From the INTRODUCTION TO the general reader, the name of the fourth dimension brings reminiscences of Flatland and The Time Machine. On hearing that to the mathematician the extension from three dimensions to four or five is trivial, he thinks he is being told that a study of mathematics, if reasonably intense, creates physical faculties or powers of visualisation with which the uninitiated are not endowed. Learning that Minkowski and Einstein combine space and time into a single continuum, he tries to believe in the existence of a state of mind in which the sensations of space and time are confused, and naturally he fails. The position of students of mathematical physics, and of all but a fortunate few of the students of pure mathematics, is little better. Accustomed to regard a Cartesian frame of axes as a scaffolding erected in the real space around them, they can attach no meaning to a fourth coordinate, but having used complex electromotive forces with success in the theory of alternating currents, and having treated a symbol of differentiation as a detachable algebraic variable even to the extent of resolving operators into partial fractions for the solution of differential equations, these students are prepared to give pragmatical sanction to the most fantastic language. The pure mathematician makes no attempt to imagine a space of four dimensions; he lays no claim to visualizing a world that is inconceivable to other men. Only he finds that certain notions in algebra are discussed most readily in terms adopted from geometry and given a meaning entirely algebraic, and since it is to the mathematician alone that algebraic problems are of concern in themselves, fear lest the man in the street should mistake the very subject of a mathematical conversation he might overhear has not prevented the mathematician from using the vocabulary he finds best suited to his own needs. Now it has happened that the talk of a few mathematicians has suddenly become of universal and absorbing interest, and a dictionary explaining the meanings they are in the habit of giving to some familiar words is required. It is this dictionary that I have tried to write, and I have written it in the simplest terms I could find, in the hope that it will prove intelligible to anyone familiar with elementary trigonometry and with the solution of simultaneous linear equations in algebra; for this reason, I have not treated the point as indefinable, I have supposed the numbers used always to be real, and I have avoided Frege-Russell definitions. The reader's first feeling will be one of disillusion. Are Einstein and Eddington talking not about a new heaven and a new earth but about linear algebraic equations? To discuss the question is beyond the province of a lexicographer." Perhaps even the mathematical student, if he can overcome a reasonable irritation at the restrictions, from his point of view arbitrary, to four dimensions and to real numbers, and at the absence of certain obvious forms of abbreviation, may derive some help from the pamphlet. The possibility of constructing an abstract 'space' is always assumed, but the details of the construction, even for two or three dimensions only, are either taken for granted or disguised as theorems on matrices or on linear equations. The idea of direction and the measurement of angles in a constructed plane demand careful consideration. The nature of pure rotation in four dimensions is by no means obvious; on the contrary, rotation is the most difficult of the elementary notions used in the theory of relativity, and with an account of rotation our formal work comes to a natural end.