Hi Kim,
I assume they are flat shanked needles rather than round shanked. Davis 00, 0, 1, & 2 are sizes 11, 14, 16, & 18 respectively. The #8 Boye tube or cap is not correct for the Davis needles which go by "Davis Long", 1x2, Boye 10, and other less common names.
Most common applications are Davis made Sears Minnesota machines, Vertical Feed and Davis Badged machines that do NOT have "Use this length needle" (some exceptions).
The Davis Round Shank,"Old Style" needles or 1x1 were used on pre early 1870 "Improved" Davis models and on early White's, White Peerless. and quite a few European models.
Jon