89[12]
fall beyond U, but for a more particular Determination ſee Lemma VII.
following] and the IId Case of the IId Problem is to be uſed, and then O,
as likewiſe o, will fall beyond U.
following] and the IId Case of the IId Problem is to be uſed, and then O,
as likewiſe o, will fall beyond U.
Case VII.
The given ratio being inæqualitatis minoris, let the point ſought
be required to lie beyond either of the aſſigned ones, i. e. beyond either ex-
treme, the ſame Conſtruction ſerving for both. Here AE is to be the diſ-
ference of the terms of the given ratio, and L to be ſet off backwards beyond
A; and the IVth Case of the IId Problem uſed, that ſo O being made to
fall beyond U, it will appear, by the IId Corollary from the IVth Case of
the ſaid Problem, that o will alſo fall beyond A.
be required to lie beyond either of the aſſigned ones, i. e. beyond either ex-
treme, the ſame Conſtruction ſerving for both. Here AE is to be the diſ-
ference of the terms of the given ratio, and L to be ſet off backwards beyond
A; and the IVth Case of the IId Problem uſed, that ſo O being made to
fall beyond U, it will appear, by the IId Corollary from the IVth Case of
the ſaid Problem, that o will alſo fall beyond A.
Epitagma III.
Case VIII.
Let the aſſigned points A and U be now
one an extreme, and the other the point next it: and let the point ſought be re-
quired to fall between the two aſſigned ones. Here AE muſt be the ſum of
the terms of the given ratio, and the IId Case of the IId Problem uſed.
And ſo O, as likewiſe o, being made to fall between L and U, they will
much more fall between A and U.
one an extreme, and the other the point next it: and let the point ſought be re-
quired to fall between the two aſſigned ones. Here AE muſt be the ſum of
the terms of the given ratio, and the IId Case of the IId Problem uſed.
And ſo O, as likewiſe o, being made to fall between L and U, they will
much more fall between A and U.
The Limitation is, that VN (found in the ſame ratio to UI as AL to AE)
muſt not exceed LE + EU - √4 LEU*.
muſt not exceed LE + EU - √4 LEU*.
Case IX.
The given ratio being inæqualitatis minoris, let the point ſought
be required between the ſecond aſſigned U and the third in order E, or elſe
beyond the firſt aſſigned A, the ſame Conſtruction ſerving for both. Here AE
is to be the difference of the terms of the given ratio, and L to be ſet off be-
yond A, and the Iſt Case of the IId Problem uſed: and ſo O being made
to fall between E and U, o will fall beyond L, and much more be-
yond A.
be required between the ſecond aſſigned U and the third in order E, or elſe
beyond the firſt aſſigned A, the ſame Conſtruction ſerving for both. Here AE
is to be the difference of the terms of the given ratio, and L to be ſet off be-
yond A, and the Iſt Case of the IId Problem uſed: and ſo O being made
to fall between E and U, o will fall beyond L, and much more be-
yond A.
Case X.
The given ratio being inæqualitatis majoris, let the point ſought
be required between the ſecond aſſigned U and the third in order E, or elſe
beyond the laſt in order I, the ſame conſtruction ſerving for both. Here AE
is to be the difference of the terms of the given ratio, and L to be ſet off be-
yond E, and the IVth Case of the IId Problem uſed, ſo that O being
made to fall between U and 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 third in order E, or elſe
beyond the laſt in order I, the ſame conſtruction ſerving for both. Here AE
is to be the difference of the terms of the given ratio, and L to be ſet off be-
yond E, and the IVth Case of the IId Problem uſed, ſo that O being
made to fall between U and E, o will fall beyond I, as any one will ſee who
conſiders the conſtruction of that Case with due attention.
Case XI.
As to the three Anomalous Caſes, in which the IId Problem is
of no uſe, and which I ſaid before might be reduced to one, they are theſe:
whereas in the IId and IIId Cases, as alſo in the VIth and VIIth, and like-
wiſe in the IXth and Xth the ratio given was that of inequality, let us
of no uſe, and which I ſaid before might be reduced to one, they are theſe:
whereas in the IId and IIId Cases, as alſo in the VIth and VIIth, and like-
wiſe in the IXth and Xth the ratio given was that of inequality, let us
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