ON THE MISSISSIPPI. 201 



the limited surface of the river is insufficient to account for so 

 great a dissipation, but we know that the spongy texture of 

 the alluvial soil is remarkably pervious to the waters of the 

 river : from the rlat and humid surface of the Delta, a perpe- 

 tual evaporation exists, the lateral pressure of the waters of the 

 river must supply the waste by exhalation, and this immense 

 expence of fresh water, is to be accounted for by filtration and 

 evaporation. 



No. XXXIII. 



Demonstration of a Geometrical Theorem; by Joseph Clay Esq. 



of Philadelphia. 



Read July 20th, 1804. 



THE following proposition was mentioned to me, some years 

 since, as one which had been proposed by Mr. Simpson some 

 time before his death. I do not know that any demonstration 

 has hitherto been published. 



From the angles at the base of any triangle, let two right 

 lines be drawn cutting each other in any point within the tri- 

 angle, and cutting the sides of the triangle, the segments of 

 the sides and of the lines so drawn will form a trapezium; 

 draw and bisect the diagonals, the right line joining the points 

 of bisection, will, if produced, bisect the base of the triangle. 



In the triangle ABC, (Fig. 6, Plate V.) draw CD, BE, cut- 

 ting each other in F, and the sides of the triangle E and D. 

 Draw AF and DE, and bisect them in G and H; draw GH, 

 which if produced, will bisect the base of the triangle in K, 

 making BK equal to KC. 



Through F, draw LFM, NFO, parallel to AB and AC cut- 

 ting the sides in M and O and the base in L and N : now be- 

 cause of the similar triangles, as CF is to CD so is FL to BD 

 and LM to AB. Therefore by alternation as FL is to LM so 

 is BD to AB. But as FL is to LM so is FN to CM ; Therefore 

 as BD is to AB so is FN to CM and the rectangle under BD, 

 CM is equal to the rectangle under AB, FN. Again, as BF 



