196 Method of corretting the DISTANCE 



half fum and the difference between the half fum and the ze- 

 nith diftance of the Moon. Half the fum of thefe four loga- 

 rithms will be the log. cofine of half the angle, which, being 

 doubled, will give the angle at the Star between the zenith and 

 the Moon. 



THE angles being found, the feveral corrections of diftance 

 are to be computed and applied according to the rules already 

 given. 



INVESTIGATION. 



IN the fpherical triangle MZS, let Z reprefent the zenith, 

 M the apparent place of the Moon, S the apparent place of the 

 Star, and MS the apparent diftance of the Star from the Moon's 

 centre. Let Zp be a perpendicular arc let fall from Z upon- 

 MS, produced if necefiary, and let m be the middle of the bafe, 

 fo that Mm or Sm be equal to half the diftance of the objects. 

 If the zenith diftances MZ and SZ are equal, the triangle will 

 be ifofceles, and the angles ZMS and ZSM will alfo be equal, 

 and Zp will fall upon MS in m ; but if MZ and SZ are un- 

 equal, Zp will fall upon MS at fome diftance from tn, either 

 within or without the triangle, and the angles ZMS and ZSM 

 will alfo be unequal. Then (by cafe 1 1. obliq.fpber. triang.) pm 

 will be the Jirjl arc, equal to the diftance between the perpen- 

 dicular and die middle of the bafe ; half the bafe added to pm 

 will be the fecond arc, equal to the diftance of the perpendicu- 

 lar from the lower object ; and the difference between half the 

 bafe and pm will be arc third,, equal to the diftance of the per- 

 pendicular from the higher object. It is evident that, when 

 pm is lefs than half the bafe, the perpendicular muft fall within 

 the triangle, and the angles, both at M and S muft be acute ; 

 on the contrary, when pm is greater than half the bafe, the per- 

 pendicular muft fall without the triangle, and the angle at that 

 object which is next the perpendicular will be obtufe ; and 



ZSp 



