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FIRST AND SECOND BOOKS OF EUCLID.

BY THE SAME AUTHOR..
Seventeenth Edition, Revised and Enlarged. Price 38., cloth.

ENGLISH GRAMMAR.
INOLUDING THE PRINCIPLES OF GRAMMATICAL ANALYSIS.

Price 28. 6d., cloth. THE ANALYSIS OF SENTENCES APPLIED TO

LATIN.

Price 38. 6d., cloth.
ANALYTICAL LATIN EXERCISES.

Troelfth Thousand. Price 9d., cloth.
FIRST STEPS IN ENGLISH GRAMMAR.

LONDON : BELL & DALDY, YORK STREET, COVENT GARDEN.

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LONDON :
BELL & DALDY, YORK ST., COVENT GARDEN.

1872.

183. q. 14.

LONDON : PARDON AND SON, PRINTERS,

PATERNOSTER ROW.

PREFACE.

ATTENTION has been strongly directed of late to the mode in which geometry is usually taught in this country, and several vigorous attempts have been made to supersede Euclid as a text-book. Euclid's method has been objected to, as being needlessly rigid, and consequently needlessly difficult for beginners, some of the most serious difficulties unfortunately being encountered in the earliest propositions. It has been supposed that by extending the field embraced in the postulates, allowing the use of hypothetical constructions, and adding two or three axioms, the truth of which no one will dispute, many, if not all of these difficulties may be avoided. It may be conceded that Euclid's method is difficult; but those who object to this should remember that the object of the study of geometry is not merely to obtain a knowledge of the relations and properties of geometrical figures, but to train the logical and perceptive faculties by a course of the most rigid and exact reasoning. The elimination of all the difficulties would deprive the study of nearly all its utility. It takes a beginner a long time to master the first six or seven propositions in Euclid, but it is quite a mistake to suppose that his real progress in geometry, as regards its higher and more important uses, is to be measured by the number of propositions thus learnt. The first six propositions in

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