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CONSISTING OF

THE FIRST FOUR, AND THE SIXTH, BOOKS

OF

EUCLID

CHIEFLY FROM

THE TEXT OF DR. ROBERT SIMSON;

WITH THE

PRINCIPAL THEOREMS IN PROPORTION,

AND

A COURSE OF PRACTICAL GEOMETRY ON THE GROUND.

ALSO,

FOUR TRACTS RELATING TO

CIRCLES, PLANES, AND SOLIDS;

WITH ONE ON SPHERICAL GEOMETRY.

For the Use of the Royal Military College.

BY

JOHN NARRIEN, F.R.S. & R.A.S.

PROFESSOR OF MATHEMATICS, ETC., IN THE INSTITUTION.

BIBL.

LONDON:

PRINTED FOR

LONGMAN, BROWN, GREEN, AND LONGMANS.

PATERNOSTER-ROW.

1842.

LONDON: Printed by A. SPOTTISWOODE,

New-Street-Square.

ADVERTISEMENT.

The following Treatise on the Elements of Geometry is the second of a series which is to constitute a General Course of Mathematics for the use of the gentlemen cadets and the officers in the senior department of this Institution. The course, when completed, will comprehend the subjects whose titles are subjoined: I. Arithmetic and Algebra. II. Geometry.

III. Plane and Spherical Trigonometry, with Mensuration. IV. Analytical Geometry and the Differential and Integral Calculus, with the Properties of the Conic Sections. V. Practical Astronomy and Geodesy. VI. The Principles of Mechanics ; and VII. Physical Astronomy.

ROYAL MILITARY COLLEGE,

1842.

PREFACE.

1

GEOMETRY is the science in which are exhibited and proved various properties of finite magnitude under its three primary characteristics : — lines, planes, and solids, without regard to its physical nature.

It is well known from the testimony of ancient writers that many geometrical properties of magnitude had been discovered and demonstrated by learned men among the Greeks at a very early period; and it is scarcely probable that, almost immediately upon some important properties being discovered, there should not have been attempted an arrangement of the propositions according to their dependence upon one another and upon certain axioms, that is, upon such

propositions as are considered self-evident.

But if any works of that nature existed previously to the age of Euclid, they must have been entirely superseded by the “ Elements of Geometry” which bear the name of that mathematician, who is said to have been a disciple of the school of Plato, and to have cultivated the science at Alexandria between the years 323 and 284 B. C. This work is not only the most ancient treatise of geometry in existence, but the greater part of it has, down to the present moment, maintained the character of being the most proper introduction to the mathematical sciences. It is divided into thirteen books, of which the six first constitute what is called plane geometry: of these the four first exhibit the construction and principal properties of figures bounded by straight lines, together with some of the most elementary properties of circles: the fifth book treats of proportion among magnitudes, and in the sixth book the general theory of proportion is applied to geometrical plane figures. The seventh, eighth, and ninth books contain the properties of numbers; their greatest common measures and least common multiples; continued and mean proportionals. The tenth book relates to incommensurable quantities. The eleventh

era.

and twelfth books treat of the intersections of planes, the properties of parallelepipeds, prisms, and pyramids, and of cylinders and cones: they contain also demonstrations of the ratios that circles and spheres, respectively, bear to one another. Lastly, the thirteenth book relates to equilateral and equiangular figures inscribed in circles, and to what are called the five regular bodies.

A fourteenth and a fifteenth book, both of which treat of the inscriptions of the five regular bodies in one another and in spheres, are frequently added to the Elements of Euclid; but they were written by Hypsicles of Alexandria. The same subjects have since been continued in a work which is sometimes called the sixteenth book of the Elements.

It will be for ever impossible now to decide whether the arrangements and demonstrations of the propositions in the thirteen books are entirely the work of Euclid ; whether he merely arranged the propositions and rendered consistent the demonstrations which had been given before his time; or, finally, whether part of the work may not be ascribed to Theon of Alexandria, who lived in the fourth century of our

All these opinions have been advanced; but Dr. Robert Simson, who was distinguished for his profound acquaintance with the works of the Greek geometers, after a comparison of the existing copies with one another and with the works of Archimedes and Apollonius, has arrived at the conclusion (Preface to his edition of the Elements) that “Theon, or whoever was the editor of the present Greek text, by adding some things, suppressing others, and mixing his own with Euclid's demonstrations, has changed many things to the worse.” Dr. Simson's first edition was published in 1750; and in that, as well as in all that have subsequently appeared under his name, the inaccuracies which he supposes the ancient editors to have put in place of the genuine demonstrations of Euclid have been corrected : it is even probable that, at the same time, there were removed some errors which

may have existed in the original text.

A work in which the purity of the ancient science has been so carefully maintained deserves to be, and in nearly all that regards plane geometry, excepting the fifth book it has been, ever since its publication, very generally used as a text-book for the study of this branch of science. In fact, the editions of Euclid's plane geometry which have since been published by other mathematicians differ from that of Simson almost wholly in the language being abbreviated.

But the extent to which mathematical science has been carried since the time of Euclid, and the numerous applica

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