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?

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]

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    $\begingroup$ Georges, going way back, perhaps it was Newton? But it would be nice to hear from a real historical expert. $\endgroup$ – Donu Arapura Feb 21 '12 at 14:05
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    $\begingroup$ One of the first may be Newton, when he discovered the Puiseux expansion of univariate algebraic functions. All is about handling singularities. $\endgroup$ – Lierre Feb 21 '12 at 14:06
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    $\begingroup$ Off-topic, but I don't think "singular" is a negative adjective; rather the opposite. $\endgroup$ – user5117 Feb 21 '12 at 15:02
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    $\begingroup$ I am no history expert and you'll need one to see what Newton did. As an upper bound, Plucker certainly had to consider singularities (for the dual at least) when working out his formulas for plane curves. $\endgroup$ – Felipe Voloch Feb 21 '12 at 17:20
  • $\begingroup$ The original tags seem to have been lost in the edit, so I restored them. $\endgroup$ – user5117 Feb 21 '12 at 21:18

Gauss' Disquisitiones generales circa superficies curvas, read at the Royal Society in Goettingen on 8 Oct. 1827, contain many termini like punctis singularibus or singulis punctis. Already on the first page we read cuius singula puncta repraesentare.

Dirichlet's collected works, edited by Kronecker and Fuchs, contain often the phrase "singular cases", and on pag. 365 punctum singulare, (written around 1850).

Riemann's collected works, edited by Weber and Dedekind, contain the first mentioning of "singulärer Punkt" on page 389 that I know of in German. It is taken from a paper about linear differential equations of 1857.

The Earliest Known Uses of Some of the Words of Mathematics supply the following dates:

SINGULAR POINT appears in a paper by George Green published in 1828. The paper also contains the synonymous phrase "singular value" [James A. Landau].

Singular point appears in 1836 in the second edition of Elements of the Differential Calculus by John Radford Young.

In An Elementary Treatise on Curves, Functions and Forces (1846), Benjamin Peirce writes, "Those points of a curve, which present any peculiarity as to curvature or discontinuity, are called singular points."

Additional remark

Hermann, a correspondent of Leibniz, wrote about singulis locis and singulis punctis in letters to Leibniz of 11 Jan. 1711 and Jun 1712, respectively. [C. I. Gerhardt (ed.): "Leibnizens mathematische Schriften", Halle (1859) p. 364 and 368]

And we should not forget that l'Hospital's famous theorem (1696) has been designed for singular points of functions only.

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  • $\begingroup$ The page to which you link seems to indicate that the original sources give no good reason for the choice of the word "singular". Do you have any better idea about this? $\endgroup$ – Paul Taylor May 9 '13 at 19:57
  • $\begingroup$ @Paul: Well, Hermann used this word. l'Hospital would have had far better a reason, but according to my research he did not use that word. By the way, as far as I have looked through some of Newton's writings, neither does he. $\endgroup$ – Rhett Butler May 9 '13 at 20:40

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