Having considered Euclid's definitions of Book I, it is proper to turn to some of those terms that have been added from time to time to his list, and are now usually incorporated in American textbooks. It will be seen that most of these were assumed by Euclid to be known by his mature readers. They need to be defined for young people, but most of them are not basal, that is, they are not used in the proofs of propositions. Some of these terms, such as magnitudes, curve line, broken line, curvilinear figure, bisector, adjacent angles, reflex angles, oblique angles and lines, and vertical angles, need merely a word of explanation so that they may be used intelligently. If they were numerous enough to make it worth the while, they could be classified in our textbooks as of minor importance, but such a course would cause more trouble than it is worth. Other terms have come into use in modern times that are not common expressions with which students are familiar. Such a term is "straight angle," a concept not used by Euclid, but one that adds so materially to the interest and value of geometry as now to be generally recognized. There is also the word "perigon," meaning the whole angular space about a point. This was excluded by the Greeks because their idea of angle required it to be less than a straight angle. The word means "around angle," and is the best one that has been coined for the purpose. "Flat angle" and "whole angle" are among the names suggested for these two modern concepts. The terms "complement," "supplement," and "conjugate," meaning the difference between a given angle and a right angle, straight angle, and perigon respectively, have also entered our vocabulary and need defining. There are also certain terms expressing relationship which Euclid does not define, and which have been so changed in recent times as to require careful definition at present. Chief among these are the words "equal," "congruent," and "equivalent." Euclid used the single word "equal" for all three concepts, although some of his recent editors have changed it to "identically equal" in the case of congruence. In modern speech we use the word "equal" commonly to mean "like-valued," "having the same measure," as when we say the circumference of a circle "equals" a straight line whose length is 2[pi]_r_, although it could not coincide with it. Of late, therefore, in Europe and America, and wherever European influence reaches, the word "congruent" is coming into use to mean "identically equal" in the sense of superposable. We therefore speak of congruent triangles and congruent parallelograms as being those that are superposable.