Daniel da Silva (mathematician)

The inclusion–exclusion principle was stated and proved for the first time by Daniel da Silva in his memoir Propriedades geraes e resolução das congruências binomias: Introducção ao estudo da theoria dos numeros (General properties and direct resolution of binomial congruences), presented to the Lisbon Academy of Sciences in 1852 and published in 1854.^[3][4]

In this memory, Daniel da Silva also proved the following generalisation of Euler's theorem: let ${\displaystyle n=a_{1}a_{2}\ldots a_{k}}$ be an integer number where a₁, ... , a_k are pairwise coprime. Then^[3][4]

${\displaystyle a_{1}^{\phi (n/a_{1})}+a_{2}^{\phi (n/a_{2})}+\cdots +a_{k}^{\phi (n/a_{k})}\equiv k-1\quad ({\text{mod}}\,n)}$

where ${\displaystyle \phi }$ denotes the Euler's totient function.

Another important publication of Daniel da Silva is Memória sobre a rotação das forças em torno dos pontos de aplicação (Memoir on the rotation of forces about their points of application) on classical mechanics, presented in 1850 to Lisbon Academy of Sciences. In this work, Daniel da Silva worked on the problem of the equilibrium of a system of forces turning around their points of application, but maintaining their relative angles while rotating. He worked on this problem without knowledge of the results of August Ferdinand Möbius on this problem, and many of the results present in Daniel da Silva's memoir anticipated Jean Gaston Darboux results on Statics, including a correction of a Möbius mistake.^[1][3]

• 1850 - Memória sobre a rotação das forças em torno dos pontos de aplicação, Comunicações da Academia Real das Sciencias de Lisboa, 1.ª Classe.
• 1854 - Propriedades geraes e resolução das congruências binomias: Introducção ao estudo da theoria dos numeros, Memórias da Academia Real das Sciencias de Lisboa, 1.ª Classe.