Almost all problems in mechanics that are typically formulated mathematically using a linear differential equation are, strictly speaking, nonlinear, meaning they should actually be expressed by a nonlinear differential equation. The problems have only been linearized to avoid the mathematical difficulties posed by such differential equations and to apply the well-established and clear theory of linear differential equations. This method of linearization has been particularly consistently applied to two important branches of mechanics: elastostatics and vibration theory. Thus, two frameworks of the linearized theory have been established, whose structure is largely and in many details already definitively shaped today. Gradually, however, the inadequacies have also become apparent.