113[36]
OY will be to OV as OI is to OE, and (Eu.
V.
15.
16.)
the rectangle
contained by VO and YO or its equal, the rectangle contained by AO and
OU, is to the rectangle contained by EO and OI as the ſquare on OV is
to the ſquare on EO, as the ſquaree on AU is to the ſquare on KY.
contained by VO and YO or its equal, the rectangle contained by AO and
OU, is to the rectangle contained by EO and OI as the ſquare on OV is
to the ſquare on EO, as the ſquaree on AU is to the ſquare on KY.
Q.
E.
D.
Scholium.
It might be obſerved that in the two Caſes of this Epitagma
where the points A and U are means, the limiting ratio will be a maxi-
mum inſtead of a minimum; and that ratio will be the ſame with that which
the ſquare on KY bears to the ſquare on EI, as is plain from what hath
been advanced above.
where the points A and U are means, the limiting ratio will be a maxi-
mum inſtead of a minimum; and that ratio will be the ſame with that which
the ſquare on KY bears to the ſquare on EI, as is plain from what hath
been advanced above.
PROBLEM II. (Fig. 46 to 57.)
Where O is ſought between a mean and an extream point:
and here,
as in the firſt Problem, there are three Epitagmas. Firſt when the points
A and U, which bound the ſegments containing the antecedent rectangle,
are one an extreme, and the other an alternate mean; ſecondly when they
are both means, or both extremes; thirdly when they are one an extreme,
and the other an adjacent mean.
as in the firſt Problem, there are three Epitagmas. Firſt when the points
A and U, which bound the ſegments containing the antecedent rectangle,
are one an extreme, and the other an alternate mean; ſecondly when they
are both means, or both extremes; thirdly when they are one an extreme,
and the other an adjacent mean.
Epitagma I.
There are here eight Caſes, but they are conſtructed at
four times, becauſe it is indifferent whether the given ratio be of a leſs to a
greater, or of a greater to a leſs.
four times, becauſe it is indifferent whether the given ratio be of a leſs to a
greater, or of a greater to a leſs.
Case I.
The order of the given points being A, E, U, I, as in Fig.
46,
make B to fall between A and I, C between U and E, and draw DH
through the center of the circle on BC; and O will fall between A and
E, becauſe AB is to BO as CO is to CE, and therefore, if AB be greater
than BO, CO muſt be greater than CE, and if leſs, leſs; but this cannot
be the caſe if O falls either beyond A or E: and the like abſurdity fol.
lows if o be ſuppoſed to fall otherwiſe than between I and U.
make B to fall between A and I, C between U and E, and draw DH
through the center of the circle on BC; and O will fall between A and
E, becauſe AB is to BO as CO is to CE, and therefore, if AB be greater
than BO, CO muſt be greater than CE, and if leſs, leſs; but this cannot
be the caſe if O falls either beyond A or E: and the like abſurdity fol.
lows if o be ſuppoſed to fall otherwiſe than between I and U.
Case II.
Wherein the order of the points is U, I, A, E;
and it is con-
ſtructed in the very ſame manner that Caſe I. is, as appears by barely
inſpecting Fig. 47.
ſtructed in the very ſame manner that Caſe I. is, as appears by barely
inſpecting Fig. 47.
Case III.
If the order of the given points be A, I, U, E (Fig.
48.)
the
points B and C muſt be made to fall as in the two preceding Caſes; but
DH muſt be drawn parallel to BC, and O will fall as required. For
points B and C muſt be made to fall as in the two preceding Caſes; but
DH muſt be drawn parallel to BC, and O will fall as required. For