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

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            <s xml:id="echoid-s4187" xml:space="preserve">
              <pb o="112" file="124" n="124" rhead=""/>
            complementi dati arcus BD, auferatur ex ſinu toto AB, notus relinquetur ſinus verſus EB,
              <lb/>
            dati arcus BD. </s>
            <s xml:id="echoid-s4188" xml:space="preserve">Pari ratione, ſi ſinus rectus DE, hoc eſt, AF, dati arcus BD, dematur ex ſinu
              <lb/>
            toto AC, notus relinquetur ſinus verſus CF, complementi dati arcus BD.</s>
            <s xml:id="echoid-s4189" xml:space="preserve"/>
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        <div xml:id="echoid-div355" type="section" level="1" n="156">
          <head xml:id="echoid-head182" xml:space="preserve">THEOR. 2. PROPOS. 4.</head>
          <p>
            <s xml:id="echoid-s4190" xml:space="preserve">SINVS rectus cuiuſlibet arcus quadrante mi
              <lb/>
              <note position="left" xlink:label="note-124-01" xlink:href="note-124-01a" xml:space="preserve">Cuiuſuisar
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              cꝰ quadrá-
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              te minoris
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              finꝰ rectus
                <lb/>
              medioloco
                <lb/>
              {pro} portiona
                <lb/>
              lis eſt inter
                <lb/>
              ſemiſsé ſi-
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              nus totius,
                <lb/>
              & ſinũ ver-
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              ſum alteriꝰ
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              arcꝰ, ꝗ prio
                <lb/>
              ris duplus
                <lb/>
              eſt, & qua-
                <lb/>
              drante quo
                <lb/>
              queminor.</note>
            noris medio loco proportionalis eſt inter ſemiſ-
              <lb/>
            ſem ſemidiametri, ſeu ſinus totius, & </s>
            <s xml:id="echoid-s4191" xml:space="preserve">ſinum ver-
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            ſum arcus alterius, qui prioris arcus duplus eſt, & </s>
            <s xml:id="echoid-s4192" xml:space="preserve">
              <lb/>
            quadrante quoque minor.</s>
            <s xml:id="echoid-s4193" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4194" xml:space="preserve">SIT arcus quicunque CE, quadrante minor, cuius dimidium ſit CD. </s>
            <s xml:id="echoid-s4195" xml:space="preserve">Di-
              <lb/>
            uiſa autem ſemidiametro AC, bifariã in G, ducatur ex E, ad AC, perpendi-
              <lb/>
            cularis EF, iungaturque recta AD, quæ ductã chordam CE, ſecabit in H, bifa-
              <lb/>
            riam, ex lemmate à nobis ad definitiones ſupra demonſtrato, atque adeo & </s>
            <s xml:id="echoid-s4196" xml:space="preserve">ad
              <lb/>
            angulos rectos. </s>
            <s xml:id="echoid-s4197" xml:space="preserve">Erit igitur CH, ſinus rectus arcus CD, & </s>
            <s xml:id="echoid-s4198" xml:space="preserve">CF, ſinus verſus
              <lb/>
              <note position="left" xlink:label="note-124-02" xlink:href="note-124-02a" xml:space="preserve">3. tertij.</note>
              <figure xlink:label="fig-124-01" xlink:href="fig-124-01a" number="121">
                <image file="124-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/124-01"/>
              </figure>
            arcus CE, qui duplus eſt arcus CD, cum EF, ſit
              <lb/>
            eiuſdem arcus CE, ſinus rectus: </s>
            <s xml:id="echoid-s4199" xml:space="preserve">vt ex definitionibus
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            conſtat. </s>
            <s xml:id="echoid-s4200" xml:space="preserve">Dico CH, ſinum rectum arcus CD, medio
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            loco eſſe proportionalẽ inter CG, dimidiũ ſinus to-
              <lb/>
            tius, & </s>
            <s xml:id="echoid-s4201" xml:space="preserve">CF, ſinum verſum arcus CE, qui arcus CD,
              <lb/>
            duplus eſt. </s>
            <s xml:id="echoid-s4202" xml:space="preserve">Quoniã enim duo anguli ACH, AHC,
              <lb/>
            trianguli ACH, æquales ſunt duobus angulis ECF,
              <lb/>
            EFC, trianguli ECF, quod angulus C, vtrique
              <lb/>
            triangulo ſit communis, & </s>
            <s xml:id="echoid-s4203" xml:space="preserve">anguli H, F, recti; </s>
            <s xml:id="echoid-s4204" xml:space="preserve">ęquian
              <lb/>
            gula erunt triangula ACH, ECF. </s>
            <s xml:id="echoid-s4205" xml:space="preserve">Igitur erit, vt
              <lb/>
              <note position="left" xlink:label="note-124-03" xlink:href="note-124-03a" xml:space="preserve">32. primi.</note>
            AC, ad CH, ita EC, ad CF: </s>
            <s xml:id="echoid-s4206" xml:space="preserve">Et permutando, vt AC, ad CE, ita CH, ad
              <lb/>
              <note position="left" xlink:label="note-124-04" xlink:href="note-124-04a" xml:space="preserve">4.ſexti.</note>
            CF. </s>
            <s xml:id="echoid-s4207" xml:space="preserve">Vt autem AC, ad CE, ita eſt CG, dimidium ipſius AC, ad CH, dimi-
              <lb/>
              <note position="left" xlink:label="note-124-05" xlink:href="note-124-05a" xml:space="preserve">15. quinti.</note>
            dium ipſius CE. </s>
            <s xml:id="echoid-s4208" xml:space="preserve">Igitur erit quoque vt CG, ad CH, ita CH, ad CF; </s>
            <s xml:id="echoid-s4209" xml:space="preserve">ac pro-
              <lb/>
            pterea CH, ſinus rectus arcus CD, medio loco proportionalis eſt inter CG,
              <lb/>
            ſemiſſem ſinus totius, & </s>
            <s xml:id="echoid-s4210" xml:space="preserve">CF, ſinum verſum arcus CE, qui arcus CD, duplus
              <lb/>
              <note position="left" xlink:label="note-124-06" xlink:href="note-124-06a" xml:space="preserve">Ex ſinu re-
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              cto cuiuſ-
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              uis arcus
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              cognito no
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              tus fit ſinus
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              rectus alte-
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              rius arcus,
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              qui illius
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              dimidiũ ſit
                <lb/>
              33. primi.
                <lb/>
              3. huius,
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              27. ſexti.</note>
            eſt. </s>
            <s xml:id="echoid-s4211" xml:space="preserve">Igitur ſinus rectus cuiuſlibet arcus quadrante minoris, &</s>
            <s xml:id="echoid-s4212" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4213" xml:space="preserve">Quod de-
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            monſtrandum erat.</s>
            <s xml:id="echoid-s4214" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div358" type="section" level="1" n="157">
          <head xml:id="echoid-head183" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s4215" xml:space="preserve">COLLIGITVR hinc, ſi ſinus rectus alicuius arcus cognitus ſit, notum etiam ſieri
              <lb/>
            ſinum rectum alterius arcus, qui illius di midium ſit: </s>
            <s xml:id="echoid-s4216" xml:space="preserve">ita vt ex EF, ſinu recto arcus CE,
              <lb/>
            cognito cognoſcatur etiam CH, ſinus rectus arcus CD, qui dimidium eſt arcus CE. </s>
            <s xml:id="echoid-s4217" xml:space="preserve">Nam
              <lb/>
            ex noto ſinu recto EF, notus fiet ſinus EI, complementi: </s>
            <s xml:id="echoid-s4218" xml:space="preserve">quo ablato ex ſinu toto AC,
              <lb/>
            (æqualis enim eſt ſinus EI, rectæ AF.) </s>
            <s xml:id="echoid-s4219" xml:space="preserve">notus relinquetur ſinus verſus CF, arcus CE, vt in
              <lb/>
            coroll. </s>
            <s xml:id="echoid-s4220" xml:space="preserve">præcedentis propoſ. </s>
            <s xml:id="echoid-s4221" xml:space="preserve">dictum eſt. </s>
            <s xml:id="echoid-s4222" xml:space="preserve">Cum ergo ſinus CH, ſit medio loco proporti onalis
              <lb/>
            inter medietatem ſinus totius, & </s>
            <s xml:id="echoid-s4223" xml:space="preserve">ſinum verſum CF, vt oſtendimus; </s>
            <s xml:id="echoid-s4224" xml:space="preserve">erit rectangulum ſub di
              <lb/>
            midio ſinus totius, & </s>
            <s xml:id="echoid-s4225" xml:space="preserve">ſinu verſo CF, contentum æquale quadrato ſinus CH. </s>
            <s xml:id="echoid-s4226" xml:space="preserve">Quare ſi
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            multiplicetur medietas ſinus totius in ſinum verſum CF, producetur quadratus </s>
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