We hope that the additional matter introduced into the book will provide sufficient exercise for pupils whose study of Euclid is preliminary to a mathematical education.

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.

To Volumes containing Books I-III., and Books I.-IV. an Appendix is added, giving an elementary account of the properties of Pole and Polar, and Radical Axis.

Throughout the book we have italicised those deductions on which we desired to lay special stress as being in themselves 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.



CLIFTON, December, 1886.

In the present edition the text has received further revision, and the notes have been for the most part re-written, with a view to greater clearness and simplicity.

References to the Definitions being frequent in the text of Book I., the convenience of a standard order has been pointed out to us by many elementary teachers. We have therefore thought it advisable to re-number the Definitions in accordance with Simson's edition. This has involved the insertion of certain definitions hitherto omitted as of slight importance : such insertions have now been printed in subordinate type.

A few typographical improvements have been introduced : notably the italicising of Particular Enunciations. Some changes in pagination have also been effected for the purpose of presenting the whole of a proposition at one view, or of bringing notes and exercises into closer connection with the text to which they refer. Further, the symbols “:: for therefore,

» for is equal to are now introduced from the 35th Proposition of Book I.

Groups of Test Questions for Revision have been inserted at various stages. These may be useful to beginners, and suggestive to teachers in framing examination papers, which so often consist of mere monotonous lists of propositions and examples.

One important change has been made. The algebraical treatment of the subject-matter of Book V. has been entirely separated from the stricter general treatment, so as to present in the simplest form such Definitions and Theorems of Proportion as are necessary before entering upon Book VI. This Introduction will be found immediately preceding Book VI. in a chapter called The Elementary Principles of Proportion.

H. S. H.
F. H. S.

and “

February, 1900.

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