The method of ellipsoids for solving systems of linear inequalities [\textit{L. G. Khachiyan}, Sov. Math., Dokl. 20, 191-194 (1979; Zbl 0414.90086)] is closely related to the ''modified center of gravity method (MCGM) for convex programming proposed by \textit{D. B. Yudin} and \textit{A. S. Nemirovskij} [Ehkon. Mat. Metody 12, 357-369 (1976; Zbl 0329.90054)] and, independently, by the second author in several papers [see Kibernetika 1970, No.1, 6-12 (1970; Zbl 0243.90037)]. In the present article the method of ellipsoids and the MCGM are presented as ''just a very particular case of gradient-type algorithms using a space-stretching operation'' which were first developed by the second author for the purpose of accelerating the convergence of subgradient methods for convex programming [see ibid. 1970, No.2, 80-85 (1970; Zbl 0243.90038)]. In these algorithms the subgradient descent method is modified at each iteration by stretching the space in the direction of the subgradient or in the direction of the difference of two consecutive subgradients. A fairly detailed description of the first variant (stretching in the direction of the subgradient) is given in this article with the method of ellipsoids appearing as a particular case. However, to obtain fast convergence the optimal value of the objective functional in the convex program is assumed to be known. If such is not the case ''the awkwardness of the algorithm for obtaining a definite accuracy in the minimization of the functional naturally grows considerably''. To remedy this defect the variant of stretching in the direction of the difference of two consecutive subgradients was initiated by the second author and \textit{N. G. Zhurbenko} [ibid. 1971, No.3, 51-59 (1971; Zbl 0272.90071)]. The authors of this article also elaborate on the evolution of the MCGM and its predecessor, the ''center of gravity (polyhedral) method'' of \textit{A. Yu. Levin} [Dokl. Akad. Nauk SSSR 160, 1244-1247 (1965; Zbl 0154.450)] and, in the last part of the article, they comment on the efficiency of the theoretical and computational applications of the method of ellipsoids to a variety of subjects: combinatorics, graph theory and others. They emphasize that ''the theoretical vaue of the ellipsoid methods began to be acknowledged only after some time'' while, simultaneously, ''doubts were raised about their practical efficiency''. Whatever the final result of this variety of developments may eventually be, this is an informative article, in particular with respect to papers which are less known, no doubt because they remain untranslated from their original Russian versions. A list of twenty-four references which contains some inaccuracies closes the paper.