This paper examines the relationship between sunshine duration and solar radiation received on the earth's surface. Sixty-nine thousand pairs of sunshine radiation readings from 670 sites were analyzed. A generalization of the Angstrom-Prescott equation of the form (K) over bar = (K) over bar (clear) [beta + (1 - beta)S-gamma] was found to most efficiently fit the data and suggests the relationship between the average daily atmospheric transmittance (K) over bar and the sunshine fraction S is non-linear. The suggested reason for this non-linearity is that a reduced sunshine fraction not only decreases the clear sky radiation duration, but also the radiation transmitted through clouds, i.e. clouds get optically thicker with decreasing S. This finding is supported on theoretical grounds and by analyzing instantaneous solar radiation measurements from Australia and Germany. Representing the sunshine fraction in terms of the proportion of beam radiation reaching the earth's surface S = (H) over bar (b)vertical bar(H) over bar (b,clear) leads to a fundamental connection between the monthly average diffuse fraction and the sunshine radiation relationship. Moreover, it confirms the non-linearity of the latter relationship, which was previously questioned because of limited data and/or poor quality sunshine measurements. (C) 2013 Elsevier Ltd. All rights reserved.