# Joaquim Gomes de Souza

### Quick Info

Born
15 February 1829
Itapecuru-Mirim, Maranhao, Brazil
Died
1 June 1864
London, England

Summary
Joaquim Gomes de Souza was the first Brazilian to be awarded a doctorate in mathematics. He had great difficulty in getting his memoirs published since leading mathematicians of the time failed to report on them.

### Biography

Joaquim Gomes de Souza was the son of Inácio José Gomes de Souza and Antonia Carneiro de Brito e Sousa. He was the grandson of José Antônio Gomes de Souza. The Gomes de Souza were of Portuguese origin and by the end of the 18th and beginning of the 19th century they played an important role in the politics and the economy of the Brazilian State of Maranhao. José Antônio Gomes de Souza, Joaquim's grandfather, was a wealthy man involved in the export trade. He had a military career becoming a lieutenant colonel and was also an alderman of the Chamber of São Luís. He was the official responsible for gauging weights and measures, taxation and distribution of foodstuffs. Joaquim's father, Inácio José Gomes de Souza, was a major in the army, owned land, made investments and held a high social position which made him financially well off and able to give his children a high quality education. Joaquim was born in Itapecuru-Mirim into a large family, the seventh of nine children. Itapecuru-Mirim is in the northeast of Brazil about 110 km south of San Luís. He was given the nickname of Souzinha when he was still a child and he was widely known by that name for the rest of his life. We shall refer to him as Souzinha in this biography.

Souzinha began his schooling in San Luís, the capital and largest city in the state of Maranhao. In 1841, when he was twelve years old, his father sent him to Olinda, a city on the northeast corner of Brazil which is on the Atlantic coast. It was an important city partly because of the Catholic Seminary there, which was built at the time of the Portuguese colonisation, and also for the Faculty of Law of Olinda which is now part of the Federal University of Pernambuco. Souzinha was sent there to join an elder brother who was studying in the city. The intention was that Souzinha would go on to study law at the Faculty of Law of Olinda but these plans all had to change when, after one year there, his brother died. His father thought he was too young to live in Olinda on his own so in 1842 he returned to San Luís to continue his education.

As we have seen, the Gomes de Souza family had strong connections with the military so, in 1843, when he was fourteen years old, Souzinha was sent to continue his studies at the Escola Militar of Rio de Janeiro. This College dates back to 1792, and was modelled on the Military Academy in Lisbon. It became the Royal Military Academy in 1811, then the Imperial Military Academy in 1822 when Brazil gained its independence. It became the Academia Militar da Corte in 1832 and, in 1840, it was renamed the Escola Militar. At the Escola Militar, Souzinha excelled academically in the engineering course, showing outstanding abilities in mathematics. The military training, however, did not go well since he was a thin boy with a poor physique. After two years at the Escola Militar, he realised that he did not want a career in the army and left the Escola Militar. In 1845 he took examinations in Latin and Philosophy and gained entry to the Faculty of Medicine in Rio de Janeiro. The Faculty of Medicine had been founded in 1808 and today is part of the Federal University of Rio de Janeiro.

In his first year at the Faculty of Medicine he enjoyed the courses he took on physics and chemistry. Keen to increase his knowledge of mathematics he began to study the subject from books, particularly differential and integral calculus, mechanics and astronomy. His passion for mathematics was such that after two years studying medicine he decided to give up his medical studies and concentrate on mathematics. Although he had never registered for a degree in Mathematical and Physical Sciences at the Escola Militar, the request he made in 1847 to be allowed to take the examinations for that course was granted. He took a series of examinations and was awarded a bachelor's degree in Mathematical and Physical Sciences on 10 June 1848. Souzinha believed that he had taken his studies of mathematics to the level of a doctorate since he had been undertaking research for several years so he now requested the Escola Militar to be allowed to undertake a public defence of a doctoral thesis. This was allowed and, on 14 October 1848, he defended his 55-page thesis Dissertação Sobre o Modo de Indagar Novos Astros Sem o Auxílio das Observações Diretas .

See the title page of the thesis at THIS LINK.

Souzinha's thesis had been motivated by the work of John Couch Adams and Urbain Le Verrier who had used the way that Uranus deviated from its expected orbit to predict the position of a previously unknown planet to account for the deviations. Their predictions were made known in September 1845 and June 1846 respectively and led to the discovery of Neptune on 23 September 1846 by Galle at the Berlin Observatory. Souzinha had clearly been studying Laplace's Mécanique Céleste and this forms the basis for the work of his thesis. Souzinha writes:-
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.
The thesis is not a great work for the scientific results it contains but it is a remarkable work by someone who was essentially self-taught in mathematics. It is also important for it is the first mathematics doctorate to be awarded by Brazil and it marks the beginning of mathematical research in Brazil.

After obtaining his doctorate, he was successful in a competition for a position on the teaching staff of the Escola Militar, being also appointed lieutenant colonel and honorary captain of the Escola Militar. Before taking up the position he returned to Maranhao where he studied languages, political economy and constitutional law. He took up his position on the teaching staff at the Escola Militar in 1849. He continued to undertake research in mathematics and he published a paper Resolution of numerical equations in number 5 of Guanabara, a monthly artistic, scientific and literary magazine. The paper presents a method which he claims is simpler than that of Lagrange but he admits that it may not give as accurate results. He ends the paper with this comment:-
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.
Between 1851 and 1854 Souzinha published a series of three articles entitled First memoir on general methods of integration. He described existing methods, in particular those given by Euler, Laplace, Liouville, Fourier and Jacobi. It is clear that these three papers constitute around half of what Souzinha intended to publish in this series. In the third part, for instance, he states explicitly that the series would continue but it never did. The magazine Guanabara ceased publication in 1856.

Souzinha believed that he had produced some important mathematical results but with no colleagues undertaking mathematical research in Brazil, he was very limited both in undertaking research and in making his results known. In 1854 he requested the government of Brazil for permission to travel to Europe. His request was granted and he was asked to undertake two missions, first to study the penitentiary system of European countries with the aim of improving the system in use in the Rio de Janeiro prison, the Casa de Correção da Corte, and secondly to obtain information about European astronomical observatories. If the first of these missions seems a strange one to give Souzinha in fact it was not since he was the secretary of the Casa de Correção da Corte. He spent 1855 and 1856 partly in London and partly in Paris at the Sorbonne where he attended courses. He presented papers to the French Academy of Sciences and enrolled at the Faculty of Medicine in Paris, obtaining the degree of Doctor of Medicine in 1856. He also submitted the paper On the Determination of Unknown Functions Which Are Involved under Definite Integrals to the Royal Society of London in 1856 and Stokes, the secretary of the Royal Society, published the report [32] while waiting for a referee to report on the paper. Stokes' report begins:-
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.
A committee consisting of Liouville, Lamé and Bienaymé was set up by the French Academy of Sciences to report on the memoirs Souzinha had submitted to the Academy and later Cauchy was asked to join the committee. There is no record that the committee ever met.

We have not mentioned the fact that Souzinha had been working on making an anthology of poetry. While on his European trip he visited Germany in 1856 and discussed the publication of his anthology with the Brazilian poet Antônio Gonçalves Dias who was in Leipzig on a mission from the Brazilian government to study the state of public instruction in the educational institutions of Europe. Souzinha's Anthologie Universalle consists of 944 pages and was published in Leipzig in 1859. The anthology contains poems in seventeen languages, each poem in its original language, German, English, French, Italian, Portuguese, Spanish, Russian, Polish, Serbian, Bohemian, Hungarian, Dutch, Danish, Swedish, Modern Greek, Latin, and Ancient Greek. Why would anyone buy such a book consisting of poems most of which were in languages they could not understand? Even more puzzling, why would a publisher take on such a publication? Souzinha begins the Preface with the following justification:-
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?
When Charles Henry wrote the Preface [16] to the posthumous publication of Souzinha's mathematical work [15] he spends more time giving details of this anthology of poetry than he does of the mathematics. You can read the Preface at THIS LINK.

While he was in Germany, Souzinha received the news from one of his brothers that he had been nominated for the position of deputy general for the province of Maranhao. He thought it would be appropriate for a member of the Brazilian parliament to be married and be the head of a family, so he returned to England to propose marriage to Rosa Edith Hamber, the daughter of the Rev Thomas Hamber (1790-1875) and Elizabeth Marsh (1803-1871) whom he had met on his visit to England one year before. Rosa had been born on 18 August 1838 in Islington, London, and baptised in Islington Holy Trinity Church on 14 April 1839. Souzinha wanted to return quickly to Brazil so the marriage on 18 March 1857 took place in the Holy Trinity Church, Brompton, Middlesex, only eight days after he had asked the Rev Hamber for the hand of his daughter. Souzinha immediately returned to Brazil leaving his new wife in England.

Back in Brazil, Souzinha was sworn in as Deputy on 19 May 1857 and made his first speech on 25 June 1857. In fact he was elected as a representative of the province of Maranhao by the Liberal Party in three consecutive legislatures (1857-1860, 1861-1864, and 1864-1867). Sadly, tragedy soon struck him and his family. In 1858 he brought his wife from England to Brazil and they had a son. On 18 February 1861, his wife died of typhoid fever contracted on a trip he made to the countryside of Maranhao to introduce his wife to his family and to meet his electorate, as he was running for a second term of deputy. Two years later, in 1863, his son died in São Luís, a victim of a sudden illness which, despite his medical training, he had not realised was serious. Souzinha's own health was also deteriorating and, suffering from the lung disease hemoptysis, he had returned to Maranhao for health reasons in 1862. He wanted to continue his duties despite his health problems and returned to Rio de Janeiro taking up residence on the Santa Teresa hill. By March 1864, when he married again, he was seriously ill. His second wife was Paulina Guerra, his neighbour on Santa Teresa. Hoping for good medical treatment in England, he made the trip in March 1864, but he died two months later.

Some of Souzinha's memoirs have been lost but eight were published in Leipzig in 1882 ([15] is a modern reprinting of the 1882 edition). The eight memoirs are:
(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.
In addition to their mathematical content, these memoirs contain information about Souzinha's efforts to get his work published. For example Souzinha writes in (6):-
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.
In his Memoir on sound, Souzinha states how he believes he has improved on the work of others:-
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.
Let us quote from Souzinha himself, the reasons he gives for attacking the mathematical problems that interested him (see for example [16]):-
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.
Souzinha was hailed as a genius by most Brazilians following his death. Only the book [9] written in 1878 by Frederico José Correia (1817-1881) is highly critical, calling Souzinha a charlatan and saying that his only talent was in deceiving public opinion by overestimating himself. For a modern look at Souzinha's mathematics see [12] where the authors write:-
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.

### References (show)

1. U D'Ambrosio, Joaquim Gomes de Souza, o "Souzinha" (1829-1864), in R A Martins, L A C P Martins, C C Silva and J M H Ferreira (eds.), Filosofia e história da ciência no Cone Sul: 3 o Encontro (AFHIC, Campinas, 2004)
2. U D'Ambrosio, Uma história concisa da matemática no Brasil (Vozes, Petrópolis, RJ, 2008).
3. B Basseches, Achegas para uma bio-bibliografia de Joaquim Gomes de Souza, Curitiba: Anuário da Sociedade Paranaense de Matemática 2 (1955), 18-25.
4. A V A S Blake, Dicionário bibliográfico brasileiro, Imprensa Nacional Rio de Janeiro 4 (1889).
5. H de Campos, Souzinha, o matemático, in Malba Tahan, Antologia da Matemática (Editora Saraiva, São Paulo, SP, 1964), 185-190.
6. I Coelho de Araujo, Joaquim Gomes de Souza (1829-1864): a construção de uma imagem de Souzinha, Doctor of Education Thesis (Pontifícia Universidade Católica de São Paulo, São Paulo, 2012).
https://tede2.pucsp.br/bitstream/handle/10930/1/Irene%20Coelho%20de%20Araujo.pdf
7. I Coelho de Araujo, Os Diferentes Discursos na Formação De Uma Imagem de Joaquim Gomes de Souza (1829-1864), Proceedings of the Fifth International Seminar on Research in Mathematics Education, Petrópolis, Rio de Janeiro, Brazil (2012).
http://sbem.iuri0094.hospedagemdesites.ws/files/v_sipem/PDFs/GT05/CC63707578100_A.pdf
8. L Corrêa, Biografia de Joaquim Gomes de Souza (Littera Maciel, Belo Horizonte-MG, 1984).
9. F J Correia, Um livro de crítica (Tip. do Frias, São Luís, 1878).
10. E Cunha, À Margem da História (Martins Fontes, São Paulo, 1999).
11. D L Fernandez, Joaquim Gomes de Souza: O Primeiro Matemático Brasileiro, International Mathematical Union (2018).
12. C S Fernandez and C Monteiro de Souza, Joaquim Gomes de Souza e as Controvérsias Sobre o Uso das Séries Divergentes no Século XIX, Ideaçao, Feira de Santana 3 (1999), 131-157.
13. I Francisco da Silva, Diccionario Bibliographico Portuguez tomo décimo segundo, quinto do suplemento J (Imprensa Nacional, Lisbon, 1884).
14. L Freire, Joaquim Gomes de Souza. Sua vida e sua obra, Revista Brasileira de Matemática 3 (1) (1963), 1-8.
15. J Gomes de Souza, Mélanges de Calcul Intégral (Wentworth Press, 2018).
16. C Henry, Preface, in Joaquim Gomes de Souza, Mélanges de Calcul Intégral (Leipzig, 1882), V-VIII.
17. Joaquim Gomes de Souza, ANE Brazil Academia Nacional de Engenharia.
http://anebrasil.org.br/membros/joaquim-gomes-de-souza/
18. Joaquim Gomes de Souza: Feb 15, 1829 - Jun 1, 1864, Google Arts and Culture.
19. A H Leal, Pantheon Maranhense: Ensaios Biográficos dos Maranhenses ilustres já falecidos (Editorial Alhambra, Rio de Janeiro-RJ, 1987).
20. L J Lopes, Joaquim Gomes de Souza. Ciência e sociedade, Centro Brasileiro de Pesquisas Físicas Rio de Janeiro 5 (1989).
21. R A F Martins, Atenienses e fluminenses: a invenção do cânone nacional, Doctoral Thesis (Universidade Estadual de Campinas, Instituto de Estudos da Linguagem. Campinas, SP, 2009).
22. C Monteiro de Souza, O Newton do Brasil: a biografia do cientista brasileiro Joaquim Gomes de Souza (Editora da UFRPE, Recife, 2008).
23. C O S Nascimento, Alguns Aspectos da Obra Matemática de Joaquim Gomes de Souza, Master's Thesis (Universidade Estadual de Campinas, 2008).
24. S Neto, Joaquim Gomes de Souza e sua proposta de reforma do currículo da Escola Central, Master's Thesis (Pontifícia Universidade Católica de São Paulo. Programa de Estudos Pós Graduados em História da Ciência, São Paulo, SP, 2008).
25. C Pereira da Silva, Matemática no Brasil. Uma história de seu desenvolvimento (Editora da UNISINOS, São Leopoldo, 1999).
26. J B Portela, Gomes de Souza e Sua Obra (Editora da UFMA, São Luís - MA, 1975).
27. Prêmio Joaquim Gomes de Souza, Fundação Universitária José Bonifáciom (2020).
http://www.fujb.ufrj.br/premio-joaquim-gomes-de-souza/
28. Review: Joaquim Gomes de Souza - Souzinha: Entre o Cálculo Integral e os Poemas Universais, by Gastao Rúbio de Sá Weyne, Portal do Escritor.
https://www.portalescritor.com.br/lermais_materias.php?cd_materias=2707&friurl=_-JOAQUIM-GOMES-DE-SOUZA---SOUZINHA--Gastao-Rubio-de-Sa-Weyne
29. G Rúbio de Sá Weyne, Joaquim Gomes de Souza - Souzinha: Entre o Cálculo Integral e os Poemas Universais (Scortecci Editora, 2012).
30. C M Souza, Joaquim Gomes de Souza: discursos parlamentares de um matemático do Império (Editora da Universidade Federal do Maranhão, São Luís, 1996).
31. C M Souza and F C Sanches, Joaquim Gomes de Souza e as controvérsias sobre o uso das séries divergentes no século XIX, in Ideaçao - Revista do núcleo interdisciplinar de estudos e pesquisas em Filosofia da Universidade Estadual de Feira de Santana (UEFS, NEF, Feira de Santana, 1999), 137-157.
32. G G Stokes, Report on the paper 'On the Determination of Unknown Functions Which Are Involved under Definite Integrals' by J Gomes de Souza, Proceedings of the Royal Society of London 8 (1856-1857), 146-149.
33. J Teixeira de Oliveira, O famoso Dr Souzinha (Centro Brasileiro de Pesquisas Físicas, 1989).