of comparison, and must be always perfect measures of each other. The unit may be infinitely varied in size—it being the name or representative of any assumed magnitude to which it is applied—but it always represents a magnitude of a definite form, and hence a magnitude which has an extension in every direction from its centre, and consequently represents not only one in length, but also one in breadth, and one in thickness. One inch, for example, in pure geometry, is always one cubic inch, but when used to measure a line, or extension in one direction, we take only one dimension of the unit, namely, the linear edge of the cube, and thus the operation not demanding either the breadth or the thickness of the unit, geometers have fallen into the error of supposing that a line is length without any breadth. These are the leading principles on which Mr. SMITH attempts the audacious task of rearing a new fabric of geometrical science, without regard to the wisdom of antiquity or the universal traditions of the schools. To us outside barbarians in the mysteries of mathematics, we confess that the work has the air of an ingenious paradox; but we must leave it to the professors to decide upon its claims to be a substitute for Euclid, Playfair, and Legendre. Every one who has a fondness for dipping into these recondite subjects will perceive in Mr. SMITH’S volume the marks of profound research, of acute and subtle powers of reasoning, and of genuine scientific enthusiasm, combined with a noble freedom of thought, and a rare intellectual honesty. For these qualities, it is certainly entitled to a respectful mention among the curiosities of literature, whatever verdict may be pronounced on the scientific claims of the author by a jury of his peers.