 Consider the numbers generated by juxtaposing each row, column, or main diagonal with its reverse image. For instance, 123321 or 159951. Every such number (there are sixteen in total) is divisible by 37. Is this a coincidence? Notice that you can check each of the sixteen numbers and prove that 37 is a factor. But this provides no explanation. It turns out there is an explanation which is connected to the fact that each such number can be expressed in terms of arithmetic progressions of the form a, a+d, a+2d (and their reverse). A theorem by Nummela (1987) shows that divisibility by 37 follows from a common feature of the sixteen numbers (i.e. they can all be expressed in the same form). Appealing to this and other examples, Lange analyzes at length what distinguishes coincidences from non-coincidences and argues that non-coincidences admit of an explanatory proof. The analysis brings into play several interesting features, such as unification, saliency, natural properties in mathematics, fruitfulness etc.  In the process, Lange also argues for the claim that mathematical explanations are context-dependent since ‘in different contexts different features might be salient’ (p. 299). Among the claims defended in chapter 8 is that there are some clear cases that show that purity and explanation come apart. The topics of purity, fruitfulness, coincidence, unification, natural properties etc. come together in chapter 9, which is devoted to Desargues’ plane theorem in projective geometry. Lange discusses and contrasts proofs of the theorem given in the context of synthetic geometry, analytic geometry and projective (synthetic) geometry. He argues that the projective proof, which obtains the result by projective means from a spatial version of the same result, provides the required explanation which eludes all the other proofs. Noteworthy are the connections Lange establishes between explanation, natural properties, and fruitfulness (see also Tappenden 2008a and 2008b).