These remarks appear to me to be well founded, and I must state why it is that I have ventured to publish an extensive work on logic, without properly making up my mind as to the fundamental nature of the reasoning process. The fault after all is one of omission rather than of commission. It is open to me on a future occasion to supply the deficiency if I should ever feel able to undertake the task. But I do not conceive it to be an essential part of any treatise to enter into an ultimate analysis of its subject matter. Analyses must always end somewhere. There were good treatises on light which described the laws of the phenomenon correctly before it was known whether light consisted of undulations or of projected particles. Now we have treatises on the Undulatory Theory which are very valuable and satisfactory, although they leave us in almost complete doubt as to what the vibrating medium really is. So I think that, in the present day, we need a correct and scientific exhibition of the formal laws of thought, and of the forms of reasoning based on them, although we may not be able to enter into any complete analysis of the nature of those laws. What would the science of geometry be like now if the Greek geometers had decided that it was improper to publish any propositions before they had decided on the nature of an axiom? Where would the science of arithmetic be now if an analysis of the nature of number itself were a necessary preliminary to a development of the results of its laws? In recent times there have been enormous additions to the mathematical sciences, but very few attempts at psychological analysis. In the Alexandrian and early mediæval schools of philosophy, much attention was given to the nature of unity and plurality chiefly called forth by the question of the Trinity. In the last two centuries whole sciences have been created out of the notion of plurality, and yet speculation on the nature of plurality has dwindled away. This present treatise contains, in the eighth chapter, one of the few recent attempts to analyse the notion of number itself. If further illustration is needed, I may refer to the differential calculus. Nobody calls in question the formal truth of the results of that calculus. All the more exact and successful parts of physical science depend upon its use, and yet the mathematicians who have created so great a body of exact truths have never decided upon the basis of the calculus. What is the nature of a limit or the nature of an infinitesimal? Start the question among a knot of mathematicians, and it will be found that hardly two agree, unless it is in regarding the question itself as a trifling one. Some hold that there are no such things as infinitesimals, and that it is all a question of limits. Others would argue that the infinitesimal is the necessary outcome of the limit, but various shades of intermediate opinion spring up.