KEY TO ELEMENTS OF GEOMETRY

Plane and SolidThe first and indispensable requisite towards facility in the solution of geometrical exercises is familiarity with the propositions from which their solutions may be deduced. It is chiefly such a familiarity that enables the practiced geometrician to see at a glance upon what propo sitions an exercise probably depends that may present considerable difficulty to a beginner. The latter frequently fails to perceive the short and easy path to a result which he reaches, perhaps, after tak ing a very circuitous route. This difficulty is obviated to a great extent when the exercise is given as a rider, that is, a proposition to be deduced mainly from a given proposition, as is the case with the exercises at the end of Book I. While this relieves the beginner from his most formidable difficulty, it must be remembered that a given exercise depends seldom, if ever, upon a single proposition, but usually also upon others with which the mind has to be familiar in order to see this dependence. The majority of the propositions of Book I. Are of such importance that pains should be taken to make the truths they express as present to the mind as those of the multiplication table, advantage being taken of such aids to mental association as may prove available. Thus the first five propositions, stating the general properties of angles, form an easily remembered group by themselves. The general propositions concerning equal triangles may be associated with the numbers 63, 66, 69, the two cases of equal right triangles with 72 and 73, the three double propositions concem ing parallels with 110, 112, 113, and so on. It is largely in order to promote this all-important familiarity with leading principles that so many easy exercises are given in the course of each book, and ques tions at its end. The latter may, according to circumstances, be taken part passu with the progress of the class through the book, or be left for review.Assuming the student s mind to be duly pre