Alvarus, Thomas, Liber de triplici motu, 1509

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              <p xml:id="N10305">
                <s xml:id="N1038E" xml:space="preserve">
                  <pb chead="Prime partis" file="0007" n="7"/>
                tas ter ſumpta: adequate conſtituit ternarium
                  <lb/>
                et quater ſumpta: quaternarium. </s>
                <s xml:id="N10398" xml:space="preserve">et dualitas eſt
                  <lb/>
                pars aliquota numeri octonarii. </s>
                <s xml:id="N1039D" xml:space="preserve">quoniam duali­
                  <lb/>
                tas quater ſumpta adequate numerū octonariuꝫ
                  <lb/>
                conſtituit. </s>
                <s xml:id="N103A4" xml:space="preserve">¶ Ex quo patet /  dualitas non eſt ꝑrs
                  <lb/>
                aliquota numeri ſeptenarii quoniam non aliquo­
                  <lb/>
                ties ſumpta: reddit illud totum adequate. </s>
                <s xml:id="N103AB" xml:space="preserve">¶ Pro­
                  <lb/>
                portio autem irrationalis: eſt illa que nõ immedi­
                  <lb/>
                ate ab aliquo numero denominatur. </s>
                <s xml:id="N103B2" xml:space="preserve">Alio modo
                  <lb/>
                proportio irrationalis: eſt duarum quantitatum
                  <lb/>
                ita ſe habentiū:  nulla pars aliquota vnius eſt
                  <lb/>
                ꝑs aliq̊ta alteriꝰ vt ꝓportio q̄ ē īter diametrū et co­
                  <lb/>
                ſtã ſui q̈drati. </s>
                <s xml:id="N103BD" xml:space="preserve">nã diameṫ excedit coſtã et nõ aliq̊ties
                  <lb/>
                nec ꝑ aliquã ꝑtem aliquotã. </s>
                <s xml:id="N103C2" xml:space="preserve">vel per aliq̈s ꝑtes ali­
                  <lb/>
                quotas. </s>
                <s xml:id="N103C7" xml:space="preserve">vt inferius probabitur in capitulo de ꝓ-
                  <lb/>
                portione irrationali.
                  <note position="left" xlink:href="note-0007-01a" xlink:label="note-0007-01" xml:id="N10458" xml:space="preserve">Diuiſio
                    <lb/>
                  ꝓportio­
                    <lb/>
                  nū rõna-
                    <lb/>
                  lium.</note>
                </s>
                <s xml:id="N103D1" xml:space="preserve">¶ Proportionum auteꝫ ra-
                  <lb/>
                tionalium .5. ſunt ſpecies tres ſimplices: et due cõ­
                  <lb/>
                poſite. </s>
                <s xml:id="N103D8" xml:space="preserve">¶ Simplices ſunt iſte. </s>
                <s xml:id="N103DB" xml:space="preserve">multiplex: ſuperpar­
                  <lb/>
                ticularis: et ſuprapartiēs. </s>
                <s xml:id="N103E0" xml:space="preserve">¶ Compoſite vero ſunt
                  <lb/>
                multiplex. </s>
                <s xml:id="N103E5" xml:space="preserve">multiplex ſuperparticularis: mĺtiplex
                  <lb/>
                ſuprapartiens </s>
                <s xml:id="N103EA" xml:space="preserve">¶ Unde proportio multiplex: eſt ꝓ­
                  <lb/>
                portio qua maius continet minus aliquoties ta-
                  <lb/>
                tū vt dupla, tripla .4. enim continent .2. bis. / et .6.
                  <lb/>
                continent .2. ter tantum </s>
                <s xml:id="N103F3" xml:space="preserve">Et ideo inter illos nume-
                  <lb/>
                ros eſt ꝓportio multiplex. </s>
                <s xml:id="N103F8" xml:space="preserve">¶ Proportio vero ſu-
                  <lb/>
                perparticularis. </s>
                <s xml:id="N103FD" xml:space="preserve">eſt proportio qua maius cõtinet
                  <lb/>
                minus ſemel tãtū: et aliquam partem eius aliquo­
                  <lb/>
                tã adeq̈te. </s>
                <s xml:id="N10404" xml:space="preserve">vt ꝓportio ſex ad .4. nã .6. cõtinet .4. ſe-
                  <lb/>
                mel tm̄ et medietatē q̄ eſt pars aliquota ipſoꝝ .4.
                  <lb/>
                </s>
                <s xml:id="N1040A" xml:space="preserve">¶ Proportio autem ſuprapartiēs: eſt proportio
                  <lb/>
                qua maius continet minus ſemel tantū: et aliquot
                  <lb/>
                partes eius aliquotas: que ſimul non faciunt ali­
                  <lb/>
                quam eius partem aliquotam. </s>
                <s xml:id="N10413" xml:space="preserve">vt ꝓportio que eſt
                  <lb/>
                inter .7. et .5. </s>
                <s xml:id="N10418" xml:space="preserve">Nam .7. continent .5. ſemel tantum: et
                  <lb/>
                duas partes eius aliquotas: puta duas vnitates
                  <lb/>
                </s>
                <s xml:id="N1041E" xml:space="preserve">¶ Sed proportio multiplex ſuperparticularis eſt
                  <lb/>
                illa qua maius continet minus aliquotiens: et
                  <lb/>
                cum hoc aliquam eius partem aliquotam tantuꝫ
                  <lb/>
                vt proportio que eſt inter nouem et .4. </s>
                <s xml:id="N10427" xml:space="preserve">Nã .9. con-
                  <lb/>
                tinent .4. bis. / et vnam partem numeri quaternarii
                  <lb/>
                puta vnitatem. </s>
                <s xml:id="N1042E" xml:space="preserve">¶ Proportio autem multiplex ſu­
                  <lb/>
                prapartiens: eſt illa qua maius continent minus
                  <lb/>
                aliquotiens et aliquot partes eiꝰ aliquotas: que
                  <lb/>
                non faciunt vnam eius partem aliquotam vt pro­
                  <lb/>
                portio que eſt inter .11. et .4. </s>
                <s xml:id="N10439" xml:space="preserve">Nã .11. continent .4. bis /
                  <lb/>
                et tres partes aliquotas ipſorum .4. et ille nõ fa-
                  <lb/>
                ciunt aliquam partem aliquotam ipſorum .4.</s>
              </p>
              <note position="left" xml:id="N10464" xml:space="preserve">Sufficiē-
                <lb/>
              cia quī
                <lb/>
              numeri ꝓ­
                <lb/>
              portiõis
                <lb/>
              rõaĺ ma­
                <lb/>
              ioris ine­
                <lb/>
              q̈litatis.</note>
              <p xml:id="N10474">
                <s xml:id="N10475" xml:space="preserve">¶ Harum autem proportionum: ſiue ſpecierum ꝓ­
                  <lb/>
                portionum ſufficientia: talis ratione haberi põt
                  <lb/>
                vt adducit Albertus de ſaxonia ī ſuo tractatu de
                  <lb/>
                proportionibus poſt alios mathematicos. </s>
                <s xml:id="N1047E" xml:space="preserve">Qm̄
                  <lb/>
                oīs numerus: ſiue quantitas ad aliam quantitatē
                  <lb/>
                habens rationalem proportiouem: aut excedit
                  <lb/>
                eam: aut exceditur ab illa. </s>
                <s xml:id="N10487" xml:space="preserve">Si excedit eam: aut
                  <lb/>
                continet ipſam aliquoties. </s>
                <s xml:id="N1048C" xml:space="preserve">aut ſemel tantū: et ali­
                  <lb/>
                quid vltra. </s>
                <s xml:id="N10491" xml:space="preserve">aut pluries et aliquid vltra. </s>
                <s xml:id="N10494" xml:space="preserve">Si primū /
                  <lb/>
                tunc erit proportio multiplex </s>
                <s xml:id="N10499" xml:space="preserve">Si ſecūdū / aut illud
                  <lb/>
                aliquid vltra eſt vna pars eius aliquota adequa-
                  <lb/>
                te: aut ē plures partes aliquote que nõ faciūt vnã
                  <lb/>
                partem aliquotam. </s>
                <s xml:id="N104A2" xml:space="preserve">Si primum: ſic eſt ꝓportio ſu­
                  <lb/>
                perparticularis. </s>
                <s xml:id="N104A7" xml:space="preserve">Si ſecundum / eſt proportio ſuꝑ-
                  <lb/>
                partiens. </s>
                <s xml:id="N104AC" xml:space="preserve">Si vero maior quantitas continet mi-
                  <lb/>
                norē pluries. </s>
                <s xml:id="N104B1" xml:space="preserve">et aliquid vltra. </s>
                <s xml:id="N104B4" xml:space="preserve">vel illud quod vltra
                  <lb/>
                continet eſt pars aliquota adequate aut: plures
                  <lb/>
                partes aliquote: que non faciunt vnã. </s>
                <s xml:id="N104BB" xml:space="preserve">Si primum /
                  <lb/>
                ſic eſt proportio multiplex ſuperparticulares. </s>
                <s xml:id="N104C0" xml:space="preserve">Si
                  <cb chead="Capitulum ſecundum"/>
                ſecundum ſic eſt proportio multiplex ſupraparti-
                  <lb/>
                ens. </s>
                <s xml:id="N104C8" xml:space="preserve">Et quia quantitas maior habens proportio­
                  <lb/>
                nē rationalem ad quantitatem minorē nõ poteſt
                  <lb/>
                pluribus modis ad illam referri
                  <gap/>
                ſiue compara-
                  <lb/>
                ri. </s>
                <s xml:id="N104D3" xml:space="preserve">quam his quin modis conſequens eſt /  non
                  <lb/>
                poſſunt eſſe plures ſpecies proportionis rationa­
                  <lb/>
                lis his .5. </s>
                <s xml:id="N104DA" xml:space="preserve">Quãdoquidem eodem modo venari po­
                  <lb/>
                teſt minoris inequalitatis proportionum ſuffici­
                  <lb/>
                entia. </s>
                <s xml:id="N104E1" xml:space="preserve">Sola enim ratione: proportio maioris ine­
                  <lb/>
                qualitatis: et minoris differunt) </s>
                <s xml:id="N104E6" xml:space="preserve">De irrationali
                  <lb/>
                autem poſterius dicetur.</s>
              </p>
            </div>
            <div xml:id="N104EB" level="3" n="2" type="chapter" type-free="capitulum">
              <head xml:id="N104F0" xml:space="preserve">Cpitulum ſecundum / in quo agitur de ſpe­
                <lb/>
              ciebus horum quin generum proportionū
                <lb/>
              et de ipſarum generatione.</head>
              <p xml:id="N104F7">
                <s xml:id="N104F8" xml:space="preserve">OMnis proportio ſiue omne ge­
                  <lb/>
                nus proportiõis: infinitas habet ſpecies
                  <lb/>
                </s>
                <s xml:id="N104FE" xml:space="preserve">Unde genus multiplicis: habet infinitas
                  <lb/>
                ſpecies denominatas a naturali ſerie numerorū
                  <lb/>
                puta duplã denominatã a binario triplã a terna­
                  <lb/>
                rio: milleculpam a millenario: centuplam a cen-
                  <lb/>
                tenario. </s>
                <s xml:id="N10509" xml:space="preserve">et ſic in infinitū. </s>
                <s xml:id="N1050C" xml:space="preserve">¶ Proportio em̄ dupla:
                  <lb/>
                eſt illa qua maius continet minus: bis adequate
                  <lb/>
                vt .4. cum .2. et tripla qua maius continet minus:
                  <lb/>
                ter adequate. </s>
                <s xml:id="N10515" xml:space="preserve">et quadrupla quater adequate. </s>
                <s xml:id="N10518" xml:space="preserve">et ſic
                  <lb/>
                in infinitum. </s>
                <s xml:id="N1051D" xml:space="preserve">¶ Generãtur autem omnes ꝓportio­
                  <lb/>
                nes duple que infinite ſunt iſto modo. </s>
                <s xml:id="N10522" xml:space="preserve">Diſpona-
                  <lb/>
                tur / primo ſeries naturalis numeroꝝ in vna linea
                  <lb/>
                et in alia linea inferiori diſponantur omnes nu-
                  <lb/>
                meri excedentes ſe binario: incipiendo a binario
                  <lb/>
                in infinitum. </s>
                <s xml:id="N1052D" xml:space="preserve">Et iſto modo cõparando primum ſu-
                  <lb/>
                perioris linie primo inferioris: et ſecundū ſecūdo
                  <lb/>
                et tertiū tertio.
                  <note position="right" xlink:href="note-0007-02a" xlink:label="note-0007-02" xml:id="N10547" xml:space="preserve">gñatio ꝓ­
                    <lb/>
                  portõnū
                    <lb/>
                  duplarū</note>
                </s>
                <s xml:id="N10539" xml:space="preserve">et ſic in infinitum inuenientur infi-
                  <lb/>
                nite ꝓportiõis duple. </s>
                <s xml:id="N1053E" xml:space="preserve">in preſenti figura clare hoc
                  <lb/>
                poteris conſpicere.</s>
              </p>
              <xhtml:table xml:id="N10551">
                <xhtml:tr xml:id="N10552">
                  <xhtml:td xml:id="N10553" xml:space="preserve"/>
                </xhtml:tr>
              </xhtml:table>
              <p xml:id="N10555">
                <s xml:id="N10556" xml:space="preserve">Per naturalem ſeriē numerorum: intelligas ordi­
                  <lb/>
                ne numerorū incipiēdo ab vnitate nullū numeruꝫ
                  <lb/>
                omittendo. </s>
                <s xml:id="N1055D" xml:space="preserve">vt .1.2.3.4. etc̈. </s>
                <s xml:id="N10560" xml:space="preserve">¶ Sed infinite ꝓportio-
                  <lb/>
                nes triple: iſto modo generantur </s>
                <s xml:id="N10565" xml:space="preserve">Diſponatur / oēs
                  <lb/>
                nūeri ſcḋm ſeriē naturalē nūerorū incipiendo ab
                  <lb/>
                vnitate ī vna linea et ī linea īferiori diſponãt̄̄ oēs
                  <lb/>
                nūeri excedētes ſe ṫnario. </s>
                <s xml:id="N1056E" xml:space="preserve">et tūc cõparãdo ṗmū īfe­
                  <lb/>
                rioris ordinis prīo ſuperioris et ſecūdū ſecūdo et
                  <lb/>
                tertiū tertio:
                  <note position="right" xlink:href="note-0007-03a" xlink:label="note-0007-03" xml:id="N1057E" xml:space="preserve">gñatio ꝓ­
                    <lb/>
                  portõnū
                    <lb/>
                  triplarū</note>
                habebunt̄̄ infinite ꝓportiões triple.</s>
              </p>
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                </xhtml:tr>
              </xhtml:table>
              <note position="right" xml:id="N1058C" xml:space="preserve">gñatio ꝓ­
                <lb/>
              portõnū
                <lb/>
              q̈drupla­
                <lb/>
              rum:</note>
              <p xml:id="N10596">
                <s xml:id="N10597" xml:space="preserve">Si vero velis gñare oēs ꝓportiões quadruplas:
                  <lb/>
                capias nūeros excedentes ſe q̈ternario. </s>
                <s xml:id="N1059C" xml:space="preserve">incipiēdo
                  <lb/>
                a nūero q̈ternario cū ſerie naturali nūeroꝝ.
                  <note position="right" xlink:href="note-0007-04a" xlink:label="note-0007-04" xml:id="N105BE" xml:space="preserve">Gñatio
                    <lb/>
                  quītupla­
                    <lb/>
                  rum.</note>
                </s>
                <s xml:id="N105A6" xml:space="preserve">¶ Si
                  <lb/>
                aūt quītuplã: capias oēs excedētes ſe q̇nario
                  <note position="right" xlink:href="note-0007-05a" xlink:label="note-0007-05" xml:id="N105C8" xml:space="preserve">Gñatio
                    <lb/>
                  ſextupla­
                    <lb/>
                  rum.</note>
                </s>
                <s xml:id="N105B0" xml:space="preserve">¶ Si
                  <lb/>
                ſextuplã ſenario. </s>
                <s xml:id="N105B5" xml:space="preserve">et ſic in infinitū vt facile eſt vide-
                  <lb/>
                re in figuris ſequentibus.</s>
              </p>
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                  <xhtml:td xml:id="N105D4" xml:space="preserve"/>
                </xhtml:tr>
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              <p xml:id="N105D6">
                <s xml:id="N105D7" xml:space="preserve">¶ Suꝑparticularis autē ꝓportio etiam infinitas
                  <lb/>
                habet ſpecies denoīatas a partibus aliquotis: et
                  <lb/>
                vnitate. </s>
                <s xml:id="N105DE" xml:space="preserve">puta a medietate: a tertia quarta quinta /
                  <lb/>
                et ſic in infinitū. </s>
                <s xml:id="N105E3" xml:space="preserve">Et ideo prima ſpecies eiꝰ et maxīa
                  <lb/>
                dicitur ſexquialtera. ſecūda vero ſexquitertia. ſex­ </s>
              </p>
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