16th July, 1776 = (1775 years + Period from 1.1.1776 to 16.7.1776). Counting of odd days: Number of odd days in 1600 years = 0. Number of odd days in 100 years = 5. 75 years = 18 leap years + 57 ordinary years = (18 × 2 + 57 × 1) odd days = 93 odd days = (13 weeks + 2 days) ≡ 2 odd days. ∴ 1775 years have (0 + 5 + 2) odd days = 7 odd days ≡ 0 odd day. Jan. + Feb. + March + April + May + June + July = (31 + 29 + 31 + 30 + 31 + 30 + 16) = 198 days = (28 weeks + 2 days) ≡ 2 odd days. ∴ Total number of odd days = (0 + 2) = 2. Hence, the required day is Tuesday.
Explanation as extracted from the printed page; notation may be imperfect.