text: While the observations of Bouguer and Beer have a similar form in the Beer-Lambert law, their areas of observation were very different.  For both experimenters, the incident beam was well collimated, with a light sensor which preferentially detected directly transmitted light.      
  Beer specifically looked at solutions.  Solutions are homogeneous and do not scatter light (Ultraviolet, visible, Infrared) of wavelengths commonly used in analytical spectroscopy (except upon entry and exit).  The attenuation of a beam of light within a solution is assumed to be only due to absorption.  In order to approximate the conditions required for the Beer Lambert law to hold, often the intensity of transmitted light through a reference sample  consisting of pure solvent is measured, and compared to the intensity of light transmitted through a sample , with the absorbance of the sample taken as:  .  It is for this case that the common mathematical formulation (see below) applies:     
  Bouguer looked at astronomical phenomena where the size of a detector is very small compared to the distance traveled by the light.  In this case, any light that is scattered by a particle, either in the forward or backward direction, will not strike the detector. The loss of intensity to the detector will be due to both absorption and scatter.  Consequently, the total loss is called attenuation (rather than absorption).  A single measurement cannot separate the two, but conceptually the contribution of each can be separated in the attenuation coefficient.  If  is the intensity of the light at the beginning of the travel and  is the intensity of the light detected after travel of a distance , the fraction transmitted, , is given by: , where  is called an attenuation constant or coefficient. The amount of light transmitted is falling off exponentially with distance. Taking the natural logarithm in the above equation, we get:  .    For scattering media, the constant is often divided into two parts, , separating it into a scattering coefficient, , and an absorption coefficient, .