172 DETERMINATION OF AN ORBIT FROM [BOOK II. 



termination are not to be picked out at random, but care is to be taken, first, that 

 tliey be not too near each other, but tJicn, also, that they be not too distant from 

 each other ; for in the first case, the calculation of elements satisfying the obser 

 vations would certainly be most expeditiously performed, but the elements them 

 selves Avould be entitled to little confidence, and might be so erroneous that they 

 could not even be used as an approximation : in the other case, we should aban 

 don the artifices which are to be made use of for an approximate determination 

 of the unknown quantities, nor could we thence obtain any other determination, 

 except one of the rudest kind, or wholly insufficient, without many more hypoth 

 eses, or the most tedious trials. But how to form a correct judgment concerning 

 these limits of the method is better learned by frequent practice than by rules : 

 the examples to be given below will show, that elements possessing great accu 

 racy can be derived from observations of Juno, separated from each other only 22 

 days, and embracing a heliocentric motion of 7 35 ; and again, that our method 

 can also be applied, with the most perfect success, to observations of Ceres, which 

 are 260 days apart, and include a heliocentric motion of 62 55 ; and can give, 

 with the use of four hypotheses or, rather, successive approximations, elements 

 agreeing excellently well with the observations. 



124. 



We proceed now to the enumeration of the most suitable methods based upon 

 the preceding principles, the chief parts of which have, indeed, already been ex 

 plained in the first book, and require here only to be adapted to our purpose. 



The most simple method appears to be, to take for x, y, the distances of the 

 heavenly body from the earth in the two observations, or rather the logarithms 

 of these distances, or the logarithms of the distances projected upon the ecliptic 

 or equator. Hence, by article 64, V., will be derived the heliocentric places and 

 the distances from the sun pertaining to those places ; hence, again, by article 110, 

 the position of the plane of the orbit and the heliocentric longitudes in it ; and 

 from these, the radii vectofes, and the corresponding times, according to the prob 

 lem treated at length in articles 85-105, all the remaining elements, by which, 

 it is evident, these observations will be exactly represented, whatever values may 



