## Determinatio attractionis: quam in punctum quodvis positionis datae exerceret planeta, si eius massa per totam orbitam, ratione temporis, quo singulae partes describuntur, uniformiter esset dispertita.

Göttingen: Dieterich, 1818.

First edition, very rare separately-paginated offprint, preceding the journal appearance in *Commentationes Societatis Regiae Scientiarum Gottingensis* (vol. 4, pp. 21-48) by two years. “In his first years at Göttingen, Gauss experienced a second upsurge of ideas and publications in various fields of mathematics. Among the latter were several notable papers inspired by his work on the tiny planet Pallas, perturbed by Jupiter … [including] *Determinatio attractionis quam in punctum quodvis positionis datae exerceret planeta si eius massa per totam orbitam ratione temporis quo singulae partes describuntur uniformiter esset dispertita* (1818)” (DSB). In this paper, Gauss’s last important contribution to theoretical astronomy, he showed “that the secular variations which the elements of the orbit of a planet would experience from another planet which disturbs it are the same as if the mass of the disturbing planet were distributed in an elliptic ring coincident with its orbit, in such a manner that equal masses of the ring would correspond to portions of the orbit described in equal times” (Dunnington, p. 111). (The ‘secular variations’ are those which occur over a period of time that is long compared to the orbital period of the planet; the ‘elements’ of the orbit are the parameters which determine it, such as its eccentricity – the extent to which it differs from a circle – and the inclination of the orbit to the plane of the ecliptic.) Gauss showed that the attraction caused by such a ‘Gaussian ring’ could be expressed in terms of elliptic integrals, and he related the evaluation of these integrals to the ‘arithmetic-geometric mean’ (see below). “In a letter, dated April 16, 1816, to a friend, H. C. Schumacher, Gauss confided that he discovered the arithmetic-geometric mean in 1791 at the age of 14. At about the age of 22 or 23, Gauss wrote a long paper describing his many discoveries on the arithmetic-geometric mean. However, this work, like many others by Gauss, was not published until after his death … Gauss obviously attached considerable importance to his findings on the arithmetic-geometric mean, for several of the entries in his diary, in particular, from the years 1799 to 1800, pertain to the arithmetic-geometric mean” (Almkvist & Berndt, p. 586). *Determinatio attractionis* was the only work Gauss published on elliptic integrals, although much more remained in manuscript and was published after his death (this showed that he had anticipated some of the later work of Abel and Jacobi in this field). It perfectly illustrates the great breadth of Gauss’s interests and expertise, combining as it does studies in both pure and applied mathematics: Gauss did not recognise the barrier between these two disciplines that exists today. “In 1813 on a single sheet appear notes relating to parallel lines, declinations of stars, number theory, imaginaries, the theory of colors, and prisms” (DSB). No copy of the offprint listed on ABPC/RBH.

Already well known for contributions to algebra and number theory, Gauss (1777-1855) made his dramatic entry into the astronomical world at the age of 24. Having graduated from the University of Göttingen some three years earlier, he had returned to his native city of Brunswick to continue the intense mathematical studies for which the Duke of Brunswick had been supporting the precocious youth since he had been in his mid-teens.

“In January 1801 Giuseppe Piazzi had briefly observed and lost a new planet [the asteroid Ceres]. During the rest of that year the astronomers vainly tried to relocate it. In September, as his *Disquisitiones* *arithmeticae *was coming off the press, Gauss decided to take up the challenge. To it he applied both a more accurate orbit theory (based on the ellipse rather than the usual circular approximation) and improved numerical methods (based on least squares). By December the task was done, and Ceres was soon found in the predicated position. This extraordinary feat of locating a tiny, distant heavenly body from seemingly insufficient information appeared to be almost superhuman, especially since Gauss did not reveal his methods. With the *Disquisitiones* it established his reputation as a mathematical and scientific genius of the first order” (DSB).

Just one year after the discovery of Ceres, on March 28, 1802, another new planet (asteroid) was discovered serendipitously by Wilhelm Olbers; it was later named Pallas. “By now, Gauss was aware of the fact that his assumption that these celestial bodies move in elliptical orbits was not strictly true. This was revealed by the distribution of the errors between the new observational data being steadily accumulated, and by the theoretical predictions based on the best elements that he was able to compute from those data. He rightly interpreted these perturbations as being due to the gravitational attraction of other planets – particularly Jupiter, the nearest and most massive – pulling Ceres and Pallas out of their otherwise elliptical paths. His first efforts to develop a new theoretical approach, which made use of some of his early mathematical researches, had to be abandoned because it gave rise to an impossibly large computational burden. An alternative involving interpolation of the perturbation function, which he developed in 1805, proved to be more tractable and became the foundation of his logically coherent and mathematically elegant theory of the motion of the celestial bodies which was ultimately published in Latin four years later [*Theoria motus corporum coelestium in sectionibus conicis solem ambientium*, 1809) … After completing that treatise, Gauss continued working on the general theory of Pallas’s perturbations and delivered a disquisition on it to the Royal Society of Sciences in Göttingen on 25 November 1810. This research presented the greater challenge to his mathematical ingenuity because Pallas’s orbit was more elliptical than that of Ceres, and inclined at a greater angle to that of Jupiter” (Forbes, pp. 173-4).

In his calculation of the orbit of Ceres, and in his detailed exposition of his methods of orbit calculation, the *Theoria motus corporum coelestium* (1809), Gauss used the standard method of determining the secular variations of the elements of any planet, which is based upon the development of the perturbing function into an infinite series whose successive terms involve continually higher powers of the eccentricities and the mutual inclination. “This method possesses two advantages. The first is that when an extreme degree of accuracy is not required, so that higher terms of the development may be disregarded, it is the simplest method available; and, in the second place, since the coefficients of all terms are general literal expressions, the change produced in the value of any variation by a change in the assumed values of one or more of the elements can readily be ascertained by a simple substitution of the more accurate values. On the other hand, this method possesses the disadvantage that the complexity of the expansion grows rapidly greater as the order of the included terms is increased, so that a slight increase in the desired accuracy greatly increases the labor of the computation. The integral methods, founded upon the celebrated theorem of Gauss [that established in the present paper], are wholly free from this latter disadvantage, for if it is desired to include all terms to the twenty fourth order this can be done by a computation which is less than twice as long as that required when the approximation is stopped at terms of the eleventh order …

“In the original memoir of Gauss the determination of the secular terms of these expressions was given a geometrical aspect … the expressions giving the secular variations are seen to be the same whether these are derived from the moving planet or from the elliptic ring … The work of Gauss contains no application to the determination of secular variations nor are all the formulas necessary for this purpose there developed; the first integration alone is effected, and it is shown that … each of the complicated integrals may be made to depend upon elliptic integrals whose values Gauss obtained by the introduction of a new algorithm called by him the arithmetico-geometrical mean” (Doolittle, pp. 39-41).

The theory of perturbations that Gauss described in this paper was cast into a more explicit form, more useful for astronomers, by George W. Hill (On Gauss's method of computing secular perturbations. *Astronomical Papers of the American Ephemeris and Nautical Almanac* I (1882), pp. 317-361). Hill writes: “Gauss investigates the expressions for the components of the attraction of a certain species of elliptic ring on a point, which can be advantageously employed in computing the secular perturbations of a planet … This method merits attention because with it we can secure almost absolute accuracy, at the cost of a comparatively small amount of labour. Moreover, it is capable of being applied with success to all asteroids, and even to such refractory cases as the periodic comets.” Hill, whom Simon Newcomb called ‘the greatest master of mathematical astronomy during the last quarter of the nineteenth century,’ himself applied Gauss’s theorem to the perturbations of Venus on the orbit of Mercury.

As already noted, the calculation of the gravitational effects of a ‘Gaussian ring’ led to elliptic integrals which Gauss evaluated using the arithmetic-geometric mean, which he discovered in his early teens. If *m* and *n *are two positive numbers, their arithmetic and geometric means are

*m’* = ½(*m + n*) and *n’* = √*mn*,

respectively. One can then form the arithmetic and geometric means of *m’* and *n’*, denoted by *m”* and *n”*; and this can be continued *ad infinitum*. It is not difficult to show that the two sequences *m, m’, m”,* … and *n, n’, n”,* … both converge to the *same* limiting value, which is called the ‘arithmetic-geometric mean’ (AGM) of *m* and *n*, and which Gauss denoted in the present paper by *μ*(*m,n*).

Just before his 19^{th} birthday, Gauss began a ‘mathematical diary,’ in which he recorded his mathematical discoveries over the next 20 years. The 98^{th} entry, dated May 30, 1799, relates the AGM to the value of a certain elliptic integral, namely the integral of

1/√(1 – *x*^{4})

taken from *x *= 0 to *x* = 1, which Gauss denoted by *ϖ*/2 (it is called an ‘elliptic integral’ because it also arises in the calculation of the arc-length of an ellipse). The entry reads:

‘We have established that the arithmetic-geometric mean between 1 and √2 is *π/ϖ* to the eleventh decimal place; the demonstration of this fact will surely open an entirely new field of analysis.’

Incredibly, Gauss had noticed, by means of a direct numerical calculation (performed by hand, of course), that both *μ*(1,√2) and *π/ϖ* are equal to 1.19814023473 … to 11 decimal places – he used power series techniques to calculate the approximate value of the lemniscate integral, and hence *ϖ* = 2.662057055429 …, and of course he knew the value of *π* = 3.141592653589 …

The integral of 1/√(1 – *x*^{4}) is the arc-length of one-fourth of the ‘lemniscate,’ the curve in the *xy*-plane with equation

(*x*^{2} + *y*^{2})^{2} = *x*^{2} – *y*^{2};

this is a curve shaped like the modern symbol for infinity. Interest in the lemniscate and its arc-length dates back to the earliest days of the calculus, when the two Bernoulli brothers, Jacob and Johann, studied them in the late 1690s. From his notebooks, we know that Gauss studied this integral and the lemniscate through the works of Euler, which he began to read by the age of 20.

Gauss gives a proof of the *exact *equality *μ*(1,√2) = *π/ϖ* in the present paper. By means of the substitution *x* = sin *θ*, the lemniscate integral is transformed into the integral of

1/√(cos^{2}*θ* + 2 sin^{2}*θ*)

taken from *θ* = 0 to *θ = π*/2. This suggests considering the integral of

1/√(*m*^{2}cos^{2}*θ* + *n ^{2}*sin

^{2}

*θ*)

over the same range, for any two positive numbers *m, n*; denote this integral by *I*(*m,n*). Then we have to show that *μ*(1,√2) = *π*/2*I*(1,√2).

Gauss now shows, by a very ingenious substitution of *θ* by another variable *θ’*, that *I*(*m,n*) = *I*(*m’,n’*), in the notation introduced earlier. Iterating, it follows immediately that

*I*(*m,n*) = *I*(*m’,n’*) = *I*(*m”,n”*) = …

Passing to the limit in the sequences *m, m’, m”,* … and *n, n’, n”,* …, we see that each of these integrals is equal to *I*(*μ,μ*), where *μ* is the AGM *μ*(*m,n*). But it is trivial that *I*(*μ,μ*) = *π*/2*μ*. We have proved that *I*(*m,n*) = *π*/2*μ*(*m,n*). The case *m* = 1, *n* = √2 gives what we wanted.

As Gauss predicted, his discovery that *μ*(1,√2) = *π/ϖ* went well beyond this one numerical relationship. The ‘new field of analysis’ that opened up in connection with this proof led him well beyond the study of elliptic functions of a single real-valued variable, and into the realm of functions of several complex-valued variables. These include the special class of such functions known as ‘theta functions,’ which provide a powerful tool used in a wide range of applications throughout mathematics.

This paper was read on January 18, 1817, and published in Vol. 4 of the *Commentationes Societatis Regiae Scientiarum Gottingensis*, which appeared in 1820.

Almkvist & Berndt, ‘Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, π, and the *Ladies Diary*,’ *The American Mathematical Monthly* 95 (1988), pp. 585-608. Doolittle, ‘The Secular Variations of the Elements of the Orbits of the Four Inner Planets Computed for the Epoch 1850.0 G.M.T.’ *Transactions of the American Philosophical Society*, New Series, 22 (1912), pp. 37-189. Dunnington, *Carl Friedrich Gauss. Titan of Science*, 2004. Forbes, ‘The astronomical work of Carl Friedrich Gauss (1777-1855),’ *Historia Mathematica* 5 (1978), pp. 167-181.

4to, pp. 30, uncut (variable spotting, heavier on the first few leaves). Original plain wrappers (a bit frayed at edges). Untouched in completely original condition.

Item #5312

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