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

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          <p>
            <s xml:id="echoid-s4585" xml:space="preserve">
              <pb o="119" file="131" n="131" rhead=""/>
            ſuperantium, reſpectu Sinus totius particularum 100000. </s>
            <s xml:id="echoid-s4586" xml:space="preserve">quotnimirum com
              <lb/>
            muniter ab omnibus, & </s>
            <s xml:id="echoid-s4587" xml:space="preserve">ànobis etiam conſtituitur, vt ſupra diximus. </s>
            <s xml:id="echoid-s4588" xml:space="preserve">Vt auté
              <lb/>
            accuratior ſiat ſupputatio, ponemus in hiſce ſupputationibus Sinum totum
              <lb/>
              <note position="right" xlink:label="note-131-01" xlink:href="note-131-01a" xml:space="preserve">Sinus totꝰ
                <lb/>
              in cuiꝰ par
                <lb/>
              tibꝰ alij ſi-
                <lb/>
              nus ompu
                <lb/>
              tátur, quoe
                <lb/>
              particula
                <lb/>
              rú ponédus
                <lb/>
              ſit, vt alij
                <lb/>
              ſinus inue-
                <lb/>
              niátur ma-
                <lb/>
              gis exquiſi
                <lb/>
              ti reſp ectu
                <lb/>
              Sinus to-
                <lb/>
              tius pattiũ
                <lb/>
              100000. vel
                <lb/>
              pauciotum
                <lb/>
              pluriúve.</note>
            partium 10000000. </s>
            <s xml:id="echoid-s4589" xml:space="preserve">Ita enim fiet, vt abiectis duabus primis figuris ad dexte-
              <lb/>
            rá ex ſingulis ſinubus inuentis, (addita tamen vnitate, ſi duæ figuræ abiectæ nu
              <lb/>
            merũ 50. </s>
            <s xml:id="echoid-s4590" xml:space="preserve">ſuperent) relinquãtur ſinus magis exquiſiti reſpectu ſinus totius par
              <lb/>
            tium 100000. </s>
            <s xml:id="echoid-s4591" xml:space="preserve">Quòd ſi quis ſinus deſideret plurium particularum, poſito ni-
              <lb/>
            mirũ ſinu toto partiũ 10000000. </s>
            <s xml:id="echoid-s4592" xml:space="preserve">quot eum Ioan. </s>
            <s xml:id="echoid-s4593" xml:space="preserve">Regiom. </s>
            <s xml:id="echoid-s4594" xml:space="preserve">poſuit, & </s>
            <s xml:id="echoid-s4595" xml:space="preserve">nos in ſe
              <lb/>
            quẽti tabula ſtatuemus, conſtituendus erit Sinus totus partiũ 1000000000.
              <lb/>
            </s>
            <s xml:id="echoid-s4596" xml:space="preserve">Nam hac ratione, abiectis duabus primis figuris ad dexteram ex ſingulis ſinu-
              <lb/>
            bus inuentis, vt dictum eſt, remanebunt Sinus magis exquiſiti reſpectu Sinus
              <lb/>
            totius partium 10000000. </s>
            <s xml:id="echoid-s4597" xml:space="preserve">Ratio huius rei eſt, quòd in ſinuum inueſtigatio-
              <lb/>
            ne error ſolum contingere poteſt in vna aut altera figura ad dextram: </s>
            <s xml:id="echoid-s4598" xml:space="preserve">quare,
              <lb/>
            abiectis duabus figuris ad dextram, relinquentur ſinus reſpectu ſinus totius
              <lb/>
            minoris exquiſrtiſsimi. </s>
            <s xml:id="echoid-s4599" xml:space="preserve">Id quod in ſupputationibus ſinuum quiuis facile ex pe-
              <lb/>
            rietur, & </s>
            <s xml:id="echoid-s4600" xml:space="preserve">nos infra demonſtrabimus. </s>
            <s xml:id="echoid-s4601" xml:space="preserve">Pari ratione, ſi inueſtigandi ſint ſinus re-
              <lb/>
            ſpectu alterius Sinus totius, qui plures, aut pauciores particulas contineat,
              <lb/>
            quàm 100000. </s>
            <s xml:id="echoid-s4602" xml:space="preserve">vel 10000000. </s>
            <s xml:id="echoid-s4603" xml:space="preserve">conſtituendus erit in ſupputatione Sinus to-
              <lb/>
            tus, qui ad dexteram illum ſuperet duabus figuris his, 00; </s>
            <s xml:id="echoid-s4604" xml:space="preserve">adeo vt illius ſit
              <lb/>
            centuplus, quemadmodum & </s>
            <s xml:id="echoid-s4605" xml:space="preserve">hic 10000000. </s>
            <s xml:id="echoid-s4606" xml:space="preserve">quem in calculo aſſumimus, cen-
              <lb/>
            tuplus eſt illius 100000. </s>
            <s xml:id="echoid-s4607" xml:space="preserve">quem nos cum alijs Aſtronomis in vſu recipimus.</s>
            <s xml:id="echoid-s4608" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4609" xml:space="preserve">SIT igitur in quadrante ABC, arcus CD, grad. </s>
            <s xml:id="echoid-s4610" xml:space="preserve">15. </s>
            <s xml:id="echoid-s4611" xml:space="preserve">CE, 30. </s>
            <s xml:id="echoid-s4612" xml:space="preserve">CF, 45. </s>
            <s xml:id="echoid-s4613" xml:space="preserve">ac
              <lb/>
            proinde EB, grad. </s>
            <s xml:id="echoid-s4614" xml:space="preserve">60. </s>
            <s xml:id="echoid-s4615" xml:space="preserve">& </s>
            <s xml:id="echoid-s4616" xml:space="preserve">DB, 75. </s>
            <s xml:id="echoid-s4617" xml:space="preserve">vtpote complementa arcuum grad. </s>
            <s xml:id="echoid-s4618" xml:space="preserve">30. </s>
            <s xml:id="echoid-s4619" xml:space="preserve">& </s>
            <s xml:id="echoid-s4620" xml:space="preserve">15.
              <lb/>
            </s>
            <s xml:id="echoid-s4621" xml:space="preserve">Horum ergo arcuum ſinus rectos ita ſupputabimus. </s>
            <s xml:id="echoid-s4622" xml:space="preserve">Ducantur EH, DI, ad AB,
              <lb/>
            perpendiculares, quæ ſinus recti erunt arcuum grad. </s>
            <s xml:id="echoid-s4623" xml:space="preserve">60. </s>
            <s xml:id="echoid-s4624" xml:space="preserve">& </s>
            <s xml:id="echoid-s4625" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s4626" xml:space="preserve">75. </s>
            <s xml:id="echoid-s4627" xml:space="preserve">Ductam
              <lb/>
            autem chordam BC, ſecet recta AF, in L, bi-
              <lb/>
              <figure xlink:label="fig-131-01" xlink:href="fig-131-01a" number="128">
                <image file="131-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/131-01"/>
              </figure>
            fariam, ex lemmate in definitionibus demon-
              <lb/>
            ſtrato; </s>
            <s xml:id="echoid-s4628" xml:space="preserve">ac proinde ad angulos rectos: </s>
            <s xml:id="echoid-s4629" xml:space="preserve">eritq̀ue
              <lb/>
              <note position="right" xlink:label="note-131-02" xlink:href="note-131-02a" xml:space="preserve">3. tertij.</note>
            BL, ſinus rectus grad. </s>
            <s xml:id="echoid-s4630" xml:space="preserve">45. </s>
            <s xml:id="echoid-s4631" xml:space="preserve">hoc eſt, arcus BF.
              <lb/>
            </s>
            <s xml:id="echoid-s4632" xml:space="preserve">Ducatur rurſus EG, ad AC, perpendicula-
              <lb/>
            ris pro ſinu grad. </s>
            <s xml:id="echoid-s4633" xml:space="preserve">30. </s>
            <s xml:id="echoid-s4634" xml:space="preserve">Item ductam chordã CE,
              <lb/>
              <note position="right" xlink:label="note-131-03" xlink:href="note-131-03a" xml:space="preserve">Supputatio
                <lb/>
              ſinuum ar
                <lb/>
              cuũ gradi-
                <lb/>
              bus 15. ſeſe
                <lb/>
              ſuperantiũ
                <lb/>
              quales ſunt
                <lb/>
              arcus grad.
                <lb/>
              15. 30. 45.
                <lb/>
              60. 75. &
                <lb/>
              90.</note>
            ſecet recta AD, in K, bifariam, ex dicto lem-
              <lb/>
            mate; </s>
            <s xml:id="echoid-s4635" xml:space="preserve">ac propterea ad angulos rectos; </s>
            <s xml:id="echoid-s4636" xml:space="preserve">eritq́ue
              <lb/>
            CK, ſinus rectus grad. </s>
            <s xml:id="echoid-s4637" xml:space="preserve">15. </s>
            <s xml:id="echoid-s4638" xml:space="preserve">Denique rectæ iun-
              <lb/>
            gantur AE, EB. </s>
            <s xml:id="echoid-s4639" xml:space="preserve">Quoniam igitur arcus BE,
              <lb/>
            grad. </s>
            <s xml:id="echoid-s4640" xml:space="preserve">60. </s>
            <s xml:id="echoid-s4641" xml:space="preserve">ſexta pars eſt totius circunferentiæ
              <lb/>
            circuli, cum ſexies 60. </s>
            <s xml:id="echoid-s4642" xml:space="preserve">ſaciant 360. </s>
            <s xml:id="echoid-s4643" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s4644" xml:space="preserve">erit
              <lb/>
            recta BE, latus Hexagoni; </s>
            <s xml:id="echoid-s4645" xml:space="preserve">atque adeo, ex
              <lb/>
            coroll. </s>
            <s xml:id="echoid-s4646" xml:space="preserve">propof. </s>
            <s xml:id="echoid-s4647" xml:space="preserve">15. </s>
            <s xml:id="echoid-s4648" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4649" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4650" xml:space="preserve">Euel. </s>
            <s xml:id="echoid-s4651" xml:space="preserve">ſemidiametro AE, æqualis. </s>
            <s xml:id="echoid-s4652" xml:space="preserve">Anguliergo EAB,
              <lb/>
            EBA, æquales erunt: </s>
            <s xml:id="echoid-s4653" xml:space="preserve">Sunt autem & </s>
            <s xml:id="echoid-s4654" xml:space="preserve">anguli ad H, æquales, vtpote recti. </s>
            <s xml:id="echoid-s4655" xml:space="preserve">Igi-
              <lb/>
              <note position="right" xlink:label="note-131-04" xlink:href="note-131-04a" xml:space="preserve">5. primi.</note>
            tur cum duo anguli EAH, EHA, trianguli AEH, æquales ſint duobus an-
              <lb/>
            gulis EBH, EHB, trianguli BEH, latusq́ue AE, lateri BE, æquale; </s>
            <s xml:id="echoid-s4656" xml:space="preserve">erit la-
              <lb/>
            tus AH, lateri BH, æquale; </s>
            <s xml:id="echoid-s4657" xml:space="preserve">ac proinde AH, medietas erit ſemidia metri AB.
              <lb/>
            </s>
            <s xml:id="echoid-s4658" xml:space="preserve">
              <note position="right" xlink:label="note-131-05" xlink:href="note-131-05a" xml:space="preserve">26. primi.</note>
            Quare cum EG, ſinus rectus grad. </s>
            <s xml:id="echoid-s4659" xml:space="preserve">30. </s>
            <s xml:id="echoid-s4660" xml:space="preserve">ſit ipſi AH, æqualis, erit ſinus rectus
              <lb/>
              <note position="right" xlink:label="note-131-06" xlink:href="note-131-06a" xml:space="preserve">34. primi.</note>
            grad. </s>
            <s xml:id="echoid-s4661" xml:space="preserve">30. </s>
            <s xml:id="echoid-s4662" xml:space="preserve">medietati ſemidiametri, ſiue ſinus totius, æqualis. </s>
            <s xml:id="echoid-s4663" xml:space="preserve">Cum ergo ſinus
              <lb/>
              <note position="right" xlink:label="note-131-07" xlink:href="note-131-07a" xml:space="preserve">Sinus rectꝰ
                <lb/>
              grad. 30. æ-
                <lb/>
              qualis eſt
                <lb/>
              medietati
                <lb/>
              ſinus totiꝰ;</note>
            totus ponatur 10000000. </s>
            <s xml:id="echoid-s4664" xml:space="preserve">erit EG, ſinus grad. </s>
            <s xml:id="echoid-s4665" xml:space="preserve">30. </s>
            <s xml:id="echoid-s4666" xml:space="preserve">talium particularum
              <lb/>
            5000000. </s>
            <s xml:id="echoid-s4667" xml:space="preserve">nempe medietas ſinus totius.</s>
            <s xml:id="echoid-s4668" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4669" xml:space="preserve">EX hoc ſinu, per propof. </s>
            <s xml:id="echoid-s4670" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4671" xml:space="preserve">cognoſcetur ſinus complementi arcus grad. </s>
            <s xml:id="echoid-s4672" xml:space="preserve">30.</s>
            <s xml:id="echoid-s4673" xml:space="preserve"/>
          </p>
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