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THE SCHOOL EUCLIN.

APPROVED EDUCATIONAL WORKS

BY THE SAME AUTHOR.

The GEOMETRICAL COPY-BOOK: Being an Attempt, by

means of a New Arrangement of the Figures and Demonstrations, to simplify and fix in the Memory the Propositions of Euclid, and to impart the habit of writing them out logically and rapidly, for Examinations, in the abbreviated and symbolical form adopted in the Universities and Public Schools. For the use of Candidates for the Military and Civil Service, London University, College of Preceptors, Oxford and Cambridge Middle-Class and other Examinations. Two Quarto Copy-Books, price 6d. each.

The great extension which has recently been given to the system of written examinations makes it desirable that some uniform system of writing out the propositions of Euclid should be adopted in schools. The object of the present Series of Copy-Books is to embrace a uniform logical arrangement of the propositions with ease and rapidity in writing them out. They may be used with any edition of Euclid, being ruled and arranged to receive the Proposi

tions of any of the First Six Books. EUCLID arranged for Examinations and for Beginners. Forming a Text-book for the above.

[In the press. The ELEMENTS of BOOK-KEEPING, by Single and Double

Entry. 12mo. price 9d. SET of EIGHT ACCOUNT BOOKS, adapted to the above.

Oblong 4to. 6d. each.

Also, preparing for publication, CÆSAR'S COMMENTARIES on the GALLIC WAR.

With Notes, Critical and Explanatory, a Lexicon of all the Words in the Text, and an Introductory Delectus, in which every sentence in the Earlier Chapters of the First Book is analysed, and its structure exhibited and

explained for Beginners. 12mo. The COLLEGE EUCLID. Comprising the First Six and the

Eleventh and Twelfth Books; with Notes, Illustrations, and Geometrical Exercises.

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LONDON:
LONGMAN, GREEN, LONGMAN, ROBERTS, & GREEN.

1862.

1831. e. 5.

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PREFACE.

THERE
JHERE are few persons engaged in tuition who have not

experienced the difficulty of teaching Euclid to young pupils, more especially to those who have not acquired habits of close application, or who have no natural aptitude for mathematical studies. One reason for this arises obably from the nature of the subject itself, which requires a more sustained attention, and a greater concentration of thought and power of abstraction, than beginners can ordinarily be induced to apply to any subject at the age when the study of Euclid is commonly begun in schools. But another, and undoubtedly the chief cause of the difficulty, is the absence in the ordinary editions of the Elements of those aids to the learner which are so plentifully supplied in every other department of instruction.

Such assistance it is the special design of the present edition to afford, partly by the use of a peculiar type in those parts of a proposition which require to be distinguished from each other, and partly by a new arrangement of the figures and demonstrations, which it is hoped will be found to embody some important improvements on the ordinary method of presenting the subject to beginners. The chief features of this arrangement may be summed

up as follows: 1. Immediately following the enunciation are given, in each case, the references,' or elements, of the proposition — the definitions, axioms, and previous propositions — on which the successive steps of the reasoning depend. These must be thoroughly mastered by the pupil; and they should in all cases be required to be written out, or repeated aloud, before either

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