324 I'Ri:SKXTMi:XT AND PK» i( )1" IX (;!:()M !:tk>". 



DX is the mean proportional between DL and Di' ; this, there- 

 fore, is the property we necessarily use. Now, when the triangle 

 is reversed along AL (the angle-bisector), the tangent LX 

 becomes LX\ and being now antiparallel, is, of course, at an 

 angle C-B to its former self. Now take the three points X\. L 

 and P (characteristic points respectively on the In-circle, the 

 triangle and the orthocentric twin-point circle), and describe a 

 circle through them, cutting the In-circle again at H. 



Fig- 1.3- 



Now IIX^ will meet BC at its mid-point; for DX\ DH = 

 DX- (for one circle), and = DL. DP (for the other). There- 

 fore DX is the mean proportional between DL and DP, and 

 consequently D is the mid-point of BC. Again, the angle IMIP 

 being in a cyclic quadrilateral = the angle DLX^, i.e., = C-B. 

 But C-B is the angle which DP subtends in the orthocentric 

 twin-point circle (a well-known property, easily proved). 

 Therefore H is on both circles. And because XI X^ at the centre 

 = C-B, the angle DHP at the circumference is bisected by HX. 

 and therefore HX bisects the arc DP. Hence the joins of the 

 ends of two sets of parallel radii meet in H, which is therefore the 

 centre of similittide, which, being on the circles, must be a point 

 of tangency. 



Here, again, we see how the triangle subdues the whole 

 plane to itself. There are three of these auxiliary triangles and 

 circles, one for each side of ABC. H, being a centre of 

 homology, may therefore be regarded as a point generating four 

 infinite series of circles, each series being in contact at the start- 

 ing-]:)oint. And as the same holds for the t^-circles, we have 

 sixteen such series sweeping fan-shaped through the plane. It 

 is a kind of geometrical picture or analogy of the Avay in which 

 each individual mind polarises the whole universe to its own 

 personality. 



Comet I9I6a. — C)n the 2-ith February an apparentl\- 

 short-j^eriod comet was discovered at the Simeis Observatory. 

 Crimea, by M. Neujmin. It was then 6° south-west of Mars 

 and of the eleventh magnitude. 



