have also attempted to render many of the proofs, as, for instance, those of Propositions 2, 13, and 35 in Book I., and those of 7, 8, and 13 in Book II., less confusing to the learner.

In Propositions 4, 5, 6, 7, and 8 of the Second Book I have ventured to make an important change in Euclid's mode of exposition, by omitting the diagonals from the diagrams and the gnomons from the text.

In the Third Book I have deviated with even greater boldness from the precise line of Euclid's method. Thus I have given new proofs of the Propositions relating to the Contact of Circles : I have used Superposition to prove Propositions 26 to 29, so as to make each of those theorems independent of the others; and I have directed the attention of the learner to the Intersection of Loci, and to the conception of an Angle as a magnitude capable of unlimited increase.

In the Fourth Book I have made no change of importance.

My treatment of the Fifth Book was suggested by the method first proposed, explained, and defended by Professor De Morgan in his Treatise on the Connexion of Number and Magnitude. The

hod is simple and rigorous, presenting Euclid's

reasoning in a clear and concise form, by means of a system of notation, to which, I think, no valid objection can be taken. I have altered the order of the Propositions in this Book, so as to give prominence to those which are of chief importance.

The only changes in the Sixth Book to which I desire to call the reader's special attention, are the applications of Superposition in the proofs of Propositions 4 and 19.

The diagrams in Book XI. form an important feature of this Edition. For them I am indebted to the kindness of Mr. Hugh Godfray, of St. John's College, Cambridge.

The Exercises have been selected with considerable care, chiefly from the University and College Examination Papers. They are intended to be progressive and easy, so that a learner may be induced from the first to work out something for himself.

A complete series of the Euclid Papers set in the Cambridge Mathematical Tripos from 1848 to 1872 will be found on pp. 198-210 and 342-349.

I have made but little allusion to Projections, because that part of the subject is fully explained by Mr. Richardson in his work on Conic Sections treated Geometrically, forming a part of RIVINGTON'S MATHEMATICAL SERIES,

During the two years in which I have been engaged on this work, I have received from Teachers of Geometry in all parts of the country so much encouragement to proceed, and so much assistance at each step of my progress, that I feel justified in asserting that no text-book on Elementary Geometry is likely to meet with general support in England, if it involve any wide departure from the Euclidean model.

It only remains for me to offer my thanks to the friends who have improved this work by their advice, and to assure each reader of the book that any suggestion for its further improvement will be thankfully received by me.




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