332 Account of Prof. Gauss's Paper intitled 



comparison of the different directions which occur. If the 

 directions of lines normal to the curve surface, drawn from 

 every point of the same, be designated by the points of the 

 sphere, corresponding to them according to the proceeding 

 above explained, so that each point of the curve surface has 

 its corresponding point on the auxiliary sphere, then, generally 

 speaking, every line on the curve surface will have a corre- 

 sponding one on the auxiliary sphere, and every portion of 

 surface of the former will have its corresponding portion on 

 the latter. The smaller the deviation of a part is from a plane, 

 the smaller will be the corresponding part of the sphere ; and 

 it is therefore a very natural proceeding to employ, as mea- 

 sure of the total curvature, the area of the corresponding por- 

 tion of the sphere. The author, accordingly, calls this area 

 the entire curvature of the corresponding portion of curve sur- 

 face. Besides this quantity, the position of the part comes into 

 consideration, which, independently of the relation of magni- 

 tude, may be a similar or a reversed one : these two cases may 

 be distinguished by the positive or negative sign being put 

 before the expression of the total curvature. This distinction, 

 however, has only a definite signification, so far as the figures 

 are assumed to be on definite sides of the surfaces : the author 

 assumes them in the sphere on the exterior, and in the curve 

 surface on that side on which the normal line is supposed to 

 be erected ; and it follows that the positive sign belongs to con- 

 vex-convex and to concave-concave surfaces (which are not 

 essentially different), and the negative sign to concave-convex 

 ones. If the portion of the curve surface in question consists 

 of parts dissimilar in this respect, further specifications be- 

 come necessary, which must here be passed over. 



The comparison of the areas of the portion of the curve 

 surface and its corresponding portion of the auxiliary sphere, 

 leads to a new notion (in the same manner as the comparison 

 of volume and mass produces the notion of density). The 

 author calls measure of curvature in any point of the curve 

 surface, the value of the fraction whose denominator is the 

 area of an infinitely small part of the curve surface at this 

 point ; and the numerator the area of the corresponding part of 

 the auxiliary sphere, or the entire curvature of the element. 

 It is clear that in the sense of the author, entire curvature and 

 measure of curvature are in curve surfaces analogous to what 

 in curve lines is respectively called amplitude and curvature ; 

 he doubted of the propriety of transferring to curve surfaces 

 the latter expressions, which, though sanctioned by use, are 

 not very appropriate terms. It is however of less consequence 



how 



