308 NEWTON S PRINCIPIA. 



orbit of a comit. One of these may be found in his little 

 treatise De Systemate Mundi, and the other in the Third 

 Book of his Principia. The first is by far the simplest 

 of the two. The small part of the comet s path is con 

 sidered as a straight line, the two parts of which are 

 described by the comet with the velocity it had at the 

 beginning of those parts. These velocities are assumed 

 to be V times that in a circle at the same distance. Let 

 a first approximation to the distance of the comet from 

 the earth at the time of the middle observation be sup 

 posed known, and let us represent it by the letter x. 

 Newton then shows how to draw a straight line which 

 will be divided by the directions in which the comet is 

 seen at the times of the three observations in the ratio of 

 the intervals between those observations. If this straight 

 line, thus found, can be described in the given time with 

 the velocity given by our assumed value of V, the value 

 of x that corresponds to V has been found ; if not, a se 

 cond value of x must be assumed ; and then by an applica 

 tion of the rule of false, a new value may be determined, 

 which is nearer the truth, and by repeating this process, 

 we can arrive at any degree of accuracy. The value of 

 V being therefore supposed known, the velocity, position, 

 and direction of motion of the comet are also known at a 

 given time. By the First Book we can describe the orbit 

 that a body projected in a given manner from a given 

 point will describe round a given centre of force. Begin, 

 then, with an approximate value of V, and construct the 

 orbit. Make a fourth observation on the comet when at 

 a great distance from its position at the three first obser 

 vations. The longitude thus found should agree with 

 that in the orbit just constructed. If not, a new value of 

 V must be found, and by the rule of false, the true value 

 may be continually approximated to. 



