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

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      <text xml:lang="it" type="free">
        <div xml:id="echoid-div524" type="section" level="1" n="144">
          <p style="it">
            <s xml:id="echoid-s9654" xml:space="preserve">
              <pb o="151" file="0171" n="171" rhead="LIBER SECVNDVS."/>
            di, neceſſe est, vt Sol exiſtens in @, hora 4. </s>
            <s xml:id="echoid-s9655" xml:space="preserve">à mer. </s>
            <s xml:id="echoid-s9656" xml:space="preserve">proijciat radium per centrum A, & </s>
            <s xml:id="echoid-s9657" xml:space="preserve">punctũ P, vſq; </s>
            <s xml:id="echoid-s9658" xml:space="preserve">ad
              <lb/>
            planũ horologij, atq; </s>
            <s xml:id="echoid-s9659" xml:space="preserve">adeo ſectio, quã facit ſemicirculus horarius per P, ductus in plano horologij, indicet
              <lb/>
            horã 4. </s>
            <s xml:id="echoid-s9660" xml:space="preserve">à mer. </s>
            <s xml:id="echoid-s9661" xml:space="preserve">vt perſpicuum eſt exijs, quę ſupra in hoc ſcholio ſcripſimus. </s>
            <s xml:id="echoid-s9662" xml:space="preserve">Vnde non immerito punctum
              <lb/>
            P, pro hora 4. </s>
            <s xml:id="echoid-s9663" xml:space="preserve">à mer. </s>
            <s xml:id="echoid-s9664" xml:space="preserve">ſumi poteſt, quandoquidem ſemicir culus horarius per ipſum ductus facit in horolo-
              <lb/>
            gio lineam horæ 4. </s>
            <s xml:id="echoid-s9665" xml:space="preserve">à mer. </s>
            <s xml:id="echoid-s9666" xml:space="preserve">Eodem modo ſumi poterit punctum O, pro hora 8. </s>
            <s xml:id="echoid-s9667" xml:space="preserve">à med. </s>
            <s xml:id="echoid-s9668" xml:space="preserve">noc. </s>
            <s xml:id="echoid-s9669" xml:space="preserve">quamuis reuera
              <lb/>
            in Aequatore indicet horam 8. </s>
            <s xml:id="echoid-s9670" xml:space="preserve">à mer. </s>
            <s xml:id="echoid-s9671" xml:space="preserve">Intelligatur iam Cylindrus, cuius baſes ſint circulus θ Y B Z, & </s>
            <s xml:id="echoid-s9672" xml:space="preserve">
              <lb/>
            alter a e b π, ei æqualis, & </s>
            <s xml:id="echoid-s9673" xml:space="preserve">oppoſitus, axis autem idem, qui axis mundi A H λ. </s>
            <s xml:id="echoid-s9674" xml:space="preserve">Et quoniam omnes circu
              <lb/>
            li horarij à mer. </s>
            <s xml:id="echoid-s9675" xml:space="preserve">vel med. </s>
            <s xml:id="echoid-s9676" xml:space="preserve">noc. </s>
            <s xml:id="echoid-s9677" xml:space="preserve">per axem mundi, ſiue Cylindri ducuntur, facient eorum plana in cylindro pa
              <lb/>
            rallelogrãma, ita vt duo latera cuiuſq; </s>
            <s xml:id="echoid-s9678" xml:space="preserve">ſint diametri baſium cylindri, reliqua vero duo in ſuperficie cylin
              <lb/>
            dri deſcribantur, vt à Sereno demonſtratur lib. </s>
            <s xml:id="echoid-s9679" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9680" xml:space="preserve">de ſectione cylindri, propoſ. </s>
            <s xml:id="echoid-s9681" xml:space="preserve">2. </s>
            <s xml:id="echoid-s9682" xml:space="preserve">Vt v. </s>
            <s xml:id="echoid-s9683" xml:space="preserve">g. </s>
            <s xml:id="echoid-s9684" xml:space="preserve">circulus horæ 4.
              <lb/>
            </s>
            <s xml:id="echoid-s9685" xml:space="preserve">
              <note position="left" xlink:label="note-0171-01" xlink:href="note-0171-01a" xml:space="preserve">10</note>
            à mer. </s>
            <s xml:id="echoid-s9686" xml:space="preserve">vel med. </s>
            <s xml:id="echoid-s9687" xml:space="preserve">noc. </s>
            <s xml:id="echoid-s9688" xml:space="preserve">ductus per puncta ε, P, in circulo θ γ B Z, faciet parallelogrammum, cuius vnum la-
              <lb/>
            tus eſt ε P, diameter baſis cylindri, duo vero ducuntur ex punctis ε, P, in ſuperficie cylindri, & </s>
            <s xml:id="echoid-s9689" xml:space="preserve">ita de
              <lb/>
            reliquis. </s>
            <s xml:id="echoid-s9690" xml:space="preserve">Dico latera hæc in ſuperficie cylindri deſcripta cadere in puncta ellipſis in plano horologij de-
              <lb/>
            ſcriptę nempe latera ex punctis Z, Y, circuli horæ 6. </s>
            <s xml:id="echoid-s9691" xml:space="preserve">àmer. </s>
            <s xml:id="echoid-s9692" xml:space="preserve">uel med. </s>
            <s xml:id="echoid-s9693" xml:space="preserve">noc. </s>
            <s xml:id="echoid-s9694" xml:space="preserve">ducta cadere in puncta R, X,
              <lb/>
            & </s>
            <s xml:id="echoid-s9695" xml:space="preserve">latus ex puncto θ, Meridiani circuli ductum cadere in punctum V, & </s>
            <s xml:id="echoid-s9696" xml:space="preserve">latera ex punctis ε, P, ducta
              <lb/>
            cadere in puncta α L, & </s>
            <s xml:id="echoid-s9697" xml:space="preserve">ſic de cæteris. </s>
            <s xml:id="echoid-s9698" xml:space="preserve">Ducatur enim per Y Z, diametrum, quæ diametrum θ B, ad an-
              <lb/>
            gulos rectos ſecet, & </s>
            <s xml:id="echoid-s9699" xml:space="preserve">per axem A H, circulus horæ 6. </s>
            <s xml:id="echoid-s9700" xml:space="preserve">occurrens plano horologij in H. </s>
            <s xml:id="echoid-s9701" xml:space="preserve">Et quoniam tam
              <lb/>
            planum horologij, quàm planum huius circuli horę 6. </s>
            <s xml:id="echoid-s9702" xml:space="preserve">rectum eſt ad Meridianum, erit quoque eorum com-
              <lb/>
            munis ſectio, nempe linea horæ 6. </s>
            <s xml:id="echoid-s9703" xml:space="preserve">ad eundem recta, ac idcirco ad meridianam lineam H B, in Meridiano
              <lb/>
              <note position="right" xlink:label="note-0171-02" xlink:href="note-0171-02a" xml:space="preserve">19. vndec.</note>
            exiſtentẽ, per defin. </s>
            <s xml:id="echoid-s9704" xml:space="preserve">3. </s>
            <s xml:id="echoid-s9705" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s9706" xml:space="preserve">11. </s>
            <s xml:id="echoid-s9707" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s9708" xml:space="preserve">perpendicularis in puncto H. </s>
            <s xml:id="echoid-s9709" xml:space="preserve">Diameter igitur S T, quam ad meridia-
              <lb/>
              <note position="left" xlink:label="note-0171-03" xlink:href="note-0171-03a" xml:space="preserve">20</note>
            nam lineam H B, duximus perpendicularem, communis ſectio eſt plani horologij & </s>
            <s xml:id="echoid-s9710" xml:space="preserve">circuli horæ 6. </s>
            <s xml:id="echoid-s9711" xml:space="preserve">hoc
              <lb/>
            eſt, linea horæ 6. </s>
            <s xml:id="echoid-s9712" xml:space="preserve">à mer. </s>
            <s xml:id="echoid-s9713" xml:space="preserve">vel med. </s>
            <s xml:id="echoid-s9714" xml:space="preserve">noc. </s>
            <s xml:id="echoid-s9715" xml:space="preserve">Ac propterea latera parallelogrammi, quod circulus horæ 6. </s>
            <s xml:id="echoid-s9716" xml:space="preserve">in cy-
              <lb/>
            lindro facit, ducta ex punctis Z, Y, cadent in rectam S T; </s>
            <s xml:id="echoid-s9717" xml:space="preserve">quandoquidem & </s>
            <s xml:id="echoid-s9718" xml:space="preserve">linea horæ 6. </s>
            <s xml:id="echoid-s9719" xml:space="preserve">& </s>
            <s xml:id="echoid-s9720" xml:space="preserve">latera hæc
              <lb/>
            in plano circuli horę 6. </s>
            <s xml:id="echoid-s9721" xml:space="preserve">exiſtunt. </s>
            <s xml:id="echoid-s9722" xml:space="preserve">Quoniam vero recta S T, ad Meridianum oſtenſa perpendicularis, per-
              <lb/>
            pendicularis quoque eſt, per defin. </s>
            <s xml:id="echoid-s9723" xml:space="preserve">3. </s>
            <s xml:id="echoid-s9724" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s9725" xml:space="preserve">11. </s>
            <s xml:id="echoid-s9726" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s9727" xml:space="preserve">ad axem A H, in Meridiano exiſtentem; </s>
            <s xml:id="echoid-s9728" xml:space="preserve">Eſt autem & </s>
            <s xml:id="echoid-s9729" xml:space="preserve">
              <lb/>
            diameter Z Y, ad eundem axem perpendicularis, (Cum enim & </s>
            <s xml:id="echoid-s9730" xml:space="preserve">circulus horæ 6. </s>
            <s xml:id="echoid-s9731" xml:space="preserve">& </s>
            <s xml:id="echoid-s9732" xml:space="preserve">Aequator rectus ſit
              <lb/>
            ad Meridianum, erit quoque corum communis ſectio Z Y, ad eundem recta, ac proinde ad axem in Meri-
              <lb/>
              <note position="right" xlink:label="note-0171-04" xlink:href="note-0171-04a" xml:space="preserve">19. vndec.</note>
            diano exiſtentem, per defin. </s>
            <s xml:id="echoid-s9733" xml:space="preserve">3. </s>
            <s xml:id="echoid-s9734" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s9735" xml:space="preserve">11. </s>
            <s xml:id="echoid-s9736" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s9737" xml:space="preserve">perpendicularis) erunt rectæ S T, Z Y, in eodem plano circu
              <lb/>
            li horę 6. </s>
            <s xml:id="echoid-s9738" xml:space="preserve">exiſtentes, cum ad axem ſint perpendiculares, inter ſe parallelæ. </s>
            <s xml:id="echoid-s9739" xml:space="preserve">Quocirca parallelogrammum
              <lb/>
              <note position="right" xlink:label="note-0171-05" xlink:href="note-0171-05a" xml:space="preserve">28. primi.</note>
            erit quadrilaterum, cuius latera ſunt axis A H, ſemidiameter A Z, latus cylindri ex Z, ductum, & </s>
            <s xml:id="echoid-s9740" xml:space="preserve">por
              <lb/>
              <note position="left" xlink:label="note-0171-06" xlink:href="note-0171-06a" xml:space="preserve">30</note>
            tio rectæ H T, inter H, & </s>
            <s xml:id="echoid-s9741" xml:space="preserve">dictum latus ex Z, ductum. </s>
            <s xml:id="echoid-s9742" xml:space="preserve">Eſt enim & </s>
            <s xml:id="echoid-s9743" xml:space="preserve">latus cylindri ex Z, ductum, axi A H,
              <lb/>
              <note position="right" xlink:label="note-0171-07" xlink:href="note-0171-07a" xml:space="preserve">33. primi.</note>
            parallelum, quòd illud latus & </s>
            <s xml:id="echoid-s9744" xml:space="preserve">axis, ſi producãtur, coniungant ſemidiametros æquales A Z, λ ξ, baſium
              <lb/>
            cylindri æqualium, quæ quidem ſemidiametri æquidiſtantes ſunt, vtpote ſectiones baſium æquidiſtantium
              <lb/>
              <note position="right" xlink:label="note-0171-08" xlink:href="note-0171-08a" xml:space="preserve">16. vndec.</note>
            factæ à plano circuli horæ 6. </s>
            <s xml:id="echoid-s9745" xml:space="preserve">à mer. </s>
            <s xml:id="echoid-s9746" xml:space="preserve">vel med. </s>
            <s xml:id="echoid-s9747" xml:space="preserve">noc. </s>
            <s xml:id="echoid-s9748" xml:space="preserve">Quare recta A Z, æqualis erit lateri oppoſito in dicto
              <lb/>
            parallelogrammo, hoc eſt, portioni rectæ H T, inter H, & </s>
            <s xml:id="echoid-s9749" xml:space="preserve">latus cylindri ex Z, ductum. </s>
            <s xml:id="echoid-s9750" xml:space="preserve">Eſt autem H R,
              <lb/>
            ſemidiameter circuli C R X, ſemidiametro A Z, circuli θ Y B Z, æqualis: </s>
            <s xml:id="echoid-s9751" xml:space="preserve">poſita enim eſt H C, æqualis
              <lb/>
            ſemidiametro A B, vel A Z. </s>
            <s xml:id="echoid-s9752" xml:space="preserve">Igitur latus cylindri ex Z, ductum cadit in punctum R. </s>
            <s xml:id="echoid-s9753" xml:space="preserve">Eodem{q́ue} pacto
              <lb/>
            latus ex Y, ductum in punctum X, cadet.</s>
            <s xml:id="echoid-s9754" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9755" xml:space="preserve">RVRSVS, quia latus cylindri ex θ, ductum cadit in lineam meridianam, cum exiſtat in Meridia-
              <lb/>
            no, parallelum{q́ue} eſt axi A H; </s>
            <s xml:id="echoid-s9756" xml:space="preserve">quod demonſtr abitur ea ratione, qua paulo ante oſtendimus, latus ex Z,
              <lb/>
              <note position="left" xlink:label="note-0171-09" xlink:href="note-0171-09a" xml:space="preserve">40</note>
              <note position="right" xlink:label="note-0171-10" xlink:href="note-0171-10a" xml:space="preserve">33. primi.</note>
            ductum eidem axi eſſe parallelum, quia nimirum latus illud ex θ, ductum, & </s>
            <s xml:id="echoid-s9757" xml:space="preserve">axis coniungunt ſemidia-
              <lb/>
            metros. </s>
            <s xml:id="echoid-s9758" xml:space="preserve">A θ, λ π, æquales & </s>
            <s xml:id="echoid-s9759" xml:space="preserve">æquidiſtantes; </s>
            <s xml:id="echoid-s9760" xml:space="preserve">erit vt B A, ad A θ, ita B H, ad portionem meridianæ lineæ
              <lb/>
              <note position="right" xlink:label="note-0171-11" xlink:href="note-0171-11a" xml:space="preserve">2. ſexti.</note>
            inter H, & </s>
            <s xml:id="echoid-s9761" xml:space="preserve">latus ex θ, ductum: </s>
            <s xml:id="echoid-s9762" xml:space="preserve">Eſt autem recta B A, rectæ A θ, æqualis. </s>
            <s xml:id="echoid-s9763" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s9764" xml:space="preserve">B H, portioni di-
              <lb/>
            ctæ meridianæ lineæ æqualis erit; </s>
            <s xml:id="echoid-s9765" xml:space="preserve">ac propterea, cum H V, ipſi B H, ſit æqualis, cadet latus cylindri ex
              <lb/>
            θ, ductum in punctum V.</s>
            <s xml:id="echoid-s9766" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9767" xml:space="preserve">POSTREMO, quoniam iuncta recta O P, parallela eſt rectæ Y Z, ex ſcholio propoſ. </s>
            <s xml:id="echoid-s9768" xml:space="preserve">27. </s>
            <s xml:id="echoid-s9769" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s9770" xml:space="preserve">3.
              <lb/>
            </s>
            <s xml:id="echoid-s9771" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s9772" xml:space="preserve">Nam arcus Y O, Z P, æquales ſunt, quòd vtrumque punctum O, P, quatuor horis diſtare pona-
              <lb/>
            tur à puncto B, at que adeo duabus horis à punctis Y, Z; </s>
            <s xml:id="echoid-s9773" xml:space="preserve">ſi per P O, concipiatur duci planum æquidiſtans
              <lb/>
            parallelogrammo per Z Y, & </s>
            <s xml:id="echoid-s9774" xml:space="preserve">axem ducto, faciet hoc planum in cylindro parallelogrammum, per pro-
              <lb/>
            poſ. </s>
            <s xml:id="echoid-s9775" xml:space="preserve">3. </s>
            <s xml:id="echoid-s9776" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s9777" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9778" xml:space="preserve">Sereni de ſectione cylindri, cuius duo latera ſunt recta O P, & </s>
            <s xml:id="echoid-s9779" xml:space="preserve">alia recta a b, in oppoſita
              <lb/>
              <note position="left" xlink:label="note-0171-12" xlink:href="note-0171-12a" xml:space="preserve">50</note>
            baſi ei reſpondens, reliqua autẽ duo in ſuperficie cylindri ex punctis O, P, ducta, quorum illud, quod ex P,
              <lb/>
            demittitur, est etiam latus parallelogrammi per ε P, ducti, adeo vt per ipſum ducatur circulus horæ 4.
              <lb/>
            </s>
            <s xml:id="echoid-s9780" xml:space="preserve">Nam ex vno puncto P, vnum tantum latus cylindri duci potest, vt facile probari poteſt ex propoſ. </s>
            <s xml:id="echoid-s9781" xml:space="preserve">8. </s>
            <s xml:id="echoid-s9782" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s9783" xml:space="preserve">
              <lb/>
            1. </s>
            <s xml:id="echoid-s9784" xml:space="preserve">Sereni de ſectione cylindri. </s>
            <s xml:id="echoid-s9785" xml:space="preserve">Nam recta quæcunque ducta ex puncto P, ad aliud punctum, quod in late
              <lb/>
            re cylindri non ſit, cadit, per dictam propoſ. </s>
            <s xml:id="echoid-s9786" xml:space="preserve">intra cylindrum. </s>
            <s xml:id="echoid-s9787" xml:space="preserve">Huius ergo parallelogrammi per O P, du
              <lb/>
            cti, & </s>
            <s xml:id="echoid-s9788" xml:space="preserve">Meridiani cõmunis ſectio ſit recta E Q, quæ axi A H, parallela erit, propterea quòd E Q, A H,
              <lb/>
              <note position="right" xlink:label="note-0171-13" xlink:href="note-0171-13a" xml:space="preserve">16. vndec.</note>
            ſectiones ſunt planorum parallelorum, nempe illorum parallelogrammorum, quæper E Q, A H, ducun-
              <lb/>
            tur, factę à Meridiano. </s>
            <s xml:id="echoid-s9789" xml:space="preserve">Quocirca erit vt ſemidiameter B H, ad B A, ſemidiametrum, ita B Q, ad B E;
              <lb/>
            </s>
            <s xml:id="echoid-s9790" xml:space="preserve">
              <note position="right" xlink:label="note-0171-14" xlink:href="note-0171-14a" xml:space="preserve">4. ſexti.</note>
            ac proinde arcus circulorum θ Y B Z, B T V S, reſpondentes ſinubus verſis B Q, B E, ſimiles erunt, ex
              <lb/>
            lemmate propoſ. </s>
            <s xml:id="echoid-s9791" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9792" xml:space="preserve">ſuperioris lib. </s>
            <s xml:id="echoid-s9793" xml:space="preserve">Eſt autem arcus B F, ſimilis arcui B P, quòd vterq; </s>
            <s xml:id="echoid-s9794" xml:space="preserve">quatuor horas com-
              <lb/>
            plectatur; </s>
            <s xml:id="echoid-s9795" xml:space="preserve">eſt{q́ue} B E, ſinus verſus arcus B P. </s>
            <s xml:id="echoid-s9796" xml:space="preserve">Igitur B Q, ſinus quoque verſus erit arcus B F; </s>
            <s xml:id="echoid-s9797" xml:space="preserve">ac </s>
          </p>
        </div>
      </text>
    </echo>
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