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

Table of contents

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[81.] LEMMA IV.
Page: 105 ([28])
[82.] LEMMA V.
Page: 106 ([29])
[83.] PROBLEM VII. (Fig. 32, 33, 34, &c.)
Page: 108 ([31])
[84.] PROBLEM I. (Fig. 32 to 45.)
Page: 109 ([32])
[85.] PROBLEM II. (Fig. 46 to 57.)
Page: 113 ([36])
[86.] PROBLEM III.
Page: 116 ([39])
[87.] THE END.
Page: 117 ([40])
[88.] A SYNOPSIS OF ALL THE DATA FOR THE Conſtruction of Triangles, FROM WHICH GEOMETRICAL SOLUTIONS Have hitherto been in Print.
Page: 133
[89.] By JOHN LAWSON, B. D. Rector of Swanscombe, in KENT. ROCHESTER:
Page: 133
[90.] MDCCLXXIII. [Price One Shilling.]
Page: 133
[91.] ADVERTISEMENT.
Page: 135
[92.] AN EXPLANATION OF THE SYMBOLS made uſe of in this SYNOPSIS.
Page: 137
[93.] INDEX OF THE Authors refered to in the SYNOPSIS.
Page: 139 ([i])
[94.] Lately was publiſhed by the ſame Author; [Price Six Shillings in Boards.]
Page: 140 ([ii])
[95.] SYNOPSIS.
Page: 141 ([1])
[96.] Continuation of the Synopsis, Containing ſuch Data as cannot readily be expreſſed by the Symbols before uſed without more words at length.
Page: 149 ([9])
[97.] SYNOPSIS
Page: 152 ([12])
[98.] FINIS.
Page: 156 ([16])
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          <p>
            <s xml:id="echoid-s1549" xml:space="preserve">Here AE is to be made the ſum of the terms of the given ratio, and we are
              <lb/>
            to uſe the IVth
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Problem</emph>
            , whereby O falling between
              <lb/>
            L and E, o will fall beyond U; </s>
            <s xml:id="echoid-s1550" xml:space="preserve">and that it will fall ſhort of I appears from
              <lb/>
            the Iſt
              <emph style="sc">Corollary</emph>
            from the IVth
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Problem</emph>
            .</s>
            <s xml:id="echoid-s1551" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1552" xml:space="preserve">
              <emph style="sc">Case</emph>
            II. </s>
            <s xml:id="echoid-s1553" xml:space="preserve">The given ratio being inæqualitatis minoris, let the point ſought
              <lb/>
            be required between the ſecond aſſigned U and the ſecond in order E, or be-
              <lb/>
            yond the firſt A, which ariſes from the ſame Conſtruction. </s>
            <s xml:id="echoid-s1554" xml:space="preserve">Here AE is to be
              <lb/>
            made the difference of the terms of the given ratio, and we are to uſe the
              <lb/>
            IVth
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Problem</emph>
            , where O being made to fall between U and E,
              <lb/>
            o will fall beyond L, much more beyond A.</s>
            <s xml:id="echoid-s1555" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1556" xml:space="preserve">
              <emph style="sc">Case</emph>
            III. </s>
            <s xml:id="echoid-s1557" xml:space="preserve">The given ratio being inæqualitatis majoris, let the point ſought
              <lb/>
            be required between the ſecond aſſigned U and the ſecond in order E, or be-
              <lb/>
            yond the laſt I, which ariſes from the ſame Conſtruction. </s>
            <s xml:id="echoid-s1558" xml:space="preserve">Here AE is to be
              <lb/>
            made the difference of the terms of the given ratio, and L is to be ſet off the
              <lb/>
            contrary way to what it was in the laſt
              <emph style="sc">Case</emph>
            ; </s>
            <s xml:id="echoid-s1559" xml:space="preserve">and we are to uſe the Iſt
              <emph style="sc">Case</emph>
              <lb/>
            of the IId
              <emph style="sc">Problem</emph>
            , whereby O being made to fall between E and L, or
              <lb/>
            between E and U, according as L or U is neareſt to the point E, o will fall
              <lb/>
            beyond I, as any one will ſee who conſiders the Conſtruction of that
              <emph style="sc">Case</emph>
            with
              <lb/>
            due attention.</s>
            <s xml:id="echoid-s1560" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1561" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            II. </s>
            <s xml:id="echoid-s1562" xml:space="preserve">
              <emph style="sc">Case</emph>
            IV. </s>
            <s xml:id="echoid-s1563" xml:space="preserve">Let the aſſigned points now be the extremes
              <lb/>
            A and U, and let O the point ſought be required now between the firſt
              <lb/>
            aſſigned A and the next to it E, or, which is effected by the ſame Conſtruction,
              <lb/>
            between the ſecond aſſigned U and the next to it I. </s>
            <s xml:id="echoid-s1564" xml:space="preserve">Here AE is to be made
              <lb/>
            the ſum of the terms of the given ratio, and the IVth
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Pro-</emph>
              <lb/>
              <emph style="sc">BLEM</emph>
            is to be uſed, ſo that of the three points L, E, U, O being made to fall
              <lb/>
            beyond L one of the extremes, and o within U the other extreme, it will appear
              <lb/>
            from the Iſt
              <emph style="sc">Corollary</emph>
            from the IVth
              <emph style="sc">Case</emph>
            of the ſaid
              <emph style="sc">Problem</emph>
            that O
              <lb/>
            will fall between A and E, and o between U and I.</s>
            <s xml:id="echoid-s1565" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1566" xml:space="preserve">
              <emph style="sc">Case</emph>
            V. </s>
            <s xml:id="echoid-s1567" xml:space="preserve">The given ratio being inæqualitatis majoris, let the point ſought be
              <lb/>
            required between the ſecond and third in order, viz. </s>
            <s xml:id="echoid-s1568" xml:space="preserve">between E and I. </s>
            <s xml:id="echoid-s1569" xml:space="preserve">Here
              <lb/>
            AE muſt be the difference of the terms of the given ratio, and L ſet off to-
              <lb/>
            wards I, and the IId
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Problem</emph>
            uſed, and then O, as like-
              <lb/>
            wiſe o, will fall between E and I, if the
              <emph style="sc">Problem</emph>
            be poſſible.</s>
            <s xml:id="echoid-s1570" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1571" xml:space="preserve">As to the
              <emph style="sc">Determination</emph>
            , ſee
              <emph style="sc">Lemma</emph>
            VII. </s>
            <s xml:id="echoid-s1572" xml:space="preserve">following.</s>
            <s xml:id="echoid-s1573" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1574" xml:space="preserve">
              <emph style="sc">Case</emph>
            VI. </s>
            <s xml:id="echoid-s1575" xml:space="preserve">The given ratio being inæqualitatis majoris, let the point ſought
              <lb/>
            be required beyond the laſt aſſigned, that is the laſt in order, U. </s>
            <s xml:id="echoid-s1576" xml:space="preserve">Here AE
              <lb/>
            muſt be the difference of the terms of the given ratio, [and L muſt </s>
          </p>
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