418 Olbers' Essai/ on Comets, 



A" — R"' sin. A'"), and by comparison with the equations in 



§ 21, t' : r = (^' COS. at' — { cos. «") : (^" cos. a" — {' COS. a!") 



= (^' sin. a! — {' sin. «") : (^" sin. a" — (" sin. a") =: (^' tang. 

 iS' — ^" tang. jS") : (^" tang. /3" — {' tang. ^'"), which obviously 

 ndicate only the proportions of ^', ^", and §"', to each other, and 

 not their actual value. Hence we may understand the remark 

 of Lambert, that Bouguer had attempted to find the distance 

 of the comet by means of the minute verse sine of the earth's 

 orbit, and Lagrange's observation, that Bouguer's solution is 

 not correct, even for infinitely small portions of the orbit ; for 

 both the magnitudes, compared with each other in their evanes- 

 cent state, are of the same order. But it is not correct to infer, 

 as this great geometrician, and Ping re' after him, have done, 

 that no portion of a comet's orbit must be assumed as straight, 

 even for the purposes of approximation ; since Boscovich's con- 

 struction, for example, proceeding on this supposition, afibrds a 

 result approaching to the truth, and becomes even perfectly ac- 

 curate when the interval is evanescent. In fact, Boscovich 

 supposes only that the verse sine vanishes in comparison with 

 the length of the arc, and the difference of the velocity in com- 

 parison with the whole velocity, which is perfectly justifiable. 

 Nor does Laplace's objection to this method appear to be 

 much more important, which is, that it may sometimes indicate 

 a retrograde motion instead of a direct one, or the reverse ; for 

 since the equation of the sixth degree, on which the solution 

 depends, may have several real roots, and must have two, the 

 ambiguity is inseparable from the nature of the problem, and 

 Laplace himself has only avoided it by means of a supernu- 

 merary equation, which he calls an equation of security. We 

 may easily understand how it happened that Bouguer was so 

 fortunate in applying his method to the comet of 1729 ; for this 

 comet having been much more remote from the sun than the 

 earth, its orbit was much less curved than that of the earth, so 

 that it might, without any great inaccuracy, be considered as 

 comparatively straight; and it is only in such cases as this, 

 when, the comet is very remote, and the arc which it describes 

 comparatively short and little curved, that Bouguer*s method 



