384 THE NINE-POINTS CIRCLE. 



pedagogic theory, and I am also in hopes of helping teachers of 

 geometry who do not venture to call themselves mathematicians. 

 I am only an amateur myself. I shall therefore gain my end, 

 without, I hope, losing the interest and helpful criticism of 

 mathematicians, if I put forward the whole manner in 'which 

 the right presentment developed in my mind. 



The figure of the triangle with its scribed circles may be 

 regarded as a network of straight lines touched by circles, or as 

 a network of circles touched by straight lines. It is strange 

 that there should be four circles and only three straight lines. 

 The dualism which runs through all geometry makes us suspect 

 that something is wanting. And there is. The missing link is 

 the Nine-points Circle, and Feuerbach's theorem simply states 

 that fact. 



For the whole system may be regarded as a special case of 

 a general problem — 8 conies (in this instance circles) touching 

 one another four and four. Four of the circles are quite isolated 

 from each other : the other four (three of which have their 

 /centres at infinity, and are therefore straight) not only touch 

 the rest, but also intersect one another. 



Looked at from this point of vievv^, the Nine-points Circle 

 may, Hibernically speaking, be called the fourth side of the 

 triangle. The more we consider the functions of this circle, 

 the more inept does its " Nine-points " name appear. I wish 

 we might call it the Link Circle. I myself, in my previous 

 paper, suggested the name Orthocentric Twin-point Circle; but, 

 besides being somewhat abstruse, this was obviously too heavy. 

 Marsano called it the circolo medioscriUo',. and Mackay tried to 

 naturalise the term in English ; but this name classifies it with 

 the scribed circles, whereas it should be classified with the sides. 



If we consider this system in a state of flux, we see the 

 Link circle grow smaller and smaller as the triangle approaches 

 regularity. It reaches its minimum when it collapses on the 

 in-circle, unites the points DLXP, etc., on each of the three 

 sides, and brings all three e-scribed circles, already in contact 

 with itself and the sides, into contact also with the in-circle, 

 all in perfect symmetry. 



At the first infinitesimal change from this regularity, we see 

 the points DLXP, etc., separate in the proportion DL.DP = DX- 

 (we shall presently see why), and we see the four scribed circles 

 take an infinitesimal list each in its own direction : we see also 

 the Link circle take a corresponding expansion pari passu in all 

 these four directions. Each further infinitesimal side movement 

 of the scribed circles is accompanied by an expansion of the 

 Link circle in the same direction, and apparently of the same 

 order of infinitesimal magnitude. The calculus could, I sup- 

 pose, easily show the truth of our impression ; but already, from 

 the mental visualising, we see that Feuerbach's theorem is only 

 " what we might have expected." 



If we follow the movement till one of the vertices goes to 

 infinity (Fig. i), we see that it takes with it six of the Link 



