10649on PERSPECTIVE.
The Demonſtration of the Problem.
66.
The Torus of the Column muſt be conceiv’d as
made up of an Infinite Number of Circular Planes,
lying one upon another. And it is evident that
the Reaſon why each of thoſe Circles cannot be wholly
ſeen, is becauſe that which is immediately under it
hides a Part thereof; from whence it follows, that
if the Plane of one oſ theſe Circles be every way
continu’d, and the Circle immediately under it, be
thrown into Perſpective upon it, (which 118. is alſo a Circle) the two Points of Interſection of this
Repreſentation, and the Circle in the Plane, will deter-
mine the viſible Part of the ſaid Repreſentation; and
conſequently if the Repreſentation of theſe two Points
of Interſection be found upon the Perſpective Plane, we
ſhall have two Points of the Perſpective of the Torus of
the propoſed Column. This is what I have done in the
Solution of the Problem, as we ſhall now Analytically
demonſtrate.
made up of an Infinite Number of Circular Planes,
lying one upon another. And it is evident that
the Reaſon why each of thoſe Circles cannot be wholly
ſeen, is becauſe that which is immediately under it
hides a Part thereof; from whence it follows, that
if the Plane of one oſ theſe Circles be every way
continu’d, and the Circle immediately under it, be
thrown into Perſpective upon it, (which 118. is alſo a Circle) the two Points of Interſection of this
Repreſentation, and the Circle in the Plane, will deter-
mine the viſible Part of the ſaid Repreſentation; and
conſequently if the Repreſentation of theſe two Points
of Interſection be found upon the Perſpective Plane, we
ſhall have two Points of the Perſpective of the Torus of
the propoſed Column. This is what I have done in the
Solution of the Problem, as we ſhall now Analytically
demonſtrate.
Let O be the Eye, A M a part of the Torus of
22Fig. 35. the Column, A P a Perpendicular to the Baſe paſſing
through the Center of the Column, and A B a
Parallel to the Baſe, drawn thro’ the Center B of
the Semicircle Concavity of the Torus. Let M P be
a Semidiameter of one of the Circles ſpoken of in the
the beginning of this Demonſtration. Then if the
Line m p be drawn parallel and infinitely near C M P
and the Lines m O and p O are drawn cutting M P
in D and T, it is evident that D T, which is in the
Plane of the Circle paſſing thro’ M P, will be the
Semidiameter of the Perſpective of the Circle imme-
diately underneath.
22Fig. 35. the Column, A P a Perpendicular to the Baſe paſſing
through the Center of the Column, and A B a
Parallel to the Baſe, drawn thro’ the Center B of
the Semicircle Concavity of the Torus. Let M P be
a Semidiameter of one of the Circles ſpoken of in the
the beginning of this Demonſtration. Then if the
Line m p be drawn parallel and infinitely near C M P
and the Lines m O and p O are drawn cutting M P
in D and T, it is evident that D T, which is in the
Plane of the Circle paſſing thro’ M P, will be the
Semidiameter of the Perſpective of the Circle imme-
diately underneath.
Now let fall the Perpendicular O S from the Eye
to the Line A B, and continue the Lines M P and
m p, till they meet the ſaid Perpendicular in the Points
Q and q. Moreover, continue the Line M P to
to the Line A B, and continue the Lines M P and
m p, till they meet the ſaid Perpendicular in the Points
Q and q. Moreover, continue the Line M P to