Alvarus, Thomas, Liber de triplici motu, 1509

Table of Notes

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              <pb chead="Prime partis" file="0033" n="33"/>
              <p xml:id="N13206">
                <s xml:id="N13207" xml:space="preserve">Tertia ſuppoſitio. </s>
                <s xml:id="N1320A" xml:space="preserve">Oēs proportiões
                  <lb/>
                ſūt eq̈les quarū denoīationes ſunt eq̈les et illa ma­
                  <lb/>
                ior cuiꝰ denoīatio ē maior: et illa mīor: cuiꝰ denoīa­
                  <lb/>
                tio mīor. </s>
                <s xml:id="N13213" xml:space="preserve">Illa autem denoīatio dicitur maior / que
                  <lb/>
                ſumitur a maiori numero cū fractione vel ſine: vel
                  <lb/>
                ab vnitate cū maiori fractione.
                  <note position="left" xlink:href="note-0033-01a" xlink:label="note-0033-01" xml:id="N1323F" xml:space="preserve">Ior. ſcḋo
                    <lb/>
                  ele.</note>
                </s>
                <s xml:id="N1321F" xml:space="preserve">Hec nõ demõſtra-
                  <lb/>
                tur / q2 diffinitio eſt / et a iordauo petitur in princi-
                  <lb/>
                pio ſecūdi elemētoꝝ. </s>
                <s xml:id="N13226" xml:space="preserve">Exemplū / vt ꝓportio que eſt
                  <lb/>
                8. ad .4. eſt equalis ꝓportioni que eſt .2. ad .1. quia
                  <lb/>
                vtra illarū denominatur dupla. </s>
                <s xml:id="N1322D" xml:space="preserve">Sexquialtera
                  <lb/>
                autē maior eſt ſexquitertia: q2 denominatio eius
                  <lb/>
                maior eſt: denominatur em̄ ab vnitate cū medieta­
                  <lb/>
                te: altera vero ab vnitate cum tertia. </s>
                <s xml:id="N13236" xml:space="preserve">Modo plus
                  <lb/>
                eſt vnitas cū medietate quã cū tertia.</s>
              </p>
              <p xml:id="N13247">
                <s xml:id="N13248" xml:space="preserve">Quarta ſuppoſitio. </s>
                <s xml:id="N1324B" xml:space="preserve">Omne totum ex
                  <lb/>
                quantolibet minori eo cõponitur: et diſtribuat ly
                  <lb/>
                quãtolibet pro generibus ſingnloꝝ. </s>
                <s xml:id="N13252" xml:space="preserve">Probat̄̄ hec
                  <lb/>
                ſuppoſitio / q2 quãtūlibet minus aliquo maiori eo
                  <lb/>
                eſt pars illius: ergo ex quãtolibet tali cõponitur.
                  <lb/>
                </s>
                <s xml:id="N1325A" xml:space="preserve">Probatur antecedens / q2 capto vno pedali: quã-
                  <lb/>
                talibet mīor quãtitas pedali eſt ꝑs eiꝰ / vt ptꝫ ex ſe.</s>
              </p>
              <p xml:id="N1325F">
                <s xml:id="N13260" xml:space="preserve">Quinta ſuppoſitio. </s>
                <s xml:id="N13263" xml:space="preserve">Omne cõpoſitū
                  <lb/>
                ex duobus equalibus adequate: eſt preciſe duplū
                  <lb/>
                ad vtrū illoꝝ: et omne cõpoſitū ex tribus equali-
                  <lb/>
                bus adequate eſt triplum ad quodlibet illoꝝ: et ex
                  <lb/>
                quattuor quadruplū: et ex quin quintuplum .etc̈.
                  <lb/>
                </s>
                <s xml:id="N1326F" xml:space="preserve">Patet hec ſuppoſitio ex diffinitione dupli, tripli
                  <lb/>
                quadrupli, et ſic ſine termino.</s>
              </p>
              <p xml:id="N13274">
                <s xml:id="N13275" xml:space="preserve">Sexta ſuppoſitio. </s>
                <s xml:id="N13278" xml:space="preserve">Omne cõpoſituꝫ
                  <lb/>
                ex duobus inequalibus eſt maius quã duplum ad
                  <lb/>
                minꝰ illoꝝ: et minus quã duplū ad maius illoꝝ: et
                  <lb/>
                ſi cõponatur ex tribus inequalibus: eſt maius quã
                  <lb/>
                triplū ad minimū illoꝝ: et minꝰ quã triplū ad ma-
                  <lb/>
                ximū: et ſi ex quattuor eſt maius quã quadruplum
                  <lb/>
                ad minimū illoꝝ: et minus quã quadruplū ad ma­
                  <lb/>
                ximū: et ſic conſequēter: ſi cõponatur ex quin, ex
                  <lb/>
                ſex .etc̈. </s>
                <s xml:id="N1328B" xml:space="preserve">Probatur prima pars: q2 illud cõpoſitum
                  <lb/>
                continet minus illorū duorū bis: et aliquid vltra:
                  <lb/>
                ergo eſt maius quã duplū ad illud. </s>
                <s xml:id="N13292" xml:space="preserve">Cõſequētia eſt
                  <lb/>
                nota: et antecedens ꝓbatur: q2 ſi ↄ̨tineret minꝰ bis
                  <lb/>
                adequate iam illud eſſet ſua medietas: et per con-
                  <lb/>
                ſequens reſiduū etiã eſſet medietas: et ſic illa duo
                  <lb/>
                eſſent equalia / quod eſt contra hypotheſim. </s>
                <s xml:id="N1329D" xml:space="preserve">Alia
                  <lb/>
                pars huius partis ſimiliter ꝓbatur / q2 ſi eſſet du-
                  <lb/>
                plū ad maius illoꝝ / iã illud eſſet ſua medietas / qḋ
                  <lb/>
                modo eſt īpugnatū. </s>
                <s xml:id="N132A6" xml:space="preserve">Secūda pars probatur / quia
                  <lb/>
                illud cõpoſitū continet minimū illoꝝ triū ter et a-
                  <lb/>
                liquid vltra: ergo eſt pluſquã triplū ad illud. </s>
                <s xml:id="N132AD" xml:space="preserve">Con­
                  <lb/>
                ſequētia patet et antecedens ꝓbatur / q2 ſi cõtineret
                  <lb/>
                eū ter adequate iã illud eſſet vna tertia eius / vt ptꝫ
                  <lb/>
                ex ſe et ꝑ cõſequēs alie due partes eſſent due tertie /
                  <lb/>
                et ſic aggregatū ex eis eſſet dupluꝫ ad illud mini-
                  <lb/>
                mū: ſed hoc eſt falſum: q2 alterū illoꝝ duoꝝ eſt ma­
                  <lb/>
                ius iſto minimo: et aliud equale vel maius / vt con-
                  <lb/>
                ſtat: igitur aggregatū ex iſtis duobꝰ eſt maiꝰ quã
                  <lb/>
                duplū ad illud minimū. </s>
                <s xml:id="N132C0" xml:space="preserve">Alia pars huius partis
                  <lb/>
                ꝓbatur / q2 maximū illoꝝ triū eſt maius quã tertia /
                  <lb/>
                ergo cõpoſitū ex illis eſt minꝰ quã triplū ad illud.
                  <lb/>
                </s>
                <s xml:id="N132C8" xml:space="preserve">Cõſequentia patet et antecedens ꝓbatur / q2 ſi eſſet
                  <lb/>
                adeq̈te tertia iã alie due ꝑtes eſſent due tertie: et ſic
                  <lb/>
                aggregatū ex eis eſſet duplū ad illud / qḋ eſt falſuꝫ /
                  <lb/>
                q2 aggregatū ex aliis duobus componitur ex vno
                  <lb/>
                minori illo: et alio equali vel minori: igitur aggre­
                  <lb/>
                gatū ex eis nõ eſt duplū ad illud. </s>
                <s xml:id="N132D5" xml:space="preserve">Et ſic ꝓbabis ali­
                  <lb/>
                as partes. </s>
                <s xml:id="N132DA" xml:space="preserve">Patet igitur ſuppoſitio.</s>
              </p>
              <p xml:id="N132DD">
                <s xml:id="N132DE" xml:space="preserve">Septima ſuppoſitio. </s>
                <s xml:id="N132E1" xml:space="preserve">Quãdo aliqua
                  <lb/>
                latitudo ſiue exceſſus additur alicui maiorē ꝓpor­
                  <cb chead="Capitulū ſequartū."/>
                tionē acquirit quã quãdo eidē additur minor ex-
                  <lb/>
                ceſſus ſiue latitudo: vt quando quaternario addi­
                  <lb/>
                tur quaternarius maiorē ꝓportionē acquirit quã
                  <lb/>
                quando ei additur binarius: </s>
                <s xml:id="N132EF" xml:space="preserve">Et ex conſequenti ſe-
                  <lb/>
                quitur /  quãdo aliq̇d deperdit aliquã latitudinē
                  <lb/>
                ſiue quantitatē maiorē ꝓportionē deperdit quaꝫ
                  <lb/>
                quando deperdit minorē latitudinē. </s>
                <s xml:id="N132F8" xml:space="preserve">Hec ſuppoſi­
                  <lb/>
                tio cū ſuo correlario propter ſui euidentiã nõ pro­
                  <lb/>
                batur: ſed ſimpliciter petitur.</s>
              </p>
              <p xml:id="N132FF">
                <s xml:id="N13300" xml:space="preserve">Octaua ſuppoſitio. </s>
                <s xml:id="N13303" xml:space="preserve">Quãdocū idē
                  <lb/>
                exceſſus ſiue latitudo additur maiori et mīori: ma­
                  <lb/>
                iorē ꝓportionē acquirit minꝰ quã maius. </s>
                <s xml:id="N1330A" xml:space="preserve">Et cum
                  <lb/>
                maius et minus deperdūt eandē latitudinē ſiue ex­
                  <lb/>
                ceſſum maiorē ꝓportionē deperdit minus quã ma­
                  <lb/>
                ius: vt ſi quaternarius et octonarius perdant bi-
                  <lb/>
                nariū maiorē ꝓportionē deperdit quaternarius
                  <lb/>
                quã octonarius. </s>
                <s xml:id="N13317" xml:space="preserve">Quaternarius em̄ perdit ꝓpor-
                  <lb/>
                tionē duplã: octonarius vero ſexquitertiã: vt con-
                  <lb/>
                ſtat. </s>
                <s xml:id="N1331E" xml:space="preserve">Et ſi binarius et ſenarius binariū acquirant
                  <lb/>
                binariꝰ eadē ratione maiorē ꝓportionē acquirit
                  <lb/>
                quam ſenarius: vt cõſtat. </s>
                <s xml:id="N13325" xml:space="preserve">Probatur / ſint a.b. due
                  <lb/>
                quantitates ſine numeri ſiue que vis alie latitudi-
                  <lb/>
                nes a. maior et b. minor que ſe habeant in ꝓporti-
                  <lb/>
                one f. et acquirat tam a. quã b.d. exceſſum ſiue lati-
                  <lb/>
                tudinē: tunc dico /  b. maiorē ꝓportionē acquirit
                  <lb/>
                quã a. </s>
                <s xml:id="N13332" xml:space="preserve">Quod ſic ꝓbatur: et volo /  quãdo a. acqui-
                  <lb/>
                rit d. antea quã b. acquirat ipſum d. acquirat vnã
                  <lb/>
                quantitatē ad quã d. ſe habet in ꝓportione f. et ſit
                  <lb/>
                illa quantitas e. / et arguitur ſic / a. et b. ſe habent in
                  <lb/>
                ꝓportione f. et quantitas acquiſita ipſi a ſe habet
                  <lb/>
                etiã in eadē ꝓportione ad quantitatē acquiſitam
                  <lb/>
                ipſi b. / ergo continuo a. et b. manent in eadē ꝓpor-
                  <lb/>
                tione f. in qua ſe habebant ante talē acquiſitionē.
                  <lb/>
                </s>
                <s xml:id="N13344" xml:space="preserve">Patet hec cõſequentia ex quīto correlario quīte
                  <lb/>
                concluſionis ſecūdi capitis huiꝰ: et per cõſequens
                  <lb/>
                tantã ꝓportionē acquiſiuit b. ſupra ſe quantam a
                  <lb/>
                ſupra ſe. </s>
                <s xml:id="N1334D" xml:space="preserve">Si em̄ b. acquiſiuiſſet minorē iã ꝓportio
                  <lb/>
                inter a. et b. fuiſſet augmentata: et ſi maiorem iam
                  <lb/>
                fuiſſet diminuta: qm̄ quantã ꝓportionē acquirit
                  <lb/>
                numerus minor vltra numeꝝ maiorē tantã deꝑdit
                  <lb/>
                ꝓportio inter illos numeros: et quantã numerus
                  <lb/>
                maior acquirit vltra minorē tãtã acq̇rit ꝓportio
                  <lb/>
                inṫ illos nūeros ſiue q̄uis alia latitudo: vt ↄ̨ſtat ex
                  <lb/>
                ſuꝑioribꝰ et ex ↄ̨ñti quantã ꝓportionē acq̇ſiuit b. ꝑ
                  <lb/>
                acquiſitionē e. latitudinis tantã adequate acqui-
                  <lb/>
                ſiuit a. per additionē d. latitudinis et eocõtra. </s>
                <s xml:id="N13362" xml:space="preserve">igit̄̄
                  <lb/>
                quando b. acquirit d. maiorē latitudinē quã ſit e.
                  <lb/>
                maiorē ꝓportionē acquirit: et per cõſequens ma-
                  <lb/>
                iorē ꝓportionē acquirit b: acquirendo d. quam a.
                  <lb/>
                acquirendo d. / quod fuit probandū. </s>
                <s xml:id="N1336D" xml:space="preserve">Patet tamen
                  <lb/>
                conſequentia ex ſeptima ſuppoſitione huiꝰ capi-
                  <lb/>
                tis. </s>
                <s xml:id="N13374" xml:space="preserve">Et ſic patet prima pars: et ſecunda facile ꝓba-
                  <lb/>
                tur / qm̄ ſi quando a. et b. acquirūt d. latitudinē ma­
                  <lb/>
                iorē ꝓportionē acquirit b. quã a. / ſequitur /  cū de­
                  <lb/>
                perdunt eandē d. latitudinē maiorē ꝓportionem
                  <lb/>
                deperdit b. quã a. </s>
                <s xml:id="N1337F" xml:space="preserve">Nam adequate perdit illã quã
                  <lb/>
                acquiſiuit et maiorē acquiſiuit: ergo maiorem de-
                  <lb/>
                perdit. </s>
                <s xml:id="N13386" xml:space="preserve">Et ſic patet ſuppoſitio.</s>
              </p>
              <p xml:id="N13389">
                <s xml:id="N1338A" xml:space="preserve">His iactis fundamentis ſit prima cõ­
                  <lb/>
                cluſio. </s>
                <s xml:id="N1338F" xml:space="preserve">Oīs ꝓportio multiplex, multiplex ſuꝑpar-
                  <lb/>
                ticularis, vel multiplex ſuprapartiens eſt maior
                  <lb/>
                ꝓportione ſuperparticulari vel ſuprapartiente.
                  <lb/>
                </s>
                <s xml:id="N13397" xml:space="preserve">Probatur: q2 cuiuſlibet ꝓportionis multiplicis,
                  <lb/>
                multiplicis ſuꝑparticularis, vel multiplicis ſu-
                  <lb/>
                prapartiens, denominatio eſt maior quã alicu-
                  <lb/>
                ius ſuperparticularis vel ſuprapartientis: igitur
                  <lb/>
                quelibet ꝓportio multiplex, aut multiplex ſuper-
                  <lb/>
                particularis, aut multiplex ſuprapartiēs, eſt ma­ </s>
              </p>
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