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

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          <p>
            <s xml:id="echoid-s5271" xml:space="preserve">
              <pb o="127" file="139" n="139" rhead=""/>
            ſinus grad. </s>
            <s xml:id="echoid-s5272" xml:space="preserve">1? </s>
            <s xml:id="echoid-s5273" xml:space="preserve">Inuenies enim differentiã 43624. </s>
            <s xml:id="echoid-s5274" xml:space="preserve">quæ addita ad 130896. </s>
            <s xml:id="echoid-s5275" xml:space="preserve">ſinũ Min.
              <lb/>
            </s>
            <s xml:id="echoid-s5276" xml:space="preserve">45. </s>
            <s xml:id="echoid-s5277" xml:space="preserve">faciat 174520. </s>
            <s xml:id="echoid-s5278" xml:space="preserve">ſinũ grad. </s>
            <s xml:id="echoid-s5279" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5280" xml:space="preserve">qui licet minor aliquanto ſit illo, quem alio mo
              <lb/>
            do inuenimus, abiectis tamẽ duabus primis ſiguris 20. </s>
            <s xml:id="echoid-s5281" xml:space="preserve">relinquetur ſinus 1745. </s>
            <s xml:id="echoid-s5282" xml:space="preserve">
              <lb/>
            exquiſitiſsimus grad. </s>
            <s xml:id="echoid-s5283" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5284" xml:space="preserve">reſpectu ſinus totius 100000. </s>
            <s xml:id="echoid-s5285" xml:space="preserve">Dic pręterea. </s>
            <s xml:id="echoid-s5286" xml:space="preserve">Iiſdẽ poſi-
              <lb/>
            tis, quantam differentiam poſcunt Minuta 30. </s>
            <s xml:id="echoid-s5287" xml:space="preserve">addendam eidẽ ſinui Min. </s>
            <s xml:id="echoid-s5288" xml:space="preserve">45. </s>
            <s xml:id="echoid-s5289" xml:space="preserve">
              <lb/>
            vt ſiat ſinus Min. </s>
            <s xml:id="echoid-s5290" xml:space="preserve">75. </s>
            <s xml:id="echoid-s5291" xml:space="preserve">hoc eſt, ſinus grad. </s>
            <s xml:id="echoid-s5292" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5293" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5294" xml:space="preserve">15? </s>
            <s xml:id="echoid-s5295" xml:space="preserve">Inuenies enim differentiam
              <lb/>
            87249. </s>
            <s xml:id="echoid-s5296" xml:space="preserve">quæ addita ad 130896. </s>
            <s xml:id="echoid-s5297" xml:space="preserve">ſinum Min. </s>
            <s xml:id="echoid-s5298" xml:space="preserve">45. </s>
            <s xml:id="echoid-s5299" xml:space="preserve">faciet 218145. </s>
            <s xml:id="echoid-s5300" xml:space="preserve">ſinum grad. </s>
            <s xml:id="echoid-s5301" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5302" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5303" xml:space="preserve">
              <lb/>
            15. </s>
            <s xml:id="echoid-s5304" xml:space="preserve">qui licet ſit aliquanto minor illo, quem prior ille modus exhibet, tamen
              <lb/>
            abiectis duabus primis figuris 45. </s>
            <s xml:id="echoid-s5305" xml:space="preserve">remanebit ſinus 2181. </s>
            <s xml:id="echoid-s5306" xml:space="preserve">ex quiſitiſsimus grad. </s>
            <s xml:id="echoid-s5307" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5308" xml:space="preserve">
              <lb/>
            Min. </s>
            <s xml:id="echoid-s5309" xml:space="preserve">15. </s>
            <s xml:id="echoid-s5310" xml:space="preserve">reſpectu ſinus totius 100000. </s>
            <s xml:id="echoid-s5311" xml:space="preserve">Item dic. </s>
            <s xml:id="echoid-s5312" xml:space="preserve">Si Minuta 45. </s>
            <s xml:id="echoid-s5313" xml:space="preserve">quibus arcus
              <lb/>
            grad. </s>
            <s xml:id="echoid-s5314" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5315" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5316" xml:space="preserve">30. </s>
            <s xml:id="echoid-s5317" xml:space="preserve">differt ab arcu ſequentigrad 2. </s>
            <s xml:id="echoid-s5318" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5319" xml:space="preserve">15. </s>
            <s xml:id="echoid-s5320" xml:space="preserve">requirunt differẽtiam
              <lb/>
            130829. </s>
            <s xml:id="echoid-s5321" xml:space="preserve">addendã ſinuigrad. </s>
            <s xml:id="echoid-s5322" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5323" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5324" xml:space="preserve">30 vt ſiat ſinus grad. </s>
            <s xml:id="echoid-s5325" xml:space="preserve">2. </s>
            <s xml:id="echoid-s5326" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5327" xml:space="preserve">15. </s>
            <s xml:id="echoid-s5328" xml:space="preserve">quantã diffe
              <lb/>
            rentiam poſcunt Minuta 15. </s>
            <s xml:id="echoid-s5329" xml:space="preserve">addendam eidẽ ſinui grad. </s>
            <s xml:id="echoid-s5330" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5331" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5332" xml:space="preserve">30. </s>
            <s xml:id="echoid-s5333" xml:space="preserve">vt ſiat ſinus
              <lb/>
            grad. </s>
            <s xml:id="echoid-s5334" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5335" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5336" xml:space="preserve">45? </s>
            <s xml:id="echoid-s5337" xml:space="preserve">quantam præterea differentiã ſlagitant Minuta 30. </s>
            <s xml:id="echoid-s5338" xml:space="preserve">addendam
              <lb/>
            eidem ſinui grad. </s>
            <s xml:id="echoid-s5339" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5340" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5341" xml:space="preserve">30. </s>
            <s xml:id="echoid-s5342" xml:space="preserve">vt fiat ſinus grad. </s>
            <s xml:id="echoid-s5343" xml:space="preserve">2? </s>
            <s xml:id="echoid-s5344" xml:space="preserve">Reperies enim priorem diffe-
              <lb/>
            rentiã eſſe 43610. </s>
            <s xml:id="echoid-s5345" xml:space="preserve">quę addita ad 261769. </s>
            <s xml:id="echoid-s5346" xml:space="preserve">ſinum grad. </s>
            <s xml:id="echoid-s5347" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5348" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5349" xml:space="preserve">30. </s>
            <s xml:id="echoid-s5350" xml:space="preserve">efficiet 305379. </s>
            <s xml:id="echoid-s5351" xml:space="preserve">
              <lb/>
            ſinum grad. </s>
            <s xml:id="echoid-s5352" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5353" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5354" xml:space="preserve">45. </s>
            <s xml:id="echoid-s5355" xml:space="preserve">Poſteriorem vero differentiam inuenies eſſe 87219. </s>
            <s xml:id="echoid-s5356" xml:space="preserve">quæ
              <lb/>
            addita ad 261769. </s>
            <s xml:id="echoid-s5357" xml:space="preserve">ſinum eundem grad. </s>
            <s xml:id="echoid-s5358" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5359" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5360" xml:space="preserve">30. </s>
            <s xml:id="echoid-s5361" xml:space="preserve">componet 348988. </s>
            <s xml:id="echoid-s5362" xml:space="preserve">ſinũ grad. </s>
            <s xml:id="echoid-s5363" xml:space="preserve">
              <lb/>
            2. </s>
            <s xml:id="echoid-s5364" xml:space="preserve">qui duo ſinus quamuis minores ſint, quàm illi, quos prior ille modus exhi-
              <lb/>
            bet, reiectis tamen duabus figuris primis ex vtroque, reliquierunt ſinus 3053. </s>
            <s xml:id="echoid-s5365" xml:space="preserve">
              <lb/>
            3489. </s>
            <s xml:id="echoid-s5366" xml:space="preserve">exquiſitiſsimi reſpectu ſinus totius 100000. </s>
            <s xml:id="echoid-s5367" xml:space="preserve">Hac eadem arte progredien
              <lb/>
            dum erit in cæteris, donec inuenti ſint ſinus omnium arcuum per 15. </s>
            <s xml:id="echoid-s5368" xml:space="preserve">Minuta
              <lb/>
              <note position="right" xlink:label="note-139-01" xlink:href="note-139-01a" xml:space="preserve">Sinus eut
                <lb/>
              magis ex-
                <lb/>
              quiſite per
                <lb/>
              regulã pro-
                <lb/>
              portionum
                <lb/>
              inueniátur
                <lb/>
              prope qua-
                <lb/>
              drantis ini
                <lb/>
              tiũ quàm
                <lb/>
              prope finé.</note>
            extenſorum vſque ad arcum grad. </s>
            <s xml:id="echoid-s5369" xml:space="preserve">45. </s>
            <s xml:id="echoid-s5370" xml:space="preserve">Vltra hunc etenim progredi hac via nõ
              <lb/>
            expedit, cum magis exquiſite ſinus complementorum arcuũ illorum, per pro-
              <lb/>
            poſ. </s>
            <s xml:id="echoid-s5371" xml:space="preserve">3. </s>
            <s xml:id="echoid-s5372" xml:space="preserve">inueſtigari poſsint, reſpectu ſinus totius 10000000. </s>
            <s xml:id="echoid-s5373" xml:space="preserve">quàm per regulam
              <lb/>
            proportionum; </s>
            <s xml:id="echoid-s5374" xml:space="preserve">propterea quòd, vt ſupra oſtendimus in coroll. </s>
            <s xml:id="echoid-s5375" xml:space="preserve">propof. </s>
            <s xml:id="echoid-s5376" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5377" xml:space="preserve">dif-
              <lb/>
            ferentiæ ſinuum verſus ſinem quadrantis minores ſunt, quàm prope initium
              <lb/>
            quadrantis; </s>
            <s xml:id="echoid-s5378" xml:space="preserve">ac proinde fæpius differentiæ ſinuum mutantur prope ſinem qua-
              <lb/>
            drantis, quàm iuxta principium. </s>
            <s xml:id="echoid-s5379" xml:space="preserve">Ex quo ſit rectius prope quadrantis initium
              <lb/>
            reperiri ſinus per regulam proportionum, quàm prope ſinem: </s>
            <s xml:id="echoid-s5380" xml:space="preserve">quandoquidem
              <lb/>
            ibi vna eademq́ue differentia pluribus ſinubus deſeruit, quàm hic, vt dictũ eſt.
              <lb/>
            </s>
            <s xml:id="echoid-s5381" xml:space="preserve">Id quod experientia a pertiſsime quoque demonſtrat.</s>
            <s xml:id="echoid-s5382" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5383" xml:space="preserve">QVOD ſi omnes ſinus arcuum per 15. </s>
            <s xml:id="echoid-s5384" xml:space="preserve">Minuta progredientium, initio fa-
              <lb/>
              <note position="right" xlink:label="note-139-02" xlink:href="note-139-02a" xml:space="preserve">Supputatio
                <lb/>
              ſinuum ar-
                <lb/>
              cuum per
                <lb/>
              5. Minuta,
                <lb/>
              & perſingu
                <lb/>
              la Minura
                <lb/>
              extenſorũ.</note>
            cto ab arcu Min. </s>
            <s xml:id="echoid-s5385" xml:space="preserve">0. </s>
            <s xml:id="echoid-s5386" xml:space="preserve">in ordinem redigantur, vna cum eorum differentijs; </s>
            <s xml:id="echoid-s5387" xml:space="preserve">repe-
              <lb/>
            riemus eodem artificio ſinus omnium arcuum per 5. </s>
            <s xml:id="echoid-s5388" xml:space="preserve">minuta extenſorum: </s>
            <s xml:id="echoid-s5389" xml:space="preserve">Ex
              <lb/>
            quibus demum eadem ratione ſinus omnium arcuum per ſingula minuta pro-
              <lb/>
            gredientium explorabimus; </s>
            <s xml:id="echoid-s5390" xml:space="preserve">ac proinde tabulam ſinuum conficiem us vſque ad
              <lb/>
            arcum gra. </s>
            <s xml:id="echoid-s5391" xml:space="preserve">45. </s>
            <s xml:id="echoid-s5392" xml:space="preserve">Per ſinus autem horũ arcuũ eliciemus, per propof. </s>
            <s xml:id="echoid-s5393" xml:space="preserve">3. </s>
            <s xml:id="echoid-s5394" xml:space="preserve">ſinus com-
              <lb/>
            plementorum arcuum eorundem. </s>
            <s xml:id="echoid-s5395" xml:space="preserve">Quare tota ſinuum tabula conſecta erit: </s>
            <s xml:id="echoid-s5396" xml:space="preserve">ac
              <lb/>
            proinde ſinus rectos omnium arcuum Quadrantis ſeſe ordine ſuperantium
              <lb/>
            vno Minuto, in partibus Sinus totius in quotcunque particulas diſtributi,
              <lb/>
            ſupputauimus. </s>
            <s xml:id="echoid-s5397" xml:space="preserve">Quod faciendum erat.</s>
            <s xml:id="echoid-s5398" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div397" type="section" level="1" n="167">
          <head xml:id="echoid-head193" xml:space="preserve">SCHOLIVM.</head>
          <note position="right" xml:space="preserve">Demonſtr@
            <lb/>
          tio ſupputa
            <lb/>
          tiõis ſinuũ
            <lb/>
          per regulã
            <lb/>
          proportio-
            <lb/>
          num.</note>
          <p style="it">
            <s xml:id="echoid-s5399" xml:space="preserve">_CAETERVM_ vt rationem ſupputationis ſinuum per proportionum regulam
              <lb/>
            videas, ſint in Quadrante _ABC_, arcuum _BD_, _BE_, ſinus _DF_, _EG_, noti, ex quibus
              <lb/>
            propoſitum ſit elicere ſinum _HI_, arcus _BH_, inter duos illos arcus poſiti Ducta recta
              <lb/>
            _DK_, perpendiculari ad _EG_; </s>
            <s xml:id="echoid-s5400" xml:space="preserve">erunt rectæ _LI_, _KG_, ſinui _DF_, æquales s atque </s>
          </p>
        </div>
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