9645on PERSPECTIVE.
and conſequently to V P.
Therefore if the before-
mentioned Plane be ſuppoſed to revolve upon the Line
V I, as an Axis, until it coincides with the Per-
ſpective Plane, the Center of the Sphere will meet
the Perſpective Plane in Q, and the Eye in F;
whence the Part G E of the Line I V is the tranſ-
verſe Diameter of the Ellipſis.
mentioned Plane be ſuppoſed to revolve upon the Line
V I, as an Axis, until it coincides with the Per-
ſpective Plane, the Center of the Sphere will meet
the Perſpective Plane in Q, and the Eye in F;
whence the Part G E of the Line I V is the tranſ-
verſe Diameter of the Ellipſis.
Again let G D E in Figure 30, and g e f, in
11Fig. 30,
31. Figure 31 repreſent the Points denoted with the ſame
Letters in the foregoing Figure. Now if the Cone,
whoſe Profile is denoted by the Lines f g and fe be ſup-
poſed to be compleated, and to be cut by a Plane paſ-
ſing through the Line g e perpendicular to the Plane
of the Figure; we ſhall have an Ellipſis g 4 e 3
ſimilar to that which is the ſought Repreſentation
of the Sphere. Further if the ſaid Cone be conceived
to be cut by a Plane 14 m 3 parallel to its Baſe,
and biſecting g e in n, it is manifeſt, that 3 4, the
common Section of the Circle 14 m 3, and the Ellip-
ſis g 4 e 3, is the conjugate Axis of the Ellip-
ſis. And therefore this conjugate Axis is equal
to the Line 3 4, Perpendicular in the Point n to the
Diameter 1 m of the Circle 14 m 3. Now draw
the Lines E O and G Y in Figure 30, parallel to
L M, then the Triangles E G Y and E N M are
ſimilar, whence
11Fig. 30,
31. Figure 31 repreſent the Points denoted with the ſame
Letters in the foregoing Figure. Now if the Cone,
whoſe Profile is denoted by the Lines f g and fe be ſup-
poſed to be compleated, and to be cut by a Plane paſ-
ſing through the Line g e perpendicular to the Plane
of the Figure; we ſhall have an Ellipſis g 4 e 3
ſimilar to that which is the ſought Repreſentation
of the Sphere. Further if the ſaid Cone be conceived
to be cut by a Plane 14 m 3 parallel to its Baſe,
and biſecting g e in n, it is manifeſt, that 3 4, the
common Section of the Circle 14 m 3, and the Ellip-
ſis g 4 e 3, is the conjugate Axis of the Ellip-
ſis. And therefore this conjugate Axis is equal
to the Line 3 4, Perpendicular in the Point n to the
Diameter 1 m of the Circle 14 m 3. Now draw
the Lines E O and G Y in Figure 30, parallel to
L M, then the Triangles E G Y and E N M are
ſimilar, whence
EG: EN:: GY: NM.
But E G is twice E N;
wherefore G Y is alſo the
double of N M, and ſo N M equal to G Z. After
the ſame manner we demonſtrate, that L N is equal
to X E; whence it follows, that G D is equal to
L M, and is ſo cut in z as L M is in N; and there-
fore R L or G T of Figure 29, is equal to 34 in
Figure 31; and conſequently equal to the conjugate
Axis of the Ellipſis to be drawn. On the other
Hand, it is manifeſt by Conſtruction, that ſome one
of the Perpendiculars m n, Figure 29, viz. that
which paſſes through the Center of the
double of N M, and ſo N M equal to G Z. After
the ſame manner we demonſtrate, that L N is equal
to X E; whence it follows, that G D is equal to
L M, and is ſo cut in z as L M is in N; and there-
fore R L or G T of Figure 29, is equal to 34 in
Figure 31; and conſequently equal to the conjugate
Axis of the Ellipſis to be drawn. On the other
Hand, it is manifeſt by Conſtruction, that ſome one
of the Perpendiculars m n, Figure 29, viz. that
which paſſes through the Center of the