This paper is a somewhat revised version of a Russian preprint (1988) [the author, The concept of randomness from Aristotle to Poincaré (Moskva 1988; Zbl 0652.01002)]. The title should be now “…up to Poincaré”. The work of Poincaré will be treated in an extra paper. The modern view of randomness, the ideas of R. von Mises (1918, 1931, 1964), of Kolmogorov (1933), and of Kolmogorov and Uspensky (1987) are also mentioned in this paper. Now 101 references.