# Clairaut's proposed correction (reported as “D'Alembert's, Clairaut's and Euler's corrections”) to the Newtonian inverse-square law of gravity

From A.P. Yushkevich, "Leonhard Euler, his life and work", in "Development of Leonhard Euler's ideas and contemporary science", Nauka, Moscow, 1988, 15--46 (translation from Russian is mine):

"One should note the Euler role in the theory of movement of the moon the moon's motion, which attracted attention of the scientific world because of the discrepancy between the observed orbit and the orbit calculated by Newton on the basis of his law of gravity. The discrepancy between observations and theoretical calculations was so large that for a while D'Alembert, Clairaut and Euler thought to introduce corrections to the law of gravity".

Update: Many thanks to Mauro ALLEGRANZA for an interesting reference, and to terry-s for a beautifully detailed answer.

This answers (with some explanation and references) the two questions, (a) Were any concrete corrections proposed (in the 1740s, to the law of gravitation)? and (b) Where can one read about it?

(a) Clairaut (but not Euler or d'Alembert) proposed in 1747 a correction to the inverse-square law, that is, an additive term depending on a higher power of the distance than the square, which he discussed in one place (p.337 of his 1747 paper cited below) as possibly an inverse fourth power, and in another (p.362) as possibly an inverse cube. The proposal was semi-quantitative and not fully 'concrete', as it still needed at least a coefficient (and a definite conclusion about the exponent), which Clairaut did not give. (The source of the quotation given in the question is mistaken as a matter of history: only Clairaut, not the other two, made any proposal for 'correction' of the gravitation law.) Clairaut later (1749) reconsidered and retracted his correction proposal. D'Alembert stated his attitude (retrospectively) in his 'Recherches sur differens points importans du systeme du monde' (part 1, 1754) (in the Discours preliminaire): that he had thought some extra force such as magnetism might be involved, but would not propose changing the the law of gravity on the basis of a single phenomenon.

(b) One can find original sources (below) for Clairaut's 1747 proposal, for part of the controversy that it evoked, and his 1749 retraction (the primary sources are only in French, see below for comment on English secondary sources).

b.1 -- for Clairaut's paper with his proposal to correct the law of gravitation (proposal itself starts at p.337):

15 Nov. 1747: "Du système du monde dans les principes de la gravitation universelle", Mém.Acad.Roy.Sci. 1745 (1749) pp. 329-364

(The date-marks of the paper are strange: it was read in 1747, but judged sufficiently important to be inserted into the memoirs for the year 1745 -- which were then still in the press -- and the whole eventually appeared in 1749.)

b.2 -- for some of the controversy that resulted: Clairaut's reply to a critique from Buffon.

17 Feb.1748: "Réponse aux réflexions de M. de Buffon sur la loi de l'attraction, et sur le mouvement des apsides", Mém.Acad.Roy.Sci. 1745 (1749) pp. 529-548

b.3 -- for Clairaut's statement of retraction of his proposal to correct the law of gravitation:

b.4 -- for Clairaut's theory of the moon's motion in which he explained that he had reached the revised conclusion (p.83) that "it would not be useful to seek any other cause of the inequalities of the movement of the Moon than solely the attraction that is inversely proportional to the squares of the distances":

'Théorie de la Lune déduite du seul principe de l'attraction réciproquement proportionelle aux quarrés des distances, par M. Clairaut' ('Theory of the moon deduced solely from the principle of attraction inversely proportional to the squares of the distances').

It is notable that many mis-statements have been in circulation for a long time in secondary accounts and commentaries, about what Clairaut and the others actually did and what they claimed. I have not yet seen any account in English that avoids all misrepresentation of the original sources.

Amongst the better brief accounts is a summary of the 1740s-1752 controversy about the lunar apogee on pages 11-12 of C A Wilson's book of 2010 about the 'Hill-Brown theory of the Moon's motion'. Many of the points made in this summary are historically good, but it does appear to misrepresent Clairaut's own account of what he actually had done to convince not only himself, but others later on, that the lunar problem did not after all justify any correction to Newton's gravitational law.

C A Wilson, like some other commentators, described Clairaut's improved calculation as a 'second-order' calculation. But from pages 29-31 of his paper on the theory of the moon, it can be seen that Clairaut expressed his results numerically, and not as a power series of terms of increasing order (which was really a contribution of d'Alembert). Clairaut's improved calculation, as indicated on the same pages, did not concern the 'order' of approximation or adding terms of higher order/power, rather it proceeded by using first a naive approximation for the moon's radius vector, based only an expression for an ellipse with a rotating axis, and then a better approximation that also took into account the effect of the main perturbations of the radius vector.

It may be of interest to add, that the reason why such an approximation had to be made at all, lay in the form of differential equation that Clairaut set out to solve. It was a form that can be given schematically as $$d^2 u / dv^2 + k u = f(u, \Omega, \Pi)$$, where u is the reciprocal radius vector, v is the moon's (angle of) anomaly, k is a constant, and $$\Omega$$ and $$\Pi$$ are perturbing forces, expressed in terms of u and other quantitities.

The methods of the time for solving such equations required that references to variables except in the derivatives had to be known and expressed in terms of independent variable v and no other (see, for early-19th-c. methods of solution, Airy's 'Mathematical Tracts', editions from 1826 onwards). But the relation between u and v was the very relation (unknown) that was intended to be found by solving the equation. So the overall solution could only be approached iteratively by starting from a rough account or guess for u in terms of v , and then incorporating an improved solution into the next iteration. Clairaut's paper (e.g. pp.29-31) shows two such iterations, giving an improvement in the coefficient of the apogee motion. But it did not give the apogee motion to full accuracy, and Clairaut at that point announced that he would dispense with the work of further repetition of the 'same calculations' (i.e. further iterations) by adopting straight away for the apogee motion the 'true value given by the observations'. This can be read at page 31 of his paper on the theory of the moon.