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

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        <div xml:id="echoid-div397" type="section" level="1" n="167">
          <p style="it">
            <s xml:id="echoid-s5400" xml:space="preserve">
              <pb o="128" file="140" n="140" rhead=""/>
            _HL_, _EK_, differentiæ inter ſinus _HI_, _EG_, & </s>
            <s xml:id="echoid-s5401" xml:space="preserve">ſinum _DT_. </s>
            <s xml:id="echoid-s5402" xml:space="preserve">Quoniam vero arcus _DE_,
              <lb/>
            ſi e xiguus fuerit, qualem nos ponimus, nempe Minutorum _45_. </s>
            <s xml:id="echoid-s5403" xml:space="preserve">vel _15_. </s>
            <s xml:id="echoid-s5404" xml:space="preserve">vel _5_. </s>
            <s xml:id="echoid-s5405" xml:space="preserve">(Ex his
              <lb/>
            enim ſolis intermediorum arcuum ſinus inquirimus) à recta linea, quoad ſenſum, non
              <lb/>
              <figure xlink:label="fig-140-01" xlink:href="fig-140-01a" number="133">
                <image file="140-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/140-01"/>
              </figure>
            differt, ac proinde multo minus arcus _DH_; </s>
            <s xml:id="echoid-s5406" xml:space="preserve">erunt trianæ
              <lb/>
            gula _DEK_, _DHL_, quodammodo rectilinea. </s>
            <s xml:id="echoid-s5407" xml:space="preserve">& </s>
            <s xml:id="echoid-s5408" xml:space="preserve">æquian-
              <lb/>
            gula inter ſe. </s>
            <s xml:id="echoid-s5409" xml:space="preserve">Quamobrem erit, vt _DE_, differentia inter
              <lb/>
              <note position="left" xlink:label="note-140-01" xlink:href="note-140-01a" xml:space="preserve">4. fexti.</note>
            arcum _BD_, & </s>
            <s xml:id="echoid-s5410" xml:space="preserve">arcum _BE_, ad _EK_, differentiam inter ſi-
              <lb/>
            num _DF_, & </s>
            <s xml:id="echoid-s5411" xml:space="preserve">ſinum _EG_, ita _DH_, differentia inter ar-
              <lb/>
            cum _BD_, & </s>
            <s xml:id="echoid-s5412" xml:space="preserve">arcum _BH_, ad _HL_, differentiam inter ſi-
              <lb/>
            num _DF_, & </s>
            <s xml:id="echoid-s5413" xml:space="preserve">ſinum _HI_. </s>
            <s xml:id="echoid-s5414" xml:space="preserve">Cum ergo priores tres magnitu-
              <lb/>
            dines ſint cognitæ, (nempe arcus _DE_, quo arcus _BE_, arcũ
              <lb/>
            _BD_, ſuperat; </s>
            <s xml:id="echoid-s5415" xml:space="preserve">& </s>
            <s xml:id="echoid-s5416" xml:space="preserve">recta _EK_, qua ſinus _EG_, ſinum _DF_,
              <lb/>
            excedit; </s>
            <s xml:id="echoid-s5417" xml:space="preserve">nec non arcus _DH_, quo arcus _BH_, cuius ſinus
              <lb/>
            queritur, ſuperat arcum _BD_.) </s>
            <s xml:id="echoid-s5418" xml:space="preserve">cognita etiam erit quarta magnitudo, id eſt, recta _HL_,
              <lb/>
            quæ addenda eſt ſinui _DF_, vt fiat ſinus _HI_. </s>
            <s xml:id="echoid-s5419" xml:space="preserve">Conſtat igitur, ſinus per regulam pro-
              <lb/>
            portionum inuentos ſenſibiliter non differre à veris ſinubus, præſertim quando arcus
              <lb/>
            _DE_, quo arcus cognitorum ſinuuminter ſe differunt, valde exiguus eſt, ita vt à recta
              <lb/>
            linea vix differat.</s>
            <s xml:id="echoid-s5420" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5421" xml:space="preserve">_MAGNVM_ quoque compendium in hoc negotio nobis afferet propoſitio octaua.
              <lb/>
            </s>
            <s xml:id="echoid-s5422" xml:space="preserve">
              <note position="left" xlink:label="note-140-02" xlink:href="note-140-02a" xml:space="preserve">Compédiũ
                <lb/>
              miri ficum
                <lb/>
              pro inuen-
                <lb/>
              rione pluti
                <lb/>
              morum ſi-
                <lb/>
              nuum.</note>
            Ex ea enim plurimos ſinus ex alijs inuentis per ſolam additionem, ſubtractionemue
              <lb/>
            nanciſcemur. </s>
            <s xml:id="echoid-s5423" xml:space="preserve">_N_am ſi ſinum cuiuſuis arcus, qui maior non ſit, quàm grad. </s>
            <s xml:id="echoid-s5424" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5425" xml:space="preserve">addamus
              <lb/>
            ſinui arcus, qui ab arcugrad. </s>
            <s xml:id="echoid-s5426" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5427" xml:space="preserve">ſuperatur ſumpto illo arcu, componemus ſinum arcus
              <lb/>
            qui eodemillo arcu aſſumpto arcũgrad. </s>
            <s xml:id="echoid-s5428" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5429" xml:space="preserve">ſuperat: </s>
            <s xml:id="echoid-s5430" xml:space="preserve">propterea quod differentia inter
              <lb/>
            ſinus duorum horum arcuum maiorum æqualis eſt ſinui arcus illius aſſumpti, qui ma
              <lb/>
            ior nõ ponitur, quàm grad. </s>
            <s xml:id="echoid-s5431" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5432" xml:space="preserve">vt ibi demonſtrauimus. </s>
            <s xml:id="echoid-s5433" xml:space="preserve">Vt ſi _3461171_. </s>
            <s xml:id="echoid-s5434" xml:space="preserve">ſinũ arcus grad.
              <lb/>
            </s>
            <s xml:id="echoid-s5435" xml:space="preserve">_20_. </s>
            <s xml:id="echoid-s5436" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5437" xml:space="preserve">_15_. </s>
            <s xml:id="echoid-s5438" xml:space="preserve">adijciamus ad _6394390_. </s>
            <s xml:id="echoid-s5439" xml:space="preserve">ſinum arcus grad. </s>
            <s xml:id="echoid-s5440" xml:space="preserve">_39_. </s>
            <s xml:id="echoid-s5441" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5442" xml:space="preserve">_45._</s>
            <s xml:id="echoid-s5443" xml:space="preserve">. </s>
            <s xml:id="echoid-s5444" xml:space="preserve">qui abarcu grad. </s>
            <s xml:id="echoid-s5445" xml:space="preserve">
              <lb/>
            _60_. </s>
            <s xml:id="echoid-s5446" xml:space="preserve">ſuperatur dicto arcu grad. </s>
            <s xml:id="echoid-s5447" xml:space="preserve">_20_. </s>
            <s xml:id="echoid-s5448" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5449" xml:space="preserve">_15_. </s>
            <s xml:id="echoid-s5450" xml:space="preserve">conficiemus _9855561_. </s>
            <s xml:id="echoid-s5451" xml:space="preserve">ſinum arcus grad. </s>
            <s xml:id="echoid-s5452" xml:space="preserve">
              <lb/>
            _80_. </s>
            <s xml:id="echoid-s5453" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5454" xml:space="preserve">_15_. </s>
            <s xml:id="echoid-s5455" xml:space="preserve">qui arcum gra. </s>
            <s xml:id="echoid-s5456" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5457" xml:space="preserve">eodem arcu grad. </s>
            <s xml:id="echoid-s5458" xml:space="preserve">_20_. </s>
            <s xml:id="echoid-s5459" xml:space="preserve">Min _15_. </s>
            <s xml:id="echoid-s5460" xml:space="preserve">ſuperat. </s>
            <s xml:id="echoid-s5461" xml:space="preserve">Sic ſi _5000000_. </s>
            <s xml:id="echoid-s5462" xml:space="preserve">
              <lb/>
            ſinum arcus grad. </s>
            <s xml:id="echoid-s5463" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5464" xml:space="preserve">addamus ſinui _5000000_. </s>
            <s xml:id="echoid-s5465" xml:space="preserve">arcus grad _30_. </s>
            <s xml:id="echoid-s5466" xml:space="preserve">quem arcus grad. </s>
            <s xml:id="echoid-s5467" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5468" xml:space="preserve">
              <lb/>
            fuperat dicto illo arcu grad. </s>
            <s xml:id="echoid-s5469" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5470" xml:space="preserve">componemus _10000000_. </s>
            <s xml:id="echoid-s5471" xml:space="preserve">ſinum arcus grad. </s>
            <s xml:id="echoid-s5472" xml:space="preserve">_90_. </s>
            <s xml:id="echoid-s5473" xml:space="preserve">qui
              <lb/>
            arcum grad. </s>
            <s xml:id="echoid-s5474" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5475" xml:space="preserve">eodem illo arcu aſſumpto grad. </s>
            <s xml:id="echoid-s5476" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5477" xml:space="preserve">ſuperat.</s>
            <s xml:id="echoid-s5478" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5479" xml:space="preserve">_ITEM_ ſi ſinum cuiuslibet arcus, qui arcu grad. </s>
            <s xml:id="echoid-s5480" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5481" xml:space="preserve">maior non ſit, ſubducamus ex
              <lb/>
            ſinu arcus, qui arcum grad _60_. </s>
            <s xml:id="echoid-s5482" xml:space="preserve">ſumpto illo arcu ſuperat, relinquetur ſinus arcus, qui
              <lb/>
            eodem illo arcu aſſumpto ab arcu grad. </s>
            <s xml:id="echoid-s5483" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5484" xml:space="preserve">ſuperatur. </s>
            <s xml:id="echoid-s5485" xml:space="preserve">Vt ſi _3502075_. </s>
            <s xml:id="echoid-s5486" xml:space="preserve">ſinum arcus
              <lb/>
            grad. </s>
            <s xml:id="echoid-s5487" xml:space="preserve">_20_. </s>
            <s xml:id="echoid-s5488" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5489" xml:space="preserve">_30_ detrahamus ex _9862856_. </s>
            <s xml:id="echoid-s5490" xml:space="preserve">ſinu arcus grad _80_. </s>
            <s xml:id="echoid-s5491" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5492" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5493" xml:space="preserve">qui arcum
              <lb/>
            grad. </s>
            <s xml:id="echoid-s5494" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5495" xml:space="preserve">dicto arcu grad. </s>
            <s xml:id="echoid-s5496" xml:space="preserve">_20_. </s>
            <s xml:id="echoid-s5497" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5498" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5499" xml:space="preserve">ſuperat, reliquus erit _6360781_. </s>
            <s xml:id="echoid-s5500" xml:space="preserve">ſinus arcus gra.
              <lb/>
            </s>
            <s xml:id="echoid-s5501" xml:space="preserve">_39_. </s>
            <s xml:id="echoid-s5502" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5503" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5504" xml:space="preserve">quem arcus grad. </s>
            <s xml:id="echoid-s5505" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5506" xml:space="preserve">eodemillo arcu grad. </s>
            <s xml:id="echoid-s5507" xml:space="preserve">_20_. </s>
            <s xml:id="echoid-s5508" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s5509" xml:space="preserve">_30_ ſuperat.</s>
            <s xml:id="echoid-s5510" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5511" xml:space="preserve">_RVRSVS_, ſi ex ſinu cuiuſuis arcus, qui maior ſit arcu grad. </s>
            <s xml:id="echoid-s5512" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5513" xml:space="preserve">detrahatur ſinus
              <lb/>
            arcus, quitanto minor ſit arcu grad. </s>
            <s xml:id="echoid-s5514" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5515" xml:space="preserve">quanto ille maior eſt, relinquetur ſinus ar-
              <lb/>
            cus, quo vteruis illorũ ab arcu grad. </s>
            <s xml:id="echoid-s5516" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5517" xml:space="preserve">differt. </s>
            <s xml:id="echoid-s5518" xml:space="preserve">Vt ſi ex _9781476_. </s>
            <s xml:id="echoid-s5519" xml:space="preserve">ſinu arcus gra. </s>
            <s xml:id="echoid-s5520" xml:space="preserve">_78_.
              <lb/>
            </s>
            <s xml:id="echoid-s5521" xml:space="preserve">auferatur _6691306_. </s>
            <s xml:id="echoid-s5522" xml:space="preserve">ſinus arcus gra. </s>
            <s xml:id="echoid-s5523" xml:space="preserve">_42_. </s>
            <s xml:id="echoid-s5524" xml:space="preserve">reliquus erit _3090170_. </s>
            <s xml:id="echoid-s5525" xml:space="preserve">ſinus arcus grad. </s>
            <s xml:id="echoid-s5526" xml:space="preserve">_18_. </s>
            <s xml:id="echoid-s5527" xml:space="preserve">
              <lb/>
            quo vterq; </s>
            <s xml:id="echoid-s5528" xml:space="preserve">illorũ ab arcu grad. </s>
            <s xml:id="echoid-s5529" xml:space="preserve">_60_. </s>
            <s xml:id="echoid-s5530" xml:space="preserve">differt. </s>
            <s xml:id="echoid-s5531" xml:space="preserve">Quæ omnia ex dicta propeſ. </s>
            <s xml:id="echoid-s5532" xml:space="preserve">_8_. </s>
            <s xml:id="echoid-s5533" xml:space="preserve">colliguntur.</s>
            <s xml:id="echoid-s5534" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5535" xml:space="preserve">_ITAQVE_ ſatis eſt, vt inueniantur per regulam proportionum ſinus ommum ar-
              <lb/>
            cuum à principio quadrantis vſque ad arcum grad. </s>
            <s xml:id="echoid-s5536" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5537" xml:space="preserve">Si enim ex his eliciantur ſinus
              <lb/>
            complementorum, & </s>
            <s xml:id="echoid-s5538" xml:space="preserve">ex his inuentis detrahãtur priores illi ſinus, reli qui erunt ſinus
              <lb/>
            omnium arcuum inter arcum grad. </s>
            <s xml:id="echoid-s5539" xml:space="preserve">_30_ & </s>
            <s xml:id="echoid-s5540" xml:space="preserve">arcum grad. </s>
            <s xml:id="echoid-s5541" xml:space="preserve">_90_. </s>
            <s xml:id="echoid-s5542" xml:space="preserve">Item ſicogniti eſſent ſinus
              <lb/>
            arcuũ omniũ ab arcu grad. </s>
            <s xml:id="echoid-s5543" xml:space="preserve">_30_. </s>
            <s xml:id="echoid-s5544" xml:space="preserve">vſq; </s>
            <s xml:id="echoid-s5545" xml:space="preserve">ad finem quadrantis, & </s>
            <s xml:id="echoid-s5546" xml:space="preserve">ſinus omniũ arcuum, </s>
          </p>
        </div>
      </text>
    </echo>
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