The diplomatic HTML and XML were then checked for the passages on which the assessment depends.

Displayed e-text label(s)	Normalized anchors read	Diplomatic/TEI check	Relevance
3v, with embedded 3r	paras 8-9; lines 36-87	passage checked	Fluxions as finite motions; moments as infinitely small generated parts; `o` in demonstration; geometry versus investigation
2v	para 11; lines 107-145	normalized only	Principia Book II Lemma II Scholium and priority narrative
6v-7v	paras 36-37, 41, 54, 56-58; lines 255-351	selected wording checked	Definitions of fluent, fluxion, moment; series terms; synthetic demonstration
16r-20v	paras 110, 112-113, 116, 141, 149, 155; lines 570-791	normalized only	Retrospective chronology; analysis/composition claims; Book II Lemma II
27v-37r	paras 216, 223, 226, 235, 334; lines 1401-1413, 1480-1522, 1531-1584, 2073-2105	para 334 revisions checked	Infinitely small hierarchy attributed to Leibniz; fluxions/moments; first ratios; geometric status
79r-83r	paras 460, 484-485, 489-490; lines 2918-2933, 3086-3186	paras 489-490 checked in diplomatic and XML	Strongest definitions and nascent/evanescent-ratio claims
90r	paras 520 and 522; lines 3505-3527	both checked	Analytical discovery versus synthetic presentation in the Principia
94r-94v	paras 558, 564-565; lines 3766-3847	normalized only	Writing the Principia in 1684; publication in 1687; priority narrative
106v-108v	paras 686 and 701; lines 4832-4844, 4993-5006	para 686 and para 701 checked in diplomatic and XML	Discouraged publication of methods; avoiding a priority controversy
117v-118r	paras 793 and 796; lines 5571-5666	selected wording checked	Writing the book in 1685; moments as infinitely short/narrow generated parts
127r	para 878; lines 6006-6028	normalized only	Retrospective claim about synthetic demonstration in Book II
142v	paras 974-975; lines 6450-6456	normalized only	The sole `delay*` hit concerns Leibniz’s alleged suppression of an invention
156r-158v	paras 1093, 1097-1098; lines 7313-7435	para 1093 revision context checked	1671 methods treatise not published; 1687 date; later priority defence

Reading and extraction status

The audit is a passage-level reading of the listed runs, with selected diplomatic/TEI revision checks. Normalized HTML hides 7,123 encoded deletions and silently incorporates additions. Full deletion-stratum reading, manuscript image comparison, equation verification and individual-leaf dating remain open.

Plain-text dumps served as search aids; consequential wording was checked against HTML/TEI anchors. The broad catalogue date leaves the chronology of individual passages unresolved. The findings above are project interpretations of the electronic edition. Exact source bytes and source-facing coverage are preserved in the companions and checksum register.