Based on a definition of “abductive insight” and a critical discussion of G. Schurz’s (2008) distinction of eleven “patterns of abduction” that he organizes in four groups, I suggest an even more comprehensive classification that distinguishes 15 forms in an alternative structure. These forms are organized, on the one hand, with regard to what is abductively inferred—singular facts; types; laws; theoretical models; or representation systems—and, on the other, with regard to the question whether the abductive procedure is selective or creative (including a distinction between “psychologically creative,” as in school learning, or “historically creative”). Moreover, I argue that theoretical-model abduction—which seems to be the most important form of abduction—depends on two preconditions: first on the availability of an adequate system of representation, and second on finding a new “perspective” on a given problem, as Peirce described it with the notion of a “theoric transformation.” To understand the significance of theoric transformations—especially in mathematics—it is necessary to analyze in some detail Peirce’s main example for a theoric transformation: the proof of Desargues’s theorem.