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-s104" xml:space="preserve">
              <pb o="(2)" file="0014" n="14"/>
            and hence the angle FEG is equal to EGH, but FEG is a right one by Con-
              <lb/>
            ſtruction. </s>
            <s xml:id="echoid-s105" xml:space="preserve">Let now HI be drawn from H perpendicular to AB: </s>
            <s xml:id="echoid-s106" xml:space="preserve">then the two
              <lb/>
            triangles EHI and EHG having two angles in one HEI and EIH reſpectively
              <lb/>
            equal to two angles in the other HEG and EGH, and alſo the ſide EH com-
              <lb/>
            mon, by Euc. </s>
            <s xml:id="echoid-s107" xml:space="preserve">I. </s>
            <s xml:id="echoid-s108" xml:space="preserve">26. </s>
            <s xml:id="echoid-s109" xml:space="preserve">HI will be equal to HG, and therefore the circle will
              <lb/>
            touch alſo the other line AB: </s>
            <s xml:id="echoid-s110" xml:space="preserve">and HG or HI equals the given line Z, becauſe
              <lb/>
            EF was made equal to Z, and HG and EF are oppoſite ſides of a paral-
              <lb/>
            lelogram.</s>
            <s xml:id="echoid-s111" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div9" type="section" level="1" n="9">
          <head xml:id="echoid-head14" xml:space="preserve">PROBLEM III.</head>
          <p>
            <s xml:id="echoid-s112" xml:space="preserve">
              <emph style="sc">Having</emph>
            two circles given whoſe centers are A and B, it is required to draw
              <lb/>
            another, whoſe Radius ſhall be equal to a given line Z, which ſhall alſo touch
              <lb/>
            the two given ones.</s>
            <s xml:id="echoid-s113" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s114" xml:space="preserve">
              <emph style="sc">This</emph>
            Problem has various Caſes, according to the various poſition of the
              <lb/>
            given circles, and the various manner of deſcribing the circle required: </s>
            <s xml:id="echoid-s115" xml:space="preserve">but there
              <lb/>
            are ſix principal ones, and to the conditions of theſe all the reſt are ſubject.</s>
            <s xml:id="echoid-s116" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s117" xml:space="preserve">
              <emph style="sc">Case</emph>
            1ſt. </s>
            <s xml:id="echoid-s118" xml:space="preserve">Let the circle to be deſcribed be required to be touched outwardly
              <lb/>
            by the given circles.</s>
            <s xml:id="echoid-s119" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s120" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s121" xml:space="preserve">Then it is neceſſary that 2Z, or the given Diameter, ſhould
              <lb/>
            not be leſs than the ſegment of the line joining the centers of the given circles
              <lb/>
            which is intercepted between their convex circumferences, viz. </s>
            <s xml:id="echoid-s122" xml:space="preserve">not leſs than CD
              <lb/>
            in the Figure belonging to Caſe 1ſt.</s>
            <s xml:id="echoid-s123" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s124" xml:space="preserve">
              <emph style="sc">Case</emph>
            2d. </s>
            <s xml:id="echoid-s125" xml:space="preserve">Let the circle to be deſcribed be required to be touched inwardly by
              <lb/>
            the given circles.</s>
            <s xml:id="echoid-s126" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s127" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s128" xml:space="preserve">Then it’s Diameter muſt not be given leſs than the right line,
              <lb/>
            which drawn through the centers of the given circles, is contained between their
              <lb/>
            concave circumferences; </s>
            <s xml:id="echoid-s129" xml:space="preserve">viz. </s>
            <s xml:id="echoid-s130" xml:space="preserve">not leſs than CD.</s>
            <s xml:id="echoid-s131" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s132" xml:space="preserve">
              <emph style="sc">Case</emph>
            3d. </s>
            <s xml:id="echoid-s133" xml:space="preserve">Let the circle to be deſcribed be required to be touched outwardly
              <lb/>
            by one of the given circles, and inwardly by the other.</s>
            <s xml:id="echoid-s134" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s135" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s136" xml:space="preserve">Then it’s Diameter muſt not be given leſs than the ſegment
              <lb/>
            of the right line, joining the centers of the given circles, which is intercepted
              <lb/>
            between the convex circumference of one and the concave circumference of the
              <lb/>
            other; </s>
            <s xml:id="echoid-s137" xml:space="preserve">viz. </s>
            <s xml:id="echoid-s138" xml:space="preserve">not leſs than CD.</s>
            <s xml:id="echoid-s139" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s140" xml:space="preserve">
              <emph style="sc">Case</emph>
            4th. </s>
            <s xml:id="echoid-s141" xml:space="preserve">Let one of the given circles include the other, and let it be re-
              <lb/>
            quired that the circle to be deſcribed be touched outwardly by them both.</s>
            <s xml:id="echoid-s142" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s143" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s144" xml:space="preserve">Then it’s Diameter muſt not be given greater than the greater
              <lb/>
            ſegment of the right line, joining the centers of the given circles, which is in-
              <lb/>
            tercepted between the concave circumference of one and the convex circumference
              <lb/>
            of the other; </s>
            <s xml:id="echoid-s145" xml:space="preserve">nor leſs than the leſſer ſegment; </s>
            <s xml:id="echoid-s146" xml:space="preserve">viz. </s>
            <s xml:id="echoid-s147" xml:space="preserve">not greater than CD, nor
              <lb/>
            leſs than MN.</s>
            <s xml:id="echoid-s148" xml:space="preserve"/>
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