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<span class="headingNumber">71.</span>
<span class="head">DETERMINATE SECTION.</span>
<br />
<span class="head">BOOK I.</span>
<br />
<span class="head">PROBLEM I. (Fig. 1.)</span>
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In any indefinite ſtraight line, let the Point A be aſſigned; it is required
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to cut it in ſome other point O, ſo that the ſquare on the ſegment AO
<br />
may be to the ſquare on a given line, P, in the ratio of two given ſtraight
<br />
lines R and S.
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<span class="caps" style="font-size: 75%">Analysis</span>
. Since, by Hypotheſis, the ſquare on AO muſt be to the
<br />
ſquare on P as R is to S, the ſquare on AO will be to the Square on P as
<br />
the ſquare on R is to the rectangle contained by R and S (
<span class="caps" style="font-size: 75%">Eu</span>
. V. 15.) Let there be taken AD, a mean proportional between AB (R) and AC
<br />
(S); then the Square on AO is to the ſquare on P as the ſquare on R is
<br />
to the ſquare on AD, or (
<span class="caps" style="font-size: 75%">Eu</span>
. VI. 22) AO is to P as R to AD; conſe-
<br />
quently, AO is given by
<span class="caps" style="font-size: 75%">Eu</span>
. VI. 12.
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<span class="caps" style="font-size: 75%">Synthesis</span>
. Make AB equal to R, AC equal to S, and deſcribe on
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BC a ſemi-circle; erect at A the indefinite perpendicular AF, meeting the
<br />
circle in D, and take AF equal to P; draw DB, and parallel thereto FO,
<br />
meeting the indefinite line in O, the point required.
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For, by reaſon of the ſimilar triangles ADB, AFO, AO is to AF (P) as
<br />
AB (R) is to AD; therefore (
<span class="caps" style="font-size: 75%">Eu</span>
. VI. 22.) the ſquare on AO is to the
<br />
ſquare on P as the ſquare on R is to the ſquare on AD; but the ſquare on
<br />
AD is equal to the rectangle contained by AB (R) and AC (S) by
<span class="caps" style="font-size: 75%">Eu</span>
. VI. 13. 17; and ſo the ſquare on AO is to the ſquare on P as the ſquare on R
<br />
is to the rectangle contained by R and S; that is (
<span class="caps" style="font-size: 75%">Eu</span>
. V. 15.) as R is to S.
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<p xmlns="http://www.w3.org/1999/xhtml">Q. E. D.</p>
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<span class="caps" style="font-size: 75%">Scholium</span>
. Here are no limitations, nor any precautions whatever to be
<br />
obſerved, except that AB (R) muſt be ſet off from A that way which O
<br />
is required to fall.
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