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Ir is generally conceded that the final aim of mathematical teaching should be not only the acquisition of practical knowledge, but that training of the student's mind which gives a distinct gain of mental power. In recognition of this principle nearly all college entrance examinations in geometry require some original work, and most text-books devote considerable space to exercises. Comparatively little, however, has been done to introduce the student systematically into original geometrical work.

No teacher of physics or chemistry would ask a student to discover a law without so guiding his work as to enable him to reach the desired result; many. text-books and teachers expect the pupil to invent geometrical proofs and to solve problems, entirely new to him, without offering any assistance further than a knowledge of the well-established theorems of all text-books. Some writers give a description of the analysis of propositions, which is entirely logical and of great advantage to a person of some mathematical knowledge, but which is usually too abstract to be of any practical value to the beginner. In this book the attempt is made to introduce the student systematically to the solution of geometrical exercises. In the beginning the exercises given in a certain group are of similar kind and related to the preceding proposition ; later some general principles are developed which are of fundamental importance for original work, as,

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