418 Mr. J. P. Henuessy on some Demonstrations in Geometry. 



most eminent editor of Euclid remarks, "only proves that a 

 thing must be so, but fails in showing why it must be so; 

 whereas direct proof not only shows that the thing is so, but 

 why it is so." 



To render the first book of the Elements in this respect perfect, 

 it would be necessary to alter the proof often propositions ; viz. 

 the VI., VIII., XIV., XIX., XXV., XXVI., XXVII., XXIX., 

 XXXIX. and XL. Dr. Lardner has given direct proofs to the 

 VIII. and XXV., and I have done so to the XL.*, leaving seven 

 still proved indirectly ; of these I will proceed to show how the 

 VI., XIX., and XXVI. may receive direct demonstrations ; of 

 the remaining four I believe the XXXIX. will at some other 

 period be proved directly, but that the XIV., XXVII. and XXIX. 

 never will. I am led to form this opinion of the three last 

 because they rest on defective premises ; the XIV., on the defi- 

 nition of a right line, which is unintelligible ; and the XXVII. 

 and XXIX. on the theory of parallels. 



According to all geometricians, the fundamental rule in geo- 

 metry is, " that the truth of a proposed principle is to be deduced 

 from the axioms and definitions or other truths previously and 

 independently established." We may therefore place the VI., 

 XIX., XX., XXI. and XXVI. after the XXXII., because that 

 proposition and all others before it are proved independently 

 of these five. 



With this arrangement, the following direct demonstrations of 

 the VI., XIX. and XXVI. can be given :— 



VI. " If two angles (B and C) of a triangle (BAG) be equal, 

 the sides (AC and AB) opposed to them are also equal." 



Find a point D which is equidistant from a 



the three vertical points of the triangle, (X., 

 XII. and IV.). 



The angles BCD and CBD are equal (V.), 

 therefore DCA and DBA are equal ; but these 



are respectively equal to CAD and BAD, there- ^'^ ^C 



fore CAD and BAD are equal ; therefore the remaining angles 

 ADC and ADB are equal (XXXII.). In these two triangles the 

 sides AD and DC are equal to AD and DB, and the included 

 angles equal, therefore AC and AB are equal (IV.). 



XIX. If in any triangle (BAC) one angle (B) be greater than 

 another (C), the side (AC) opposite the greater angle is greater 

 than the side (AB) opposite the less. y^ 



From the point C draw CD parallel to AB and 

 equal to AC. 



As the angle ABC is greater than BCA, the 

 angle DCB which is equal to ABC (XXIX.) 

 must be greater than BCA, therefore the line 

 * Phil. Mag. S. 3. October 1850. 



