﻿OF THE FIRST BOOK OF EUCLID. 



44$ 



Prop. 5, Fig. 8, 9, 10. 



Fis:. 8. 



If two rlo'lit lines, 



a e, do, stand upon a 



riglit Vint a (I, so that 



the two internal an- 



_r,ies e a d, o d b are 



a bed less tlian two riglit 



angles, the lines a e, do, produced on the same s,de 



of ab, shall meet. N. B. This is the liith axiom of 



the 1st book of Euclid. 



1st. Let one of the lines, a e, be perpendicular to 

 ad, and consequently the angle id a acute (bv sup- 

 position) from any point .r, in the line d i, let fall a 

 perpendicular .v c, meeting ad in c; take xo zi d,v; 

 produce c.r; draw the perpendiculars o^, on. Now 

 in the triangles oxu^dxc, the angles at .v are vertical, 

 and those at u and c right angles, and the siVie o.v 

 equal to .vd, icd — on. (25. \.) — bc (prop, ad.) 

 therefore, if from the line da be taken pn.rts equal to 

 cd, till the whole be exhausted, and from d i pro- 

 duced be taken the same number of parts =: to da:, and 

 right lines be drawn from the several points of divi- 

 sion in di to the corresponding points of division in 

 ad, these lines will be all perpendicular Xoaa, but tiie 

 last of them either coincides with ae or fails beyond 

 it : di must meet ae. 



Fig. 9. 



If both the lines a e, 

 di form acute angles 

 Avith a d, erect the per- 

 pendicular r/???.- by the 

 last ca e 4 l must meet 



d am, atid tbei etore must 

 first meet a e^ 



G G Fig. 



