Viviani, who was born at Florence in Italy, was an associate and pupil of Galileo, although his chief interest was in mathematics rather than in physics. After the condemnation of Galileo's ideas by the Catholic Church it was unsafe for Viviani to pursue his work on Galileo's mathematics. Accordingly Viviani devoted himself to the thorough study of Greek mathematics, in particular geometry, and in this field of work he achieved wide fame. In 1696 he was elected a fellow of the Royal Society of London. Viviani was particularly interested in trying to reconstruct lost sections of works by ancient Greek mathematicians, such as the missing fifth book of Apollonius's Conics. He also published Italian translations of the works of classical mathematicians including Euclid and Archimedes. He was an associate of the physicist Evangelista Torricelli and collaborated with him in his work on atmospheric pressure and in the invention of the mercury barometer.