1842.] Remarks on the Theory of Angular Geometry. 241 



The definition has other faults: instead of saying "bounded in one 

 direction, unlimited in the other," he should have said bounded in two 

 directions, unbounded otherwise, (or elsewhere) : for surely the thing 

 meant is bounded by two straight lines, and therefore in two directions. 

 Moreover, not unlimited, but unbounded, is the opposite of bounded. If any 

 where, surely in geometrical definitions, it is indespensible that words 

 should be used with strict propriety, so as to avoid confusion. With 

 equal propriety it might be said, that the angle ABC + the angle 

 BCD = the corner A B D. 



As I hold that the definition of angle here proposed is a failure, so 

 likewise is the demonstration of the 

 property in Prop. VI., that the sum of 

 the three angles of the triangle are equal 

 to two right angles ; and for the same 

 reason. The space E F below the line 

 D B C G may belong to the angle A, 

 or to any thing else, as in the annexed figure. 



Instead of systematizing and refining till we get our ideas into an 

 atmosphere too sublime for them to be of any use, we may take in com- 

 mon sense view of the subject. In the triangle A B C lay a ruler on 

 the line A B, marking the ends towards A and B with the corresponding 

 letters. Turn the ruler about A, till the end marked B come into the 

 direction A C ; then let it turn about C, till the end A come into the 

 direction C B ; and finally let it turn about B, till the end B come into 

 the direction B A. The ruler has thus turned about each of the three 

 angles, and the ends marked A and B have changed places, shewing, 

 that the sum of the three angles of the triangle are equal to quantity 

 formed by turning a straight line half round, or to two right angles. 



In like manner, if we measure the exterior angles of the triangle of any 

 polygon, the ruler at last will have the same direction as at first, after 

 having gone completely once round ; or after having described four right 

 angles. 



If I were to write a Treatise on Geometry, I should without hesitation 

 introduce these as demonstrations of the theorems regarding the interior 

 angles of a triangle, and the exterior angles of any polygon. They have 

 long appeared to me to be quite as evident and satisfactory as any 

 principles in Geometry. A good treatise should be something like a 



