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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              <pb o="[viii]" file="0066" n="73"/>
            tion is at the ſecond Epitagma; </s>
            <s xml:id="echoid-s1004" xml:space="preserve">and farther, the Limiting Ratio is
              <lb/>
            therein a maximum, as it ought. </s>
            <s xml:id="echoid-s1005" xml:space="preserve">Again, the Problem, wherein
              <lb/>
            it is propoſed to make the ſquare on AO in a given ratio to the rect-
              <lb/>
            angle contained by EO and P, has its limiting ratio a minimum
              <lb/>
            when the required point is ſought beyond (E) that of the given ones
              <lb/>
            which bounds the ſegment concerned in the conſequent term of
              <lb/>
            the ratio; </s>
            <s xml:id="echoid-s1006" xml:space="preserve">which, therefore, I apprehend muſt have been the third
              <lb/>
            Epitagma, and if ſo, this of courſe muſt have been the third Pro-
              <lb/>
            blem: </s>
            <s xml:id="echoid-s1007" xml:space="preserve">and as there remains only one wherein the number of given
              <lb/>
            pointsare two, I make that the fourth. </s>
            <s xml:id="echoid-s1008" xml:space="preserve">With reſpect to the fifth and
              <lb/>
            ſixth Problems, in which three points are given, it ſhould ſeem
              <lb/>
            that that would be the firſt in order, wherein there is a given ex-
              <lb/>
            ternal line concerned.</s>
            <s xml:id="echoid-s1009" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1010" xml:space="preserve">But it ſhould, by no means, be diſſembled that objections may be
              <lb/>
            brought againſt the identity, and arrangment of ſome of theſe Pro-
              <lb/>
            blems. </s>
            <s xml:id="echoid-s1011" xml:space="preserve">For firſt,
              <emph style="sc">Pappus</emph>
            no where expreſsly ſays that
              <emph style="sc">Apollo</emph>
            -
              <lb/>
              <emph style="sc">NIUS</emph>
            compared together two ſquares, wherefore, if this cannot be
              <lb/>
            implied, the identity of the fourth Problem is deeply ſtruck at:
              <lb/>
            </s>
            <s xml:id="echoid-s1012" xml:space="preserve">and moreover, this fourth Problem perhaps cannot with propriety,
              <lb/>
            be ſaid to have its limiting ratio either maximum or minimum, un-
              <lb/>
            leſs the ratio of equality, can be admitted as ſuch. </s>
            <s xml:id="echoid-s1013" xml:space="preserve">Laſtly, in the
              <lb/>
            fifth Problem, the ſaid limiting ratio is a minimum, and not a maxi-
              <lb/>
            mum as it is ſaid to have been by
              <emph style="sc">Pappus</emph>
            : </s>
            <s xml:id="echoid-s1014" xml:space="preserve">either, therefore, </s>
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