ません

From ませぬ (masenu), from まらせぬ (marasenu), from まゐらせぬ (mairasenu), the negative form of 参(まゐ)らす (mairasu, “to do, to give, to offer”). Negative -ぬ (nu) regularly contracts to -ん (n).

ません • (-masen) 

1. A suffix used to form the polite negative present of a verb. The negative form of the polite suffix -ます (-masu).

   出来(でき)る (dekiru, “to be able”) → 出来(でき)ます (dekimasu, “to be able”, formal) → 出来(でき)ません (dekimasen, “to be unable”, formal)

From ませむ (masemu), from まらせむ (marasemu), from まゐらせむ (mairasemu), the volitional form of 参(まゐ)らす (mairasu, “to do, to give, to offer”). Classical volitional auxiliary -む (mu) regularly contracts to -ん (n).

The same root also serves as a basis for modern ましょう (mashō), from ませう (maseu), based off of an alternative regular contraction of classical volitional auxiliary -む (mu) to -う (u).

ません • (-masen) 

1. (archaic, Edo period) A suffix used to form the polite volitional of a verb. The volitional form of the polite suffix -ます (-masu).

• ましょう (mashō)
• ませう (maseu)