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="[vi]" file="0064" n="71"/>
          <p>
            <s xml:id="echoid-s977" xml:space="preserve">
              <emph style="sc">Pappus</emph>
            , in his preface to the ſeventh Book of Matbematical
              <lb/>
            Collections, tells us that this Tract of
              <emph style="sc">Apollonius</emph>
            was divided into
              <lb/>
            two Books; </s>
            <s xml:id="echoid-s978" xml:space="preserve">that the firſt Book contained ſix Problems, and the
              <lb/>
            ſecond three: </s>
            <s xml:id="echoid-s979" xml:space="preserve">now
              <emph style="sc">Snellius'</emph>
            whole work contains but four; </s>
            <s xml:id="echoid-s980" xml:space="preserve">and
              <lb/>
            it ſeemed to me difficult to ſhew how thoſe could contain the ſub-
              <lb/>
            ſtance of nine, and yet the ſix firſt have ſixteen Epitagmas, or ge-
              <lb/>
            neral Caſes, and the three laſt nine. </s>
            <s xml:id="echoid-s981" xml:space="preserve">I firſt, therefore began with
              <lb/>
            inquiring whether, or no, other Problems could not be found,
              <lb/>
            wherein the ſection of an indefinite ſtraight line is propoſed to be
              <lb/>
            effected, “So, that of the ſegments contained between the point of
              <lb/>
            ſection ſought, and given points in the ſaid line, either the ſquare
              <lb/>
            on one of them, or the rectangle contained by two of them, may
              <lb/>
            have a given ratio to the rectangle contained by one of them and a
              <lb/>
            given external line, or to the rectangle contained by two of them;</s>
            <s xml:id="echoid-s982" xml:space="preserve">“
              <lb/>
            as is deſcribed by P
              <emph style="sc">APPUS.</emph>
            </s>
          </p>
          <p>
            <s xml:id="echoid-s983" xml:space="preserve">In this inquiry it ſoon occurred to me, that the three problems
              <lb/>
            which make my firſt, ſecond and fourth, come within the account
              <lb/>
            given by P
              <emph style="sc">APPUS</emph>
            ;</s>
            <s xml:id="echoid-s984" xml:space="preserve">and therefore are properly Problems in Determi-
              <lb/>
            ate Section, to be added to the four given by
              <emph style="sc">Snellius</emph>
            : </s>
            <s xml:id="echoid-s985" xml:space="preserve">and it does
              <lb/>
            not appear to me that more can be found which ſhould. </s>
            <s xml:id="echoid-s986" xml:space="preserve">Hence I
              <lb/>
            concluded, that there were in theſe, ſome, more general than thoſe
              <lb/>
            of
              <emph style="sc">Apollonius</emph>
            , which ought therefore to be divided.</s>
            <s xml:id="echoid-s987" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s988" xml:space="preserve">My next buſineſs was, if poſſible, to find out the order in which
              <lb/>
              <emph style="sc">Apollonius</emph>
            had arranged them: </s>
            <s xml:id="echoid-s989" xml:space="preserve">and here, with reſpect to the
              <lb/>
            firſt Book, I had no other information to guide me, but what is </s>
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