350 Mr. Bappace on the Influence of Signs 
notice is taken of the fact that all the three roots are possible, 
nor am I aware of its being noticed by any author who has 
treated of this question: had it been observed and enquired into, 
the existence of three species of heptagons answering: strictly 
to the definition, and the knowledge of the star-shaped polygons 
which were discovered by M. Poinsot, could not have remained 
so long unknown. 
If x = OP, denote part of the diameter intercepted between the 
centre, and a perpendicular from the extremity of the first side 
of the heptagon, then the usual trigonometrical formule give 
5 3 
ee eas! i rh) 
x ar +3 > 
this contains six roots, of which the three positive are (abstract- 
ing the signs) equal to the three negative: it may be resolved 
into the two factors 
1 1 i 1 1 1 
ees eave oS = 3 st —ir—=)=0 
(« Q” ptt) (« ar Be Ya > 
the second of which is only the first with the signs of its reots 
changed. The three roots are real and are represented by 
xz = OP, x= OS, and z = OR, 
the first of these gives 4B for the side of the heptagon, this is 
the same as that which has long been known; the other two 
roots give AC and AD, as the sides of the polygon, and by carrying 
them round the circle, the two star-shaped heptagons, (Figures 
2 and 3,) are produced which have no re-entering angles, and 
