"The essay of Bernhard Riemann, "On the Hypotheses which lie at the Base of Geometry," owes its great celebrity to the fact that he was a mathematical analyst of the first order, one of the favorite pupils of Gauss, under the inspiration of whose teachings, if not at his suggestion, the essay was written—by whom, in fact, it was presented, in 1854, shortly before his (Gauss's) death to the philosophical faculty of Goettingen, and by whom its cardinal propositions were expressly indorsed as an exposition of his own speculative opinions. Every intelligent reader of this essay will agree... that its intrinsic merit is not at all commensurate with the attention with which it was received and the interest with which it is still generally considered. Not only are its statements, both of the problem and of the proposed methods of solution, crude and confused, but they bear the impress throughout of Riemann's very imperfect acquaintance with the nature of logical processes and even with the import of logical terms. It is apparent... that its author was an utter stranger to the discussions respecting the nature of space which have been so vigorously carried on by the best thinkers of our time ever since the days of Kant, and that he was so little familiar with the history of logic as to be without the faintest suspicion of the manifold ambiguity of such terms as "concept" and "quantity," and of the necessity of their exact definition preliminary to an inquiry respecting the very foundations of human knowledge."