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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.

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LONDON:
W. KENT & CO. 51 & 52, PATERNOSTER ROW.

1855.

183. c. f.

PRE FACE.

This edition of Euclid's Elements of Geometry differs from the common 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, especially in the First Six Books, new Corollaries, Exercises, and Anno. tations 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; ana 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.

The necessity of some method of this kind, to keep the mind alive to the march of the argument, was felt by the editor of one of our best recent editions of Euclid, Mr. Potts, of Cambridge. The plan adopted by him was to print every sentence or half sentence that contained a new step in the reasoning, in a separate line; thus giving the book the appearance of a book of limping poetry, and unnecessarily extending the letter-press over a much greater space; whereas, the great object of a reasoner should be to bring the words of the argument as closely together as possible, in order that the eye may help the mind. It was on this ground that algebraic signs were lately introduced into some editions of Euclid, and that a variety of other signs were employed to denote words and names of figures of frequent repetition. It was found, however, that by this plan nothing was gained, for the memory was now partially burdened by a new sort of language, a species of shorthand, more troublesome to write and read than the ordinary language of the science. Hence 6 Symbolical Euclids” have gradually and wisely given place to those editions in which the ordinary style of language is used, as much improved and abridged as the nature of the reasoning and the consistency of the argument will allow.

The above-mentioned simplifications, amendments, and additions, and other improvements which will be found on perusal, have been introduced into this edition, because it is chiefly intended for the use of the People. Euclid is now placed within the reach of all who are desirous of making themselves acquainted with this masterpiece of reasoning, with the foundation of all the sciences, with the basis of all the arts of design and machinery, and with the origin of all the processes relating to the measurement and calculation of surfaces and solids, required both in the arts of life and in the arts of production. Our space will not permit us to say a few words concerning the ancient author of this work, and its able translator, Dr. Robert Simson, of the University of Glasgow, but our readers may judge of the accuracy of an edition hitherto read in all our schools and universities, which, though it has passed through the press many times, declares on its title-page that Dr. Robert Simson belonged to the University of Oxford.

EUCLID'S

ELEMENTS

GEOMETRY.

BOOK 1.

DEFINITIONS.

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, fcr

instance, the beginning of a line.

II.

A line is length without breadth.
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 Elemezts is considered to be of indefinite length, unless otherwise expressed.

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A superficies is that which hath only length and breadth. A superficies or surface, is extension in any two directions, uniform or variable ; 23, 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 or

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 consider.d to be of indefinite extent, unless otherwise expressed.

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