1) Geometry. A mode of linear measurement, magnitude, or extension, in a particular direction; usually as co-existing with similar measurements or extensions in other directions.
2) Algebra. Since the product of two, or of three, quantities, each denoting a length (i.e. a magnitude of one dimension), represents an area or a volume (i.e. a magnitude of two, or of three, dimensions), such products themselves are said to be of so many dimensions; and generally, the number of dimensions of a product is the number of the (unknown or variable) quantities contained in it as factors (known or constant quantities being reckoned of no dimensions); any power of a quantity being of the dimensions denoted by its index.