Convert from palm [US, Roman major] to gnat's eye

Unit Definition (palm [US, Roman major])
A palm, when used as a unit of length, is usually four digits or three inches, i.e. 7.62 cm (for the international inch). The width of an adult human male palm is indeed about 10 centimetres.In English this unit has mostly fallen out of use, as do others based on the human arm: digit (¼ palm), finger (7/24 palm), hand (4/3 palms), shaftment (2 palms), span (3 palms), cubit (6 palms) and ell (15 palms).

Unit Definition (gnat's eye)
The gnat's eye is an idiomatic "unit" of distance; it's common to hear that something is "as small as a gnat's eye". In fact, the eyes of typical gnats tend to have diameters similar in size to a hair's breadth: roughly 100-150 micrometers (0.10-0.15 millimeters).