0.5

1 cm

GEOMETRICAL CONSTRUCTIONS—DRAWING OF ELEMENTARY FORMS.

19

For example, vertical lines are perpendicular to horizontal

lines; and the edge of the blade of a T square is perpen-

dicular to the edge of the stock.

(Fig. 31.) (Fig. 32.)

Right Angle. Acute Augle.

(Fig. S3.)

An acute angle {Fig. 32), is less than a right angle.

An obtuse angle {Fig. 33) is greater than a right angle.

A plane triangle, or simply a triangle, is a flat surface

bounded by three straight lines or

sides. When the sides are equal

{Fig. 34), the triangle is equila-

teral; when one of the angles is a

right angle, the triangle is right-

angled.

An angle is designated by a letter A

at the apex, as, A {Fig. 34), or by ^ Equaateral Triangle.

the letters used to denote its sides, as B A C.

For convenience, any one side of a triangle may be

distinguished as the base, the remaining two sides being

properly the sides.

A quadrilateral figure is a surface bounded by four

straight lines. When the opposite sides are parallel, it is

called a parallelogram {Figs. 35, 36, 37); if it have right

(Fig. 31.)

c

(Fig. 3G.)

Rectangle.

Square.

(Fg. 37.)

Parallelogram.

angles, it is a rectangle {Fig. 35); if all the sides also be

equal, it is a square {Fig. 36).

A diagonal is a straight line joining two opposing

angles of a figure.

Plane figures of more than four sides are called poly-

gons. When the sides are equal, they are regular poly-

gons ; of which {Figs. 38-41) are examples, annexed to

which are their respective designations.

Pentagon, five sides.

A circle {Fig. 42) is a plane figure bounded by one

continuous curve line, bed, called the circumference, and

$*9- 42•) (Fig. 43.) (Fig. 44.)

is such that the circumference is at all points equally

distant from the centre, a, within it. Any straight line,

b a, or b c {Fig. 43), drawn from the centre to the cir-

cumference, is termed a radius ; and a line a b {Fig. 44),

drawn through the centre and terminated by the circum-

ference, is a diameter. The magnitude of a circle is

usually designated in terms of its diameter; the radius is

half the diameter.

An arc of a circle is any part of the circumference, as

ced {Fig. 45).

A chord of a circle is a straight line joining the extre-

mities of an arc, as c cl {Fig. 45).

A sector of a circle is the space or area enclosed by two

radii, as abc {Fig. 43). When the radii are at right

angles, the space is called also a quadrant, as it is one-

fourth of a circle.

A segment of a circle is the space cut off by a chord, as

the space ced {Fig. 45).

(Fg. 45.) (Fg. 40.) (Fig. 47.)

O)

A semicircle, or half-circle, is one of the two equal

parts into which a circle is divided by a diameter, as in

{Fig. 44).

A tangent to a circle or other curve line, is a straight

line which touches it, meeting it only at one point, as a b

{Fig. 46), touching at c.

Circles are concentric when they have the same centre,

as in {Fig. 47); if they have different centres, they are ec-

centric, as in (Fig. 48); where also they are shown to touch

at their circumferences, or to be tangential to each other.

(Fig. 48.) (Fig. 49.) (Fig. 50.)

A triangular or other figure with a greater number of

sides, is inscribed in a circle, or circumscribed by it,

when all its angles touch the circumference, as in {Fig. 49).

Inversely, a circle is inscribed in a straight-sided figure,

when it touches all the sides {Fig. 50).

All regular polygons may be inscribed in circles, and

circles may be inscribed in polygons; hence the facility

with which polygons may be constructed.

Of solids. As surfaces are bounded by lines, and derive

their names from the character and disposition of these

lines—so solids are bounded by surfaces, and also derive

their titles therefrom.

A prism is a solid of which the ends are equal, similar,

and parallel straight-sided figures, and of which the other

sides are parallelograms. When all the sides are squares,

it is called a cube {Fig. 51).

A pyramid is a solid having a straight-sided base,

and triangular sides terminating in one point or vertex

{Fig. 52).

Prisms and pyramids are distinguished as triangular,

19

For example, vertical lines are perpendicular to horizontal

lines; and the edge of the blade of a T square is perpen-

dicular to the edge of the stock.

(Fig. 31.) (Fig. 32.)

Right Angle. Acute Augle.

(Fig. S3.)

An acute angle {Fig. 32), is less than a right angle.

An obtuse angle {Fig. 33) is greater than a right angle.

A plane triangle, or simply a triangle, is a flat surface

bounded by three straight lines or

sides. When the sides are equal

{Fig. 34), the triangle is equila-

teral; when one of the angles is a

right angle, the triangle is right-

angled.

An angle is designated by a letter A

at the apex, as, A {Fig. 34), or by ^ Equaateral Triangle.

the letters used to denote its sides, as B A C.

For convenience, any one side of a triangle may be

distinguished as the base, the remaining two sides being

properly the sides.

A quadrilateral figure is a surface bounded by four

straight lines. When the opposite sides are parallel, it is

called a parallelogram {Figs. 35, 36, 37); if it have right

(Fig. 31.)

c

(Fig. 3G.)

Rectangle.

Square.

(Fg. 37.)

Parallelogram.

angles, it is a rectangle {Fig. 35); if all the sides also be

equal, it is a square {Fig. 36).

A diagonal is a straight line joining two opposing

angles of a figure.

Plane figures of more than four sides are called poly-

gons. When the sides are equal, they are regular poly-

gons ; of which {Figs. 38-41) are examples, annexed to

which are their respective designations.

Pentagon, five sides.

A circle {Fig. 42) is a plane figure bounded by one

continuous curve line, bed, called the circumference, and

$*9- 42•) (Fig. 43.) (Fig. 44.)

is such that the circumference is at all points equally

distant from the centre, a, within it. Any straight line,

b a, or b c {Fig. 43), drawn from the centre to the cir-

cumference, is termed a radius ; and a line a b {Fig. 44),

drawn through the centre and terminated by the circum-

ference, is a diameter. The magnitude of a circle is

usually designated in terms of its diameter; the radius is

half the diameter.

An arc of a circle is any part of the circumference, as

ced {Fig. 45).

A chord of a circle is a straight line joining the extre-

mities of an arc, as c cl {Fig. 45).

A sector of a circle is the space or area enclosed by two

radii, as abc {Fig. 43). When the radii are at right

angles, the space is called also a quadrant, as it is one-

fourth of a circle.

A segment of a circle is the space cut off by a chord, as

the space ced {Fig. 45).

(Fg. 45.) (Fg. 40.) (Fig. 47.)

O)

A semicircle, or half-circle, is one of the two equal

parts into which a circle is divided by a diameter, as in

{Fig. 44).

A tangent to a circle or other curve line, is a straight

line which touches it, meeting it only at one point, as a b

{Fig. 46), touching at c.

Circles are concentric when they have the same centre,

as in {Fig. 47); if they have different centres, they are ec-

centric, as in (Fig. 48); where also they are shown to touch

at their circumferences, or to be tangential to each other.

(Fig. 48.) (Fig. 49.) (Fig. 50.)

A triangular or other figure with a greater number of

sides, is inscribed in a circle, or circumscribed by it,

when all its angles touch the circumference, as in {Fig. 49).

Inversely, a circle is inscribed in a straight-sided figure,

when it touches all the sides {Fig. 50).

All regular polygons may be inscribed in circles, and

circles may be inscribed in polygons; hence the facility

with which polygons may be constructed.

Of solids. As surfaces are bounded by lines, and derive

their names from the character and disposition of these

lines—so solids are bounded by surfaces, and also derive

their titles therefrom.

A prism is a solid of which the ends are equal, similar,

and parallel straight-sided figures, and of which the other

sides are parallelograms. When all the sides are squares,

it is called a cube {Fig. 51).

A pyramid is a solid having a straight-sided base,

and triangular sides terminating in one point or vertex

{Fig. 52).

Prisms and pyramids are distinguished as triangular,