111[34]
the ratio of a greater to a leſs;
but theſe, I ſhall ſhew, may be reduced
to four.
to four.
Case I.
The order of the given points being A, U, E, I;
and the given
ratio a leſs to a greater, the conſtruction will be eſſected by Fig. 40,
wherein B is made to fall beyond A with reſpect to I and C beyond U, and
DH is drawn through the center of the circle on BC. And O will fall
between U and E for reaſons ſimilar to thoſe urged in the ſirſt Caſe of Epi-
tagma I. It is moreover obvious that the conſtruction will not be eſſentially
different ſhould the points E and I change places, and therefore need not
here be made a new Caſe.
ratio a leſs to a greater, the conſtruction will be eſſected by Fig. 40,
wherein B is made to fall beyond A with reſpect to I and C beyond U, and
DH is drawn through the center of the circle on BC. And O will fall
between U and E for reaſons ſimilar to thoſe urged in the ſirſt Caſe of Epi-
tagma I. It is moreover obvious that the conſtruction will not be eſſentially
different ſhould the points E and I change places, and therefore need not
here be made a new Caſe.
Case II.
The order of the points being the ſame as in the laſt Caſe, let
the given ratio be of a greater to a leſs; then, as in Fig. 41, B muſt fall
beyond I, and C beyond E; but DC muſt ſtill be drawn through the
center of the circle on BC. It is manifeſt that this conſtruction will ſerve
for that Caſe wherein the points A and U change ſituations, if the ratio be,
as here, of a greater to a leſs.
the given ratio be of a greater to a leſs; then, as in Fig. 41, B muſt fall
beyond I, and C beyond E; but DC muſt ſtill be drawn through the
center of the circle on BC. It is manifeſt that this conſtruction will ſerve
for that Caſe wherein the points A and U change ſituations, if the ratio be,
as here, of a greater to a leſs.
Case III.
Here, let the order of the points be U, A, I, E, and the
given ratio of a leſs to a greater, and the Conſtruction will be aſſected by
Fig. 42, in which B falls beyond A, and C beyond U with reſpect to I and
E: and the ſame conſtruction will ſerve if I and E change places, but the
ratio remain the ſame.
given ratio of a leſs to a greater, and the Conſtruction will be aſſected by
Fig. 42, in which B falls beyond A, and C beyond U with reſpect to I and
E: and the ſame conſtruction will ſerve if I and E change places, but the
ratio remain the ſame.
Case IV.
If the poſition of the points be retained, but the ratio be
made of a leſs to a greater; then muſt B fall beyond I (Fig. 43.) and C
beyond E; but DH drawn as before. That O muſt fall as was required,
in theſe three laſt caſes, is obvious enough from what has been ſaid be-
fore on the like occaſion: and it is alſo plain that the conſtruction will
not be materially diſſerent though A and U change places.
made of a leſs to a greater; then muſt B fall beyond I (Fig. 43.) and C
beyond E; but DH drawn as before. That O muſt fall as was required,
in theſe three laſt caſes, is obvious enough from what has been ſaid be-
fore on the like occaſion: and it is alſo plain that the conſtruction will
not be materially diſſerent though A and U change places.
Scholium.
That none of the Caſes of theſe two Epitagmas are ſubject
to Limitations, might be proved with the utmoſt rigour of geometrical
reaſoning was it not ſuſſiciently manifeſt from conſidering that as the point
O approaches points A, or U, the ratio of the rectangle AO, OU to the
rectangle EO, OI will become very ſmall, and as it approaches the points
E, or I the ſaid ratio will become very great: and nothing hinders that
the ſaid point may ſall any where between thoſe.
to Limitations, might be proved with the utmoſt rigour of geometrical
reaſoning was it not ſuſſiciently manifeſt from conſidering that as the point
O approaches points A, or U, the ratio of the rectangle AO, OU to the
rectangle EO, OI will become very ſmall, and as it approaches the points
E, or I the ſaid ratio will become very great: and nothing hinders that
the ſaid point may ſall any where between thoſe.
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