It is familiar knowledge that the line, the surface and ordinary Euclidean space are to be regarded as spaces of one, two and three dimensions respectively and readers of this journal are aware that a hypothetical space of four dimensions has been closely investigated. The most convenient space to study is the surface or two-space, since we can regard it as embedded in a three-space. If a surface is curved it is generally impossible to draw a straight line on it, for as we see clearly, the "straightest" line is changing its direction at every point. To describe this property accurately it is necessary to ascribe to each point a magnitude which expresses what happens to the direction of a short line in the region when displaced a short distance parallel to itself. This is called the direction-defining magnitude. Different sets of values of this magnitude relate to surfaces of different curvatures. A second fundamental property has recently been pointed out. There is inherent in every part of a space a measure of length peculiar to that particular region and which in general varies from region to region. To describe this variation accurately it is necessary to ascribe to each point another magnitude called the length-defining magnitude, which expresses the change from each point to the next of the unit of length. These two magnitudes define the surface completely. Similarly, a space of any number of dimensions is defined completely by a similar pair of magnitudes. A space is the "field" of such a magnitude-pair and the nature of these magnitudes defines the dimensions of the space. The four-space usually described is the Euclidean member of an infinity of four-spaces. When we look into a mirror we see a space differing from ordinary space in that right and left are interchanged and this is described mathematically by saying that if we locate points as usual by specifying three distances $X_1$, $X_2$, $X_3$ of the point from three mutually perpendicular planes, then a point $X_1$, $X_2$, $X_3$, in actual space corresponds with a point $X_1$, $X_2$, $ -X_3$ in the mirrored space: in other words the mirrored space is derived from the real space by multiplying the $X_3$ coordinates by $-1$. If we were to multiply by $\sqrt{-1}$ instead of $-1$ we should derive a different space; in this case, however, we have no mirror to show us what it looks like. Such a space is said to have one negative dimension and it has the peculiar property that in the figure derived from the right triangle of ordinary space the square of the "hypotenuse" equals the difference and not the sum of the squares of the other two sides, so that the length of a line may sometimes have to be represented by the square-root of a negative number, a "complex" number.