Salvemini was born on 15 January 1708 in Castiglione del Valdarno (currently in the province of Arezzo), Grand Duchy of Tuscany.
[3] On 8 January 1732 he was appointed as a sub-chancellor at the Opera di Santa Maria del Fiore, for a five-year term.
[3] Unable to find employment in Switzerland, in 1751 he obtained a job teaching mathematics and astronomy at the University of Utrecht where he began work on a second doctorate, which he received in 1754 under the name Johann Castillon.
[1] In 1745, he married Elizabeth du Fresne with whom he had three children, the only surviving child was Maximilian Friedrich Gustav Adolf Salvemini (see below).
[4] His son Frederick Salvemini de Castillon (1747-1814) also became a member Berlin Academy and was a writer for Diderot's Encyclopédie.
Succeeding Joseph-Louis Lagrange, he was appointed Director of the Mathematics Section of the Berlin Academy, a role he held until his death.