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

Table of figures

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          <p>
            <s xml:id="echoid-s13324" xml:space="preserve">
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            quadrans CG, & </s>
            <s xml:id="echoid-s13325" xml:space="preserve">arcus CB, producatur vſque ad H, vt & </s>
            <s xml:id="echoid-s13326" xml:space="preserve">CH, quadrans ſit,
              <lb/>
            deſcribaturq́ue per puncta G, H, arcus circuli maximi GH, ſecans arcum AB,
              <lb/>
              <note position="right" xlink:label="note-401-01" xlink:href="note-401-01a" xml:space="preserve">20.1 Theod.</note>
            in I; </s>
            <s xml:id="echoid-s13327" xml:space="preserve">erit vterque angulus G, H, rectus. </s>
            <s xml:id="echoid-s13328" xml:space="preserve">Cum ergo & </s>
            <s xml:id="echoid-s13329" xml:space="preserve">angulus A, ponatur re-
              <lb/>
              <note position="right" xlink:label="note-401-02" xlink:href="note-401-02a" xml:space="preserve">25. huius.</note>
            ctus, erunt in triangulo AGI, duo anguli A, G, recti. </s>
            <s xml:id="echoid-s13330" xml:space="preserve">Quare vterque arcus
              <lb/>
            AI, GI, quadrans eſt; </s>
            <s xml:id="echoid-s13331" xml:space="preserve">atq́ue adeo arcus AB, quadrante maior. </s>
            <s xml:id="echoid-s13332" xml:space="preserve">quod eſt con-
              <lb/>
              <note position="right" xlink:label="note-401-03" xlink:href="note-401-03a" xml:space="preserve">25. huius.</note>
            tra hypotheſim.</s>
            <s xml:id="echoid-s13333" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13334" xml:space="preserve">POSSVMVS tamen aliter demonſtrare, angulum A, non poſſe eſſe
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            rectum, licet non abſcindatur quadrans CG, &</s>
            <s xml:id="echoid-s13335" xml:space="preserve">c. </s>
            <s xml:id="echoid-s13336" xml:space="preserve">Si enim angulus A, rectus
              <lb/>
              <note position="right" xlink:label="note-401-04" xlink:href="note-401-04a" xml:space="preserve">26. huius.</note>
            concedatur, erit arcus DE, quadrans. </s>
            <s xml:id="echoid-s13337" xml:space="preserve">Cum ergo & </s>
            <s xml:id="echoid-s13338" xml:space="preserve">EA, quadrans ſit, erit
              <lb/>
              <note position="right" xlink:label="note-401-05" xlink:href="note-401-05a" xml:space="preserve">25. huius.</note>
            vterque angulus A, D, rectus; </s>
            <s xml:id="echoid-s13339" xml:space="preserve">& </s>
            <s xml:id="echoid-s13340" xml:space="preserve">E, polus arcus AC, propterea quòd vter-
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            que arcus DE, AE, per polum arcus AD, tranſit, ob angulos rectos A,
              <lb/>
              <note position="right" xlink:label="note-401-06" xlink:href="note-401-06a" xml:space="preserve">13. 1. Theod.</note>
            D. </s>
            <s xml:id="echoid-s13341" xml:space="preserve">Eodem modo D, polus erit arcus AB. </s>
            <s xml:id="echoid-s13342" xml:space="preserve">Quoniam igitur punctum F, eſt
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            intra peripheriam circuli AB, & </s>
            <s xml:id="echoid-s13343" xml:space="preserve">præter eius polum, duciturq́ue arcus FE,
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            per polum circuli AB, nempe per D, erit arcus FB, maior arcu FE. </s>
            <s xml:id="echoid-s13344" xml:space="preserve">Ea-
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            dem ratione arcus FC, maior erit arcu FD, cum FD, ducatur per E, polum
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            circuli AC. </s>
            <s xml:id="echoid-s13345" xml:space="preserve">Totus igiturarcus BC, quadrante DE, maior erit. </s>
            <s xml:id="echoid-s13346" xml:space="preserve">quod eſt
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              <note position="right" xlink:label="note-401-07" xlink:href="note-401-07a" xml:space="preserve">Schol. 21.
                <lb/>
              2. Theod.</note>
            abſurdum, cum minor quadrante ponatur. </s>
            <s xml:id="echoid-s13347" xml:space="preserve">Nullus ergo angulorum A, B,
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            C, rectus eſt. </s>
            <s xml:id="echoid-s13348" xml:space="preserve">Quamobrem, In quolibet triangulo ſphærico, &</s>
            <s xml:id="echoid-s13349" xml:space="preserve">c. </s>
            <s xml:id="echoid-s13350" xml:space="preserve">Quod de-
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            monſtrandum erat.</s>
            <s xml:id="echoid-s13351" xml:space="preserve"/>
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        <div xml:id="echoid-div1056" type="section" level="1" n="528">
          <head xml:id="echoid-head563" xml:space="preserve">THEOR. 29. PROPOS. 31.</head>
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            <s xml:id="echoid-s13352" xml:space="preserve">CVIVSCVNQVE trianguli ſphærici tres
              <lb/>
            anguli duobus quidem rectis ſunt maiores, ſex ve-
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            rò rectis minores.</s>
            <s xml:id="echoid-s13353" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13354" xml:space="preserve">SIT triangulum ſphæricum ABC. </s>
            <s xml:id="echoid-s13355" xml:space="preserve">Dico tres angulos A, B, C, maiores
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            quidem eſſe duobus rectis, minores verò ſex rectis. </s>
            <s xml:id="echoid-s13356" xml:space="preserve">Si enim omnes tres angu-
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            lirecti ſint, vel obtuſi; </s>
            <s xml:id="echoid-s13357" xml:space="preserve">vel duo tantũ recti, vel obtuſi; </s>
            <s xml:id="echoid-s13358" xml:space="preserve">vel vnus tantum rectus,
              <lb/>
            & </s>
            <s xml:id="echoid-s13359" xml:space="preserve">reliquorum alter obtuſus, perſpicuum eſt, omnes tres duobus eſſe rectis
              <lb/>
            maiores. </s>
            <s xml:id="echoid-s13360" xml:space="preserve">In quolibet autem triangulo hæc erit demonſtratio. </s>
            <s xml:id="echoid-s13361" xml:space="preserve">Producto late-
              <lb/>
            re BC, ad D, erit angulus ACD, vel æqualis, vel mi-
              <lb/>
              <figure xlink:label="fig-401-01" xlink:href="fig-401-01a" number="247">
                <image file="401-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/401-01"/>
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            nor, vel maior angulo B. </s>
            <s xml:id="echoid-s13362" xml:space="preserve">Sit primum æqualis. </s>
            <s xml:id="echoid-s13363" xml:space="preserve">Erunt
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            igitur arcus AB, AC, ſimul ſemicirculo æquales; </s>
            <s xml:id="echoid-s13364" xml:space="preserve">atq;
              <lb/>
            </s>
            <s xml:id="echoid-s13365" xml:space="preserve">
              <note position="right" xlink:label="note-401-08" xlink:href="note-401-08a" xml:space="preserve">15. huius.</note>
            adeò duo anguli ABC, ACB, duobus rectis æquales.
              <lb/>
            </s>
            <s xml:id="echoid-s13366" xml:space="preserve">
              <note position="right" xlink:label="note-401-09" xlink:href="note-401-09a" xml:space="preserve">16. huius.</note>
            Tres ergo anguli A, B, C, duobus rectis maiores erũt.
              <lb/>
            </s>
            <s xml:id="echoid-s13367" xml:space="preserve">Sit deinde angulus ACD, minor angulo B. </s>
            <s xml:id="echoid-s13368" xml:space="preserve">Erunt
              <lb/>
            igitur arcus AB, AC, ſimul maiores ſemicirculo; </s>
            <s xml:id="echoid-s13369" xml:space="preserve">ac
              <lb/>
              <note position="right" xlink:label="note-401-10" xlink:href="note-401-10a" xml:space="preserve">15. huius.</note>
            propterea duo anguli ABC, ACB, duobus rectis ma-
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            iores. </s>
            <s xml:id="echoid-s13370" xml:space="preserve">Multo ergo magis tres anguli A, B, C, duobus
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            rectis maiores erũt. </s>
            <s xml:id="echoid-s13371" xml:space="preserve">Sit denique angulus ACD, maior
              <lb/>
            angulo B, & </s>
            <s xml:id="echoid-s13372" xml:space="preserve">fiat angulus DCE, angulo B, æqualis, occurratq́ue arcus CE,
              <lb/>
              <note position="right" xlink:label="note-401-11" xlink:href="note-401-11a" xml:space="preserve">10. huius.</note>
            arcui BA, producto in E: </s>
            <s xml:id="echoid-s13373" xml:space="preserve">& </s>
            <s xml:id="echoid-s13374" xml:space="preserve">tandem arcus CA, protrahatur ad F. </s>
            <s xml:id="echoid-s13375" xml:space="preserve">Erunt igi-
              <lb/>
            tur arcus EB, EC, ſimul æquales ſemicirculo; </s>
            <s xml:id="echoid-s13376" xml:space="preserve">ac propterea arcus EA, EC,
              <lb/>
              <note position="right" xlink:label="note-401-12" xlink:href="note-401-12a" xml:space="preserve">15. huius.</note>
            ſimul ſemicirculo minores. </s>
            <s xml:id="echoid-s13377" xml:space="preserve">Angulus igitur EAF, hoc eſt, angulus </s>
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