After a planet has been found that satisfies the perturbations of the known planet, is it not possible to find another that produces the same as it does? Is it not possible to find a system of planets that replaces the other? If this takes place when trying to solve the question exactly, isn't there a mistake when it comes to approximate formulas? To what degree of approximation should we bring our formulas so that the mistake disappears? These are the questions that mainly form the object of this dissertation. We will deal first with the movement of the centre of gravity of the stars, then with their figures and with their movement around the same centre.

Our method does not give limits as close as those of Lagrange. To remedy this, when one wants to approximate the roots, it will be convenient to use Daniel Bernoulli's method (that of recurrent series) as a first approximation. The method we have presented, notwithstanding this minor inconvenience, is, I believe, the simplest imaginable. Considering the advantages of so expeditious a method, I shall perhaps treat this subject in more detail on another occasion for the benefit of practical engineers and the students of the Escola Militar.

The author, after referring to a previous memoir on the same subject, presented by him to the French Academy, proposes to himself the problem of determining the function $\phi(x)$ which (f, F being given functions, and the limits α, β of the integration being also given) satisfies the equation

$\int _{\alpha}^{\beta}f(x, \theta) \phi(x + \theta) d\theta = Fx$.

He observes, that, unlike the methods employed in his former memoir, and the solutions there employed, which are quite rigorous, the methods of the present memoir depend upon developments into series, the strictness of which has been contested by some mathematicians; but that passing over these difficulties, he has solved the famous problem, the solution of which has been vainly sought after for the last two hundred years, because on the above-mentioned equation depends the integration of the generally linear equation of any order whatever of two variables, and consequently the whole Integral Calculus.

With the increasing development of civilisation and the extension that our knowledge has taken, such a bond of brotherhood has been established between nations, that there is no longer anyone who can be content with the literature of their own country, and that, crossing the physical limits of the place in which they were born, do not endeavour to gather, in other climates and under other skies, the flowers and fruits of the tree of science which the human spirit has sown, everywhere on earth, as traces of its divine origin. How many times has man truly fond of literary art felt, in moments of solitude, when their mind needs a diversion after very serious reflections, and a quiet occupation, the desire to have in hand a meeting of the most perfect literatures all nations possess, in order to let their thoughts wander and find feelings that correspond to the dispositions of their spirit?

From the questions written above, as well as from more general ones, I have given a large number of solutions, based either on convergent series, or on entirely series-independent methods, always deducing as particular cases of my formulas the solutions by Liouville. This memoir (perhaps the least important of the ones I have written) was presented to the Institute in France, which has not yet wished to give an opinion, with Liouville as the rapporteur; what I believe I have the right to attribute to 'petite jalousie', having the celebrated Lamé, one of the commissioners whom I urged that the commission give his opinion, wrote a letter to me in which he said:

I have read your memoir: it proves that you are a good analyst; I greet you as such and think that my colleagues will not be of a different opinion.

No one until today has tried to determine the movement or vibration of the air, considering things as they are in Nature: everyone has abstracted from gravity except Poisson, who considers it in case where air moves in a straight tube. He then tried to apply his solution to the case where air has three dimensions, but as I demonstrate, his reasoning is entirely erroneous. His reasoning is this one - the speed of sound is proportional to the square root of the quotient of the elasticity and density of air, in the case where forces are not considered; now, in the case of the latter, he says, this relationship does not change; so the speed of sound is constant. But whence did he deduce the above law of the propagation of sound? From a certain differential equation, the form of which changes profoundly from the other, to which he intends, applicable to the conclusions drawn from the first, which is a monstrous absurdity, which, if only by an absurdity, he could have escaped. So, for the first time, I am dealing with the problem with all rigour. Geometricians who have dealt with the theory of sound, in order to simplify the formulas, assume that all vibrations of air molecules are very small. I assume any: the equations are no longer linear, but I completely resolve the issue. They also assume that the air temperature is always uniform. Poisson was the only one who considered motion in the case of variable temperature, as it exists in nature, but he deals only with motion in a straight tube. I, however, consider the case of curvilinear tubes and any laws of heat variation.

Loving above all else the sciences which have as their object the study of nature, I have determined study mathematics in order to know them better. But when one begins this study, one constantly stops before the insurmountable difficulties offered by the integral calculus. If there is however something really attractive, it is the study of this branch of Analysis. Do you want to know the theory of the distribution of heat on the surface of conducting bodies? You stop before the obstacles presented to you by the integral calculus. Do you want to know the movement of heat in the interior of solid bodies of any figure whatsoever? Here again it is the integral calculus which obliges you to stop almost at the beginning of your study. Do you want to know the propagation of motion inside bodies? the vibrational state of their molecules? tidal theory? the figure of the planets which deviate significantly from a spherical shape? the law of variation of their densities, etc. etc.? You find integral calculus before you immense, impassive, insurmountable, resisting the combined efforts of all the distinguished geometers of Europe, not a single one of whom has been able to prevent himself from struggling, at least for some time, hand to hand with it. When you see all these theories depending on this calculation and this calculation itself reduced to a single problem, there is something that pushes you, which pulls you along almost in spite of yourself.

He was audacious and fought with insistence for his scientific recognition in Europe. His effort was fruitless, though. This paper is concerned with the relations between Gomes de Souza's mathematical works and the controversy about the legitimate use of divergent series in the nineteenth century. It brings a brief introduction of Gomes de Souza. Then, it succinctly describes the characteristics of the polemic about the use of divergent series. Finally, it reveals the critical considerations of these authors concerning Gomes de Souza's Works.