On the Theory of parallel Lines in Geometry. 425 

 defect alluded to will be more readily seen by extracting the 

 hrst paragraph and annexing a diagram somewhat different 

 rrom the original. 



" Prop. III. The three angles of a triangle are equal to 

 two right angles." ^ 



." ^! '^^i^^i.i^ affirmed be not true, let the three angles of the 

 triangle A C B be less than two right angles, and let the defect 

 trom two right angles be equal to the angle x. Let P stand 

 lor a right angle, and find a multiple of the angle .r, viz. m xx, 

 such that 4P-?«.r, or the excess of four right angles above the 

 multiple angle, shall be less than the sum of the two angles 

 , ' 5 S ^^ ^^^ proposed triangle. Produce the side C B, 

 and cut off BE EF, FG, &c. each equal to BC, so that the 

 whole CG shall contain CBm times; and construct the tri- 

 angles BHE, EKF, FLG, &c. having their sides equal to 

 the sides of the triangle A C B ; and consequently their angles 

 equal to the angles of the same triangle. In G A produced 

 toke any point M, and draw HM, KM, LM, &c. ; AH, 

 rl K, is^L,, &c. 



Having thus given the construction, Mr. Ivory proceeds 

 prematurely with the investigation; for it ought to have been 

 previously demonstrated, that in such a construction the points 

 rrl- I -1 ^'^spectively below the lines KM, LM, &c. 

 Ihis he tacitly assumes without proof; and it is this assump- 

 tion which stands opposed to the angle x, and enables him to 

 bring out an absurd conclusion; for if no such assumption is 

 ventured on, the investigation, so far as I can see, comes to no- 

 thing at all, as is abundantly evident from what follows. 



The triangles 

 ABH, HEK,&c. 

 having by con- 

 struction two sides, 

 and the contained 

 angles in each equal 

 to those in another, 

 are equal : hence 

 the angles A H K, 

 H K L, &c. are 

 equal; and the three angles of each of the triangles ABC, 

 B H E, &c. being less thaii two right angles by the angle x, 

 the angles A H K, II KL, &c. are each short of two right an- 

 gles by a quantity not less than x, but perhaps by one much 

 greater. If therefore x be a right angle, each of the angles 

 A 1 1 K, II K L cannot exceed a right angle, but, for any thing 

 we are yet supposed to know, they may be far short of right 

 angles; because the triangles A B H, HEK, &c. may each 

 have ihc siun ol' its .-mgles much less than two right angles. 

 \ oi. h'O. No. 29G. Dec. I 8'22. .'} H If 



