Your button's body is constructed like an 1820s-thru-early-1830s "Jacksonian" button... a 1-piece flat brass disc with an applied ring-like rim, and soldered/brazed-on wire loop shank. Yes, I know, it's true that on typical Jacksonian button's the back does not show the emblem seen on the button's front. Nonetheless, this button's body-construction is the same as a Jacksonian. Also, note that in the Jacksonian Buttons group-photo below, the 6th button in the 2nd row shows that some Jacksonians have a "decorated" (not plain) applied rim, like yours. So, in summary, I believe the emblem on your button represents a famous man from prior to about 1835.
Sidenote:
The majority of Jacksonian buttons were made in Britain... which somewhat implies the "famous man" is a European of some kind (British, or historical Roman, or Greek), not an American.