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
101	[24]
102	[25]
103	[26]
104	[27]
105	[28]
106	[29]
107	[30]
108	[31]
109	[32]
110	[33]
111	[34]
112	[35]
113	[36]
114	[37]
115	[38]
116	[39]
117	[40]
118
119
120
121
122
123
124
125
126
127
128
129
130

110[33]

Case II. If the order oſ the given points be retained, but the ratio be of
a greater to a leſs, B muſt ſall beyond I with reſpect to A, and C beyond
E, as in Fig. 33, and DH muſt be drawn as before: and it may be proved
by reaſonings ſimilar to thoſe uſed in the ſirſt Caſe that O will fall between
I and U, as was required.

Cases III and IV. Theſe caſes are conſtructed exactly in the ſame man-
ner as Caſes I and II reſpectively; and the reaſonings to prove that O will
fall as it ought arethe ſame with thoſe made uſe of in Caſe I, as will appear
by inſpecting Fig. 34. and 35.

Case V. Let now the order of the given points be A, E, U, I, and the
given ratio of a leſs to a greater. Then muſt B be made to fall beyond A
and C beyond U, as in Fig. 36, and DH is here to be drawn parallel to
to BC: and that O will fall as required may be made to appear thus. Draw
EF and UG perpendicular to BC. Now (Eu. VI. 13. 17.) the ſquare on
EF is equal to the rectangle contained by EC and EB, the ſquare on UG
to the rectangle contained by UC and UB; and, by conſtruction, the
ſquare on HO to the rectangle contained by EC and AB, or to the rect-
angle contained by UC and IB; and ſince, by ſuppoſition, EB is greater
than AB, and UB leſs than IB, the rectangle EC, AB, or its equal, the
rectangle UC, IB will be leſs than the rectangle EC, EB, and greater than
the rectangle UC, UB; and conſequently the ſquare on HO will be leſs
than the ſquare on EF and greater than the ſquare on UG; and therefore
HO leſs than EF and greater than UG: but this could not be the Caſe
unleſs O fell between E and U, as was was required.

Case VI. If the order of the points be retained; but the given ratio be
of a greater to a leſs, B muſt then fall beyond I, and C beyond E; and
DH is drawn as in the preceding Caſe (See Fig. 37.) moreover, that O
will fall between E and U may be made to appear, by reaſonings ſimilar to
thoſe there made uſe on the like occaſion.

Case VII. Is conſtructed exactly in the ſame manner as Caſe V. and,

Case VIII. As Caſe VI: the truth of which will appear by. barely in-
ſpecting Fig. 38 and 39.

Epitagma II. In this Epitagma there are alſo eight Caſes, viz. when the
order of the given points is A, U, E, I; A, U, I, E; U, A, E, I; or
U, A, I, E; and the given ratio of a leſs to a greater: and there are
four others wherein the order of the points are the ſame as in thoſe,

Page concordance

Text layer

Text normalization

Search