In this paper, the research method of large deformations of hyperelastic solids using the Finger strain measure is considered. The kinematics of continua motion is given; the stress state is described by the Cauchy – Euler tensor of the true stresses. Much attention is given to the algorithm of the linearized constitutive equations in terms of the Cauchy – Euler stresses. Also, there is an example of obtaining the linearized physical ratio in the Cauchy – Euler stresses for material, which is described by Mooney – Rivlin potential. The numerical implementation is based on a finite-element method within the framework of incremental methods. To verify efficiency of the technique, several problems were solved: the problem of plane strain of the square strip and the problem of elastic deformation of the plate under uniform pressure. The results obtained do not contradict those that have been published before.