360 THE POPULAR SCIENCE MONTHLY. 



solutions -upon the blackboard. The problems admit of such 

 graphical representation on the board, and there are always so 

 many different forms of solution, that pupils delight in this bold 



drawing of what they have wrought out at their desks. Explana- 

 tion follows the drawing, in which the pupil accounts for each 

 step and every point he has used in constructing. At the start 



he does not see what points he may take, nor what are given. For 

 instance, the simple problem, " Can you place two circles to touch 

 each other at a particular point ? " will likely be drawn as shown 



