434 PROFESSOR KELLAND ON THE LIMITS OF OUR KNOWLEDGE 



novelty which it would otherwise present. In truth, many of the following pro- 

 positions are due to my students. 



For the sake of clearness, I shall divide the different steps of the argument 

 into propositions. Legendee, in the 12th volume of the Memoires deV Institute 

 proved, — 



Prop. I. The sum of the angles of a triangle can never' exceed two right angles. 



Prop. IT. If the angles of any one triangle can he proved to be equal to two right 

 angles^ then the angles of every triangle can. 



These two propositions reduce the difficulty to the very narrow requirement 

 of proving that some one triangle has the sum of its angles equal to two right 

 angles. As a limit, it is evidently true. For if CD be perpendicular to CA ; and 

 if CD be very great and CA indefinitely small, the angles of the triangle CAD 

 approach two right angles as their limit. But this, as we shall see presently, 

 proves nothing. For although they approach two right angles when CA is in- 

 definitely diminished, they may, for anything that appears, approach a right angle 

 and a half, or any similar magnitude, when CA is indefinitely increased. 



Further, Mr Meikle has proved in the 36*th volume of the Edinburgh Philo- 

 sophical Journal — 



Prop. III. Triangles which ham their areas equal have the sum of their anghi^ 

 the same. 



We shall consider these three propositions as established beyond question, and 

 refer to them in the order here given. 



Some years ago, there appeared in Crelle's Journal, a notice of a work entitled 

 " Imaginary or Impossible Geometry," viz., a discussion of the conclusions which 

 would follow from the assumption as an axiom, of the hypothesis that " the three 

 angles of a triangle are together less than two right angles." I have never met 

 with any statement of the propositions which the author deduced from this hypo- 

 thesis ; but I have been accustomed, from time to time, to draw conclusions from 

 the same hypothesis, and to induce my students to follow my example, so that, 

 from one source and another, I believe I am in possession of most of those 

 conclusions which are likely to bear on the theory of parallels. And as these 

 conclusions are both curious in themselves, from their connection with the 

 properties of the circle, and appear to point to the limits of our knowledge of the 

 doctrine of parallels, I have thrown them together in the form of a sequence to 

 the three propositions above enunciated. 



It must be premised, that all the definitions and axioms of Euclid are retained, 



