The algorithmic medium is an infrastructure of communication augmenting the powers of previous media by adding the automation of symbolic operations. Its emergence in the middle of the twentieth century results from a long scientific and technological history that I summarize at the beginning of the paper. Then I recall in this article the main stages of its development (mainframes, Internet and PC, social Web, cloud augmented by artificial intelligence and the blockchain) as well as their socio-cognitive consequences. I evoke in the end the future developments of this medium, in the perspective of a reflexive collective intelligence based on a new form of semantic computation.