Johann Castillon studied mathematics at Pisa then went to Switzerland. While in Switzerland he changed his name so that he took the name of the town of his birth. He was born Giovanni Francesco Melchiore Salvemini.
Castillon taught at Lausanne and also at Bern. In 1751 he went to the University of Utrecht to lecture on mathematics and astronomy. He obtained a doctorate from Utrecht in 1754, becoming a professor there in 1755. Three years later Castillon became rector of the University.
In 1764 he went to Berlin becoming Royal Astronomer at the Berlin Observatory in 1765.
His first two papers are on the cardioid curve which he named. He also studied conic sections, cubic equations and artillery problems.
Castillon published the Leibniz - Johann Bernoulli correspondence. He edited works of Euler and published a commentary on Newton's Arithmetica Universalis. He also translated Locke's Elements of Natural Philosophy into French.
He was made a Fellow of the Royal Society of London in 1753.