Newtonian uses one concave and one flat mirror (or just one concave).
Gregorian uses two concave mirrors, and
Cassegrain uses one concave and one convex mirror.
All these mirrors should be parabolic in an ideal situation. First Gregorian was made by Hooke in 1673.
This was 5 years before John Hadley was born. Hadley made a parabolic mirror in 1721, for a Newtonian telescope btw.
There was certainly known that for Newtonian telescope optics, at some point when the f/no becomes small enough spherical aberration will limit the resolution rather than diffraction, and that a more parabolic shape can avoid this particular problem.
When grinding optical surfaces, near-spheres are the easiest, the tend to come naturally. There are ways to make them aspherical known to present-day amateur telescope makers.
But I don't know anything about how Hadley's first parabolic mirror was made parabolic, nor how it was tested to verify its asphericity, nor what it was used for; a Newtonian telescope or a more complex system with two or more curved surfaces.
Question: How did Hadley do it and why exactly? How did Hadley make the first parabolic telescope mirror, verify its asphericity, and for what kind of telescope was this work done?