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

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          <p style="it">
            <s xml:id="echoid-s5546" xml:space="preserve">
              <pb o="129" file="141" n="141" rhead=""/>
            minores ſint arcu grad. </s>
            <s xml:id="echoid-s5547" xml:space="preserve">60. </s>
            <s xml:id="echoid-s5548" xml:space="preserve">detraherentur ex ſinubus omnium arcuum maiorũ, quàm
              <lb/>
            grad. </s>
            <s xml:id="echoid-s5549" xml:space="preserve">60. </s>
            <s xml:id="echoid-s5550" xml:space="preserve">remanerent ſinus omnium arcuum à principio quadrantis vſque ad arcum
              <lb/>
            grad. </s>
            <s xml:id="echoid-s5551" xml:space="preserve">30. </s>
            <s xml:id="echoid-s5552" xml:space="preserve">Denique ſi ſinus omnium arcuum à principio quadrantis vſque ad arcum
              <lb/>
            grad. </s>
            <s xml:id="echoid-s5553" xml:space="preserve">60. </s>
            <s xml:id="echoid-s5554" xml:space="preserve">inuenti eſſent, & </s>
            <s xml:id="echoid-s5555" xml:space="preserve">ſinus omnium arcuum minorum, quàm grad. </s>
            <s xml:id="echoid-s5556" xml:space="preserve">30. </s>
            <s xml:id="echoid-s5557" xml:space="preserve">ſinubus
              <lb/>
            omnium arcuum maiorum, quàm grad. </s>
            <s xml:id="echoid-s5558" xml:space="preserve">30. </s>
            <s xml:id="echoid-s5559" xml:space="preserve">adijcerentur, componerentur ſinus omnium
              <lb/>
            arcuum maiorum, quàm grad. </s>
            <s xml:id="echoid-s5560" xml:space="preserve">60.</s>
            <s xml:id="echoid-s5561" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5562" xml:space="preserve">_SED_ iam ſubijciamus tabulam ſinuum omnium arcuum quadrantis per ſingula
              <lb/>
            Minuta extenſorumà Ioanne Regiom. </s>
            <s xml:id="echoid-s5563" xml:space="preserve">ſupputatam, quam tamen pleriſque in locis ab
              <lb/>
            erroribus, qui incuria typographorum irrepſerant, purgauimus. </s>
            <s xml:id="echoid-s5564" xml:space="preserve">Cõtinet autem in hac
              <lb/>
            tabula ſinus totus particulas 10000000. </s>
            <s xml:id="echoid-s5565" xml:space="preserve">ratione cuius omnes alij ſinus inuenti ſunt.
              <lb/>
            </s>
            <s xml:id="echoid-s5566" xml:space="preserve">Quòd ſi à ſingulis reijciantur primæ duæ figuræ ad dexteram, (addita tamen vnita-
              <lb/>
            te, ſi duæ figuræ abiectæ numerum maiorem conſtituant, quàm 50.) </s>
            <s xml:id="echoid-s5567" xml:space="preserve">relinquetur ſinus
              <lb/>
            reſpectu ſinus totius 100000. </s>
            <s xml:id="echoid-s5568" xml:space="preserve">Si tamen quis eandem tabulam per præcepta tradita
              <lb/>
              <note position="right" xlink:label="note-141-01" xlink:href="note-141-01a" xml:space="preserve">Quid ageu
                <lb/>
              dum, vt ſe-
                <lb/>
              quens tabu
                <lb/>
              la ſinuum
                <lb/>
              exquiſite có
                <lb/>
              ſtruat, poſi-
                <lb/>
              to ſinu to-
                <lb/>
              to partium
                <lb/>
              10000000.</note>
            proprio Marte conſtrnere velit, poſito ſinu toto partium 10000000. </s>
            <s xml:id="echoid-s5569" xml:space="preserve">conſtituendus
              <lb/>
            ei erit ſinus totus in ſupputatione partium 1000000000. </s>
            <s xml:id="echoid-s5570" xml:space="preserve">_N_am ſi ex ſingulis ſinubus
              <lb/>
            inuentis abijciantur duæ primæ figuræ ad dexteram, vt diximus, reliqui erunt ſinus
              <lb/>
            reſpectu ſinus totius 10000000. </s>
            <s xml:id="echoid-s5571" xml:space="preserve">ijdem omnino, qui in tabula Ioannis Regiom. </s>
            <s xml:id="echoid-s5572" xml:space="preserve">de-
              <lb/>
            ſcripti ſunt. </s>
            <s xml:id="echoid-s5573" xml:space="preserve">_H_oc idcirco dico, ne mireris, non omnes ſinus per regulam proportionũ
              <lb/>
            inuentos ex ſinubus arcuum Minutis 45. </s>
            <s xml:id="echoid-s5574" xml:space="preserve">ſe ordine ſuperantium, poſito ſinu toto
              <lb/>
            10000000. </s>
            <s xml:id="echoid-s5575" xml:space="preserve">ad vnguem reſpondere ſinubus huius tabulæ. </s>
            <s xml:id="echoid-s5576" xml:space="preserve">Vt enim ſinus exquiſitere-
              <lb/>
            periatur reſpectu alicuius ſinus totius, conſtituendus eſt in ſupputatione ſinus totus
              <lb/>
            centies maior, vt ſupra dictum eſt.</s>
            <s xml:id="echoid-s5577" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5578" xml:space="preserve">_QVOD_ autẽ, abiectis duabus primis figuris ad dexterã ex ſingulis ſinubus, rema
              <lb/>
              <note position="right" xlink:label="note-141-02" xlink:href="note-141-02a" xml:space="preserve">Ratio,
                <lb/>
              cur abtectis
                <lb/>
              duabus pri
                <lb/>
              mis figuris
                <lb/>
              ex ſinu
                <lb/>
              quocũque
                <lb/>
              reſpectu ſi-
                <lb/>
              nus totius
                <lb/>
              10000000.
                <lb/>
              relinqua -
                <lb/>
              tut idé ſi-
                <lb/>
              nus reſpe-
                <lb/>
              ctu ſinꝰ to-
                <lb/>
              tiꝰ 100000.
                <lb/>
              licet prior
                <lb/>
              ille Ron ſit
                <lb/>
              exquiſitein
                <lb/>
              uentus.</note>
            neant ſinus exquiſiti reſpectu ſinus totius 100000. </s>
            <s xml:id="echoid-s5579" xml:space="preserve">quàmuis priores illi nõ ſint exqui
              <lb/>
            ſite inuenti, manifeſtũ eſt. </s>
            <s xml:id="echoid-s5580" xml:space="preserve">Quoniã enim ſinus totus, ſiue ſemidiameter ad ſinum rectũ
              <lb/>
            quemcumq; </s>
            <s xml:id="echoid-s5581" xml:space="preserve">determinatã quandã proportionẽ habet; </s>
            <s xml:id="echoid-s5582" xml:space="preserve">fit, vt omnes partes illius, quot-
              <lb/>
            cunq; </s>
            <s xml:id="echoid-s5583" xml:space="preserve">illæ ſint, ad partes huius inuẽtas reſpectu illarũ partiũ ſinus totius eandẽ habeãt
              <lb/>
            proportionem, quam omnes partes eiusdẽ ſinus totius pauciores, quàm illæ priores, ha
              <lb/>
            bent ad partes eiuſdem ſinus reſpectu illarum partium ſinus totius pauciorum : </s>
            <s xml:id="echoid-s5584" xml:space="preserve">alio-
              <lb/>
            quin ſinus totus non haberet ſemper ad eundem ſinum eandem proportionem, ſed ali-
              <lb/>
            quando eſſet maioris quantitatis reſpectu illius, & </s>
            <s xml:id="echoid-s5585" xml:space="preserve">ali quando minoris : </s>
            <s xml:id="echoid-s5586" xml:space="preserve">quod eſt ab-
              <lb/>
            ſurdum. </s>
            <s xml:id="echoid-s5587" xml:space="preserve">Quocirca ſi ſinus cuiuslibet arcus, vt v. </s>
            <s xml:id="echoid-s5588" xml:space="preserve">g illius, qui continet grad. </s>
            <s xml:id="echoid-s5589" xml:space="preserve">28. </s>
            <s xml:id="echoid-s5590" xml:space="preserve">inuen-
              <lb/>
            tus ſit reſpectu ſinus totius in quotuis particulas diſtributi, facile per regulam pro-
              <lb/>
            portionum inueniemus eundem ſinũ reſpectu eiuſdem ſinus totius in pauciores partes
              <lb/>
            diuiſi, ſi ita dicamus. </s>
            <s xml:id="echoid-s5591" xml:space="preserve">Si ſinus totus partium 10000000. </s>
            <s xml:id="echoid-s5592" xml:space="preserve">dat ſinum arcus grad. </s>
            <s xml:id="echoid-s5593" xml:space="preserve">28. </s>
            <s xml:id="echoid-s5594" xml:space="preserve">par
              <lb/>
            tium 4694716. </s>
            <s xml:id="echoid-s5595" xml:space="preserve">Idem ſinus totus partium 100000. </s>
            <s xml:id="echoid-s5596" xml:space="preserve">quot partium dabit ſinam eiuſdẽ
              <lb/>
            arcus grad. </s>
            <s xml:id="echoid-s5597" xml:space="preserve">28? </s>
            <s xml:id="echoid-s5598" xml:space="preserve">Inueniemus enim ſinum partium 46947 {1600000/10000000}. </s>
            <s xml:id="echoid-s5599" xml:space="preserve">Omiſ-
              <lb/>
            ſa autem hac fractione, (quòd minor ſit, quàm vnitas.) </s>
            <s xml:id="echoid-s5600" xml:space="preserve">continebit idem ſinus partes
              <lb/>
            duntaxat 46947. </s>
            <s xml:id="echoid-s5601" xml:space="preserve">quemadmodum in tabulis ſinuum ponitur, in quibus ſinus totus
              <lb/>
            continet partes 100000. </s>
            <s xml:id="echoid-s5602" xml:space="preserve">Vbi vides, ſinum hunc relinqui, ſi ex illo duæ primæ figuræ
              <lb/>
            ad dexteram abijciantur. </s>
            <s xml:id="echoid-s5603" xml:space="preserve">Ratio huius abiectionis eſt; </s>
            <s xml:id="echoid-s5604" xml:space="preserve">quia vt ſinus ille grad. </s>
            <s xml:id="echoid-s5605" xml:space="preserve">28. </s>
            <s xml:id="echoid-s5606" xml:space="preserve">mul-
              <lb/>
            tiplicatus per ſinum totum 100000 (quæ multiplicatio fit per appoſitionem quinque
              <lb/>
            cifrarũ, hoc modo; </s>
            <s xml:id="echoid-s5607" xml:space="preserve">469471600000. </s>
            <s xml:id="echoid-s5608" xml:space="preserve">vt in cap. </s>
            <s xml:id="echoid-s5609" xml:space="preserve">4. </s>
            <s xml:id="echoid-s5610" xml:space="preserve">Arithmeticæ diximus.) </s>
            <s xml:id="echoid-s5611" xml:space="preserve">diuidatur per
              <lb/>
            ſinum totum 10000000. </s>
            <s xml:id="echoid-s5612" xml:space="preserve">vt regula proportionnm præcipit; </s>
            <s xml:id="echoid-s5613" xml:space="preserve">ſatis eſt, ſi ex numero pro-
              <lb/>
            ducto reijciantur ſeptem figuræ, quotnimirum cifræ ſunt in diuiſore 10000000.
              <lb/>
            </s>
            <s xml:id="echoid-s5614" xml:space="preserve">vt in cap. </s>
            <s xml:id="echoid-s5615" xml:space="preserve">5. </s>
            <s xml:id="echoid-s5616" xml:space="preserve">Arithmeticæ docuimus. </s>
            <s xml:id="echoid-s5617" xml:space="preserve">Quare reijciendæ ſunt quing; </s>
            <s xml:id="echoid-s5618" xml:space="preserve">illæ cifræ appoſitæ,
              <lb/>
            & </s>
            <s xml:id="echoid-s5619" xml:space="preserve">præterea duæ figuræ primæ, nẽpe hic numerus 1600000. </s>
            <s xml:id="echoid-s5620" xml:space="preserve">qui cum diuiſore hanc </s>
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