The fourth dimension the time traveler tells his dinner guests in chapter 1 of H. G. Wells' The Time Machine, is time itself. Consistent with the author's materialist philosophy (Wells was a Fabian Socialist when he wrote The Time Machine), time is another dimension of space added to the three human beings already perceive. The reason that we fail to notice this attribute of reality, the time traveler explains against the objections of his guests, is that "our consciousness moves along it." As with much of his other works, Wells uses scientific hypotheses to comment on the political developments of the society of his time. The Time Machine is no exception. The time traveler visits the future only to discover that socialism - or a corrupt version of it - is the state to which human society has evolved.
In Chapter 1 of The Time Machine the reader is introduced to Wells' definition of the Fourth Dimension. In Wells' story, the Time Traveller introduces the idea of time as the Fourth Dimension (current science now holds that time, in space-time, provides a fourth dimension to the universe). The Time Traveller's audience, the Medical Man, the Psychologist, the Very Young Man, the Provincial Mayor, Fillby and Hillyer all protest that time cannot be a dimension because it is impossible to willfully move through it in the way that altitude, latitude and longitude can be intentionally moved through. The Time Traveller quiets their protestations by having the Psychologist pull a lever on a small box. When the Psychologist does so, the box vanishes and the group is told the the box has left them and is traveling through space, an announcement followed by his declaration to similarly travel through the Fourth Dimension of time by using a bigger version of the small time traveling box.