Let N be a positive integer. We are concerned with the sum

Thus GN(N) is the ordinary Gauss sum. Previous methods of estimating such exponential sums have not brought to light the peculiar behaviour of GN(m) for m < N/2, namely that, for almost all values of m, GN(m) is in the vicinity of the point . A sharp estimate is given for max |GN(m)|, depending on the residue of N modulo 4. The results were suggested by graphs of GN(m) made for N near 1000. The analysis employs the Fresnel integrals and the Cornu spiral whose curvature is proportional to its arc length.