324 



SCIENTIFIC THOUGHT. 



the best we can attain to in our results. 1 An entirely 

 new branch of investigation springs up viz., the theory 

 of error, the doctrine of probability, and the investigation 



and calculation were invented to 

 deal practically with the problem. 

 Up to 1781, when the new planet 

 Uranus was discovered by Her- 

 schel, the interest centred mainly 

 in the determination of the orbits 

 of comets, which were assumed to 

 be parabolic. Halley was the first 

 to calculate these by means of ten- 

 tative methods given by Newton in 

 the ' Principia.' After 1781 the ne- 

 cessity arose of determining closed 

 orbits, and a first attempt was made 

 to do so by assuming circular orbits 

 (neglecting the ellipticity) and ne- 

 glecting the inclination of the plane 

 of the orbit to that of the earth. 

 But in the first year of this century 

 neither the parabolic nor the circular 

 figure of the orbits seemed to an- 

 swer in the case of the new planet 

 Ceres, nor could the inclination" of 

 the orbit be neglected. It required 

 all the skill of Gauss to tackle the 

 entire, unabbreviated problem, and 

 this was done in his fundamental 

 work ' Theoria motus corporum 

 ccelestium.' As the 'Principia' 

 form the foundation of all physical, 

 so does the ' Theoria motus 'of all 

 calculating astronomy. A similar 

 fundamental work which should take 

 the next important step, solving 

 generally the problem of the motion 

 of a body which is attracted from 

 more than one fixed or movable 

 centre (the problem of three bodies), 

 would mark the next great era in 

 calculating astronomy. Hitherto 

 this problem has only been treated 

 under the assumption that the third 

 attracting body disturbs the real 

 orbit which has been calculated. 

 The necessity of solving the prob- 

 lem of three bodies has made itself 

 felt in the theory of the moon and 

 other satellites, which stand under 



the influence of the main planet as 

 well as the sun, and where therefore 

 the ellipsis of Kepler cannot even 

 be taken as a first approximation. 

 And here again the necessity of tak- 

 ing into account the volume and 

 the figures of the attracting bodies 

 still further complicates the prob- 

 lem. On them depend the preces- 

 sion of the equinoxes and the ir- 

 regularity of the precession known 

 under the name of nutation. 



1 According to Wolf (' Handbuch 

 der Astronomic,' vol. i. p. 128 sqq.) 

 the merit of having first considered 

 the best methods of dealing with 

 errors of observation belongs to 

 Picard (1670) and Roger Cotes 

 (' Aestimatio errorum in mixta 

 mathesi,' 1722). The former seems 

 to have first used the apparently so 

 obvious rule of taking the arith- 

 metical mean of a number of ob- 

 servations, the latter introduced 

 the notion of attributing to each 

 observation its value or weight. 

 Cotes accordingly found that the 

 centre of gravity of a number of 

 weighted points distributed over a 

 plane coincided with the position of 

 greatest probability. Gauss sus- 

 pected that Tobias Mayer had 

 already employed modern methods 

 in his calculation of long series of 

 observations, and he himself used 

 what is termed after Legendre the 

 " method of least squares " as early 

 as 1795. It was not published till 

 1806 by Legendre, in his memoir 

 ' Nouvelles methodes pour la de" ter- 

 mination des orbites des cometes. ' 

 Gauss published his methods in 1809 

 in the celebrated ' Theoria motus 

 corporum ccelestium.' This method 

 of finding the most probable result 

 when a larger number of equations 

 is given than unknown quantities 



