DEMONSTRATED. 203 



angles GZH, HPK, have equal angles, GZH and HPK, be- 

 cause GZ is parallel to HP and ZH to KW, and the sides ZH, 

 ZG, KP, PH which are about the equal angles proportional, 

 therefore the remaining angles HGZ, GHZ of the triangle 

 GZH are equal to the remaining angles PHK, PKH of the trian- 

 gle HPK, each to each which are opposite to the homologous 

 sides, so the angle IIGZ is equal to the angle PHK and the angle 

 GHZ is equal to the angle PKH. The angle ZHP is equal to 

 the angle HPK, because ZH is parallel to PK and PH falls 

 upon them; and the three angles GHZ, ZHP, and PHK ta- 

 ken together are equal to the three angles HKP, HPK, and 

 PHK taken together, that is to two right angles. So to the 

 point II in the right line ZH are drawn two right lines KH 

 and GH on opposite sides, making the two angles KHZ and 

 GHZ taken together equal to two right angles; therefore the two 

 right lines form one straight line ; But BC is bisected in K by- 

 construction, and the right line GHK drawn through G and H 

 bisects BC. Therefore in the triangle ABC, CD and BE being- 

 drawn, cutting each other in F, and the sides of the triangle 

 in D and E, and the diagonals AF DE of the trapezium 

 ADFE being drawn and bisected in G and H, the right line 

 GH joining the points of bisection being produced bisect the 

 base. Q. E. D. 



No. XXXIV. 



An Account and description of a temporary rudder, invented 

 by Captain William Mugford, of Salem, (Massachusetts) and 

 for which the Society uiourdcd to him a Gold Medal, from the 

 Extra-MagcltaiW fund. 



Motto. Nil desperandum — eras iterabimus ccquor. 



Read November :6th 1804. 



THE Ship Ulysses of Salem (Massachusetts) under the com 

 mand of Captain William Mugford, sailed from that port 01 

 the 2d day of January 1S04, bound to Marseilles. On the 



