﻿44() DEMOXSTRATrON- Of THE 12tH AXI05/ 



e are all right angles, m a will coincide with c o, 7ib 

 Avith kd, and o c with ^e, but m a^nc o^zls e by con- 

 struction ; therefore the point 7n will coincide with 

 0, and the point o with s, therefore the line mo will 

 coincide with os; but the line b n coincides \v\\\\d k\ 

 therefore the point n will coincide with k : therefore 

 ?i:zid k. Now if ;^ A be supposed greater or less 

 than ?Ji a, take bpzzamzzoc^iidr, take oi:=.pn, draw 

 the right lines p i, p o, r iy ro: no'.v it is obvious, 

 from the construction, that the figure p b d r i may 

 be placed upon the figure ocamn, so that the points 

 p k. 0, b 8c c\ d 8c a, r 8c m, and i 8c ?i shall coincide ; 

 but the points o, ?i, m, are in one right line ; there- 

 fore the points p, i, r, are also in one right line ; 

 therefore/? /, i ?', form one right line: now produce 

 p 0, and it must meet i r in some point as u ; and 

 therefore two right lines p i u, p o u would include 

 a space, which is absurd. Therefore n b cannot be 

 2:reater nor less than 7ii a : • therefore 71 b^zima Q E D. 



Prop. 2d, Fig. Sd, 4th, 5th. 



Iv two equal right lines a c, b d, 

 be perpendicular to the same right 

 line a b, and a riglit line c d be 

 P/2*. 3. X drawn joining their terms; 1st. 



The angles a c d, b d c will be 

 equal ; 12dly, tlie angles acd, bdc 

 will be right angles; and 3dly, the 

 right line c d will be equal to a b. 



Draw the right lines a d, b c : in the right angle 

 triangles c a b, d b a the sides containing the right 

 angles are equal by construction, therefore (by 4. i.) 

 ad—b c, therefore" the triangles a c d, b a c are mu- 

 tually equilateral, therefore the angles acd, bdc, 

 which are opposed, to the equal sides a d, b c, are 

 equal (by 8, 1.) 



2dly. 



