Reviewed by: Salomon Maimon's Theory of Invention. Scientific Genius, Analysis and Euclidian Geometry by Idit Chikurel Peter Thielke Idit Chikurel. Salomon Maimon's Theory of Invention. Scientific Genius, Analysis and Euclidian Geometry. Berlin: Walter de Gruyter, 2020. Pp. x + 168. Cloth, $97.99. What role does genius play in scientific invention and discovery? This was a question at the fore of many discussions in the seventeenth and eighteenth centuries, especially as the Romantic movement came to a head. It also serves as a frame for Idit Chikurel's focused and fascinating account of Salomon Maimon's views about mathematical invention and discovery. As with many of his positions, Maimon adopted a rather iconoclastic theory of [End Page 689] genius and invention, and Chikurel does an excellent job of charting the ways in which Maimon sought to categorize the means by which we can make inventions and discoveries in mathematics. The project is fairly narrow and assumes a good deal of familiarity with Maimon, so it is probably not the best introduction to his thought as a whole, but anyone conversant with Maimon, and in particular his views of mathematics, will find the book valuable and rewarding. While her study ranges across his works, Chikurel focuses on two essays Maimon wrote in 1795, "The Genius and the Methodical Inventor" and "On the Use of Philosophy for Expanding Cognition," neither of which has received much previous scholarly attention. Chikurel makes clear, though, that there is much of interest to be found in them concerning Maimon's views on mathematical and philosophical methodology. Unlike many of his contemporaries, who celebrated the insights genius provides, Maimon harbored some reservations about such "inspired" methods. The problem, as Chikurel points out, is that the genius frequently cannot give a rational account for the conclusions she reaches—she gets the right results but without explanation. Maimon instead urges his readers to follow the lead of the "methodical inventor" (6), who reaches the same conclusions as the genius, but is able to provide the rules that he follows. This "true philosopher" follows three criteria: he must "be able not only to have a method but also to understand it and the connections between its principles" (23); he should incorporate the views of others as if they were his own, "thus making the work of others the basis for his own work" (24); and the true philosopher "should be able to follow his explanations with examples" (25). And, importantly, the best platform for such methodical invention is found in mathematics. In four compact chapters, Chikurel examines (1) how the methodical inventor differs from both the genius and the mere imitator, (2) the difference Maimon draws between invention and discovery, (3) the ways in which invention can involve both synthesis and analysis, and finally, (4) a detailed discussion of Maimon's account of how invention proceeds. The last involves a careful examination of the methods of analysis Maimon draws from Euclidean geometry, and Chikurel here does an excellent job of extracting Maimon's views from his brief and often cryptic allusions to Euclid's proofs and solutions. These include the analysis of the conditions of a problem; breaking down a complex problem into simpler ones; the analysis of the various cases a problem might have; an analysis of the mathematical object, including the various transformations it can undergo; the ways in which a solution can have different cases; an analysis of the different ways in which a problem can be solved; and logical analysis, including regressive analysis and syllogisms. Chikurel provides a detailed discussion of each of these methods, and her account is supplemented by various examples drawn from Euclid. She also considers Maimon's methods of conversion, generalization, and assuming a problematic proposition as true, here too presenting examples from Euclid and other previous mathematicians to flesh out Maimon's cases. It is an impressive display of drawing out a very sophisticated account of analysis from the rather sparse and elusive claims Maimon makes in the 1795 essays. Chikurel acknowledges that Maimon's account of invention is not entirely new, since many of the analyses he presents "are similar to the problem-solving practices of Greek geometricians," and...