63, 



praise and the angels, through fear of Him ! He sends the 

 thunderbolts to strike therewith whomever He pleases, even 

 while they are disputing- concerning God ; indeed. He is 

 of immense power ! His name is Cayeeala Tacamamu 

 Daburim Buaburim Jawburim Masseeyayaramu. O God ! 

 in Salvation ; if it is the will of God." 



Mr. Wilkinson also laid before the meeting the following 

 Geometrical Theorems. 



If from the angular points of any triangle, ABC, lines be 

 drawn making the same constant angles with the adjacent 

 sides, four triangles, AiBiQ, A2B2C2, A3B3C3, A4B4C1, will 

 be formed which possess the following properties : — 



1. The triangles A,B,Ci, A.B^Co, A3B3C3, A,B,C„ are all 

 similar amongst themselves and to the triangle ABC. 



2. If circles be described about the triangles A2C A, BoA B, 

 C2B C, they w^ill all pass through a common point P. 



3. Circles described about the triangles A3BA, B3C B, 

 C^A C, will all intersect in another common point P. 



4. If Oj, O2, Oo, be the centres of the circles in (2) ; and 

 O4, O5, Og, the centres of those in (3) ; then the triangles 

 OjOoOs, OiOgOg are similar to each other and to the original 

 triangle ABC. 



5. The triangles O1O2O3, 040g06, are coaxial, sinc6 their 

 vertices lie two and two on the radials O Oi, O Oo, O O3 ; 

 they are therefore also copolar, or the intersections of their 

 opposite sides range in the same straight line. 



6. The triangle OiOoOgziithe triangle O4O5O6; for 

 16 ABCx 0,Oo03=16 AB C X 0,0,0o= aH' -\-a^c' 



7. If ri, ^2, r^i Ti, 7*5, re, be the radii of the circles in (2) 

 and (3) respectively, and R = the radius of the circle circum- 

 scribing ABC; then R^ = r^r.^'z = 7\r^rc. 



8. Let f, g, h, be the intersections of A C3, C A,; A B3, 

 A Bo; CB3, BC2; then the lines A//, B/, Cg, will all 

 intersect in the same point P2. 



