Apollonius <Pergaeus>; Lawson, John - The two books of Apollonius Pergaeus, concerning tangencies, as they have been restored by Franciscus Vieta and Marinus Ghetaldus : with a supplement to which is now added, a second supplement, being Mons. Fermat's Treatise on spherical tangencies

Scan	Original
71	[vi]
72	[vii]
73	[viii]
74	[ix]
75	[x]
76	[xi]
77
78
79	[2]
80	[3]
81	[4]
82	[5]
83	[6]
84	[7]
85	[8]
86	[9]
87	[10]
88	[11]
89	[12]
90	[13]
91	[14]
92	[15]
93	[16]
94	[17]
95	[18]
96	[19]
97	[20]
98	[21]
99	[22]
100	[23]

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.

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.

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.

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.

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

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