Euler vs. Newton Cubic Species
Summary: Euler’s 16 cubic species reduce all 72 of Newton’s; the discrepancy comes entirely from Newton’s added attention to bounded-region shape, which Euler proposes calling varieties.
Sources: chapter9 (§§236, 238).
Last updated: 2026-04-30.
§236 — Two correct counts of the same set
Euler arrives at sixteen species (see cubic-species-classification); Newton’s classification of the same set reaches seventy-two. Euler explicitly accepts that this is not a contradiction. He used “only the criteria of branches going to infinity and their species,” whereas Newton “considered also that part of the curve which is in a bounded region.” Both classifications cover the same curves — Newton’s just makes finer cuts.
“Although this criterion for classification may seem arbitrary, still Newton was able to find many more species by following his criteria, while using my method I am able to classify no more nor less curves than he.”
The 72-into-16 mapping is laid out via the simplest representative equations in §237 (see simplest-cubic-equations).
§238 — Genus / species / variety
Euler closes the chapter with a nomenclature suggestion. Within each of his sixteen classes the curves typically take “notable varieties” of bounded-region shape; Newton’s species are precisely those varieties under another name. Euler’s proposal:
- genus — what he has been calling species. Sixteen for cubics.
- species — Newton’s finer divisions. Seventy-two for cubics.
- variety — yet finer sub-divisions, useful “if one wishes to classify the lines of the fourth or higher order. In that case there is so much more variety in each species thus found.”
The proposal is offered as nomenclature only; Euler’s own usage in the surrounding text continues to call the sixteen classes species.
Why the criterion choice matters at higher order
Euler hints that the choice will scale poorly past third order if Newton’s bounded-region distinctions are to be tracked: the variety in each branch-at-infinity class grows quickly with . The branches-at-infinity criterion, by contrast, remains a finite combinatorial enumeration grounded in the principal member’s factor structure.
Related pages
- chapter-9-on-the-species-of-third-order-lines
- cubic-species-classification — Euler’s 16-class derivation.
- simplest-cubic-equations — the §237 table that exhibits the 72 → 16 mapping.
- hyperbolic-and-parabolic-branches — the species-of-branch labels Euler counts.