﻿OF THE FIRST BOOK OF EUCLID. 



44.7 



Fig 



Fi 



2(.lly. Fro:m any point m, in the line c a\ let fall 



c 7/1 d a perpendicular to the line ah: 



~ bv tlie 1st proposition, m n~.a c 



4. ^=:.b d ; therefore, by tlie fore- 



g(;in^ part, n m c^iia c m':=ih d m 



~:n 7)1 d:-n m c, n m d are right 



angles : consequently a c d, b d c 



arc also right angles, 



Stlly. Draw the right line d a ; the angle a c dh 

 c d [X right angle by the 2cl 



part, and therefore equal to 

 a h c ; and the sides a c,bd 

 are equal by construction ; 

 now if a b bt not equal to 

 c d, take b m cither greater 

 or less than a b, v»hich shall be equal to c d ; and 

 draw the right line dm, and since a c d is a right 

 angle, by the foregoing part, and therefore equal to 

 a b d, and a c:=:b d by construction, and also d c:z: 

 b m by supposition ; d m will be equal to d a (4, 1,) 

 and therefore the angle dmazzdam (5, 1,) but dma is ' 

 an obtuse angle (l6\ 1,) therefore two angles of a 

 triangle would be greater than two right angles, 

 contrary to 17, 1, of the Elements; therefore ^ « 

 cannot be greater nor less than d c: c d'^a b. 

 Q. E. D. 



Prop. 3d. Fig. Q. 



If two right lines a c, 

 h d, be perpendicular to 

 tbc same right line a b ; 

 and from any point r, in 

 one line, be drawn c d, per- 

 pendicular to the other ; 

 a c'=zb d, and therefore c d~a b, and the angle a c d 



Suppose 



a right angle. 



