Continually to advance, continually to attempt to make mathematics fascinating, always to conserve the best of the old and to sift out and use the best of the new, to believe that "mankind is better served by nature's quiet and progressive changes than by earthquakes,"[3] to believe that geometry as geometry is so valuable and so interesting that the normal mind may rightly demand it,--this is to ally ourselves with progress. Continually to destroy, continually to follow strange gods, always to decry the best of the old, and to have no well-considered aim in the teaching of a subject,--this is to join the forces of reaction, to waste our time, to be recreant to our trust, to blind ourselves to the failures of the past, and to confess our weakness as teachers. It is with the desire to aid in the progressive movement, to assist those who believe that real geometry should be recommended to all, and to show that geometry is both attractive and valuable that this book is written. FOOTNOTES: [1] And really, though not nominally, in the United States, where the first concepts are found in the kindergarten, and where an excellent course in mensuration is given in any of our better class of arithmetics. That we are wise in not attempting serious demonstrative geometry much earlier seems to be generally conceded. [2] The third stage of geometry as defined in the recent circular (No. 711) of the British Board of Education, London, 1909. [3] The closing words of a sensible review of the British Board of Education circular (No. 711), on "The Teaching of Geometry" (London, 1909), by H. S. Hall in the _School World_, 1909, p. 222. CHAPTER II WHY GEOMETRY IS STUDIED With geometry, as with other subjects, it is easier to set forth what are not the reasons for studying it than to proceed positively and enumerate the advantages. Although such a negative course is not satisfying to the mind as a finality, it possesses definite advantages in the beginning of such a discussion as this. Whenever false prophets arise, and with an attitude of pained superiority proclaim unworthy aims in human life, it is well to show the fallacy of their position before proceeding to a constructive philosophy. Taking for a moment this negative course, let us inquire as to what are not the reasons for studying geometry, or, to be more emphatic, as to what are not the worthy reasons.