Joaquim Gomes de Souza's Miscellaneous Integral Calculus

Joaquim Gomes de Souza's Miscellaneous Integral Calculus: Plus a Memoir by the Author on sound and a foreword by M Charles Henry was published in Leipzig in 1882. Written in French, it contains the following eight memoirs by Gomes de Souza:

(1) Dissertation on general methods of integration.

(2) Addition to the Memoir on general methods of integration.

(3) On the determination of the constants, which, in the problems of mathematical physics, enter into the integrals of the partial differential equations, according to the initial state of the system.

(4) Proof of some general theorems for the comparison of new transcendental functions.

(5) Dissertation on a theorem of integral calculus and its applications to the solution of problems in mathematical physics.

(6) Memoir on the determination of the unknown functions which enter under the sign of definite integration.

(7) On the analogy between linear differential equations and ordinary algebraic equations.

(8) Memoir on sound.

We give a translation of the Preface by Charles Henry.


For most scholars life is just one long line of reasoning. They know neither the seductions of the world, nor the intoxications of art, nor the generous utopias. A date, a few titles of work and functions, then another date and their history is made. Their life is such a desert that one is sometimes tempted to curse the passion that has withered them.

It is not the same with the eminent geometer whose posthumous work we are publishing today: Joaquim Gomes de Souza inspires both through his work and his personality.

Born in Brazil on 15 February 1829, in the province of Maranhào, where his father, Major Ignacio José de Souza, owned properties, he showed from his earliest childhood a decided taste for psychological and physical studies. Unfortunately his family destined him for a career in arms and he had, without the slightest vocation, to enlist as a cadet after having attended the Military School for a year. It was a torture for the young scholar, but a torture that did not last long; unable to bear the fatigues of the profession, he finally obtained permission from his father to devote himself to medicine.

In 1844, he therefore entered the School of Rio de Janeiro. Predictably, he shone with off-line talent there. However, he was studying mathematics at the same time and with such ardour that after having passed the medical school examinations brilliantly, he asked to take the engineering examination. Its success was also brilliant.

That's not all. Engineer and doctor, he adored art and not only the art of a country, a literature of a few parochialisms, but literature in all its forms and in all its universality. Later in the preface to a book, to which we will return, it began thus: "With the increasing progress of civilisation and the extent that our knowledge has taken, there has been established such a bond of fraternity between the nations that there is no longer anyone who can be satisfied with the literature of his own country and who, crossing the physical limits of the place where he was born, does not strive to pick in other climates or under other skies those flowers and those fruits of the tree of science which the human spirit has strewn all over the earth as traces of its divine origin." It was in the hours of leisure torn from these hard studies that he acquired on the monuments of French, English, German and Italian literature this erudition of which he was to leave the written testimony.

In 1854, he came to Europe: there he made illustrious friendships, received praise and some encouragement. Professor Stokes presented on his behalf to the Royal Society of London, in the session of 12 June 1856, a short note on the determination of the unknown functions which come under the sign of definite integration. It is the rapid summary of a memoir which had been submitted on 18 June 1855, to the Academy of Sciences of Paris and sent back for the examination of MM. Liouville, Lamé and Bienaymé. The same commission, to which M Cauchy was attached, also had to judge a thesis on mathematical analysis and a booklet on sound, presented on 16 July of the same year. On 9 June 1856, the author was allowed to read an addition to his first memoir; he could not, however, despite his requests, obtain, before his departure from France, a report on his work. The following year, he sent from Rio de Janeiro a reworking and an extract of his first work: the extract was deemed too long to find a place in the Comptes-Rendus; from then on there is no longer any trace of Gomes de Souza in the publications of the Academy.

Most of these works will be corrected and supplemented in this publication. Its origin has been traced by the author himself in this quote which is a piece of high philosophy: "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."

This unique and important problem of finding the integral of any differential equation is the subject of the first memoir.

All the others, except one, are only developments, clarifications or applications of this first work, applications which, in the plans of the author, were to exceed the limits of this collection, since we do not find there, for example, a Memoir on the state of vibration of bodies, to which reference is made on page 163 of this book. The penultimate work entitled: On the analogy between linear differential equations and ordinary algebraic equations is a development of the ideas that Libri expounded on this subject in Crelle's Journal and in Liouville's Journal.

Probably one does not expect the publisher to provide a formal critique of these works. A skill proven by long researches in mathematical physics could alone allow such a judgment and still it should bring to its appreciation an extreme circumspection. Who better than MM. Liouville and Lamé could formulate a definitive opinion on this work? We will therefore leave it to the reader to judge, contenting ourselves with a few formal observations.

The style is not without flaws. There are somewhat bold inversions for mathematical language and after the formulas are very often set aside explanations which would be, for the reader, better placed in front of the equation. One can also note a certain inexperience with conventions of language (for example, constants taken at will replace everywhere in the first sheets the given expression of arbitrary constants) finally philological incorrectnesses, all faults which very rarely harm the understanding of the text and which will moreover be corrected for the most part in the Errata.

These years 1857-1858 mark a singularly active period in the life of our scholar. While preparing the publication of his scientific memoirs, he taught at the Faculty of Mathematics of Rio de Janeiro and sat as a deputy in the Legislative General Assembly. Distinguishing himself here and there as elsewhere, he delivered during the course of the two parliamentary sessions remarkable speeches on scientific, economic and legislative questions. These speeches were printed in the Annals of the Brazilian Parliament; we will be allowed to refer to this collection which we have not been able to consult.

It was also while in Rio de Janeiro that he published, in 1859, in Leipzig, under the title of Universal Anthology, a highly original compilation prepared in Germany during his previous trip, a "choice of the best lyric poetry of various nations in the original languages." The collection does not have a thousand pages and yet seventeen literatures are represented in it. English literature occupies the largest place; then come in order of importance Germany, France, Italy, Portugal; Spain, Poland, Russia, modern Greece, Hungary and Serbia, Sweden and ancient Greece, Holland, Denmark, Bohemia, and finally Latin literature. The idea of choosing poetry was good; Is not poetry a much more living incarnation of the soul of a people than prose? The idea of choosing more particularly lyric poetry, this highest form of poetic inspiration, was excellent. But this framework excluded some famous poets: we must be grateful to the author for having made an exception in favour of Racine, Corneille, Molière, Caldéron. It is also believed that the Brazilian scholar could neither understand nor appreciate by himself the masterpieces of so many languages; he had recourse to competent collaborations and his choices, which were thereby less likely to be marked by individual preferences, are quite generally stamped with impersonality.

For the French anthology, it is from the work of Victor Hugo that the author borrows the most; then comes Béranger, of which there are no less than fourteen songs, among others the King of Yvetot, the Beggars, the Farewells of Marie Stuart, the Marquis of Carabas, the Good Old Woman, the God of Good People, the Memories of people, etc.: that's a lot. Then there are the most beautiful fables by La Fontaine, a few well-chosen pieces by Lamartine, typical scenes by Racine, the Epistle to Madame Du Châtelet and the Systems by Voltaire, a few fragments by Boileau, the Tableaux de la Nature and the Peinture de Dieu by Châteaubriand, the Young Captive and the last verses by André Chénier, Newton and the Dog by Abbé Delille, a piece from Cid, a piece from Tartufe, Falling Leaves by Millevoye, Memmo by Casimir Delavigne, in there is, beside real pearls, a little tinsel, and among masterpieces of passion, a few pieces of rather artificial enthusiasm. Then, without wishing to create here the Anthology of the Rejected, is it not permissible to deplore the absence, at this family celebration, of Dante's closest relative, Charles Baudelaire?

Goethe, Schiller, Lessing, Klopstock, Tieck, Bürger, Herder, Heine, Uhland, de Platen, Freiligrath, Lenau, Eichendorff, Geibel, Rückert, Chamisso represent Germany: the Song of Mignon, the King of Thule, the Bell, Lénore and how many other famous pieces in universal literature, are in their place to respond to the impatience of the reader.

English literature begins with a rich harvest in the work of Byron (The Bride of Abydos, Parisina, Manfred, etc.): then Thomas Moore, Robert Burns, Goldsmith, Walter Scott, Coleridge, Thomas Campbell, Wordsworth, Robert Southey, Longfellow, Milton, Tennyson, Shakespeare, Gray, Shelley each bring to the texture of this delightful poetic embroidery the characteristic of their geniuses.

The universally classic fragments of Dante (Ugolin and Françoise of Rimini) open Italian literature: Ariosto, Tasso, Petrarch, Manzoni, Casti, Parini, Ugo Foscolo, Grossi, Leopardi, Pindemonte, Guarini, Monti, Metastasio, Silvio Pellico, Maria Boiardo connect by an unbroken chain the modern feeling to the terrible passions of the thirteenth century.

In Portuguese and Spanish literature the choice was more easy and agreeable to the author than elsewhere. We will simply cite the names: Almeida Garrett, Francisco Manuel, Garçao, Bocage, António Diniz da Cruz e Silva, Caldas, Gonzaga, A Gonçalves Dias, J Bazilio da Gama, Camoens; in the Spanish part, among other anonymous authors, Jorge Montemayor, Espronceda, Lope de Véga, Caldéron de la Barca, José Zorrilla.

Russia is worthily represented by Pouchkine (Recklessness of the Bird), Lermontoff (the Lullaby), Lomonosoff, Joukowski (Song of the poor), Derzhavin (God), Batuschkoff, Delwick (Invocation to the Night), Ivan Koslof.

We find in the Polish Anthology the most beautiful ballads of Adam Mickiewicz, among other things Zaleski's famous Prayer for Poland, Slowacki's Song of the Lithuanian Legion and Gaszinski's Epistle to a Polish Mother.

One cannot congratulate the author too highly for having made a large place for the Popular Songs of Serbia and Modern Greece, without however neglecting poets like Const Rigas, Alexander Rangabis, the Soutsos brothers Alexandros and Panagiotis, Christopoulos etc.

The bohemian songs must have been for many a piquant revelation; there is in particular by Karel Winaricky a piece, L'Amant de Milina, of an unforgettable simplicity of rhythm.

Horace, Ovid, Catullus, Propertius, Martial, Tibullus, Theognis, Theocritus, Bion, Moschus, Meleager, Anacreon and some judicious extracts from the Anthology evoke the ancient world.

We will go no further in these arid enumerations; they are enough to recommend the collection as a truly useful and original book: perhaps they have taken us too far; but it seems to us that however diminished it may be, the personality of the author always appears behind these names and think of what price would be for scholars an anthology signed by La Fontaine, Diderot, Heine or Leopardi!

We are advancing rapidly towards the end of our task. At the end of one of the sessions of the Chamber of Deputies, Gnomes de Souza crossed the Atlantic again. He resumed his medical studies and regularly attended the Hôtel-Dieu Clinics in Paris. It was no longer activity, but fever. In 1863, already very ill, he embarked a third time for Europe. It was his last trip; he was never to see his native country again; on the 1st of June of this year, death came to strike him in London, in the midst of his labours and his friends, in the flower of age and talent.

About two-thirds of the present volume was printed, but rather imperfectly. We can judge by our Errata that the correction of the proofs leaves something to be desired: the author's exhaustion and weakened eyesight did not allow him to do better. Listen to this touching and grandiose declaration: "General methods of integration have been found, it is true, but my health is destroyed and my organisation is exhausted, the condition of my eyes perhaps no longer permitting me to devote myself to research where the deepest attention and the most assiduous perseverance are absolutely necessary. But if I am forced to stop here and if I am not allowed to see unrolled before my eyes the scene where my imagination had so often thrown away, I shall at least have the pleasure of having opened the way for others and of knowing that man will now be permitted to read more deeply into the bosom of the Creator." The rest of the work remained in proofs which continued to deteriorate and to make matters worse, the manuscript was lost. The last memoir, the one we have already had the opportunity from which to quote on the theory of sound, is published by us from the manuscript kept in the Archives of the Academy of Sciences in Paris.

Gomes de Souza also had in his portfolio a remarkable book on the social and philosophical sciences. Unfortunately we have no document on this work, which was, it seems, very advanced. It would be very much to be desired that some pious hand collect the fragments and shed some light on the intimate life of the author.

These few pages that the lack of information did not allow us to make more numerous are perhaps enough to show what a void the death of Joaquim Gomes de Souza created in the aristocracy of knowledge. A mathematician, he tackled the most difficult and urgent problem in science; a mathematician in love with experience and observation, he tasted the enchantments of art; the brutality of the social problem imposed itself on his heart and the poignant complexities of the philosophical problem did not leave him indifferent. Child prodigy, ideal nature, complex and sickly, it is from this family of intelligences which seem to have been created to mark the fundamental identity of all the varieties of knowledge, of these souls removed too early from their work, that the poetry of antiquity said dear to the gods and which across the distance of centuries and background evoke the melancholy figure of Pascal.

Last Updated November 2022