text: Bouguer-Lambert law: This law is based on observations made by Pierre Bouguer before 1729. It is often attributed to Johann Heinrich Lambert, who cited Bouguers  (Claude Jombert, Paris, 1729) – and even quoted from it – in his Photometria' in 1760. Lambert expressed the law, which states that the loss of light intensity when it propagates in a medium is directly proportional to intensity and path length, in the mathematical form used today.  
  Lambert began by assuming that the intensity  of light traveling into an absorbing body would be given by the differential equation:  which is compatible with Bouguer's observations. The constant of proportionality  was often termed the "optical density" of the body.  Integrating to find the intensity at a distance  into the body, one obtains:    For a homogeneous medium, this reduces to:   from which follows the exponential attenuation law:  
  Beers law:  Much later, in 1852, the German scientist August Beer studied another attenuation relation.  In the introduction to his classic paper, he wrote: "The absorption of light during the irradiation of a colored substance has often been the object of experiment; but attention has always been directed to the relative diminution of the various colors or, in the case of crystalline bodies, the relation between the absorption and the direction of polarization. Concerning the absolute magnitude of the absorption that a particular ray of light suffers during its propagation through an absorbing medium, there is no information available."  By studying absorption of red light in colored aqueous solutions of various salts, he concluded that "the transmittance of a concentrated solution can be derived from a measurement of the transmittance of a dilute solution".  It is clear that he understood the exponential relationship, as he wrote: "If  is the coefficient (fraction) of diminution, then this coefficient (fraction) will have the value  for double this thickness."  Furthermore Beer stated: "We shall take the absorption coefficient to be the coefficient giving the diminution in amplitude suffered by a light ray as it passes through a unit length of an absorbing material. We then have, according to theory, and as I have found verified by experiment,    where  is the absorption coefficient and D the length of the absorbing material traversed in the experiment."    This is the relationship that might properly be called Beers law.  There is no evidence that Beer saw concentration and path length as symmetrical variables in an equation in the manner of the Beer-Lambert law.  
  Beer-Lambert law:   The modern formulation of the Beer–Lambert law combines the observations of Bouguer and Beer into the mathematical form of Lambert. It correlates the absorbance, most often expressed as the negative decadic logarithm of the transmittance, to both the concentrations of the attenuating species and the thickness of the material sample.  An early, possibly the first, modern formulation was given by Robert Luther and Andreas Nikolopulos in 1913.