115 



On Dr, Matthew Steivart's Method. 



His firfl: approximation is as before mentioned, the fame as Caffiui's. 

 Let AD be the firll approximated excentric anomaly, and AB the mean ^'S> S- '' 

 anomaly ; then it readily appears that the area DSG = the fegment 

 BGD AC, being the excentric anomaly. Inftead of double the former area, 

 Stewart takes its near value DG X perpendicular let fall from S on the 

 tangent at D, and thence deduces a correftion to his firft approximation, viz. 

 DG : BD-s, BD : : t, ODS : s, BOD. 



To find the error of DG refulting from this proportion. Draw DL and 

 Gm perpendicular to the tangent DC, and SL and GR parallel to the^'^'^-'- 

 fame. Now the area DGS=DSC— GDC=LCD-GDC, and LCD== 



GD^ 



5LDxCD=;LDxGR-C«, but C^;=RDx^ mGc=.-^xt,m Gc (AC 



2 



being unity) omitting the powers of GD above the fecond, alfo GR=GD 

 omitting the powers of GD above the fecond. Therefore LCD='-LDXGD 

 -^LDxiGD^X/,OTG..TheareaGCD,asiseafilyfhewn, depends upon the 

 powers of GD above the fecond; hence, omitting the powers of GD above 

 the fecond, DGS^-LDxCD^-lGD^/T^iTCC. Therefore inftead of GD 

 being found by the above proportion, the quantity GD— iGD'xi, mGC 



is deduced. It has been fhewn before that GD=.'/s \ ^+&c. Hence it 

 follows that the firfl term of the feries expreffing the error ofDG will be 



6 



3. s, mxt,mGC. It is eaCly fhewn that the tangent of /«GC may be of 



( P O the 



