109[32]
CU as IO is to BO;
and ſo, by compound ratio, the rectangle AO, UO
is to the rectangle BO, CU as the rectangle EO, IO is to the rectangle
BO, CE; by permutation, the rectangle AO, UO is to the rectangle EO,
IO, as the rectangle BO, CU is to the rectangle BO, CE; or (Eu. V. 15.)
as CU is to CE; that is, by conſtruction as R to S.
is to the rectangle BO, CU as the rectangle EO, IO is to the rectangle
BO, CE; by permutation, the rectangle AO, UO is to the rectangle EO,
IO, as the rectangle BO, CU is to the rectangle BO, CE; or (Eu. V. 15.)
as CU is to CE; that is, by conſtruction as R to S.
Q.
E.
D.
Scholium.
In enumerating the ſeveral Caſes of this Problem I ſhall en-
deavour to follow the method which I conceive Apollonius did: and there-
fore, notwithſtanding the preceding Analyſis and Conſtruction are general
for the whole, divide it into three Problems, each Problem into three Epi-
tagmas, or general Caſes, and theſe again into their ſeveral particular ones.
deavour to follow the method which I conceive Apollonius did: and there-
fore, notwithſtanding the preceding Analyſis and Conſtruction are general
for the whole, divide it into three Problems, each Problem into three Epi-
tagmas, or general Caſes, and theſe again into their ſeveral particular ones.
PROBLEM I. (Fig. 32 to 45.)
Here O is ſought between the two mean points of the four given ones:
and the three Epitagmas are, firſt, when A and U, the points which bound
the ſegments containing the antecedent rectangle, are one an extreme, and
the other an alternate mean; ſecondly, when thoſe points are one an ex-
treme and the other an adjacent mean; thirdly, when they are both means,
or both extremes.
and the three Epitagmas are, firſt, when A and U, the points which bound
the ſegments containing the antecedent rectangle, are one an extreme, and
the other an alternate mean; ſecondly, when thoſe points are one an ex-
treme and the other an adjacent mean; thirdly, when they are both means,
or both extremes.
Epitagma I, Conſiſts of eight Caſes, viz.
when the order of the given
points is A, I, U, E; U, E, A, I; A, E, U, I; or U, I, A, E, and the
given ratio of a leſs to a greater, and four others wherein the order of the
points is the ſame as in thoſe, but the ratio of R to S, the ratio of a greater
to a leſs.
points is A, I, U, E; U, E, A, I; A, E, U, I; or U, I, A, E, and the
given ratio of a leſs to a greater, and four others wherein the order of the
points is the ſame as in thoſe, but the ratio of R to S, the ratio of a greater
to a leſs.
Case I.
Let the order of the given points be A, I, U, E, and the given
ratio of a leſs to a greater; and the Conſtruction will be as in Fig. 32, where
B is made to fall beyond A, with reſpect to I, and C beyond U with re-
ſpect to E, and DH is drawn through F, the center of the circle on BC.
That O, when this conſtruction is uſed, will fall between I and U is
plain, becauſe CO is to CU as IB is to BO; and therefore if CU be
greater than CO, BO will be greater than IB, and if leſs, leſs; but this,
it is manifeſt, cannot be the Caſe if O falls either beyond I or U, and
therefore it falls between them.
ratio of a leſs to a greater; and the Conſtruction will be as in Fig. 32, where
B is made to fall beyond A, with reſpect to I, and C beyond U with re-
ſpect to E, and DH is drawn through F, the center of the circle on BC.
That O, when this conſtruction is uſed, will fall between I and U is
plain, becauſe CO is to CU as IB is to BO; and therefore if CU be
greater than CO, BO will be greater than IB, and if leſs, leſs; but this,
it is manifeſt, cannot be the Caſe if O falls either beyond I or U, and
therefore it falls between them.