Six lessons plus two interactive alternates, in a new standalone-lesson library. The graph does not appear until lesson six, and when it does it is explicitly demoted to a transcription of things already proved.
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f5b15ed, 32 files in . Tracked at. Every table entry
and exercise answer was checked numerically; both interactive pages were driven in Chrome
and screenshotted. Nothing has been in front of a student.
show
Reopens this page. Works whether or not anything is running.
Ptolemaic surveyors at Edfu, around 100 BC, assessed a four-sided field by averaging each pair of opposite sides and multiplying. Apply it to fields with two square corners and sides x,\ 500-x,\ 500-x,\ x — boundary always 1000.
The x cancels. The assessor charges the same for every field in the family — the splinter and the square alike.
Brahmagupta's exact rule, with s=500, hands over the requested factorization directly:
And the gap between what you are charged and what you own is:
So the error is a perfect square. The assessor never undercharges; the rule is exact for exactly one field, the square one; and therefore the largest field holds 62500. The optimization is never asked for. It falls out of asking whether a tax rule is fair. The assessed value turns out to be the area of the best field you could have had for that boundary — the scribes were charging everyone the maximum.
Heron's formula cannot produce this: matching s(s-a)(s-b)(s-c) to two pairs contradicts the sum of the factors either way. The four-sided field is forced, not chosen.
Click to open. Each markdown lesson has its own ANSWERS.md
alongside it, with commentary on what to press students on.
| lesson | the move |
|---|---|
| quad-01 species |
Powers of the unknown are different kinds. Carries your Syversen quotation — three coins of unknown denomination. |
| quad-02 parts-of-500 |
Name the deviation, not the parts: 250\pm d, product 62500-d^2. Maximum, symmetry, solving and impossibility all fall out of one re-parameterization. |
| quad-02b the-table interactive |
Same concepts, numeric-first. Build the ledger by hand; the deviation columns unlock at six cuts. No graph, on purpose. |
| quad-03 assessors-rule |
The anchor above, in full, with the honest complication about floppy quadrilaterals. |
| quad-04 do-not-expand |
What the factored form says and the expansion does not — plus the one-line step that collapses the fourth degree to the second. |
| quad-05 completing-the-square |
al-Khwārizmī's dissection, and the discovery that his fifth case is this problem. Closes on Euclid VI.27. |
| quad-06 the-turn |
The picture arrives, matched feature by feature to the fact it transcribes. |
| quad-06b ledger-turned interactive |
Watch the ledger become the curve. Overlays for the assessor's line, the overcharge, the mirror at 250, and the rescaled quartic. |
al-Khwārizmī's fifth case is the two-parts problem. x^2+21=10x is "two parts of ten with product twenty-one." It is the only one of his six types where he reports two roots — and the only one that is secretly a question about splitting something. His own impossibility condition is the discriminant, stated in the ninth century.
Euclid VI.27 is this maximum theorem, and VI.28 is the solving. He places the maximum immediately before the construction as its solvability condition. Maximum first, then solving, the maximum being what tells you which problems are solvable. That is Euclid's own order, not a retrofit — and it is the order this arc uses.
The pedagogical rule and the historical constraint coincide. "Do not expand until you know what the expansion will tell you" is usually a heuristic. Here it has teeth: expanding x^2(500-x)^2 gives a zenzizenzic, less a thousand cubes, plus 250000 squares — three species, the first with no natural name. Expanding destroyed the only description that meant anything.
: standalone lessons, own formats, an index that grows. The contract
is four required keys long. Lessons assume concepts, never
lessons, so nothing may refer to another by position — links are derived from the
shared vocabulary, which is what lets this become a tutor program without rewriting any
lesson.
Multiple explanations are first-class. Two lessons teaching the same concept are alternates and differ in register, not wording. Five concepts have alternates now; two formats are live and nothing validates lesson content.
cd && python3 build_index.py
Prints the derived graph: which lessons are alternates of which, and which concepts nothing teaches yet. Adding a lesson means adding a directory.
Worth reading: DESIGN.md (the whole argument, and the alternates held in reserve) · README.md (the contract) · SOURCES.md (every historical claim, tagged by how far it was checked) · PROVENANCE.md
Lesson six tells students, in plain terms, that the graph proves nothing new — that every feature is a transcription of something already established. That is true, and it is the honest payoff of having withheld it for five lessons. It also cuts against how every textbook frames coordinates, and a teacher using this material may not want that fight. Softening it is a two-paragraph edit.
Written at roughly late-Algebra-I reading level, but the register is adult — it assumes a student who will sit still for a paragraph of history. If the target is the population, lessons one and six are the two that need cutting, and they are the two carrying the argument.
The quotation in lesson one came from YouTube's automatic captions, which mangle every proper noun in that video — "Deart" for Descartes, "law geometry" for La Géométrie. The two passages actually quoted are ordinary prose with one marked artifact, but nobody has checked them against the audio, and "looks clean" is not "checked." This is the one thing blocking publication.
Ben Syversen, Why Did it Take 1,900 Years to Invent the Exponent? — listen from 9:40 to 11:15.
Also unbuilt and worth knowing about: the half-chord astronomy alternate (Āryabhaṭa's versed sine gives h^2=x(500-x) and the largest half-chord is the radius) — the strongest unwritten lesson here; a tutor-app JSON version of lesson three, which needs only widget types that already exist; and figures for the lesson-five dissection. All noted in DESIGN.md and on.