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Transactions Texas Academy of Science. 
sides. Naturally, he wishes to draw regular polygons of any number of 
sides and his first problem is to discover a method of constructing a reg- . 
ular pentagon. When he succeeds in this he' can go on to the regular 
heptagon (where he would fail). 
Our Greek, therefore, first attacks the construction of a regular penta- « 
gon, using only the unmarked straight edge and compasses in accordance 
with the traditional limitations. That he could easily solve his problem 
by what we call in practical life fudge work, and in mathematics suc- 
cessive approximations, is obvious. He could draw a circle and spread- 
ing his compasses a little wider than the radius of his circle, step around 
the circle, taking five steps. If at the fifth step the steps closed the 
problem is solved. If not, the compasses may be readjusted in accord- 
ance with the size and nature of the failure to close. With the read- 
justed compasses he can again step around the circle when the failure to 
close will again indicate the readjustment necessary. In this way a reg- 
ular pentagon can be obtained with great accuracy, as accurately as one 
can draw. This is distinctly an applied mathematics way, one might even 
say an Anglo-Saxon way, of solving the problem. It is a way that is 
even invading pure mathematics more and more in widely diverse regions, 
and which is easily applicable to polygons other than regular pentagons. 
But our Greek, like the rest of his race, is a pure and not an applied 
mathematician, and seeks not practical but ideal accuracy. 
After looking at the matter several days, let us say, he sees that one 
way at least of solving the problem is to draw a triangle, isosceles and 
with base angles double the vertical angle. Let us suppose that he con- 
siders other ways of solving the problem, but fails, and finally comes 
back to the triangle as the most feasible solution. His problem now is : . 
To draw an isosceles triangle having the base angles each double the ver- 
tical angle. 
Now our text-books all have this problem, giving first the construction, 
then the proof that it affords the desired triangle. Too often our stu- 
dents memorize both construction and proof. But our Greek, free from 
text-books, is unable at once to see how to construct such a triangle. 
What is a very natural human thing for him to do? It is to assume it 
somehow constructed and then to investigate its properties. Our Greek 
