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I am reading the following short paper:

Here, Tait writes

The closed curves contemplated are supposed to have nothing higher than double points. By infinitesimal changes of position of the branches intersecting in it, a triple point is decomposable into $3$ double points, a quadruple point into $6$, and generally an $x$-ple point into $\dfrac{x(x-1)}{1.2}$ double points.

I'm having trouble figuring out what exactly Tait intends by the term "double" points here. The description of how to change a "triple", "quadruple", or more generally an "$x$-ple" point into double points has not illuminated to me the definition of a double point itself.

From the diagrams in Plate 2 (specifically, Fig. 11–17), I think I intuitively understand what Tait is saying, but the conversion of triple points (and so on) into double points is still completely opaque to me.

Plate II

  • Does Tait discuss his definition of double points anywhere else in greater detail? What is a modern definition of his $x$-ple points?
  • What precisely is meant by "infinitesimal changes of position" when decomposing a triple point, for instance, into three double points?
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    $\begingroup$ Doesn't the quoted text define a double point as a point of intersection. Then a double point is traced twice by the curve; a triple point is traced three times; etc... This fits with his reducing a triple point to three double points, for example. $\endgroup$
    – nwr
    Jun 17 at 18:46
  • $\begingroup$ @nwr "Doesn't the quoted text define a double point as a point of intersection." I couldn't find an explicit statement of that form, but that's what the diagrams in Plate 2 seem to suggest. However, I'm still not clear on how Tait decomposes a triple point into three double points, for instance. $\endgroup$ Jun 17 at 19:07
  • $\begingroup$ @nwr So, something like this would be the right way to decompose a triple point into three double points —? i.stack.imgur.com/cgFWx.jpg. Why exactly would something like this be incorrect, then —? i.stack.imgur.com/loaaD.jpg $\endgroup$ Jun 17 at 19:47
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    $\begingroup$ These are standard notions of algebraic/differential geometry, see Ordinary Double Point. When three or more plane curves (or branches of the same curve) intersect at a single point a small perturbation of them near it puts them into "general position" where only double points remain. You can even do it on straight lines they are tangent to at the point, as in attached image. See Gibson, Elementary Geometry of Algebraic Curves. $\endgroup$
    – Conifold
    Jun 17 at 19:58
  • $\begingroup$ @Conifold Ah, I see. Thank you for the reference, I'll follow up on that. I suppose this question is not really appropriate for this site, then? Should I remove it? $\endgroup$ Jun 17 at 20:03

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