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

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        <div xml:id="echoid-div476" type="section" level="1" n="240">
          <head xml:id="echoid-head266" xml:space="preserve">EXPLICATIO, ATQVE VSVS TABVLAE</head>
          <head xml:id="echoid-head267" xml:space="preserve">præcedentis Sinuum rectorum.</head>
          <p style="it">
            <s xml:id="echoid-s5732" xml:space="preserve">_IN_vertice præcedentis tabulæ ordine deſcripti ſunt 90. </s>
            <s xml:id="echoid-s5733" xml:space="preserve">gradus Quadrantis, & </s>
            <s xml:id="echoid-s5734" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-180-01" xlink:href="note-180-01a" xml:space="preserve">Expoſitio
                <lb/>
              partium ta
                <lb/>
              bulęSinuũ.</note>
            ad ſiniſtram deorſum verſus, 60. </s>
            <s xml:id="echoid-s5735" xml:space="preserve">Minuta. </s>
            <s xml:id="echoid-s5736" xml:space="preserve">In infimo deinde latere ijdem 90. </s>
            <s xml:id="echoid-s5737" xml:space="preserve">gradus
              <lb/>
            Quadrantis repoſiti ſunt ordine retrogrado, & </s>
            <s xml:id="echoid-s5738" xml:space="preserve">ad dexteram ſurſum verſus, 60. </s>
            <s xml:id="echoid-s5739" xml:space="preserve">Mi-
              <lb/>
            nuta. </s>
            <s xml:id="echoid-s5740" xml:space="preserve">Quod ideofactum est à nobis, vt illico cuiuslibet arcus complementum cognoſca-
              <lb/>
            tur. </s>
            <s xml:id="echoid-s5741" xml:space="preserve">Nam quilibet gradus in vertice tabulæ poſitus cum quouis Minuto ad ſiniſtram
              <lb/>
              <note position="left" xlink:label="note-180-02" xlink:href="note-180-02a" xml:space="preserve">Comple
                <lb/>
              mẽtum cu
                <lb/>
              iuſuis arcꝰ
                <lb/>
              quo pacto
                <lb/>
              ex hac ta-
                <lb/>
              bula elicia
                <lb/>
              tur.</note>
            collocato, habet pro complemento gradum in infimo latere gradui accepto in vertice
              <lb/>
            reſpondentem cum Minuto, quod ad dextram Minuto ad ſiniſtram accepto reſpondet.
              <lb/>
            </s>
            <s xml:id="echoid-s5742" xml:space="preserve">Vt quomam gradui 46. </s>
            <s xml:id="echoid-s5743" xml:space="preserve">in vertice, & </s>
            <s xml:id="echoid-s5744" xml:space="preserve">Minuto O ad ſiniſtram poſito, reſpondet in infi-
              <lb/>
            mo latere gradus 43. </s>
            <s xml:id="echoid-s5745" xml:space="preserve">& </s>
            <s xml:id="echoid-s5746" xml:space="preserve">Minutum 60. </s>
            <s xml:id="echoid-s5747" xml:space="preserve">ad dexteram collocatum; </s>
            <s xml:id="echoid-s5748" xml:space="preserve">erit arcus grad. </s>
            <s xml:id="echoid-s5749" xml:space="preserve">43. </s>
            <s xml:id="echoid-s5750" xml:space="preserve">
              <lb/>
            Min. </s>
            <s xml:id="echoid-s5751" xml:space="preserve">60. </s>
            <s xml:id="echoid-s5752" xml:space="preserve">hoceſt, arcus grad. </s>
            <s xml:id="echoid-s5753" xml:space="preserve">44. </s>
            <s xml:id="echoid-s5754" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5755" xml:space="preserve">o. </s>
            <s xml:id="echoid-s5756" xml:space="preserve">complementum arcus grad. </s>
            <s xml:id="echoid-s5757" xml:space="preserve">46. </s>
            <s xml:id="echoid-s5758" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5759" xml:space="preserve">o. </s>
            <s xml:id="echoid-s5760" xml:space="preserve">Sic
              <lb/>
            quoque arcus grad. </s>
            <s xml:id="echoid-s5761" xml:space="preserve">43. </s>
            <s xml:id="echoid-s5762" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5763" xml:space="preserve">13. </s>
            <s xml:id="echoid-s5764" xml:space="preserve">complementum habebit arcum grad. </s>
            <s xml:id="echoid-s5765" xml:space="preserve">46 Min. </s>
            <s xml:id="echoid-s5766" xml:space="preserve">47. </s>
            <s xml:id="echoid-s5767" xml:space="preserve">Ea-
              <lb/>
            dem ratione quilibet gradus in infimo latere poſitus cum quouis Minuto ad dexteram
              <lb/>
            collocato, habet pro complemento gradum in vertice gradui acceptoin infimo latere
              <lb/>
            reſpondentem cum Minuto, quod ad ſiniſtram Minuto ad dextram accepto reſpondet. </s>
            <s xml:id="echoid-s5768" xml:space="preserve">
              <lb/>
            Poſtremo ſub gradibus in vertice tabulæ deſcriptis poſiti ſunt ſinus recti omnium ar-
              <lb/>
            cuũ per ſingula Quadrãtis Minuta progredientiũ, quatenus ſinus totus eſt 10000000. </s>
            <s xml:id="echoid-s5769" xml:space="preserve">
              <lb/>
            Quòd ſi ex ſingulis ſinubus binæ priores figuræ ad dexter am abijciãtur, (addita tamen
              <lb/>
            vnitate, ſi duæ figuræ abiectæ numerum 50. </s>
            <s xml:id="echoid-s5770" xml:space="preserve">excedunt) reliqui erunt ſinus eorundem
              <lb/>
            arcuum, quatenus ſinus totus eſt 100000. </s>
            <s xml:id="echoid-s5771" xml:space="preserve">vt ſupra diximus. </s>
            <s xml:id="echoid-s5772" xml:space="preserve">Vnde quicquid in vſu hu-
              <lb/>
            ius tabulæ præcipiemus de ſinubus reſpectu ſinus totius 10000000. </s>
            <s xml:id="echoid-s5773" xml:space="preserve">intelligendum quo-
              <lb/>
            que erit de ſinubus reſpectu ſinus totius 100000. </s>
            <s xml:id="echoid-s5774" xml:space="preserve">abiectis nimirum duabus primis fi-
              <lb/>
              <note position="left" xlink:label="note-180-03" xlink:href="note-180-03a" xml:space="preserve">Vſus tabu-
                <lb/>
              bulę ſinuũ
                <lb/>
              duplex.</note>
            guris ad dexteram, vt diximus.</s>
            <s xml:id="echoid-s5775" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5776" xml:space="preserve">_HVIVS_tabulæ vſus duplex eſt. </s>
            <s xml:id="echoid-s5777" xml:space="preserve">Nam in ea vel cuiuslibet arcus inquiritur ſi-
              <lb/>
            nus, vel cuiuſuis ſinus cogniti arcus inueſtigatur. </s>
            <s xml:id="echoid-s5778" xml:space="preserve">Quando ergo dati arcus Quadrante
              <lb/>
              <note position="left" xlink:label="note-180-04" xlink:href="note-180-04a" xml:space="preserve">Sinus tectꝰ
                <lb/>
              cuiuſuis ar
                <lb/>
              cus quadrã
                <lb/>
              te minoris
                <lb/>
              quo pacto í
                <lb/>
              tabula repe
                <lb/>
              riatur.</note>
            minoris ſinum rectum quæris, ſume gradus illius in vertice tabulæ, Minuta vero ad ſini
              <lb/>
            ſtram. </s>
            <s xml:id="echoid-s5779" xml:space="preserve">In communi enim angulo, procedendo nimirum à Minutis dextram verſus, do-
              <lb/>
            nec adlocum ſub gradibus acceptis peruenias, illico ſinum rectum inuenies. </s>
            <s xml:id="echoid-s5780" xml:space="preserve">Ita ſinum
              <lb/>
            rectum arcus grad 25. </s>
            <s xml:id="echoid-s5781" xml:space="preserve">ſub grad 25. </s>
            <s xml:id="echoid-s5782" xml:space="preserve">è regione Minuti o ad ſinistram collocati repe-
              <lb/>
            ries 4226183. </s>
            <s xml:id="echoid-s5783" xml:space="preserve">Sinum vero rectum arcus grad. </s>
            <s xml:id="echoid-s5784" xml:space="preserve">25. </s>
            <s xml:id="echoid-s5785" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5786" xml:space="preserve">19. </s>
            <s xml:id="echoid-s5787" xml:space="preserve">inuenies 4276209. </s>
            <s xml:id="echoid-s5788" xml:space="preserve">Sinum
              <lb/>
            denique rectum arcus grad. </s>
            <s xml:id="echoid-s5789" xml:space="preserve">25. </s>
            <s xml:id="echoid-s5790" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5791" xml:space="preserve">50. </s>
            <s xml:id="echoid-s5792" xml:space="preserve">offendes 4357549. </s>
            <s xml:id="echoid-s5793" xml:space="preserve">Si vero ſinum rectum ar-
              <lb/>
              <note position="left" xlink:label="note-180-05" xlink:href="note-180-05a" xml:space="preserve">Sinꝰ rectus
                <lb/>
              cuiuſuis ar
                <lb/>
              cus quadrã
                <lb/>
              te maioris,
                <lb/>
              qua rõne
                <lb/>
              inueniatur</note>
            cus Quadrante maioris, ſed ſemicirculo minoris deſideras, detrabe arcum datum ex
              <lb/>
            ſemicirculo, & </s>
            <s xml:id="echoid-s5794" xml:space="preserve">reſidui arcus ſinum rectum cape, vt prius. </s>
            <s xml:id="echoid-s5795" xml:space="preserve">Hic enim ſinus erit etiam
              <lb/>
            ſinus rectus arcus quadrante maioris: </s>
            <s xml:id="echoid-s5796" xml:space="preserve">propterea quòd duo arcus ſemicirculum con-
              <lb/>
            ficientes eundem ſinũ rectum habent, vtin definitionũ expoſitione diximus. </s>
            <s xml:id="echoid-s5797" xml:space="preserve">Vt ſi da-
              <lb/>
            tus ſit arcus grad. </s>
            <s xml:id="echoid-s5798" xml:space="preserve">138. </s>
            <s xml:id="echoid-s5799" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5800" xml:space="preserve">47. </s>
            <s xml:id="echoid-s5801" xml:space="preserve">detrabe eum ex ſemicirculo, hoc eſt, ex grad. </s>
            <s xml:id="echoid-s5802" xml:space="preserve">180.
              <lb/>
            </s>
            <s xml:id="echoid-s5803" xml:space="preserve">Sinus namq; </s>
            <s xml:id="echoid-s5804" xml:space="preserve">rectus 6589082. </s>
            <s xml:id="echoid-s5805" xml:space="preserve">reſidui arcus grad. </s>
            <s xml:id="echoid-s5806" xml:space="preserve">41. </s>
            <s xml:id="echoid-s5807" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5808" xml:space="preserve">13. </s>
            <s xml:id="echoid-s5809" xml:space="preserve">eſt quoq; </s>
            <s xml:id="echoid-s5810" xml:space="preserve">ſinus rectus
              <lb/>
            arcus grad. </s>
            <s xml:id="echoid-s5811" xml:space="preserve">138. </s>
            <s xml:id="echoid-s5812" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5813" xml:space="preserve">47. </s>
            <s xml:id="echoid-s5814" xml:space="preserve">cum illo arcu grad. </s>
            <s xml:id="echoid-s5815" xml:space="preserve">41. </s>
            <s xml:id="echoid-s5816" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5817" xml:space="preserve">13. </s>
            <s xml:id="echoid-s5818" xml:space="preserve">ſemicirculũ coficientis.</s>
            <s xml:id="echoid-s5819" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5820" xml:space="preserve">_QVOD_ſiarcus datus præter gradus, ac minuta habeat etiam Secunda, inquiren-
              <lb/>
              <note position="left" xlink:label="note-180-06" xlink:href="note-180-06a" xml:space="preserve">Sinꝰ rectus
                <lb/>
              cuiuſuis ar
                <lb/>
              cus habẽtis
                <lb/>
              Secũda, prę
                <lb/>
              ter Minuta
                <lb/>
              quomodo
                <lb/>
              eliciat̃ per</note>
            da erit pars proportionalis, hoc modo. </s>
            <s xml:id="echoid-s5821" xml:space="preserve">Accipe differentiam inter ſinum rectum arcus
              <lb/>
            proxime minoris, & </s>
            <s xml:id="echoid-s5822" xml:space="preserve">ſinũ arcus proxime maioris, & </s>
            <s xml:id="echoid-s5823" xml:space="preserve">dic. </s>
            <s xml:id="echoid-s5824" xml:space="preserve">Si 60. </s>
            <s xml:id="echoid-s5825" xml:space="preserve">ſecunda (qui-
              <lb/>
            bus ſinguli arcus proximiin hae tabula inter ſe differunt) requirũt tota eam differen
              <lb/>
            tiam addendam ſinui arcus proxime minoris, vt componatur ſinus arcus proxime
              <lb/>
            maioris, quantam differentiam requirunt propoſita ſecunda addendam eidem </s>
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