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bears therefrom, and DH drawn through F, the center of the circle on EQ:
none of theſe Caſes are ſubject to any Limitations.
none of theſe Caſes are ſubject to any Limitations.
Epitagma II.
Wherein A is the middle point, and the Caſes, when O
is ſought beyond E, between E and A, between A and I or beyond I. The
firſt and third of which are conſtructed at once by Fig. 11, wherein IQ is
ſet off from I towards A and DH drawn through F, the center of the circle
on EQ. The ſecond and fourth are conſtructed at once, alſo, by Fig. 12.
where IQ is ſet off from I the contrary way to that which A lies, and DH
drawn parallel to EQ. There are no Limitations to any of theſe Caſes.
is ſought beyond E, between E and A, between A and I or beyond I. The
firſt and third of which are conſtructed at once by Fig. 11, wherein IQ is
ſet off from I towards A and DH drawn through F, the center of the circle
on EQ. The ſecond and fourth are conſtructed at once, alſo, by Fig. 12.
where IQ is ſet off from I the contrary way to that which A lies, and DH
drawn parallel to EQ. There are no Limitations to any of theſe Caſes.
Epitagma III.
Here, E being the middle point, the Caſes are, when O
muſt lie beyond A, or between E and I; and the ſame Caſes occur when
I is made the middle point. The firſt is conſtructed by Fig. 13, the ſecond
by Fig. 14, the third by Fig. 15, and the fourth by Fig. 16: in every one
of which IQ is ſet off from I towards A, and DH drawn parallel to EQ.
The Limits are that the given ratio of R to S, muſt not be leſs than the ratio
which the rectangle AE, P bears to the ſquare on half the Sum, or half the
difference of AE, and a fourth propor tional to R, S and P; that is, to the
ſquare on half EQ: ſince if it ſhould, the rectangle contained by AE and
the ſaid fourth proportional will be greater than the ſquare on half EQ;
and of courſe ED (a mean proportional between them) greater than half
EQ, in which Caſe DH can neither cut nor touch the circle on EQ, and
ſo the problem be impoſſible. It is farther obſervable in the two laſt caſes,
that to have the former of them poſſible, AE muſt be leſs, and to have
the latter poſſible, EI muſt be greater than the above-mentioned half
ſum; for if this latter part of the Limitation be not obſerved, theſe caſes
are changed into one another.
muſt lie beyond A, or between E and I; and the ſame Caſes occur when
I is made the middle point. The firſt is conſtructed by Fig. 13, the ſecond
by Fig. 14, the third by Fig. 15, and the fourth by Fig. 16: in every one
of which IQ is ſet off from I towards A, and DH drawn parallel to EQ.
The Limits are that the given ratio of R to S, muſt not be leſs than the ratio
which the rectangle AE, P bears to the ſquare on half the Sum, or half the
difference of AE, and a fourth propor tional to R, S and P; that is, to the
ſquare on half EQ: ſince if it ſhould, the rectangle contained by AE and
the ſaid fourth proportional will be greater than the ſquare on half EQ;
and of courſe ED (a mean proportional between them) greater than half
EQ, in which Caſe DH can neither cut nor touch the circle on EQ, and
ſo the problem be impoſſible. It is farther obſervable in the two laſt caſes,
that to have the former of them poſſible, AE muſt be leſs, and to have
the latter poſſible, EI muſt be greater than the above-mentioned half
ſum; for if this latter part of the Limitation be not obſerved, theſe caſes
are changed into one another.
PROBLEM VI.
(Fig. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.)
(Fig. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.)
In any indefinite ſtraight line let there be aſſigned the points A, E and I;
it is required to cut it in another point O, ſo that the rectangle contained
by the ſegments AO, EO may be to the ſquare on IO in the ratio of two
given ſtraight lines, R and S.
it is required to cut it in another point O, ſo that the rectangle contained
by the ſegments AO, EO may be to the ſquare on IO in the ratio of two
given ſtraight lines, R and S.