Since the teeth on each gear must be the same size in order to mesh properly, the arclengths that are turned by the two gears must be equal in magnitude. The two gears, however, turn in opposite directions.

If we equate the arclengths and account for the opposite rotation directions, we obtain:

${\mathrm{R}}_{\mathrm{large}} {\mathrm{\varphi }}_{\mathrm{large}}=-{\mathrm{R}}_{\mathrm{small} }{\mathrm{\varphi }}_{\mathrm{small}}$