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

Table of figures

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          <p style="it">
            <s xml:id="echoid-s5620" xml:space="preserve">
              <pb o="130" file="142" n="142" rhead=""/>
            nutiam {1600000/10000000}. </s>
            <s xml:id="echoid-s5621" xml:space="preserve">conſtituit, quæ vnitate minor eſt; </s>
            <s xml:id="echoid-s5622" xml:space="preserve">proptered quòd n@
              <lb/>
            merator cotinet ſeptem figuras, denominator autem octo. </s>
            <s xml:id="echoid-s5623" xml:space="preserve">Eademq́; </s>
            <s xml:id="echoid-s5624" xml:space="preserve">ratio eſt in omnibus
              <lb/>
            alijs ſinubus. </s>
            <s xml:id="echoid-s5625" xml:space="preserve">Hinc fit, ſinum relictum poſt abiectionem duarum primarum figurarum
              <lb/>
            ſatis exquiſitum eſſe reſpectu ſinus totius 100000. </s>
            <s xml:id="echoid-s5626" xml:space="preserve">etiamſi ille, a quo duæ figuræ abij
              <lb/>
            ciuntur, reſpectu ſinus totius 10000000. </s>
            <s xml:id="echoid-s5627" xml:space="preserve">non eſſet exquiſite inuentus. </s>
            <s xml:id="echoid-s5628" xml:space="preserve">Cum enim to-
              <lb/>
            tus error, qui in ſupputatione contingere poteſt, (quando nimirum conſtituendus eſt
              <lb/>
            ſinus inter duos numeros, quorum vnus vero ſinu maior eſt, & </s>
            <s xml:id="echoid-s5629" xml:space="preserve">alter minor; </s>
            <s xml:id="echoid-s5630" xml:space="preserve">Vel quan-
              <lb/>
            do per regulam proportionum ſinus inquiritur: </s>
            <s xml:id="echoid-s5631" xml:space="preserve">hic enim maius periculum errandi eſ-
              <lb/>
            ſe poteſt. </s>
            <s xml:id="echoid-s5632" xml:space="preserve">Nam quando ſinus inuenitur per extractionem radicis quadratæ, error
              <lb/>
            vnitatem non excedit.) </s>
            <s xml:id="echoid-s5633" xml:space="preserve">conſiſtat vel in prima ſola figura ad dexteram, vel in duabus
              <lb/>
            primis, ita vt ad ſummum error ſit in 99. </s>
            <s xml:id="echoid-s5634" xml:space="preserve">vnitatibus, quibus ſinus inuentus verum ſi-
              <lb/>
            num excedat, vel ab eo deficiat; </s>
            <s xml:id="echoid-s5635" xml:space="preserve">(quis enim pluribus vnitatibus à ſcopo aberret, niſi
              <lb/>
            plane rerum Geometricarum, at que Arithmeticarum ſit ignarus?) </s>
            <s xml:id="echoid-s5636" xml:space="preserve">Duæ vero primæ
              <lb/>
            figuræ cum quinque cifris appoſitis conſtituant numeratorem fractionis, quam diui-
              <lb/>
            ſio exhibet, minorem denominatore, ita vt fractio minor ſit, quàm vnitas; </s>
            <s xml:id="echoid-s5637" xml:space="preserve">liquet,
              <lb/>
            ſatis exquiſitum ſinum relinqui.</s>
            <s xml:id="echoid-s5638" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">Siad ſinũ
            <lb/>
          quemcúq;
            <lb/>
          reſpectu ſi-
            <lb/>
          nus totius
            <lb/>
          100000.
            <lb/>
          adijciãtut
            <lb/>
          duæ cifræ
            <lb/>
          ad dexterã,
            <lb/>
          fitidẽ ſinꝰ
            <lb/>
          reſpectu ſi-
            <lb/>
          nus totius
            <lb/>
          10000000.</note>
          <p style="it">
            <s xml:id="echoid-s5639" xml:space="preserve">_E A D E M_ ratione, ſicuilibet ſinui reſpectu ſinus totius partium 100000. </s>
            <s xml:id="echoid-s5640" xml:space="preserve">inuen-
              <lb/>
            to apponãtur duæ cifræ ad dexteram, habebitur idem ſinus reſpectu ſinus totius par-
              <lb/>
            tium 10000000. </s>
            <s xml:id="echoid-s5641" xml:space="preserve">Nam ſi dicamus verbi gratia; </s>
            <s xml:id="echoid-s5642" xml:space="preserve">Sinus totus partium 100000. </s>
            <s xml:id="echoid-s5643" xml:space="preserve">dat
              <lb/>
            ſinum arcus grad. </s>
            <s xml:id="echoid-s5644" xml:space="preserve">28. </s>
            <s xml:id="echoid-s5645" xml:space="preserve">partium 46947: </s>
            <s xml:id="echoid-s5646" xml:space="preserve">Sinus ergo totus partiũ 10000000. </s>
            <s xml:id="echoid-s5647" xml:space="preserve">quot par-
              <lb/>
            tium dabit eundẽ ſinũ arcus grad. </s>
            <s xml:id="echoid-s5648" xml:space="preserve">28? </s>
            <s xml:id="echoid-s5649" xml:space="preserve">reperiemus ſinũ partium 4694700. </s>
            <s xml:id="echoid-s5650" xml:space="preserve">Vbi vides,
              <lb/>
            ſinum hunc procreari, ſi illi duæ ciſræ ad dexteram adijciantur. </s>
            <s xml:id="echoid-s5651" xml:space="preserve">Ratio huius adiectio-
              <lb/>
            nis eſt; </s>
            <s xml:id="echoid-s5652" xml:space="preserve">quia vt ſinus ille grad. </s>
            <s xml:id="echoid-s5653" xml:space="preserve">28. </s>
            <s xml:id="echoid-s5654" xml:space="preserve">multiplicatus per ſinum totum 10000000. </s>
            <s xml:id="echoid-s5655" xml:space="preserve">(quæ
              <lb/>
            multiplicatio fit per appoſitionem ſeptem cifrarum, vt in cap 4. </s>
            <s xml:id="echoid-s5656" xml:space="preserve">Arithmeticæ tradidi-
              <lb/>
            mus) diuidatur per ſinum totum 100000. </s>
            <s xml:id="echoid-s5657" xml:space="preserve">vt regula proportionum præcipit, ſatis eſt,
              <lb/>
            ſi ex numero producto 469470000000. </s>
            <s xml:id="echoid-s5658" xml:space="preserve">auferantur quinque cifræ, vt ex cap. </s>
            <s xml:id="echoid-s5659" xml:space="preserve">5. </s>
            <s xml:id="echoid-s5660" xml:space="preserve">no
              <lb/>
            ſtræ Arithmeticæ conſtat. </s>
            <s xml:id="echoid-s5661" xml:space="preserve">Quocirca relinquetur prior ſinus cum duabus cifris ad dex-
              <lb/>
            tram appoſitis. </s>
            <s xml:id="echoid-s5662" xml:space="preserve">Quòd autẽ ex ſinu 46947. </s>
            <s xml:id="echoid-s5663" xml:space="preserve">non sit inuentus sinus 4694716. </s>
            <s xml:id="echoid-s5664" xml:space="preserve">ille idẽ, ex
              <lb/>
            quo prius illum elicuimus, ſed ſolum hic 4694700. </s>
            <s xml:id="echoid-s5665" xml:space="preserve">cauſa eſt, quòd sinus 46947. </s>
            <s xml:id="echoid-s5666" xml:space="preserve">non eſt
              <lb/>
            omnino exquisitus reſpectu sinus totius 100000. </s>
            <s xml:id="echoid-s5667" xml:space="preserve">Deberet namq; </s>
            <s xml:id="echoid-s5668" xml:space="preserve">eſſe 46947. </s>
            <s xml:id="echoid-s5669" xml:space="preserve">& </s>
            <s xml:id="echoid-s5670" xml:space="preserve">inſu-
              <lb/>
            per {1600000/10000000}. </s>
            <s xml:id="echoid-s5671" xml:space="preserve">vt ex dictis patet, ex quo præciſe inuenietur sinus ille
              <lb/>
            4694716. </s>
            <s xml:id="echoid-s5672" xml:space="preserve">Sed licet hæ 16. </s>
            <s xml:id="echoid-s5673" xml:space="preserve">vnitates negligantur, accipiaturq́; </s>
            <s xml:id="echoid-s5674" xml:space="preserve">sinus 4694700. </s>
            <s xml:id="echoid-s5675" xml:space="preserve">qua-
              <lb/>
            lem inuenimus, non tamen fit error notabilis, cum 16. </s>
            <s xml:id="echoid-s5676" xml:space="preserve">vnitates reſpectu sinus totius
              <lb/>
            sint {16/10000000}. </s>
            <s xml:id="echoid-s5677" xml:space="preserve">quæ minutia multo minor eſt, quàm {1/216000}.
              <lb/>
            </s>
            <s xml:id="echoid-s5678" xml:space="preserve">hoc eſt, quàm vnũ ſecundũ reſpectu sinus totius 60. </s>
            <s xml:id="echoid-s5679" xml:space="preserve">vt merito negligi poßit. </s>
            <s xml:id="echoid-s5680" xml:space="preserve">Ad ſum
              <lb/>
            mũ poterit aliquãdo cõtingere error, quàmuis valde raro, in {99/10000000}. </s>
            <s xml:id="echoid-s5681" xml:space="preserve">
              <lb/>
            quæ minutia licet sit ali quanto maior, quàm {1/216000}. </s>
            <s xml:id="echoid-s5682" xml:space="preserve">hoc eſt, quàm vnum
              <lb/>
            Secundum reſpectu sinus totius 60 eſt tamen multo minor, quàm {1/3600}. </s>
            <s xml:id="echoid-s5683" xml:space="preserve">hoc eſt,
              <lb/>
            quàm vnum Minutum reſpectu sinus totius 60. </s>
            <s xml:id="echoid-s5684" xml:space="preserve">Id vero, quod de sinubus totis partiũ
              <lb/>
            10000000 & </s>
            <s xml:id="echoid-s5685" xml:space="preserve">100000. </s>
            <s xml:id="echoid-s5686" xml:space="preserve">diximus, intelligendum quoq; </s>
            <s xml:id="echoid-s5687" xml:space="preserve">eſt de alijs sinubus totis quot-
              <lb/>
              <note position="left" xlink:label="note-142-02" xlink:href="note-142-02a" xml:space="preserve">Quo pacto
                <lb/>
              ex ſinubus
                <lb/>
              maioribus
                <lb/>
              ſiant mino
                <lb/>
              res, & con-
                <lb/>
              tra, quotcũ
                <lb/>
              que parti-
                <lb/>
              cularum ſi
                <lb/>
              nus totus
                <lb/>
              @atuatur.</note>
            cunq; </s>
            <s xml:id="echoid-s5688" xml:space="preserve">partium siue plurium, siue pauciorum. </s>
            <s xml:id="echoid-s5689" xml:space="preserve">Semper enim ex sinubus re ſpectu sinus
              <lb/>
            totius maioris inuentis abijciendæ ſunt tot figuræ, vt relinquantur sinus reſpectu si-
              <lb/>
            nus totius minoris, quot ci fris sinus totus maior sinum totum minorem ſuperat: </s>
            <s xml:id="echoid-s5690" xml:space="preserve">Item
              <lb/>
            sinubus reſpectu sinus totius minoris inuentis adijciẽ dæ ſunt tot cifræ, vt fiãt sinus re
              <lb/>
            ſpectu sinus totius maioris, quot cifris minor sinus totus à sinu toto maiore ſupera-
              <lb/>
            tur. </s>
            <s xml:id="echoid-s5691" xml:space="preserve">Quod eodem modo demonſtrabitur. </s>
            <s xml:id="echoid-s5692" xml:space="preserve">Vt si ſupputentur sinus reſpectu sinus totius
              <lb/>
            100000000. </s>
            <s xml:id="echoid-s5693" xml:space="preserve">& </s>
            <s xml:id="echoid-s5694" xml:space="preserve">ex singulis abijciantur tres primæ figuræ ad dexteram, reliqui </s>
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