A definition names a quantity and fixes its measure. One force gets three measures, separated on purpose so they cannot be confused. That is the distinction four chapters written this week each ran into with no name for it.
This series is called Principia Orthogona. The word “Principia” occurs in 814 tracked files and 712 chapters of it. Measured at HEAD, entity-aware, with this page classified out:
| pattern | files | chapters |
|---|---|---|
| Principia | 814 | 712 |
| definition | 260 | 233 |
| lemma | 142 | 123 |
| corollary | 79 | 67 |
| Newton | 28 | 25 |
| measure of | 24 | 19 |
| centripetal | 1 | 1 |
| Scholium | 0 | 0 |
| Rules of Reasoning / Regulae | 0 | 0 |
| quantity of matter | 0 | 0 |
| vis insita | 0 | 0 |
| motive quantity | 0 | 0 |
| absolute … accelerative | 0 | 0 |
| hypotheses non fingo | 0 | 0 |
The series borrows the name and the outer furniture — definitions, lemmas, corollaries — and has never opened the apparatus underneath. This page is not an homage. One piece of that apparatus turns out to be the missing vocabulary for something four chapters written this week each hit from a different side.
Read Definitions I–VIII for their shape rather than their content. Every one of them has the same one:
Two things follow, and the second is the one this corpus needed. First: a definition is not finished until the measure is fixed. Newton never names a quantity and leaves you to guess how it is read off. Second: one quantity can have several measures — a centripetal force has three, and he separates them by number, in consecutive definitions, and then says he will call them motive, accelerative and absolute “for brevity's sake … and for distinction's sake”. The distinction is the point of the definitions.
Block [1] checks the relation between them on his own terms: motion is mass times velocity, so motive = accelerative × mass, and three bodies of masses 1, 2 and 3½ at one place have three different motive measures and one accelerative measure. The accelerative measure does not know the test body. The motive measure does.
λ⊥ = e−4π is an accelerative measure. The Conley index is an absolute one. They are two measures of one quantity, and the reason WP-82 §3b found the first one moving is that it is the kind of measure that moves.
The test is Newton's own: change the test body and see which measure follows it. Here the “test body” is the parametrisation. Take ṙ = k·r(1−r²) for k > 0 — the same Γ = {r = 1}, the same stability type, the same isolating block N = [½, 2], whose boundary signs do not depend on k at all. Block [2]:
Fifty-plus orders of magnitude in one column and nothing at all in the other, under a change that alters neither the orbit nor its type. That is Definition VII against Definition VI. ch-conley reached the same wall by showing the multiplier is a monotone function of the base point and closing the candidate; ch-smale reached it by showing no homotopy invariant can be a monotone function of a parameter it does not feel. Newton had the vocabulary for the confusion in 1687 — not the theorem, the vocabulary, which is what stops the confusion from being available.
ch-gelfand asked whether “the operator algebra of C → K → F → U” is an algebra and found a composition monoid: no sum, no involution, no norm. In the definitional form that is a quantity named with no measure fixed, and it is refused for the same reason Newton would refuse it. The chapter then supplies one — dim = n·p, with p the period of the gate mask — and the quantity becomes usable. The form does not supply the mathematics. It says which sentence is missing.
Newton's Scholium to the Definitions separates absolute quantities from the relative measures we take of them. ch-feigin is an instance, and block [3] checks it: α(m³ − m) is a cocycle for every α, so the class is the absolute thing — one-dimensional, fixed — while the 12 is a unit and c(1) = 0 is the convention that picks the representative. The −m is absolute; the 12 is relative. Stating which is which is the whole of that block.
Book III opens with four Rules of Reasoning, and they read as a standard rather than a method. Set against the seventeen-odd standing rules in this repository and its five tier tags, the correspondence is close enough to be useful and is offered as a reading, not a result:
| Newton | in substance | what this corpus already calls it |
|---|---|---|
| Rule I | admit no more causes than are both true and sufficient | the ASSUME tag — a choice declared as a choice, not smuggled |
| Rule II | to the same effects assign the same causes | R14, a mismatch is a defect only where something is distributed; and the demand that one number be one number wherever it is printed |
| Rule III | what holds throughout the reach of our experiments is held universal | the DATA / MODEL split, and every [HONESTY] block that says an exhaustion over a finite box is evidence and not proof |
| Rule IV | propositions from induction stand until other phenomena make them more accurate or liable to exceptions | the dated measurement — R15, never report absence from a single search, and the rule that a published number is recomputed rather than remembered |
Rule IV is the one that earned itself here this week. WP-82's second column was a correct measurement that stopped being current inside a single day, because six of its twelve rows moved as the corpus was written to. Rule IV does not call that an error; it calls it the ordinary fate of a proposition collected from phenomena, and it says what to do — make it more accurate, or record the exception. The column now names a commit.
The other half of Newton's standard is the sentence he added in the General Scholium, on what
he would not claim: he had not assigned a cause for gravity, and would not feign a
hypothesis for it. That phrase occurs nowhere in this corpus, and it is the exact register of
every OPEN tag in it — and of four statements made this week that could have
been dressed up and were not: no compact invariant set, no horseshoe, not an algebra, no
W-algebra built.
| term | as used from here on |
|---|---|
| quantity | the thing being claimed about. Named, and not finished until its measure is named. |
| measure | how the quantity is read off. A quantity may have several; which one is meant is stated. |
| absolute measure | independent of the test parametrisation. An index, a homotopy type, a cohomology class, a dimension. |
| accelerative measure | a rate generated in a given time. A multiplier, an exponent, a Lyapunov number. Moves when the parametrisation moves. |
| unit | the relative part of a measure — the 12 in c/12, the normalisation that a convention fixes. |
| restriction | a property of a whole family, traced to a stated feature of it. Not a defect and not a failure — the reach of a result. |
| no-go | a proof that a named instrument does not apply, with the reason it does not. Closes a question rather than leaving it. |
| Scholium | a remark that belongs beside a result and is not part of it. Where scope, convention and the limits of an instrument go. |
| Newton 1687 | I. Newton, Philosophiae Naturalis Principia Mathematica. Quoted here from the Motte translation as revised by Chittenden (public domain): Definitions I–VIII and their Scholium, the Axioms or Laws of Motion, the Rules of Reasoning in Philosophy and the General Scholium (Book III). Passages located by OCR of a scanned copy; nothing is quoted at length. |
| in-corpus | ch-conley · ch-smale · ch-gelfand · ch-feigin · WP-82 · WP-122 |
| verification | book7/ch-newton-verify.py — six blocks, standard library only. Every number on this page is printed by it. |
No new mathematics is claimed. Blocks [2] and [3] re-run computations already published in ch-conley, ch-smale and ch-feigin; what this page adds is a count and a vocabulary. The reading of Newton is a reading — the Definitions and the Rules are quoted from a translation and the mapping onto this corpus's tier tags is proposed, not proved. It earns its place only if it stops a measure from going unnamed again, which is a claim about future pages and cannot be checked here. No priority is claimed for anything: the Principia is 1687, and the point is that the apparatus was available the whole time.