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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        <div xml:id="echoid-div14" type="section" level="1" n="14">
          <head xml:id="echoid-head19" xml:space="preserve">PROBLEM VI.</head>
          <p>
            <s xml:id="echoid-s230" xml:space="preserve">
              <emph style="sc">Having</emph>
            a right line given BC, and alſo a circle whoſe center is A, it is re-
              <lb/>
            quired to draw another circle, whoſe Radius ſhall be equal to a given right line
              <lb/>
            Z, and which ſhall touch both the given line and alſo the given circle.</s>
            <s xml:id="echoid-s231" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s232" xml:space="preserve">
              <emph style="sc">This</emph>
            Problem has alſo three caſes, each of which is ſubject to a Limitation.</s>
            <s xml:id="echoid-s233" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s234" xml:space="preserve">
              <emph style="sc">Case</emph>
            Iſt, Let the circle to be deſcribed be required to be touched outwardly
              <lb/>
            by the given circle.</s>
            <s xml:id="echoid-s235" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s236" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s237" xml:space="preserve">Then the Diameter of the circle required muſt not be given
              <lb/>
            leſs than the ſegment of a line, drawn from the center of the given circle, per-
              <lb/>
            pendicular to the given line, which is intercepted between the ſaid line and the
              <lb/>
            convex circumference; </s>
            <s xml:id="echoid-s238" xml:space="preserve">viz. </s>
            <s xml:id="echoid-s239" xml:space="preserve">not leſs than BD.</s>
            <s xml:id="echoid-s240" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s241" xml:space="preserve">
              <emph style="sc">Case</emph>
            2d. </s>
            <s xml:id="echoid-s242" xml:space="preserve">Let the circle to be deſcribed be required to be touched inwardly by
              <lb/>
            the given circle.</s>
            <s xml:id="echoid-s243" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s244" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s245" xml:space="preserve">Then the given line muſt not be in the given circle, neither
              <lb/>
            muſt the Diameter of the circle required be given leſs than that portion of the
              <lb/>
            perpendicular, drawn from the center of the given circle to the given line, which
              <lb/>
            is intercepted between the ſaid line and the concave circumference; </s>
            <s xml:id="echoid-s246" xml:space="preserve">viz. </s>
            <s xml:id="echoid-s247" xml:space="preserve">not leſs
              <lb/>
            than BD.</s>
            <s xml:id="echoid-s248" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s249" xml:space="preserve">
              <emph style="sc">Case</emph>
            3d. </s>
            <s xml:id="echoid-s250" xml:space="preserve">Let the circle to be deſcribed be required to be both touched and
              <lb/>
            included in the given circle.</s>
            <s xml:id="echoid-s251" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s252" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s253" xml:space="preserve">Then the given right line muſt be in the given circle, and
              <lb/>
            when a Diameter of this given circle is drawn cutting the given line at right an-
              <lb/>
            gles, the Diameter of the circle required muſt not be given greater than the
              <lb/>
            greater ſegment ; </s>
            <s xml:id="echoid-s254" xml:space="preserve">viz. </s>
            <s xml:id="echoid-s255" xml:space="preserve">not greater than BD.</s>
            <s xml:id="echoid-s256" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div15" type="section" level="1" n="15">
          <head xml:id="echoid-head20" xml:space="preserve">
            <emph style="sc">The general</emph>
            <emph style="sc">Solution</emph>
          .</head>
          <p>
            <s xml:id="echoid-s257" xml:space="preserve">
              <emph style="sc">From</emph>
            A draw AB perpendicular to BC, cutting the given circumference in D;
              <lb/>
            </s>
            <s xml:id="echoid-s258" xml:space="preserve">and in this perpendicular let BG and DF be taken each equal to the given line
              <lb/>
            Z; </s>
            <s xml:id="echoid-s259" xml:space="preserve">and through G draw GE parallel to BC; </s>
            <s xml:id="echoid-s260" xml:space="preserve">and with center A and diſtance
              <lb/>
            AF let an arc be ſtruck, which by the Limitations will neceſſarily either touch
              <lb/>
            or cut GE; </s>
            <s xml:id="echoid-s261" xml:space="preserve">let the point of concourſe be E, let AE be joined, and, if neceſſary,
              <lb/>
            be produced to meet the given circumference in H; </s>
            <s xml:id="echoid-s262" xml:space="preserve">then with E center and
              <lb/>
            EH diſtance deſcribe a circle, and I ſay it will be the required circle; </s>
            <s xml:id="echoid-s263" xml:space="preserve">it is evi-
              <lb/>
            dent it will touch the given circle: </s>
            <s xml:id="echoid-s264" xml:space="preserve">and becauſe AD and AH are equal, as alſo
              <lb/>
            AF and AE, therefore DF (which was made equal to Z) will be equal to HE: </s>
            <s xml:id="echoid-s265" xml:space="preserve">
              <lb/>
            let now EC be drawn perpendicular to BC, then GBCE will be a Parallelogram,
              <lb/>
            and EC will be equal to GB, which was alſo made equal to Z: </s>
            <s xml:id="echoid-s266" xml:space="preserve">hence the
              <lb/>
            circle will alſo touch the given line BC, becauſe the angle ECB is a right
              <lb/>
            one.</s>
            <s xml:id="echoid-s267" xml:space="preserve"/>
          </p>
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