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

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          <p>
            <s xml:id="echoid-s7775" xml:space="preserve">
              <pb o="192" file="204" n="204" rhead=""/>
            arcus CD. </s>
            <s xml:id="echoid-s7776" xml:space="preserve">Item vt AF, ſinus rectus arcus BD, ad AD, ſinum totum, ita AC,
              <lb/>
            ſinus totus ad AE, ſecantem arcus CD, qui complementum eſt eiuſdem arcus
              <lb/>
            BD. </s>
            <s xml:id="echoid-s7777" xml:space="preserve">Quare ſinus totus medius proportionalis eſt inter AF, ſinũ cõ plementi
              <lb/>
            arcus CD, & </s>
            <s xml:id="echoid-s7778" xml:space="preserve">AE, ſecantem eiuſdem arcus CD:
              <lb/>
            </s>
            <s xml:id="echoid-s7779" xml:space="preserve">
              <figure xlink:label="fig-204-01" xlink:href="fig-204-01a" number="151">
                <image file="204-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/204-01"/>
              </figure>
            Item inter AF, ſinum rectum arcus BD, & </s>
            <s xml:id="echoid-s7780" xml:space="preserve">AE,
              <lb/>
            ſecantem complementi eiuſdem arcus BD. </s>
            <s xml:id="echoid-s7781" xml:space="preserve">quod
              <lb/>
            eſt tertium.</s>
            <s xml:id="echoid-s7782" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7783" xml:space="preserve">POSTREMO, quia triangula ACE;
              <lb/>
            </s>
            <s xml:id="echoid-s7784" xml:space="preserve">GBA, æquiangula ſunt, quòd anguli C, B, ſint
              <lb/>
            recti; </s>
            <s xml:id="echoid-s7785" xml:space="preserve">& </s>
            <s xml:id="echoid-s7786" xml:space="preserve">alterni CAE, BGA, nec non alterni
              <lb/>
              <note position="left" xlink:label="note-204-01" xlink:href="note-204-01a" xml:space="preserve">29. primi.
                <lb/>
              4. ſexti.</note>
            AEC, GAB, æquales: </s>
            <s xml:id="echoid-s7787" xml:space="preserve">erit vt CE, tangens ar-
              <lb/>
            cus CD, ad AC, ſinum totum, ita AB, ſinus to-
              <lb/>
            tus ad BG, tangentem complementi eiuſdem ar-
              <lb/>
            cus CD. </s>
            <s xml:id="echoid-s7788" xml:space="preserve">Pari ratione erit, vt BG, tangens arcus
              <lb/>
            BD, ad AB, ſinum totum, ita AC, ſinus totus
              <lb/>
            ad CE, tangentem complementi eiuſdem arcus BD. </s>
            <s xml:id="echoid-s7789" xml:space="preserve">quod eſt quartum.
              <lb/>
            </s>
            <s xml:id="echoid-s7790" xml:space="preserve">Quã ergo proportionem habet ſinus complementi arcus cuiuſuis, &</s>
            <s xml:id="echoid-s7791" xml:space="preserve">c. </s>
            <s xml:id="echoid-s7792" xml:space="preserve">Quod
              <lb/>
            erat demonſtrandum.</s>
            <s xml:id="echoid-s7793" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div553" type="section" level="1" n="259">
          <head xml:id="echoid-head286" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s7794" xml:space="preserve">ITAQVE per ea, quæ primo loco demonſtrata ſunt in hac propoſ. </s>
            <s xml:id="echoid-s7795" xml:space="preserve">ſi fiat, vt ſin{us} com
              <lb/>
              <note position="left" xlink:label="note-204-02" xlink:href="note-204-02a" xml:space="preserve">Quo pacto
                <lb/>
              tangentes
                <lb/>
              omniú ar-
                <lb/>
              cuum repe-
                <lb/>
              tiantur.</note>
            plementi cuiuſuis arc{us} ad ſinũ rectũ eiuſdẽ arc{us}, ita ſin{us} tot{us} ad aliud; </s>
            <s xml:id="echoid-s7796" xml:space="preserve">hoc eſt, ſi ſi-
              <lb/>
            n{us} arc{us} cui{us}libet in ſinum totũ multiplicetur, (quod fiet facile, ſi ei ad dextrã præ-
              <lb/>
            ponas tot ci fras, quot in ſinu toto continentur, nempe ſeptem, ſi ſin{us} tot{us} fuerit
              <lb/>
            10000000. </s>
            <s xml:id="echoid-s7797" xml:space="preserve">vel quinque, ſi ſinus totus fuerit 100000.) </s>
            <s xml:id="echoid-s7798" xml:space="preserve">productusq́; </s>
            <s xml:id="echoid-s7799" xml:space="preserve">numerus per ſi-
              <lb/>
            num complementi eiuſdem arc{us} diuidatur: </s>
            <s xml:id="echoid-s7800" xml:space="preserve">inuenietur Tangens illi{us} arc{us}, cui{us} ſi-
              <lb/>
            num complementi accepiſti, vel cuius ſinum rectum in ſinum totum multiplicaſti. </s>
            <s xml:id="echoid-s7801" xml:space="preserve">Vt
              <lb/>
            ſi tangens arc{us} gra. </s>
            <s xml:id="echoid-s7802" xml:space="preserve">30. </s>
            <s xml:id="echoid-s7803" xml:space="preserve">quæratur, adiungemus eius ſinui recto 5000000 ſeptem ci-
              <lb/>
            fras, hoc modo. </s>
            <s xml:id="echoid-s7804" xml:space="preserve">50000000000000. </s>
            <s xml:id="echoid-s7805" xml:space="preserve">& </s>
            <s xml:id="echoid-s7806" xml:space="preserve">hunc numerum per 8660254 ſinumcomple-
              <lb/>
            menti eiuſdem arcus grad. </s>
            <s xml:id="echoid-s7807" xml:space="preserve">30. </s>
            <s xml:id="echoid-s7808" xml:space="preserve">partiemur. </s>
            <s xml:id="echoid-s7809" xml:space="preserve">Nam quotiens numerus 5773503. </s>
            <s xml:id="echoid-s7810" xml:space="preserve">dabit
              <lb/>
            tangentem arcus grad. </s>
            <s xml:id="echoid-s7811" xml:space="preserve">30. </s>
            <s xml:id="echoid-s7812" xml:space="preserve">quatenus ſinus totus eſt 10000000. </s>
            <s xml:id="echoid-s7813" xml:space="preserve">Hinc fit, tangentes
              <lb/>
              <note position="left" xlink:label="note-204-03" xlink:href="note-204-03a" xml:space="preserve">Sola diui-
                <lb/>
              ſione om
                <lb/>
              nes tangen
                <lb/>
              tes eliciun-
                <lb/>
              tur.</note>
            emnes per ſolam diuiſionem inueniri. </s>
            <s xml:id="echoid-s7814" xml:space="preserve">Nam ſiomnium arcuum ſinubus, initio facto à
              <lb/>
            principio quadr antis, ſeptem ci fr{as} appon{as}, & </s>
            <s xml:id="echoid-s7815" xml:space="preserve">compoſitos numeros per ſinus comple
              <lb/>
            mentorum eorundem arcuum, quemlibet per ſuum correſpondentem, diuid{as}, prodi-
              <lb/>
            bunt omnium arcuum tangentes, vt ex demonſtratis liquido cõſtat. </s>
            <s xml:id="echoid-s7816" xml:space="preserve">Quamobrem per-
              <lb/>
            facilis eſt conſtructio tabulæ Tangentium.</s>
            <s xml:id="echoid-s7817" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7818" xml:space="preserve">RVRSVS per ea, quæ tertio loco in hac propoſ. </s>
            <s xml:id="echoid-s7819" xml:space="preserve">ſunt demonſtrata, ſi fiat, vt ſi-
              <lb/>
              <note position="left" xlink:label="note-204-04" xlink:href="note-204-04a" xml:space="preserve">Qua tõne
                <lb/>
              ſecátes om-
                <lb/>
              niú arcuũ
                <lb/>
              inueſtigen-
                <lb/>
              tur.</note>
            nus complementi cuiuſuis arcus ad ſinum totum, ita ſinus totus ad aliud: </s>
            <s xml:id="echoid-s7820" xml:space="preserve">hoc eſt, ſi ſi-
              <lb/>
            nus totus in ſeipſum multiplicetur, (quod facile fiet, ſi ei ad dextram præpon{as} tot
              <lb/>
            cifras, quot ſunt in ſinu toto, puta ſeptem, vel quinque, prout ſinus totus habuerit
              <lb/>
            ſeptem, aut quinque cifras.) </s>
            <s xml:id="echoid-s7821" xml:space="preserve">numerusq́; </s>
            <s xml:id="echoid-s7822" xml:space="preserve">productus per ſinum complementi cuiuſuis ar-
              <lb/>
            cus diuidatur: </s>
            <s xml:id="echoid-s7823" xml:space="preserve">reperietur Secans illius arcus, per cuius ſinum complementi diuiſio fa
              <lb/>
            cta eſt. </s>
            <s xml:id="echoid-s7824" xml:space="preserve">Vt ſi ſecans arcus grad. </s>
            <s xml:id="echoid-s7825" xml:space="preserve">30. </s>
            <s xml:id="echoid-s7826" xml:space="preserve">deſideretur, diuidemus 100000000000000. </s>
            <s xml:id="echoid-s7827" xml:space="preserve">(pro-
              <lb/>
            ductum ſcilicet numerum ex ſinu toto in ſeipſum) per 8660254. </s>
            <s xml:id="echoid-s7828" xml:space="preserve">ſinum complementi
              <lb/>
            arcus grad. </s>
            <s xml:id="echoid-s7829" xml:space="preserve">30. </s>
            <s xml:id="echoid-s7830" xml:space="preserve">Numerus enim Quotiens 11547005. </s>
            <s xml:id="echoid-s7831" xml:space="preserve">dabit ſecantem arcus grad. </s>
            <s xml:id="echoid-s7832" xml:space="preserve">30.
              <lb/>
            </s>
            <s xml:id="echoid-s7833" xml:space="preserve">quatenus ſinus totus est 10000000. </s>
            <s xml:id="echoid-s7834" xml:space="preserve">Sola ergo diuiſ@one huius ſemper </s>
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