Daniel Augusto da Silva (1814-1878)

Mathematician and Navy officer, he was born in Lisbon on 16 May 1814. He studied at the Academia Real da Marinha and at the Academia Real dos Guardas-Marinhas, concluding the corresponding courses in 1832 and 1835, respectively.

Daniel da Silva then completed a Bachelor’s degree at the Faculty of Mathematics of the University of Coimbra in 1839. In the meantime, the Academia Real dos Guardas-Marinhas gave rise to the Naval School in 1845, while the Academia Real da Marinha had been integrated into the Escola Politécnica in 1837. He was nominated substitute lector at the recently created Naval School, ascending three years later to proprietary lector, a position he occupied until his jubilee in 1865. At the end of 1868, he retired as a navy officer, from the position of Navy Commander.

He became a free member of the Academia Real das Ciências de Lisboa on19 February 1851, a permanent member on 7 January 1852 and an honorary member on 20 January 1859.

He died on 6 October 1878.

Scientific Activity

A His first memoir was titled Da transformação e redução dos binários de fôrças. This memoir deals with the theory of the binaries that Daniel da Silva simplified in many points and especially as regards the decomposition of the binaries into other binaries placed in oblique coordinated planes, using a new geometrical representation of these groups of forces.

On 27 February 1850, Daniel da Silva presented the Academia das Ciências his Memória sobre a rotação das forças em torno dos pontos de aplicação, a treaty that is a new chapter in Rational Mechanics, Astaticism, that would come to have important applications in certain Physics problems. In this memoir, Daniel da Silva introduces the term binary in the Portuguese scientific terminology, which in French is written couple, and which had been introduced in Statistics by the French mathematician Louis Poinsot (1777-1859). It shows how the effects of the forces applied to a body vary when these forces revolve around their points of application, with the angles they make between themselves remaining constant however, and determines the various circumstances that follow this change in orientation of the same forces.

In 1877, Daniel da Silva complained that his work had been ignored internationally, despite being published in the Academy’s Memoirs. Concerning a memoir presented by Jean Gaston Darboux (1842-1917) to the Academy of Sciences of Paris, in which almost all of his propositions had been presented in Daniel da Silva’s memoir 25 years earlier, he stated that “Almost all of Darboux’ propositions were published twenty-five years ago in the Academy’s Memoirs in my work on the rotation of forces around their own points of application!… My Memoir, which has many other things, besides that which Möbius recalled, including even the correction of one of his errors, in whose correction Darboux glories, has lain ignored for almost twenty-five years, in the libraries of almost all of the Academies in the world. Which he takes the opportunity to write in Portuguese!” (cit. in Dionísio, 170).

In his third memoir, Propriedades gerais e resolução das congruências binómias, presented to the Academy in 1852, he deals with problems related to the Theory of Numbers. According to Gomes Teixeira, Daniel da Silva was the first person to create a method to resolve the systems of linear congruencies – an honour which has been unduly attributed to the English arithmetician H. J. S. Smith (1826-1883), who only concerned himself with this issue in 1861 – and also the first to make a general study of binomial congruencies.

In this memoir, Daniel da Silva presented new demonstrations of formulas given by Euler and Poinsot to determine the number of prime numbers inferior to a given number, a new formula to determine the sum of these numbers, a generalisation of a famous theorem of Fermat and Euler, the direct demonstration of a formula of Gauss, which Daniel da Silva reached via an indirect path, research on properties and calculation of modular roots, among other interesting aspects.

He also carried out several studies in the area of Physics and Chemistry related to his lecturing activities in the Naval School, where he lectured theoretic and practical Artillery, inter alia. A study of his on the speed of transmission of a gas flame in the bluish part and in the brilliant part where carbon is found in a state of incandescence won him the acknowledgement of the German chemist Heumann of Daniel da Silva’s priority over the results that Heumann announced in the Annalen der Chemie in Leipzig. In these studies on the flame, Daniel da Silva had the collaboration of António Augusto Aguiar (1838-1887), professor at the Escola Politécnica.

Daniel da Silva also dedicated himself to demographic and actuarial research related to the financial structuring of pension funds. In this field he developed a scientific analysis model of a concrete problem of the real, its mathematisation and resolution.

“Admitting, as a work premises, the validity of four postulates – which he calls questions of normality (for example, the constancy of the average age of the members) –, introduces an essential greatness, which he calls the average value of the amortization rate; and obtains an algebraic equation of n for the calculation, where n is the number of years along which the observations extend (…); an equation which then teaches how to resolve problems using successive approximations”. (Dionísio, p. 187)

Gomes Teixeira wrote the following about Daniel da Silva: “Some mathematicians use all their efforts to explore new regions of the world of numbers or to study those which others opened up in the past. Daniel da Silva is an example of this. Other mathematicians seek to logically secure previously explored fields. Anastácio da Cunha is an example of this. Others, still, seek in the world of numbers the elements which they need in order to study the physical world. An example of this is Monteiro da Rocha. And there are still others who at the same time contribute with their work towards the progress of the study of the physical world and of the world of numbers. An example of such a mathematician is Pedro Nunes. Daniel da Silva, the poet of Mathematics, looked for the beautiful side of these sciences; Monteiro da Rocha, a realist, searched for their useful side; the spirit of Pedro Nunes, inspired simultaneously by the theories of the Greek science and by the needs of the Portuguese mariners, searched simultaneously for what is beautiful and useful”. (Gomes Teixeira, 1925).


“Memória sobre a rotação das forças em torno dos pontos de aplicação”, Memórias da Academia das Ciências, tomo III, parte I da 2ª série, 1851.

“Propriedades gerais e resolução directa das congruências binómias”, Memórias da Academia das Ciências, t. I, parte I das, nova série, classe I, 1854.

“Da transformação e redução dos binários”, Memórias da Academia das Ciências, t. III, parte II, 1856.

“Nota sobre alguns teoremas novos de Estática”, Jornal de Ciências Matemáticas...etc., nº 1, 1866, pp. 1-5.

“De várias fórmulas novas de Geometria analítica relativas aos eixos coordenados oblíquos”, Memórias da Academia Real das Ciências de Lisboa, nova série, Tomo V, 1ª parte, 1875, pp. 1-20.

“Considerações e experiências àcerca da chama”, Jornal de Ciências Matemáticas, Físicas e Naturais, T. IV, 1873, pp. 113-137.

“Amortização anual média das pensões nos principais montepios de sobrevivência portugueses”, Jornal de Ciências Matemáticas, Físicas e Naturais, T. I, 1866, pp. 175-187.

“Contribuição para o estudo comparativo do movimento da população em Portugal”, Jornal de Ciências Matemáticas, Físicas e Naturais, T. II, 1868, pp. 175-187.

Fernando Reis


DIONÍSIO, José Joaquim, “No Centenário de Daniel Augusto da Silva”, Memórias da Academia das Ciências de Lisboa, Classe de Ciências, XXII, 1978/79, 167-188.

OLIVEIRA J. Silva, ”Daniel Augusto da Silva", Boletim da Sociedade Portuguesa de Matemática, 2, 1979, 3-15.

TEIXEIRA, Francisco Gomes, ”Elogio Histórico de Daniel Augusto da Silva", in: Panegíricos e Conferências, Coimbra: Imprensa da Universidade, 1925, pp. 155-193.

TEIXEIRA, Francisco Gomes, História das Matemáticas em Portugal, Lisboa, Academia das Ciências de Lisboa, 1934.

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