Mr. Graves on the Porismatic Arrangement of Points. 133 

 the same lengths have the same form. For instance, the cir- 



C) ,-P fff 



cular arc -7— is not a model of the arc ■;— , nor of any other 



arc belonging to the same circle, and shorter or longer than 

 itself. No mere magnifying nor diminishing will make any 

 such shorter or longer arc agree in shape as well as length 



with — - That the condition of the preceding definition 



contains in itself no contradiction, or, in other words, that 

 straightness, so defined, is possible ; that there is but one form 

 of line which admits so much as one division possessing the 

 required characteristic property ; and that every division of 

 the unique form equally possesses that property, — these are 

 three true propositions, which seem to me to admit of proof, 

 if we assume it as fact, or can prove independently, that there 

 is a unique shortest distance between two given points. 



Being once asked to arrange 19 trees in 9 rows, having 5 

 trees in each row, I found, upon trying to place in directo 

 certain previously formed combinations which seemed feasible, 

 that the feat might be performed porismatically with 18 trees 

 only, thus: 



Fig. 3, 



Fig. 3 is in fact fig. 1., with every line produced to meet 

 every other line. A tree is to be supposed planted at every 

 intersection. 



Mr. Sylvester has investigated very curious practical me- 

 thods (which I hope he will publish) of forming numerical 

 combinations which shall possess certain assigned properties, 

 and he has also long entertained the opinion that geometry, so 

 far as it relates to position, is dependent on a higher or more 



