ADVERTISEMENT

THERE are already so many and such excellent editions of Euclid that to offer another, though only of Books I. and II., may be considered both superfluous and presuming. Yet, it would be somewhat rash, on the other hand, to say that we have arrived at perfection in the matter, and that no further attempts at improvement in the publication of Euclid-for beginners, at least-need be made.

Every teacher of Euclid knows, and a long experience has confirmed the knowledge in myself, that the better the mere typical arrangement of the text, the more attractive the study of Euclid in itself is, and the quicker, and more complete, is the progress which the beginner makes. If, further, the text can be expressed more distinctly, and be brought more fully within the boy's comprehension, so that he can readily perceive what he really has to do in 'learning a Proposition,' as it is called, and be led to put a definite value upon the statement of the abstract truths he finds demonstrated, then every possible aid is afforded him in his often difficult and uninteresting work. These two points are aimed at in this edition of Euclid, as strictly a book for beginners. If the way can be smoothed thus far, the rest of the road is fully open.

As regards the first requisite, the typical arrangement of the text, it is hoped that the distinct expression, in red ink, of the particular enunciation with reference to the figure employed, and the special statement of the point or points to

be proved in the Proposition, will be found to contribute ma

terially to the advantages spoken of. provement is sought in the further inclusive, of red ink, to denote the 'construction' of the several figures.

A corresponding imuse, to Prop. XXVI.

lines employed in the

And with reference to the second requisite, the language of the text itself, it is believed that by a very simple deviation from the usual phraseology of the demonstration, without any sacrifice of geometrical or logical truth, a great help will be afforded to the scholar, both in learning and remembering the several Propositions.

The alteration is chiefly this :-Through the whole of the First Book there runs, as a thread, the frequent comparison of two triangles. This comparison is usually made in a manner which, to a boy, seems unnecessarily cumbrous and puzzling. Of the three parts to be taken in each of the two triangles, as equal to each other, each to each, two are first taken, and their respective equalities stated; then the third in each is taken, and their equality asserted. By this a kind of break is made in the argument, which acts as a hindrance to a learner, unimportant as it may appear to be. Now, surely, to take the three parts in each triangle, and to compare them, each with each, once for all, is just as correct, geometrically and logically, and it is certainly by far the simpler plan. A reference to the proof of Proposition V. will explain my meaning, and show, I think, the advantage claimed.

This point allowed, the application of the principle through the whole Book tends greatly to simplify it, for the purposes stated. Again, another common defect in the text is that, after the necessary comparison of two triangles has been made, while often only one of the consequences deducible is required in the argument of the Proposition, all the consequences are stated. Surely this also is cumbrous and puzzling.

For these, and other reasons which will speak for them

selves in the several Propositions, I presume to issue this edition of Books I. and II. My original purpose would have been answered by the publication of Book I. only, but the addition of Book II., which has also its especial features, will make the whole available for some of the examinations which have to be undergone, especially since all symbols, or abbreviations, which the several examining bodies disallow, are carefully excluded. I do not suppose that I have found out the 'royal road.' I shall be more than satisfied if I do but indicate another step in that direction.

I have not thought it necessary to introduce additional Problems, or Riders. In doing so I should be departing somewhat from the object I have chiefly in view, of preparing a book especially suitable to beginners, who, when they are sufficiently advanced for such work, will find excellent and ample material for it elsewhere. I have, however, appended some very simple Exercises, generally variations authorised by the Propositions under which they are placed. These, if they are not thought superfluous, will contribute to a more thorough understanding of the Propositions themselves, and help to train the scholar for the higher efforts of the kind which he may afterwards have to make.

Any favourable testimony that the use of this book may warrant will be appreciated, and suggestions and even hostile criticism shall have a hearty welcome, and the fullest attention.

CHEDDINGTON RECTORY,

TRING.

F. B. HARVEY.