n8 DEMONSTRATIONS of 



2 (D A*+ 4-DB^+ -7- DC») == 2 (^~) (ED*+EX*) = 



(^)(DX*+DY*), (Prop. i.). Or, 

 DA*+ ~ DB*-f -^DO = (^tf)(DX*+DY*). 



THEOREM X. Fig. V. 



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



given by pofition, a Jlraight line XY may be found parallel to tbem y 



fucby 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, 



GAM- GCH-GE* &c. =^GX 2 +A% A 2 being a given J pace. 



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 2 -f-GCM-GE* == XA*+XO-f XE*+3GX a (Theor. 6.), and 

 XAM-XCM-XE 2 is a given fpace. 



THEOREM XI. Fig. VI. 



Let there be any number, m t 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 (EBM-ECH-ED 2 & c .) ,= w.EA 2 . 



This 



