]62 DETERMINATION OF AN ORBIT FROM [BOOK II. 



But in the special case, in which the plane of the orbit coincides with the 

 ecliptic, and thus both the heliocentric and geocentric latitudes, from their nature, 

 vanish, the three vanishing geocentric latitudes cannot any longer be considered 

 as three data independent of each other: then, therefore, this problem would 

 remain indeterminate, and the three geocentric places might be satisfied by an 

 infinite number of orbits. Accordingly, in such a case, four geocentric longitudes 

 must, necessarily, be given, in order that the four remaining unknown elements 

 (the inclination of the orbit and the longitude of the node being omitted) may be 

 determined. But although, from an indiscernible principle, it is not to be ex 

 pected that such a case would ever actually present itself in nature, nevertheless, 

 it is easily imagined that the problem, which, in an orbit exactly coinciding with 

 the plane of the ecliptic, is absolutely indeterminate, must, on account of the 

 limited accuracy of the observations, remain nearly indeterminate in orbits very 

 little inclined to the ecliptic, where the very slightest errors of the observations 

 are sufficient altogether to confound the determination of the unknown quan 

 tities. Wherefore, in order to examine this case, it will be necessary to select 

 six data : for which purpose we will show in section second, how to determine an 

 unknown orbit from four observations, of which two are complete, but the other 

 two incomplete, the latitudes or declinations being deficient. 



Finally, as all our observations, on account of the imperfection of the instru 

 ments and of the senses, are only approximations to the truth, an orbit based 

 only on the six absolutely necessary data may be still liable to considerable 

 errors. In order to diminish these as much as possible,, and thus to reach the 

 greatest precision attainable, no other method will be given except to accumulate 

 the greatest number of the most perfect observations, and to adjust the elements, 

 not so as to satisfy this or that set of observations with absolute exactness, but 

 so as to agree with all in the best possible manner. For which purpose, we will 

 show in the third section how, according to the principles of the calculus of 

 probabilities, such an agreement may be obtained, as will be, if in no one place 

 perfect, yet in nil the places the strictest possible. 



The determination of orbits in this manner, therefore, so far as the heavenly 

 bodies move in them according to the laws of KEPLER, will be carried to the 



