2 7 6, 277 Legendre and Gauss 359 



observations of its position, which was published in his 

 Thtorin Motus (1809). As we have seen (chapter XL, 

 236), the complete determination of a planet's orbit 

 depends on six independent elements : any complete ob- 

 servation of the planet's position in the sky, at any time, 

 gives two quantities, e.g. the right ascension and declination 

 (chapter n., 33) ; hence three complete observations 

 give six equations and are theoretically adequate to de- 

 termine the elements of the orbit ; but it had not hitherto 

 been found necessary to deal with the problem in this 

 form. The orbits of all the planets but Uranus had been 

 worked out gradually by the use of a series of observations 

 extending over centuries ; and it was feasible to use ob- 

 servations taken at particular times so chosen that certain 

 elements could be determined without any accurate know- 

 ledge of the others ; even Uranus had been under observa- 

 tion for a considerable time before its path was determined 

 with anything like accuracy ; and in the case of comets 

 not only was a considerable series of observations generally 

 available, but the problem was simplified by the fact that 

 the orbit could be taken to be nearly or quite a parabola 

 instead of an ellipse (chapter ix., 190). The discovery 

 of the new planet Ceres on January ist, 1801 ( 294), and 

 its loss when it had only been observed for a few weeks, 

 presented virtually a new problem in the calculation of an 

 orbit. Gauss applied his new methods including that 

 of least squares to the observations available, and with 

 complete success, the planet being rediscovered at the 

 end of the year nearly in the position indicated by his 

 calculations. 



277. The theory of the "reduction" of observations 

 (chapter x., 218) was first systematised and very much 

 improved by Friedrich Wilhelm Bessel (1784-1846), who 

 was for more than thirty years the director of the new 

 Prussian observatory at Konigsberg. His first great work 

 was the reduction and publication of Bradley's Greenwich 

 observations (chapter x., 218). This undertaking involved 

 an elaborate study of such disturbing causes as precession, 

 aberration, and refraction, as well as of the errors of Bradley's 

 instruments. Allowance was made for these on a uniform and 

 systematic plan, and the result was the publication in 1818, 



