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

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          <p>
            <s xml:id="echoid-s4755" xml:space="preserve">
              <pb o="121" file="133" n="133" rhead=""/>
            coroll. </s>
            <s xml:id="echoid-s4756" xml:space="preserve">propof. </s>
            <s xml:id="echoid-s4757" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4758" xml:space="preserve">ex ſinu recto cuiuſuis arcus noto cognoſcitur quoque ſinus
              <lb/>
            rectus dimidij illius arous; </s>
            <s xml:id="echoid-s4759" xml:space="preserve">cognofcemus ex ſinu a rcus grad. </s>
            <s xml:id="echoid-s4760" xml:space="preserve">15. </s>
            <s xml:id="echoid-s4761" xml:space="preserve">ſinum arcus
              <lb/>
            grad. </s>
            <s xml:id="echoid-s4762" xml:space="preserve">7. </s>
            <s xml:id="echoid-s4763" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s4764" xml:space="preserve">30. </s>
            <s xml:id="echoid-s4765" xml:space="preserve">Atque ex hoc ſinum arcus grad. </s>
            <s xml:id="echoid-s4766" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4767" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s4768" xml:space="preserve">45. </s>
            <s xml:id="echoid-s4769" xml:space="preserve">qui arcus amplius
              <lb/>
            bifariam ſecari non poteſt ſine Secundis, (continet enim eius medietas grad. </s>
            <s xml:id="echoid-s4770" xml:space="preserve">1.
              <lb/>
            </s>
            <s xml:id="echoid-s4771" xml:space="preserve">
              <note position="right" xlink:label="note-133-01" xlink:href="note-133-01a" xml:space="preserve">Supputatia
                <lb/>
              ſinuum ac-
                <lb/>
              cuum gta-
                <lb/>
              dibus 3.
                <lb/>
              Min. 45. ſe-
                <lb/>
              ſe ſupetan-
                <lb/>
              tium.</note>
            Min. </s>
            <s xml:id="echoid-s4772" xml:space="preserve">52. </s>
            <s xml:id="echoid-s4773" xml:space="preserve">Sec. </s>
            <s xml:id="echoid-s4774" xml:space="preserve">30.) </s>
            <s xml:id="echoid-s4775" xml:space="preserve">quæ in Sinuum tractatione negliguntur. </s>
            <s xml:id="echoid-s4776" xml:space="preserve">Deinde quia per pro
              <lb/>
            poſ. </s>
            <s xml:id="echoid-s4777" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4778" xml:space="preserve">ex ſinu recto cuiuſlibet arcus cognito notus quoque efficitur ſinus
              <lb/>
            complementi illius arcus, cognoſcemus ex ſinu arcus grad. </s>
            <s xml:id="echoid-s4779" xml:space="preserve">7. </s>
            <s xml:id="echoid-s4780" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s4781" xml:space="preserve">30. </s>
            <s xml:id="echoid-s4782" xml:space="preserve">ſi-
              <lb/>
            num arcus grad. </s>
            <s xml:id="echoid-s4783" xml:space="preserve">82. </s>
            <s xml:id="echoid-s4784" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s4785" xml:space="preserve">30. </s>
            <s xml:id="echoid-s4786" xml:space="preserve">Ex hoc autem, per coroll. </s>
            <s xml:id="echoid-s4787" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s4788" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4789" xml:space="preserve">ſinum ar-
              <lb/>
            cus grad. </s>
            <s xml:id="echoid-s4790" xml:space="preserve">41. </s>
            <s xml:id="echoid-s4791" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s4792" xml:space="preserve">15. </s>
            <s xml:id="echoid-s4793" xml:space="preserve">Atque ex hoc, per propof. </s>
            <s xml:id="echoid-s4794" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4795" xml:space="preserve">ſinum arcus grad. </s>
            <s xml:id="echoid-s4796" xml:space="preserve">48. </s>
            <s xml:id="echoid-s4797" xml:space="preserve">Min.
              <lb/>
            </s>
            <s xml:id="echoid-s4798" xml:space="preserve">45. </s>
            <s xml:id="echoid-s4799" xml:space="preserve">Item ex ſinu arcus gra. </s>
            <s xml:id="echoid-s4800" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4801" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s4802" xml:space="preserve">45. </s>
            <s xml:id="echoid-s4803" xml:space="preserve">cognoſcemus, per propoſ. </s>
            <s xml:id="echoid-s4804" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4805" xml:space="preserve">ſinum arcus
              <lb/>
            grad. </s>
            <s xml:id="echoid-s4806" xml:space="preserve">86. </s>
            <s xml:id="echoid-s4807" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s4808" xml:space="preserve">15. </s>
            <s xml:id="echoid-s4809" xml:space="preserve">Quòd ſi alij arcus, quorum ſinus in uenti funt, bifariam quo
              <lb/>
            que fecentur, & </s>
            <s xml:id="echoid-s4810" xml:space="preserve">eorum medietates rurſus bifariam, & </s>
            <s xml:id="echoid-s4811" xml:space="preserve">ita deinceps, donec ad
              <lb/>
            Minuta numero imparia, quæ amplius bifariã diuidi ſine Secundis nequeunt,
              <lb/>
            peruentum ſit; </s>
            <s xml:id="echoid-s4812" xml:space="preserve">Itemque harum medietatum complementa accipiantur, quæ
              <lb/>
            rurſus, eodem modo continue bifariam ſecẽtur, donec ad Minuta numero im-
              <lb/>
            paria ſit peruentum; </s>
            <s xml:id="echoid-s4813" xml:space="preserve">& </s>
            <s xml:id="echoid-s4814" xml:space="preserve">medietatum complementa ſumantur, &</s>
            <s xml:id="echoid-s4815" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4816" xml:space="preserve">cognoſce-
              <lb/>
            mus, per coroll. </s>
            <s xml:id="echoid-s4817" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s4818" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4819" xml:space="preserve">& </s>
            <s xml:id="echoid-s4820" xml:space="preserve">per propof. </s>
            <s xml:id="echoid-s4821" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4822" xml:space="preserve">ſinus omnium harum medieta-
              <lb/>
            tum, & </s>
            <s xml:id="echoid-s4823" xml:space="preserve">complementorum. </s>
            <s xml:id="echoid-s4824" xml:space="preserve">Qui ſinus cum illis ſex primò inuentis conſtituent
              <lb/>
            ſinus 24. </s>
            <s xml:id="echoid-s4825" xml:space="preserve">arcuum fefe ordine ſuperantium gradibus 3. </s>
            <s xml:id="echoid-s4826" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s4827" xml:space="preserve">45. </s>
            <s xml:id="echoid-s4828" xml:space="preserve">vt in hac ta-
              <lb/>
            bula vides.</s>
            <s xml:id="echoid-s4829" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">
            <lb/>
          ## Arcus # Sinus # ## Arcus # Sinus # ## Arcus # Sinus # ## Arcus # Sinus
            <lb/>
          G # M # # G # M # # G # M # # G # M
            <lb/>
          3 # 45 # 654031 # 26 # 15 # 4422887 # 48 # 45 # 7518398 # 71 # 15 # 9469301
            <lb/>
          7 # 30 # 1305262 # 30 # 0 # 5000000 # 52 # 30 # 7933533 # 75 # 0 # 9659258
            <lb/>
          11 # 15 # 1950903 # 33 # 45 # 5555702 # 56 # 15 # 8314696 # 78 # 45 # 9807853
            <lb/>
          15 # 0 # 2588190 # 37 # 30 # 6087614 # 60 # 0 # 8660254 # 82 # 30 # 9914449
            <lb/>
          18 # 45 # 3214395 # 41 # 15 # 6593458 # 63 # 45 # 8968727 # 86 # 15 # 9978589
            <lb/>
          22 # 30 # 3826834 # 45 # 0 # 7071068 # 67 # 30 # 9238795 # 90 # 0 # 10000000
            <lb/>
          </note>
          <p>
            <s xml:id="echoid-s4830" xml:space="preserve">POST hæc, per ea, quæ propof. </s>
            <s xml:id="echoid-s4831" xml:space="preserve">2. </s>
            <s xml:id="echoid-s4832" xml:space="preserve">de inuentione lateris Decagoni, & </s>
            <s xml:id="echoid-s4833" xml:space="preserve">pen-
              <lb/>
              <note position="right" xlink:label="note-133-03" xlink:href="note-133-03a" xml:space="preserve">Supputatio
                <lb/>
              ſinus arcus
                <lb/>
              grad. 36.</note>
            tagoniæquilateri in eodem circulo demon ſtrauimus, inquiremus ſinum arcus
              <lb/>
            grad. </s>
            <s xml:id="echoid-s4834" xml:space="preserve">36. </s>
            <s xml:id="echoid-s4835" xml:space="preserve">hac ratione. </s>
            <s xml:id="echoid-s4836" xml:space="preserve">Repetatur figura propoſitionis 2. </s>
            <s xml:id="echoid-s4837" xml:space="preserve">vbi demonſtratum eſt,
              <lb/>
            DF, eſſe latus Decagoni, & </s>
            <s xml:id="echoid-s4838" xml:space="preserve">BF, latus
              <lb/>
              <figure xlink:label="fig-133-01" xlink:href="fig-133-01a" number="130">
                <image file="133-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/133-01"/>
              </figure>
            Pentagoni ęquilateri. </s>
            <s xml:id="echoid-s4839" xml:space="preserve">Et quoniam qua-
              <lb/>
            drata rectarum BD, DE, notarum (Eſt
              <lb/>
            enim BD, ſinus totus, & </s>
            <s xml:id="echoid-s4840" xml:space="preserve">DE, eius me-
              <lb/>
            dietas.) </s>
            <s xml:id="echoid-s4841" xml:space="preserve">nota ſunt, & </s>
            <s xml:id="echoid-s4842" xml:space="preserve">æqualia quadrato
              <lb/>
              <note position="right" xlink:label="note-133-04" xlink:href="note-133-04a" xml:space="preserve">47. primi.</note>
            rectæ EB; </s>
            <s xml:id="echoid-s4843" xml:space="preserve">notum erit quadratum rectæ
              <lb/>
            EB; </s>
            <s xml:id="echoid-s4844" xml:space="preserve">ac propterea & </s>
            <s xml:id="echoid-s4845" xml:space="preserve">ipſa recta EB, hoc
              <lb/>
            eſt, recta EF, illi æqualis, nota erit, nem
              <lb/>
            pe partium fere 11180339. </s>
            <s xml:id="echoid-s4846" xml:space="preserve">Ex qua ſi
              <lb/>
            detrahatur recta ED, medietas ſinus to
              <lb/>
            tius partiũ 5000000. </s>
            <s xml:id="echoid-s4847" xml:space="preserve">nota fiet recta DF,
              <lb/>
            partium 6180339. </s>
            <s xml:id="echoid-s4848" xml:space="preserve">cuius quadratum ſi addatur quadrato ſinus totius BD, no
              <lb/>
            tum fiet aggregatum quadratorum ex rectis DF, BD, deſcriptorum; </s>
            <s xml:id="echoid-s4849" xml:space="preserve"/>
          </p>
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