Clavius, Christoph, Gnomonices libri octo, in quibus non solum horologiorum solariu[m], sed aliarum quo[quam] rerum, quae ex gnomonis umbra cognosci possunt, descriptiones geometricè demonstrantur

Table of contents

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[41.] COROLLARIVM.
Page: 60 (40)
[42.] SCHOLIVM.
Page: 60 (40)
[43.] THEOREMA 9. PROPOSITIO 11.
Page: 62 (42)
[44.] SCHOLIVM.
Page: 63 (43)
[45.] COROLLARIVM PRIMVM.
Page: 63 (43)
[46.] COROLLARIVM II.
Page: 63 (43)
[47.] THEOREMA 10. PROPOSITIO 12.
Page: 63 (43)
[48.] COROLLARIVM.
Page: 64 (44)
[49.] THEOREMA 11. PROPOSITIO 13.
Page: 64 (44)
[50.] THEOREMA 12. PROPOSITIO 14.
Page: 65 (45)
[51.] SCHOLIVM.
Page: 66 (46)
[52.] THEOREMA 13. PROPOSITIO 15.
Page: 66 (46)
[53.] LEMMA.
Page: 69 (49)
[54.] COROLLARIVM.
Page: 69 (49)
[55.] THEOREMA 14. PROPOSITIO 16.
Page: 69 (49)
[56.] COROLLARIVM.
Page: 70 (50)
[57.] THEOREMA 15. PROPOSITIO 17.
Page: 70 (50)
[58.] LEMMA.
Page: 72 (52)
[59.] SCHOLIVM.
Page: 72 (52)
[60.] THEOREMA 16. PROPOSITIO 18.
Page: 72 (52)
[61.] THEOREMA 17. PROPOSITIO 19.
Page: 73 (53)
[62.] SCHOLIVM.
Page: 74 (54)
[63.] THEOREMA 18. PROPOSITIO 20.
Page: 75 (55)
[64.] SCHOLIVM.
Page: 75 (55)
[65.] Linea horæ 24. ab ortu vel occaſu. Vel horizontalis linea.
Page: 76 (56)
[66.] Linea horæ 12. ab ortu vel occaſu.
Page: 76 (56)
[67.] Linea horæ ſextæ à meridie vel media nocte.
Page: 77 (57)
[68.] Linea horæ 12. à meridie vel media nocte.
Page: 77 (57)
[69.] Linea horæ 23. ab ortu vel occaſu.
Page: 81 (61)
[70.] Linea horæ 22. ab ortu vel occaſu.
Page: 81 (61)
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        <div xml:id="echoid-div80" type="section" level="1" n="24">
          <head xml:id="echoid-head27" xml:space="preserve">THEQREMA 4. PROPOSITIQ 5.</head>
          <p>
            <s xml:id="echoid-s1514" xml:space="preserve">SECTIO communis earundem ſuperficierum conicarum, & </s>
            <s xml:id="echoid-s1515" xml:space="preserve">pla-
              <lb/>
              <note position="left" xlink:label="note-0042-01" xlink:href="note-0042-01a" xml:space="preserve">Planum horo-
                <lb/>
              logij æquidiſtãs
                <lb/>
              maximo circu-
                <lb/>
              lo baſes conica
                <lb/>
              rum ſuperficie-
                <lb/>
              rum tãgenti fa-
                <lb/>
              cit in altera ſu-
                <lb/>
              per@cierum Pa
                <lb/>
              tabolen.</note>
            ni horologij æquidiſtantis circulo maximo, qui baſes conicarum ſuper-
              <lb/>
            ficierum tangit, Parabole eſt.</s>
            <s xml:id="echoid-s1516" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1517" xml:space="preserve">SINT in eadem Sphæra duæ conicæ ſuperficies, quæ prius; </s>
            <s xml:id="echoid-s1518" xml:space="preserve">& </s>
            <s xml:id="echoid-s1519" xml:space="preserve">E F, maximus circulus tangens
              <lb/>
            baſes oppoſitas in punctis E, & </s>
            <s xml:id="echoid-s1520" xml:space="preserve">F. </s>
            <s xml:id="echoid-s1521" xml:space="preserve">Huic autem circulo æquidiſtet horologij planum H I, faciens
              <lb/>
            in conica ſuperficie A F G, ſectionem K L M. </s>
            <s xml:id="echoid-s1522" xml:space="preserve">Dico K L M, Parabolen eſſe. </s>
            <s xml:id="echoid-s1523" xml:space="preserve">Ducatur per paralle-
              <lb/>
              <note position="left" xlink:label="note-0042-02" xlink:href="note-0042-02a" xml:space="preserve">10</note>
              <figure xlink:label="fig-0042-01" xlink:href="fig-0042-01a" number="20">
                <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0042-01"/>
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            lorum polos B, C, & </s>
            <s xml:id="echoid-s1524" xml:space="preserve">per conta-
              <lb/>
            ctum E, circulus maximus B D-
              <lb/>
            C G, per ꝓpoſ. </s>
            <s xml:id="echoid-s1525" xml:space="preserve">20. </s>
            <s xml:id="echoid-s1526" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s1527" xml:space="preserve">1. </s>
            <s xml:id="echoid-s1528" xml:space="preserve">Theod.
              <lb/>
            </s>
            <s xml:id="echoid-s1529" xml:space="preserve">qui neceſſario quoque per po-
              <lb/>
            los circuli F E, per propoſ. </s>
            <s xml:id="echoid-s1530" xml:space="preserve">6. </s>
            <s xml:id="echoid-s1531" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s1532" xml:space="preserve">
              <lb/>
            2. </s>
            <s xml:id="echoid-s1533" xml:space="preserve">Theod. </s>
            <s xml:id="echoid-s1534" xml:space="preserve">atque adeo per polos
              <lb/>
            circuli H I, quem in Sphæra ex
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s1535" xml:space="preserve">1. </s>
            <s xml:id="echoid-s1536" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s1537" xml:space="preserve">1. </s>
            <s xml:id="echoid-s1538" xml:space="preserve">Theod. </s>
            <s xml:id="echoid-s1539" xml:space="preserve">efficit
              <lb/>
            planum horologij, (cũ ex pro-
              <lb/>
            poſ. </s>
            <s xml:id="echoid-s1540" xml:space="preserve">1. </s>
            <s xml:id="echoid-s1541" xml:space="preserve">eiuſdem, eoſdem habeant
              <lb/>
              <note position="left" xlink:label="note-0042-03" xlink:href="note-0042-03a" xml:space="preserve">20</note>
            polos paralleli F E, H I,) tranſi-
              <lb/>
            bit, ideoq́; </s>
            <s xml:id="echoid-s1542" xml:space="preserve">per propoſ. </s>
            <s xml:id="echoid-s1543" xml:space="preserve">15 lib. </s>
            <s xml:id="echoid-s1544" xml:space="preserve">1.
              <lb/>
            </s>
            <s xml:id="echoid-s1545" xml:space="preserve">Theodoſ. </s>
            <s xml:id="echoid-s1546" xml:space="preserve">& </s>
            <s xml:id="echoid-s1547" xml:space="preserve">circulum F G, & </s>
            <s xml:id="echoid-s1548" xml:space="preserve">cir
              <lb/>
            culum H I, per rectas F G, H I,
              <lb/>
            ſe mutuo in N, ſecantes, (ſeca-
              <lb/>
            bunt enim ſe ſe rectæ F G, H I,
              <lb/>
            mutuo, quòd in eodem plano
              <lb/>
            circuli B D C G, exiſtant) bifa-
              <lb/>
            riam & </s>
            <s xml:id="echoid-s1549" xml:space="preserve">ad angulos rectos ſeca-
              <lb/>
            bit, facietq́; </s>
            <s xml:id="echoid-s1550" xml:space="preserve">communes ſectio-
              <lb/>
              <note position="left" xlink:label="note-0042-04" xlink:href="note-0042-04a" xml:space="preserve">30</note>
            nes planorum parallelorũ F E,
              <lb/>
              <note position="left" xlink:label="note-0042-05" xlink:href="note-0042-05a" xml:space="preserve">16. vndec.</note>
            H I, parallelas. </s>
            <s xml:id="echoid-s1551" xml:space="preserve">Faciat quoque
              <lb/>
            idem circulus B D C G, cũ per
              <lb/>
            axem B C, incedat, triangulum
              <lb/>
            per axem A F G, ſecans rectam
              <lb/>
            H I, & </s>
            <s xml:id="echoid-s1552" xml:space="preserve">ſectionẽ conicã in k: </s>
            <s xml:id="echoid-s1553" xml:space="preserve">Secet etiã planũ H I, per rectam K N, tranſiens, circulum F G, per rectã
              <lb/>
            M N L, per punctum N, tranſeuntẽ. </s>
            <s xml:id="echoid-s1554" xml:space="preserve">Nam cum planum per K N, ductum per punctum N, quod
              <lb/>
            in plano G F, eſt, tranſeat, trãſibit quoque L M, communis ſectio planorum H I, F G, per punctum
              <lb/>
            N. </s>
            <s xml:id="echoid-s1555" xml:space="preserve">Quoniam igitur plana F G, H I, recta ſunt ad planum circuli B D C G; </s>
            <s xml:id="echoid-s1556" xml:space="preserve">erit quoq; </s>
            <s xml:id="echoid-s1557" xml:space="preserve">eorum com
              <lb/>
            munis ſectio L M, ad idem recta in puncto N, atque adeò & </s>
            <s xml:id="echoid-s1558" xml:space="preserve">ad rectam F G, baſ@m trianguli per
              <lb/>
              <note position="left" xlink:label="note-0042-06" xlink:href="note-0042-06a" xml:space="preserve">19. vndec.</note>
              <note position="left" xlink:label="note-0042-07" xlink:href="note-0042-07a" xml:space="preserve">40</note>
            axem, perpendicularis erit, ex definitione 3. </s>
            <s xml:id="echoid-s1559" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s1560" xml:space="preserve">11. </s>
            <s xml:id="echoid-s1561" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s1562" xml:space="preserve">Quare cum conus A F G, ſecetur plano
              <lb/>
            B D C G, per axem, ſecetur autem & </s>
            <s xml:id="echoid-s1563" xml:space="preserve">altero plano H I, quod baſim coni ſecat per rectam lineam
              <lb/>
            L M, perpendicularem ad F G, baſim trianguli per axem, ſitq́; </s>
            <s xml:id="echoid-s1564" xml:space="preserve">K N, ſectionis diameter lateri A F,
              <lb/>
            trianguli per axem parallela; </s>
            <s xml:id="echoid-s1565" xml:space="preserve">erit, per propoſ. </s>
            <s xml:id="echoid-s1566" xml:space="preserve">11. </s>
            <s xml:id="echoid-s1567" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s1568" xml:space="preserve">1. </s>
            <s xml:id="echoid-s1569" xml:space="preserve">Apollonij, ſectio K L M, parabole. </s>
            <s xml:id="echoid-s1570" xml:space="preserve">Sectio
              <lb/>
              <note position="left" xlink:label="note-0042-08" xlink:href="note-0042-08a" xml:space="preserve">Planum horo-
                <lb/>
              logii horizonta-
                <lb/>
              lis cuiuſque, &
                <lb/>
              Verticalis ad la
                <lb/>
              titudinẽ gr. 45.
                <lb/>
              immo & æqui-
                <lb/>
              diſtantis cuili-
                <lb/>
              bet circulo ho-
                <lb/>
              rarum ab ortu
                <lb/>
              vel occaſu, facit
                <lb/>
              in altera ſuper-
                <lb/>
              ficierum coni-
                <lb/>
              carum quarum
                <lb/>
              baſes ſunt pa-
                <lb/>
              rallelus ſemper
                <lb/>
              apparentiũ ma-
                <lb/>
              ximus & maxi-
                <lb/>
              mus ſemper la-
                <lb/>
              tentium, Para-
                <lb/>
              bolam.</note>
            ergo communis earundem ſuperficierum, &</s>
            <s xml:id="echoid-s1571" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1572" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s1573" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div83" type="section" level="1" n="25">
          <head xml:id="echoid-head28" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s1574" xml:space="preserve">ITAQVE, cum Horizon quilibet obliquus tangat duos parallelos, quorum alter eſt maximus eo-
              <lb/>
            rum, qui ſemper apparent, alter uerò maximus eorum, qui ſemper ſub terra occultantur; </s>
            <s xml:id="echoid-s1575" xml:space="preserve">erit communis
              <lb/>
              <note position="left" xlink:label="note-0042-09" xlink:href="note-0042-09a" xml:space="preserve">50</note>
            ſectio plani horologii Horizontalis, & </s>
            <s xml:id="echoid-s1576" xml:space="preserve">coni, cuius baſis parallelus eſt maximus eorum, qui deliteſcunt,
              <lb/>
            (neque enim alter conus, cuius baſis ſemper apparet, ſecatur, cum totus extet ſupra Horizontem) Parabo-
              <lb/>
            le. </s>
            <s xml:id="echoid-s1577" xml:space="preserve">Idem continget in horologio Verticali ad latitudinem graduum 45. </s>
            <s xml:id="echoid-s1578" xml:space="preserve">Item in horologio, cuius planum
              <lb/>
            circulo horæ cuiuslibet ab ortu, vel occaſu æquidiſtat. </s>
            <s xml:id="echoid-s1579" xml:space="preserve">Nam & </s>
            <s xml:id="echoid-s1580" xml:space="preserve">Verticalis circulus latitudinis graduum
              <lb/>
            45. </s>
            <s xml:id="echoid-s1581" xml:space="preserve">& </s>
            <s xml:id="echoid-s1582" xml:space="preserve">circulus cuiuslibet horæ ab ortu, uel occaſu, in omni Horizonte tangit maximum parallelum eo-
              <lb/>
            rum, qui toti ſupra Horizontem extant, vt propoſ. </s>
            <s xml:id="echoid-s1583" xml:space="preserve">10. </s>
            <s xml:id="echoid-s1584" xml:space="preserve">huius lib. </s>
            <s xml:id="echoid-s1585" xml:space="preserve">demonſtrabitur.</s>
            <s xml:id="echoid-s1586" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1587" xml:space="preserve">DENIQVE communis ſectio cuiuſcunque horologii, & </s>
            <s xml:id="echoid-s1588" xml:space="preserve">coni, cuius baſis tantum ab Aequatore de-
              <lb/>
            clinat ad Auſtrum, quantum eſt complementum altitudinis poli arctici ſupra circulum maximum, cui pla
              <lb/>
              <note position="left" xlink:label="note-0042-10" xlink:href="note-0042-10a" xml:space="preserve">Quæ horologia
                <lb/>
              faciant in coni-
                <lb/>
              eis ſuperfic@eb
                <emph style="sub">9</emph>
              ,
                <lb/>
              quarum baſes
                <lb/>
              ſunt quicũque
                <lb/>
              paralleli Solis,
                <lb/>
              Parabolas.</note>
            num horologii æquidiſtat, Parabole erit. </s>
            <s xml:id="echoid-s1589" xml:space="preserve">Talis erit ſectio coni baſim habentis parallelum ♑, & </s>
            <s xml:id="echoid-s1590" xml:space="preserve">horo-
              <lb/>
            logii horizontalis ad latitudinem ſeptentrionalem grad. </s>
            <s xml:id="echoid-s1591" xml:space="preserve">66. </s>
            <s xml:id="echoid-s1592" xml:space="preserve">min. </s>
            <s xml:id="echoid-s1593" xml:space="preserve">30. </s>
            <s xml:id="echoid-s1594" xml:space="preserve">Nam complementum huius lati-
              <lb/>
            tudinis continet grad. </s>
            <s xml:id="echoid-s1595" xml:space="preserve">23. </s>
            <s xml:id="echoid-s1596" xml:space="preserve">min. </s>
            <s xml:id="echoid-s1597" xml:space="preserve">30. </s>
            <s xml:id="echoid-s1598" xml:space="preserve">quanta nimirum eſt declinatio paralleli ♑. </s>
            <s xml:id="echoid-s1599" xml:space="preserve">Talis etiam erit ſectio
              <lb/>
            coni baſim habentis parallelum ♒ & </s>
            <s xml:id="echoid-s1600" xml:space="preserve">♐, & </s>
            <s xml:id="echoid-s1601" xml:space="preserve">horologii horizontalis ad latitudinem ſeptentrionalem
              <lb/>
            graduum 69. </s>
            <s xml:id="echoid-s1602" xml:space="preserve">min. </s>
            <s xml:id="echoid-s1603" xml:space="preserve">48. </s>
            <s xml:id="echoid-s1604" xml:space="preserve">Complementum enim latitudinis iſtius, nempe grad. </s>
            <s xml:id="echoid-s1605" xml:space="preserve">20. </s>
            <s xml:id="echoid-s1606" xml:space="preserve">min. </s>
            <s xml:id="echoid-s1607" xml:space="preserve">12. </s>
            <s xml:id="echoid-s1608" xml:space="preserve">ęquale eſt </s>
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