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Numerous instances, too, occur in the Elements, where the demonstrations might be improved. For instance, in the demonstration of the Twenty-third Proposition of the First Book, it is proved that the given rectilineal angle is equal to the rectilineal angle required to be made; but this is precisely the reverse of what should have been demonstrated; because the angle which is made should be proved equal to the angle which is given, and the demonstration is equally simple in either way. In one or two cases of this kind, the necessary modification of the demonstration has been attended to, whilst in others the demonstrations have been allowed to remain, in deference to the general usage of geometers; but they may easily be demonstrated by the student, in strict conformity with the language of the enunciation of the Proposition.

The demonstration of the First Proposition of the Third Book has been long known to be defective, and it is here rendered valid by the addition of a line or two to the original text. Besides, wherever a looseness of the phraseology of Simson exists, the Editor has endeavoured to supply the defect by a slight alteration of the language.

To the Elements have been added an Appendix, containing a selection of Propositions for exercise on the first six Books; besides a number of critical questions on Euclid, which, it is hoped, will be found useful in pro

moting accuracy and perspicuity, and in improving the active investigating powers of intellect.

With a view to greater clearness, and to arrest the attention of the student at every step of the reasoning, marginal references have been added where they appeared to be wanting; and, as utility rather than novelty has been the Editor's principal object throughout the work, it is presumed that the changes which have been made in this edition will be found to be real improvements.

WILLIAM RUTHERFORD.

ROYAL MILITARY ACADEMY,

Woolwich, March, 1841.

THE

ELEMENTS OF EUCLID.

BOOK I.

DEFINITIONS.

I. A point is that which hath no parts, or which hath no magnitude.

II.
A line is length without breadth.

III.
The extremities of a line are points.

IV. A straight line is that which lies evenly between its extreme points.

V. A superficies is that which hath only length and breadth.

VI.
The extremities of a superficies are lines.

VII.
A plane superficies is that in which any two points

being taken, the straight line between them lies wholly in that superficies.

VIII. A plane angle is the inclination of two lines to one

B

“another in a plane, which meet together, but are “not in the same direction."

IX. A plane rectilineal angle is the inclination of two

straight lines to one another, which meet together, but are not in the same straight line.

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N. B. When several angles are at one point B, any one of them is expressed by three letters, of which • the letter that is at the vertex of the angle, that is,

at the point in which the straight lines that contain • the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other upon the

other line: thus the angle which is contained by the straight lines AB, CB, is named the angle ABC, or

CBA; that which is contained by AB, BD, is named the angle ABD, or DBA; and that which is con'tained by BD, CB, is called the angle DBC, or CBD; • but, if there be only one angle at a point, it may be expressed by a le :ter placed at that point; as the angle at E,

X.
When a straight line standing on ano-

ther straight line makes the adjacent
angles equal to one another, each of
the angles is called a right angle;
and the straight line which stands on the other is
called a perpendicular to it.

XI.
An obtuse angle is that which is

greater than a right angle.

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