198 Method of correcting the DISTANCE 



for the fame reafon, SL and Sb will be equal to one another : 

 Then the feveral corrections and corrected diftances will be as 

 follows : 



ifl correction Ma, diftance once corrected Sa. 



zd correction ab, diftance twice corrected Sb, equal to SL. 



2>d correction Sc, diftance thrice corrected Lc. 



4tb correction cd, diftance four times corrected Ld, equal to 

 LR the true diftance. Radius i. 



THE firjl correction Ma is equal ML multiplied by the co- 

 fme of LMS. 



THE arc La being perpendicular to MS, the two triangles 

 MaL and SaL will both be right-angled at a. It has been 

 proved, that SL and Sb are equal ; therefore ab, which is the 

 fecond correction, will be the difference between SL and Sa ; and 

 is found by the following proportion : The rectangle under the 

 ^tangents of half the fum, and half the difference of LM, the 

 hypothenufe, and aM, one of the legs, is equal to the fquare 

 tangent of half La, the other leg ; and the rectangle, under the 

 tangents of half the fum and half the difference of SL and Sa, 

 is alfo equal to the fquare tangent of half La, (HEATH'S Aftron. 

 p. 330.) Hence the fquare tangent of half La, divided by the tan- 

 gent of half the fum of SL and Sa, will be equal to the tangent 

 of half the difference of SL and Sa, that is, to the tangent of 

 half ab ; and twice the fquare tangent of half La, divided by the 

 tangent of half the fum of SL and Sa, will be equal to the tan- 

 gent of ab. The proper divifor, therefore, for finding this fe- 

 cond correction juftly, is the tangent of Sa increafed by half ab. 

 But, as this will not make the correction |" lefs than when Sa 

 is made the divifor, fuppofing the diftance not under 20, it 

 was thought needlefs to mention it in the precepts. 



IN fuch a fmall triangle, where the corrected parallax is the 

 hypothenufe, which can never exceed 56'^, the difference be- 

 tween the arcs themfelves and their fines or tangents is fo very 

 inconfiderable, that they may be taken indifferently for one ano- 

 ther 



