Experiments on the Regular Pentagon. 
107 
metician who introduces algebra problems into his texts without the 
appropriate algebraic apparatus. Elementary mathematics will become 
more concrete. 
Amid all this, some, unable to appreciate in detail the value of the 
changes now going on, will fear that something of value is being lost, 
that the rigor of geometry will be injured, that the benefit to be derived 
from studying it will be impaired, that courses in geometry may fall 
into the general culture group. 
By means of a concrete problem, treated somewhat after the manner 
of the reformers, it is hoped to show that such fears are groundless. 
Recall as much of the method of the geometry of your school days as 
you can and compare it with the method presented as we go forward. 
It is hoped that you will see that the new is not radically different from 
the old ; that it is equally logical ; that it gives a certain systematic v iew 
of geometric constructions that was before lacking; that it makes geom- 
etry almost as experimental as chemistry; that it asks the student to 
systematically look over old knowledge in attacking new problems; that 
it involves an attempt to follow the historical order ; that it removes two 
just objections often made to the study of what are called “geometric 
originals/ 5 (The first objection is that such originals are, for most 
pupils, mere puzzles ; while the second is the lack of a general method 
on the part of elementary synthetic geometry.) 
A dozen lectures would scarcely suffice to show the value of what may 
be called the experimental way of studying geometric originals. If from 
what follows you get some notion of the method it is all that can be 
expected. 
Let us imagine a Greek geometer living about 400 B. C., before the 
science had acquired text-books, axioms, postulates, and all the impedi- 
menta of the later deductive formalism. This Greek would have known, 
let us say, eight or ten of the familiar theorems about triangles, includ- 
ing the Pythagorean theorem, a theorem or two about parallels, a few 
about polygons, six or eight about the circle, along with a knowledge of 
the elementary constructions, such as erecting perpendiculars and bisect- 
ing angles. 
With this limited equipment, let us suppose that he becomes interested 
in the construction of a regular pentagon, which no one before him has 
succeeded in constructing and which even today constitutes a problem 
of moderate difficulty, judged by high school standards. Now the con- 
struction of the regular pentagon is a problem that naturally arises in 
the mind of our Greek geometer. He already knows how to draw an 
equilateral triangle and a square and, by aid of a circle and angle bisec- 
tions, to draw regular hexagons, octagons, dodecagons, etc. In other 
words, he knows how to draw regular polygons of 3, 4, 6, 8, 12, 16, etc.. 
