11353on PERSPECTIVE.
Demonstration.
If a Plane be conceiv’d to paſs thro’ the Eye,
perpendicular to the Geometrical Plane, and paral-
lel to the given Lines; it is evident, that the ſaid
Plane will cut the Horizontal Plane in the Line
O D, and the perſpective Plane in D F. It is,
moreover, manifeſt, that a Line drawn thro’ the
Eye, parallel to the given Line, is in the ſaid
Plane, and (with the Line O D) makes an An-
gle, equal to the Angle E C P, below the Hori-
zontal Plane, if the Lines be inclin’d towards
the perſpective Plane, and above it, if they in-
cline to the oppoſite ſide; whence this laſt Line
makes a right-angled Triangle with O D and
D F, whoſe Angle at the Point O, is equal to
the Angle C E P. But D G F is likewiſe a
right-angled Triangle, as having the Angle at the
Point G, equal to ECP; therefore theſe two
Triangles are ſimilar. And ſince the Side D G
is equal to the Side D O, the Triangles are alſo
equal: Therefore the Line D F, being common
to theſe two Triangles; the Point F, is the
Point wherein the Line, paſſing thro’ the Eye
parallel to the given Line, meets the Per-
ſpective Plane: And this Point is the acciden- tal one ſought.
perpendicular to the Geometrical Plane, and paral-
lel to the given Lines; it is evident, that the ſaid
Plane will cut the Horizontal Plane in the Line
O D, and the perſpective Plane in D F. It is,
moreover, manifeſt, that a Line drawn thro’ the
Eye, parallel to the given Line, is in the ſaid
Plane, and (with the Line O D) makes an An-
gle, equal to the Angle E C P, below the Hori-
zontal Plane, if the Lines be inclin’d towards
the perſpective Plane, and above it, if they in-
cline to the oppoſite ſide; whence this laſt Line
makes a right-angled Triangle with O D and
D F, whoſe Angle at the Point O, is equal to
the Angle C E P. But D G F is likewiſe a
right-angled Triangle, as having the Angle at the
Point G, equal to ECP; therefore theſe two
Triangles are ſimilar. And ſince the Side D G
is equal to the Side D O, the Triangles are alſo
equal: Therefore the Line D F, being common
to theſe two Triangles; the Point F, is the
Point wherein the Line, paſſing thro’ the Eye
parallel to the given Line, meets the Per-
ſpective Plane: And this Point is the acciden- tal one ſought.
Note, This Demonſtration as well regards
1113, 14. inclin’d Lines entirely ſeparate from the Geo-
metrical Plane, as thoſe that meet it in one of
their Extremes only.
1113, 14. inclin’d Lines entirely ſeparate from the Geo-
metrical Plane, as thoſe that meet it in one of
their Extremes only.
Problem X.
69.
To find the Repreſentation of one or more
Lines, inclin’d to the Geometrical Plane.
22Fig. 36.
Lines, inclin’d to the Geometrical Plane.
22Fig. 36.
Let A be a Point given in the Geometrical
Plane; whereon ſtands a Line, whoſe Length,
Direction, and Angle of Inclination is known.
Plane; whereon ſtands a Line, whoſe Length,
Direction, and Angle of Inclination is known.
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