## Gomes Teixeira on Daniel da Silva

In his famous Portuguese book

*History of Mathematics in Portugal*, which was published in 1934, the year after he died, Gomes Teixeira wrote about one of his own teachers Daniel Augusto da Silva. We give below an English version of Teixeira's text:Daniel Augusto da Silva, who wrote his main works in the mid-nineteenth century, will be the last mathematician we will speak of here. By virtue of his receiving the title of Member of Merit of the Academy of Sciences of Lisbon, we pronounced his historical Eulogy in 1916 in a solemn session of this Academy, held in his noble room, a Eulogy which was published in the

*Memoirs*of this Academy and transcribed in our

*Panegyrics and Conferences*. Here we will summarize the most essential things we have said about the life and writings of the illustrious Portuguese geometer in the Eulogy we mentioned, and thus we will finish the fourth of the periods in which we divide the History of Mathematics in Portugal, and with this we end this book.

Daniel Augusto da Silva was born in Lisbon on 16 May 1814. He was a Marine Officer, a Bachelor of Mathematics from the University of Coimbra, a professor at the Naval School and a Member of Merit from the Academy of Sciences of Lisbon. His scientific activity, which was great, began in 1845, the year he joined the faculty of the Naval School. From that time on, he fulfilled his duties as a teacher with the study of the classics of Mathematics and with deep meditations, whose fruits were three remarkable Memoirs that he presented to the Academy of Sciences of Lisbon in the period from 1850 to 1852.

Let's talk about these Memoirs.

Daniel da Silva's main inspiration in these early works was Louis Poinsot, and I believe I can say that the Portuguese geometer reveals himself to be a worthy continuator of the eminent French mathematician. Sometimes in these works he applied the methods of the eminent French mathematician with remarkable intelligence, at other times he used these methods for his own investigations, handling them with great skill.

The Memoir he composed first is entitled

*Transforming and reducing couples of forces.*

It is well known that Louis Poinsot, in his beautiful treatise on Statics, replaced the moments of forces, employed before him by geometers as subsidiary means for deducing the equilibrium conditions of bodies, by pairs of equal forces, in parallel and opposite directions (couples), and in this way managed to simplify and illuminate most of the theories of Mechanics.

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.

The second work, composed by Daniel da Silva, is entitled

*Memoir on the rotation of forces on their application points*. Presented to the Academy of Sciences of Lisbon, it was published in 1851 and opened to the author the doors of this house, which he entered that same year as a corresponding member.

This work is much more important than the one we first talked about. In the first work there are only new demonstrations of known results, while the second is a study, full of originality and depth, of a question that he himself had proposed.

In this beautiful and important Memoir, in which 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. The important theory to which this work is applied is nowadays a chapter of rational mechanics which has been given the name of Astatics, which has today remarkable applications to some questions of physics.

We can summarize the history of Astatics in the following terms:

August Möbius occupied himself with it in his

*Lehrbuch der Statik*, published in 1837, but he merely began it, and Ferdinand Minding enriched it with a remarkable theorem, published in Volume XV of Crelle's

*Journal*. Daniel da Silva, without knowing those works, studied the same subject, and, penetrating deeply into it, organized it in a complete way. There is a fundamental proposition in Möbius's work that we must point out because it is false and the Memoir of Daniel contains one that must replace it. I want to refer to the determination of the orientation of the forces of a system to which its equilibrium corresponds. Möbius believed that the whole system of forces which is in equilibrium in four different orientations must be in equilibrium in all other orientations.

Daniel da Silva came to a different result, showing that there are in general four equilibrium positions and only four. This last statement, which should replace that of Möbius, was confirmed by the authors who, after the Portuguese geometer, dealt with this subject.

Twenty-five years after the publication of this Memoir by our Mathematician, Gaston Darboux, without knowing this Memoir, dealt with the same question in a communication made in 1876 to the Academy of Sciences of Paris and in a work published in 1877 in

*Memoirs of the Society of Physical and Natural Sciences of Bordeaux*, but the propositions he gave are almost all contained in the Memoir of our Portuguese geometer, which contains still other interesting results that are given neither in the work of Möbius nor in the work of Darboux.

The impression which was created in the mind of our geometer by the fact that he was the inventor of an important theory with two eminent foreign mathematicians, was expressed in a touching way in a letter he addressed to me in 1877 and which can be read in our

*Panegyrics and Conferences.*

The methods used by Daniel and Darboux to study Astronomy are different. Both used geometric and analytical means simultaneously, predominately the first one of these in the Memoir of the Portuguese geometrician, predominately the second one in the Memoir of Darboux. Both studied the subject in depth, and both exposed it in a serious, clear and elegant style.

In conclusion the chapter of Mechanics called Statics is mainly Portuguese work. Möbius began it, but erred in a fundamental proposition, while Daniel da Silva gave the proposition that should replace the fundamental proposition of Möbius and organized it completely.

The comparison of the works of Möbius, Minding, Darboux and Daniel was carefully made by Fernando de Vasconcelos in a remarkable Memoir published in volume VII of the

*Annals of the Polytechnic Academy of Porto*.

The third of Daniel da Silva's Memoirs mentioned above is entitled

*General properties and direct resolution of binomial congruences. Introduction to the study of number theory*, and was presented to the Academy of Sciences of Lisbon in 1852. It is a work on the theory of numbers in which the author does not shine any less in handling calculations than he had shone in the previous works in the handling of the methods of pure Geometry.

The subjects studied by our mathematician in this Memoir belong to Arithmetic. 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 in 1861 dealt with this subject, and was also the one who first undertook the general study of congruences.

The same Memoir contains still other notable results concerning higher number theory.

The author presents in it new proofs of the formulas given by Euler and Poinsot to determine the number positive integers prime to a given number and smaller than that number, a new formula to determine the sum of those numbers; a generalization of a celebrated theorem of Fermat and Euler, the direct proof of a Gaussian formula, to which this great geometer came in an indirect way and thought it difficult to obtain by direct means, interesting investigations of properties and calculation of modular roots, etc.

This important Memoir remained, like the Memoir on Mechanics just mentioned, unjustly forgotten for about half a century, until, in 1903, an Italian mathematician of great merit, Christoforo Alasia published in the Journal of Physics, Mathematics and Natural Sciences of Pavia an appreciation of it which was very well developed and very well done.

We just talked about the main works of Daniel da Silva. After writing them, a serious illness prevented him from continuing to work. Later he improved a bit and was able to continue his scientific inquiries, but he had to lower himself to more modest matters. Then he presented to the Lisbon Academy of Sciences a memoir entitled

*On several new formulas of Analytical Geometry relative to the axes of oblique coordinates*and notes on various questions of Geometry, Mechanics, Actuary and Physics, in all of which there is something original.

His life was a persistent struggle with a disease that would not allow him to devote as much as he wanted the science of his predilection, until on 6 October 1878, his soul departed from the scene of the world, and his name passed into the history of Portuguese science, where he remains to occupy a distinct place.