THE FIVE REGULAR SOLIDS 
161 
constructed the regular pentagon ? The answer must, I think, 
be yes. If ABODE be a regular pentagon, and AC, AD, GE 
be joined, it is easy to prove, from the (Pythagorean) proposi 
tions about the sum of the internal angles of a polygon and 
the sum of the angles of a triangle, that each of the angles 
BAG, DAE, ECD is fths of a right angle, whence, in the 
triangle A CD, the angle CAD is fths of a right angle, and 
each of the base angles AGD, ADC is fths of a right angle 
or double of the vertical angle CAD; and from these facts 
it easily follows that, if CE and AD meet in F, CDF is an 
isosceles triangle equiangular, and therefore similar, to ACD, 
and also that AF = FG = CD. Now, since the triangles 
ACD, GDF are similar, 
AG: CD — CD: DF, 
or AD: AF = AF: FD] 
that is, if AD is given, the length of AF, or GD, is found by 
dividing AD at Fin ' extreme and mean ratio ’ by Eucl. II. 11. 
This last problem is a particular case of the problem of 
‘ application of areas ’, and therefore was obviously within 
the power of the Pythagoreans. This method of constructing 
a pentagon is, of course, that taught in Eucl. IV. 10, 11. If 
further evidence is wanted of the interest of the early Pytha 
goreans in the regular pentagon, it is furnished by the fact, 
attested by Lucian and the scholiast to the Clouds of Aristo 
phanes, that the ‘ triple interwoven triangle, the pentagon 
i. e. the star-pentagon, was used by the Pythagoreans as a 
symbol of recognition between the members of the same school, 
and was called by them Plealth. 1 Now it will be seen from the 
separate diagram of the star-pentagon above that it actually 
1 Lucian, Piv lapsu in saint. § 5 (vol. i, pp. 447-8, Jacobitz) ; schol. on 
Clouds 609. 
152S M