PREFACE. Xi 



paved the way to new and most brilliant triumphs in analysis; and the 

 comets, up to that time wholly unmanageable, or soon breaking from the 

 restraints to which they seemed to be subjected, having now submitted to 

 control, and being transformed from enemies to guests, moved on in the 

 paths marked out by the calculus, scrupulously conforming to the same eter 

 nal laws that govern the planets. 



In determining the parabolic orbits of comets from observation, difficul 

 ties arose far greater than in determining the elliptic orbits of planets, and 

 principally from this source, that comets, seen for a brief interval, did not 

 afford a choice of observations particularly suited to a given object : but the 

 geometer was compelled to employ those which happened to be furnished 

 him, so that it became necessary to make use of special methods seldom 

 applied in planetary calculations. The great NEWTON himself, the first geome 

 ter of his age, did not disguise the difficulty of the problem: as might have 

 been expected, he came out of this contest also the victor. Since the time 

 of NEWTON, many geometers have labored zealously on the same problem, 

 with various success, of course, but still in such a manner as to leave but 

 little to be desired at the present time. 



The truth, however, is not to be overlooked that in this problem the 

 difficulty is very fortunately lessened by the knowledge of one element of 

 the conic section, since the major-axis is put equal to infinity by the very 

 assumption of the parabolic orbit. For, all parabolas, if position is neg 

 lected, differ among themselves only by the greater or less distance of the 

 vertex from the focus; while conic sections, generally considered, admit of 

 infinitely greater variety. There existed, in point of fact, no sufficient reason 

 why it should be taken for granted that the paths of comets are exactly 



