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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          <p>
            <s xml:id="echoid-s1576" xml:space="preserve">
              <pb o="[12]" file="0082" n="89"/>
            fall beyond U, but for a more particular
              <emph style="sc">Determination</emph>
            ſee
              <emph style="sc">Lemma</emph>
            VII.
              <lb/>
            </s>
            <s xml:id="echoid-s1577" xml:space="preserve">following] and the IId
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Problem</emph>
            is to be uſed, and then O,
              <lb/>
            as likewiſe o, will fall beyond U.</s>
            <s xml:id="echoid-s1578" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1579" xml:space="preserve">
              <emph style="sc">Case</emph>
            VII. </s>
            <s xml:id="echoid-s1580" xml:space="preserve">The given ratio being inæqualitatis minoris, let the point ſought
              <lb/>
            be required to lie beyond either of the aſſigned ones, i. </s>
            <s xml:id="echoid-s1581" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1582" xml:space="preserve">beyond either ex-
              <lb/>
            treme, the ſame Conſtruction ſerving for both. </s>
            <s xml:id="echoid-s1583" xml:space="preserve">Here AE is to be the diſ-
              <lb/>
            ference of the terms of the given ratio, and L to be ſet off backwards beyond
              <lb/>
            A; </s>
            <s xml:id="echoid-s1584" xml:space="preserve">and the IVth
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Problem</emph>
            uſed, that ſo O being made to
              <lb/>
            fall beyond U, it will appear, by the IId
              <emph style="sc">Corollary</emph>
            from the IVth
              <emph style="sc">Case</emph>
            of
              <lb/>
            the ſaid
              <emph style="sc">Problem</emph>
            , that o will alſo fall beyond A.</s>
            <s xml:id="echoid-s1585" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1586" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            III. </s>
            <s xml:id="echoid-s1587" xml:space="preserve">
              <emph style="sc">Case</emph>
            VIII. </s>
            <s xml:id="echoid-s1588" xml:space="preserve">Let the aſſigned points A and U be now
              <lb/>
            one an extreme, and the other the point next it: </s>
            <s xml:id="echoid-s1589" xml:space="preserve">and let the point ſought be re-
              <lb/>
            quired to fall between the two aſſigned ones. </s>
            <s xml:id="echoid-s1590" xml:space="preserve">Here AE muſt be the ſum of
              <lb/>
            the terms of the given ratio, and the IId
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Problem</emph>
            uſed.
              <lb/>
            </s>
            <s xml:id="echoid-s1591" xml:space="preserve">And ſo O, as likewiſe o, being made to fall between L and U, they will
              <lb/>
            much more fall between A and U.</s>
            <s xml:id="echoid-s1592" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1593" xml:space="preserve">The
              <emph style="sc">Limitation</emph>
            is, that VN (found in the ſame ratio to UI as AL to AE)
              <lb/>
            muſt not exceed LE + EU - √4 LEU*.</s>
            <s xml:id="echoid-s1594" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1595" xml:space="preserve">
              <emph style="sc">Case</emph>
            IX. </s>
            <s xml:id="echoid-s1596" xml:space="preserve">The given ratio being inæqualitatis minoris, let the point ſought
              <lb/>
            be required between the ſecond aſſigned U and the third in order E, or elſe
              <lb/>
            beyond the firſt aſſigned A, the ſame Conſtruction ſerving for both. </s>
            <s xml:id="echoid-s1597" xml:space="preserve">Here AE
              <lb/>
            is to be the difference of the terms of the given ratio, and L to be ſet off be-
              <lb/>
            yond A, and the Iſt
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Problem</emph>
            uſed: </s>
            <s xml:id="echoid-s1598" xml:space="preserve">and ſo O being made
              <lb/>
            to fall between E and U, o will fall beyond L, and much more be-
              <lb/>
            yond A.</s>
            <s xml:id="echoid-s1599" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1600" xml:space="preserve">
              <emph style="sc">Case</emph>
            X. </s>
            <s xml:id="echoid-s1601" xml:space="preserve">The given ratio being inæqualitatis majoris, let the point ſought
              <lb/>
            be required between the ſecond aſſigned U and the third in order E, or elſe
              <lb/>
            beyond the laſt in order I, the ſame conſtruction ſerving for both. </s>
            <s xml:id="echoid-s1602" xml:space="preserve">Here AE
              <lb/>
            is to be the difference of the terms of the given ratio, and L to be ſet off be-
              <lb/>
            yond E, and the IVth
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Problem</emph>
            uſed, ſo that O being
              <lb/>
            made to fall between U and E, o will fall beyond I, as any one will ſee who
              <lb/>
            conſiders the conſtruction of that
              <emph style="sc">Case</emph>
            with due attention.</s>
            <s xml:id="echoid-s1603" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1604" xml:space="preserve">
              <emph style="sc">Case</emph>
            XI. </s>
            <s xml:id="echoid-s1605" xml:space="preserve">As to the three Anomalous Caſes, in which the IId
              <emph style="sc">Problem</emph>
            is
              <lb/>
            of no uſe, and which I ſaid before might be reduced to one, they are theſe:
              <lb/>
            </s>
            <s xml:id="echoid-s1606" xml:space="preserve">whereas in the IId and IIId
              <emph style="sc">Cases</emph>
            , as alſo in the VIth and VIIth, and like-
              <lb/>
            wiſe in the IXth and Xth the ratio given was that of inequality, let us </s>
          </p>
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