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EUCLID'S

ELEMENTS OF GEOMETRY,

THE FIRST SIX BOOKS, AND THE PORTIONS OF THE
ELEVENTH AND TWELFTH BOOKS READ

AT CAMBRIDGE:

CHIEFLY FROM THE TEXT OF DR. SIMSON, WITH
EXPLANATORY NOTES AND QUESTIONS:

TOGETHER WITH A SELECTION OF GEOMETRICAL EXERCISES
FROM THE SENATE-HOUSE AND COLLEGE
EXAMINATION PAPERS:

DESIGNED FOR THE USE OF THE HIGHER FORMS IN PUBLIC SCHOOLS
AND STUDENTS IN THE UNIVERSITIES.

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LONGMAN, GREEN, LONGMAN, ROBERTS, & GREEN.

M.DCCC.LXV.

1965

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INTERNATIONAL EXHIBITION,

1862.

A Medal has been awarded to R. Potts,
"For the excellence of his Works on Geometry."

Jury Awards Class xxix, p. 313.

99

PREFACE.

THE favourable notices of the First Edition of this work which appeared on its publication, and the reception it has met with in the Principal Schools and Universities of this Country and the Colonies, claim the Editor's grateful acknowledgements, and afford the satisfaction, that this his first though imperfect essay in Editorship has not been made in vain.

One of the Reviewers was pleased to observe in reference to the Selection of the Geometrical Exercises, "As a series of judicious exercises, we do not think there exists one at all comparable to it in our language, viewed either in reference to the student or teacher". This opinion may receive some confirmation from the fact, that a Translation in German by Hans H. von Aller, of the Geometrical Exercises, was published at Hanover in 1860, with a preface by Dr. Wittstein.

In this Edition, the text of Dr. Simson is retained as the authorised English Version of Euclid's Elements, with the exception of a few verbal emendations. Those portions of the Eleventh and Twelfth Books which are now read at Cambridge are retained, and the rest omitted.

The explanatory notes have been considerably augmented and improved; and Algebraical proofs of the propositions of the Fifth Book have been added to the notes on that Book.

A Selection of questions on the Elements, chiefly taken from the College Examination Papers, has been added to each of the first Six Books after the notes on each Book.

Considerable additions have been made to the Geometrical Exercises, and those which involve the consideration of Loci, Maxima and Minima, and Tangencies have been arranged in three separate classes.

As the design of the Editor has been to exhibit the teaching at Cambridge of Elementary Geometry, as it might be learned from the Exercises proposed in the College and University Examinations, he considers it advisable in this New Edition to replace many of the Geometrical Exercises by others which have appeared in recent examinations since the publication of the First Edition.

The "hints, &c., for the solution of the Problems, &c." have been included in the Volume itself.

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It was intended to add some account of the Extensions of the Euclidean Geometry, including the Porisms, Transversals, Poles and Polars, &c. &c., but as the volume has grown to larger dimensions than was anticipated, the Editor is compelled to reserve that part of his work for a future publication. The sketch of the History of Geometry prefixed to the First Edition is also reserved; but the Editor trusts that the following brief notices and remarks, may not be without interest to the student.

Euclid's Elements of Geometry are invested with an interest which belongs to no other Elementary work on Pure Science. Having survived the age in which it was composed for the long period of upwards of twenty centuries, it has maintained its superiority in Schools and Universities, as the best introduction to the Science of Pure Geometry. In ancient times this work sustained its reputation in the Schools of Athens and Alexandria, the chief places of resort of philosophers and their hearers. In the age succeeding that of Euclid, the Science was greatly advanced by Archimedes, Apollonius, Theodosius, and others; and during the long stationary period which followed, down to the times of Proclus in the fifth century of the Christian era, no superior work appeared to take the place of Euclid's Elements. The appropriate opinion of Professor De Morgan may be here quoted: "There never has been, and till we see it, we never shall believe there can be, a system of Geometry worthy of the name, which has any material departures (we do not speak of corrections, or extensions or developements) from the plan laid down by Euclid. If there be one worthy of consideration, it is the commencing with a strict theory of proportion. But it may very well be doubted whether any complete treatment of the Fifth Book of Euclid could be made intelligible to students of our day, before they have had some familiarity with demonstration applied to particular species of magnitude. We say of our day, because it is impossible to foresee what the advance of education may do. It is perfectly conceivable that the rapid advance of demonstrative Arithmetic as a study preliminary to that of Geometry, may ultimately render the change desirable".

The rise of the Mahommedan power in the Seventh Century, and the rapid and desolating effects which followed, hastened the extinction of Grecian Science. In the conquest of Egypt, the great library of Alexandria was committed to the flames by the ignorance and fanaticism of the Arabian conquerors; and the

learned men there congregated for the cultivation of Science and Philosophy, either fell by the sword, or escaped by flight, carrying with them some remains of the Sciences.

It is remarkable that in little more than a century after this infatuated deed of destruction, the Arabians became most zealous cultivators of the Science and Philosophy of the Greeks. The few Manuscripts of the Mathematical and Philosophical writings of the Greeks which had escaped the general ruin, were diligently sought and translated into Arabic, and commentaries were written to elucidate and explain them.

About the middle of the Twelfth Century, in the reign of Henry I., the Elements of Euclid were introduced into England through a Latin translation from the Arabic. Manuscript copies of this most ancient translation of Euclid's Elements are preserved at Oxford in the Library of Trinity College, and in the Bodleian Library. It was long after this period, the fact became known in Western Europe, that Euclid's Elements were originally written in Greek.

The revival of ancient literature in Europe about the middle of the Fifteenth Century, contributed to extend the knowledge of the Mathematical writings of the Greeks; and the discovery of the art of printing, was the commencement of a new era in literature and science. The writings of the Greeks were no longer confined to the few who had the means of purchasing or procuring manuscript copies; and both the Greek text of Euclid's Elements and Latin translations of it were printed in different countries.

The Latin translation of the Elements made from the Arabic by Campanus of Novara, was printed at Venice in 1482, and was the earliest Edition printed in that language. The original Greek was first printed at Basle in 1533. Translations also of the Elements soon after appeared in the modern languages of Western Europe. The first English translation of Euclid's Elements was made by Henry Billingsley, a citizen of London, and was published in London in the year 1570. A learned preface was prefixed to the translation by a Fellow of Trinity College, one of those who were first elected at the foundation of the College by Henry VIII. It does not appear that this English Version of Euclid's Elements was adopted at either of the English Universities. Latin being at that time the language of the learned, translations in that language were preferred, as text-books in Elementary Geometry, as in all other subjects of Academical study.

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