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118 equal." This definition of equality contains a circle and thus is meaningless. For what is the meaning of "not equal to a part of the other" in this addition? To answer this question, we must first know when two numbers are equal or unequal. Thus, apart from the arbitrariness of his definition of equality, it presupposes a definition of equality, and this again presupposes a definition of equality, in which we must know again what equal and unequal are, and so on ad infinitum. After Veronese has, so to speak, given up of his own free will the indispensable foundation for the comparison of numbers, we ought not to be surprised at the lawlessness with which, later on, he operates with his pseudo-transfinite numbers, and ascribes properties to them which they cannot possess simply because they themselves, in the form imagined by him, have no existence except on paper. Thus, too, the striking similarity of his "numbers" to the very absurd "infinite numbers" in Fontenelle's Géométrie de l'Infini (Paris, 1727) becomes comprehensible. Recently, W. Killing has given welcome expression to his doubts concerning the foundation of Veronese's book in the Index lectionum of the Münster Academy for 1895-1896.