0.5

1 cm

20

ENGINEER AND MACHINIST’S DRAWING-BOOK.

quadrangular, pentagonal, hexagonal, &c., according as

the base has three, four, five, six sides, &c.

A sphere or globe (Fig. 53), is a solid bounded by a

uniformly curved surface, every point of which is equally

distant from the centre, a point within the sphere. A

fine passing through the centre, and terminating both

ways at the surface, is a diameter.

(Fig. 54.) (Fig. 55.) (Fig. 56.)

Cylinder. Cone, Tetrahedron.

A cylinder is a round solid of uniform thickness, of

which the ends are equal and parallel circles (Fig. 54).

A cone is a round solid, with a circle for its base, and

tapering uniformly to a point at the top (Fig. 55).

When a solid is cut through transversely by a plane

parallel to the base, the part cut off is a segment, and the

part remaining is a frustrum of the solid. The latter

term is usually limited to pyramids and cones.

(Fig. 57.) (Fig. 58.) (Fig. 59.) (Fig. 60.)

Hexahedron. Octahedron. Dodecahedron. Icosahedron.

A regidar solid is bounded by similar and regular

plane figures. There are five regular solids (Figs. 56, 57,

58, 59, 60), namely:—

The tetrahedron, bounded by four equilateral triangles.

The hexahedron, or cube, bounded by six squares.

The octahedron, bounded by eight equilateral triangles.

The dodecahedron, bounded by twelve pentagons.

The icosahedron, bounded by twenty equilateral tri-

angles.

Regular solids may be circumscribed by spheres, and

spheres may be inscribed in regular solids.

A problem is something to be done.

Geometrical construction signifies the series of opera-

tions performed with mathematical instruments, in deter-

mining the points and lines involved in the solution of a

geometrical problem.

Lines of construction are the lines used in solving

problems.

A given point; a given line. A point or line of fixed

position or length.

Describe an arc signifies that an arc of a circle is to be

drawn by the dividers, or in pencil or ink.

To set off or lay off a distance or length, is to mark on

the drawing a given distance by the dividers, the com-

passes, or the pricker ; and, with the compasses, it is done

most conveniently by describing a short arc.

Lines cut each other, when they intersect or cross.

To set off or lay out an angle is to draw a line making

a given angle with another.

To produce a line is to lengthen a given line as far as

may be necessary.

The following problems are in the first place worked

out independently of the assistance derivable from the

drawing board and squares, to embrace geometrical opera-

tions on the largest scale, in which the use of these

instruments would be inadmissible. Their use and appli-

cation will be subsequently illustrated.

SECTION I.

Problems on Straight Lines.

Problem I.—To draiv a straight line through given

points.

Let A and B be two given points, represented by the

intersection of two lines, or pricked into the surface.

Surround the points by small circles, when advisable for

(Fig. 61.)

assisting to define their locality, as thus ©; place the

straight edge at, or so near the points, that the point of

the pen or pencil may pass through them, and draw the

line firmly and steadily.

Problem II.—To bisect, or divide into two equal parts,

a given straight line, or circular arc.

With a radius greater than half the given line A B,

describe arcs of circles Rom the ex-

tremities A and B, cutting each other

at the points C and D; and draw C D.

This line will cut the given line or

arc into two equal parts, at the point

E or F.

Note.—The longer the radius taken

for describing the circular arcs, the

smaller is the chance of error in the

construction, as it places the points C, D farther apart,

and facilitates the accurate drawing of the line C D in the

required position.

It is not necessary in practice to draw the complete

arcs C D. An experienced eye can readily anticipate

the points of intersection of the arcs, within narrow

limits. Nor is it necessary to do more than apply a

straight edge to these points, and to tick the point E or F.

Problem III.—To draiv a perpendicular to a straight

line.

Operate as in the foregoing problem. The line C D is

perpendicular to A B.

(Fig. 62.)

\ i p ;

X

*' j D '*•

ENGINEER AND MACHINIST’S DRAWING-BOOK.

quadrangular, pentagonal, hexagonal, &c., according as

the base has three, four, five, six sides, &c.

A sphere or globe (Fig. 53), is a solid bounded by a

uniformly curved surface, every point of which is equally

distant from the centre, a point within the sphere. A

fine passing through the centre, and terminating both

ways at the surface, is a diameter.

(Fig. 54.) (Fig. 55.) (Fig. 56.)

Cylinder. Cone, Tetrahedron.

A cylinder is a round solid of uniform thickness, of

which the ends are equal and parallel circles (Fig. 54).

A cone is a round solid, with a circle for its base, and

tapering uniformly to a point at the top (Fig. 55).

When a solid is cut through transversely by a plane

parallel to the base, the part cut off is a segment, and the

part remaining is a frustrum of the solid. The latter

term is usually limited to pyramids and cones.

(Fig. 57.) (Fig. 58.) (Fig. 59.) (Fig. 60.)

Hexahedron. Octahedron. Dodecahedron. Icosahedron.

A regidar solid is bounded by similar and regular

plane figures. There are five regular solids (Figs. 56, 57,

58, 59, 60), namely:—

The tetrahedron, bounded by four equilateral triangles.

The hexahedron, or cube, bounded by six squares.

The octahedron, bounded by eight equilateral triangles.

The dodecahedron, bounded by twelve pentagons.

The icosahedron, bounded by twenty equilateral tri-

angles.

Regular solids may be circumscribed by spheres, and

spheres may be inscribed in regular solids.

A problem is something to be done.

Geometrical construction signifies the series of opera-

tions performed with mathematical instruments, in deter-

mining the points and lines involved in the solution of a

geometrical problem.

Lines of construction are the lines used in solving

problems.

A given point; a given line. A point or line of fixed

position or length.

Describe an arc signifies that an arc of a circle is to be

drawn by the dividers, or in pencil or ink.

To set off or lay off a distance or length, is to mark on

the drawing a given distance by the dividers, the com-

passes, or the pricker ; and, with the compasses, it is done

most conveniently by describing a short arc.

Lines cut each other, when they intersect or cross.

To set off or lay out an angle is to draw a line making

a given angle with another.

To produce a line is to lengthen a given line as far as

may be necessary.

The following problems are in the first place worked

out independently of the assistance derivable from the

drawing board and squares, to embrace geometrical opera-

tions on the largest scale, in which the use of these

instruments would be inadmissible. Their use and appli-

cation will be subsequently illustrated.

SECTION I.

Problems on Straight Lines.

Problem I.—To draiv a straight line through given

points.

Let A and B be two given points, represented by the

intersection of two lines, or pricked into the surface.

Surround the points by small circles, when advisable for

(Fig. 61.)

assisting to define their locality, as thus ©; place the

straight edge at, or so near the points, that the point of

the pen or pencil may pass through them, and draw the

line firmly and steadily.

Problem II.—To bisect, or divide into two equal parts,

a given straight line, or circular arc.

With a radius greater than half the given line A B,

describe arcs of circles Rom the ex-

tremities A and B, cutting each other

at the points C and D; and draw C D.

This line will cut the given line or

arc into two equal parts, at the point

E or F.

Note.—The longer the radius taken

for describing the circular arcs, the

smaller is the chance of error in the

construction, as it places the points C, D farther apart,

and facilitates the accurate drawing of the line C D in the

required position.

It is not necessary in practice to draw the complete

arcs C D. An experienced eye can readily anticipate

the points of intersection of the arcs, within narrow

limits. Nor is it necessary to do more than apply a

straight edge to these points, and to tick the point E or F.

Problem III.—To draiv a perpendicular to a straight

line.

Operate as in the foregoing problem. The line C D is

perpendicular to A B.

(Fig. 62.)

\ i p ;

X

*' j D '*•