to deal with more complicated cases in which two or more unknown quantities have to be determined from observations of different quantities, as, for example, when the elements of the orbit of a planet (chapter xi., § 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 favourable 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 priori; 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 affecting astronomy. Gauss on the other hand, was for nearly half a century head of the observatory of Göttingen, 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 astronomical observations, the best known being a method of calculating the orbit of a planet from three complete
