Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/109

the auxiliary sphere parallel to the normal, so the aggregate of the points on the auxiliary sphere, which correspond to all the points of a line on the curved surface, forms a line which will correspond to the line on the curved surface. And, likewise, to every finite figure on the curved surface will correspond a finite figure on the auxiliary sphere, the area of which upon the latter shall be regarded as the measure of the amplitude of the former. We shall either regard this area as a number, in which case the square of the radius of the auxiliary sphere is the unit, or else express it in degrees, etc., setting the area of the hemisphere equal to ${\textstyle 360^{\circ }.}$

The comparison of the area upon the curved surface with the corresponding amplitude leads to the idea of what we call the measure of curvature of the surface. If the former is proportional to the latter, the curvature is called uniform; and the quotient, when we divide the amplitude by the surface, is called the measure of curvature. This is the case when the curved surface is a sphere, and the measure of curvature is then a fraction whose numerator is unity and whose denominator is the square of the radius.

We shall regard the measure of curvature as positive, if the boundaries of the figures upon the curved surface and upon the auxiliary sphere run in the same sense; as negative, if the boundaries enclose the figures in contrary senses. If they are not proportional, the surface is non-uniformily curved. And at each point there exists a particular measure of curvature, which is obtained from the comparison of corresponding infinitesimal parts upon the curved surface and the auxiliary sphere. Let ${\textstyle d\sigma }$ be a surface element on the former, and ${\textstyle d\Sigma }$ the corresponding element upon the auxiliary sphere, then

$${\frac {d\Sigma }{d\sigma }}$$

will be the measure of curvature at this point.

In order to determine their boundaries, we first project both upon the ${\textstyle xy}$-plane. The magnitudes of these projections are ${\textstyle Z\,d\sigma ,}$ ${\textstyle Z\,d\Sigma .}$ The sign of ${\textstyle Z}$ will show whether the boundaries run in the same sense or in contrary senses around the surfaces and their projections. We will suppose that the figure is a triangle; the projection upon the ${\textstyle xy}$-plane has the coordinates

$$x,\ y;\qquad x+dx,\ y+dy;\qquad x+\delta x,\ y+\delta y.$$

Hence its double area will be

$$2Z\,d\sigma =dx.\delta y-dy.\delta x.$$

To the projection of the corresponding element upon the sphere will correspond the coordinates: