NEWTCXN S PRINCJPIA. 



309 



The other method begins by assuming the orbit to be 

 parabolic. 



Three observations of a comet are supposed to be made 

 at given times; then the positions of the earth T, t, T, being 

 known, we can draw the directions of the straight lines 

 T A, tB, T C, joining the comet and the earth at those 



times. Suppose now that by 

 some means a first approxima 

 tion to the distance of the 

 comet at the times of the second 

 observation has been dis 

 covered, then we can mark off 

 a distance t B, and B will be 

 one point in the comet s orbit. 

 Suppose also that from some 

 properties of the parabola we 

 can determine approximately, 

 First, the length of the sagitta 

 B E drawn from the point B 

 of a comet s path in a known 



direction and cut off by the chord A C joining the positions 

 of the comet at the first and last observations ; Secondly, 

 the ratio in which the chord A C is divided in the point 

 E : Thirdly, the length of the chord A C. Then, by means 

 of the first two properties, we can construct the chord A C, 

 and if its length be the same as that given by the third 

 property, our assumption as to the value of t B was correct. 

 If not, by making several trials, and by applying the &quot; rule 

 of false,&quot; we can correct our assumed distance t B, and, 

 therefore, ultimately obtain three points A, B, C in the 

 orbit. By a simple construction given in the First Book, a 

 parabola can be described which passes through two points 

 and has its focus in the centre of the sun. This will be the 

 comet s orbit. 



x 3 



