338 ^'^"^ PROPERTY OF TANGENTS. 



The face c is produced by a decrease of two rows of atoms 

 at the corners of the cube, and the angle it forms with the 

 face6is = 125« 15' 52^'. 



The face h being produced like the face a. forms the same 

 angle M'ith the face m. 



No crystal I possess has enabled me to measure the incli- 

 nations of the faces o-, rf, or/; should the face g, as is presum- 

 able, result from a decrease of one row of atoms at the cor- 

 ners of the cube, it will form with the Jace b an angle of 

 144° 44' 8'^; and if the faces d and / are, as is also proba- 

 ble, produced by a decrease of two rows of atoms along 

 the edges of the cube, the first will form an angle of 116* 

 33' 54'', and the latter one of 153^ 26' Q", with the face^/e. 

 This differs The angles assigned here differ considerably from those 



from the former gj^gj^ j^ ^]^g former account of these crystals: but the angles 

 account of the » j > a 



rrystals. there given have not only appeared to me to be contradicted 



by observation, but, crystallographically considered, are in- 

 consistent with each other, as the tetraedral prism of dimen- 

 sions to produce an angle of 135° by a decrement along its 

 edge would not afford angles of 140'^ and 120'' by decre- 

 ments at its corners. 



The sum of the faces of these crystals is 50. 



III. 



On a nezo Property of the Tangents of the three Angles of 

 a Plane Triangle. By Mr. AVilliam Garrard, Quarter 

 jy^ister of Instructio7i at the Roi/al Naval Asylum at 

 Greenxicich. Commumcated by the Astronomer Royal*. 



Sum of three 1 PROPOSITION I. In every acute angled plane tri- 



tangents of a ansrle, the sum of the three tangents of the three angles 

 plane triangle ^ ^ ° ° 



multiplied by multiplied by the square of the radius is equal to the conti- 

 square of radius j^„g^ product of the tangents. 



equal to their ^ ^ 



continued j^ro- Demonstration. — Let AH, HI, and IB, Plate 9 Fig. 3, 



duct. Ijg |.jjg arches to represent the given angles ; and AG, HK» 



in an acute an- and BT bc their tangentSj put r the radius, AG = a, and 



gled triangle : BT=r& 



Philos. Trans, for 1807, Part I, p. 120. 



Then 



