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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 contained 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 letter placed at that point; as the angle at E.”

[Observe that BAC is a greater angle than DEF,

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because the inclination of BA to AC is greater than that of DE to EF. This may be seen by placing EF upon AC, so that the point E shall coincide with A. Observe that DE will fall nearer to AC or EF than BA does.]

B 2

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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.

[That there may be two such angles equal, may be exhibited by the following figures :

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DC meets the right line ACB, and makes two angles DCB and DCA.

Suppose DC to be moveable round the fixed point C in the plane of the paper, and from being coincident with CB, gradually to move round till it coincides with AC. The angle DCB, which is at first less than DCA, becomes after some time greater than it, and the increase of the one and the BOOK I. decrease of the other being perfectly gradual, there is some place at which the angle DCB has become not less than DCA, nor greater than it; or, in other words, equal 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.


“ A term or boundary is the extremity of any thing."


A figure is that which is inclosed by one or more boundaries.

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Fig. 3. [Fig. 1. a figure contained by one boundary.

Fig. 2. and fig. 3. figures contained by more than one boundary.]


A circle is a plane figure contained by one line, which is

called the circumference, and is such that all straight lines


drawn from a certain point within the figure to the circumference, are equal to cne another.

And this point is called the centre of the circle.

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[In the above figures A is the centre of the circle; B is not the centre.]

XVII. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.


A semicircle is the figure con

tained by a diameter and the
part of the circumference
cut off by the diameter.

[A is called the centre of the
semicircle as well as of the


XIX. “ A segment of a circle is the figure contained by a straight line, and the circumference it cuts off.”


Rectilineal figures are those which are contained by straight


Trilateral figures, or triangles, by three straight lines.

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Of three-sided figures, an equilateral triangle is that which
has three equal sides.

An isosceles triangle is that which has two sides equal.

[Observe that an equilateral triangle must be isosceles,

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