Account of Prof. Gauss's " Disquisitiones generates, &c" 331 



sions than by a selection of the most decisive observations. 

 As this subject has, since the first part of these remarks was 

 written, been treated of by Mr. Ivory, in a late Number of the 

 Phil. Mag., much more ably than I could pretend to, it will 

 be unnecessary for me to say any thing more at present upon 

 the subject. 

 Edinburgh, March 12, 1828. W. G. 



LII. Account of a Paper by Prof. Gauss, intitled " Disquisi- 

 tiones generales circa Superficies Curvas :" communicated 

 to the Royal Society of Gbttingen on the 8th of October 

 1827*. 

 A LTHOUGH geometricians have much occupied them- 

 -**• selves with general investigations on curve surfaces, the 

 results of which form a considerable portion of the higher de- 

 partment of geometry, this subject is so far from being ex- 

 hausted, that it may be safely asserted that as yet a small part 

 only of a most fertile field of inquiry has been cultivated. The 

 author has already endeavoured some years ago to take a new 

 view of this subject in the solution of the problem : To find all 

 representations of a given surface on another surface, in which 

 the smallest parts shall remain similar. The object of the 

 present paper is to open new views, and to unfold a portion of 

 the new truths which are thereby rendered accessible. We 

 shall explain as much as can be rendered intelligible without 

 entering too deeply into the subject ; but we must remark, that 

 the new definitions as well as the theorems, in order to be ge- 

 neralized, will require some restrictions and qualifications 

 which must be omitted in this place. 



In investigations which involve a variety of directions of 

 straight lines in space, it is advantageous to designate these 

 directions by those points on the surface of an invariable sphere 

 at which the radii drawn parallel to the same, terminate : the 

 radius and centre of this auxiliary sphere are entirely arbi- 

 trary ; for the latter, the unity of linear dimension may be 

 chosen. This proceeding agrees, in fact, with the one con- 

 stantly used in astronomy, where all directions are referred to 

 a fictitious celestial sphere of an infinite radius. Spherical tri- 

 gonometry, and some other theorems to which the author 

 has added one of frequent application, are then employed for 

 solving the various problems that present themselves by a 



* From the Gottinguche gelehrte Anzcigen. — This abstract is probably 

 from the pen of the distinguished Author. 



2 U 2 comparison 



