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PROFESSOR KELLAND ON THE LIMITS OF OUR KNOWLEDGE 



Prop. XVL If the angular defects of two triangles are equal, the areas of the 

 triangles are equal. 



Let ABC, DEF, be two triangles which have the same angular defect ; 

 they have the same area. ^; A 



If not, let the area of ABC be greater 

 than that of DEF. Cut off ABG = DEF 



(Post. 1), then MBG = ^DEF (Prop. III.), ^^- ^c f 'f 



but 5ABC=MBG + 5AGC (Prop. XV.), 

 = 5DEF + MGC; 



Therefore 5AGC = 0, which is contrary to the hypothesis (axiom). The areas are 

 therefore equal. 



Peop. XVII. The areas of triangles have to one another the same ratio as their 

 angular defects. 



Let ABCi DEF, be two triangles, area ABC : area DEF : : ^ABC : ^DEF. 

 Let ABC be greater than DEF : Cut off from ABC the part ABG = DEF (Post 

 1). Suppose ABG, ABG, commensurable ; and let ABC and ABG be divided 

 into triangles of the same area (Post. 2) : let ABC contain m such triangles 



and ABG 7i of them, 

 ^ABC = sum of all the angular defects of the triangles into which it is divided 



—m times the angular defect of one of them 

 8Pi3(ji—n times the same 



.-. ^ABC : ^ABG : : m : n : : area ABC : area ABG. 

 Therefore (Prop. XVI.) ^ABC : 8DEF : : area ABC : : area DEF. 



This demonstration is only applicable when the larger triangle infinite. 



If we desire to extend the demonstration to the case in which the larger 

 triangle is unlimited, and the smaller triangle finite, we require to introduce a 

 third postulate, viz., that a triangle of which the vertex is given may be increased 

 indefinitely by increasing its base. This postulate (if admitted) would lead at 

 once to the conclusion that ^ABC : ^ABG : : area ABC : area ABG, whilst 

 ABG is made as large as we please; and, consequently, if ^ABC be a finite 

 quantity, ^ABG might be made to exceed two right angles by making ABG large 

 enough. But as this is manifestly impossible, we should have a proof of the fact 

 that ^ABC=0, or the three angles of a triangle taken together equal to two right 

 angles. The postulate, however, is not legitimate as a logical canon. It evi- 

 dences, however, the extreme narrowness of the limits towards the received 

 doctrine to which the inquiry is pushed. 



