I07 



3- SD : SO=DI-DT : : rad : s, SDO. Hence, by computing SD, 

 a more accurate value of SDO will be had, and therefore a more ac- 

 curate value of ODC=DCI. 



TO obtain the fecond approximation, Cafllni gives a table containing 

 the difference of the arches and their fines for a few of the firft de- 

 grees of the quadrant, a table which is eafily extended. 



The refpeftivc degrees of accuracy of thefe approximations may be 

 inveftigated as follows. 



I. We have CD-t-SC : CD— SC : : /, i^DCA : /, ^DSC— ^SDC 



2 



or I +f : I — e : : t, '^m : t, |w 



which is the fame proportion as that in the preceding theorem, 



2 3 



whence w=m — ?^es, m-\-\e s, zm — -j s, ^m+kc. 



Indeed it is evident the angle aDSC— aSDC= Seth Ward's anomaly. 

 For the triangle SCD is fimilar to the triangle formed by the diflance be- 

 tween the foci and the axis major, including an angle equal to the mean 

 anomaly according to Seth Ward's hypothefis. 



« 3 



Hence SDC=f^, m — y s, im+\e s, yn — &c. 

 But in Caffini's firft approximation DCI is taken equal to SDC there- 

 fore according to that approximation ACI the excentric anomaly = /«— 



2 3 



es,}n-\-'^es, 2m— ^e s, ^m-j-kc. or the excentric anomaly=i the mean ano- 

 maly 4- ^^ Seth Ward's anomaly. Taking the difference between this and 

 the true excentric anomaly wc have the error of Caffmi's firft approxima- 



3 ' 3 3 4 . , . , 



tion =J.e . ym — s, 3m (i)? s , m) -\-e . ^'is. Am — [s, im &c. m which 

 the third and higher powers of the excentricity only are eoncerned. 



This quantity, when a maximum in the orbit of Mercury, is 5^ 

 nearly, in the orbit of Mars only 10" &c. 



The facility with which this near approximation may be obtained, 

 renders it highly valuable, when combined with the method of extend- 

 ing at pleafure the approximation. It alfo deferves notice, from the 



( O 2 ) elegance 



