THIRD SECTION. 



THE DETERMINATION OF AN ORBIT SATISFYING AS NEARLY AS POSSIBLE ANY 

 NUMBER OF OBSERVATIONS WHATEVER. 



172. 



IF the astronomical observations and other quantities, on which the computa 

 tion of orbits is based, were absolutely correct, the elements also, whether deduced 

 from three or four observations, would be strictly accurate (so far indeed as the 

 motion is supposed to take place exactly according to the laws of KEPLER), and. 

 therefore, if other observations were used, they might be confirmed, but not cor 

 rected. But since all our measurements and observations are nothing more than 

 approximations to the truth, the same must be true of all calculations resting 

 upon them, and the highest aim of all computations made concerning concrete 

 phenomena must be to approximate, as nearly as practicable, to the truth. But 

 this can be accomplished in no other way than by a suitable combination of 

 more observations than the number absolutely requisite for the determination of 

 the unknown quantities. This problem can only be properly undertaken when 

 an approximate knowledge of the orbit has been already attained, which is after 

 wards to be corrected so as to satisfy all the observations in the most accurate 

 manner possible. 



It then can only be worth while to aim at the highest accuracy, when the 

 final correction is to be given to the orbit to be determined. But as long as it 

 appears probable that new observations will give rise to new corrections, it will 

 be convenient to relax more or less, as the case may be, from extreme precision, 

 if in this way the length of the computations can be considerably diminished. 

 We will endeavor to meet both cases. 



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