Theodosius <Bithynius>; Clavius, Christoph, Theodosii Tripolitae Sphaericorum libri tres

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        <div xml:id="echoid-div529" type="section" level="1" n="254">
          <p style="it">
            <s xml:id="echoid-s7451" xml:space="preserve">
              <pb o="186" file="198" n="198" rhead=""/>
            ferrentur ex chordis omnium arcuũ maiorũ, quàm grad. </s>
            <s xml:id="echoid-s7452" xml:space="preserve">120. </s>
            <s xml:id="echoid-s7453" xml:space="preserve">reliquæ fierent chordæ
              <lb/>
            mnium arcuum à principio ſemicirculi vſque ad arcum grad. </s>
            <s xml:id="echoid-s7454" xml:space="preserve">60. </s>
            <s xml:id="echoid-s7455" xml:space="preserve">Deniq; </s>
            <s xml:id="echoid-s7456" xml:space="preserve">ſi chordæ om
              <lb/>
            nium arcuum à principio ſemicirculi vſque ad arcum grad. </s>
            <s xml:id="echoid-s7457" xml:space="preserve">120. </s>
            <s xml:id="echoid-s7458" xml:space="preserve">inuentæ eſſent, & </s>
            <s xml:id="echoid-s7459" xml:space="preserve">
              <lb/>
            chordæ omnium arcuum minorum, quàm 60. </s>
            <s xml:id="echoid-s7460" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s7461" xml:space="preserve">chordis omnium arcuum maiorum
              <lb/>
            quàm grad. </s>
            <s xml:id="echoid-s7462" xml:space="preserve">60. </s>
            <s xml:id="echoid-s7463" xml:space="preserve">adijcerentur, componerentur chordæ omnium arcuum maiorum,
              <lb/>
            quàm grad. </s>
            <s xml:id="echoid-s7464" xml:space="preserve">120.</s>
            <s xml:id="echoid-s7465" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7466" xml:space="preserve">PORRO ſi chordæ omnium arcuum ſemicir culi in tabulam redigantur, & </s>
            <s xml:id="echoid-s7467" xml:space="preserve">chor
              <lb/>
              <note position="left" xlink:label="note-198-01" xlink:href="note-198-01a" xml:space="preserve">Qua rõne
                <lb/>
              ſinus oium
                <lb/>
              a@cuum ex
                <lb/>
              cho@dis in-
                <lb/>
              ueniantur.</note>
            dæ omnium arcuum, qui ſecontinuè Minutis 2. </s>
            <s xml:id="echoid-s7468" xml:space="preserve">excedunt, ſecentur bifariam, inuenti
              <lb/>
            erunt ſinus omnium arcuum per ſingula Minuta progredientiũ, vt ex defin. </s>
            <s xml:id="echoid-s7469" xml:space="preserve">3. </s>
            <s xml:id="echoid-s7470" xml:space="preserve">conſtat.</s>
            <s xml:id="echoid-s7471" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7472" xml:space="preserve">HINC conſtat, rectius feciſſe recentiores, qui ſinuum tabulã confecerunt, quàm
              <lb/>
            Ptolemæum & </s>
            <s xml:id="echoid-s7473" xml:space="preserve">veteres, qui chordas in tabulam redegerunt. </s>
            <s xml:id="echoid-s7474" xml:space="preserve">Pro ſinubus enim ſatis eſt,
              <lb/>
              <note position="left" xlink:label="note-198-02" xlink:href="note-198-02a" xml:space="preserve">Rectiꝰ face
                <lb/>
              re eos, ꝗ @a
                <lb/>
              bulã ſinuũ
                <lb/>
              cõſtruunt,
                <lb/>
              quàm qu@
                <lb/>
              chordarũ.</note>
            ſi Quadrans circuli per ſingula Minuta extendatur: </s>
            <s xml:id="echoid-s7475" xml:space="preserve">at pro chordis neceſſe eſt totum
              <lb/>
            ſemicir culum per Minuia ſingula extendere: </s>
            <s xml:id="echoid-s7476" xml:space="preserve">ita vt chordarum tabula ſit duplo maior
              <lb/>
            quàm tabula ſinuum. </s>
            <s xml:id="echoid-s7477" xml:space="preserve">Taceo, multo expeditiorem, breuiorem, fac@ lioremq́; </s>
            <s xml:id="echoid-s7478" xml:space="preserve">eſſe ſinuum,
              <lb/>
            vjum in rebus Aſtronomicis, & </s>
            <s xml:id="echoid-s7479" xml:space="preserve">Geometricis, quàm chordarum: </s>
            <s xml:id="echoid-s7480" xml:space="preserve">vt ijs eſt manifeſtum,
              <lb/>
            qui ſeſe in bu@@ ſmodi vſu exercue; </s>
            <s xml:id="echoid-s7481" xml:space="preserve">unt aliquando.</s>
            <s xml:id="echoid-s7482" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7483" xml:space="preserve">QVEMADMODVM autem ſinuũ differentiæ ſenſim decreſcunt à principie
              <lb/>
              <note position="left" xlink:label="note-198-03" xlink:href="note-198-03a" xml:space="preserve">Differẽ@iæ
                <lb/>
              chordarũ à
                <lb/>
              prin c@pio
                <lb/>
              ſemici@culi
                <lb/>
              vſq; ad e@@s
                <lb/>
              finé ſen ſim
                <lb/>
              decreſcunt@
                <lb/>
              ita vt chor-
                <lb/>
              dæ minorũ
                <lb/>
              arcuũmaio
                <lb/>
              res habeant
                <lb/>
              differentias
                <lb/>
              quàm chor
                <lb/>
              dæ arcuum
                <lb/>
              maiorũ dũ
                <lb/>
              modo arcꝰ
                <lb/>
              habeãt dif-
                <lb/>
              ferentias æ
                <lb/>
              quales.</note>
            quad@antis vſque ad eius finem; </s>
            <s xml:id="echoid-s7484" xml:space="preserve">ita vt, poſitis pluribus arcubus æqualiter ſeſe exc@-
              <lb/>
            dentibus, minorum ſinus habeani maiores differentias, quàm ſinus maiorum, vt in co-
              <lb/>
            roll. </s>
            <s xml:id="echoid-s7485" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s7486" xml:space="preserve">1 oſtendimus@ ita quoque chordarum differentiæ paulatim decreſcunt à
              <lb/>
            principio ſemicirculiad eius finem vſq. </s>
            <s xml:id="echoid-s7487" xml:space="preserve">Nam poſitis pluribus arcubus, quorum æqua-
              <lb/>
            les ſint differentiæ, minorum chordæ maiores habent differentias, quàm chordæ ma-
              <lb/>
            iorum. </s>
            <s xml:id="echoid-s7488" xml:space="preserve">quod ita demonſtrabimus. </s>
            <s xml:id="echoid-s7489" xml:space="preserve">Sint in ſemicirculo
              <emph style="sc">ABCDe</emph>
            , arcus AB, AC,
              <lb/>
            AD, quorum differentiæ
              <emph style="sc">B</emph>
            C, CD, æquales ſint,
              <lb/>
              <figure xlink:label="fig-198-01" xlink:href="fig-198-01a" number="147">
                <image file="198-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/198-01"/>
              </figure>
            chordæ autem eorundem AB, AC, AD: </s>
            <s xml:id="echoid-s7490" xml:space="preserve">abſeinda-
              <lb/>
            turq; </s>
            <s xml:id="echoid-s7491" xml:space="preserve">recta
              <emph style="sc">Af</emph>
            , chordæ AB, æqualis; </s>
            <s xml:id="echoid-s7492" xml:space="preserve">& </s>
            <s xml:id="echoid-s7493" xml:space="preserve">recta AG,
              <lb/>
            chordæ AC, æqualis. </s>
            <s xml:id="echoid-s7494" xml:space="preserve">Dico FC, differentiam inter
              <lb/>
            chordas AB, AC, maiorem eſſe, quàm GD, diffe-
              <lb/>
            rentiam inter chordas AC, AD. </s>
            <s xml:id="echoid-s7495" xml:space="preserve">Abſeiſſa enim re-
              <lb/>
            cta AH, æquali ipſi AF, vel ipſi AB, iunctiſque re-
              <lb/>
            ctis BC, CD, CG, CH: </s>
            <s xml:id="echoid-s7496" xml:space="preserve">quoniam latera BA, AC,
              <lb/>
            lateribus HA, AC, æqualia ſunt, angulosq́; </s>
            <s xml:id="echoid-s7497" xml:space="preserve">con-
              <lb/>
            tinent æquales ad A, propter æquales arcus BC, CD; </s>
            <s xml:id="echoid-s7498" xml:space="preserve">erunt baſes BC, CD; </s>
            <s xml:id="echoid-s7499" xml:space="preserve">æqua-
              <lb/>
              <note position="left" xlink:label="note-198-04" xlink:href="note-198-04a" xml:space="preserve">27. tertij.</note>
            les. </s>
            <s xml:id="echoid-s7500" xml:space="preserve">Eſt autem recta BC, rectæ CD, æqualis, ob æquales arcus eoſdem BC, CD. </s>
            <s xml:id="echoid-s7501" xml:space="preserve">Igi-
              <lb/>
              <note position="left" xlink:label="note-198-05" xlink:href="note-198-05a" xml:space="preserve">29.tertij.</note>
            tur & </s>
            <s xml:id="echoid-s7502" xml:space="preserve">recta CH, rectæ CD, æqualis erit. </s>
            <s xml:id="echoid-s7503" xml:space="preserve">Anguli ergo CHD, CDH, æquales quo-
              <lb/>
              <note position="left" xlink:label="note-198-06" xlink:href="note-198-06a" xml:space="preserve">5. primi.</note>
            que erunt: </s>
            <s xml:id="echoid-s7504" xml:space="preserve">qui cum ſint duobus rectis minores; </s>
            <s xml:id="echoid-s7505" xml:space="preserve">erit vterque eorum acutus. </s>
            <s xml:id="echoid-s7506" xml:space="preserve">Eodem pa-
              <lb/>
              <note position="left" xlink:label="note-198-07" xlink:href="note-198-07a" xml:space="preserve">17.primi.</note>
            cto erit vterque angulorum ACG,
              <emph style="sc">AGc</emph>
            , acutus: </s>
            <s xml:id="echoid-s7507" xml:space="preserve">propterea quod inter ſe etiam
              <lb/>
            æqu@les ſunt ob æquales rectas AC, AG. </s>
            <s xml:id="echoid-s7508" xml:space="preserve">Quia igitur in triangulo CGH, anguli
              <lb/>
              <note position="left" xlink:label="note-198-08" xlink:href="note-198-08a" xml:space="preserve">5.primi.</note>
            ad G, H, acuti ſunt; </s>
            <s xml:id="echoid-s7509" xml:space="preserve">fit, vt CI, ducta ad DH, perpendicularis cadat intra triangu-
              <lb/>
            lum in r@ctam GH, vt in ſeholio propoſ. </s>
            <s xml:id="echoid-s7510" xml:space="preserve">13. </s>
            <s xml:id="echoid-s7511" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7512" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7513" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s7514" xml:space="preserve">demonſtrauimus. </s>
            <s xml:id="echoid-s7515" xml:space="preserve">Itaque quia
              <lb/>
            duo anguli CHI, CIH, trianguli CHI, duobus angulis CDI, CID, æquales
              <lb/>
            ſunt, ſuntq; </s>
            <s xml:id="echoid-s7516" xml:space="preserve">duo latera CH, CD, æqualibus rectis angulis oppoſita æqualia, vel
              <lb/>
            certe latus CI, commune eſt; </s>
            <s xml:id="echoid-s7517" xml:space="preserve">erunt quoque latera IH, ID, æqualia. </s>
            <s xml:id="echoid-s7518" xml:space="preserve">Cum ergo ID,
              <lb/>
              <note position="left" xlink:label="note-198-09" xlink:href="note-198-09a" xml:space="preserve">26.primi.</note>
            maior ſit, quàm GD, erit quoq; </s>
            <s xml:id="echoid-s7519" xml:space="preserve">HI, & </s>
            <s xml:id="echoid-s7520" xml:space="preserve">à fortiori
              <emph style="sc">Hg</emph>
            , maior, quam GD. </s>
            <s xml:id="echoid-s7521" xml:space="preserve">Eſt au-
              <lb/>
            tem HG, ipſi FC, æqualis, propterea quòd & </s>
            <s xml:id="echoid-s7522" xml:space="preserve">rectæ AC, AG, inter ſe, & </s>
            <s xml:id="echoid-s7523" xml:space="preserve">rectæ
              <emph style="sc">Af</emph>
            ,
              <lb/>
            AH, inter ſe æquales ſunt. </s>
            <s xml:id="echoid-s7524" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s7525" xml:space="preserve">FC, differentia chordarum AB, AC, maior
              <lb/>
            erit, quàm GD, differentia chordarum AC, AD. </s>
            <s xml:id="echoid-s7526" xml:space="preserve">Quod eſt propoſitum.</s>
            <s xml:id="echoid-s7527" xml:space="preserve"/>
          </p>
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