110 
Transactions Texas Academy of Science. 
when applied to the three triangles (one giving nothing new). Note 
that p is common to all three. It is to be remembered that onr Greek 
knows practically no algebra and thinks these results in geometrical 
form as actual squares. Nevertheless, with him geometrically, as with 
ns algebraically, it is mere routine work to deduce from these equations 
that 
4m 2 -— ( 1— m) 2 =41 2 — ( 1-f-m) 2 , 
and hence, 
m 2 =l(l — m). 
It is now obvious that if our geometer can take a sect l and divide it 
into two parts, x and y, such that 
x 2 =y(x+y), 
then the regular pentagon can be constructed. 
Let us now assume that our Greek (I am not aware of any known his- 
torical facts to contradict us here), after trying a day or so to divide a 
sect in this way, gave it up and tried by other ways to draw his regular 
pentagon. Let us suppose that he tried in vain many years to get the 
solution in other ways and finally, almost in despair, came back to try to 
divide a sect into two parts, x, y, such that the square on one part was 
equal to rectangle of the whole sect on the other part. 
Our Greek now has a sect, nothing more, with which to start ; he seeks 
a point of division having a certain property. Some construction lines 
are obviously necessary. The whole plane is before him upon which to 
experiment. How shall the first construction line be drawn? 
In order to get the right point of view let us make a rather violent 
assumption. Let us suppose that there was one and only one way of 
reaching this desired division point, and that this way consists (1) in 
producing the given sect by °t its own length (this length 
being the only one that brings success and discoverable only by trial). 
Suppose it further essential to success (2) to lay off an angle equal to 
tWAtV °t a right angle (one arm being the sect already drawn and 
produced and the fraction again discoverable only upon trial) and (3) 
to lay off on this second arm from the vertex a sect yWr °f the original 
sect, when (4) a perpendicular at the end point of the last sect cuts the 
