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

Table of figures

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        <div xml:id="echoid-div1291" type="section" level="1" n="600">
          <head xml:id="echoid-head635" xml:space="preserve">SCHOLIVM. II.</head>
          <p style="it">
            <s xml:id="echoid-s16091" xml:space="preserve">CATERVM ex hac propoſ. </s>
            <s xml:id="echoid-s16092" xml:space="preserve">58. </s>
            <s xml:id="echoid-s16093" xml:space="preserve">colligemus ſequens theorema ad calculum trian
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            gulorum ſphæricorum non rectangulorum perutile, videlicet.</s>
            <s xml:id="echoid-s16094" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16095" xml:space="preserve">IN omni triangulo ſphærico, cuius duo arcus ſint inæquales: </s>
            <s xml:id="echoid-s16096" xml:space="preserve">ſi-
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            nus totus ad quantitatem, quæ ſinui toti, & </s>
            <s xml:id="echoid-s16097" xml:space="preserve">duobus ſinubus arcuum
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            inæqualium quarto loco proportionalis eſt, eandem proportionem
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            habet, quam ſinus verſus anguli ſub dictis arcubus comprehenſi ad
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            differentiam inter ſinum verſum reliqui tertij arcus, & </s>
            <s xml:id="echoid-s16098" xml:space="preserve">ſinum ver-
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            ſum arcus, quo duo inæquales arcus inter ſe differunt.</s>
            <s xml:id="echoid-s16099" xml:space="preserve"/>
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          <p style="it">
            <s xml:id="echoid-s16100" xml:space="preserve">IN triangulo ſphærico ABC, proxime antecedenti
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              <figure xlink:label="fig-466-01" xlink:href="fig-466-01a" number="334">
                <image file="466-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/466-01"/>
              </figure>
            ſit arcus AB, maior arcu AC, & </s>
            <s xml:id="echoid-s16101" xml:space="preserve">ex polo A, ad interual
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            lum AC, deſcribatur arcus circuli CD, vt arcus AC,
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            AD, per defin. </s>
            <s xml:id="echoid-s16102" xml:space="preserve">poli, ſint æquales, atque adeo arcus BD,
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            exceſſus ſit, ſeu differentia arcuum AB, AC. </s>
            <s xml:id="echoid-s16103" xml:space="preserve">Fiat iam,
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            vt ſinus totus ad ſinum arcus AB, ita ſinus arcus AC, ad
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            aliud, quod quantitas quarta proportionalis vocetur,
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            vt hic vides:
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            </s>
            <s xml:id="echoid-s16104" xml:space="preserve">
              <note position="right" xlink:label="note-466-01" xlink:href="note-466-01a" xml:space="preserve">
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              Sinus \ṫotus. # ſinus arcus \\ AB. # ſinus arcus \\ AC. # quantitas quarta \ṗroportionalis.
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              </note>
            Dico ita eſſe ſinum totum ad quantitatem quartam ſinui toti, & </s>
            <s xml:id="echoid-s16105" xml:space="preserve">duobus ſinubus ar-
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            cuum inæqualium proportionalem, vt eſt ſinus verſus anguli A, ad differentiam inter
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            ſinum verſum arcus BC, & </s>
            <s xml:id="echoid-s16106" xml:space="preserve">ſinum verſum arcus BD, quo inter ſe arcus
              <emph style="sc">AB</emph>
            ,
              <emph style="sc">Ac</emph>
            ,
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            differunt. </s>
            <s xml:id="echoid-s16107" xml:space="preserve">Quoiniam enim proportio ſinus totius ad quantitatem illam quartam pro-
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            portionalem componitur (poſito ſinu arcus AB, medio) ex proportione ſinus totius ad
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            ſinum arcus AB, & </s>
            <s xml:id="echoid-s16108" xml:space="preserve">ex proportione ſinus arcus AB, ad quantitatem quartam pro-
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            portionalem: </s>
            <s xml:id="echoid-s16109" xml:space="preserve">Et proportio quadrati ſinus totius ad rectangulum ſub ſinubus rectis ar-
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            cuum AB, AC, componitur ex eiſdem proportionibus, nempe ex proportione ſinus to
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            tius ad ſinum arcus AB, & </s>
            <s xml:id="echoid-s16110" xml:space="preserve">ex proportione ſinus totius ad ſinum arcus AC, quæ ea-
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              <note position="left" xlink:label="note-466-02" xlink:href="note-466-02a" xml:space="preserve">23. ſexti.</note>
            dem eſt, quæ proportio ſinus arcus AB, ad quantitatem quartam proportionalem:
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            </s>
            <s xml:id="echoid-s16111" xml:space="preserve">(Nam cum ſit, vt ſinus totus ad ſinum arcus
              <emph style="sc">A</emph>
            B, ita ſinus arcus AC, ad quantita-
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            tem quartam proportionalem; </s>
            <s xml:id="echoid-s16112" xml:space="preserve">erit permutando, vt ſinus totus ad ſinum arcus AC,
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            ita ſinus arcus AB, ad quantitatem quartam proportionalem.) </s>
            <s xml:id="echoid-s16113" xml:space="preserve">erit, vt ſinus totus
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            ad quantitatem quartam proportionalem, ita quadratum ſinus totius ad rectangu-
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            lum ſub ſinubus arcuum AB, AC, contentum. </s>
            <s xml:id="echoid-s16114" xml:space="preserve">Cum ergo ſit, vt quadratum ſinus to-
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            tius ad rectangulum ſub ſinubus arcuum AB, AC, ita ſinus verſus anguli A, ad dif-
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              <note position="left" xlink:label="note-466-03" xlink:href="note-466-03a" xml:space="preserve">38. huius.</note>
            ferentiam ſinuum verſorum arcuum BC, BD; </s>
            <s xml:id="echoid-s16115" xml:space="preserve">erit quoque, vt ſinus totus ad quan-
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            titatem quartam proportionalem, ita ſinus verſus anguli A, ad differentiam inter ſi-
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            nus verſos arcuum BC, BD. </s>
            <s xml:id="echoid-s16116" xml:space="preserve">quod eſt propoſitum,</s>
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        <div xml:id="echoid-div1293" type="section" level="1" n="601">
          <head xml:id="echoid-head636" xml:space="preserve">THEOR. 57. PROPOS. 59.</head>
          <p>
            <s xml:id="echoid-s16117" xml:space="preserve">SI duo triangula ſphærica duos angulos duo-
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            bus angulis æquales habeant, vtrumque vtrique:</s>
            <s xml:id="echoid-s16118" xml:space="preserve"/>
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