SECTION IX. 
ON THE CURVATURES OF CURVES IN SPACE. 
197. Preparatory to finding the radius of curvature, 
and evolutes of a curve in space, consider figure (59), where 
for the curve is substituted an equilateral polygon mm'm"..., 
and through the middle points of its sides are drawn planes 
respectively perpendicular to them, which intersect, two and 
two, in the lines kh, k'h', k"h", &c. Then the plane which 
contains the tw’o consecutive sides mm, m'm", is perpendicular 
to each of the planes gh, g h!, and therefore to their common 
intersection kh ; let kh meet this plane in the point q, then 
q is the center of a circle passing through the three angles 
m, m!, m"; and every point in the line kh is likewise equi 
distant from the same three angles. 
The lines kh-, k'h', he. will be parallel only when the 
sides of the polygon mm m"...are in the same plane; in other 
cases, if they be produced till each meets its consecutive, they 
will form a polygon hop,.., the angular points of which are 
equidistant from four consecutive angles of the first polygon 
mm m"... The point o for instance, since it is situated in kh, is 
equidistant from m, m, m"; and again, being situated in k'h', 
it is equidistant from m, m", m"; that is, it is the center of a 
sphere passing through four consecutive angles, m, m, m", m". 
198. The preceding resvdts being true when the number 
of sides of the polygon is indefinitely increased, it follows that 
the normal planes of a curve generate, by their perpetual 
intersection, a curve surface; also, since of the lines of 
intersection kh,k'h', he. every two consecutive ones are in the 
same plane, the surface which they generate is developable.