The quantitative representation and analysis of astigmatism present difficulties for the researcher. The case is made that the difficulties arise because of the way astigmatism is conceived and defined. In most cases astigmatism is regarded as cylinder. Cylinder, however, is not invariant under spherocylindrical transposition and, hence, cannot strictly be regarded as meaningful. The purpose of the paper is to find a rational, context-free, invariant and universally-applicable definition of astigmatism for all quantitative analyses. One is led to definitions of astigmatism and its components some of which have already appeared in the literature but which are not in use in analyses of astigmatism. Astigmatism is defined with respect to pure sphere. In the case of thin systems (including keratometric measurements and refraction) astigmatism turns out to be Jacksonian power, that is, the power of a Jackson crossed cylinder. The power of every thin system can be regarded as consisting of two orthogonal components, sphere and astigmatism. The astigmatism of thin systems itself further decomposes naturally into two orthogonal components called ortho- and oblique astigmatism. In the case of thick systems, like the eye itself, astigmatism decomposes naturally into three orthogonal components, ortho-, oblique and antisymmetric astigmatism. The approach is based on the general definition of power in paraxial optics, the dioptric power matrix, and leads to useful graphical representations. Because of its mathematical foundation the analysis can claim completeness and contextual independence. Furthermore it is also directly applicable to the four fundamental paraxial properties of optical systems.