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

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          <p>
            <s xml:id="echoid-s4673" xml:space="preserve">
              <pb o="120" file="132" n="132" rhead=""/>
            nempe ſinus EH, grad. </s>
            <s xml:id="echoid-s4674" xml:space="preserve">60. </s>
            <s xml:id="echoid-s4675" xml:space="preserve">ſi nimirum quadratũ ſinus 5000000. </s>
            <s xml:id="echoid-s4676" xml:space="preserve">ex quadrato
              <lb/>
            ſinus totius 10000000. </s>
            <s xml:id="echoid-s4677" xml:space="preserve">auferatur, & </s>
            <s xml:id="echoid-s4678" xml:space="preserve">reliqui numeri radix quadrata accipia-
              <lb/>
            tur, quę eſt 8660254. </s>
            <s xml:id="echoid-s4679" xml:space="preserve">fere.</s>
            <s xml:id="echoid-s4680" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4681" xml:space="preserve">DEINDE, quoniam recta AF, fecans
              <lb/>
              <figure xlink:label="fig-132-01" xlink:href="fig-132-01a" number="129">
                <image file="132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/132-01"/>
              </figure>
            arcum BC, bifariam, fecat quoque, ex lem-
              <lb/>
            mate definitionum, rectam BC, bifariam, at-
              <lb/>
            que adeo & </s>
            <s xml:id="echoid-s4682" xml:space="preserve">ad angulos rectos; </s>
            <s xml:id="echoid-s4683" xml:space="preserve">erit CL, ſinus
              <lb/>
              <note position="left" xlink:label="note-132-01" xlink:href="note-132-01a" xml:space="preserve">3. tertij.</note>
            arcus CF, grad. </s>
            <s xml:id="echoid-s4684" xml:space="preserve">45. </s>
            <s xml:id="echoid-s4685" xml:space="preserve">quem ita inueniemus.
              <lb/>
            </s>
            <s xml:id="echoid-s4686" xml:space="preserve">Cum in triangulo CAL, angulus L, rectus
              <lb/>
            ſit, & </s>
            <s xml:id="echoid-s4687" xml:space="preserve">angulus CAL, ſemirectus, erit & </s>
            <s xml:id="echoid-s4688" xml:space="preserve">angu-
              <lb/>
              <note position="left" xlink:label="note-132-02" xlink:href="note-132-02a" xml:space="preserve">32. primi.</note>
            lus ACL, ſemirectus, atque adeo angulo
              <lb/>
            CAL, æqualis. </s>
            <s xml:id="echoid-s4689" xml:space="preserve">Igitur rectæ AL, CL, æqua-
              <lb/>
              <note position="left" xlink:label="note-132-03" xlink:href="note-132-03a" xml:space="preserve">6 primi.</note>
            les erunt. </s>
            <s xml:id="echoid-s4690" xml:space="preserve">Cum ergo quadratum rectæ AC,
              <lb/>
            æquale ſit quadratis rectarum AL, CL, ſi-
              <lb/>
              <note position="left" xlink:label="note-132-04" xlink:href="note-132-04a" xml:space="preserve">47. primi</note>
            mul; </s>
            <s xml:id="echoid-s4691" xml:space="preserve">erit quadratum ſinus totius AC, du-
              <lb/>
            plum quadrati ſinus CL, grad. </s>
            <s xml:id="echoid-s4692" xml:space="preserve">45. </s>
            <s xml:id="echoid-s4693" xml:space="preserve">Medietas
              <lb/>
            igitur quadrati ſinus totius erit quadratũ rectæ CL, cuius radix quadrata da-
              <lb/>
            bit ſinum CL, 7071068. </s>
            <s xml:id="echoid-s4694" xml:space="preserve">fere pro arcu grad. </s>
            <s xml:id="echoid-s4695" xml:space="preserve">45. </s>
            <s xml:id="echoid-s4696" xml:space="preserve">Qui etiam hoc modo re-
              <lb/>
            perietur. </s>
            <s xml:id="echoid-s4697" xml:space="preserve">Quoniam quadratum rectæ BC, æquale eſt quadratis rectarum AB,
              <lb/>
              <note position="left" xlink:label="note-132-05" xlink:href="note-132-05a" xml:space="preserve">47. primi.</note>
            AC; </s>
            <s xml:id="echoid-s4698" xml:space="preserve">atque adeo duplum quadrati ſinus totius AC, ſi quadratum ſinus to-
              <lb/>
            tius duplicetur, habebitur quadratum rectæ BC, cuius quadrati radix dabit re
              <lb/>
            ctam BC, partium 14142136. </s>
            <s xml:id="echoid-s4699" xml:space="preserve">fere, & </s>
            <s xml:id="echoid-s4700" xml:space="preserve">huius dimidium 7071068. </s>
            <s xml:id="echoid-s4701" xml:space="preserve">dabit ſinum
              <lb/>
            CL, grad. </s>
            <s xml:id="echoid-s4702" xml:space="preserve">45.</s>
            <s xml:id="echoid-s4703" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4704" xml:space="preserve">RVRSVS, quia recta AD, ſecans arcum CE, bifariam, ſecat quoque
              <lb/>
            rectam CE, bifariam in K, ex lemmate definitionum, atque adeo & </s>
            <s xml:id="echoid-s4705" xml:space="preserve">ad angu-
              <lb/>
              <note position="left" xlink:label="note-132-06" xlink:href="note-132-06a" xml:space="preserve">3. tertij.</note>
            los rectos; </s>
            <s xml:id="echoid-s4706" xml:space="preserve">erit CK, ſinus arcus CD, grad. </s>
            <s xml:id="echoid-s4707" xml:space="preserve">15. </s>
            <s xml:id="echoid-s4708" xml:space="preserve">quem ſic inueniemus. </s>
            <s xml:id="echoid-s4709" xml:space="preserve">Quoniã
              <lb/>
            ex propoſ. </s>
            <s xml:id="echoid-s4710" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4711" xml:space="preserve">ſinus CK, medio loco proportionalis eſt inter medietatem ſi-
              <lb/>
            nus totius, & </s>
            <s xml:id="echoid-s4712" xml:space="preserve">finum verfum CG; </s>
            <s xml:id="echoid-s4713" xml:space="preserve">(qui quidem habetur, ſi EH, ſinus grad. </s>
            <s xml:id="echoid-s4714" xml:space="preserve">60.
              <lb/>
            </s>
            <s xml:id="echoid-s4715" xml:space="preserve">ex ſinu toto AC, detrahatur, vt in coroll. </s>
            <s xml:id="echoid-s4716" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s4717" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4718" xml:space="preserve">diximus) ſit vt, per co-
              <lb/>
            roll. </s>
            <s xml:id="echoid-s4719" xml:space="preserve">propof. </s>
            <s xml:id="echoid-s4720" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4721" xml:space="preserve">notus ſiat ſinus CK, arcus CD, qui dimidium eſt arcus CE. </s>
            <s xml:id="echoid-s4722" xml:space="preserve">
              <lb/>
            Nam ſi medietas ſinus totius multiplicetur in ſinum verſum CG, cognitum,
              <lb/>
            producetur quadratum rectæ CK; </s>
            <s xml:id="echoid-s4723" xml:space="preserve">quòd rectangulum ſub medietate ſinus to-
              <lb/>
            tius, & </s>
            <s xml:id="echoid-s4724" xml:space="preserve">ſinu verſo CG, contentum, æquale ſit quadrato mediæ proportiona-
              <lb/>
              <note position="left" xlink:label="note-132-07" xlink:href="note-132-07a" xml:space="preserve">17. ſexti.</note>
            lis Ck. </s>
            <s xml:id="echoid-s4725" xml:space="preserve">Si igitur quadrati rectæ CK, radix eruatur, habebitur ſinus CK, par-
              <lb/>
            tium 2588190. </s>
            <s xml:id="echoid-s4726" xml:space="preserve">vt in dicto coroll. </s>
            <s xml:id="echoid-s4727" xml:space="preserve">propof. </s>
            <s xml:id="echoid-s4728" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4729" xml:space="preserve">docuimus. </s>
            <s xml:id="echoid-s4730" xml:space="preserve">Quem ſinum hoc etiam
              <lb/>
              <note position="left" xlink:label="note-132-08" xlink:href="note-132-08a" xml:space="preserve">47. primi.</note>
            modo inueſtigabimus. </s>
            <s xml:id="echoid-s4731" xml:space="preserve">Quoniam quadratis rectarum notarum EG, GC, æqua
              <lb/>
              <note position="right" xlink:label="note-132-09" xlink:href="note-132-09a" xml:space="preserve">
                <lb/>
              Arcus. # Sinus
                <lb/>
              G.
                <lb/>
              15 # 2588190
                <lb/>
              30 # 5000000
                <lb/>
              45 # 7071068
                <lb/>
              60 # 8660254
                <lb/>
              75 # 9659258
                <lb/>
              90 # 10000000
                <lb/>
              </note>
            le eſt quadratum rectæ EC; </s>
            <s xml:id="echoid-s4732" xml:space="preserve">fiet notum quadratum
              <lb/>
            rectæ EC; </s>
            <s xml:id="echoid-s4733" xml:space="preserve">cuius quadrati radix quadrata dabit re-
              <lb/>
            ctam EC, notam, & </s>
            <s xml:id="echoid-s4734" xml:space="preserve">huius medietas erit ſinus CK,
              <lb/>
            cognitus.</s>
            <s xml:id="echoid-s4735" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4736" xml:space="preserve">EX ſinu autem grad. </s>
            <s xml:id="echoid-s4737" xml:space="preserve">15. </s>
            <s xml:id="echoid-s4738" xml:space="preserve">cognito cognofcetur
              <lb/>
            quoque, per propof. </s>
            <s xml:id="echoid-s4739" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4740" xml:space="preserve">ſinus DI, complementi arcus
              <lb/>
            CD, hoc eſt, ſinus arcus BD, grad. </s>
            <s xml:id="echoid-s4741" xml:space="preserve">75. </s>
            <s xml:id="echoid-s4742" xml:space="preserve">qui quidem
              <lb/>
            deprehendetur partium 9659258. </s>
            <s xml:id="echoid-s4743" xml:space="preserve">Itaq; </s>
            <s xml:id="echoid-s4744" xml:space="preserve">inuenti ſunt
              <lb/>
            hactenus ſinus recti arcuum continentium grad.
              <lb/>
            </s>
            <s xml:id="echoid-s4745" xml:space="preserve">15. </s>
            <s xml:id="echoid-s4746" xml:space="preserve">30. </s>
            <s xml:id="echoid-s4747" xml:space="preserve">45. </s>
            <s xml:id="echoid-s4748" xml:space="preserve">60. </s>
            <s xml:id="echoid-s4749" xml:space="preserve">75. </s>
            <s xml:id="echoid-s4750" xml:space="preserve">& </s>
            <s xml:id="echoid-s4751" xml:space="preserve">90. </s>
            <s xml:id="echoid-s4752" xml:space="preserve">vt in hac formula apparet.</s>
            <s xml:id="echoid-s4753" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4754" xml:space="preserve">HORVM autem ſinuũ beneficio ad aliorum
              <lb/>
            inueſtigationem ita progrediemur. </s>
            <s xml:id="echoid-s4755" xml:space="preserve">Quoniam </s>
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