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PROFESSOR KELLAND ON THE LIMITS OF OUR KNOWLEDGE 



It is required to find a point H 



which is therefore less than a right angle 

 in BC produced, so that the triangle 

 ACH may be equal in area and in the 

 sum of its angles to ABC ; whenever 

 it is possible. 



Upon AC describe the triangle 

 DAC = BAC (Euc. i. 23.) Bisect DA, 

 DC in E and F ; join EF, and produce it 

 till it meets BC in G, when it is possible. 



Make GH = CG, CAH is the triangle required, For (Prop. XX. Cor. 8) AO = OH ; 

 therefore (Cor. 9) the triangle ACH is equal in area to ADC, that is, to ABC. 



Cor. By carrying on this process, we can multiply the triangle ABC to some 

 extent, as is done in Euclid vi. 1. But here there is a limit to the multiplication. 

 It must terminate when EF fails to meet BC produced. That this will happen 

 after a finite number of equal triangles has been found, we now proceed to show. 

 First, the further angle of the triangle constantly diminishes (Euc. I. 16). Let 

 ACB (to save the trouble of drawing another figure), be any one of the equal 

 triangles. The construction shows that the perpendiculars from A and C on EG 

 are equal ; therefore (Prop. XVI 11.) 



-=i:ACB-^-rEGB = 5 area included by AC, EG, CG, and the common perpen- 

 dicular. 



Now, the common perpendicular bisects the area included between the two 

 perpendiculars from A and C on EG (Prop. VI. Cor. 2), and this area is equal to 

 the area of the gwen triangle ABC ; therefore -:^ACB- EGB:p^^^ABC. It follows, 

 therefore, that for the addition of every new triangle equal to ACB, the angle in 

 advance suffers a diminution of more than half the angular defect of the given 

 triangle. It will therefore be reduced to zero, or a negative quantity, by taking 

 a finite multiple of the triangles ; after which no point can be found in BC 

 produced which will yield another triangle equal to the original triangle. 



Again, let ACB be a triangle of which C is an obtuse angle ; and let AX be 

 drawn perpendicular to BC produced ; then, since both the triangles ABC, ACX 

 are finite, a finite multiple of ABC will exceed ACX. Let ACY be the first 

 multiple of ABC that exceeds ACX ; ARY the last of the triangles, each equal 

 to ABC, which constitute their multiple. Then the triangle ARY may be treated 

 as the triangle ABC is treated above ; and the conclusion is general : that only 

 a finite miiltiple of a given triangle can he formed hy joining the vertex mith succes- 

 sive points in the base j^^'oduced. 



This very curious result is important, as maintaining the consistency of the 

 results mentioned in Prop. XVIL, and it serves as a strong caution against hasty 

 inferences. 



To retain Euclid's definition of a parallelogram, it is requisite to combine with 



