THE

ELEMENTS OF GEOMETRY;

OR,

THE FIRST SIX BOOKS, WITH THE ELEVENTH AND

TWELFTH,

OF

EUCLID.

FROM THE TEXT OF ROBERT SIMSON, M.D.

Emeritus Professor of Mathematics in the University of Glasgow.

WITH CORRECTIONS, ANNOTATIONS, AND EXERCISES,

BY PROFESSOR WALLACE, M.A.

Of the same University,

And sometime Collegiate Tutor of the University of London.

1

PREFACE.

THIS edition of Euclid's Elements of Geometry differs from the com. mon editions with Dr. Simson's name attached to them, in several important particulars. First, the style has been simplified and modernised as much as possible, by removing much of its technicality; and in places where this was necessarily retained, numerous explanations have been added, especially in the Definitions. Secondly, many new Demonstrations of the Propositions have been given, in addition to those of Euclid, in order to bring the subject within the comprehension of different capacities. In not a few, while the spirit of the demonstration has been preserved, the original verbosity of the Greek, which was often retained by Dr. Simson in his translation, has been greatly curtailed; and in others, it has been altogether replaced by a new and better demonstration. Thirdly, to almost the whole of the propositions there have been added new Corollaries, Exercises, and Annotations of various kinds, tending to render the additions a species of short and running commentary on the immortal work of Euclid.

Explanations of all difficult terms in the science of Geometry have been given wherever they occur; and a style of punctuation in the different sentences of a proposition, and especially in the demonstration, has been adopted, which, it is believed, will be found of the greatest advantage to the student. This advantage will be discovered by comparison with other editions, and its utility will be seen from the following consideration. In reading the Demonstrations, the student is obliged to pause at every step, in order to make himself sure of the reasoning before he advances to the next step. This assurance may at once be supplied by his recollection of a previous proposition, a definition, a postulate, or an axiom; but, if not, the reference is generally given, not at the margin, as in the common editions, but in the body of the text, just at the place where it is wanted. In either case, time is required to bring the reference vividly to the mind, and to assure it of the accuracy of the reasoning. In general, the time of a full period is not too much to enable the student to bring the memory to the aid of his judgment; and where the memory fails, to refer at once to the places in the book actually cited in the course of the argument. Every new period, therefore, has always the advantage of indicating a new step in the argument, and keeping the student awake to its progress.

EUCLID'S

ELEMENTS OF GEOMETRY.

BOOK L

DEFINITIONS.

I.

A POINT is that which hath no parts, or which hath no magnitude. "A point is" more clearly defined to be "the beginning of magnitude;" as, for instance, the beginning of a line.

A line is length without breadth.

II.

A line is extension in any one direction, uniform or variable; as, the unbroken contour or outline of any given surface.

III.

The extremities of a line are points.

By the extremities of a line, are here meant, the beginning and the end of the line.

IV.

A straight line is that which lies evenly between its extreme points. "A straight line is " more clearly defined to be "that in which, if any two points be taken, the part intercepted between them is the shortest that can be drawn." This shows that every straight line in the Elements is considered to be of indefinite length, unless otherwise expressed.

V.

A superficies is that which hath only length and breadth.

A superficies or surface, is extension in any two directions, uniform or variable; as, the continuous boundary of any given solid. Def. I. Book XI.

VI.

The extremities of a superficies are lines.

By the extremities of a superficies or surface, are here meant, the boundaries of edges of the surface.

VII.

A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

A plane superficies, or simply, a plane, is a surface in which a straight line can any where be drawn. This shows that every plane in the Elements is considered to be of indefinite extent, unless otherwise expressed.

B