John Leigh Smeathman Hatton (1865-1933) was a British mathematician and educator. He worked for 40 years at a pioneering educational project in East London that began as the People's Palace and eventually became Queen Mary College in the University of London. Hatton served as its Principal from 1908 to 1933. This book, published in 1920, explores the relationship between imaginary and real non-Euclidean geometry through graphical representations of imaginaries under a variety of conventions. This relationship is of importance as points with complex determining elements are present in both imaginary and real geometry. Hatton uses concepts including the use of co-ordinate methods to develop and illustrate this relationship, and concentrates on the idea that the only differences between real and imaginary points exist solely in relation to other points. This clearly written volume exemplifies the type of non-Euclidean geometry research current at the time of publication.