Tone curves are one of the simplest techniques for image enhancement. Specified as a function, a tone curve is a transformation that maps pixel levels of an input image to new output levels. Tone curves are the basis of many contrast enhancement algorithms, including Contrast Limited Histogram Equalisation (CLHE), which derives a tone curve from a modification of the image histogram. While these methods can provide good enhancement, they are generally non-linear. In this paper we show the surprising result that a tone curve generated by the non-linear CLHE method (and HE) can be calculated by applying a linear transform to the histogram of the input image. Experiments validate our method.