It is rare that a theory of such fundamental mathematical importance is expressed so simply and clearly: the reader with a good grasp of college mathematics will be able to understand most of the basic ideas and many of the proofs.Cantor first develops the elementary definitions and operations of cardinal and ordinal numbers and analyzes the concepts of "canlinality" and "ordinality." He covers such topics as the addition, multiplication, and exponentiation of cardinal numbers, the smallest transfinite cardinal number, the ordinal types of simply ordered aggregates, operations on ordinal types, the ordinal type of the linear continuum, and others. He then develops a theory of well-ordered aggregates, and investigates the ordinal numbers of well-ordered aggregates and the properties and extent of the transfinite ordinal numbers.An 82-page introduction by the eminent mathematical historian Philip E. B. Jourdain first sketches the background of Cantor's theory, discussing the contributions of such predecessors as Veicrstrass, Cauchy, Dedekind, Dirichlet, Riemann, Fourier, and Hankel; it then traces the development of the theory by summarizing and analyzing Cantor's earlier work. A bibliographical note provides information on further investigations in the theory of transfinite numbers by Frege, Peano, Whitehead, Russell, etc."Would serve as well as any modern textto initiate a student in this exciting branch of mathematics." — Mathematical Gazette.

    You must conceive, too, that this plateau is cut by gulches and canons in many directions, and that beautiful valleys are scattered about at different altitudes. The first series of canons we are about to explore constitutes a river channel through such a range of mountains. The canon is cut nearly half-way through the range, then turns to the east, and is cut along the central line, or axis, gradually crossing it to the south. Keeping this direction for more than fifty miles, it then turns abruptly to a southwest course, and goes diagonally through the southern slope of the range. This much we knew before entering, as we made a partial exploration of the region last fall, climbing many of its peaks, and in a few places reaching the brink of the canon walls, and looking over the precipices, many hundreds of feet high, to the water below. Here and there the walls are broken by lateral canons, the channels of little streams entering the river; through two or three of these, we found our way down to the Green in early winter, and walked along the low water-beach at the foot of the cliffs for several miles. Where the river has this general easterly direction, the western part only has cut for itself a canon, while the eastern has formed a broad valley, called, in honor of an old-time trapper, Brown's Park, and long known as a favorite winter resort for mountain men and Indians. May 30.?This morning we are ready to enter the mysterious canon, and start withsome anxiety. The old mountaineers tell us that it cannot be run; the Indians say, Water ...