A central concept in the calibration of instruments is that of the mathematical error model. An adequately derived, physically related, error model provides for the correction in the data reduction process for measurable systematic errors. Error interactions and the nature of their propagation into the data as a function of instrument parameters are provided. Such a physically related mathematical error model is presented. While specifically intended for two axis Cinetheodolites, it is quite applicable to other types of mounts, such as radars. Techniques are indicated for its extension to instruments with more than two axes. Conceptually, the error model divides into three parts: a nodal point-image vector which may be perturbed to correct for certain errors; three or more matrices which represent the basic rotations and certain errors; and point by point corrections to the observed angular values used in the matrices. Vectors and matrices are used freely in the model. A brief tutorial presentation is provided on the mathematical methods employed. The model is useful in calibration of instruments, error analysis, acceptance testing, error correction, and provides a basis for a new philosophy of instrument design which potentially could result in cost savings.