The Affine cipher has no single inventor and no dramatic debut: it is
the generalisation of the Caesar shift that mathematics offers for
free. Where Caesar slides the alphabet (x + b), the affine rule
multiplies before sliding (a·x + b) — and the name comes from the
affine function of school algebra, f(x) = a·x + b.
Ciphers of this shape were certainly used in the manuscript age, but the systematic understanding came with number theory. Working modulo 26, asking which multipliers can be undone — which have a modular inverse — is a question Euler and Gauss would have found elementary. The cipher became a textbook exercise: the smallest system where encryption is arithmetic and the key is a formula.
Set a = 1 and the Affine cipher is Caesar. Set a = 1, b = 13 and
it is ROT13's shift. Set a = 7, b = 0 and you have the
multiplicative cipher the Arab cryptographers had already catalogued.
The Affine cipher is the family tree drawn as a single formula — which
is why teachers love it and cryptanalysts shrug at it.
The real inheritance is conceptual. Affine makes the key a piece of mathematics rather than a table or a disk, and makes "the key has to be undoable" a provable statement (coprimality). Public-key cryptography — Diffie and Hellman's 1976 breakthrough, RSA a year later — runs on the same instinct, with exponentiation in place of multiplication and moduli of 2048 bits in place of 26.