436 PROFESSOR KELLAND ON THE LIMITS OF OUR KNOWLEDGE 



Prop. IV. Given that the angles of every triangle are together equal to two right 



angles, to j)rove Euclid's 12th axiom. 



Let AB, CD make with EF the angles BEF, EFD less than two right angles ; 

 AB, CD, shall meet, if produced, towards B, D. 



Let EFG be not greater than a right angle. 

 Draw EG perpendicular to FD. Set off equal 

 distances along EB, viz., EH, HI, &c., and draw 

 HK, 10, &c., perpendicular to EG; and HP per- 

 pendicular to 10. 



The angles BEG, GEE, EFG, are less than two right angles, but GEE, EFG, 

 make up a right angle ; therefore BEG is less than a right angle. Now EHK 

 and HEK make up a right angle, and EIO, lEO make up a right angle, therefore 

 EHK = EIO; and the triangles EHK, HIP, are equal (Euc. L 26), therefore HP 

 = EK; but the triangles KHO, HOP, formed by joining HO, are equal (Euc. I. 

 26), therefore KO = HP = EK. It follows, therefore, that as we advance by 

 equal distances along EB, we also advance by equal distances along EG; so 

 that by going far enough along EB, we must at last advance beyond G ; hence 

 there is some point in EB produced, from which, if a perpendicular be drawn to 

 EG produced, it shall cut EG produced beyond G; AB, therefore, meets CD, 

 towards B, D. 



Let us now examine the consequences of assuming as true the following 



Axiom. The three angles of every triangle are together less than tno right 

 angles. 



It may be remarked, that it would have sufficed to have assumed that the 

 three angles of some one triangle are together less than two right angles. For, 

 that being admitted, the axiom as enunciated follows at once ; inasmuch as no 

 triangle can have the sura of its angles greater than two right angles (Prop. I.), 

 neither can any triangle have them equal to two right angles, for then the same 

 would be true of every triangle (Prop. II.) 



Cor. 1. The exterior angle of every triangle is greater than the sum of the 

 two interior opposite angles. 



Cor. 2. The angles of a quadrilateral are together less than four right angles. 



We shall designate the deficiency of the angles of a given triangle from two 

 right angles by the term angular defect of that triangle ; and shall abbreviate the 

 phrase "angular defect of the triangle ABC" by ^ABC : hence ^ABC = 180'- 

 (A + B + C.) 



