102[25]
IB, as EI is to IC:
and hence, by compound ratio, the ſquare on the
abovementioned ſum or difference is to the rectangle contained by R and S
as the rectangle contained by AI and EI is to the rectangle contained by
IB and EC, alſo by permutation, AI is to EI as IB is to IC; wherefore,
by compoſition or diviſion, AE is to AI as BC is to IB, by permutation,
AE is to BC as AI is to IB, therefore by equality, the ſum or difference
of R and S is to S as AE is to BC; or (Eu. V. 15.) as half AE is to
half BC; conſequently (Eu. VI. 22.) the ſquare on the above mentioned
ſum or diſſerence is to the ſquare on S as the ſquare on half AE is to the
ſquare on half BC, or to the rectangle contained by IB and EC. Hence
exæquo perturbaté, the rectangle contained by R and S is to the ſquare on
S as the ſquare on half AE is to the rectangle contained by AI and EI,
or (Eu. V. 15.) R is to S as the ſquare on half AE is to the rectangle
contained by AI and EI; and which is therefore the greateſt ratio which
R can have to S in thoſe Caſes.
abovementioned ſum or difference is to the rectangle contained by R and S
as the rectangle contained by AI and EI is to the rectangle contained by
IB and EC, alſo by permutation, AI is to EI as IB is to IC; wherefore,
by compoſition or diviſion, AE is to AI as BC is to IB, by permutation,
AE is to BC as AI is to IB, therefore by equality, the ſum or difference
of R and S is to S as AE is to BC; or (Eu. V. 15.) as half AE is to
half BC; conſequently (Eu. VI. 22.) the ſquare on the above mentioned
ſum or diſſerence is to the ſquare on S as the ſquare on half AE is to the
ſquare on half BC, or to the rectangle contained by IB and EC. Hence
exæquo perturbaté, the rectangle contained by R and S is to the ſquare on
S as the ſquare on half AE is to the rectangle contained by AI and EI,
or (Eu. V. 15.) R is to S as the ſquare on half AE is to the rectangle
contained by AI and EI; and which is therefore the greateſt ratio which
R can have to S in thoſe Caſes.
It ought farther to be remarked, that to have Caſe III poſſible, where
O is ſought beyond A, and the ratio of a greater to a leſs, it is neceſſary
that AI be leſs than IE, and to have Caſe IV. poſſible, that it be greater.
For it is plain from the Conſtruction, that IB muſt in the former caſe be
leſs, and in the latter greater than I C; but as R is to S ſo is AB to
IB, and ſo is EC to IC, wherefore by diviſion, the exceſs of R above S
is to S as AI is to IB, and as EI is to IC; and ſo by permutation AI is
to EI as IB is to IC: conſequently when IB is greater than IC, AI will
be greater than EI; and when leſs, leſs.
O is ſought beyond A, and the ratio of a greater to a leſs, it is neceſſary
that AI be leſs than IE, and to have Caſe IV. poſſible, that it be greater.
For it is plain from the Conſtruction, that IB muſt in the former caſe be
leſs, and in the latter greater than I C; but as R is to S ſo is AB to
IB, and ſo is EC to IC, wherefore by diviſion, the exceſs of R above S
is to S as AI is to IB, and as EI is to IC; and ſo by permutation AI is
to EI as IB is to IC: conſequently when IB is greater than IC, AI will
be greater than EI; and when leſs, leſs.
With reſpect to thoſe caſes wherein the given ratio is that of equality,
it may be ſufficient to remark, that none of the Caſes of Epitagma II.
are poſſible under that ratio: that one of Caſes III. and IV. Epitagma III.
is always impoſſible when the given ratio of R to S is the ratio of equality;
and both are ſo if AI be at the ſame time equal to IE. Laſtly Caſes V.
and VI. are never poſſible under the ratio of equality, unleſs the ſquare on
half AE be equal to, or exceed the rectangle contained by AI and EI;
all which naturally follows from what has been delivered above.
1it may be ſufficient to remark, that none of the Caſes of Epitagma II.
are poſſible under that ratio: that one of Caſes III. and IV. Epitagma III.
is always impoſſible when the given ratio of R to S is the ratio of equality;
and both are ſo if AI be at the ſame time equal to IE. Laſtly Caſes V.
and VI. are never poſſible under the ratio of equality, unleſs the ſquare on
half AE be equal to, or exceed the rectangle contained by AI and EI;
all which naturally follows from what has been delivered above.
THE END OF BOOK I.
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