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

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            <s xml:id="echoid-s1014" xml:space="preserve">
              <pb o="[ix]" file="0067" n="74"/>
            miſtake muſt be admitted in this Author, or the fifth Problem is
              <lb/>
            wrong placed. </s>
            <s xml:id="echoid-s1015" xml:space="preserve">I am not prepared farther to obviate theſe objec-
              <lb/>
            tions, and only mention them to ſhew that although I ſaw them
              <lb/>
            in their full force, I could by no means agree, that they are pow-
              <lb/>
            erful enough to overturn thoſe already advanced for what I have
              <lb/>
              <handwritten xlink:label="hd-0067-01" xlink:href="hd-0067-01a" number="1"/>
              <handwritten xlink:label="hd-0067-01" xlink:href="hd-0067-01a" number="2"/>
            done.</s>
            <s xml:id="echoid-s1016" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1017" xml:space="preserve">I come now to Book II, which if I am not much miſtaken, was
              <lb/>
            entirely employed about what
              <emph style="sc">Snellius</emph>
            makes his fourth Problem.
              <lb/>
            </s>
            <s xml:id="echoid-s1018" xml:space="preserve">In this I am confirmed not only by the account which
              <emph style="sc">Pappus</emph>
            gives
              <lb/>
            in his Preface, but much more by the Lemmas of
              <emph style="sc">Apollonius</emph>
              <lb/>
              <handwritten xlink:label="hd-0067-01" xlink:href="hd-0067-01a" number="1"/>
              <handwritten xlink:label="hd-0067-01" xlink:href="hd-0067-01a" number="2"/>
            which he hath left us. </s>
            <s xml:id="echoid-s1019" xml:space="preserve">For we there find that
              <emph style="sc">Lemma</emph>
            21, where-
              <lb/>
            in is aſſigned the leaſt ratio which the rectangle contained by AO
              <lb/>
            and UO can bear to that contained by EO and IO, when O is ſought
              <lb/>
            between the two mean points of the four given ones, is ſaid to be
              <lb/>
            concerned in determining the μοναχὴ, or ſingle Caſe , of Problem 1.</s>
            <s xml:id="echoid-s1020" xml:space="preserve"> This Problem therefore of
              <emph style="sc">Apollonius</emph>
            contained only thoſe
              <lb/>
            Caſes of the general one, where O is ſought between the two mean
              <lb/>
            points. </s>
            <s xml:id="echoid-s1021" xml:space="preserve">In like manner, we gather from Lemma 22, that his ſe-
              <lb/>
            cond Problem was concerned in determining the point O when ſought
              <lb/>
            between a mean point, and an extreme one. </s>
            <s xml:id="echoid-s1022" xml:space="preserve">And laſtly, from
              <lb/>
            Lemma 24, that the third Problem of Book II. </s>
            <s xml:id="echoid-s1023" xml:space="preserve">determined
              <lb/>
            the point O when required without all the given ones.
              <lb/>
            </s>
            <s xml:id="echoid-s1024" xml:space="preserve">
              <note symbol="*" position="foot" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">So called, I conceive, becauſe in every other Caſe of the third Epitagma,
                <lb/>
              except this extreme, or limiting one, there are two points which will ſatisſy
                <lb/>
              the Problem.</note>
            </s>
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