ii8 DEMONSTRATIONS of 



2(DA^+4DB^+-fDC0 = 2(^')(ED'+EX0 = 

 i^-^)[DX^+m% (Prop. I.). Or, 



THEOREM X. Fig. V. 



Let there be any number, m, of parallel Jlraight lines AB,CD, EF,&c. 



given by pojition, a Jlraight line XY ?nay be found parallel to them^ 



fuch, that if from any point G, perpendiculars GA, GC, GE, &c. 



be drawn to AB, CD, EF, &c. and the line GX perpendicular to 



XY, 



GA'+GC^+GE^ &c. =: wGX^+AS A^ being a given fpace. 



This propofition is one of the fimpleft cafes of theor. 6. A 



line XY parallel to AB, drawn through X, the centre of gravity 



of the points A, C, E, where a perpendicular from G meets 



the parallels AB, CD, EF, will be the line required. For, 



GA=+GC'+GE^ - XA^+XC'+XE'+3GX^ (Theor. 6.), and 



XA'+XC^+XE= is a given fpace. 



THEOREM XI. Fig. VI. 



Let there be any number, m, of Jlraight lines AB, AC, AD, &c. 

 interfering in a point A, fo as to make all the angles round it equal j 

 and from any point E, let perpendiculars EB, EC, ED, &c. be drawn 

 to AB, AC, AD, &c. and let AE be Joined, 



2(EB^+EC^+ED» &c.) .= w.EA'. 



This 



