Alvarus, Thomas, Liber de triplici motu, 1509

Page concordance

< >
Scan Original
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
< >
page |< < of 290 > >|
    <echo version="1.0">
      <text xml:lang="la">
        <div xml:id="N10132" level="1" n="1" type="body">
          <div xml:id="N1194D" level="2" n="2" type="other" type-free="pars">
            <div xml:id="N11E85" level="3" n="2" type="chapter" type-free="capitulum">
              <p xml:id="N12C75">
                <s xml:id="N12DAA" xml:space="preserve">
                  <pb chead="Secunde partis" file="0031" n="31"/>
                dratus: inter tales numeros reperitur medium ꝓ­
                  <lb/>
                portionabile ꝓportione rationali ita  primi ad
                  <lb/>
                ipſum ſit ea proportio rationalis que eſt ipſiꝰ ad
                  <lb/>
                tertium. </s>
                <s xml:id="N12DBE" xml:space="preserve">et illius numeri quadrati tale medium eſt
                  <lb/>
                vnum latus. </s>
                <s xml:id="N12DC3" xml:space="preserve">Probatur prima pars huius corre-
                  <lb/>
                larii / quia illa pars eſt vna cõditionalis ex cuiꝰ op­
                  <lb/>
                poſito conſequentis / ſequitur oppoſitum antece-
                  <lb/>
                dentis: vt patet ex ſecundo correlario: igitur illa
                  <lb/>
                pars vera. </s>
                <s xml:id="N12DCE" xml:space="preserve">Secunda probatur ex correlario īme-
                  <lb/>
                diate precendenti. </s>
                <s xml:id="N12DD3" xml:space="preserve">¶ Sequitur quīto /  inter ṗmos
                  <lb/>
                numeros ꝓportionis duple: triple: octuple: ſexq̇-
                  <lb/>
                altere etc̈. non inuenitur medium ꝓportionabile ꝓ­
                  <lb/>
                portione rationali </s>
                <s xml:id="N12DDC" xml:space="preserve">Probatur primo de dupla / q̄
                  <lb/>
                eſt inter iſtos terminos .4.2. quoniam numerus q̇
                  <lb/>
                fit ex ductu vnius extremi in alterum puta .4. in .2.
                  <lb/>
                non eſt quadratus / igitur inter illa extrema non ī­
                  <lb/>
                uenitur medium ꝓportionabile proportione ra-
                  <lb/>
                tionali </s>
                <s xml:id="N12DE9" xml:space="preserve">Añs patet intelligenti diffinitionem nu-
                  <lb/>
                meri quadrati. </s>
                <s xml:id="N12DEE" xml:space="preserve">et conſequentia patet ex ſecundo
                  <lb/>
                correlario. </s>
                <s xml:id="N12DF3" xml:space="preserve">Et eodē modo ꝓbabis reliquas ꝑtes.
                  <lb/>
                </s>
                <s xml:id="N12DF7" xml:space="preserve">¶ Et ex hoc habes pulchrū documentuꝫ ab cogno­
                  <lb/>
                ſcendū quãdo aliqua ꝓportio īeq̈litatꝪ habet ſub­
                  <lb/>
                duplam proportionem ad eam rationalem. </s>
                <s xml:id="N12DFE" xml:space="preserve">Quã­
                  <lb/>
                do enim numerus reſultans ex ductu vnius extre-
                  <lb/>
                mi in alterum non eſt quadratus / tunc talis ꝓpor­
                  <lb/>
                tio non habet ꝓportionem rationalem ſubduplã
                  <lb/>
                ad illam cum non habeat medium ꝓportionabile
                  <lb/>
                ꝓportione rationali. </s>
                <s xml:id="N12E0B" xml:space="preserve">et ſic tale medium inter ter-
                  <lb/>
                minos illius ꝓportionis non ſe habet vt numerꝰ
                  <lb/>
                reſpectu alicuius extremi illius ꝓportionis. </s>
                <s xml:id="N12E12" xml:space="preserve">Si eī
                  <lb/>
                ſe haberet vt numerus: maioris extremi ad ipſum
                  <lb/>
                eſſet aliqua ꝓportio rationalis: et ipſius ad mini­
                  <lb/>
                mum extremum eſſet eadem ꝓportio rationalis: et
                  <lb/>
                ſic iam ibi eſſent tres numeri continuo ꝓportiona­
                  <lb/>
                biles in hac medietate geometrica: et ſic numerus
                  <lb/>
                qui fit ex ductu extremi in extremū eſſet quadratꝰ /
                  <lb/>
                vt patet ex primo correlario / quod eſt oppoſitū da­
                  <lb/>
                ti.
                  <note position="left" xlink:href="note-0031-01a" xlink:label="note-0031-01" xml:id="N12E4B" xml:space="preserve">irrõnaliſ
                    <lb/>
                  ꝓportio
                    <lb/>
                  alio mõ
                    <lb/>
                  ponenda
                    <lb/>
                  oñditur.</note>
                </s>
                <s xml:id="N12E2A" xml:space="preserve">Et ex hoc facile elicitur ꝓportionem irrationa-
                  <lb/>
                lem neceſſario ponendã eſſe: quod nota.</s>
              </p>
              <p xml:id="N12E59">
                <s xml:id="N12E5A" xml:space="preserve">Gratia ordinis obſeruandi medieta­
                  <lb/>
                tis harmonice aliquas proprietates ponã quas
                  <lb/>
                non intendo demonſtrare: quia huic operi paruꝫ
                  <lb/>
                conducunt.
                  <note position="left" xlink:href="note-0031-02a" xlink:label="note-0031-02" xml:id="N12EEF" xml:space="preserve">ṗma ꝓṗe­
                    <lb/>
                  tas medi­
                    <lb/>
                  etatꝪ har­
                    <lb/>
                  monice.</note>
                </s>
                <s xml:id="N12E68" xml:space="preserve">¶ Prima proprietas </s>
                <s xml:id="N12E6B" xml:space="preserve">Medietas har-
                  <lb/>
                monica in maioribus terminis maiorem ſeruat ꝓ­
                  <lb/>
                portionē quam in minoribus. </s>
                <s xml:id="N12E72" xml:space="preserve">Hoc eſt dicere /  ca­
                  <lb/>
                ptis tribus terminis hac medietate ꝓportionabi­
                  <lb/>
                libus: maior eſt proportio maximi ad mediū: quã
                  <lb/>
                medii ad minimū. </s>
                <s xml:id="N12E7B" xml:space="preserve">vt conſtitutis his terminis .12.8
                  <lb/>
                6. maior eſt proportio .12. ad .8. que eſt ſexquialte­
                  <lb/>
                ra quã .8. ad .6. que eſt ſexquitertia.
                  <note position="left" xlink:href="note-0031-03a" xlink:label="note-0031-03" xml:id="N12EFB" xml:space="preserve">ſcḋa ꝓṗe­
                    <lb/>
                  tas medi­
                    <lb/>
                  etatꝪ har­
                    <lb/>
                  monice.</note>
                </s>
                <s xml:id="N12E87" xml:space="preserve">¶ Secunda ꝓ-
                  <lb/>
                prietas. </s>
                <s xml:id="N12E8C" xml:space="preserve">tribus terminis in hac medietate conſtitu­
                  <lb/>
                tis medius terminus in collectas extremitates du­
                  <lb/>
                ctus dupluꝫ numero qui fit ex extremo in extremū
                  <lb/>
                ꝓducit. </s>
                <s xml:id="N12E95" xml:space="preserve">vt conſtitutis predictis terminis .12.8.6. et
                  <lb/>
                collectis extremis puta .6. et .12. que .18. conſtituūt
                  <lb/>
                numerus qui fit ex ductu medii puta octonarii in
                  <lb/>
                collectas extremitates puta ī .18. eſt duplus ad nu­
                  <lb/>
                merum qui fit ex ductu extremorum .12. ſcilicet ī .6
                  <lb/>
                </s>
                <s xml:id="N12EA1" xml:space="preserve">Quod patet / quia ille eſt .144. hic vero .72. mõ con­
                  <lb/>
                ſtat illū eſſe dupluꝫ ad hunc.
                  <note position="left" xlink:href="note-0031-04a" xlink:label="note-0031-04" xml:id="N12F07" xml:space="preserve">3. ꝓṗetas
                    <lb/>
                  medieta­
                    <lb/>
                  tis har-
                    <lb/>
                  monice.</note>
                </s>
                <s xml:id="N12EAB" xml:space="preserve">¶ Tertia proprietas
                  <lb/>
                in hac medietate determinatis extremis medius
                  <lb/>
                terminus reperitur ſi per extremorum coniuncto-
                  <lb/>
                rum numerum: numerus qui ex differentia extre-
                  <lb/>
                morum in minimū conſurgit diuiditur. </s>
                <s xml:id="N12EB6" xml:space="preserve">iſ qui
                  <lb/>
                ex diuiſiõe relinquit̄̄ accipiat̄̄: at minimo extre-
                  <lb/>
                mo aggregatur. </s>
                <s xml:id="N12EBD" xml:space="preserve">vt determinatis his terminis .6.
                  <lb/>
                et .3. / ſi vis inuenire medium harmonicum inter il-
                  <lb/>
                los addas extremū extrēo puta .3. ip̄is .6 et erūt 9. /
                  <lb/>
                deiñ ducas dnr̄aꝫ inter .6. et .3. in .3. mīmū extremū:
                  <cb chead="Capitulum tertiū."/>
                et quia illa differentia eſt .3. ex ductu eius in .3. fi-
                  <lb/>
                unt .9. diuidas / igitur .9. per .9. et relictū ex diuiſio­
                  <lb/>
                ne erit vnitas: addas igitur vnitatem ternario: et
                  <lb/>
                aggregatum ex illa vnitate et ternario eſt mediuꝫ
                  <lb/>
                harmonicum inter ſex. et tria: eſt enim aggregatū
                  <lb/>
                illud quaternarius numerus. </s>
                <s xml:id="N12ED3" xml:space="preserve">Modo .6.4.3: ꝓpor­
                  <lb/>
                tionantur harmonice. </s>
                <s xml:id="N12ED8" xml:space="preserve">¶ Et hic aduerte /  quibuſ-
                  <lb/>
                cū duobus numeris inequalibus cõſtitutis hac
                  <lb/>
                doctrina mediante reperies medium terminū in-
                  <lb/>
                ter eos: et hoc cum fractione aut ſine inter .4. enim
                  <lb/>
                et .3. medium harmonicū eſt .3. cuꝫ tribus ſeptimis
                  <lb/>
                </s>
                <s xml:id="N12EE4" xml:space="preserve">Quomodo autem inueniatur medium geometri-
                  <lb/>
                cum partim ex his / que dicta ſunt / patet et comple­
                  <lb/>
                te in poſterum dicetur.</s>
              </p>
            </div>
            <div xml:id="N12F13" level="3" n="3" type="chapter" type-free="capitulum">
              <head xml:id="N12F18" xml:space="preserve">Capitulum tertium / in quo
                <lb/>
              agitur de quibuſdam propor­
                <lb/>
              tionalitatibus et modis argu­
                <lb/>
              endi in eis.</head>
              <p xml:id="N12F21">
                <s xml:id="N12F22" xml:space="preserve">SEx modos argumentandi pro­
                  <lb/>
                portionabiliter ſiue in ꝓportionalitati-
                  <lb/>
                bus quibus nonun̄. </s>
                <s xml:id="N12F29" xml:space="preserve">et philoſophi et cal­
                  <lb/>
                culatores phiſici vtūtur ponit Euclides ſexto ele-
                  <lb/>
                mentorum et recentiores mathematici poſt eum.
                  <lb/>
                </s>
                <s xml:id="N12F31" xml:space="preserve">¶ Iſtarum autem argumentationum prima dici-
                  <lb/>
                tur conuerſa: ſecunda permutata: tertia coniun-
                  <lb/>
                cta. </s>
                <s xml:id="N12F38" xml:space="preserve">quarta diſiuncta. </s>
                <s xml:id="N12F3B" xml:space="preserve">quinta euerſa: et ſexta equa.
                  <lb/>
                </s>
                <s xml:id="N12F3F" xml:space="preserve">¶ Pro intelligentia primi modi arguendi aduer­
                  <lb/>
                tendum eſt /  in propoſito antecedens alicuius ꝓ­
                  <lb/>
                portionis dicitur terminus / qui ad alterum com-
                  <lb/>
                paratur et conſequens terminus cui aliquis com­
                  <lb/>
                paratur / vt cum dicitur quatuor ad duo ille termi­
                  <lb/>
                nus quatuor eſt antecedens et duo conſequens / et
                  <lb/>
                ſi dicamus duo ad quatuor duo dicuntur antece-
                  <lb/>
                dens et quatuor conſequens
                  <note position="right" xlink:href="note-0031-05a" xlink:label="note-0031-05" xml:id="N1300B" xml:space="preserve">ꝓportõa­
                    <lb/>
                  litas con­
                    <lb/>
                  uerſa</note>
                </s>
                <s xml:id="N12F55" xml:space="preserve">¶ Iſto ſuppoſito pro­
                  <lb/>
                portionalitas conuerſa eſt quando ex anteceden-
                  <lb/>
                tibus fiunt conſequētia: et eocontra. </s>
                <s xml:id="N12F5C" xml:space="preserve">Uel aliter eſt
                  <lb/>
                proportionalis illatio in qua ex proportionibus
                  <lb/>
                maioris inequalitatis concluduntur proportio-
                  <lb/>
                nes minoris ineq̈litatis eis correſpondentes. </s>
                <s xml:id="N12F65" xml:space="preserve">ſic
                  <lb/>
                arguendo ſicut ſe habet octo ad quatuor ita duo a­
                  <lb/>
                d vnum / igitur ſicut ſe habet vnum ad duo ita qua­
                  <lb/>
                tuor ad octo. </s>
                <s xml:id="N12F6E" xml:space="preserve">Et etiã econuerſo cõcludēdo ex pro­
                  <lb/>
                portionibus minoris inequalitatis ꝓportiones
                  <lb/>
                maioris īeq̈litatꝪ eis correſpõdētes.
                  <note position="right" xlink:href="note-0031-06a" xlink:label="note-0031-06" xml:id="N13015" xml:space="preserve">ꝑmutata</note>
                </s>
                <s xml:id="N12F7A" xml:space="preserve">¶ Permuta-
                  <lb/>
                ta ꝓportiõalitas dicit̄̄ / cū ex ãtecedēte ſcḋe ꝓporti-
                  <lb/>
                onis ſit ↄ̨ñs prime et ex ↄ̨ñti prime ſit añs ſcḋe. </s>
                <s xml:id="N12F81" xml:space="preserve">Uel
                  <lb/>
                aliter eſt diſpoſitis quatuor terminis geometri-
                  <lb/>
                ce proportionalibus primi ad tertium. </s>
                <s xml:id="N12F88" xml:space="preserve">et ſecundi
                  <lb/>
                ad quartum proportionalis illatio ſic arguendo
                  <lb/>
                ſicut ſe habet .8. ad .4. ita .2. ad .1. / igitur ſicut ſe ha­
                  <lb/>
                bent .8. ad .2. ita .4. ad vnū. </s>
                <s xml:id="N12F91" xml:space="preserve">Et iſto modo arguen-
                  <lb/>
                endi vtitur philoſophus in pleriſ locis vt in fi-
                  <lb/>
                ne ſecundi perihermenias: in tertio topi. </s>
                <s xml:id="N12F98" xml:space="preserve">et in pri­
                  <lb/>
                mo celi et mundi in tractatu de infinito.
                  <note position="right" xlink:href="note-0031-07a" xlink:label="note-0031-07" xml:id="N1301B" xml:space="preserve">Cõiūcta.</note>
                </s>
                <s xml:id="N12FA2" xml:space="preserve">¶ Coniun­
                  <lb/>
                cta proportionalitas eſt a diſiunctis terminis geo­
                  <lb/>
                meteice proportionabilibus ad coniunctos pro-
                  <lb/>
                portionalis illatio. </s>
                <s xml:id="N12FAB" xml:space="preserve">tali modo arguendo: ſicut ſe
                  <lb/>
                habent .8. ad .4. ita .2. ad .1. / igitur ſicut ſe habent.
                  <lb/>
                </s>
                <s xml:id="N12FB1" xml:space="preserve">octo et quatuor ad quatuor ita duo et vnū ad vnū
                  <lb/>
                  <note position="right" xlink:href="note-0031-08a" xlink:label="note-0031-08" xml:id="N13021" xml:space="preserve">diſiūcta.</note>
                </s>
                <s xml:id="N12FBB" xml:space="preserve">¶ Diſiuncta proportionalitas eſt a cõiunctis ter-
                  <lb/>
                minis geometrice proportionabilibus ad diſiun­
                  <lb/>
                ctos proportionalis illatio. </s>
                <s xml:id="N12FC2" xml:space="preserve">tali modo arguendo /
                  <lb/>
                ſicut ſe habent 8. et .4. ad .4. ita duo et vnū ad vnū /
                  <lb/>
                igitur ſicut ſe habent octo ad quatuor ita duo ad
                  <lb/>
                vnum.
                  <note position="right" xlink:href="note-0031-09a" xlink:label="note-0031-09" xml:id="N13027" xml:space="preserve">Euerſa.</note>
                </s>
                <s xml:id="N12FD0" xml:space="preserve">¶ Euerſa ꝓportionalitas eſt a diuiſis ter-
                  <lb/>
                minis geometrice proportionabilibus ad coniun­
                  <lb/>
                ctos ordine conuerſo ad coniunctam proportio- </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum