He was educated at the school of Thame, Oxfordshire, and entered Merton College, Oxford, in January 1639.
On 13 June 1648 he was elected Gresham Professor of Geometry, and was at the same time Linacre reader at Oxford.
In 1653, during the First Anglo-Dutch War, Whistler was assisted by Elizabeth Alkin in setting up a network of casualty reception stations in Portsmouth and East Anglia.
At the College of Physicians he delivered the Harveian oration in 1659, was twelve times censor, registrar from 1674 to 1682, treasurer in 1682, and in 1683 president.
He married in 1657, and died on 11 May 1684, while president, of pneumonia, and was buried in Christ Church, Newgate Street.