 The second development in the refounding of the concept of infinitesimal took place in the 1970s with the emergence of synthetic differential geometry, also known as smooth infinitesimal analysis. Based on the ideas of the American mathematician F. W. Lawvere, and employing the methods of category theory, smooth infinitesimal analysis provides an image of the world in which the continuous is an autonomous notion, not explicable in terms of the discrete. It provides a rigorous framework for mathematical analysis in which every function between spaces is smooth (i.e., differentiable arbitrarily many times, and so in particular continuous) and in which the use of limits in defining the basic notions of the calculus is replaced by nilpotent infinitesimals, that is, of quantities so small (but not actually zero) that some power—most usefully, the square—vanishes. Smooth infinitesimal analysis embodies a concept of intensive magnitude in the form of infinitesimal tangent vectors to curves. A tangent vector to a curve at a point \(p\) on it is a short straight line segment \(l\) passing through the point and pointing along the curve. In fact we may take \(l\) actually to be an infinitesimal part of the curve. Curves in smooth infinitesimal analysis are “locally straight” and accordingly may be conceived as being “composed of” infinitesimal straight lines in de L’Hôpital’s sense, or as being “generated” by an infinitesimal tangent vector.