182 



SCIENTIFIC THOUGHT. 



second was the invention of a new and shorter method 

 of calculating the orbit of a planet from a limited number 

 of contiguous observations. 1 This method was conimuni- 



The latter was the first addition 

 made after 2000 years to the 

 knowledge of this matter possess- 

 ed by the ancients. (See ' Disquis. 

 Arithm.,' sec. 365: " Magnopere 

 sane est mirandum, quod, quum 

 jam Euclidis temporibus circuli 

 divisibilitas geometrica in tres et 

 quinque partes nota fuerit, nihil 

 his inventis intervallo 2000 anno- 

 rum adjectum sit," &c. ; and his 

 manuscript note to this passage, 

 given by Schering, vol. i. p. 176 : 

 " Girculum in 17 partes divisibilem 

 esse geometrice, deteximus 1796, 

 Mart. 30.") It is probably owing 

 to the independent manner in which 

 Gauss approached the subject that 

 he early found the necessity of 

 treating subjects of higher arith- 

 metic (i.e., of the theory of num- 

 bers or " discrete magnitudes " as 

 distinguished from algebra, which 

 is the theory of "continuous mag- 

 nitudes") by an independent me- 

 thod, for which he invented a 

 language and an algorithm. He 

 thus raised this part of mathe- 

 matics into an independent science, 

 on which the ' Disquisitiones Arith- 

 metics ' is the first elaborate and 

 systematic treatise. Legendre's 

 'Traite des nombres' (1799) is a 

 complete thesaurus of all that was 

 at that time known and of what 

 was added by him, but it does not 

 attempt to establish the science on 

 a new basis. 



1 On the 1st January 1801 

 Piazzi at Palermo had found a 

 movable star of 8th magnitude, 

 RA. 57 47', ND. 16 8', which he 

 announced to Bode at Berlin as a 

 comet on the 24th January ; but 

 a few days later he concluded it 

 must be a planet, and named it 

 " Ceres Ferdinandea." No one be- 



sides Piazzi could find the star, but 

 several astronomers, Piazzi himself, 

 Olbers at Bremen, and Burckhardt 

 at Paris, tried to calculate the orbit 

 from the observations of the dis- 

 coverer, which were contained 

 within only 9 degrees. The at- 

 tempt to do so under the sup- 

 position of either a circular or a 

 parabolic or an elliptic orbit failed, 

 and Olbers expressed the fear that 

 with the circular or elliptic ele- 

 ments which had been published in 

 Zach's periodical, it might prove 

 impossible to find the star when 

 it should again become visible. 

 Very near the expected time, as 

 late as the beginning of December, 

 Gauss communicated his elements 

 to Von Zach, who published them 

 at once, recommending astronomers 

 to follow Dr Gauss's figures and 

 look 6 to 7 more eastward than 

 the positions of Burckhardt, Piazzi, 

 and Olbers indicated. And actu- 

 ally on the 7th December 1801 

 Zach himself, and on the 1st Janu- 

 ary 1802 Olbers, succeeded in find- 

 ing the star, "like a grain of sand 

 on the sea -shore," very near the 

 positions calculated by Gauss. 

 These results, followed soon by 

 the discovery of other planets by 

 Olbers and Harding, gave a great 

 impetus to the study of astronomy. 

 Gauss's methods were published 

 in extenso in the now celebrated 

 ' Theoria motus corporum cceles- 

 tium' in 1809. Two problems are 

 herein treated in a novel and com- 

 plete manner. The first was to 

 calculate by a simple and accurate 

 method from the necessary number 

 of observations the orbit of a planet 

 or comet on the assumption of New- 

 ton's law of gravitation, but with- 

 out any other special conditions. 



