l.JO Mr. Graves on the Porismatic Arrangement of Points. 



remaining set from any other such system. For example, 

 instead of putting the set 3" l" 3 last, as was done in the pre- 

 ceding scheme, we may put 2' 3' 2' last, and we can get an 

 arrangement which is a function ol' its constituent symbols 

 similar to the preceding, thus : 



3"r3 3 13' SM'S' 



r t' I 12 r r 2' r^ 



2'' 3'' 2 2 3 2' 2' 3' 2'' 



Theorem I. may be stated in various other ways, and fig. 1 

 will assume various appearances according to the position of 

 the arbitrarily assumed points. It is interesting to examine 

 the limits and relations of the arbitrary and the necessary ac- 

 cording to the scheme. For example, take 6 points, arbi- 

 trary except in as much as they form 2 sets, and the remain- 

 ing 3 points will be necessary, according to the scheme. 

 Again, take 5 points, no 3 of which form a set, and the re- 

 maining 4 will be determined. Theorem I. is a particular 

 case of a theorem due, I believe, to Pascal, viz. the intersec- 

 tions of the opposite sides of a hexagon inscribed in a conic 

 section lie in directo. I may add, however, that, before the more 

 limited proposition occurred to me, it had not, so far as I am 

 aware, been treated as a particular case, or otherwise noticed. 

 In fig. 1, the conic section always appears in the form of 

 two straight lines. For example, the angles of the hexagon 

 1, 2, 2'\ 2\ r\ 3 lie alternately on the lines T' 2'' 1 and 2 3 2\ 

 and the line 3' T 3'' connects the intersections of its 1st and 

 4th, 2nd and 5th, and 3rd and 6th sides. I abstain from 

 proving that those intersections, as a necessary consequence, 

 lie in directo^ because the proof would carry me away from 

 the principal purpose of this paper, and proofs of Pascal's 

 theorem are well known. The proposition relating to a hexa- 

 gon inscribed in a conic section is a great acquisition to the 

 geometry of the rule, as distinguished from the geometry of 

 the compass treated by Mascheroni. Its converse enables us, 

 being given 5 points, to trace out all the other points of a 

 conic section by the mere intersections of straight lines drawn 

 to and from given and arbitrarily assumed points without 

 measurement of angles or distances. 



Definition. — A polygon is said to circumscribe another when 

 the angles of the latter lie upon the sides of the former 

 (whether produced or not). 



In fig. 1. the triangle 1, 2, 3 circumscribes T, 2\ 3', which 

 circumscribes 1", 2", 3", which circumscribes 1, 2, 3 ! We have 

 thus a ternary circulating series of mutual and simultaneous 

 circumscription and inscription 1 , 



