234 On the Theory of Angular Geometry. [No. 123. 



" in settling his ideas of greater, less, and equal on that definition, that 

 " the whole objection to M. Bertrand's demonstration turns." 



I have been minute in these quotations, not only because they contain 

 all that to my knowledge has been developed on a very interesting sub- 

 ject, but also in the hope that they may draw further attention. Led 

 independently to similar conclusions, by attempting to trace the natural 

 affinities, if we may so term it, of geometrical truths, with the intention 

 of forming a definite arrangement of them, I was induced to trace their 

 consequences in establishing the various relations of angular space. 

 The results of the inquiry may be thrown into a connected chain of pro- 

 positions, as subjoined. 



Definition 1 . — The plane surface between two straight lines, bounded 

 in the one direction, unlimited in the other, — is called an angular space. 



Definition 2. — When an angular space is bounded on one side by the 

 intersection of the containing lines, it is called an angle. 



Definition 3. — The point of intersection is called the vertex, and the 

 containing lines are called the sides of the angle. 



Axiom. — From the definition, it will follow that two angular spaces 

 A B C D, and E F G H, must be compared thus : If placing the line F E 

 on B A, we find that, F falling on the point P, G will fall on some point 

 Q in C D, then according as the line G H falls within, upon or without 

 the line C D — is the angular space A B C D, greater than, equal to, or 

 less the angular space E F G H. (Fig. 1.) 



Definition 4, 5, 6. — Euclid's definitions of right, acute, and obtuse 

 angles. 



Proposition I. 

 Every angular space is equivalent to its angle. 



This follows from the axiom, since the sides of the angular space, and 

 of the angle are identical, and may therefore be considered to coincide. 



Proposition II. (Fig. 2. J 



All right angles are equal to one another. 



Let the right angles ABC, E F G, be made respectively by A B with 

 B C and E F with F G; they are equal. Produce C B to D and F G to H, 

 and apply the figures one to the other, so as to make F coincide with B 

 and G H with C D. If then F E do not coincide with B A, let it fall 



b 



