134- Mr. Graves vn the Porismatic Ayrangement of Points. 



abstract science of form, just as geometrical magnitude is sub- 

 ject to the science of number. Though the tree problem had 

 led me to consider combinatorial schemes for the arrange- 

 ment of points, it was from conversation with Mr. Sylvester 

 that I was induced to attribute any importance to the study 

 of the symmetrical combinations which the notation of geo- 

 metrical theorems of position presents, and to suppose it pos- 

 sible that, more generally, the study of functional forms might 

 produce a method suggestive of new geometrical theorems. 

 It would probably be well for the theories both of geometry 

 and functions, if an expressive symbolic analogy could be con- 

 trived between geometrical and functional form. In the or- 

 dinary Cartesian equations of curves, the symbols of mere 

 magnitude precede and almost wholly constitute the formulae 

 which, to a practised eye, disclose relations of curvature. I 

 desiderate rather a converse system of notation, or at least 

 a system in which more of the elements than + and — should 

 directly symbolize form. I hail the introduction into the al- 

 gebra of geometry of such things as spherical coordinates, — as 

 elliptic sines and cosines, — as r (cos 9 + i^ —1 sin 6) employed 

 to denote a line of a certain length inclined to a given ori- 

 gin at a certain angle. Let us at all events pay more atten- 

 tion to the mere functional properties of the algebraic equa- 

 tions of curves, and to the geometrical interpretation of those 

 properties, if we cannot invent a notation which shall prove a 

 more direct and obvious index to the nature of their curva- 

 ture. 



I take this opportunity of publishing the following four 

 theorems which occurred to me about fifteen years ago, and 

 have never, to my knowledge, appeared in print ; but I am 

 little acquainted with the writings on the subject, and can 

 scarcely expect that in a field so often gone over, any glean- 

 ing should be left. I will not at present further occupy the 

 pages of this Magazine with a proof p- ^ 



of them, but merely mention that the °* 



proof of them, like that of the pre- 

 ceding theorems, may be made to 

 depend on the well-known constant 

 property of the section of all trans- 

 versals drawn across the same four 

 straight lines diverging from a point. 



1. If from two given points {a . c) 

 (fig. 4.) in the base (^ c a ^) of a tri- 

 angle {e,g,h)i any two straight lines 

 (azj:,c2;r/) be drawn to the sides, so ^ 

 that the intercepts {g x, h y) from the extremities of the base 



