Leaving aside the French neo-critical School (Renouvier) and the English School (Spencer)--the first of whom deny the Infinite, thus acting in opposition to mathematical reasoning, while the second perpetuate Kant’s error of considering the Infinite, though thinkable, as unknowable (Dr. Caird has pointed out that this position is illogical)--we arrive at a moment in history which is more fruitful in result on the mathematical side, and will, no doubt, have an effect on metaphysics. For, owing to recent discoveries in Germany and England, mathematics is now in a position to give greater support than before to the intuitions of philosophy. Hitherto, philosophers have been reluctant to allow full value to the mathematical conceptions of Infinity, and with some justice, as the notion had not been sufficiently analysed. Philosophers, who never attempted the analysis, have been inclined to accept certain contradictions in their conception as inherent in the nature of Infinity. Within the last twenty-five years Cantor and Dedekind have cleared up the notion of continuity, and Russell has given greater precision to the idea, and has applied this reasoning to philosophy. Present-day metaphysicians seem to be divided into two groups; on the one side, those who consider in philosophy the value of a theory of being, and, on the other, those who chiefly consider the value of a theory of knowledge, _i.e._ the Epistemologists. The first group, devoting themselves to psychology, evolution, and history, have no _necessary_ belief in the Infinite. The Epistemologists, whose work is founded on Kant, discuss the theory of knowledge and enumerate the conditions of knowledge. Their argument may not touch, but does not exclude, the notion of the Infinite. The position of the Epistemologist has been made infinitely more secure by recent mathematical work. That of the psychologist remains almost untouched. It is necessary now to examine more closely the mathematical results to which reference has been made. In general terms it may be said that mathematics has, as a study, led immediately from the nature of the subject to the perception of the Infinite, and to a knowledge of the connection between the Infinite and the Finite. The simplest form in which the idea can be put is stated by St. Augustine, who said that numbers considered individually were finite, but considered as an aggregate were infinite.[7] Before St. Augustine, and after him down the long stream of philosophic thought, the theologian and the philosopher have turned to mathematics for illustrations of the infinitely great and infinitely little, as developed from the concrete processes of arithmetic and geometry. The recurring decimal in arithmetic, the properties of the circle and ellipse in geometry, of the cone in conic sections, and of the surd in algebra, all touch the problem of number and space on the side of Infinity.