A letter from Berlin, 14 November 1750, carrying the first number a solid ever had that did not come from measuring it. This corpus uses that number in seventy-six chapters — and filed one of its properties as a curiosity.
Euler writes to Christian Goldbach in St Petersburg. He has noticed something about solids, and he states it in words, with no symbols at all:
In the same letter he adds the remark that tells you what he thought he had found: “It astonishes me that these general properties of stereometry have not, as far as I know, been noticed by anyone else.” He was right to be astonished, and the reason is the thing worth taking. Everything known about polyhedra before 1750 was metric — volume, surface area, angles, inscribability. Nobody had looked at a solid and simply counted.
Block [1] counts nine of them, in integer arithmetic: the five Platonic solids, a prism, a pyramid, an antiprism, and a truncated icosahedron. Every one gives 2. Squash the cube, shear it, round its corners into something unrecognisable — still 2. It is the first quantity in the history of the subject that survives bending the object.
“Enclosed by plane faces” is not decoration. Glue a grid of quadrilaterals into a torus and the count changes — block [2], at four grid sizes:
Euler's own proof of the formula turns out to have a flaw in it, found and repaired by others in the century after — which is the ordinary fate of a first proof and no mark against the observation. What the zeros above say is that the number is not a fact about solids. It is a fact about surfaces, and the sphere and the torus are different ones. That is where the subject turns topological, and it is where this corpus lives: Γ sits on a cylinder.
ch-conley computed the Conley index in three cases and printed, as an aside, that all three Euler characteristics come out 0 and that χ therefore separates nothing — recorded because χ is what a reader reaches for first. The aside has an explanation, and it is Euler's.
| space | homology | χ |
|---|---|---|
| annulus A | (ℤ, ℤ, 0) | 0 |
| A/∂A — the repeller | (0, ℤ, ℤ) | 0 |
| A/L — the pair | (0, 0, 0) | 0 |
All three are built from the annulus, and χ(A) = 0. The first two differ by a shift of two degrees — the unstable dimension — and χ is an alternating sum, so an even shift leaves it exactly where it was. χ was never going to see the difference.
Euler's construction answers one question: how many cells, counted with sign. It does not answer which cells, in which degree. Reaching for χ to separate an attractor from a repeller is asking the 1750 question of a 1920s object. In ch-newton's vocabulary it is a measure, correctly computed, of the wrong quantity — and block [3] computes it by two independent routes, cell counts and Betti numbers, which agree on all six spaces tested.
Euler's count reaches this corpus's actual subject through the index of a vector field. Strogatz gives it at §6.8: go once round a closed curve and count the turns the field direction makes. Block [5] computes it by winding, and it lands on integers:
Now the corpus's own frozen field, ṙ = −(r−1)(r²+r−a), θ̇ = 1, with its two invariant circles at r = 1 and r2 = 1.884652. Block [6] asks first where the field vanishes, and the answer corrects an easy misreading:
The field does not vanish on either invariant circle. ṙ does, but θ̇ does not, and an invariant circle is not a fixed point. The only zero of the planar field is the origin — and every circle about the origin, at every radius tested from 0.3 to 10, has index exactly +1, while a small loop enclosing nothing has index 0. That is Strogatz's Theorem 6.8.2 satisfied on the corpus's own equations: a closed orbit must enclose fixed points of index sum +1, and both invariant circles enclose exactly the origin.
And it joins up with the other end. ch-conley's no-go turned on the three-dimensional flow having no fixed point anywhere, because θ̇ ≡ 1. A surface admits a nowhere-zero field only when its Euler characteristic is 0. The cylinder's is 0 — block [2] counted it. The sphere's is 2 — block [3] computed it — which is why you cannot comb a hairy ball. The corpus's flow is zero-free and its surface is a cylinder, and those are not two facts.
One more, because it is this corpus's own failure mode in someone else's ledger. The Königsberg bridges, 1736, are told everywhere as the birth of graph theory: Euler drew four vertices and seven edges and reasoned about the graph. Hopkins and Wilson, in the Bradley–Sandifer volume, put it plainly — Euler did not draw that graph. Graphs of that kind do not appear until the second half of the nineteenth century. The picture everyone credits to him was drawn by somebody else, a hundred and thirty years later, and attributed backwards.
A citation that attributes to a source something the source does not contain is exactly what Nine Pages, No Proofs is about, and this one has been repeated for a century by people in a position to check. Measured here, “Königsberg” reaches one chapter of this corpus and “graph theory” one. Neither repeats the error, which is luck rather than diligence, and worth converting into the second.
Olaf Neumann opens his chapter on cyclotomy in Leonhard Euler: Life, Work and Legacy with an attribution most readers will not expect. The formula
is universally called de Moivre's. Neumann's sentence is flat: "But in the form (1) it is due to Leonhard Euler (1707–1783), see [Euler 1748], cap. VIII." De Moivre had the content, scattered across results about roots of unity, and never wrote it down this way. Euler did, in the Introductio. The name on the theorem is not the person who put it in the form everyone quotes.
This chapter has already recorded one case of a name attached to the wrong thing — the polyhedron formula that was not a curiosity. This is the reverse: the right formula, correctly attributed to nobody in particular for two and a half centuries. Both are transmission with drift, which is the subject the Nachbin chapter ended on and the Ramanujan chapter ran into from the other side.
Dividing a circle into n equal parts is a problem in geometry. Euler's move turns it into a problem about the roots of xn − 1 = 0 — the geometry is gone, and what remains is an equation. In this series' vocabulary that is compression: many degrees of freedom reduced to their essential coordinates.
The reduction is famous because it worked. Vandermonde and then Gauss could push it through to radicals, and Gauss could construct the 17-gon, because the Galois group of the cyclotomic field is (ℤ/nℤ)* — abelian, of order φ(n), which is smaller than n. Abelian implies solvable implies radicals. The compression landed somewhere simpler than it started.
This corpus performs the same manoeuvre on its own ladder polynomials qn(x) = xn − xn−1 − … − x − 1, whose roots are the n-bonacci constants. Same kind of object: a monic integer polynomial, roots wanted. The outcome is opposite.
book4/cyclotomy-ladder-verify.py computes both sides exactly. The cyclotomic group is abelian for every n tested, and φ(n) < n always. The ladder's discriminants are computed by resultant over ℚ for n = 2…8 and not one is a perfect square, so no Galois group here sits inside An; with the irreducibility already established in Book 4 the group is generically the full Sn, of order n!, non-solvable from n = 5. At n = 8 the ratio of the two group orders is 10,080.
Euler's reduction of the circle succeeded because there was something simpler on the other side of it. The same reduction applied to the ladder arrives at S8, and no amount of effort on the equation changes that — it is a fact about the group.
What this is not. An earlier draft of this paragraph called it a limit on the series' central move. That was an overstatement and is withdrawn. Solvability by radicals is a fact about one notation, not about whether a reduction is worth making. The ladder's roots exist, are algebraic, and are computed here to any precision wanted; η ≈ 1.8393 is not less available for being unwriteable in nested surds. Hermite solved the quintic in 1858 using elliptic modular functions — the quintic is solvable, just not in radicals. The finding above is specific to this family and to that alphabet, and it does not generalise to compression as such.
Which leaves something better than the overstatement. The ladder is a case where the answer was reachable and the alphabet was not yet there — the same shape as the acoustic vessels, and as Ramanujan's radicals waiting eighty years for someone to work out what they were of. Not a limit on reduction. A limit on one way of writing the result down, which is this corpus's subject rather than its obstacle.
Verified in book4/cyclotomy-ladder-verify.py — five blocks, standard library only, exact integer and rational arithmetic throughout except the one block that checks Gauss's radical form for cos(2π/17) numerically and says so. Along the way it confirms that Φ105 is the first cyclotomic polynomial whose coefficients leave {−1, 0, 1}, which is the standard warning against believing a pattern that holds for the first hundred cases.
Measured at HEAD, entity-aware, this page classified out:
| pattern | files | chapters |
|---|---|---|
| χ | 89 | 76 |
| Euler | 64 | 56 |
| zeta | 32 | 29 |
| Runge–Kutta / RK4 | 20 | 16 |
| Euler characteristic | 19 | 17 |
| Euler product | 17 | 14 |
| Gauss–Bonnet | 9 | 7 |
| Poincaré–Hopf | 2 | 1 |
| Königsberg · graph theory | 1 each | 1 each |
| index of a vector field | 0 | 0 |
| hairy ball | 0 | 0 |
| polyhedron formula | 0 | 0 |
| Euler's method | 0 | 0 |
| Euler–Lagrange · Basel problem | 0 | 0 |
Sixteen chapters integrate with Runge–Kutta and none names the method Euler wrote first, of which Runge–Kutta is the refinement. Seventy-six use χ and none had written down where it comes from. That is the gap this page closes, and the three zeros at the bottom — Euler–Lagrange, the Basel problem, Euler's method — are the next ones, each of which the corpus has the material for and has not used.
| Euler 1750/1758 | L. Euler to C. Goldbach, 14 November 1750; E230 (statement, written 1750) and E231 (proof, written 1751), both published 1758. The theorem is quoted in the translation given by Richeson. |
| Richeson | D. Richeson, “The Polyhedral Formula”, in R. E. Bradley and C. E. Sandifer (eds.), Leonhard Euler: Life, Work and Legacy, Elsevier, 2007 — Euler's proof, its flaw, and the repair. |
| Hopkins–Wilson | B. Hopkins and R. J. Wilson, “The Truth about Königsberg”, in the same volume; first published in The College Mathematics Journal 35(3), and awarded the MAA's George Pólya Award in 2005. |
| Strogatz | S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., §6.8 pp. 179–180; Theorem 6.8.2. |
| in-corpus | ch-conley · ch-newton · ch-strogatz · WP-120 · Nine Pages, No Proofs |
| verification | book7/ch-euler-verify.py — eight blocks, standard library only. Every number on this page is printed by it. |
Poincaré–Hopf is not proved here, and neither is the hairy ball theorem. Block [6] computes χ for a cylinder and a sphere and computes indices for particular fields; the remark joining the corpus's zero-free flow to χ = 0 is a reading of two computed facts, not a derivation of one from the other. The winding numbers are numerical quadrature landing within 10−6 of integers — evidence that the integers are right, not a proof. Euler's own proof of the polyhedral formula, and the question of which hypotheses on a solid make the theorem true, are the substance of a century of later work and are not assessed here. No priority is claimed: the letter to Goldbach is dated 14 November 1750.