The article is devoted to the original mathematical models of combat operations developed in Russia at the beginning of the XX century. One of the first works in which approaches to mathematical modeling of military operations were outlined can be considered an article by Y. Karpov «Tactics of fortress artillery», published in 1906. It considered the task of defending the fortress from attacking enemy infantry chains. Based on the idea of the attackers overcoming the line of defense, mathematical relations were obtained linking the parameters of the shot of the shrapnel charge with the movements of the infantryman. Similarly, the task of using a machine gun for the defense of the fortress was considered. After analyzing the obtained ratios, Y. Karpov came to the conclusion that all means of defense of the fortress can be correlated through the length of the area defended by this means. P. Nikitin developed Y. Karpov's ideas. He considered a wide range of means of destruction. Based on the results of the research, the author made recommendations on the distribution of forces and means in the defense of fortresses. M. Osipov in 1915 published vivid and original models of two-way combat operations, a year earlier than the well-known Lanchester theory. Summing up the numbers of the fighting sides at infinitesimal intervals of time, and then moving to the limits, he obtains linear and quadratic laws of the influence of the ratio of the number of the fighting sides on their losses, and explores heterogeneous means of destruction. All this is verified by the practice of various battles. M. Osipov showed that the coefficients in the laws of losses depend on the training of personnel, terrain, the presence of fortifications, the moral and psychological state of the troops, etc. Based on the results of mathematical modeling, M. Osipov for the first time substantiated a number of provisions of the art of war. He showed that neither linear nor quadratic laws of losses in general do not correspond to the practice of the battles conducted. For ease of use at that level of computer technology development and to obtain a more reliable result, M. Osipov proposed using the degree of "three second" in the laws of losses, although he himself understood its approximate nature. Much attention is paid to the problem of authorship, the search for a prototype of the creator of the first two-sided model of combat operations, and the application of theory to solve modern applied problems.
