Experiments on the Regular Pentogon. 
Ill 
original sect at the desired point of division as (5) conld be proved in 
some way. 
It is needless to say that this construction does not give the desired 
division point. Had, however, this been the only way in which it conld 
have been obtained, how long, think yon, would our poor Greek have been 
in solving the problem ? How long would he have been in guessing these 
somewhat peculiar fractions? Even if by accident he got one of them, 
bow long would it be before he got the others to combine with it prop- 
erly? This problem, with our present hypothesis, would be as difficult 
as, and very similar to, the unlocking of a combination safe of three 
turns where the turning possibilities are something enormous in each 
case. The Greek would have failed, and today it would be an unsolved 
problem. 
As we go forward contrast the difficulty of this hypothetical condition 
with the simplicity of the actual problem. The contrast will bring out 
the fact that geometric originals are not mere puzzles, but can be solved 
by promising experiments combined with simple bits of reasoning. 
Coming back to our geometer trying to divide a sect into two parts 
x, y, such that x 2 =y(x+y), let us imagine him attentively considering 
the promising ways of beginning his construction, some construction 
being obviously necessary. On looking over his limited knowledge of 
constructions he recalls the following as constituting most of his equip- 
ment: (1) Producing lines already in the figure by amounts equal to 
sects already in the figure. (2) Joining two points in the figure by a 
straight line, unless already so joined. (3) Erecting perpendiculars to 
already existing lines at favorable points. (4) Drawing parallels under 
similar conditions. Our Greek, to begin, may either produce, join, erect, 
or draw a parallel. To produce the initial sect is feasible, to join two 
points is not, to erect a perpendicular is feasible, to draw a parallel is 
possible but not promising, there is no favorable point to draw it through. 
As a first effort our Greek either produces the sect or erects a perpendic- 
ular to it. Let us assume that he produces the given sect, and, after try- 
ing a year or so to get the problem that way, fails. (As a matter of fact 
failure would not necessarily result.) He is now driven back to erecting 
a perpendicular. At what point shall he erect it ? Many points may be 
chosen, of course, in fact the perpendicular might be erected at a random 
point. But our Greek would recall no case of a successful random per- 
pendicular in his limited geometric experience and hence would try to 
select a suitable point. As a first effort would he not select an end point ? 
If he failed there, then the mid-point could be tried, and, if failure 
resulted there, a trisection point could then be tried. As a first step in 
the desired construction, then, our Greek erects a perpendicular to the 
given sect at an end point. How long shall this perpendicular be made ? 
