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BY

FRANCIS CUTHBERTSON, M.A.

LATE FELLOW OF CORPUS CHRISTI COLLEGE, CAMBRIDGE ;
HEAD MATHEMATICAI, MASTER OF THE CITY OF LONDON SCHOOL.

London:

MACMILLAN AND CO.

w

(All Rights reserved.]

Cambridge:

PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS.

PREFACE.

BEFORE entering into a detailed account of the plan of the present work one characteristic may perhaps deserve especial prominence, namely that, while all those parts of the Elements of Euclid which are required by the Universities have been established and at the same time Problems separated from Theorems, both classified according to the nature of the subject, and demonstrations replaced by others less cumbrous, I have been careful in all cases to retain Euclid's order as far as is necessary to allow of the proofs given being substituted for those of Euclid in Examinations. How important this is must be at once apparent to all who are conversant with Examinations either at Schools or Universities.

One of the most generally acknowledged defects in the Elements of Euclid is the treatment of parallels; for a theorem is assumed as self-evident which certainly requires proof as much as those which are made to depend upon it. To avoid this difficulty a Lemma has been introduced which, while it rests on an axiomatic basis, brings out the property

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