316 Mr Meikle on the Theory of Parallel Lines. 



gendre, Memdires de VAcademiCy tome xii. p. 367, Can elaborate essay, bat 

 seems to have escaped the notice of Col. Thompson). It will be found 

 that these illustrious geometers have assumed the whole that was to be 

 proved. The following is a theorem of Legendre : — If the indefinite 

 straight lines AC, B D, (Fig. 6) be both perpendicular to A B, and if 

 from any 'point N in B D we draw N M perpendicular to A C, then 

 N M will be equal A B and perpendicular to B D. 

 Draw N I bisecting A B in I and meeting A C in P, and produce it 

 till N Q = N P. Draw also the indefinite 

 line QY, making angle NQY = PND, 

 and produce M N to meet Y Q in G. From 

 these premises, M. Legendre readily proves 

 that the triangles P I A, B N I are equal, 

 and, consequently, that area D B A C = 

 D NP C ; and that angle CPN + PND = 

 DNQ + NQY = CPQ + PQY = 180° As readily does he prove 

 the triangles P N M, N G Q to be equal, and therefore area Y Q P C = 

 YGMC. 



When any two indefinite straight lines — such as N D, P C — make, 

 with the same side of a third straight line, P N, the two interior angles 

 C P N, P N D equal to 180°, Legendre calls the figure D N P C a 

 biangle. Now it is evident that in the above construction we have other 

 four biangles. But, unfortunately, through some sad oversight, M. 

 Legendre in effect makes it to contain one or two besides. Thus, with- 

 out having proved anything whatever regarding the value of angle 

 M N D, he not only calls C M N D a biangle, but reasons upon it as 

 such, or as having two right angles, which obviously is the same as just 

 at once assuming M N D = 90° ; and it is on this ground he concludes 

 that M N = A B. Then, by reasoning in a circle from this conclusion, he 

 proves what, as just stated, he had already assumed, namely, that M N D 

 = 90°. The direct demonstration of Legendre is, therefore, a total 

 failure. 



But since area C M N D is less than D N Q Y by the sum of the tri- 

 angles P N M, G N Q, the angle G N D cannot be acute ; because then 

 C M N D, instead of being less than D N Q Y, would exceed it by the 

 infinite area of an angle equal to the excess of angle M N D over G N D, 

 which is absurd. Were reasoning upon infinite quantities liable to no 

 objection, this absurdity obviously would amount to an indirect proof 

 that the angles of the quadrilateral A B N M cannot be less than 360°, or 

 that those of a triangle cannot be less than 180°. 



MM. Bertrand and Legendre regard a biangle as an infinite area of 

 the first order, and maintain that it bears no proportion to, and so could 

 never by repeated subtraction exhaust the area of an angle, which they 

 reckon an infinite of the second order. But I shall now shew that this 

 doctrine, which is the foundation of Bertrand's demonstration, can only 

 be maintained on the assumption that the angles of a triangle amount to 



