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I was wondering if anyone knows if Foucault actually gave a mathematical proof of his observations related to the motion of the pendulum. And if he didn't prove it, who described the motion of the pendulum first? I'm doing an historical research, but in the original article I haven't found the proof.

Yes. Léon Foucault in 1851 published in the Comptes rendus a paper Démonstration physique du mouvement de rotation de la Terre au moyen du pendule detailing his experiment and the mathematical justification for it. (“Physical demonstration of the rotational movement of the Earth by means of the pendulum” is the English translation of that title.)

He did not write any equations to describe it directly but instead took it to a limiting case (if the pendulum is at the North Pole) and described that correctly without equations. Then he described how at the equator the pendulum must not rotate, and so he reasoned that therefore up in France it’s got to be somewhere in the middle. He did not derive the sine-of-the-latitude dependence in this short note which connects the two, but he did end on a quick note connecting his effect to Poisson's 1837 Recherches sur le mouvement des projectiles dans l’air, “Research on the movement of projectiles through the air,” which concerns the deviation of the movement of bullets and cannon balls by the Coriolis force, and which has real equations: if you know that this is the only cause of the precession of Foucault’s pendulum, then the sine-of-the-latitude dependence should follow from that directly. He also indicates that this not-quite-a-full-rotation-per-day effect is visible in his experiment, too.

Interestingly he also resolves a question in this letter which I myself had long had, whether the rope can torque the pendulum and thus somehow alter the plane of rotation that way. Foucault says,

L’indépendance du plan d’oscillation et du point de suspension peut être rendue évidente par une expérience qui m’a mis sur la voie et qui est très-facile à répéter. Après avoir fixé, sur l’arbre d’un tour et dans la direction de l’axe, une verge d’acier ronde et flexible, on la met en vibration en l’écartant de sa position d’équilibre et en l’abandonnant à elle-même. Ainsi l’on détermine un plan d’oscillation qui, par la persistance des impressions visuelles, se trouve nettement dessiné dans l’espace ; or on remarque qu’en faisant tourner à la main l’arbre qui sert de support à cette verge vibrante, on n’entraîne pas le plan de vibration.

translated,

The independence of the plane of oscillation at the point of suspension can be rendered evident by an experiment which put me on this path, and which is very simple to replicate. After having fixed, on the shaft of a turntable [lit. tower?] and in the direction of the axis, a rod of steel round and flexible, it is put into vibration by the displacement of its position from equilibrium and then abandoning it to move on its own. Thus it determines a plane of oscillation that, by the persistence of visual impressions, is located clearly-drawn through space; but we remark that making turns, by hand, of the shaft that supports this vibrating rod, we do not entrain the plane of vibration.

So that’s really cool.

I think this all raises an interesting point which is how much “mathematics” you consider to be “mathematics.” There are no equations in this paper but there is a beautiful piece of abstract reasoning; ‘take the situation to the North Pole, then this law of inertia must force the pendulum to swing back and forth in a fixed plane, while the Earth if it is truly rotating must rotate out-from-under it and thus someone standing on the Earth will see the pendulum rotate once per day—this effect has vanished completely at the equator but must be partially visible in the middle.’