**Daniel da Silva**'s parents were Roberto José da Silva (?-1866) and Maria do Patrocinio. Daniel was baptised in the Lisbon parish of Our Lady of the Martyrs on 19 September 1814. He was the second of his parents' children, having an older brother Carlos Bento da Silva (1812-1891) who went on the become a government minister in charge of various offices such as the Treasury, Public Works, Public Instruction, the Navy, and Foreign Affairs.

We note that da Silva grew up in Portugal at a time when the country suffered civil wars. In 1820, when da Silva was six years old, there was an uprising in the country. A more major revolution took place in the early 1830s causing the closure of courses at the University of Coimbra. It is certainly the case that da Silva's education took place against a difficult background of tensions and war in the country.

In 1829, at the age of fifteen, da Silva entered the Portuguese Royal Academy of Midshipmen in Lisbon. This university naval academy had been founded in 1792 but the Lisbon Academy ceased to exist after Napoleon invaded Portugal in 1807. It was reestablished in Lisbon in 1823, only six years before da Silva began studying there. At the Royal Academy of Midshipmen he took mathematics courses on Arithmetic, Geometry, Algebra, Plane and Spherical Trigonometry, Differential and Integral Calculus. He also took courses on Mechanics such as Astronomy, Dynamics, Hydrostatics, Hydraulics and Optics. Obviously, given that this was a naval academy, he also studied navigation. In addition, he attended courses at the Lisbon Royal Naval Observatory which had been established to give cadets practical observing experience.

Da Silva showed that he was a very able mathematician and won prizes during his first two years of study for his excellent performance in the Mathematics Courses. He was awarded a distinction in each of the three years. He also performed well in the Observatory courses on the use of instruments and on astronomical calculations. The next step in educational route he was following was to move to the Royal Academy of Marine Guards. To enter this Academy a student had to be the son of the titled nobility, the son of a senior naval officer or award winning student from the Royal Academy of Midshipmen. Da Silva did not qualify under the first two criteria, but could have entered in 1832, after three years of study at the Royal Academy of Midshipmen, because of his excellent performance. However, he chose to wait a year and applied for admission on 12 August 1833. Accepted on 28 August he then served as a commercial clerk for two months before beginning his studies at the Royal Academy of Marine Guards at the end of October 1833. He attended classes in Artillery, Naval Architecture and Operating equipment. In addition he had practical experience of seamanship on ships which travelled along the Portuguese coast and also made voyages to the Azores and Madeira.

On 17 August 1835, da Silva applied for leave from the Royal Academy of Marine Guards so that he might attend the Mathematics Course at the University of Coimbra. These courses had been closed for some time because of the civil war, but they had reopened in 1834 after Liberalism had been established in the country which led to various educational reforms. Permission was granted on 4 September 1835 for da Silva to take leave to attend the University of Coimbra but, in November 1835, the new Liberal government proposed to set up an Institute of Physical and Mathematical Sciences in Lisbon. Da Silva applied for admission to this new Institute on the grounds that it would be much more convenient to be able to study advanced mathematics courses while in Lisbon. However, despite the Liberal government's initiative for this new Institute in Lisbon, the project never got off the ground and it was cancelled on 2 December. Da Silva reverted to his original plan and enrolled in the Faculty of Mathematics at the University of Coimbra. Again da Silva was awarded prizes for his achievements in the mathematics courses and was awarded a Bachelor's Degree in Mathematics from the Faculty of Mathematics of the University of Coimbra at the age of twenty-five in 1839.

After graduating, da Silva applied for a teaching position in the Polytechnic School of Lisbon. This Polytechnic School had been founded in 1837 with the aim of providing a scientific training for young men who were aiming at a career in the army of navy. When founded, it replaced the Royal Academy of the Navy. The post was to be filled by competition but, for some reason, the competition never took place. For the next few years da Silva followed a naval career. He was promoted to Brigadier of the 2nd Brigade on 18 January 1840, and, later that year, to Second Lieutenant of the Navy on 26 November.

The Sociedade Escholastico-Philomatica of Lisbon was founded in 1839. The Society put on public courses, organised philosophical discussions and published a *Journal*. Many of the leading politicians and academics were members of the Society and, from 1843, da Silva was associated with it. In March 1844 he was appointed examiner for the chair of Artillery, Geography and Hydrography at the Royal Academy of Marine Guards in Lisbon. In the following year the Royal Academy of Marine Guards was renamed the Naval School but continued to operate from buildings in Terreiro do Paço in Lisbon training officers for the Portuguese Navy. Da Silva became a substitute lecturer at the Naval School in 1845. From 1845 to 1848 he taught Mechanics, Spherical Trigonometry, Nautical Astronomy, Optics, Use of Instruments, and Practical Astronomical Observations. From 1848 to 1852, he held the chair which taught "Theoretical and Practical Artillery", "Principles of Provisional Fortification", "Geography" and "Hydrography". His health was not good and he had to take periods away from teaching due to these problems. He was promoted to First Lieutenant of the Navy in 1851 but in 1852 his health problems became so severe that he applied for leave to go to Madeira where he hoped he would recover. From 1851 to 1859 he seems to have been unable to carry out his duties due to his poor health and on 12 July 1859 the Naval Health Board classed him "unfit for active duty." In fact da Silva published a Memoir in 1854, which he had read to the Royal Academy of Sciences of Lisbon in 1852, but the publisher added a note which [9]:-

By 1859 there appears to have been a small improvement in his health and, on the 16 April of that year, he married Zefferina d'Aguiar (1825-1913) from the town of Funchal. They had one child, a son Júlio Daniel da Silva who was born in 1866. Sadly he died aged 25 in 1891.... stated that da Silva had been seriously ill for a long time, so for that reason he had been able to revise neither the preface nor the final part(from page117onwards)of his memoir. For the same reason the last two chapters are incomplete: in the ninth some theorems are missing he would like to have added; and he was only able to write an abstract of the intended contents of the tenth chapter.

Da Silva does not appear to have been well enough to teach between 1852 and 20 October 1965 when he retired as a lecturer. He continued to hold his naval positions until 31 December 1868 when he retired from the Navy. Once retired, da Silva's health improved a little and he taught some courses at the University of Coimbra. There he taught and strongly encouraged the young student Gomes Teixeira. In [1] Teixeira writes:-

[Most of da Silva's mathematical contributions were made during the years when he was forced to give up teaching because of ill health. His publications begin in 1851 with his first Memoir and continued until 1878, the year of his death. We should now say a little about these mathematical contributions. Three important memoirs by da Silva are discussed by Gomes Teixeira in [1]. These areDa Silva]lived in a perpetual struggle with a disease which would not allow him to give as much as he wanted to the science of his predilection.

*Transforming and reducing couples of forces*(1851),

*Memoir on the rotation of forces on their application points*(1851) and

*General properties and direct resolution of binomial congruences. Introduction to the study of number theory*(1854). Concerning the first of these Teixeira writes:-

This Memoir, although interesting, only gives alternative proofs of results already in the literature. The second of his Memoirs, however, is a more important work which presents an original approach to problems posed by da Silva himself. In this Memoir [1]:-Daniel's Memoir, to which we are referring, concerns the theory of couples, a theory which our geometer has simplified in many points, and especially in the part concerning the decomposition of couples, which others placed in planes using oblique coordinates, employing for this a new geometric representation of these groups of forces.

These results by da Silva did not reach an international audience. Over twenty years later, in 1877, Gaston Darboux proved (independently) the same results in his paper... the author revealed himself for the first time as a mathematician of great merit, he shows how the effects of the forces applied to a body vary, when these forces revolve around their points of application, while preserving as constant the angles they make between themselves, and determines the various particular circumstances that accompany this change in orientation of the same forces.

*Mémoire sur l'équilibre astatique et sur l'effet que peuvent produire des forces de grandeurs et de directions constantes appliquées en des points déterminés d'un corps solide, quand ce corps change de position dans l'espace*Ⓣ. When da Silva saw that his priority had not been recognised, he was upset since he knew that the journal in which his paper had appeared was taken by the Paris Academy of Sciences. He wrote a letter "Réclamation de Priorité" to the editor of

*Deux Mondes*but, as Luis Manuel Ribeiro Saraiva writes in [9]:-

The third of da Silva's three Memoirs was on number theory. It was read by da Silva to the Royal Academy of Sciences of Lisbon at the session on 24 March 1852 but illness meant that the published version was incomplete (see the quote about the publisher's note above). Teixeira writes [1]:-This seems to have been a bad decision, as the newspaper was not widely read, and therefore da Silva's work remained unknown. He should have written directly to the Academy of Sciences in Paris, which would have had much more impact.

For further comments by Teixeira on these three Memoirs, see THIS LINK.In this area the author knew the works of Euler, Lagrange, Legendre, Gauss and Poinsot. The main subject he considered was the resolution of binomial congruences, a theory which belongs simultaneously to the domain of higher arithmetic and higher algebra, and he enriched it with such important and general results that his name deserves to be included in the list of those who founded it. It was indeed Daniel da Silva who first gave a method to solve systems of linear congruences, an honour which has been unduly attributed to the distinguished English arithmetician Henry Smith, who only in1861dealt with this subject, and was also the one who first undertook the general study of congruences.

Da Silva's health prevented him from doing any serious mathematical research in the years following 1852. However, only many later, after he had retired, his health did improve a little and he again undertook research. He published *The present and future of Mount Pius General* in 1868 which is an actuarial work on annuity theory based on mortality tables. A few years later he published *On several new formulas of Analytical Geometry relative to the axes of oblique coordinates *in 1872 which generalised certain formulas which are well-known for orthogonal axes to general non-orthogonal axes. He also published notes on physics.

For a list of all of da Silva's publications, see THIS LINK.

Da Silva's contribution were recognised with his election to the Grémio Literário (1846), the Academy of Sciences of Lisbon (1850), and the Institute of Coimbra (1855). He played a major role in the Grémio Literário, of which he was a founding member. In 1849 he gave the course 'Popular Astronomy' which was run by that Society. He was elected to the Mathematical, Physical and Natural Sciences Class of the Academy of Sciences of Lisbon on 27 January 1852. His ill health prevented him playing as large a role in the Academy as he wished but, nevertheless, the Academy proposed that he be elected a "Member of Merit" in 1858 and, at the session which took place on 20 January 1859, the honour was conferred.

Da Silva died in Oeiras following a severe bout of pneumonia. Oeiras is a coastal town to the west of Lisbon. He was buried in the Cemitério dos Prazeres in the western part of Lisbon.

Let us end this biography by quoting from Teixeira [10]:-

Some mathematicians make every effort to explore new regions of the world of numbers or to study those that others have previously initiated. This is Daniel da Silva. ... Daniel da Silva, poet of Mathematics, went to seek in these sciences what they possess that is beautiful ...

**Article by:** *J J O'Connor* and *E F Robertson*