on pain of a vicious circle. We must, to return to the above analogy, know the number of houses in Piccadilly, before we know whether a given number has a corresponding house or not; and arithmetic alone, however subtly employed, will never give us this information. Thus the distinction which is important is, not the distinction between real and imaginary quantities, but between quantities to which points correspond and quantities to which no points correspond. We can conventionally agree to denote real points by imaginary coordinates, as in the Gaussian method of denoting by the single quantity (_a_ + √-1_b_) the point whose ordinary coordinates are _a_, _b_. But this does not touch Cayley's meaning. Cayley means that it is of great utility in mathematics to regard, as points with a real existence in space, the assumed spatial correlates of quantities which, with the coordinate system employed, have no correlates in every-day space; and that this utility is supposed, by many mathematicians, to indicate the validity of so fruitful an assumption. To fix our ideas, let us consider Cartesian axes in three-dimensional Euclidean space. Then it appears, by inspection, that a point may be situated at any distance to right or left of any of the three coordinate planes; taking this distance as a coordinate, therefore, it appears that real points correspond to all quantities from -∞ to +∞. The same appears for the other two coordinates; and since elementary Geometry proves their variations mutually independent, we know that one and only one real point corresponds to any three real quantities. But we also know, from the exhaustive method pursued, that all space is covered by the range of these three variable quantities: a fresh set of quantities, therefore, such as is introduced by the use of imaginaries, possesses no spatial correlate, and can be supposed to possess one only by a convenient fiction.