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The questions distributed through the text follow very easily from the propositions to which they are attached, and we think that teachers are likely to find in them all that is needed for an average pupil reading the subject for the first time.

The Theorems and Examples at the end of each Book contain questions of a slightly more difficult type : they have been very carefully classified and arranged, and brought into close connection with typical examples worked out either partially or in full; and it is hoped that this section of the book, on which much thought has been expended, will do something towards removing that extreme want of freedom in solving deductions that is so commonly found even among students who have a good knowledge of the text of Euclid.

In the course of our work we have made ourselves acquainted with most modern English books on Euclidean Geometry: among these we have already expressed our special indebtedness to the text-book recently published by the Association for the Improvement of Geometrical Teaching; and we must also mention the Edition of Euclid's Elements prepared by Dr. J. S. Mackay, whose historical notes and frequent references to original authorities have been of the utmost service to us.

Our treatment of Maxima and Minima on page 239 is based upon suggestions derived from a discussion of the subject which took place at the annual meeting of the Geometrical Association in January 1887.

Of the Riders and Deductions some are original; but the greater part have been drawn from that large store of floating material which has furnished Examination Papers for the last 30 years, and must necessarily form the basis of any elementary collection. Proofs which have been

found in two or more books without acknowledgement have been regarded as common property.

As regards figures, in accordance with a usage not uncommon in recent editions of Euclid, we have made a distinction between given lines and lines of construction.

Throughout the book we have italicised those deductions on which we desired to lay special stress as being in them selves important geometrical results: this arrangement we think will be useful to teachers who have little time to devote to riders, or who wish to sketch out a suitable course for revision.

We have in conclusion to tender our thanks to many of our friends for the valuable criticism and advice which we received from them as the book was passing through the press, and especially to the Rev. H. C. Watson, of Clifton College, who added to these services much kind assistance in the revision of proof-sheets.

H. S. HALL,
F. H. STEVENS.

July, 1888.

PREFACE TO THE SECOND EDITION

In the Second Edition the text of Books I-VI. has been revised ; and at the request of many teachers we hare added the first twenty-one Propositions of Book XI. together with a collection of Theorems and Examples illustrating the elements of Solid Geometry.

September, 1889.

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