If you look at the cross $C\subset \mathbb A^2_k$ given by $xy=0$ in the affine plane over the field $k$, you see or compute that it is exceptional at $O=(0,0)$ for many (obviously not independent) reasons:
$\bullet$ The gradient of $xy$ vanishes at $ O$ .
$\bullet$ Two irreducible components pass through $O$.
$\bullet$ If $k=\mathbb C$, the complement of $O$ is disconnected.
$\bullet$ The tangent cone of $C$ at $O$ is not a line .
$\bullet$ The maximal ideal $(x,y)\subset \mathcal O_{C,O}$cannot be generated by just one element.
$\bullet$ The sheaf $\Omega_{C/k}$ is not locally free.
$\bullet$ The $k$-morphism $Spec (k[t]/\langle t^2\rangle) \to C$ given by $x=t,=y=t$ cannot be lifted to the overscheme $Spec (k[t]/\langle t^2\rangle) \subset Spec (k[t]/\langle t^3\rangle) $.
This exceptional character of $O$ is covered by several negative adjectives: non smooth,non-regular, non manifold-like , singular,...
Although I know that the purely algebraic condition for singularity (in terms of number generators of the maximal ideal of a local ring) is due to Zariski and that smoothness in terms of infinitesimal liftings is due to Grothendieck, I don't know the earlier history of the concept of singularity.
So my question is:
Who first considered explicitly the concept of singularity for varieties , why the interest and what was the definition?
Edit
First of all, thanks for the interesting comments. It is certainly plausible that Newton knew what a singularity was, but from what I read (very little) his preoccupation seems to have been classification of curves by degree.
I am curious about when he or others first wrote down the dichotomy between singular and non singular varieties , in analogy with Descartes's sharp distinction between mechanical (=transcendental) curves and geometric (=algebraic) curves ( see here) .
[By the way, if you know French you will be delighted by Descartes's old-fashioned but easily understandable language]
