:As far as I can tell the point with the Huygens/de Sluse analysis of the cissoid of Diocles is that the area between the cissoid and the line  is finite.  If this area for non-negative  is rotated about the line , then the resulting volume is also finite (it is  times the corresponding area) and looks like a glass tumbler or jar infinitely extended in one direction. Meanwhile the volume inside the rotated cissoid is clearly infinite, and it is the comparison of these two volumes which is interesting. The area here is not a surface area as in Gabriel's Horn, but in effect a cross-sectional area.