1842.] On the Theory of Angular Geometry. 239 



A and B being each parallel to C, B is parallel to A. For A C = 

 A B -J- B C, but A C and B C are each a zero, .*. A B is also zero, or B 

 parallel to C. 



Proposition XII. (Fig. 13 J 



If a straight line cut one of two parallels, it must cut the other. 



A C, meeting A B, not meet its parallel G D, parallel to E D, conse- 

 quently A C, A B being both parallel to E D, are parallel inter se, which 

 is not the case. 



The only other property of parallel lines not included in the above is, 

 that two straight lines which are respectively parallel to two others con- 

 tain an angle equal to the angle of those others. But there is nothing 

 peculiar in its demonstration. These thirteen propositions contain a 

 complete and homogenous geometry of position as contra- distinguished 

 from that of magnitude : I speak of course relatively to lines. It is 

 scarcely necessary to refer the student to the Third Book of Euclid, as 

 far as relates to the consideration of angles in a circle, to shew how 

 much this mode of treatment, and the introduction of reverse angles 

 would simplify the subject, as well as prepare him for analytical in- 

 quiries by generalising his ideas on it. 



Postscript. 



In looking over some of the mathematical articles of the Penny Cyclo- 

 pedia, written by Professor De Morgan, I have subsequently to the writ- 

 ing of the above, found a confirmation of my views as to the nature of 

 the angle under the heads, " Angle" and " Infinite." 



The former proposes to introduce the axiom, that " two spaces whe- 

 " ther of finite or infinite extent are equal, when one can be placed upon 

 " the other, so that the two shall coincide in all their parts." After 

 which it is remarked, that Bertrand's demonstration becomes rigorous. 

 This also considers an interparallel space viewed as an angle to be zero, 

 as I have done, since it is less than any assignable angle. 



The latter has the following passage : — 



" The comparison of such infinite spaces is therefore possible, con- 

 " sistently with perfect clearness in the meaning of the terms employed, 

 " and a simplicity of reasoning which would convince any one who is 

 " capable of the most ordinary thought. Had Euclid been accustomed 



