0.5

1 cm

GEOMETRICAL CONSTRUCTIONS—DRAWING OF ELEMENTARY FORMS.

21

(Fig. 64.)

Note.—The security effected by selecting a large radius

for the intersections, referred to in last problem, is still

more important in the case before us, as a line may be

truly bisected even though the bisecting line be obliquely

drawn; while the perpendicularity is destroyed.

In this way, a perpendicular may be drawn through

the middle of a board or a sheet of paper, by one operation,

when a base-line is laid down.

In this way, also, radial lines may be drawn to a circular

arc, without reference to the centre, as the line C D plainly

passes towards the centre of the arc A F B.

Problem IV.—To draw a perpendicular to a straight

line, from a given point in that line.

1st Method.—With any radius, from the given point

A, in the given line B C, cut (f-w. 63.)

the line at B and C; with a

greater radius describe arcs

from B and C, cutting each

other at D, and draw the per-

pendicular D A.

2 d Method. — From any

centre F, above B C, describe a circle passing through

the given point A, and cutting the given line at D; draw

D F, and produce it to cut the

circle at E; and draw A E the

perpendicular.

Note.—This second method

is useful when the point A is

at or near one end; and in prac-

tice, it is expedient in the first

place to strike out a preliminary

arc, of any convenient radius, from the point A, as any

point in that arc may be chosen for the centre F, with the

certainty that the arc from this centre will pass through

A, without the delay of adjusting the point of the compass

to it. This expedient is of general use where an arc is to

be passed through a given point, and particularly if the

point of the pencil be round or misshapen, and therefore

uncertain.

3 d Method.—From A describe an arc E C, and from E,

with the same radius, the arc A C,

cutting the other at C; through C draw

a line E C D, and set off C D equal to

C E, and through D draw the perpen-

dicular A D as required.

Note. — This method, like the pre-

vious one, is useful when the point A

is at one end.

4th Method.—From the given point A, set off a distance

A E equal to three parts, by any

scale ; and on the centres A and E,

with radii of four and five parts re-

spectively, describe arcs intersecting

at C. Draw A C for the perpendicular

required.

Note.—This method is most useful

on very large scales, where straight edges are inapplicable,

as in laying down perpendiculars on the ground. Beams,

(Fig. 65.)

/ /'

/ /

/

//

(Fig. 66.)

columns, and the like, may be set vertically and per-

pendicularly by the same method. The numbers 3, 4, 5,

are, it is to be observed, taken to measure respectively—-

the base, the perpendicular, and the slant side of the

triangle A E C. Any multiples of these numbers may

be used with equal propriety, as 6, 8, 10, or 9, 12, 15,

whether feet, yards, or any other measure of length.

Problem V.—To draw a perpendicular to a straight

line, from a point without it.

1st Method.—From the point A, with a sufficient radius,

(Fig. 67.) cut the given line at F and G;

and from these points describe

arcs cutting at E. Draw the

perpendicular A E.

Note.—If there be no room

below the line, the intersection

may be taken above; that is,

between the line and the given

point. This mode is not, however, likely to be as exact

in practice as the one given.

2d Method.—From any two points B, C, at some dis-

tance apart, in the given

line, and with radii B A,

C A, respectively, describe

arcs cutting at A D. Draw

the perpendicular A D.

Note.—This method is

most useful where the given

point is opposite the end

of the line, or nearly so. The second centre, C, may be

selected at or near G.

Problem VI.—To draw a straight line parallel to a

given line, at a given distance apart.

From the centres A, B, in the given line, with the

given distance as radius, de-

scribe arcs C D; and draw the

parallel line C D touching the

arcs.

A B Note.—The method of draw-

ing tangents will be afterwards shown ; meantime, in all

ordinary cases, the line C D may be drawn by simply

applying a straight edge by the eye.

Problem VII.—To draw a parallel through a given

point.

1st Method.—With a radius equal to the distance of

the given point C from the

given line A B, describe the

arc D from B, taken con-

siderably distant from C;

A B draw the parallel through

C to touch the arc D.

a/ (Ffy-71.)_2d Method.—From A, the

/ ! given point, describe the arc

/ / F D, cutting the given line

i / at F; from F, with the same

“e——----p-- radius, describe the arc E A;

and set off F D equal to E A. Draw the parallel through

the points A, D.

(Fig. 69.)

(Fig. 70.)

21

(Fig. 64.)

Note.—The security effected by selecting a large radius

for the intersections, referred to in last problem, is still

more important in the case before us, as a line may be

truly bisected even though the bisecting line be obliquely

drawn; while the perpendicularity is destroyed.

In this way, a perpendicular may be drawn through

the middle of a board or a sheet of paper, by one operation,

when a base-line is laid down.

In this way, also, radial lines may be drawn to a circular

arc, without reference to the centre, as the line C D plainly

passes towards the centre of the arc A F B.

Problem IV.—To draw a perpendicular to a straight

line, from a given point in that line.

1st Method.—With any radius, from the given point

A, in the given line B C, cut (f-w. 63.)

the line at B and C; with a

greater radius describe arcs

from B and C, cutting each

other at D, and draw the per-

pendicular D A.

2 d Method. — From any

centre F, above B C, describe a circle passing through

the given point A, and cutting the given line at D; draw

D F, and produce it to cut the

circle at E; and draw A E the

perpendicular.

Note.—This second method

is useful when the point A is

at or near one end; and in prac-

tice, it is expedient in the first

place to strike out a preliminary

arc, of any convenient radius, from the point A, as any

point in that arc may be chosen for the centre F, with the

certainty that the arc from this centre will pass through

A, without the delay of adjusting the point of the compass

to it. This expedient is of general use where an arc is to

be passed through a given point, and particularly if the

point of the pencil be round or misshapen, and therefore

uncertain.

3 d Method.—From A describe an arc E C, and from E,

with the same radius, the arc A C,

cutting the other at C; through C draw

a line E C D, and set off C D equal to

C E, and through D draw the perpen-

dicular A D as required.

Note. — This method, like the pre-

vious one, is useful when the point A

is at one end.

4th Method.—From the given point A, set off a distance

A E equal to three parts, by any

scale ; and on the centres A and E,

with radii of four and five parts re-

spectively, describe arcs intersecting

at C. Draw A C for the perpendicular

required.

Note.—This method is most useful

on very large scales, where straight edges are inapplicable,

as in laying down perpendiculars on the ground. Beams,

(Fig. 65.)

/ /'

/ /

/

//

(Fig. 66.)

columns, and the like, may be set vertically and per-

pendicularly by the same method. The numbers 3, 4, 5,

are, it is to be observed, taken to measure respectively—-

the base, the perpendicular, and the slant side of the

triangle A E C. Any multiples of these numbers may

be used with equal propriety, as 6, 8, 10, or 9, 12, 15,

whether feet, yards, or any other measure of length.

Problem V.—To draw a perpendicular to a straight

line, from a point without it.

1st Method.—From the point A, with a sufficient radius,

(Fig. 67.) cut the given line at F and G;

and from these points describe

arcs cutting at E. Draw the

perpendicular A E.

Note.—If there be no room

below the line, the intersection

may be taken above; that is,

between the line and the given

point. This mode is not, however, likely to be as exact

in practice as the one given.

2d Method.—From any two points B, C, at some dis-

tance apart, in the given

line, and with radii B A,

C A, respectively, describe

arcs cutting at A D. Draw

the perpendicular A D.

Note.—This method is

most useful where the given

point is opposite the end

of the line, or nearly so. The second centre, C, may be

selected at or near G.

Problem VI.—To draw a straight line parallel to a

given line, at a given distance apart.

From the centres A, B, in the given line, with the

given distance as radius, de-

scribe arcs C D; and draw the

parallel line C D touching the

arcs.

A B Note.—The method of draw-

ing tangents will be afterwards shown ; meantime, in all

ordinary cases, the line C D may be drawn by simply

applying a straight edge by the eye.

Problem VII.—To draw a parallel through a given

point.

1st Method.—With a radius equal to the distance of

the given point C from the

given line A B, describe the

arc D from B, taken con-

siderably distant from C;

A B draw the parallel through

C to touch the arc D.

a/ (Ffy-71.)_2d Method.—From A, the

/ ! given point, describe the arc

/ / F D, cutting the given line

i / at F; from F, with the same

“e——----p-- radius, describe the arc E A;

and set off F D equal to E A. Draw the parallel through

the points A, D.

(Fig. 69.)

(Fig. 70.)