of the Trapezium. 



121 



prolonging the sides of the pentagon ALSTK till they meet. 

 Draw DK cutting EL in O and BP in M, and draw EL 

 cutting PC in N. Then if R be the intersection of NB, MC, 

 the line OR will tend to A, the summit of the pentagon. 



Cor. 2. BE, CD (fig. 4) are two straight lines intersect- 

 ing in A ; let two points B, E and C, D be taken one pair 

 in each of these lines, and P a point between those lines. 

 From P draw PB, PC cutting DC in L and BE in K ; draw 

 DK cutting BP in M and LE in O, and let LE cut PC in N. 

 Lastly, draw NB, MC intersecting each other in R. Then 

 OR always tends to the same point, the intersection of the 

 lines BE; CD. 



This corollary has fifteen separate cases, all of which are 

 referable to the different cases of Prop. 6. 



Cor. 3. Let S'VT'W (fig. 5) be any trapezium whatever. 



Through 



figure, however, had been long celebrated amongst the Greeks, Romans, 

 Etruscans, and other nations, before it was introduced into this country. 

 Whether it was a Druidical figure 1 have not been able to learn, though 

 I should think it highly probable that it was. The medial section, or ex- 

 treme and mean ratio, probably obtained the name of the "divine section " 

 from its necessity in constructing the pentagonal base of this figure, and the 

 figure itself having been considered emblematical of the Deity. 



BishopWilkins, speaking concerning characters that express words, goes 

 on to say : " Of this nature was that angular figure so much used by the 

 Grecians of old, which might be resolved into the letters vyisoc. 



yy^a 



" This mark was esteemed so sacred amongst the ancients, that Antiochus 



Soter, a perpetual conqueror, did always instamp it on his coin and in- 



Vol. 68. No. 340. Aug. 1826. Q scribe 



