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THE principal alteration which has been made in this edition of the Elements of Euclid after the text of Dr. Simson, is the division of the demonstrations by the manner of printing, so as to make the succession of their steps more easily apprehended. It is hoped that hereby much of the advantage will be secured which has been found to result from the use of a Symbolical Notation in this subject, without the compression of reasoning and departure from Geometrical strictness which in some editions on that principle have been considered objectionable.
CAMBRIDGE, January 1845.
ELEMENTS OF EUCLID.
A POINT is that which hath no parts, or which hath no magnitude.
A line is length without breadth.
The extremities of a line are points.
A straight line is that which lies evenly between its extreme points.
A superficies is that which hath only length and breadth.
The extremities of a superficies are lines.
A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.
"A plane angle is the inclination of two lines to one "another in a plane, which meet together, but are not "in the same direction."
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.
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, DB, 'is named the angle ABD, or DBA; and that which is 'contained by DB, CB, is called the angle DBC, or CBD; but, if there be only one angle at a point, it may be expressed by a letter placed at that point: as the angle ' at E.'
When a straight line standing on another 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.
An obtuse angle is that which is greater than a right angle.
An acute angle is that which is less than a right angle.