358 A Short History of Astronomy [Cn. xin. 



to deal with more complicated cases in which two or more 

 unknown quantities have to be determined from observa- 

 tions of different quantities, as, for example, when the 

 elements of the orbit of a planet (chapter XL, 236) have 

 to be found from observations of the planet's position at 

 different times. The method of least squares gives a rule 

 for dealing with such cases, which was a generalisation 

 of the ordinary rule of averages for the case of a single 

 unknown quantity ; and it was elaborated in such a way 

 as to provide for combining observations of different value, 

 such as observations taken by observers of unequal skill 

 or with different instruments, or under more or less favour- 

 able conditions as to weather, etc. It also gives a simple 

 means of testing, by means of their mutual consistency, 

 the value of a series of observations, and comparing their 

 probable accuracy with that of some other series executed 

 under different conditions. The method of least squares 

 and the special case of the " average " can be deduced 

 from a certain assumption as to the general character of 

 the causes which produce the error in question ; but the 

 assumption itself cannot be justified a prior i\ on the other 

 hand, the satisfactory results obtained from the application 

 of the rule to a great variety of problems in astronomy 

 and in physics has shewn that in a large number of cases 

 unknown causes of error must be approximately of the 

 type considered. The method is therefore very widely 

 used in astronomy and physics wherever it is worth 

 while to take trouble to secure the utmost attainable 

 accuracy. 



276. Legendre's other contributions to science were 

 almost entirely to branches of mathematics scarcely affect- 

 ing astronomy. Gauss, on the other hand, was for nearly 

 half a century head of the observatory of Gottingen, and 

 though his most brilliant and important work was in pure 

 mathematics, while he carried out some researches of first- 

 rate importance in magnetism and other branches of physics, 

 he also made some further contributions of importance to 

 astronomy. These were for the most part processes of 

 calculation of various kinds required for utilising astrono- 

 mical observations, the best known being a method of 

 calculating the orbit of a planet from three complete 



