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

Table of handwritten notes

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            <s xml:id="echoid-s7638" xml:space="preserve">
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            ducta recta AD, circulum tangat, recta autem CD, circulum ſecet, conueniens cum
              <lb/>
            AD, in D, (conueniet enim neceſſario, propterea quòd duo anguli
              <emph style="sc">Ca</emph>
            C, DCA,
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            duobus rectis ſunt minores; </s>
            <s xml:id="echoid-s7639" xml:space="preserve">cum ille rectus ſit, hic autem
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            recto minor, propter arcum
              <emph style="sc">A</emph>
            B, quadrante minorem.)
              <lb/>
            </s>
            <s xml:id="echoid-s7640" xml:space="preserve">dicetur AD, Tangens arcus
              <emph style="sc">A</emph>
            B, at CD, Secans eiuſdẽ
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            arcus. </s>
            <s xml:id="echoid-s7641" xml:space="preserve">Tangentem vocant nonnulli Adſcriptam, quòd
              <lb/>
              <note position="right" xlink:label="note-201-01" xlink:href="note-201-01a" xml:space="preserve">Linea ad-
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              ſcripta, &
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              Hypotenu-
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              ſa quid.</note>
            circulo quodãmodo adſcribatur; </s>
            <s xml:id="echoid-s7642" xml:space="preserve">Secantem vero, Hypo-
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            tenuſam, propterea quòd in triangulo rectãgulo ACD,
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            (angulus enim
              <emph style="sc">A</emph>
            , apud contactum rectus eſt) angulum
              <lb/>
            rectum ſubtendit: </s>
            <s xml:id="echoid-s7643" xml:space="preserve">Semidiametrum denique
              <emph style="sc">A</emph>
            C, ſiue ſi-
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              <note position="right" xlink:label="note-201-02" xlink:href="note-201-02a" xml:space="preserve">18. tertij.</note>
            num totum, dicunt baſem eiuſdem trianguli.</s>
            <s xml:id="echoid-s7644" xml:space="preserve"/>
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            <s xml:id="echoid-s7645" xml:space="preserve">
              <emph style="sc">QVem</emph>
              <emph style="sc">ADm</emph>
              <emph style="sc">ODVm</emph>
            autemin omni triangulo
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              <note position="right" xlink:label="note-201-03" xlink:href="note-201-03a" xml:space="preserve">Si in trian-
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              gu’o rectan
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              gulo alteru
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              trum late-
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              rum circa
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              angulũ re-
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              ctum pona
                <lb/>
              tur ſinus to
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              tus, erit al-
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              terum latus
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              circa angu-
                <lb/>
              lum rectú
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              tangens an
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              gulĩ acutiſi
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              bi oppoſiti,
                <lb/>
              & latus re-
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              cto angulo
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              oppoſitum
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              eiuſdem ſe
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              cans.</note>
            rectangulo, ſilatus recto angulo oppoſitum ponatur ſinus
              <lb/>
            totus, reliqua duo latera ſunt ſinus recti reliquorum angulorum acutorum, quibus
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            opponuntur; </s>
            <s xml:id="echoid-s7646" xml:space="preserve">Item vtrumuis reliquorum laterum eſt ſinus complementi anguli ſibi
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            adiacentis, vt in definitionibus ſinuum traditum eſt: </s>
            <s xml:id="echoid-s7647" xml:space="preserve">ita quoque ſi alterutrum late-
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            rum circa angulum rectum ſtatuatur ſinus totus, erit alterum latus circa angulum
              <lb/>
            rectum Tangens anguli acuti ſibi oppoſiti, latus vero angulo recto oppoſitum Secans
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            eiuſdem anguli. </s>
            <s xml:id="echoid-s7648" xml:space="preserve">Vt in triangulo rectangulo ACD, latus CA, eſt ſinus totus, nempe
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            ſemidiameter circuli AB: </s>
            <s xml:id="echoid-s7649" xml:space="preserve">at
              <emph style="sc">A</emph>
            D, tangens anguli C, vel arcus
              <emph style="sc">Ab</emph>
            , & </s>
            <s xml:id="echoid-s7650" xml:space="preserve">CD, eiuſdem
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            ſecans. </s>
            <s xml:id="echoid-s7651" xml:space="preserve">Eodem pacto, ſt DA, ſtatuatur ſinus totus, erit AC, tangens anguli D, & </s>
            <s xml:id="echoid-s7652" xml:space="preserve">
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            DC, eiuſdem ſecans.</s>
            <s xml:id="echoid-s7653" xml:space="preserve"/>
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            <s xml:id="echoid-s7654" xml:space="preserve">ETSI autem diximus, tangentem, & </s>
            <s xml:id="echoid-s7655" xml:space="preserve">ſecantem ſumi reſpectu arcus quadrante
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            minoris, tamen eadem tangens, & </s>
            <s xml:id="echoid-s7656" xml:space="preserve">ſecans referri ſolet ad arcum etiam, qui cum illo
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            ſemicirculum complet: </s>
            <s xml:id="echoid-s7657" xml:space="preserve">adeo vt duo arcus ſemicirculum conficientes, vel duo anguli
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            duobus rectis æquales, vnam eandemq; </s>
            <s xml:id="echoid-s7658" xml:space="preserve">tangentem, atq; </s>
            <s xml:id="echoid-s7659" xml:space="preserve">ſ@cantem habeant: </s>
            <s xml:id="echoid-s7660" xml:space="preserve">quemad-
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            modum & </s>
            <s xml:id="echoid-s7661" xml:space="preserve">eundem ſinum rectum habent, vt in tractatione ſinuum tradidimus: </s>
            <s xml:id="echoid-s7662" xml:space="preserve">adeo
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              <note position="right" xlink:label="note-201-04" xlink:href="note-201-04a" xml:space="preserve">Duo arcus
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              ſem icircu-
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              lú cóficiéte
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              vel duo än
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              guli duobꝰ
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              rectis æqua
                <lb/>
              les habent
                <lb/>
              eandé tan-
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              gentem &
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              ſecantem.</note>
            vt ſi quæratur tangens & </s>
            <s xml:id="echoid-s7663" xml:space="preserve">ſecans alicuius arcus quadrante maioris, ſumenda ſit tan
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            gens, & </s>
            <s xml:id="echoid-s7664" xml:space="preserve">ſecans arcus quadrante minoris, qui cum illo ſemicireulum complet.</s>
            <s xml:id="echoid-s7665" xml:space="preserve"/>
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            <s xml:id="echoid-s7666" xml:space="preserve">PORRO qua ratione Tangentes, & </s>
            <s xml:id="echoid-s7667" xml:space="preserve">Secantes omnium arcuum quadrantis
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            reddantur cognitæ in partibus ſinus totius, ac proinde qua via tabula Tangentium,
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            tabula item Secantium componenda ſit, ſequentibus propoſitionibus, quæ ad line{as}
              <lb/>
            Tangentes, ac Secantes ſpectant, planum fiet.</s>
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        <div xml:id="echoid-div541" type="section" level="1" n="256">
          <head xml:id="echoid-head283" xml:space="preserve">THEOR. .9. PROPOS. 17.</head>
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            <s xml:id="echoid-s7669" xml:space="preserve">TANGENS dimidij quadrantis ſinui toti
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              <note position="right" xlink:label="note-201-05" xlink:href="note-201-05a" xml:space="preserve">Tangentes
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              quomodo
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              ſe habeant
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              cũ ſinu to-
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              to com pa-
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              ratæ.</note>
            æqualis eſt: </s>
            <s xml:id="echoid-s7670" xml:space="preserve">Tangens autem arcus maioris dimidi
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            o
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            quadrantis maior eſt ſinu toto: </s>
            <s xml:id="echoid-s7671" xml:space="preserve">Et Tangens mino-
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            ris arcus minor eſt. </s>
            <s xml:id="echoid-s7672" xml:space="preserve">Secans denique dimidij qua-
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            drantis dupla eſt ſinus recti eiuſdem dimidij.</s>
            <s xml:id="echoid-s7673" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7674" xml:space="preserve">IN quadrante ABC, ſit arcus CD, ſemiſsis ipſius; </s>
            <s xml:id="echoid-s7675" xml:space="preserve">CE, ſemiſſe maior, &</s>
            <s xml:id="echoid-s7676" xml:space="preserve"/>
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