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All rights reserved.

First printed March, 1889.

Reprinted January and June, 1890; April, 1891; 1892, 1893, 1894, 1895, 1896. 1897, 1898, 1899.



Hist. of sei.




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THIS volume is part of a complete edition of Euclid's Elements, as usually read at schools.

The text has been carefully revised, and special attention given to those points which experience has shown to present difficulties to beginners.

In the course of this revision the enunciations have been altered as little as possible; and very few departures have been made from Euclid's proofs; in each case changes have been adopted only where the old text has been generally found a cause of difficulty; and such changes are for the most part "in favour of well-recognised alternatives.

In Book I., for example, the ambiguity has been removed from the Enunciations of Propositions 18 and 19, and the fact that Propositions 8 and 26 establish the complete equality of the two triangles considered has been strongly urged: thus the redundant step has been removed from Proposition 34.

In Book II. Simson's arrangement of Proposition 13 has been abandoned for a well-known equivalent.

In Book III. Propositions 35 and 36 have been treated generally, and it has not been thought necessary to do more than call attention in a note to the special cases.

These are the chief deviations from the ordinary text as regards method and arrangement of proof: they are points familiar as difficulties to most teachers, and to name them indicates sufficiently, without further enumeration, the general principles which have guided our revision.

A few alternative proofs of difficult propositions are given for the convenience of those teachers who care to use them.

One purpose of the book is gradually to familiarise the student with the use of legitimate symbols and abbreviations; for a geometrical argument may thus be thrown into a form which is not only more readily seized by an advanced reader, but is useful as a guide to the way in which Euclid's propositions may be handled in written work. On the other hand, we think it very desirable to defer the introduction of symbols until the beginner has learnt that they can only be properly used in Pure Geometry as abbreviations for verbal argument: and we hope thus to prevent the slovenly and inaccurate habits which are very apt to arise from their employment before this principle is fully recognised.

Accordingly in Book I. we have used no contractions or symbols of any kind, though we have introduced verbal alterations into the text wherever it appeared that conciseness or clearness would be gained.

In Book II. abbreviated forms of constantly recurring words are used, and the phrases therefore and is equal to are replaced by the usual symbols.

In the Third and following Books, and in additional matter throughout the whole, we have employed all such signs and abbreviations as we believe to add to the clearness of the reasoning, care being taken that the symbols chosen are compatible with a rigorous geometrical method, and are recognised by the majority of teachers.

If this arrangement should be thought fanciful or wanting in uniformity, we may plead that it is the outcome of long experience in the use of various text-books. For some years, for example, we were accustomed to teach from a symbolical text, but in consequence of the frequent misconceptions and inaccuracies which too great brevity was found to generate among beginners, we were compelled to return to one of the older and unabbreviated editions. The gain to our younger boys was immediate and unmistakeable; but the change has not

been unattended with disadvantage to more advanced students, who on reaching the Third or Fourth Book may not only be safely trusted with a carefully chosen system of abbreviations, but are certainly retarded by the monotonous and lengthy formalities of the old text.

It must be understood that our use of symbols, and the removal of unnecessary verbiage and repetition, by no means implies a desire to secure brevity at all hazards. On the contrary, nothing appears to us more mischievous than an abridgement which is attained by omitting steps, or condensing two or more steps into one. Such uses spring from the pressure of examinations; but an examination is not, or ought not to be, a mere race; and while we wish to indicate generally in the later books how a geometrical argument may be abbreviated for the purposes of written work, we have not attempted to reduce the propositions to the barest skeleton which a lenient Examiner may be supposed to accept. Indeed it does not follow that the form most suitable for the page of a text-book is also best adapted to examination purposes; for the object to be attained in each case is entirely different. The text-book should present the argument in the clearest possible manner to the mind of a reader to whom it is new: the written proposition need only convey to the Examiner the assurance that the proposition has been thoroughly grasped and remembered by the pupil.

From first to last we have kept in mind the undoubted fact that a very small proportion of those who study Elementary Geometry, and study it with profit, are destined to become mathematicians in any special sense; and that, to a large majority of students, Euclid is intended to serve not so much as a first lesson in mathematical reasoning, as the first, and sometimes the only, model of formal argument presented in an elementary education.

This consideration has determined not only the full treatment of the earlier Books, but the retention of the formal, if somewhat cumbrous, methods of Euclid in many places where proofs of greater brevity and mathematical elegance are available.

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