It has been pointed out to me that the instructions to census takers specify the time frame of the census, which may not coincide with that of the census taker. The census of 1870 seems to have been for June 1, 1870, which was before Cornelius Miller's 9th birthday, so he was registered as 8 years old. The same explanation applies to 1880, when he was listed as 18.

These census records are consistent with a birth date on a day in June, 1861, later than June 1.