Commentary on Proposition IV.11¶
Back to Proposition IV.11.
De Morgan remarks that “the method of IV. 11 is not so natural as making a direct use of the angle obtained in the last.” On the other hand, if we look at the figure and notice that it shows the whole of the pentagram-star except one line (that connecting B and E), I think we shall conclude that the method is nearer to that used by the Pythagoreans, and therefore of much more historical interest.
Another method would of course be to use IV. 10 to describe a decagon in the circle, and then to join any vertex to the next alternate one, the latter to the next alternate one, and so on.
Mr H. M. Taylor gives “a complete geometrical construction for inscribing a regular decagon or pentagon in a given circle,” as follows.
“Find O the centre.
Draw two diameters AOC, BOD at right angles to one another.
Bisect OD in E.
Draw AE and cut off EF equal to OE.
Place round the circle ten chords equal to AF.
These chords will be the sides of a regular decagon. Draw the chords joining five alternate vertices of the decagon; they will be the sides of a regular pentagon.”
The construction is of course only a combination of those in II. 11 and IV. 1; and the proof would have to follow that in IV. 10.