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On the Theory of Parallel Lines. By Henry Meiklb, Esq. 

 Communicated by the Author. 



In my former article on the same subject, in this Journal 

 for April 1844, it was hinted that some doubt might be sup- 

 posed to attach to a certain part of the proof ; but a hope was 

 likewise expressed that my second proposition would lead to 

 something better. Accordingly, by means of it, I have been 

 so fortunate as to obtain the following legitimate substitute 

 for the part in question ; and here, as formerly, brevity has 

 been studied, especially in the more obvious details. 



Prop. The angles of a triangle cannot be less than 180°. 



It was proved, as a case of the second proposition, that if 

 any one triangle had the sum of its angles less than 180°, so 

 would every triangle, and the defect from 180° would always 

 be proportional to the area. Hence, proceeding on the same 

 hypothesis, a very large triangle should have the sum of its 

 angles very minute, and the two at the base still less. Now, 

 it is shewn under the first proposition, that on a base equal 

 to any side or base of any given triangle, there may always 

 be as great an isosceles triangle ; but the smaller an isosce- 

 les triangle is on an equal base, the smaller evidently will 

 its angles be at that base. 



Let ABC be a triangle in which angle B is large, say = 

 150°, and AB = BC = the greatest side of 

 the foresaid large triangle ; and let BC, 

 CD, &c., be an indefinite series of equal 

 bases in as many triangles whose com- 

 mon vertex is A, and in which the angles 

 ABC, BCD, &c., continually decrease, 

 such that those triangles shall gradu- 

 ally approximate to being isosceles, and yet not more than 

 one of them could be perfectly isosceles, otherwise some 

 of the decreasing angles ABC, BCD, &c., would not only 

 be equal, but have that equality beginning with an in- 

 crease ; thus two angles at the bases of the isosceles tri- 

 angles would exceed the sum of one and the angle just be- 



