Increasingly throughout the nineteenth century, geometry had come to rely less on the graphical devices of synthetic geometry and more on the algebra of analytic techniques.9 Unlike synthetic geometry (the diagrams of which can be considered a formal representation), in the abstract shape of equations, little of the experience of form remains. The resultant divergence between geometry and the physical space it had once sought to describe soon therefore gave rise to a new freedom of thought in mathematics. With this freedom
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came many new forms of geometry, among them n-dimensional geometry (which is the study of geometry in any number of dimensions, including the fourth) and non-Euclidean geometry (in which the basic rules that determine the structure of space are amended).10 Both these propositions use analytic geometry to reverse the process through which the mathematical properties of figures had traditionally been determined. Unlike the geometrical description of a shape such as a circle or a sphere, in these new geometries the description (the formula) precedes the visual representation, and can therefore present to the imagination a clearly defined, rule-based construction, of which there is no real equivalent in experience. In the equations and formulae of analytic geometry it was thus possible to define four-dimensional figures using a technique removed from the difficulties of envisaging four-dimensional space.