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It is one of the encouraging signs of the time that such a book as this should be written and that its publication should be assured. Geometry, like other branches of mathematics, is under critical inspection, often on the part of those who are manifestly incapable of passing judgment as to the merits of the subject. On the other hand teachers possessed of scholarship, not only in the field of education, but also in that of mathematics, occasionally are found to pass calm judgment on the real questions at issue -What is the purpose in teaching geometry, and how is that purpose best achieved? To find a general educator who can discover flaws in a subject like geometry is an easy matter—simply go to any school of education and find a man to whom the subject was poorly taught. To find a mathematician who can discover flaws in a subject like the theory of education is equally simple—simply look in any class, for example in the theory or functions of a complex variable, and find a man whose teaching is that of most college instructors. Neither one is possessed of a judicial frame of mind that qualifies him for fair criticism.