88[11]
Here AE is to be made the ſum of the terms of the given ratio, and we are
to uſe the IVth Case of the IId Problem, whereby O falling between
L and E, o will fall beyond U; and that it will fall ſhort of I appears from
the Iſt Corollary from the IVth Case of the IId Problem.
to uſe the IVth Case of the IId Problem, whereby O falling between
L and E, o will fall beyond U; and that it will fall ſhort of I appears from
the Iſt Corollary from the IVth Case of the IId Problem.
Case II.
The given ratio being inæqualitatis minoris, let the point ſought
be required between the ſecond aſſigned U and the ſecond in order E, or be-
yond the firſt A, which ariſes from the ſame Conſtruction. Here AE is to be
made the difference of the terms of the given ratio, and we are to uſe the
IVth Case of the IId Problem, where O being made to fall between U and E,
o will fall beyond L, much more beyond A.
be required between the ſecond aſſigned U and the ſecond in order E, or be-
yond the firſt A, which ariſes from the ſame Conſtruction. Here AE is to be
made the difference of the terms of the given ratio, and we are to uſe the
IVth Case of the IId Problem, where O being made to fall between U and E,
o will fall beyond L, much more beyond A.
Case III.
The given ratio being inæqualitatis majoris, let the point ſought
be required between the ſecond aſſigned U and the ſecond in order E, or be-
yond the laſt I, which ariſes from the ſame Conſtruction. Here AE is to be
made the difference of the terms of the given ratio, and L is to be ſet off the
contrary way to what it was in the laſt Case; and we are to uſe the Iſt Case
of the IId Problem, whereby O being made to fall between E and L, or
between E and U, according as L or U is neareſt to the point E, o will fall
beyond I, as any one will ſee who conſiders the Conſtruction of that Case with
due attention.
be required between the ſecond aſſigned U and the ſecond in order E, or be-
yond the laſt I, which ariſes from the ſame Conſtruction. Here AE is to be
made the difference of the terms of the given ratio, and L is to be ſet off the
contrary way to what it was in the laſt Case; and we are to uſe the Iſt Case
of the IId Problem, whereby O being made to fall between E and L, or
between E and U, according as L or U is neareſt to the point E, o will fall
beyond I, as any one will ſee who conſiders the Conſtruction of that Case with
due attention.
Epitagma II.
Case IV.
Let the aſſigned points now be the extremes
A and U, and let O the point ſought be required now between the firſt
aſſigned A and the next to it E, or, which is effected by the ſame Conſtruction,
between the ſecond aſſigned U and the next to it I. Here AE is to be made
the ſum of the terms of the given ratio, and the IVth Case of the IId Pro-
BLEM is to be uſed, ſo that of the three points L, E, U, O being made to fall
beyond L one of the extremes, and o within U the other extreme, it will appear
from the Iſt Corollary from the IVth Case of the ſaid Problem that O
will fall between A and E, and o between U and I.
A and U, and let O the point ſought be required now between the firſt
aſſigned A and the next to it E, or, which is effected by the ſame Conſtruction,
between the ſecond aſſigned U and the next to it I. Here AE is to be made
the ſum of the terms of the given ratio, and the IVth Case of the IId Pro-
BLEM is to be uſed, ſo that of the three points L, E, U, O being made to fall
beyond L one of the extremes, and o within U the other extreme, it will appear
from the Iſt Corollary from the IVth Case of the ſaid Problem that O
will fall between A and E, and o between U and I.
Case V.
The given ratio being inæqualitatis majoris, let the point ſought be
required between the ſecond and third in order, viz. between E and I. Here
AE muſt be the difference of the terms of the given ratio, and L ſet off to-
wards I, and the IId Case of the IId Problem uſed, and then O, as like-
wiſe o, will fall between E and I, if the Problem be poſſible.
required between the ſecond and third in order, viz. between E and I. Here
AE muſt be the difference of the terms of the given ratio, and L ſet off to-
wards I, and the IId Case of the IId Problem uſed, and then O, as like-
wiſe o, will fall between E and I, if the Problem be poſſible.
As to the Determination, ſee Lemma VII.
following.
Case VI.
The given ratio being inæqualitatis majoris, let the point ſought
be required beyond the laſt aſſigned, that is the laſt in order, U. Here AE
muſt be the difference of the terms of the given ratio, [and L muſt
be required beyond the laſt aſſigned, that is the laſt in order, U. Here AE
muſt be the difference of the terms of the given ratio, [and L muſt
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