In 1860, a French mining engineer named Édouard Phillips published a 54-page memoir in the Journal de mathématiques pures et appliquées that (among other things) solved a problem watchmakers had been attacking by trial and error for nearly two centuries: what shape should the ends of a balance spring have, so that a watch keeps the same time whether its mainspring is freshly wound or nearly spent? The answer — two clean geometric conditions on the center of gravity of the spring’s terminal curve — is still the design rule behind every Breguet overcoil made today, and its modern descendants are etched into silicon hairsprings.

As far as I can tell, the memoir has never been translated into English. It now has been, and this post describes both the mathematics and the somewhat instructive process of producing the translation.

The problem

Huygens invented the spiral balance spring in 1674 (built by the horologist Thuret, with priority disputed at the time by Hooke and by the abbé Hautefeuille). Paired with a balance wheel, it forms a torsional oscillator — the timekeeping heart of every mechanical watch. The trouble is isochronism: the period should not depend on the amplitude of oscillation, but in practice it does, because the spring does not behave as an ideal torsion element. Pierre Le Roy discovered empirically in the eighteenth century that isochronism could be coaxed out of a spring by choosing its endpoints suitably. Nobody knew why, and nobody had a theory: the exact equations of a deformed elastic spiral are hopeless to integrate directly.

Phillips — an elastician who came to horology from the theory of railway carriage springs — found the way around. His memoir was examined for the Académie des Sciences by a commission of Mathieu, Lamé, and Delaunay, whose report praised it for making it possible “to substitute simple rules for the trial-and-error by which the makers of chronometers seek to obtain the isochronism of the oscillations of the balance.”

What Phillips proved

The engine of the whole memoir is one identity. Clamp both ends of the spring (as the collet and stud in fact do), rotate the balance through an angle , and let be the flexural rigidity of the blade, its length. Cutting the spring at an arbitrary point and balancing moments, then integrating along the blade, Phillips obtains

where is the restoring couple, is the reaction force at the balance pivot, and is the center of gravity of the spring. If the second term vanishes, the torque law is exactly linear, , and the swing time comes out as

( = moment of inertia of the balance) — independent of amplitude. Isochronism, then, reduces to killing the term , and Phillips identifies two sufficient conditions: keep the spring’s center of gravity on the balance axis, or make the pivot reaction vanish. He then proves a lovely equivalence: the pivot reaction vanishes if and only if the change of curvature is uniform along the blade — circles deform to circles, the coils “breathe” concentrically, exactly what one observes in a good chronometer spring.

The centerpiece is the terminal-curve problem: shape the end curve of the spring so that, under this uniform-curvature deformation, the center of the body coils stays put for every working . After a page of integration by parts and a linearization in , four conditions collapse to two statements about the center of gravity of the terminal curve alone. If the curve has developed length and joins the coils (radius ) at the point :

1. — the center of gravity lies on the radius perpendicular to the junction radius;
2. — its distance from the axis is the third proportional to the curve’s length and the coil radius.

These are the Phillips conditions. Along the way he proves that a circular arc can never serve as a terminal curve (explaining why the old practice of simply using a piece of the last coil had been abandoned as “vicious”), and he shows the conditions are independent of the blade’s cross-section, of the total spring length, and of the relative placement of the two end curves — which is precisely what leaves Le Roy’s length-adjustment method available as a final regulating trim.

The memoir then keeps giving. The terminal-curve conditions automatically place the center of gravity of the entire spring on the axis, by an elegant moment cancellation ( for every opening angle ). Pushing to second order yields two further integral conditions, which Phillips converts — via the theorem of Pappus–Guldin — into statements about volumes of revolution that a draftsman can check from a drawing: a numerical-verification protocol, vintage 1860. There is a fully practical chapter on constructing compliant curves graphically, an analysis of the flat (watch) spring showing it can be isochronous only for small arcs unless terminated by a curve brought back toward the center — the mathematics of the Breguet overcoil — a proof that the work of deforming the spring depends only on its volume and the imposed strains, a temperature chapter showing the conditions are preserved under thermal dilation (and that thermal stress at the attachments vanishes if the terminal curve springs from the axis itself), and a closing analysis of Coulomb friction proving the beautiful classical result: friction shortens the outgoing half-swing and lengthens the return by exactly equal amounts, so the period between equilibrium passages is unchanged — friction decays the amplitude (linearly, a hallmark of dry friction) without perturbing the rate.

My favorite passages are the last ones, where Phillips confronts the theory with the workshop literature — Moinet, Saunier, the Tribune chronométrique — and shows the accumulated craft lore falling out as corollaries. A chronometer-maker in Dieppe, M. Jacob, found that one of Phillips’ theoretical curves was identical to a curve he had reached by trial and error a dozen years earlier and had used in instruments delivered to the navy. Theory recovering craft, then extending it with drawable rules.

The translation, and what it taught me about AI and old mathematics

The memoir is freely available on Numdam, but only in French, and only as a scan whose OCR text layer mangles the mathematics badly — 1860s typography, accents, braces spanning multiple lines. The translation was produced in collaboration with Anthropic’s Claude (the Fable 5 model), and the workflow turned out to be a case study in what such tools can and cannot do.

What the AI did well: the prose translation (Phillips’ mathematical French is formal and regular), the LaTeX typesetting, and — importantly — rederivation. Where the OCR was hopeless, the model reconstructed equations from the surrounding argument and verified the reconstructions symbolically (in SymPy); several nontrivial checks, such as the exact reduction of the two brace-grouped expressions for the moment at equation (46), or the endpoint conditions on , were confirmed this way.

What it could not do, even when told the OCR was wrong: faithfully read a multiline displayed equation from the original. Every reconstruction was mathematically coherent, but coherence is not fidelity — the model substitutes derivation for transcription, and the result can differ from what Phillips actually printed (a missing equation here, a sign convention there, a coefficient absorbed into the wrong factor). In this project, essentially every displayed equation spanning more than one line required checking against the page images and manual retyping, and several equations present in the original had to be restored outright. The division of labor that finally worked: machine for derivation, symbolic verification, and typesetting; human eyes on the actual page for everything the page says. If you are contemplating a similar rescue of an old mathematical text, budget for that human pass — it is not optional.

The current version of the translation runs to about 36 pages: all fourteen sections of the memoir, all 89 numbered equations under their original numbers, the original figures, translator’s notes where conventions need flagging, and an appendix with annotated diagrams decoding Phillips’ notation (his accents mark evaluation at the endpoints, not derivatives; the letters , , and each do double duty).

Links. The original: J. Math. Pures Appl. (2) 5 (1860), 313–366, on Numdam. The translation: PDF and a 12 page summary summary of Phillip’s memoir, including many details left out of this blog post.