Of these, _b_ is given by way of explanation in most modern textbooks. Indeed, we cannot do better than simply to define an angle as the opening between two lines which meet, and then explain what is meant by size, through the bringing in of the idea of rotation. This is a simple presentation, it is easily understood, and it is sufficiently accurate for the real purpose in mind, namely, the grasping of the concept. We should frankly acknowledge that the concept of angle is such a simple one that a satisfactory definition is impossible, and we should therefore confine our attention to having the concept understood. 10. _When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands._ We at present separate these definitions and simplify the language. 11. _An obtuse angle is an angle greater than a right angle._ 12. _An acute angle is an angle less than a right angle._ The question sometimes asked as to whether an angle of 200 deg. is obtuse, and whether a negative angle, say -90 deg., is acute, is answered by saying that Euclid did not conceive of angles equal to or greater than 180 deg. and had no notion of negative quantities. Generally to-day we define an obtuse angle as "greater than one and less than two right angles." An acute angle is defined as "an angle less than a right angle," and is considered as positive under the general understanding that all geometric magnitudes are positive unless the contrary is stated. 13. _A boundary is that which is an extremity of anything._ The definition is not exactly satisfactory, for a circle is the boundary of the space inclosed, but we hardly consider it as the extremity of that space. Euclid wishes the definition before No. 14. 14. _A figure is that which is contained by any boundary or boundaries._ The definition is not satisfactory, since it excludes the unlimited straight line, the angle, an assemblage of points, and other combinations of lines and points which we should now consider as figures. 15. _A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another._ 16. _And the point is called the center of the circle._