132 Mr. Graves oil the Porismatk Arrangement of Points. 



Theorem II. may be stated in various ways. I believe it is 

 best known in the following statement (see fig. 2). It' a vari- 

 able triangle (e. g. 2, 3', ¥) whose sides always pass respect- 

 ively through three given points (2' 4 3) lying in directo^ 

 have always two of its angles (2 and 4') on two given straight 

 lines (1 2 r and 4' 5' 1) its third angle will always lie on a 

 third straight line (5 1 3'). This proposition, which is not 

 new except in form, admits of the following converse and ex- 

 tension. If a w-given lateral figure be inscribed in 7w-lines 

 diverging from a point, while m—\ of its sides pass respect- 

 ively through given points lying in diredo, the remaining side 

 will always pass through a fixed point. 



In fig. 2. I, 2, 3, 4, 5 and T, 2\ 3\ 4\ 5' are pentagons 

 which circumscribe each other ! Fig. 2 is dotted and coloured 

 for the sake of more clearly exhibiting this to the eye. If 

 one pentagon and a point in one of its sides be assumed, the 

 remaining points of the other pentagon will be necessarily 

 determined by the scheme. It was by examining the scheme, 

 not by inspecting the geometrical diagram, that I discovered 

 the property of mutual circumscription. No triangle can be 

 picked out of fig. 2, having 3 of the 10 points for its vertices, 

 and having its 3 sides of the same colour. 



I have stated the preceding theorems in a new form, for the 

 purpose of directing attention to the symmetry and functional 

 properties of the combinations of the symbols denoting points. 

 The 9 sets in theorem I. and the 10 sets in theorem II. form 

 completed circuits of arrangement^ such that if in either case 

 every set but one were given, the mind would b}' a natural 

 induction from a perceived law of combination be led to sup- 

 ply the omitted set. This encourages the belief that, in cases 

 where position without regard to magnitude is considered, 

 recourse should be had to functional rather than algebraic 

 equations, and that a systematic and manageable notation ap- 

 plicable to geometi'ical points might be devised which should 

 not only express but suggest geometrical theorems. 



Pondering on the nature of straightness, I have arrived at 

 the following definition of a straight line. " Let a line be called 

 straight, if it be ideally possible to divide it into two parts, 

 each of which shall be of the same form as the whole." It 

 is much more than enough to say, every part of a straight 

 line is an exact model of the whole to which it belongs, and 

 of every other straight line. The simple property stated in 

 the definition is not possessed by any other line. Even cir- 

 cular and helical lines possess it not, although in the same 

 circle or helix the curvature is the same at every point, and 



