97[20]
rectangle EO, OQ is equal to the rectangle AE, IQ;
conſequently, as the
ſum or difference of EO and OQ is alſo given, thoſe lines themſelves are
given by the 85th or 86th of the Data.
ſum or difference of EO and OQ is alſo given, thoſe lines themſelves are
given by the 85th or 86th of the Data.
Synthesis.
Take IQ a fourth proportional to R, S and P, and
deſcribe on EQ a circle; erect at E the indefinite perpendicular EK, and
take therein ED a mean proportional between AE and IQ; from D draw
DH, parallel to EQ, if O muſt lie any where between the points E and Q;
but through F, the center of the circle on EQ if it muſt fall without them,
cutting the ſaid circle in H: Laſtly, draw HO perpendicular to DH, which
will meet the indeſinite line in O, the point required.
deſcribe on EQ a circle; erect at E the indefinite perpendicular EK, and
take therein ED a mean proportional between AE and IQ; from D draw
DH, parallel to EQ, if O muſt lie any where between the points E and Q;
but through F, the center of the circle on EQ if it muſt fall without them,
cutting the ſaid circle in H: Laſtly, draw HO perpendicular to DH, which
will meet the indeſinite line in O, the point required.
For it is manifeſt from the conſtruction that ED and HO are equal;
and
(Eu. VI. 17.) the rectangle AE, IQ is equal to the ſquare on ED, and
therefore equal to the ſquare on HO; but the ſquare on HO is equal to the
rectangle EO, OQ (Eu. III. 35. 36.) : therefore the rectangle AE, IQ is
equal to the rectangle EO, OQ; and hence (Eu. VI. 16.) AE is to OE as OQ
to IQ, whence, by compoſition or diviſion, AO is to EO as OI to IQ; but
IQ is to P as S to R, or inverſely, P is to IQ as R to S; and ſo, by compound
ratio, the rectangle AO, P is to the rectangle EO, IQ as the rectangle IO,
R is to the rectangle IQ, S; that is (Eu. V. 15 and 16.) the rectangle AO,
P is to the rectangle IO, R as EO is to S; or the rectangle AO, P is to the
rectangle IO, R as the rectangle IO, EO is to the rectangle IO, S
(Eu. V. 16.) the rectangle AO, P is to the rectangle EO, IO as the rectangle
IO, R is to the rectangle IO, S; that is (Eu. V. 15.) as R is to S. Q. E. D.
(Eu. VI. 17.) the rectangle AE, IQ is equal to the ſquare on ED, and
therefore equal to the ſquare on HO; but the ſquare on HO is equal to the
rectangle EO, OQ (Eu. III. 35. 36.) : therefore the rectangle AE, IQ is
equal to the rectangle EO, OQ; and hence (Eu. VI. 16.) AE is to OE as OQ
to IQ, whence, by compoſition or diviſion, AO is to EO as OI to IQ; but
IQ is to P as S to R, or inverſely, P is to IQ as R to S; and ſo, by compound
ratio, the rectangle AO, P is to the rectangle EO, IQ as the rectangle IO,
R is to the rectangle IQ, S; that is (Eu. V. 15 and 16.) the rectangle AO,
P is to the rectangle IO, R as EO is to S; or the rectangle AO, P is to the
rectangle IO, R as the rectangle IO, EO is to the rectangle IO, S
(Eu. V. 16.) the rectangle AO, P is to the rectangle EO, IO as the rectangle
IO, R is to the rectangle IO, S; that is (Eu. V. 15.) as R is to S. Q. E. D.
Scholium.
This Problem may be conſidered as having three Epitagmas,
or general Caſes, viz. when A, the point which bounds the ſegment aſſigned
for the co efficient of the given line P being an extreme, O is ſought be-
tween it and the next thereto, or beyond all the points with reſpect to A;
ſecondly, where A is the middle point; and thirdly, when A being again
an extreme, O is ſought beyond it, or between the other two points E and
I: and each of theſe is ſubdiviſible into four more particular ones.
or general Caſes, viz. when A, the point which bounds the ſegment aſſigned
for the co efficient of the given line P being an extreme, O is ſought be-
tween it and the next thereto, or beyond all the points with reſpect to A;
ſecondly, where A is the middle point; and thirdly, when A being again
an extreme, O is ſought beyond it, or between the other two points E and
I: and each of theſe is ſubdiviſible into four more particular ones.
Epitagma I.
Here the four Caſes are when E being the middle point,
O is required between A and E, or beyond I; and theſe are both con-
ſtructed at once by Fig. 9: when I is the middle point and O ſought between
A and I or beyond E; and theſe are both conſtructed at once by Fig. 10.
and in both of theſe IQ is ſet off from I contrary to that direction which
O is required between A and E, or beyond I; and theſe are both con-
ſtructed at once by Fig. 9: when I is the middle point and O ſought between
A and I or beyond E; and theſe are both conſtructed at once by Fig. 10.
and in both of theſe IQ is ſet off from I contrary to that direction which