Algebraic geometry
Solution sets of polynomial systems — varieties, and in the modern language, schemes.
Founded: 1830 — Abel & Jacobi's elliptic-integral theory, the ancestor of elliptic curves (classical algebraic geometry itself traces further back, to 17th-century curve classification)
The claim
Source: Koblitz et al., IACR ePrint 2008/390.
A 2008/2011 retrospective characterization by ECC's own historians, not a contemporaneous 19th-century dismissal. No credible dismissal of algebraic geometry more broadly — including the ultra-abstract Grothendieck/scheme-theoretic reformulation of the 1950s–60s — was found.
What happened
155 years — founding (1830) to ECC (1985) — for the elliptic-curve sub-story only
Verdict: Mixed / doesn't fit cleanly.
Mixed field: the elliptic-curve story is a strong case, but the broader, more abstract core of algebraic geometry (schemes, sheaf cohomology) has no well-documented application outside pure mathematics — its most-cited physics connection (Calabi–Yau manifolds in string theory) doesn't clear the real-world bar, since string theory is experimentally unconfirmed.
Confidence: medium.