Born near Newnan, Georgia, Bigby attended the common schools. He was graduated from Emory College, Oxford, Georgia, in 1853. He studied law. He was admitted to the bar in 1856 and commenced practice in Newnan, Georgia. He served as member of the State constitutional conventions of 1867-1868. He served as solicitor general of the Tallapoosa circuit from August 1867 to September 22, 1868. He served as judge of the superior court of the same circuit from September 22, 1868, to March 3, 1871.
Bigby was elected as a Republican to the Forty-second Congress (March 4, 1871-March 3, 1873). He was an unsuccessful candidate for reelection in 1872 to the Forty-third Congress. He resumed the practice of law in Atlanta, Georgia. He served as delegate to the Republican National Convention at Cincinnati in 1876. He became president of the Atlanta &. West Point Railroad in 1876. He died in Atlanta, Georgia, March 28, 1898. He was interred in West View Cemetery.