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

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        <div xml:id="echoid-div566" type="section" level="1" n="263">
          <p style="it">
            <s xml:id="echoid-s8425" xml:space="preserve">
              <pb o="198" file="210" n="210" rhead=""/>
            grad. </s>
            <s xml:id="echoid-s8426" xml:space="preserve">15. </s>
            <s xml:id="echoid-s8427" xml:space="preserve">qui ſemiſsis eſt complementi dati arcus grad. </s>
            <s xml:id="echoid-s8428" xml:space="preserve">60. </s>
            <s xml:id="echoid-s8429" xml:space="preserve">Remanebit enim numeras
              <lb/>
            20000006 pro ſecante dati arcus grad. </s>
            <s xml:id="echoid-s8430" xml:space="preserve">60.</s>
            <s xml:id="echoid-s8431" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8432" xml:space="preserve">HAEC, quæ hoc ſcholio tradita à nobis ſunt, vera ſunt, ſi ſinus exquiſite inuen@
              <lb/>
              <note position="left" xlink:label="note-210-01" xlink:href="note-210-01a" xml:space="preserve">Tangétes, &
                <lb/>
              Secátes ma
                <lb/>
              gis eſſe ac
                <lb/>
              curatas, per
                <lb/>
              ſinus inué
                <lb/>
              tas, q̃ per
                <lb/>
              additioné,
                <lb/>
              ſubtractio -
                <lb/>
              néue, vt in
                <lb/>
              hoc ſcholio
                <lb/>
              traditú eſt.</note>
            ti fuerint: </s>
            <s xml:id="echoid-s8433" xml:space="preserve">ſed quia non omnes ſinus accurate ſunt cogniti, maxime ſinus arcus grad.
              <lb/>
            </s>
            <s xml:id="echoid-s8434" xml:space="preserve">1. </s>
            <s xml:id="echoid-s8435" xml:space="preserve">& </s>
            <s xml:id="echoid-s8436" xml:space="preserve">alij ex hoc dependentes, quales ſunt ſinus arcuum per ſingula minuta extenſo-
              <lb/>
            rum; </s>
            <s xml:id="echoid-s8437" xml:space="preserve">fit vt neq; </s>
            <s xml:id="echoid-s8438" xml:space="preserve">tangentes, neq; </s>
            <s xml:id="echoid-s8439" xml:space="preserve">ſecantes inuẽtæ per hoſce ſinus ſint admodũ accuratæ. </s>
            <s xml:id="echoid-s8440" xml:space="preserve">
              <lb/>
            Quare ſi ex inuentis quibuſdam aliæ per ſolam additionem, ſubtractionem ve inqui-
              <lb/>
            rantur vt hoc ſcholio docuimus, non parum different ab eiſdem, ſi per ſinus inueſtiga
              <lb/>
            rentur. </s>
            <s xml:id="echoid-s8441" xml:space="preserve">Nam tangentes & </s>
            <s xml:id="echoid-s8442" xml:space="preserve">ſecantes per ſinus inuentæ ex vno ſolo principio non omni
              <lb/>
            ex parte vero, nempe ex ſinubus, gignuntur: </s>
            <s xml:id="echoid-s8443" xml:space="preserve">at eædem per ſolam additionem, ſubtra,
              <lb/>
            ctionem ve procreatæ oriuntur ex pluribus falſis principijs, nimirum ex ſinubus pr
              <lb/>
            mum, deinde vero etiá ex tangentibus, & </s>
            <s xml:id="echoid-s8444" xml:space="preserve">ſecátibus per ſinus inuentis, quæ accuratæ
              <lb/>
            eſſe
              <unsure/>
            non poſſunt, vt diximus. </s>
            <s xml:id="echoid-s8445" xml:space="preserve">Magis exquiſite ergo cognoſcentur huiuſmodi lineæ per
              <lb/>
            ſinus, vt propoſ. </s>
            <s xml:id="echoid-s8446" xml:space="preserve">18. </s>
            <s xml:id="echoid-s8447" xml:space="preserve">eiuſq́; </s>
            <s xml:id="echoid-s8448" xml:space="preserve">ſcholio traditum eſt. </s>
            <s xml:id="echoid-s8449" xml:space="preserve">Hac ratione & </s>
            <s xml:id="echoid-s8450" xml:space="preserve">tabulam Tangentium,
              <lb/>
            & </s>
            <s xml:id="echoid-s8451" xml:space="preserve">tabulam Secantium breui ſupputabimus. </s>
            <s xml:id="echoid-s8452" xml:space="preserve">Non paruos enim errores in aliorum ta-
              <lb/>
            bulis deprehendimus; </s>
            <s xml:id="echoid-s8453" xml:space="preserve">vt tutò illis fid ere non poſsimus; </s>
            <s xml:id="echoid-s8454" xml:space="preserve">propterea quòd multas tangen-
              <lb/>
            tes, & </s>
            <s xml:id="echoid-s8455" xml:space="preserve">ſecátes vel per partem proportionalem, vel per ſolam additionem aut ſubtra-
              <lb/>
            ctionem inueſtigarunt, non autem omnes per ſinus. </s>
            <s xml:id="echoid-s8456" xml:space="preserve">Subiungemus tamen paulo infra
              <lb/>
            aliorum tabulas, donec per tempus nouas conſtruere licebit.</s>
            <s xml:id="echoid-s8457" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div571" type="section" level="1" n="264">
          <head xml:id="echoid-head291" xml:space="preserve">THEOR. 13. PROPOS. 21.</head>
          <p>
            <s xml:id="echoid-s8458" xml:space="preserve">TANGENS cuiuſuis arcus eſt ad tangen-
              <lb/>
              <note position="left" xlink:label="note-210-02" xlink:href="note-210-02a" xml:space="preserve">Tangentes
                <lb/>
              duorum at
                <lb/>
              cuú quotú-
                <lb/>
              libet sút re
                <lb/>
              ciprocè {pro}-
                <lb/>
              portionales
                <lb/>
              cũ tangen-
                <lb/>
              tibꝰ comple
                <lb/>
              métorú ar-
                <lb/>
              cuú eoiun-
                <lb/>
              dem.</note>
            tem alterius arcus cuiuſlibet, vt tangens comple-
              <lb/>
            menti poſterioris arcus ad tangétem complemen-
              <lb/>
            ti prioris.</s>
            <s xml:id="echoid-s8459" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8460" xml:space="preserve">IN quadrante ABC, arcus CD, tangens ſit CE, & </s>
            <s xml:id="echoid-s8461" xml:space="preserve">ſecans AE: </s>
            <s xml:id="echoid-s8462" xml:space="preserve">Item ar-
              <lb/>
            cus CF, tangens ſit CG, & </s>
            <s xml:id="echoid-s8463" xml:space="preserve">ſecans AG: </s>
            <s xml:id="echoid-s8464" xml:space="preserve">Ducta autem recta BH, circulum tan
              <lb/>
            gente, & </s>
            <s xml:id="echoid-s8465" xml:space="preserve">vtrique ſecanti AE, AG, occurrente in I, H; </s>
            <s xml:id="echoid-s8466" xml:space="preserve">erit BI, tangens com-
              <lb/>
            plementi arcus CD; </s>
            <s xml:id="echoid-s8467" xml:space="preserve">& </s>
            <s xml:id="echoid-s8468" xml:space="preserve">BH, tangens comple-
              <lb/>
              <figure xlink:label="fig-210-01" xlink:href="fig-210-01a" number="154">
                <image file="210-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/210-01"/>
              </figure>
            menti arcus CF. </s>
            <s xml:id="echoid-s8469" xml:space="preserve">Dico ita eſſe CE, tangentem
              <lb/>
            arcus CD, ad CG, tangentem arcus CF, vt eſt
              <lb/>
            BH, tangens complementi poſterioris arcus
              <lb/>
            CF, ad BI, tangentem complementi arcus prio
              <lb/>
            ris CD. </s>
            <s xml:id="echoid-s8470" xml:space="preserve">Cum enim ſinus totus ſit medius pro-
              <lb/>
              <note position="left" xlink:label="note-210-03" xlink:href="note-210-03a" xml:space="preserve">18. huius</note>
            portionalis tam inter CE, tangenté arcus CD,
              <lb/>
            & </s>
            <s xml:id="echoid-s8471" xml:space="preserve">BI, tangentem complementi arcus eiuſdem
              <lb/>
            CD, quàm inter CG, tangentem arcus CF, & </s>
            <s xml:id="echoid-s8472" xml:space="preserve">
              <lb/>
            BH, tangentem complementi arcus eiuſdem
              <lb/>
            CF; </s>
            <s xml:id="echoid-s8473" xml:space="preserve">erit tam rectangulum ſub CE, BI, quam re-
              <lb/>
            ctangulum ſub CG, BH, quadrato ſinus totius æquale: </s>
            <s xml:id="echoid-s8474" xml:space="preserve">ac proinde rectangu-
              <lb/>
              <note position="left" xlink:label="note-210-04" xlink:href="note-210-04a" xml:space="preserve">17. ſexti.</note>
            lum ſub CE, BI, rectangulo ſub CG, BH, æquale erit. </s>
            <s xml:id="echoid-s8475" xml:space="preserve">Quare erit, vt CE,
              <lb/>
            prima ad CG, ſecundam, ita BH, tertia ad BI, quartam; </s>
            <s xml:id="echoid-s8476" xml:space="preserve">nempe vt CE, tan-
              <lb/>
              <note position="left" xlink:label="note-210-05" xlink:href="note-210-05a" xml:space="preserve">16. ſexti.</note>
            </s>
          </p>
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