Experiments on the Regular Pentagon. 
109 
examines the ideal triangle by means of a defective figure, very much as 
an entomologist examines a bug, labeling the parts. Now when one has 
a triangle in geometry one should think, sooner or later, in turn, of per- 
pendicular bisectors of sides, of perpendiculars from vertices to sides, 
of angle bisectors, etc. Thus our Greek, taking up in order various 
familiar auxiliaries to the triangle (let us assume, without result, the 
waste that inevitably accompanies all research), comes eventually to 
think of the angle bisectors of this triangle. Immediately he sees (1) 
that the one from A is perpendicular to the opposite side and (2) that 
the ones from 2A gave rise to other isosceles triangles having angles A, 
2A, 2A, and A, A, 3A. 
/ 
X 
fig 2. 
Assuming that he considers the one from A without success (and it 
may be said that a variety of these ways assumed unsuccessful here are 
not so in fact) our Greek arrives at a consideration of the bisector of 
one of the 2A angles. Again he has the paraphernalia of perpendiculars, 
angle bisectors, etc., to think of in due turn. Sooner or later, then, he 
arrives at Fig. 3. 
fig 3. 
Looking upon this, again let me say, as an entomologist upon a new 
bug, he can hardly fail to note the presence of three right angle triangles, 
two of them congruent. Now, being a sensible man (by hypothesis) and 
knowing but few theorems about right angle triangles, he could not fail 
t<J try to use these few in order. Very soon, then, he thinks of the 
Pythagorean theorem, and tries to use it. This theorem tells him that 
