268 [Recess 



(0) C 24 H 16 14 + 2HO=C 8 H 10 O 8 + C 14 H 8 



Picroerythrin. Erythroglucine. Orcin. 



2C 1C H 8 8 =2C 14 H 8 4 



Orsellinic acid. Orcin. 

 These equations have been established either by myself or other 

 observers. 



IX. " On the Theory of Parallels." By Lieut.-General T. PER- 

 RONET THOMPSON, F.R.S. Received August 4, 1862. 



More than thirty published efforts, from Ptolemy downwards, 

 attest the satisfaction with which the Theory of Parallels would be 

 seen established without merging the difficulty into an axiom. 



As many of these are certainly not elementary, it amounts to an 

 admission that any knowledge on the subject would be acceptable, 

 even though it left the necessity of beginning from the axiom with 

 freshmen. 



Believing that the generation of the straight line with the impos- 

 sibility of two enclosing a space, and of the plane with the straight 

 line joining any two points lying wholly in the surface, may be rigidly 

 demonstrated from the property of the sphere, which Plato calls its 

 "perfection," or the faculty of turning about its centre without 

 change of place, I am induced to submit whether some light may 

 not be offered by the following : 



Place two equal circles in the same plane, and let a straight line 

 rest upon them (spheres and a superincumbent plane might be taken, 

 but it is conceived the other is easier). The centres remaining at 

 rest, let the circles dilate as by inflation, preserving always the 

 equality of the diameters to one another. 



It would appear to be deducible from the Platonic property, that 

 the motion of any point in the circumference during the inflation 

 must be perpendicular to the circumference, and consequently at any 

 instant to the straight line which touches the circumference. Also 

 the touching point in that straight line is at any instant impelled in 

 a direction perpendicular to the circumference and to the touching 

 line ; out of which it seems impossible that the points of contact in 

 the circumference and in the touching line should ever separate ; for 

 that would imply a motion other than perpendicular in one or both. 



If this was supposed allowed, it would follow from making the 



