This book is intended essentially as an "Introduction" and does not aim at giving an exhaustive discussion of the problems with which it deals. It seemed desirable to set forth certain results, hitherto only available to those who have mastered logical symbolism, in a form offering the minimum of difficulty to the beginner. The utmost endeavour has been made to avoid dogmatism on such questions as are still open to serious doubt, and this endeavour has to some extent dominated the choice of topics considered.
The beginnings of mathematical logic are less deffinitely known than its later portions, but are of at leastequal philosophical interest. Much of what is set forth in the following chapters is not properly to be called "philosophy" though the matters concerned were included in philosophy so long as no satisfactory science of them existed.
The nature of infinity and continuity, for example, belonged in former days to philosophy, but belongs now to mathematics. Mathematical philosophy, in the strict sense, cannot, perhaps, be held to include such definite scientific results as have been obtained in this region; the philosophy of mathematics will naturally be expected to deal with questions on the frontier of knowledge, as to which comparative certainty is not yet attained.More info →
In the greatly awakened interest in the common-school subjects during recent years, geography has received a large share. The establishment of chairs of geography in some of our greatest universities, the giving of college courses in physiography, meteorology, and commerce, and the general extension of geography courses in normal schools, academies, and high schools, may be cited as evidence of this growing appreciation of the importance of the subject.
While physiographic processes and resulting land forms occupy a large place in geographical control, the earth in its simple mathematical aspects should be better understood than it generally is, and mathematical geography deserves a larger place in the literature of the subject than the few pages generally given to it in our physical geographies and elementary astronomies.More info →
My Dear Children,
A young monkey named Genius picked a green walnut, and bit, through a bitter rind, down into a hard shell. He then threw the walnut away, saying:
“How stupid people are! They told me walnuts are good to eat.” His grandmother, whose name was Wisdom, picked up the walnut—peeled off the rind with her fingers, cracked the shell, and shared the kernel with her grandson, saying: “Those get on best in life who do not trust to first impressions.” In some old books the story is told differently; the grandmother is called Mrs Cunning-Greed, and she eats all the kernel herself. Fables about the Cunning-Greed family are written to make children laugh. It is good for you to laugh; it makes you grow strong, and gives you the habit of understanding jokes and not being made miserable by them. But take care not to believe such fables; because, if you believe them, they give you bad dreams.
MARY EVEREST BOOLE.More info →
It is intended to have the first sixteen pages of this book simply read in the class, with such running comment and discussion as may be useful to help the beginner catch the spirit of the subject-matter, and not leave him to the mere letter of dry definitions. In like manner, the definitions at the beginning of each Book should be read and discussed in the recitation room.
There is a decided advantage in having the de_nitions for each Book in a single group so that they can be included in one survey and discussion. For a similar reason the theorems of limits are considered together. The subject of limits is exceedingly interesting in itself, and it was thought best to include in the theory of limits in the second Book every principle required for Plane and Solid Geometry.More info →
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.More info →
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.More info →
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.More info →
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.More info →
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.More info →
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.