Why is it called Linear Regression?

Edit: There are two parts to this question, to explain the words ‘linear’ and ‘regression’. I previously explained ‘linear’. The word ‘regression’ was used by Sir Francis Galton to describe the relationship between heights of parents and their children. Tall parents tend to have tall children, but shorter than themselves while short parents tend to have short children, but taller than themselves. He called this ‘regression towards mediocrity’. This is known as the ‘regression effect’ and generally applies to a regression when both explanatory and response variables are repeated measurements on the same object (in this case families). The strange thing is that we can exchange the words ‘parents’ and ‘children’ so, for example, tall children tend to have shorter parents and tall parents tend to have shorter children.

The model [math]Y = aX^2 + bX + c + \epsilon[/math] where [math]\epsilon ~ N(0, \sigma^2)[/math] is regarded as linear regression.

Linear regressions are linear as functions of the parameters. (A linear function satisfies [math]f(au + bv) = af(u) + bf(v)[/math] where [math]u[/math] and [math]v[/math] could be numbers or vectors.)