THE ASTRONOMICAL VIEW OF NATURE. 325 



of the degree of approximation which we can attain to. 

 And this does not only refer to the methods of calculation 

 which we adopt, is not only a consequence of the limits 

 of our mathematical powers ; this element of error attaches 

 likewise to our actual observations, to the imperfection of 

 our senses and of our instruments. The many sources 

 of mistake and inaccuracy which surround us may either 

 combine to produce an absolutely useless result, or may 

 be adroitly adjusted so as very largely to destroy each 

 other. 1 The arrangement of instruments of observation 

 and calculation, so as to minimise our errors, is a special 

 branch of science. Before the time of Newton few minds 



is the same as that of finding the 

 centre of gravity of a number of 

 weighted points. This centre has 

 the property that the sum of the 

 squares of its distances from these 

 points is a minimum. After the 

 method had been introduced, La- 

 place and Gauss independently tried 

 to prove it by a variety of considera- 

 tions. These have not always been 

 accepted as conclusive, though it is 

 remarkable that very different ways 

 of attacking the problem all lead to 

 the same result, and that the rule 

 is confirmed by actual trials on a 

 large scale. It has been shown that 

 the method of least squares in the 

 case of a series of observations of 

 one and the same quantity is equal 

 to taking the arithmetical mean, a 

 process which recommends itself to 

 common-sense, though it is not easy 

 to prove it mathematically to be the 

 best. On the whole, the calculus 

 of probabilities and the so-called 

 law of error are attempts to put 

 into figures and mathematical for- 

 mula} a few common-sense notions, 

 and it is interesting to see to what 

 complicated processes of reasoning 

 a combination of these simple no- 

 tions may lead. The literature of 



the subject, belonging almost en- 

 tirely to this century, is very large, 

 Laplace and Gauss heading the list. 

 Encke has summarised the scattered 

 discussions of Gauss and Bessel in 

 his memoir on the subject, reprinted 

 in Taylor's ' Scientific Memoirs ' and 

 in the 2d vol. of Encke's ' Abhand- 

 lungen,' Berlin, 1888. De Morgan, 

 Airy, and Jevons ('Principles of 

 Science,' vol. i.) in England have 

 done much to popularise the sub- 

 ject, and Bertrand (' Calcul des Pro- 

 babilites,' 1888) has very fully dis- 

 cussed the principles of the whole 

 matter and shown up the weak 

 points. The application of the cal- 

 culus to statistics will occupy us in 

 a future chapter. 



1 Not only has every instrument 

 its constant errors, but even every 

 observer himself has what is called 

 a personal equation i.e., he is sub- 

 ject to constant errors of observa- 

 tion, dependent on the peculiarity 

 of his sense organs, or his tem- 

 perament, &c. This was hardly 

 recognised at the beginning of this 

 century, when Maskelyne, the As- 

 tronomer Royal, dismissed an as- 

 sistant whose observations showed 

 a constant difference from his own. 



