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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.

and "


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.

February, 1900.

F. H. S.

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The First Proposition of Book XII. will be found at p. 364,
worked out as an Example on Proposition 20 of Book VI.,
of which it is a development. Prop. 3 of Book XII, is
briefly treated as a Corollary to Prop. 1.

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