Alvarus, Thomas, Liber de triplici motu, 1509

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          <div xml:id="N1194D" level="2" n="2" type="other" type-free="pars">
            <div xml:id="N14B3F" level="3" n="7" type="chapter" type-free="capitulum">
              <pb chead="Secunde partis." file="0050" n="50"/>
              <p xml:id="N14DA1">
                <s xml:id="N150E9" xml:space="preserve">
                  <note position="left" xlink:href="note-0052-01a" xlink:label="note-0052-01" xml:id="N1511F" xml:space="preserve">Branar­
                    <lb/>
                  dinus.</note>
                Iſta cõncluſio / vt dicit thomos branardinꝰ in ſua
                  <lb/>
                geometria in capitulo de proportionalitate con-
                  <lb/>
                cluſione quarta longã et prolixã expetit demõſtra­
                  <lb/>
                tionem.
                  <note position="left" xlink:href="note-0052-02a" xlink:label="note-0052-02" xml:id="N15127" xml:space="preserve">Eu. 6. ele</note>
                </s>
                <s xml:id="N15100" xml:space="preserve">Ideo ſufficiat ad eam euclidis auctoritas
                  <lb/>
                ſexto elementoꝝ propoſitione decima tertia.</s>
              </p>
              <p xml:id="N1512D">
                <s xml:id="N1512E" xml:space="preserve">Nona cõcluſio. </s>
                <s xml:id="N15131" xml:space="preserve">Ad inueniendã pro-
                  <lb/>
                portionē ſubduplã duple, aut alicuiꝰ alterius, cõ-
                  <lb/>
                ſtituantur due linee ſe habentes in ꝓportione illa
                  <lb/>
                cuiꝰ medietas queritur: et inueniatur media linea
                  <lb/>
                inter eas per artem precedentis cõcluſionis: et tūc
                  <lb/>
                maioris linee ad illam mediã et etiam illius medie
                  <lb/>
                ad minimã erit proportio que eſt media ſiue me-
                  <lb/>
                dietas talis proportionis. </s>
                <s xml:id="N15142" xml:space="preserve">Et ſi velis īuenire ſub-
                  <lb/>
                quadruplã proportionē īuenias lineã mediã inter
                  <lb/>
                primã, et ſecundã et vnã aliam inter ſecundã et ter-
                  <lb/>
                tiam, et tunc quelibet illarū intermediarū erit ſub­
                  <lb/>
                quadrupla: q2 erūt ibi .5. termini continuo ꝓpor-
                  <lb/>
                tionabiles: igitur proportio extremi ad extremū
                  <lb/>
                eſt quadrupla ad quãlibet intermediam. </s>
                <s xml:id="N15151" xml:space="preserve">Et ſi vis
                  <lb/>
                īuenire ſuboctuplã poſtquã īueniſti ſubq̈druplam
                  <lb/>
                inter quaſlibet duas lineas īmediate ſe habentes
                  <lb/>
                eleua vnã. </s>
                <s xml:id="N1515A" xml:space="preserve">Et ſi vis īuenire ſubſexdecuplã poſtquã
                  <lb/>
                īueniſti ſuboctuplã: īter quaſlibet duas eleua vnã
                  <lb/>
                artificio precedentis cõcluſionis / et ſic in infinitum
                  <lb/>
                duplicando. </s>
                <s xml:id="N15163" xml:space="preserve">Hec concluſio patet ex priori patro-
                  <lb/>
                cinio octaue concluſionis precedentis capitis.</s>
              </p>
              <p xml:id="N15168">
                <s xml:id="N15169" xml:space="preserve">Decima cõcluſio. </s>
                <s xml:id="N1516C" xml:space="preserve">Quãuis facile ſit
                  <lb/>
                cuilibet ꝓportioni īuenirē ſubduplã, ſubquadru-
                  <lb/>
                plam, ſuboctuplã, ſubſexdecuplã, et ſic in infinitū
                  <lb/>
                aſcendendo per numeros pariter pares: difficile
                  <lb/>
                tamen eſt ſubtriplã, ſubquintuplã, ſubſextuplam /
                  <lb/>
                et ſic in infinitū per numeros impares vel impari­
                  <lb/>
                ter pares aſcendendo īuenire. </s>
                <s xml:id="N1517B" xml:space="preserve">Prima pars patet
                  <lb/>
                ex priori concluſione: et ſecūda eſt michi experimē­
                  <lb/>
                to cõperta: quãuis nicholaꝰ horen in ſuo tractatu
                  <lb/>
                ꝓportionū capite quarto velit dare modum per
                  <lb/>
                artem medie rei inuentionis ad īueniendam pro-
                  <lb/>
                portionem et ſubduplam, et ſubtriplam, et ſubſex-
                  <lb/>
                quialteram.
                  <note position="left" xlink:href="note-0052-03a" xlink:label="note-0052-03" xml:id="N15208" xml:space="preserve">Contra
                    <lb/>
                  horeu:</note>
                </s>
                <s xml:id="N1518F" xml:space="preserve">¶ Sed ſaluo meliori indicio et aucto-
                  <lb/>
                ritate tam circuaſpecti viri ſignanter in mathe-
                  <lb/>
                mathicis ſciētiis: videtur michi /  per artē medie
                  <lb/>
                rei īuentionis nõ poſſunt īueniri quatuor linee cõ­
                  <lb/>
                tinuo proportionabiliter ſe habentes. </s>
                <s xml:id="N1519A" xml:space="preserve">Quod ſic
                  <lb/>
                oſtendo: quia captis duabus lineis ſe habentibꝰ
                  <lb/>
                in ꝓportione dupla ad īueniendã quatuor lineas
                  <lb/>
                cõtinuo ꝓpprtionabiles: oportet inter illas duas
                  <lb/>
                īuenire alias duas cõtinuo ꝓportionabiles inter
                  <lb/>
                ſe et cū extremis / vt ipſemet fatetur: ſed hoc nõ põt
                  <lb/>
                fieri per medii rei īuentionē igitur. </s>
                <s xml:id="N151A9" xml:space="preserve">Minor proba­
                  <lb/>
                tur / q2 vel prima illarū duarū linearū que īuenit̄̄
                  <lb/>
                inter illas duas īuenitur per illã artē vel nõ. </s>
                <s xml:id="N151B0" xml:space="preserve">ſi non
                  <lb/>
                habeo ꝓpropoſitū /  oportet dare aliã artē: ſi ſic tū
                  <lb/>
                manifeſtū eſt /  illa erit medio loco ꝓportionabi­
                  <lb/>
                lis inter lineas ſe habentes in ꝓportione dupla:
                  <lb/>
                et per cõſequens maioris linee ad ipſam / et etiam
                  <lb/>
                ipſius ad minimū erit proportio que eſt medietas
                  <lb/>
                duple: et tūc quero de īuentione ſecūde linee inter­
                  <lb/>
                medie: q2 vel ille īuenietur per artem medie rei in-
                  <lb/>
                uentionis vel nõ: ſi nõ habeo ꝓpoſitū: ſi ſic quero
                  <lb/>
                vel illa debet īueniri per illam artem inter illam
                  <lb/>
                mediam lineam et vltimam: vel inter primã et illã
                  <lb/>
                mediam: ſed neutrū iſtorum eſt diceudum igitur.
                  <lb/>
                </s>
                <s xml:id="N151CA" xml:space="preserve">Probatur minor: quoniã ſi inueniatur inter me-
                  <lb/>
                diam et vltimam: iam ille quatuor linee nõ erunt
                  <lb/>
                continuo proportionabiles: quoniã prime ad ſe-
                  <lb/>
                cundam erit medietas duple: et ſecunde ad tertiã
                  <lb/>
                et etiam tertie ad quartam erit ſubquadrupla du­
                  <cb chead="Capitulū octauū."/>
                ple: quia erit medietas medietatis duple: vt patet
                  <lb/>
                ex nona concluſione huius: ſi vero īueniatur inter
                  <lb/>
                primam et mediam idē ſequitur.
                  <note position="right" xlink:href="note-0052-04a" xlink:label="note-0052-04" xml:id="N15210" xml:space="preserve">Correĺ.</note>
                </s>
                <s xml:id="N151E1" xml:space="preserve">¶ Ex quo ſequi-
                  <lb/>
                tur horen non tradidiſſe doctrinam ad inuenien-
                  <lb/>
                dam proportionē compoſitam ex duabus tertiis
                  <lb/>
                proportiõis duple puta ſubſequialterã ad duplã
                  <lb/>
                </s>
                <s xml:id="N151EB" xml:space="preserve">Probatur / quia vt ſonant verba eius videtur in-
                  <lb/>
                nuere illas lineas īueniendas eſſe per artē medie
                  <lb/>
                rei īuentionis / quod ſtare nõ poteſt / vt probatū eſt
                  <lb/>
                </s>
                <s xml:id="N151F3" xml:space="preserve">Et ſi hec nõ fuit intentio et mens venerabilis ma-
                  <lb/>
                giſtri. </s>
                <s xml:id="N151F8" xml:space="preserve">Nicholai horen detur imbecillitati et par-
                  <lb/>
                uitati ingenioli mei venia. </s>
                <s xml:id="N151FD" xml:space="preserve">Eligat igitur vnuſq̇ſ-
                  <lb/>
                 / quod vult et me magis ſtudioſum quã maliuo-
                  <lb/>
                lum probet.</s>
              </p>
            </div>
            <div xml:id="N15216" level="3" n="8" type="chapter" type-free="capitulum">
              <head xml:id="N1521B" xml:space="preserve">Capitulum octauū / in quo agitur decre-
                <lb/>
              mento et decremento ꝓportionū.</head>
              <p xml:id="N15220">
                <s xml:id="N15221" xml:space="preserve">QUoniã inſequētibus plerū
                  <lb/>
                ſeſe offert diminutio proportionis ex
                  <lb/>
                augmento reſiſtentie: aut virtutis decre­
                  <lb/>
                mento / et etiam augmentatio proueniens ex decre­
                  <lb/>
                mento reſiſtētie aut virtutis augmento. </s>
                <s xml:id="N1522C" xml:space="preserve">Ideo ope­
                  <lb/>
                re precium eſt in huiꝰ ſecunde partis calce aliquid
                  <lb/>
                de augmento et decremento ꝓportionū adiicere.</s>
              </p>
              <p xml:id="N15233">
                <s xml:id="N15234" xml:space="preserve">Pro quo ſuppono primo. </s>
                <s xml:id="N15237" xml:space="preserve">Augere ſi-
                  <lb/>
                ue augmentare aliquã proportionē cõtingit mul-
                  <lb/>
                tipliciter: aut em̄ maiori numero aliquid additur
                  <lb/>
                minore īuariato: aut decreſcente: aut minori ali-
                  <lb/>
                quid demitur maiore nõ variato aut creſcēte. </s>
                <s xml:id="N15242" xml:space="preserve">aut
                  <lb/>
                vtro creſcente velocius tamen ꝓportiõabiliter
                  <lb/>
                creſcente maiore quã minore. </s>
                <s xml:id="N15249" xml:space="preserve">Aut vtro diminu-
                  <lb/>
                to velocius tamē ꝓportionabiliter diminuto mi-
                  <lb/>
                nore quã maiore. </s>
                <s xml:id="N15250" xml:space="preserve">Probat̄̄ / qm̄ capta proportione
                  <lb/>
                dupla que eſt .8. ad .4. cõtingit eã augeri ꝑ cremen­
                  <lb/>
                tū ipſoꝝ .8. ipſis .4. īuariatis vel decreſcētibus. </s>
                <s xml:id="N15257" xml:space="preserve">vt
                  <lb/>
                ſi .8: acquirãt vnitatē ipſis .4. īuariatis: manebit
                  <lb/>
                ꝓportio maior dupla: nouē ad .4. q̄ eſt dupla ſex-
                  <lb/>
                quiquarta: ſi quãdo .8. acquirūt vnitatē .4. deper­
                  <lb/>
                dūt vnitatē: etiã manebit proportio maior dupla
                  <lb/>
                puta tripla. </s>
                <s xml:id="N15264" xml:space="preserve">Itē ſi quieſcētibꝰ .8.4. deꝑdant bina­
                  <lb/>
                riū: augmentabit̄̄ ꝓportio / vt cõſtat: et ſi etiã tūc .8
                  <lb/>
                aliquid acquirãt: etiã augmētabitur ꝓportio. </s>
                <s xml:id="N1526B" xml:space="preserve">Si
                  <lb/>
                vero .8. acquirãt quaternariū numeꝝ puta ꝓpor-
                  <lb/>
                tionē ſexquialterã: et q̈ternariꝰ numerꝰ acq̇rat vni­
                  <lb/>
                tatē puta ꝓportionē ſexquiquartã: ꝓportio effi-
                  <lb/>
                cietur maior: </s>
                <s xml:id="N15276" xml:space="preserve">Efficiet̄̄ em̄ dupla ſuprabipartiens
                  <lb/>
                quītas. </s>
                <s xml:id="N1527B" xml:space="preserve">Si aūt .8: deꝑdant duo et .4. / ſiĺr duo aug-
                  <lb/>
                mētabit̄̄ etiã ꝓportio: q2 maiorē ꝓportionē deꝑ-
                  <lb/>
                dit numerꝰ mīor quã maior. </s>
                <s xml:id="N15282" xml:space="preserve">Et ſic ptꝫ ſuppoſitio.</s>
              </p>
              <p xml:id="N15285">
                <s xml:id="N15286" xml:space="preserve">Secūda ſuppoſitio. </s>
                <s xml:id="N15289" xml:space="preserve">Augmētare pro­
                  <lb/>
                portionē eſt addere ꝓportioni ꝓportionē ceteris
                  <lb/>
                paribꝰ: vt augere duplã eſt ei addere aliquã ꝓpor­
                  <lb/>
                tionē ceteris aliis manentibus paribus.</s>
              </p>
              <p xml:id="N15292">
                <s xml:id="N15293" xml:space="preserve">Ex quo ſequit̄̄ tertia ſuppoſitio ꝓpo-
                  <lb/>
                ſita vna ꝓportione quauis et duabꝰ aliis minori-
                  <lb/>
                bus: īueſtigare vtrū illa maior ex illis duabꝰ mi-
                  <lb/>
                noribꝰ adeq̈te ↄ̨ponit̄̄: vt ꝓpoſita ꝓportiõe dupla
                  <lb/>
                et ſexq̇altera, et ſeq̇tertia minoribꝰ, videre vtrum
                  <lb/>
                dupla ex ſexq̇altera et ſexq̇tertia adeq̈te cõponat̄̄.
                  <lb/>
                </s>
                <s xml:id="N152A1" xml:space="preserve">Probat̄̄ / ſit a. ꝓportio maior b: et c: mīores: et volo
                  <lb/>
                videre vtrū adeq̈te ↄ̨ponat̄̄ a. ex b. et c. </s>
                <s xml:id="N152A6" xml:space="preserve">Ad qḋ vidē-
                  <lb/>
                dū: addã c. ipſi b. / et ſi tūc ꝓportio ↄ̨poſita ex b. et c.
                  <lb/>
                adeq̈te eſt eq̈lis ipſi a. / ex illis adeq̈te cõponit̄̄ a.
                  <lb/>
                ſin minus: nõ ex his adequate componitur: ſed ex
                  <lb/>
                duabus maioribus, aut duabus minoribus.</s>
              </p>
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