The story of another candidate for numberhood, the infinitesimal, is even stranger than the story of the irrationals or the imaginaries. The infinitesimal is the limit of a process, where the method stipulates that the end of that process can never be reached. Consider: can a single mathematical point have a slope? Our intuitive answer, trained by Euclid, is: no. A point is pure position. It isn’t a line, it isn’t even a tiny part of a line, so it can’t have a slope! Can there be a smallest possible line? How short can a line get and still have a slope! That’s a question that Euclid taught us not to ask. It is akin to suggesting that we can have two adjacent points. If we could have two adjacent points, then they would presumably constitute the smallest possible line and they would have a slope. But we can’t. Between any two points, properly speaking, infinity of other points can fit. In the seventeenth century it was common for the mathematicians at its cuttin...