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

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          <p>
            <s xml:id="echoid-s4097" xml:space="preserve">
              <pb o="110" file="122" n="122" rhead=""/>
            a F. </s>
            <s xml:id="echoid-s4098" xml:space="preserve">Cum ergo ſit, vt Da, ad aF, ita b V, ad VF, erit quoque b V, maior,
              <lb/>
              <note position="left" xlink:label="note-122-01" xlink:href="note-122-01a" xml:space="preserve">@. fexti.</note>
            quàm VF, hoc eſt, GH, maior, quàm HI; </s>
            <s xml:id="echoid-s4099" xml:space="preserve">& </s>
            <s xml:id="echoid-s4100" xml:space="preserve">KL, maior quàm LM.</s>
            <s xml:id="echoid-s4101" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div346" type="section" level="1" n="152">
          <head xml:id="echoid-head178" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s4102" xml:space="preserve">CONSTAT ex hac propoſitione, ſi quotcunque arcus quadrãtis à ſemidiametro eadé
              <lb/>
              <note position="left" xlink:label="note-122-02" xlink:href="note-122-02a" xml:space="preserve">Differentię
                <lb/>
              ſinuũ recto
                <lb/>
              rũ à princi
                <lb/>
              pio quadrã
                <lb/>
              tis vſq; ad
                <lb/>
              eius finem
                <lb/>
              ſenſim de-
                <lb/>
              creſcunt:a-
                <lb/>
              deo vi ſinꝰ
                <lb/>
              minorú ar
                <lb/>
              cuũ maio-
                <lb/>
              res habeát
                <lb/>
              differétias,
                <lb/>
              q̇
                <unsure/>
              ſinꝰ arcuũ
                <lb/>
              maiorũ; dũ
                <lb/>
              modo arcꝰ
                <lb/>
              habeát dif-
                <lb/>
              ferentias ę
                <lb/>
              quales.</note>
            incipientes habeant æquales differentias, exceſſusve; </s>
            <s xml:id="echoid-s4103" xml:space="preserve">ſinus rectos minorum arcuum habe-
              <lb/>
            re maiores differentias, quàm ſinus arcuum maiorum; </s>
            <s xml:id="echoid-s4104" xml:space="preserve">adeo vt differentiæ ſinuum à prin-
              <lb/>
            cipio quadrantis ad finem vſque ſemper decreſcant. </s>
            <s xml:id="echoid-s4105" xml:space="preserve">Nam ſi in eadem figura huius propof.
              <lb/>
            </s>
            <s xml:id="echoid-s4106" xml:space="preserve">a cci piatur arcus FX, arcubus DE, EF, æ qualis, ducaturq́ue recta XY, ad ſemidiametrum
              <lb/>
            AC, perpendicularis, habebunt quatuor arcus BX, BF, BE, BD, æquales exceſlus, cum
              <lb/>
            BX, ipſum BF, ſuperetarcu FX; </s>
            <s xml:id="echoid-s4107" xml:space="preserve">& </s>
            <s xml:id="echoid-s4108" xml:space="preserve">BF, ipſum BE, arcu EF, qui arcui FX, poſitus eſt
              <lb/>
            æqualis; </s>
            <s xml:id="echoid-s4109" xml:space="preserve">& </s>
            <s xml:id="echoid-s4110" xml:space="preserve">arcus BE, arcum BD, arcu DE, qui arcui EF, æqualis eſt. </s>
            <s xml:id="echoid-s4111" xml:space="preserve">Sinus autem recti
              <lb/>
            eorum arcuum ſunt AY, AI, AH, AG, vt ſupra in ex poſitione definitionum docuimus,
              <lb/>
            cum ſint partes ſemidiametri AC, inter centrum A, & </s>
            <s xml:id="echoid-s4112" xml:space="preserve">ſinus complementorum interiectæ,
              <lb/>
            vt patet. </s>
            <s xml:id="echoid-s4113" xml:space="preserve">Et quoniam in hac propof. </s>
            <s xml:id="echoid-s4114" xml:space="preserve">demonftrauimus, rectam GH, maiorem eſſe, quàm
              <lb/>
            HI, & </s>
            <s xml:id="echoid-s4115" xml:space="preserve">HI, maiorem, quàm IY; </s>
            <s xml:id="echoid-s4116" xml:space="preserve">liquet, exceſſum GH, inter ſinus arcuum minorum BE,
              <lb/>
            BD, maiorem eſſe exceſſu HI, inter ſinus arcuum maiorum BF, BE: </s>
            <s xml:id="echoid-s4117" xml:space="preserve">Item exceſſum HI,
              <lb/>
            inter ſinus arcuum minorum BF, BE, maiorem eſſe exceſſu IY, inter ſinus maiorum ar-
              <lb/>
            cuum BX, BF. </s>
            <s xml:id="echoid-s4118" xml:space="preserve">Eademq́ue ratio eſt de cæteris. </s>
            <s xml:id="echoid-s4119" xml:space="preserve">Conſtat igitur, differentias ſinuum rectorum
              <lb/>
            ſenſim decreſcere à principio quadrantis v ſque ad eius finem: </s>
            <s xml:id="echoid-s4120" xml:space="preserve">Id quod perſpicuè ex ſinuum
              <lb/>
            tabula apparet.</s>
            <s xml:id="echoid-s4121" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div348" type="section" level="1" n="153">
          <head xml:id="echoid-head179" xml:space="preserve">PROBL. 1. PROPOS. 2.</head>
          <p>
            <s xml:id="echoid-s4122" xml:space="preserve">LATERA Decagoni, & </s>
            <s xml:id="echoid-s4123" xml:space="preserve">Pentagoni æquila-
              <lb/>
              <note position="left" xlink:label="note-122-03" xlink:href="note-122-03a" xml:space="preserve">Latera De.
                <lb/>
              cagoni, &
                <lb/>
              Penta goni
                <lb/>
              in vno eo-
                <lb/>
              demq; cir-
                <lb/>
              culo quo
                <lb/>
              pacto inue-
                <lb/>
              niantur.</note>
            teri in vno eodemq́; </s>
            <s xml:id="echoid-s4124" xml:space="preserve">circulo inueſtigare.</s>
            <s xml:id="echoid-s4125" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4126" xml:space="preserve">QVAMVIS hæc latera inueniantur per ea, quæ ab Euclide lib. </s>
            <s xml:id="echoid-s4127" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4128" xml:space="preserve">ſunt
              <lb/>
            demonſtrata: </s>
            <s xml:id="echoid-s4129" xml:space="preserve">nihilominus eadem à Ptolemæo lib. </s>
            <s xml:id="echoid-s4130" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4131" xml:space="preserve">Almageſti cap. </s>
            <s xml:id="echoid-s4132" xml:space="preserve">9. </s>
            <s xml:id="echoid-s4133" xml:space="preserve">inueſti-
              <lb/>
            gantur ratione alia, quæ ad plurimorum ſinuum inuentionem multum con-
              <lb/>
            ducit. </s>
            <s xml:id="echoid-s4134" xml:space="preserve">Eſt autem hæc ratio. </s>
            <s xml:id="echoid-s4135" xml:space="preserve">Sit circulus, vel (quod ſatis eſt) ſemicirculus ABC,
              <lb/>
              <figure xlink:label="fig-122-01" xlink:href="fig-122-01a" number="119">
                <image file="122-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/122-01"/>
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            ad cuius diametrum AC, ex D, centro
              <lb/>
            educatur perpendicularis DB. </s>
            <s xml:id="echoid-s4136" xml:space="preserve">Diuiſa
              <lb/>
            quoque ſemidiametro CD, bifariam
              <lb/>
            in E, ducatur recta EB, cui ęqualis ab-
              <lb/>
            ſcindatur EF, iungaturq́ue recta FB.
              <lb/>
            </s>
            <s xml:id="echoid-s4137" xml:space="preserve">Dico rectam BF, eſſe latus Penta-
              <lb/>
            goni, & </s>
            <s xml:id="echoid-s4138" xml:space="preserve">DF, latus Decagoni in cir-
              <lb/>
            culo ABC. </s>
            <s xml:id="echoid-s4139" xml:space="preserve">Cum enim recta CD,
              <lb/>
            ſecta ſit bifariam in E, eique addi-
              <lb/>
            ta DF; </s>
            <s xml:id="echoid-s4140" xml:space="preserve">erit rectangulum ſub CF, DF,
              <lb/>
              <note position="left" xlink:label="note-122-04" xlink:href="note-122-04a" xml:space="preserve">6.ſecundi.</note>
            vna cum quadrato rectæ DE, æquale
              <lb/>
            quadrato rectæ EF, ideoq́ quadrato rectæ EB, quæ ipſi EF, ęqualis
              <lb/>
            eſt: </s>
            <s xml:id="echoid-s4141" xml:space="preserve">Eſt autem quadratum rectæ EB, æquale quadratis rectarum BD, DE.
              <lb/>
            </s>
            <s xml:id="echoid-s4142" xml:space="preserve">
              <note position="left" xlink:label="note-122-05" xlink:href="note-122-05a" xml:space="preserve">47.primi.</note>
            Igitur rectangulum ſub CF, DF, vnà cum quadrato rectæ DE, æquale eſt
              <lb/>
            quadratis rectarum BD, DE: </s>
            <s xml:id="echoid-s4143" xml:space="preserve">Ac proinde, dempto communi quadrato rectæ
              <lb/>
              <emph style="sc">De</emph>
            , relinquetur rectangulum ſub CF, DF, æquale quadrato rectæ BD,
              <lb/>
            hoc eſt, quadrato rectæ CD. </s>
            <s xml:id="echoid-s4144" xml:space="preserve">Quamobrem erit, vt CF, ad CD, ita CD, ad
              <lb/>
              <note position="left" xlink:label="note-122-06" xlink:href="note-122-06a" xml:space="preserve">17.fexti.</note>
            DF; </s>
            <s xml:id="echoid-s4145" xml:space="preserve">proptereaq́ue recta CF, diuiſa erit in D, extrema ac media ratione. </s>
            <s xml:id="echoid-s4146" xml:space="preserve">Cum
              <lb/>
            igitur maius ſegmentum CD, ſit latus Hexagoni in circulo ABC, ex </s>
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