Mr. S. M. Drach on the Polyhedric Fan, 267 



Hence 



BPF =113 41 49839445; 

 one-half 



=FPX = 56 50 54-919722, 



BPX=170 32 44-759167. 



Now take XPY=§ of 90°=101° 15', andYPZ = 12°or 

 the central angle of a 30-gon (Euc. VI. last prop.), and X P Z 

 = 113° 15'; which, taken from BPX, leaves 



BPZ = 57° 17'44"«759167, 



being only 0"'04708 (a 4370000th) below the true value 

 44"-806247. 



Ihadpreviouslytriedothermethods^.CPD+DPE-imS' 

 gives 57° 17' 10"*58) ; but this is incomparably nearer, and is a 

 very curious effect of the peculiar polyhedral angles 



20c + Do + Ic-(f + &=M) 90". 



As to the areas of the respective triangles, APB=512, or 

 13-688 per cent.; BP = 256*/ 2, or 9-679 percent.; CPD = 

 288 */2, or 10889 per cent. ; D P E = 648, or 17-324 per cent. ; 

 E P F = 810 v/5, or 48-421 per cent, of the total area 



PABCDEFP = 1160 + 544*/2 + 810V5. 



The angle APF = 158° 41' 49"-839445. The other poly- 

 hedral angles, viz. 



35 15 52 = iBCP, 54 44 08 = D C P-JB CP, 

 110 54 19=PEF + iPFE, 



70 31 44=BCP, 125 15 52 = DCP + iBCP, 

 121 43 03 = PEF-f jPED. 



May I suggest the term CORPORA DOCTI for the five 

 regular bodies ? The second word is formed of their initial let- 

 ters, and the extremes point out the numerically related Dodeca- 

 hedron and Icosahedron, whilst the correlated Octahedron and 

 Tetrahedron guard the central Cube. 



Perhaps the stiffer framework of certain wings and leaves in 

 organic nature may have geometric fans for their types, the 

 angles being certain well-known quantities. 



I have likewise found that 



EPF + 3DPF-4CPD = 49° 59' 59"-842139 



differs from 51° = 17 x3° by 1° 00' 00"-157861, or one degree 

 increased by its 22800th part, which circle-graduators may per- 

 haps find useful. There are probably many other combinations 

 of the three angles leading to nearly accurate values of the sexa- 



