4. In the case of two congruent triangles use the letters _A_, _B_, _C_ and _A'_, _B'_, _C'_, or _X_, _Y_, _Z_, instead of letters chosen at random, like _D_, _K_, _L_. It is easier to follow a proof where some system is shown in lettering the figures. Some teachers insist that a pupil at the blackboard should not use the letters given in the textbook, hoping thereby to avoid memorizing. While the danger is probably exaggerated, it is easy to change with some system, using _P_, _Q_, _R_ and _P'_, _Q'_, _R'_, for example. 5. Use small letters for lines, as above stated, and also place them within angles, it being easier to speak of and to see [L]_m_ than [L]_DEF_. The Germans have a convenient system that some American teachers follow to advantage, but that a textbook has no right to require. They use, as in the following figure, _A_ for the point, _a_ for the opposite side, and the Greek letter [alpha] (alpha) for the angle. The learning of the first three Greek letters, alpha ([alpha]), beta ([beta]), and gamma ([gamma]), is not a hardship, and they are worth using, although Greek is so little known in this country to-day that the alphabet cannot be demanded of teachers who do not care to use it. [Illustration] 6. Also use small letters to represent numerical values. For example, write _c_ = 2[pi]_r_ instead of _C_ = 2[pi]_R_. This is in accord with the usage in algebra to which the pupil is accustomed. 7. Use initial letters whenever convenient, as in the case of _a_ for area, _b_ for base, _c_ for circumference, _d_ for diameter, _h_ for height (altitude), and so on. Many of these suggestions seem of slight importance in themselves, and some teachers will be disposed to object to any attempt at lettering a figure with any regard to system. If, however, they will notice how a class struggles to follow a demonstration given with reference to a figure on the blackboard, they will see how helpful it is to have some simple standards of lettering. It is hardly necessary to add that in demonstrating from a figure on a blackboard it is usually better to say "this line," or "the red line," than to say, without pointing to it, "the line _AB_." It is by such simplicity of statement and by such efforts to help the class to follow demonstrations that pupils are led through many of the initial discouragements of the subject. FOOTNOTES: [40] Carson, loc. cit., p. 13. CHAPTER X THE CONDUCT OF A CLASS IN GEOMETRY