10851on PERSPECTIVE.
s i by x, and i h be y;
it is manifeſt, that i 4 = a 5
being Algebraially Expreſſed, will be {ydy/dx}
being Algebraially Expreſſed, will be {ydy/dx}
Again, the ſimilar Triangles, s a 5 and s i g give
s a (e): a 5 ({ydy/dx}): : s i (x): i g ({xydx/edx}) Alſo by
the Conſtruction of Figure 32,
A S (e): A P = i h (y): : A P (y): A I ({yy/e});
Whence it follows, ſince I G = i g, that A G =
(I G - A I) = {xyd/cdx} - {yy/e}. And conſequently, H
and F are the Seats of the two Points whoſe Perſpe-
ctive is required, and thoſe Points are both in a
Plane parallel to the Geometrical Plane, which is the
height of 21 above the Geometrical Plane.
s a (e): a 5 ({ydy/dx}): : s i (x): i g ({xydx/edx}) Alſo by
the Conſtruction of Figure 32,
A S (e): A P = i h (y): : A P (y): A I ({yy/e});
Whence it follows, ſince I G = i g, that A G =
(I G - A I) = {xyd/cdx} - {yy/e}. And conſequently, H
and F are the Seats of the two Points whoſe Perſpe-
ctive is required, and thoſe Points are both in a
Plane parallel to the Geometrical Plane, which is the
height of 21 above the Geometrical Plane.
If the precedent Calculation be apply’d to the Lower
Part of the Torus, the Expreſſion {xydy/edx} - {yy/e}, will
be chang’d into this, - {xydy/edx} - {yy/e; } which ſhews that
theſe two Quantities muſt be aſſumed on the ſame Side
of A, viz. towards S. Moreover 9 q, inthe Line
9 m, is equal to {xydy/edx}; for 98 ({ydy/e}) = i 4.
Which ſhews that M and L are alſo the Seats of two
Points whoſe Perſpective muſt be found, and which are
both in a Plane parallel to the Geometrical Plane, and
above it the Height of 29.
Part of the Torus, the Expreſſion {xydy/edx} - {yy/e}, will
be chang’d into this, - {xydy/edx} - {yy/e; } which ſhews that
theſe two Quantities muſt be aſſumed on the ſame Side
of A, viz. towards S. Moreover 9 q, inthe Line
9 m, is equal to {xydy/edx}; for 98 ({ydy/e}) = i 4.
Which ſhews that M and L are alſo the Seats of two
Points whoſe Perſpective muſt be found, and which are
both in a Plane parallel to the Geometrical Plane, and
above it the Height of 29.
Remarks.
67.
This Problem may be likewiſe ſolved in
conſidering the Torus of a Column as made up of
an infinite Number of Baſes of Cones, whoſe Al-
titudes are determin’d by the concurrence of the
Tangents of the Semicircular Concavity of the
Axis of the Column; and then determining 1153.
conſidering the Torus of a Column as made up of
an infinite Number of Baſes of Cones, whoſe Al-
titudes are determin’d by the concurrence of the
Tangents of the Semicircular Concavity of the
Axis of the Column; and then determining 1153.
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