408 Dr. Adamson on Geometrical Demonstrations, 



one of this group, being the last of those which meet the given 

 base, and forming, of all these lines, the furthest from the given 

 perpendicular. Let such a straight line be drawn ; then extend 

 the base beyond it ; and it is evident, that, from a point in the 

 extension, another straight line may be drawn further from the 

 perpendicular than that which is furthest. This is a contradic- 

 tion, and therefore no other than the non-concurrent lines can be 

 at right angles to the perpendicular, or it is perpendicular to 

 both. 



Legendre's mode of obviating this difficulty has been objected 

 to, but not perhaps exactly at the point where it is objectionable. 

 We may grant as legitimate, the conclusion that an angle of a 

 triangle is, as to its magnitude, dependent on some function of 

 the other two, or that A=</>(B,C) ; but though it be a truth, the 

 certainty of which may mislead the mind in these inquiries, that 

 when B and C are given A is determined, yet that truth does 

 not necessarily result from this argument; for angles can be 

 found which are determined when other two are given, but are 

 not therefore absolutely determined to be equal. Even in the 

 case before us, 180^ — A 'and 90^— A, or the supplement and 

 complement of A, are expressible by the symbol ^ (B, C), and 

 so is any multiple or part of A. There is therefore nothing in 

 the nature of the argument, whatever there may be in the nature 

 of the facts, to exclude the position that when B and C are given 

 there may be two triangles containing them, but having the other 

 angles supplementary, or only definitely related, but not identi- 

 cal. In rectangular trihedral trigonometry (improperly called 

 spherical, for arcs may have nothing to do with the matter), any 

 one of three different angular quantities is expressible by <^(B,C), 

 therefore from the fact that A = </> (B, C) and A'=<^ (B, C), it 

 is not logical to conclude A=A'. 



All modes however of deriving anything from the properties 

 of triangles is, in regard to this subject, inadmissible, from an- 

 other consideration. If ever the vast advantage of natural ar- 

 rangements in regard to scientific treatises is adequately con- 

 ceived, then the subject of angles alone, independent of the mag- 

 nitude or ratio of sides, will form the first or commencing section 

 of elementary geometry. This will afford the subordinate sec- 

 tions of {a) angles at one point; {b) angles at two points, or 

 parallelism; (c) angles at more than two points, or those of 

 figures depending on the number of the sides. In proceeding 

 from what is estabhshed regarding angles at one point to the 

 subject of parallelism, logic ought not to fail, for there must be 

 found in the nature of things reasoned about, sufficient founda- 

 tions for all conclusions which are true regarding them. For 

 this end we require a twofold preparation, consisting in a pro- 



