Dr steward's theorems. 123 



dicular to HK. Therefore NO = LG = HM = NP. But 

 the angle OAP is given, being the fupplement of OK? ; and 

 fince the point N is given, and NO = NP, the points O and P 

 are given ; and therefore AO and AP. Therefore the Hnes GK, 

 HK, are given by pofition. 



THEOREM XIII. Fig. X. No. i. 



Let there be any number, fw, of Jira'ight lines AB, BC, CD, 

 DA, &c. given by pofition, neither all parallel nor interfering in 

 one point, two flraight lines XY, XZ, may be found, "which will be 

 given by pofition, fuch, that if from any point E, there be drawn per- 

 pendiculars EF, EG, EH, EK, &c. to AB, BC, CD, DA, &c. and 

 EY, EZ, perpendicular to XY, XZ, 



2(EF^+EG^+EH^+EK^ &c.) = «(EY^+EZO+A% 



A* being a given fpace. 



Let »2 rz 4, and from C, one of the points of inter- 

 fedion, draw Cf C/f, parallel to the lines given by pofition that 

 do not interfedt in C. Let two flraight lines CL, CM be found, 

 fuch, that 2(E/^+EG'+EH'+E/{0 = 4(EL^+EM^), (Theor. 12.). 

 Let N be the centre of gravity of the four points F, G, H, K, 

 (Theor. 6.). Through N draw YNZ to meet EL, EM in Y, Z, 

 and fo as to be bifedled in N. Through Y and Z draw YX, 

 ZX perpendicular to EL, EM, interfedling each other in X. 

 From X draw XP, XQ^ XR, XS, perpendicular, and Xa, XiJ, 

 Xf, X^, parallel to AB, BC, CD, DA ; let Xa, X^, Xr, X^, meet 

 EF, EG, EH, EK, in a, b,c,d; join XF, XG, XH, XK ; NF, 

 NG, NH, NK, NX ; and let O be the centre of gravity of the 

 four points f G, H, k, where the parallels from C, to the lines 

 given by pofition, meet the perpendiculars from E. 



q 2 By 



