Modern button sizing is based on the old French ligne (pronounced "line"; often indicated by the symbol "') system. Adapted by German button makers in the early 1700s, the system was eventually standardized: 40 lignes = one English inch, so a #40 button = 1" and so on. It's still the international industry standard for buttons and
snaps, even after hundreds of years.
Approximate metric equivalents for common American sizes are as follows:
#16 = 10.00 (10.16) mm (3/8")
#18 = 11.00 (11.43) mm (7/16")
#20 = 12.50 (12.70) mm (1/2")
#22 = 14.00 (13.97) mm (9/16")
#24 = 15.50 (15.24) mm (5/8")
#28 = 18.00 (17.78) mm (11/16") — American dime
#30 = 19.00 (19.05) mm (3/4") — American penny
#36 = 23.00 (22.86) mm (7/8") — American nickle
#40 = 25.50 (25.50) mm (1") — American quarter
#45 = 28.50 (28.50) mm (1-1/8")
#50 = 31.50 (31.75) mm (1-1/4")
#60 = 38.00 (38.10) mm (1-1/2")
#80 = 51.00 (51.00) mm (2")
Sizes are before covering; fabric will increase button diameter.