MATHEMATICAL PAPERS, 29 



In order to demonflrate this numerical method of finding 

 the area, I £hall lay down the following Lemmas. 



Lem. I ft. tn the trapezium R G H O {Plate L fig. 8,) right 

 angled at R, and Q ; if to twice G R be added O H, and this 

 fum be multiplied by G O, the produd will be equal to twice 

 the area of the trapezium. For 2 GRxGO=2 R G O Q, 

 and O HxG 0=2 GHO. Therefore 2 GR-f-OHxG O 



2 R G H Q. O. E. D. , 



Lem. 2d. In the foregoing trapezium if from twice H Q be 

 fubtraded H O, and the remainder be multiplied by G O, the 

 produdl will be equal to twice the area of the trapezium. 

 For 2HQxGO=:2RAHQ, and HOxGO=2 G AHO. 



Therefore 2HQ-- HOxG 0=2 R G H O. Q. E. D. 



Lem. 3d. In figure 9th, plate L if the triangle C F L be 

 made equal to the right angled triangle B O JC, then M L 



multiplied by K M wilFbe equal to twice the area of the tra- 

 pezium O F L M. For M LxK M=0 F L M+0 F A K 

 2OFLM. O. E.D. 



The diagram ABODE (Plate I. fig. i o,) is a geometrical 

 confiruiflion of the foregoing example. K Q is the calcula- 

 trix. K M, D T are eaflings, and IK, N D, PA are weflings. 



BK-|-MC,TE are northings, and A I-fK B, CN, EP are 

 fouthings. C M, D R, E Q, A I are the calculatral diflances 

 of the fides B C, C D, D E, E A. The operation according to 



the conftru(5lion is as follows. 



NE, 



4- ^ 



