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

Page concordance

< >
Scan Original
141 129
142 130
143 131
144 132
145 133
146 134
147 135
148 136
149 137
150 138
151 139
152 140
153 141
154 142
155 143
156 144
157 145
158 146
159 147
160 148
161 149
162 150
163 151
164 152
165 153
166 154
167 155
168 156
169 157
170 158
< >
page |< < (130) of 532 > >|
    <echo version="1.0RC">
      <text xml:lang="la" type="free">
        <div xml:id="echoid-div397" type="section" level="1" n="167">
          <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>
          </p>
        </div>
      </text>
    </echo>
Impressum