If a road _AB_ on an arc described about _O_, is to be joined to road _CD_, described about _O'_, the arc _BC_ should evidently be internally tangent to _AB_ and externally tangent to _CD_. Hence the center is on _BOX_ and _O'CY_, and is therefore at _P_. The problem becomes more real if we give some width to the roads in making the drawing, and imagine them in a park that is being laid out with drives. It will be noticed that the above problems require the erecting of perpendiculars, the bisecting of angles, and the application of the propositions on tangents. A somewhat different line of problems is that relating to the passing of a circle through three given points. It is very easy to manufacture problems of this kind that have a semblance of reality. [Illustration] For example, let it be required to plan a driveway from the gate _G_ to the porch _P_ so as to avoid a mass of rocks _R_, an arc of a circle to be taken. Of course, if we allow pupils to use the Pythagorean Theorem at this time (and for metrical purposes this is entirely proper, because they have long been familiar with it), then we may ask not only for the drawing, but we may, for example, give the length from _G_ to the point on _R_ (which we may also call _R_), and the angle _RGO_ as 60 deg., to find the radius. A second general line of exercises adapted to Book II is a continuation of the geometric drawing recommended as a preliminary to the work in demonstrative geometry. The copying or the making of designs requiring the describing of circles, their inscription in or circumscription about triangles, and their construction in various positions of tangency, has some value as applying the various problems studied in this book. For a number of years past, several enthusiastic teachers have made much of the designs found in Gothic windows, having their pupils make the outline drawings by the help of compasses and straightedge. While such work has its value, it is liable soon to degenerate into purposeless formalism, and hence to lose interest by taking the vigorous mind of youth from the strong study of geometry to the weak manipulation of instruments. Nevertheless its value should be appreciated and conserved, and a few illustrations of these forms are given in order that the teacher may have examples from which to select. The best way of using this material is to offer it as supplementary work, using much or little, as may seem best, thus giving to it a freshness and interest that some have trouble in imparting to the regular book work. The best plan is to sketch rapidly the outline of a window on the blackboard, asking the pupils to make a rough drawing, and to bring in a mathematical drawing on the following day. [Illustration]