FROM PREFACE TO THE FIRST EDITION.

At present, it is not rash to assume that, out of the vast number of our youths who are trained in Euclid, but a low percentage ever become Geometers in any true sense of the word. In the circumstances of modern times, this must be admitted to be a fair subject of serious reproach; whatever else may be accomplished by Euclidian training as it is. But, in the view of the present writer, there is no need to dispense with Euclid on this account; for his text may be thoroughly learnt for its own peculiar benefits, not only as well but better, when it is accompanied by all that original work which will make a pupil into more or less of a true Geometer. Among the most distinctive features of this work are the number and the position of the explanations, questions, and exercises. But the nature of all these, which will not be so readily obvious, is intended to be the most useful of its characteristics.

The earlier ones are not only designed to make clear the meanings of the definitions, &c., but also to take advantage of these to gradually develop the power of constructing simple geometrical arguments; so that, on reaching the propositions, the pupil will be sure to enter upon them with intelligent interest and sufficient geometrical power. He will then never be found among those who go far through the propositions with no notion of distinguishing their various parts, except by differences of type or from direct labelling; and who

know they are at the end of a proposition only by meeting with the well-known sign-post of the three capital letters.

In the further exercises which accompany the propositions, every proposition, except some of the very easiest of the later ones, meets with analytical consideration before it is entered upon. By means of a series of questions and easy exercises, the pupil is led to study each part of the proposition in turn, and then himself to build the parts together. At the same time, this is not done directly, so as to produce the exact form of the coming proposition, but with diverse features and conditions, so as to afford variety and scope for originality and mental exercise.

The value of such exercises will perhaps be admitted by all. But, in giving so much original work, it may be objected that no account is taken of the time at the teacher's disposal; and that it can never be got through. In this objection a double misconception would be involved.

A large portion of the work is set so that it shall admit of explicit answering and ready correction. For pupils who are not very young, many portions of the exercises may be taken as simply oral practice in class. There are numerous other questions and exercises whose answers are determinate and brief; and the easy deductions which often occur may be added to them by simply requiring in each case the construction and the quoting of those propositions on which the proof will be founded, full proofs being afterwards taken orally with the blackboard. Proceeding in this way, it is evident that in regard to much of the practical work of the class the teacher may correct, with or without the aid of the printed "Key," very much as he would in the case of exercises in Arithmetic or Latin. It is recommended that the most important of such examples as the teacher finds it needful to discuss with the aid of the blackboard, should be set for writing out in full with the next exercise. To glance

over portions of a coming exercise also with the aid of the blackboard, will frequently be found of advantage with younger pupils. Pursuing the course here indicated, the expenditure of time required for the thorough study of these pages is probably far less than would be estimated.

means.

On the other hand, it is quite possible to under-estimate the time required to arrive at an equal result by the usual By the method described above, a natural and continuous training may be afforded to pupils taken in classes; which is more than can be said for systems which involve the use of the ordinary sets of deductions, commonly and properly enough styled mere collections." To succeed with these, teaching power of a high quality is indispensable; and it must be very liberally expended. Indeed, a high authority on this subject, the head-master of Clifton College, has asserted that boys cannot be got to work systematically upon "deductions."

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The text of Simson is given throughout with a clear arrangement and in conspicuous type, so as to be easily followed when specially revising it. A few changes have been made in words and phraseology, for purposes of emendation or greater clearness; but this has been done with great care. By the use of Italics it has been sought to emphasize those statements whose importance learners are most apt to underestimate. The Italics are thus used with their ordinary application; and it is simply incidental that they frequently assist in the logical division of the propositions. diagrams will be found good and clear, and are repeated wherever the text requires it.

The

For those who are favourable to replacing Euclid's proofs in some cases, a series of alternative ones is given in an Appendix, consisting of those which are likely to find most. favour with teachers. They are expressed with the aid of symbols, in order to assist, at the same time, such as may

desire to initiate their pupils early into the use of symbols. Those symbols and abbreviations only are used which would not be objected to in any of the ordinary examinations.

For some pupils there may be superabundance in the materials herein provided; but in no case can this be a serious fault; for to supply the omissions of a scanty treatise is by no means so easy as to pass by portions not needed in a fuller

one.

The book is intended to be suitable for preceding the study of any good edition of Euclid.

J. W.

EUCLID-BOOK I.

INTRODUCTION.

THE name Geometry is derived from two Greek wordsge, the earth, and metron, a measure. The science of Geometry appears to have taken its rise from the art of measuring land, most probably in Egypt, and long before the time of Jesus Christ. The ancient Greeks cultivated the subject, and made great progress in increasing human knowledge of it. About 300 B.C., a famous Greek named Euclid, who lived at Alexandria in Egypt, wrote a treatise on the "Elements of Geometry." This treatise has been one of the most wonderful books ever written. Its interesting history cannot be told here; but we require to mention that the present little book is founded upon a portion of Euclid's work, and is intended to enable the beginner in Geometry to understand and learn thoroughly what that portion contains. What is here called learning will not be considered complete until it has become easy to put the knowledge acquired to practical use.

The portion of this book which is due, in the main, to Euclid, will be found printed in larger type.

DEFINITIONS.

1. WHAT ARE DEFINITIONS?

"By a definition we mean a precise statement of the qualities which are just sufficient to mark out a class, and thus to tell exactly what things belong to a class and what do not.”. -(JEVONS.)

Notice the word 'just' in this explanation, and remember that a thing

B