There is often a tendency on the part of teachers in their first years of work to overestimate the logical powers of their pupils and to introduce forms of reasoning and technical terms that experience has proved to be unsuited to one who is beginning geometry. Usually but little harm is done, because the enthusiasm of any teacher who would use this work would carry the pupils over the difficulties without much waste of energy on their part. In the long run, however, the attempt is usually abandoned as not worth the effort. Such a term as "contrapositive," such distinctions as that between the logical and the geometric converse, or between perfect and partial geometric conversion, and such pronounced formalism as the "syllogistic method,"--all these are happily unknown to most teachers and might profitably be unknown to all pupils. The modern American textbook in geometry does not begin to be as good a piece of logic as Euclid's "Elements," and yet it is to be observed that none of these terms is found in this classic work, so that they cannot be thought to be necessary to a logical treatment of the subject. We need the word "converse," and some reference to the law of converse is therefore permissible; the meaning of the _reductio ad absurdum_, of a necessary and sufficient condition, and of the terms "synthesis" and "analysis" may properly form part of the pupil's equipment because of their universal use; but any extended incursion into the domain of logic will be found unprofitable, and it is liable to be positively harmful to a beginner in geometry. A word should be said as to the lettering of the figures in the early stages of geometry. In general, it is a great aid to the eye if this is carried out with some system, and the following suggestions are given as in accord with the best authors who have given any attention to the subject: 1. In general, letter a figure counterclockwise, for the reason that we read angles in this way in higher mathematics, and it is as easy to form this habit now as to form one that may have to be changed. Where two triangles are congruent, however, but have their sides arranged in opposite order, it is better to letter them so that their corresponding parts appear in the same order, although this makes one read clockwise. [Illustration] 2. For the same reason, read angles counterclockwise. Thus [L]_A_ is read "_BAC_," the reflex angle on the outside of the triangle being read "_CAB_." Of course this is not vital, and many authors pay no attention to it; but it is convenient, and if the teacher habitually does it, the pupils will also tend to do it. It is helpful in trigonometry, and it saves confusion in the case of a reflex angle in a polygon. Designate an angle by a single letter if this can conveniently be done. 3. Designate the sides opposite angles _A_, _B_, _C_, in a triangle, by _a_, _b_, _c_, and use these letters in writing proofs.