He was a grandson of William Badcock, the London goldsmith and father of William Badcock (1772-1802), who married Sophia Cumberland, a daughter of the dramatist Richard Cumberland, (1732-1811).
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His ability brought him into communication with Richard Cumberland, his contemporary at Cambridge, and with William Whiston; but, inheriting a small property at Luffingham, Northamptonshire, he quietly pursued his mathematical studies in that county to the end of his life.