In re-writing the Solid Geometry the authors have consistently carried out the distinctive features described in the preface of the Plane Geometry. Mention is here made only of certain matters which are particularly emphasized in the Solid Geometry.

Owing to the greater maturity of the pupils it has been possible to make the logical structure of the Solid Geometry more prominent than in the Plane Geometry. The axioms are stated and applied at the precise points where they are to be used. Theorems are no longer quoted in the proofs but are only referred to by paragraph numbers; while with increasing frequency the student is left to his own devices in supplying the reasons and even in filling in the logical steps of the argument. For convenience of reference the axioms and theorems of plane geometry which are used in the Solid Geometry are collected in the Introduction.

The present work is constructed on the same plan as my treatise on Plane Trigonometry, to which it is intended as a sequel; it contains all the propositions usually included under the head of Spherical Trigonometry, together with a large collection of examples for exercise. In the course of the work reference is made to preceding writers from whom assistance has been obtained; besides these writers I have consulted the treatises on Trigonometry by Lardner, Lefebure de Fourcy, and Snowball, and the treatise on Geometry published in the Library of Useful Knowledge. The examples have been chiefly selected from the University and College Examination Papers.

In the account of Napier’s Rules of Circular Parts an explanation has been given of a method of proof devised by Napier, which seems to have been overlooked by most modern writers on the subject. I have had the advantage of access to an unprinted Memoir on this point by the late R. L. Ellis of Trinity College; Mr Ellis had in fact rediscovered for himself Napier’s own method. For the use of this Memoir and for some valuable references on the subject I am indebted to the Dean of Ely.Considerable labour has been bestowed on the text in order to render it comprehensive and accurate, and the examples have all been carefully verified; and thus I venture to hope that the work will be found useful by Students and Teachers.

The material contained in the following translation was given in substance by Professor Hilbert as a course of lectures on euclidean geometry at the University of G¨ottingen during the winter semester of 1898–1899. The results of his investigation were re-arranged and put into the form in which they appear here as a memorial address published in connection with the celebration at the unveiling of the Gauss-Weber monument at G¨ottingen, in June, 1899. In the French edition, which appeared soon after, Professor Hilbert made some additions, particularly in the concluding remarks, where he gave an account of the results of a recent investigation made by Dr. Dehn.

The Theory of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not a_ect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics.

This principle is indeed of fundamental importance; and it may with safety be affirmed, that the recent advances of pure analysis have been much assisted by the inuence which it has exerted in directing the current of investigation. But the full recognition of the consequences of this important doctrine has been, in some measure, retarded by accidental circumstances. It has happened in every known form of analysis, that the elements to be determined have been conceived as measurable by comparison with some fixed standard.

In preparing this book, the author had especially in mind classes in the upper grades of grammar schools, though the work will be found equally well adapted to the needs of any classes of beginners.

The ideas which have guided in the treatment of the subject are the following:

The study of algebra is a continuation of what the pupil has been doing for years, but it is expected that this new work will result in a knowledge of general truths about numbers, and an increased power of clear thinking.

All the differences between this work and that pursued in arithmetic may be traced to the introduction of two new elements, namely, negative numbers and the representation of numbers by letters. The solution of problems is one of the most valuable portions of the work, in that it serves to develop the thought-power of the pupil at the same time that it broadens his knowledge of numbers and their relations. Powers are developed and habits formed only by persistent, long-continued practice.

The topics in this book are arranged for primary courses in calculus in which the formal division into differential calculus and integral calculus is deemed necessary. The book is mainly made up of matter from my Infinitesimal Calculus, Changes, however, have been made in the treatment of several topics, and some additional matter has been introduced, in particular that relating to indeterminate forms, solid geometry, and motion.
The articles on motion have been written in the belief that familiarity with the notions of velocity and acceleration, as treated by the calculus, is a great advantage to students who have to take mechanics.

Analysis of a Proposition into its Elements. Numerical and Geometrical Problems. The Theory of Inference. The Construction of Problems. And many other Curiosa Logica.

In Book I, Chapter II, I have adopted a new definition of ‘Classification’, which enables me to regard the whole Universe as a ‘Class,’ and thus to dispense with the very awkward phrase ‘a Set of Things.’

It has been the author's aim to treat the subject according to the latest and most approved methods. The book is designed for the use of colleges, technical schools, normal schools, secondary schools, and for those who take up the subject without the aid of a teacher.

The contemplation of the various steps by which mankind has come into possession of the vast stock of mathematical knowledge can hardly fail to interest the mathematician. He takes pride in the fact that his science, more than any other, is an exact science, and that hardly anything ever done in mathematics has proved to be useless.

The chemist smiles at the childish efforts of alchemists, but the mathematician finds the geometry of the Greeks and the arithmetic of the Hindoos as useful and admirable as any research of today. He is pleased to notice that though, in course of its development, mathematics has had periods of slow growth, yet in the main it has been pre-eminently a progressive science.

THIS work here undertaken differs somewhat in its scope and design from systems of Logic which have hitherto been given to the world. The Aristotelian Logic is simply the method of deduction and, as such, it is complete. Subsequent works, in so far as they have been strictly logical, have closely copied the great master, and have confined them-selves to an exhibition of the deductive principles and processes.

Now, the deductive method comprehends merely the laws which govern inferences or conclusions from premises previously established.
These premises may, in their turn, be inferences from other premises, and so on, to certain extent and just so far this method is all sufficient. But it is evident that the evolution of premises and conclusions, and conclusions and premises, must have limit.