The results of fifty-three experiments in which octopuses were trained to make tactile discriminations between the members of pairs of Perspex cylinders are reported. Grooves cut into these otherwise smooth cylinders varied in number and arrangement. The proportion of errors made in distinguishing such objects depends upon the difference between the proportions of groove on the objects concerned, and is not affected by the pattern or orientation of the grooves. It has thus been possible to measure the similarity to Octopus of the objects used and to predict the errors that will be made in any such discrimination. When these results are considered in the light of the known nervous arrangements in the arms, it is possible to present a hypothesis about the mode of action of the peripheral tactile sensory system and the function of the brain. It is necessary to suppose that the latter distinguishes frequencies of nerve impulses in the sensory nerves from the arms; it is not necessary to postulate any projection of the sensory field or scanning mechanism involving the use of proprioceptive information.