100[23]
Scholium.
This Problem alſo hath three Epitagmas;
ſirſt, when O is
ſought between I, the point which bounds the ſegment whoſe ſquare is
concerned, and either of the other given ones; ſecondly, the ſaid point
being an extreme one, when O is ſought beyond it, or beyond both the
other given points with reſpect to it; and thirdly, when O is required be-
yond the next in order to the abovementioned point I: theſe are each of
them ſubdiviſible into other more particular caſes.
ſought between I, the point which bounds the ſegment whoſe ſquare is
concerned, and either of the other given ones; ſecondly, the ſaid point
being an extreme one, when O is ſought beyond it, or beyond both the
other given points with reſpect to it; and thirdly, when O is required be-
yond the next in order to the abovementioned point I: theſe are each of
them ſubdiviſible into other more particular caſes.
Epitagma I.
Here O is ſought between I, the point which bounds the
ſegment whoſe ſquare is concerned, and the next in order to it: and there
are four caſes, viz. when I is an extreme point; and the given ratio of R
to S the ratio of a greater to a leſs; when I, remaining as before, the given
ratio is of a leſs to a greater; again when I is the middle point, and O
ſought between it and either of the other given ones.
ſegment whoſe ſquare is concerned, and the next in order to it: and there
are four caſes, viz. when I is an extreme point; and the given ratio of R
to S the ratio of a greater to a leſs; when I, remaining as before, the given
ratio is of a leſs to a greater; again when I is the middle point, and O
ſought between it and either of the other given ones.
Case I.
Here the points B and C are both made to fall beyond I
(Fig. 17.) and DH is drawn through the center of the circle on BC, and O
will fall between the point I, and the next in order thereto; becauſe by
conſtruction, EC is to CO as BO is to IB, and therefore when EC is greater
than CO, BO will be greater than IB, and when leſs, leſs; but it is plain
that if O ſhould fall either beyond E or I, this could not be the Caſe. It
is farther manifeſt that ſhould the points A and E change places, the con-
ſtruction would be no otherwiſe altered than that AB would then be greater
than IC.
(Fig. 17.) and DH is drawn through the center of the circle on BC, and O
will fall between the point I, and the next in order thereto; becauſe by
conſtruction, EC is to CO as BO is to IB, and therefore when EC is greater
than CO, BO will be greater than IB, and when leſs, leſs; but it is plain
that if O ſhould fall either beyond E or I, this could not be the Caſe. It
is farther manifeſt that ſhould the points A and E change places, the con-
ſtruction would be no otherwiſe altered than that AB would then be greater
than IC.
Case II.
If the given points retain their poſition, but the ratio be made
of a leſs to a greater, the conſtruction will then be by Fig. 18, where B
muſt be made to fall beyond A, and C beyond E with reſpect to I; but
DH is ſtill drawn through the center of the circle on BC: and that O will
fall as required may be made appear by reaſonings ſimilar to thoſe uſed in
Caſe I. Moreover no change will enſue in the conſtruction when the Points
A and E change places, except that B and C will change ſituations alſo.
of a leſs to a greater, the conſtruction will then be by Fig. 18, where B
muſt be made to fall beyond A, and C beyond E with reſpect to I; but
DH is ſtill drawn through the center of the circle on BC: and that O will
fall as required may be made appear by reaſonings ſimilar to thoſe uſed in
Caſe I. Moreover no change will enſue in the conſtruction when the Points
A and E change places, except that B and C will change ſituations alſo.
Cases III and IV, Are conſtructed at once by Fig.
19, when B muſt
fall between A and I, C between I and E, and DH be drawn as before:
and it is here evident that the conſtruction will be the ſame let the given
ratio be what it will. None of thoſe Caſes admit of any Limitations.
fall between A and I, C between I and E, and DH be drawn as before:
and it is here evident that the conſtruction will be the ſame let the given
ratio be what it will. None of thoſe Caſes admit of any Limitations.
Epitagma II.
There are here only two Caſes, viz.
when O is required
beyond I, the point which bounds the ſegment whoſe ſquare is concerned;
beyond I, the point which bounds the ſegment whoſe ſquare is concerned;