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

Table of Notes

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          <p style="it">
            <s xml:id="echoid-s8295" xml:space="preserve">
              <pb o="197" file="209" n="209" rhead=""/>
            tem eiuſdem prioris illius arcus, cuius ſecantem ſubduxiſti Item ſi tangentem cuiuſ@
              <lb/>
            libet arcus ex eiuſdem ſecante detrahas, remanebit tangens ſemiſsis complementi
              <lb/>
            arcus eiuſdem. </s>
            <s xml:id="echoid-s8296" xml:space="preserve">Primum conſtat ex ſcholio propoſ. </s>
            <s xml:id="echoid-s8297" xml:space="preserve">19. </s>
            <s xml:id="echoid-s8298" xml:space="preserve">vbi oſtenſum eſt, ſecantem, & </s>
            <s xml:id="echoid-s8299" xml:space="preserve">
              <lb/>
            tangentem cuiuſuis arcus ſimul æquales eſſe tangenti arcus compoſiti ex illo, & </s>
            <s xml:id="echoid-s8300" xml:space="preserve">ex
              <lb/>
            ſemiſſe complementi eiuſdem. </s>
            <s xml:id="echoid-s8301" xml:space="preserve">H@nc enim fit, vt ſecans ex cõpoſita hac tangente ablata
              <lb/>
            relinquat alteram illá tangentem. </s>
            <s xml:id="echoid-s8302" xml:space="preserve">Secundum vero liquet ex propoſ. </s>
            <s xml:id="echoid-s8303" xml:space="preserve">hac 20. </s>
            <s xml:id="echoid-s8304" xml:space="preserve">vbi de-
              <lb/>
            monſtrauimus, ſeca ntem cuiusuis arcus æqualem eſſe tangenti eiuſdem, vna cum tan-
              <lb/>
            gente ſemiſsis complementi arcus eiuſdem. </s>
            <s xml:id="echoid-s8305" xml:space="preserve">Quare huius ſemiſsis tangens reliqua fiet
              <lb/>
            poſt ſubtractionem alterius illius tangentis ex ſecante. </s>
            <s xml:id="echoid-s8306" xml:space="preserve">V. </s>
            <s xml:id="echoid-s8307" xml:space="preserve">g. </s>
            <s xml:id="echoid-s8308" xml:space="preserve">ſi ſecantem arcus grad.
              <lb/>
            </s>
            <s xml:id="echoid-s8309" xml:space="preserve">20. </s>
            <s xml:id="echoid-s8310" xml:space="preserve">quæ eſt 10641777. </s>
            <s xml:id="echoid-s8311" xml:space="preserve">detrahamus ex 14281480. </s>
            <s xml:id="echoid-s8312" xml:space="preserve">tangente arcus grad. </s>
            <s xml:id="echoid-s8313" xml:space="preserve">55. </s>
            <s xml:id="echoid-s8314" xml:space="preserve">
              <lb/>
            compoſiti ex arcu grad. </s>
            <s xml:id="echoid-s8315" xml:space="preserve">20. </s>
            <s xml:id="echoid-s8316" xml:space="preserve">& </s>
            <s xml:id="echoid-s8317" xml:space="preserve">ſemi
              <unsure/>
            ſſe complementi eiuſdem, relinquetur tangens
              <lb/>
            3639703. </s>
            <s xml:id="echoid-s8318" xml:space="preserve">arcus eiuſdem grad. </s>
            <s xml:id="echoid-s8319" xml:space="preserve">20. </s>
            <s xml:id="echoid-s8320" xml:space="preserve">Item ſi 4244748. </s>
            <s xml:id="echoid-s8321" xml:space="preserve">tangentem arcus grad. </s>
            <s xml:id="echoid-s8322" xml:space="preserve">23. </s>
            <s xml:id="echoid-s8323" xml:space="preserve">
              <lb/>
            ex 10863603 ſecante eiuſdem arcus ſubducamus, remanebit tangens 6618855. </s>
            <s xml:id="echoid-s8324" xml:space="preserve">ar-
              <lb/>
            cus grad 33. </s>
            <s xml:id="echoid-s8325" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8326" xml:space="preserve">30. </s>
            <s xml:id="echoid-s8327" xml:space="preserve">hoc eſt, ſem ſsis complementi dati arcus grad. </s>
            <s xml:id="echoid-s8328" xml:space="preserve">23. </s>
            <s xml:id="echoid-s8329" xml:space="preserve">Rurſus
              <lb/>
            ſi 11547004. </s>
            <s xml:id="echoid-s8330" xml:space="preserve">ſecantem arcus grad. </s>
            <s xml:id="echoid-s8331" xml:space="preserve">30 ex 17320508. </s>
            <s xml:id="echoid-s8332" xml:space="preserve">tangente arcus grad. </s>
            <s xml:id="echoid-s8333" xml:space="preserve">60. </s>
            <s xml:id="echoid-s8334" xml:space="preserve">
              <lb/>
            qui ex arcu grad. </s>
            <s xml:id="echoid-s8335" xml:space="preserve">30. </s>
            <s xml:id="echoid-s8336" xml:space="preserve">& </s>
            <s xml:id="echoid-s8337" xml:space="preserve">ſemiſſe complementi eiuſdem componitur, auferamus, @elin
              <lb/>
            quetur tangens 5773504. </s>
            <s xml:id="echoid-s8338" xml:space="preserve">arcus grad. </s>
            <s xml:id="echoid-s8339" xml:space="preserve">30. </s>
            <s xml:id="echoid-s8340" xml:space="preserve">Et ſi 1763268. </s>
            <s xml:id="echoid-s8341" xml:space="preserve">tangentem arcus grad. </s>
            <s xml:id="echoid-s8342" xml:space="preserve">10. </s>
            <s xml:id="echoid-s8343" xml:space="preserve">
              <lb/>
            demamus ex 10154264. </s>
            <s xml:id="echoid-s8344" xml:space="preserve">ſecante eiuſdem arcus gra. </s>
            <s xml:id="echoid-s8345" xml:space="preserve">10. </s>
            <s xml:id="echoid-s8346" xml:space="preserve">remanebit tangẽs 8390996. </s>
            <s xml:id="echoid-s8347" xml:space="preserve">
              <lb/>
            arcus grad. </s>
            <s xml:id="echoid-s8348" xml:space="preserve">40. </s>
            <s xml:id="echoid-s8349" xml:space="preserve">qui ſemiſsis eſt complementi dicti arcus grad. </s>
            <s xml:id="echoid-s8350" xml:space="preserve">10.</s>
            <s xml:id="echoid-s8351" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8352" xml:space="preserve">IAM vero ſi per ea, quæ propoſ. </s>
            <s xml:id="echoid-s8353" xml:space="preserve">18. </s>
            <s xml:id="echoid-s8354" xml:space="preserve">eiusq́; </s>
            <s xml:id="echoid-s8355" xml:space="preserve">ſcholio tradidimus, tangentes om-
              <lb/>
            nium arcuum quadrantis per ſingula Minuta extenſorum inueſtigemus; </s>
            <s xml:id="echoid-s8356" xml:space="preserve">reperiemus
              <lb/>
            earum beneficio per ſolam additionem ſecantes omnium arcuum per bina minuta pro-
              <lb/>
            gredientium, ſi mmirum tangentem cuiuſuis arcus minuta numero paria habentis
              <lb/>
            addamus ad tangentem ſemiſsis complementi arcus eiuſdem: </s>
            <s xml:id="echoid-s8357" xml:space="preserve">propterea quòd Secans
              <lb/>
            cuiuſuis arcus æqualis eſt tangenti eiuſdem, vna cum tangente ſemiſsis complementi
              <lb/>
              <note position="right" xlink:label="note-209-01" xlink:href="note-209-01a" xml:space="preserve">20. huius.</note>
            eiuſdem; </s>
            <s xml:id="echoid-s8358" xml:space="preserve">conſtat autem omnium arcuum minuta numero paria habentium comple-
              <lb/>
            menta ſemiſſes habere, Exempli cauſa, ſi deſideretur ſecans arcus Min. </s>
            <s xml:id="echoid-s8359" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8360" xml:space="preserve">addemus eius
              <lb/>
            tangentem 5818. </s>
            <s xml:id="echoid-s8361" xml:space="preserve">ad 9994184. </s>
            <s xml:id="echoid-s8362" xml:space="preserve">tangentem arcus grad. </s>
            <s xml:id="echoid-s8363" xml:space="preserve">44. </s>
            <s xml:id="echoid-s8364" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8365" xml:space="preserve">59. </s>
            <s xml:id="echoid-s8366" xml:space="preserve">qui ſemiſsis eſt com
              <lb/>
            plementi arcus dati Min. </s>
            <s xml:id="echoid-s8367" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8368" xml:space="preserve">Numerus enim compoſitus 10000002. </s>
            <s xml:id="echoid-s8369" xml:space="preserve">erit ſecans arcus
              <lb/>
            Min. </s>
            <s xml:id="echoid-s8370" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8371" xml:space="preserve">sic etiam ſi quæratur ſecans arcus grad. </s>
            <s xml:id="echoid-s8372" xml:space="preserve">89. </s>
            <s xml:id="echoid-s8373" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8374" xml:space="preserve">58. </s>
            <s xml:id="echoid-s8375" xml:space="preserve">addemus eius tangentem
              <lb/>
            17188033689. </s>
            <s xml:id="echoid-s8376" xml:space="preserve">ad 2909 tangentem arcus Min. </s>
            <s xml:id="echoid-s8377" xml:space="preserve">1. </s>
            <s xml:id="echoid-s8378" xml:space="preserve">qui ſemiſsis eſt complementi arcus
              <lb/>
            dati grad. </s>
            <s xml:id="echoid-s8379" xml:space="preserve">89. </s>
            <s xml:id="echoid-s8380" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8381" xml:space="preserve">58. </s>
            <s xml:id="echoid-s8382" xml:space="preserve">Ná numerus conflatus 17188036598. </s>
            <s xml:id="echoid-s8383" xml:space="preserve">erit ſecans arcus grad.
              <lb/>
            </s>
            <s xml:id="echoid-s8384" xml:space="preserve">89 Min. </s>
            <s xml:id="echoid-s8385" xml:space="preserve">58. </s>
            <s xml:id="echoid-s8386" xml:space="preserve">Hac ratione conficietur dimidiata pars tabulæ Tangentium: </s>
            <s xml:id="echoid-s8387" xml:space="preserve">at Tangen-
              <lb/>
            tes arcuum minuta numero imparia habentium, quoniam eorum complementa ſemiſ-
              <lb/>
            ſes non habent, niſi ad Secunda venire velimus, inueſtigandæ erunt, vt propoſ. </s>
            <s xml:id="echoid-s8388" xml:space="preserve">18. </s>
            <s xml:id="echoid-s8389" xml:space="preserve">
              <lb/>
            eiuſq; </s>
            <s xml:id="echoid-s8390" xml:space="preserve">ſcholio præcepimus.</s>
            <s xml:id="echoid-s8391" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8392" xml:space="preserve">RVRSVS ſecantem cuiuſuis arcus inueniemus, ſi eius tangentem demamus ex
              <lb/>
            tangente arcus compoſiti ex ar cu illo, & </s>
            <s xml:id="echoid-s8393" xml:space="preserve">ſemiſſe complementi eiuſdem arcus. </s>
            <s xml:id="echoid-s8394" xml:space="preserve">Nam cũ,
              <lb/>
            vt demonſtrauimu, ſecans cuiuſuis arcus, vnà cum tangente eiuſdem æqualis ſit tan-
              <lb/>
              <note position="right" xlink:label="note-209-02" xlink:href="note-209-02a" xml:space="preserve">Schol. 19.
                <lb/>
              huius.</note>
            genti arcus compoſiti ex dato arcu, & </s>
            <s xml:id="echoid-s8395" xml:space="preserve">ſemiſſe complementi eiuſdem; </s>
            <s xml:id="echoid-s8396" xml:space="preserve">efficitur, vt tan
              <lb/>
            gens dati arcus ex tangente arcus ex eo, & </s>
            <s xml:id="echoid-s8397" xml:space="preserve">ſemiſſe complementi compoſiti ablata re-
              <lb/>
            linquat ſecantem eiuſdem dati arcus. </s>
            <s xml:id="echoid-s8398" xml:space="preserve">Vt ſi cupiamus ſecantem arcus Min. </s>
            <s xml:id="echoid-s8399" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8400" xml:space="preserve">aufere-
              <lb/>
            mus 5818. </s>
            <s xml:id="echoid-s8401" xml:space="preserve">tangentem ipſius ex 10005820. </s>
            <s xml:id="echoid-s8402" xml:space="preserve">tangente arcus grad. </s>
            <s xml:id="echoid-s8403" xml:space="preserve">45. </s>
            <s xml:id="echoid-s8404" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8405" xml:space="preserve">1 cõpoſiti ex
              <lb/>
            arcu Min. </s>
            <s xml:id="echoid-s8406" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8407" xml:space="preserve">& </s>
            <s xml:id="echoid-s8408" xml:space="preserve">ex arcu grad. </s>
            <s xml:id="echoid-s8409" xml:space="preserve">44. </s>
            <s xml:id="echoid-s8410" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8411" xml:space="preserve">59. </s>
            <s xml:id="echoid-s8412" xml:space="preserve">qui ſemiſsis eſt complementi arcus dati
              <lb/>
            Min. </s>
            <s xml:id="echoid-s8413" xml:space="preserve">2 Relictus namque numerus 10000002 erit ſecans arcus dati Min. </s>
            <s xml:id="echoid-s8414" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8415" xml:space="preserve">Ita quoq;
              <lb/>
            </s>
            <s xml:id="echoid-s8416" xml:space="preserve">ſi velimus habere ſecantem arcus grad. </s>
            <s xml:id="echoid-s8417" xml:space="preserve">60. </s>
            <s xml:id="echoid-s8418" xml:space="preserve">ſubducemus 17320508. </s>
            <s xml:id="echoid-s8419" xml:space="preserve">eius tangentem
              <lb/>
            ex 37320514. </s>
            <s xml:id="echoid-s8420" xml:space="preserve">tangente arcus grad. </s>
            <s xml:id="echoid-s8421" xml:space="preserve">75. </s>
            <s xml:id="echoid-s8422" xml:space="preserve">compoſiti ex dato arcu grad. </s>
            <s xml:id="echoid-s8423" xml:space="preserve">60. </s>
            <s xml:id="echoid-s8424" xml:space="preserve">& </s>
            <s xml:id="echoid-s8425" xml:space="preserve">ex </s>
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      </text>
    </echo>
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