At the present time, in the educational circles of the United States, questions of the following type are causing the chief discussion among teachers of geometry: 1. Shall geometry continue to be taught as an application of logic, or shall it be treated solely with reference to its applications? 2. If the latter is the purpose in view, shall the propositions of geometry be limited to those that offer an opportunity for real application, thus contracting the whole subject to very narrow dimensions? 3. Shall a subject called geometry be extended over several years, as is the case in Europe,[1] or shall the name be applied only to serious demonstrative geometry[2] as given in the second year of the four-year high school course in the United States at present? 4. Shall geometry be taught by itself, or shall it be either mixed with algebra (say a day of one subject followed by a day of the other) or fused with it in the form of a combined mathematics? 5. Shall a textbook be used in which the basal propositions are proved in full, the exercises furnishing the opportunity for original work and being looked upon as the most important feature, or shall one be employed in which the pupil is expected to invent the proofs for the basal propositions as well as for the exercises? 6. Shall the terminology and the spirit of a modified Euclid and Legendre prevail in the future as they have in the past, or shall there be a revolution in the use of terms and in the general statements of the propositions? 7. Shall geometry be made a strong elective subject, to be taken only by those whose minds are capable of serious work? Shall it be a required subject, diluted to the comprehension of the weakest minds? Or is it now, by proper teaching, as suitable for all pupils as is any other required subject in the school curriculum? And in any case, will the various distinct types of high schools now arising call for distinct types of geometry? This brief list might easily be amplified, but it is sufficiently extended to set forth the trend of thought at the present time, and to show that the questions before the teachers of geometry are neither particularly novel nor particularly serious. These questions and others of similar nature are really side issues of two larger questions of far greater significance: (1) Are the reasons for teaching demonstrative geometry such that it should be a required subject, or at least a subject that is strongly recommended to all, whatever the type of high school? (2) If so, how can it be made interesting?