, i.e. the square of
is -1. This is what we desired to secure. Thus our definitions serve all necessary purposes.
It is easy to give a geometrical interpretation of complex numbers in the geometry of the plane. This subject was agreeably expounded by W. K. Clifford in his Common Sense of the Exact Sciences, a book of great merit, but written before the importance of purely logical definitions had been realised.
Complex numbers of a higher order, though much less useful and important than those what we have been defining, have certain uses that are not without importance in geometry, as may be seen, for example, in Dr Whitehead's Universal Algebra. The definition of complex numbers of order
is obtained by an obvious extension of the definition we have given. We define a complex number of order
as a one-many relation whose domain consists of certain real numbers and whose converse domain consists of the integers from 1 to