PREFACE

TO THE SECOND EDITION.

THIS Volume contains

1o, Enunciations, numbered according to Euclid, of all the Propositions in Books i-iv and vi of his Elements of Geometry :

2o, Proofs of these Propositions in which the essentials of Euclid's methods are followed:

3o, an Abridgement of Book v, including so much only as is necessary to render valid the proofs of Book vi:

4o, Addenda at the end of each Book, arising out of its principles, in which are given

(a) All the most obvious Corollaries to the Propositions; (6) Some immediate Developments of the Propositions; (c) The proofs of many useful Additional Theorems; (d) Numerous Theorems as Exercises to be proved; accompanied by Hints towards the proofs of the more difficult :

5o, General Addenda arranged in Sections; wherein will be found most of the fundamental Propositions of Maxima and Minima, Concurrency and Collinearity, Centres of Similitude, Coaxal Circles, The Tangencies, Inversion, Harmonic Section, and Poles and Polars.

The whole Work is divided into two Parts.

Part I Plane Geometry without Proportion-contains Books i-iv, with their Addenda.

Part II Proportion, and Modern Geometry contains Books v and vi, with their Addenda; and the eight Sections entitled General Addenda.

Each Part concludes with a collection of Problems for solution.

In considering what modification of Euclid's proofs might be admissible, the question at once arose-Why is it that, while in all other Sciences, text-books seldom outlive a generation, Euclid's Elements still hold their place as the basis of Geometry; and moreover that, in spite of the weighty arguments which have been urged against them, there can be no doubt but that a strong preponderance of feeling exists in favour of their retention in that position?

To the present Editor (after much reading about, and discussion of the question) it seems that there are two substantial reasons, of expediency and convenience, out of which the feeling arises.

1o, an established order of geometric proof is expedient for examination purposes;

2o, a recognised numbering of fundamental results is convenient for reference.

As co-operative reasons may be added-the fact that there is no consensus of opinion among experts that any other scheme yet proposed is superior to Euclid's; and the sentiment of repugnance at the thought of sweeping away an institution rendered venerable by the usage of more than 2000 years.

From these considerations it becomes apparent, on the one hand, that what is essential to be retained in Euclid is his order, numbering, and general mode of proof; and, on the other hand, that what is non-essential, and of small (or no) importance, is the accidental details of his proofs—whether, for example, i. 20 is proved by bisecting an angle, or producing a side.

solely for the

It may strengthen this position to state the fact that there does not exist a modern edition which gives Euclid pure and simple. The modifications of proof have been made sake of greater brevity, clearness, and simplicity. strictly in accordance with Euclid's order and methods. making these changes the Editor has not been guided by à priori

They are all

In

considerations of what might, could, would, or should be thought improvements; but by what he himself has (in many years' experience) actually found to be clearer, and to present fewer obstacles to a large number of learners of very varied age and mental calibre.

In the first edition, proofs of four Propositions in Book i, one in Book ii, and four in Book vi, were omitted; because these Propositions are neither necessary links in the chain of proof, nor of intrinsic geometrical value. Those omitted in Books i and ii are now inserted in an Appendix, to meet the requirements of examinations.

Definitions, Axioms, and Postulates, are introduced as they are needed; and certain Axioms and Postulates, tacitly assumed by Euclid, are inserted. This plan seems preferable to that of loading the beginner's mind with a string of words, many of which will not be needed till he is far advanced in the subject; and some not at all. The Index at the end gives the means of finding any one when it is wanted.

The Abridgement of Book v is given in the notation, and according to the methods set forth by the late Professor De Morgan in his Connexion of Number and Magnitude.

The present custom of omitting Book v, though quietly assuming such of its results as are needed in Book vi, is singularly illogical; and is indefensible on any ground, excepting that this Book has been found too difficult for the average learner. Nor does it mend the matter, but the reverse, to give— as some modern writers do the arithmetical treatment of Proportion, which applies only to the exceptional case of commensurable magnitudes, as a substitute for a rigorous treatment applying to magnitudes of all kinds. The Editor has therefore taken special care to avoid that confusion of commensurable and incommensurable magnitudes, which arises from introducing purely

arithmetical processes in the treatment of the latter-a confusion most assuredly not to be found in Euclid. What is here given is at once strictly accurate, and quite within the capacity of any one who has capacity enough to understand Book vi.

Dominated as all teachers are by examination programmes, it may not be irrelevant to call special attention to the extraordinary anomaly prevalent in such programmes-that Book vi is usually named without the parts of Book v needed in its proofs. Thus the learner finds that while an iron logic is insisted on in the first four books-so that the omission of no link in the chain of proof (how simple soever) is permitted-ever after, complex principles are assumed without a hint of the incongruity. If it is necessary to prove that two sides of a triangle are greater than the third, surely it is necessary to prove ex æquali, componendo, and alternando. Every teacher admits the absurdity of the prevailing system; but the truth is that what does not 'pay' in examinations is not, and is not likely to be, taught.

Our appeal in this matter is not to teachers, but to Examining Boards.

But the main point which the Editor has aimed at is to give all demonstrations in their most compact form consistent with proof. His experience, as a teacher for twenty years, has shown him that NOTHING is so great a hindrance to the learner, especially when commencing The Elements, as Euclid's prolixity. And while tc the beginner this prolixity is a stumbling-block, to the more proficient scholar it is a nuisance. A feeble learner is lost in Euclid's maze of words: while, in an examination-hall, an able candidate discards it as quite incompatible with the amount he has to get through in a limited time. In fact the raison d'être of this book is to give, in the clear, compact, orderly form that suits the necessities of modern examinations, some such rearrangement of Euclid, as most teachers probably find themselves compelled to