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PREFACE.

In offering to students and teachers a new edition of the Elements of Euclid, it will be proper to give some account of the plan on which it has been arranged, and of the advantages which it hopes to present.

Geometry may be considered to form the real foundation of mathematical instruction. It is true that some acquaintance with Arithmetic and Algebra usually precedes the study of Geometry; but in the former subjects a beginner spends much of his time in gaining a practical facility in the application of rules to examples, while in the latter subject he is wholly occupied in exercising his reasoning faculties.

In England the text-book of Geometry consists of the Elements of Euclid; for nearly every official programme of instruction or examination explicitly includes some portion of this work. Numerous attempts have been made to find an appropriate substitute for the Elements of Euclid; but such attempts, fortunately, have hitherto been made in vain. The advantages attending a common standard of reference in such an important subject, can hardly be overestimated; and it is extremely improbable, if Euclid were once abandoned, that any agreement would exist as to the author who should replace him. It cannot be denied that defects and difficulties occur in the Elements of Euclid, and that these become more obvious as we examine the work more closely; but probably during such examination the conviction will grow deeper that these defects and difficulties are due in a great measure to the nature of the subject itself, and to the place which it occupies in a course of education; and it may be readily believed that an equally minute criticism of any other work on Geometry would reveal more and graver blemishes.

Of all the editions of Euclid that of Robert Simson has been the most extensively used in England, and the present edition substantially reproduces Simson's; but his translation has been carefully compared with the original, and some alterations have been made, which it is hoped will be found to be improvements. These alterations, however, are of no great importance; most of them have been introduced with the view of rendering the language more uniform, by constantly using the same words when the same meaning is to be expressed.

As the Elements of Euclid are usually placed in the hands of young students, it is important to exhibit the work in such a form as will assist them in overcoming the difficulties which they experience on their first introduction to processes of continuous argument. No method appears to be so useful as that of breaking up the demonstrations into their constituent parts; this was strongly recommended by Professor De Morgan more than thirty years ago as a suitable exercise for students, and the plan has been adopted more or less closely in some modern editions. An excellent example of this method of exhibiting the Elements of Euclid will be found in an edition in quarto, published at the Hague, in the French language, in 1762. Two persons are named in the title-page as concerned in the work,

Koenig and Blassiere. This edition has served as the model for that which is now offered to the student: some slight modifications have necessarily been made, owing to the difference in the size of the pages.

It will be perceived then, that in the present edition each distinct assertion in the argument begins a new line; and at the ends of the lines are placed the necessary references to the preceding principles on which the assertions depend. Moreover, the longer propositions are distributed into subordinate parts, which are distinguished by breaks at the beginning of the lines.

This edition contains all the propositions which are usually read in the Universities. After the text will be found a selection of notes; these are intended to indicate and explain the principal difficulties which have been noticed in the Elements of Euclid, and to supply the most important inferences which can be drawn from the propositions. The notes relate to Geometry exclusively; they do not introduce developments involving Arithmetic and Algebra, because these latter subjects are always studied in special works, and because Geometry alone presents sufficient matter to occupy the attention of early students. After some hesitation on the point, all remarks relating to Logic have also been excluded. Although the study of Logic appears to be reviving in this country, and may eventually obtain a more assured position than it now holds in a course of liberal education, yet at present few persons take up Logic before Geometry; and it seems therefore premature to devote space to a subject which will be altogether unsuitable to the majority of those who use a work like the present.

After the notes will be found an Appendix, consisting of propositions supplemental to those in the Elements of Euclid; it is hoped that a judicious choice has been made from the abundant materials which exist for such an Appendix. The propositions selected are worthy of notice on various grounds; some for their simplicity, some for their value as geometrical facts, and some as being problems which may naturally suggest themselves, but of which the solutions are not very obvious.

The work finishes with a collection of exercises. Geometrical deductions afford a most valuable discipline for a student of mathematics, especially in the earlier period of his course; the numerous departments of analysis which subsequently demand his attention will leave him but little time then for pure Geometry. It seems however that the habits of mind which the study of pure Geometry tends to form, furnish an advantageous corrective for some of the evils resulting from an exclusive devotion to Analysis, and it is therefore desirable to engage the attention of beginners with geometrical exercises.

Many persons whose duties have rendered them familiar with the examination of large numbers of students in elementary mathematics have noticed with regret the frequent failures in geometrical deductions. Several collections of exercises already exist, but the general complaint is that they are too difficult. Those in the present volume may be divided into two parts; the first part contains 440 exercises, which it is hoped will not be found beyond the

power of early students; the second part consists of the remainder, which may be reserved for practice at a later stage. These exercises have been principally selected from College and University examination papers, and have been tested by long experience with pupils. It will be seen that they are distributed into sections according to the propositions in the Elements of Euclid on which they chiefly depend. As far as possible they are arranged in order of difficulty, but it must sometimes happen, as is the case

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