Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N10E2D" level="3" n="5" type="chapter" type-free="capitulum">
              <pb chead="Prime partis" file="0013" n="13"/>
              <p xml:id="N10F6D">
                <s xml:id="N10F6E" xml:space="preserve">His poſitis ſit prima cõcluſio. </s>
                <s xml:id="N10F71" xml:space="preserve">Quã­
                  <lb/>
                docun aliquod corpus diuiditur quouis genere
                  <lb/>
                proportionis: totū corpus ſe debet habere ad ag­
                  <lb/>
                gregatum ex omnibus partibus proportionalibꝰ
                  <lb/>
                ſequentibus primam: in ea proportione qua cor­
                  <lb/>
                pus diuiditur. </s>
                <s xml:id="N10F7E" xml:space="preserve">Exemplum / vt ſi corpus diuidatur
                  <lb/>
                proportione ſexquialtera: oportet /  illud corpus
                  <lb/>
                ſe habeat ad aggregatum ex omnibus partibus
                  <lb/>
                proportionabilibꝰ. </s>
                <s xml:id="N10F87" xml:space="preserve">ſequentibus primam: in pro­
                  <lb/>
                portione ſexquialtera. </s>
                <s xml:id="N10F8C" xml:space="preserve">Probatur hec concluſio: et
                  <lb/>
                volo /  b. corpꝰ diuidatur in partes proportiona­
                  <lb/>
                les proportione a. in infinitum: et arguo ſic / b. cor-
                  <lb/>
                pus diuiditur in partes proportionales propor­
                  <lb/>
                tione .a. in infinitum: igitur deperdendo primam
                  <lb/>
                partem proportionalem proportione a. ipſum ef­
                  <lb/>
                ficitur in a. proportione minus: patet conſequētia
                  <lb/>
                ex ſecunda parte quarte ſuppoſitionis: et vltra il­
                  <lb/>
                lud corpus b. deperdendo primã partem propor-
                  <lb/>
                tionalem a. efficitur ſiue manet in a. proportione
                  <lb/>
                minus et non manet niſi aggregatum ex omībus
                  <lb/>
                ſequentibus primam partem proportionalē: igi­
                  <lb/>
                tur illud corpus b. ſe habet ad aggregatum ex om­
                  <lb/>
                nibus partibus proportionabilibus ſequentibus
                  <lb/>
                primam eius partem proportionalem proportio­
                  <lb/>
                ne a. in eadem proportione a. / quod fuit ꝓbanduꝫ.
                  <lb/>
                </s>
                <s xml:id="N10FAE" xml:space="preserve">Patet hec conſequentia: quia ſi illud aggregatū
                  <lb/>
                ex omnibus ſequentibus primã. etc̈. eſt minus ipſo
                  <lb/>
                b. corpore in a proportione: ſequitur /  ipſum b.
                  <lb/>
                corpus eſt maius illo aggregato ex omnibus ſe-
                  <lb/>
                quentibus primam in a. proportione.</s>
              </p>
              <p xml:id="N10FB9">
                <s xml:id="N10FBA" xml:space="preserve">Secunda cõcluſio. </s>
                <s xml:id="N10FBD" xml:space="preserve">Ad inueniendū
                  <lb/>
                reſiduū a prima parte ꝓportionali quauis ꝓpor­
                  <lb/>
                tione rationali corpus diuidatur: capiãtur primi
                  <lb/>
                numeri talis ꝓportionis: et diuidat̄̄ corpus in tot
                  <lb/>
                vnitates quotus eſt numerꝰ maior illius propor­
                  <lb/>
                tionis: et ex illis partibꝰ ꝓ reſiduo a prima parte
                  <lb/>
                capiantur tot: quotus eſt numerus minor talis ꝓ-
                  <lb/>
                portionis. </s>
                <s xml:id="N10FCE" xml:space="preserve">Exēplum / vt ſi vis diuidere corpꝰ ꝓpor-
                  <lb/>
                tione ſexquitertia: et videre quid reſtabit pro reſi-
                  <lb/>
                duo a prima parte proportionali: capias .4. et .3.
                  <lb/>
                primos numeros ꝓportionis ſexquitertie: et diui­
                  <lb/>
                das totū corpus in quatuor partes equales: quia
                  <lb/>
                numerus maior eſt quaternarius: et pro reſiduo a
                  <lb/>
                prima ꝑte ꝓportionali capias tres partes ex illis
                  <lb/>
                q2 numerus minor eſt ternarius. </s>
                <s xml:id="N10FDF" xml:space="preserve">Probat̄̄ hec con­
                  <lb/>
                cluſio et volo /  b. corpus diuidatur proportione
                  <lb/>
                a. cuius proportionis primi numeri ſint c. maior
                  <lb/>
                numerus et d. minor / et arguo ſic. </s>
                <s xml:id="N10FE8" xml:space="preserve">Iſtud corpus eſt
                  <lb/>
                diuiſum per partes ꝓportionales proportione a /
                  <lb/>
                ergo totū iſtud b. corpus ſe habet ad aggregatuꝫ
                  <lb/>
                ex oībus partibus ꝓportionabilibus ꝓportione
                  <lb/>
                a. ſequētibus primã in proportione a. </s>
                <s xml:id="N10FF3" xml:space="preserve">Patet ↄ̨ña
                  <lb/>
                ex priori concluſione: et vltra totum b. ſe habet ad
                  <lb/>
                aggregatum .etc̈. in ꝓportione a. / ergo ſequitur / 
                  <lb/>
                ipſuꝫ b. ſe habet ad illud aggregatū ſicut c. nume­
                  <lb/>
                reus ad d. numerū / vt cõſtat et d. numerꝰ eſt nume­
                  <lb/>
                rus minor: ergo ſequitur /  aggregatū ex omībꝰ
                  <lb/>
                partibus ꝓportionalibꝰ proportione a. ſequē-
                  <lb/>
                tibus primã ſe habet vt numerus mīor primorum
                  <lb/>
                numerorū proportionis a. reſpectu maioris nu-
                  <lb/>
                meri: et nõ poteſt ſic ſe habere: niſi fiat diuiſio ta-
                  <lb/>
                lis corporis modo dicto in concluſione vel equiua­
                  <lb/>
                lenti / vt conſtat: igitur ſequitur concluſio.</s>
              </p>
              <p xml:id="N1100C">
                <s xml:id="N1100D" xml:space="preserve">Tertia cõcluſio. </s>
                <s xml:id="N11010" xml:space="preserve">Ad diuidendū cor-
                  <lb/>
                pus per partes proportionales qua vis ꝓportõe
                  <cb chead="Capitulum quintū"/>
                multiplici capiēda eſt pro reſiduo a prima parte
                  <lb/>
                proportionali vna pars aliquota denoīata a nu­
                  <lb/>
                mero talē proportionē multiplicem denominante
                  <lb/>
                vt in diuiſione dupla proportione capiēda eſt vna
                  <lb/>
                medietas pro reſiduo a prima parte ꝓportionali
                  <lb/>
                et proportione tripla vna tertia et quadrupla vna
                  <lb/>
                quarta quintupla vero vna quinta et ſic ī infinitū
                  <lb/>
                </s>
                <s xml:id="N11025" xml:space="preserve">Probatur hec cõcluſio: qm̄ ſemper corpus diuiſū
                  <lb/>
                per partes proportionales aliqua proportione ſe
                  <lb/>
                debet habere ad reſiduū a prima parte ꝓportio-
                  <lb/>
                nali in eadeꝫ ꝓportione qua diuiditur: vt patet ex
                  <lb/>
                prima concluſione: ſed quodlibet corpus ſe hab3
                  <lb/>
                ad ſuã medietatē in proportiõe dupla et quodlib3
                  <lb/>
                ad ſuã tertiã in tripla: ad quartã in quadrupla: et
                  <lb/>
                ſic conſequēter: ergo in qualibet diuiſione corpo-
                  <lb/>
                ris ꝓportione dupla debet capi ꝓ reſiduo a pri-
                  <lb/>
                ma parte proportionali medietas. </s>
                <s xml:id="N1103A" xml:space="preserve">et proportione
                  <lb/>
                tripla vna tertia: et q̈drupla vna quarta et quintu­
                  <lb/>
                pla vna quīta. </s>
                <s xml:id="N11041" xml:space="preserve">et ſic in infinituꝫ: quod fuit ꝓbandū
                  <lb/>
                  <note position="right" xlink:href="note-0013-01a" xlink:label="note-0013-01" xml:id="N110DC" xml:space="preserve">Correla­
                    <lb/>
                  riū ṗmū.</note>
                </s>
                <s xml:id="N1104B" xml:space="preserve">¶ Ex hac cõcluſione ſequitur primo:  diuidendo
                  <lb/>
                corpus proportiõe dupla prima pars erit medie­
                  <lb/>
                tas, et ſecūda medietas reſidui: et tertia medietas
                  <lb/>
                reſidui, et ſic cõſequenter. </s>
                <s xml:id="N11054" xml:space="preserve">ꝓportione tripla prima
                  <lb/>
                pars eſt due tertie totius: et ſecūda due tertie reſi-
                  <lb/>
                dui, et tertia due tertie reſidui a prima et ſecunda:
                  <lb/>
                et ſic ſine termino. </s>
                <s xml:id="N1105D" xml:space="preserve">ꝓportione vero quadrupla pri­
                  <lb/>
                ma pars eſt tres quarte, et ſecunda tres quarte re­
                  <lb/>
                ſidui. </s>
                <s xml:id="N11064" xml:space="preserve">ꝓportiõe vero quītupla prima pars eſt qua­
                  <lb/>
                tuor quinte. </s>
                <s xml:id="N11069" xml:space="preserve">et ſextupla quin ſexte et ſeptupla ſex
                  <lb/>
                ſeptime: et ſic ſine termino. </s>
                <s xml:id="N1106E" xml:space="preserve">Probatur hoc correla­
                  <lb/>
                riū: quia diuidendo proportione dupla: totum re­
                  <lb/>
                ſiduū a prima parte ꝓportõali eſt vna medietas /
                  <lb/>
                vt patet ex cõcluſione: igitur prima pars erit vna
                  <lb/>
                medietas </s>
                <s xml:id="N11079" xml:space="preserve">Patet cõſequētia ex ſecūda ſuppoſitio­
                  <lb/>
                ne / qm̄ omnes partes proportionales totū corpꝰ
                  <lb/>
                abſoluūt. </s>
                <s xml:id="N11080" xml:space="preserve">Item diuidendo ꝓportione tripla reſi­
                  <lb/>
                duū a prima parte ꝓportionali eſt vna tertia / igit̄̄
                  <lb/>
                prima erit due tertie. </s>
                <s xml:id="N11087" xml:space="preserve">Itē diuidēdo quadrupla re­
                  <lb/>
                ſiduū a ṗma eſt vna quarta / igit̄̄ prima eſt 3 quar-
                  <lb/>
                te. </s>
                <s xml:id="N1108E" xml:space="preserve">Quītupla vero eſt vna quīta / igitur prima erit
                  <lb/>
                quatuor quinte. </s>
                <s xml:id="N11093" xml:space="preserve">Et ſimiliter arguēdū eſt de ꝓpor­
                  <lb/>
                tione ſextupla ſeptupla / et ſic cõſequenter. </s>
                <s xml:id="N11098" xml:space="preserve">igit̄̄ cor-
                  <lb/>
                relarium verū. </s>
                <s xml:id="N1109D" xml:space="preserve">Antecedentia harū cõſequētiarum
                  <lb/>
                patēt ex ꝓxima concluſione et ipſe conſequentie ex
                  <lb/>
                ſecunda ſuppoſitione.
                  <note position="right" xlink:href="note-0013-02a" xlink:label="note-0013-02" xml:id="N110E4" xml:space="preserve">Corelari­
                    <lb/>
                  riū ſcḋm</note>
                </s>
                <s xml:id="N110A9" xml:space="preserve">¶ Sequitur ſecūdo /  diui­
                  <lb/>
                dēdo corpus per partes proportionales ꝓportõe
                  <lb/>
                dupla: reſiduum a prima eſt equale prime parti: et
                  <lb/>
                ꝓportione tripla eſt ſubduplū ad ṗmã: et quadru­
                  <lb/>
                pla ſubtriplū: et quītupla ſubquadruplū: et ſextu-
                  <lb/>
                pla ſubquintuplū: et ſic ſine termīo. </s>
                <s xml:id="N110B6" xml:space="preserve">Patet hec cor­
                  <lb/>
                relariū facile ex priori et concluſione. </s>
                <s xml:id="N110BB" xml:space="preserve">Si em̄ diui-
                  <lb/>
                dendo ꝓportione tripla prima pars eſt due tertie
                  <lb/>
                et reſiduū vna tertia cū vna tertia ſit ſubduplū ad
                  <lb/>
                duas tertias reſiduū a prima diuidēdo ꝓportiõe
                  <lb/>
                tripla erit ſubduplū ad primã. </s>
                <s xml:id="N110C6" xml:space="preserve">Item cū diuidēdo
                  <lb/>
                corpus ꝓportione quadrupla prima pars ſit tres
                  <lb/>
                quarte et reſiduuꝫ a prima vna quarta vna: autem
                  <lb/>
                quarta eſt ſubtripla ad tres quartas: igitur reſi-
                  <lb/>
                duū a prima parte diuidendo proportõe quadru­
                  <lb/>
                pla eſt ſubtriplum ad primã partem. </s>
                <s xml:id="N110D3" xml:space="preserve">Et hoc mo-
                  <lb/>
                do de aliis probabis.</s>
              </p>
              <p xml:id="N110EC">
                <s xml:id="N110ED" xml:space="preserve">Quarta cõcluſio. </s>
                <s xml:id="N110F0" xml:space="preserve">Ad diuidendū cor­
                  <lb/>
                pus qua vis ꝓportione ſuperparticulari: capiēda
                  <lb/>
                eſt ꝓ ṗma parte ꝓportionali vna pars aliquota
                  <lb/>
                denoīata a maiori numero ṗmorū numeroꝝ talis
                  <lb/>
                ꝓportionis. </s>
                <s xml:id="N110FB" xml:space="preserve">puta diuidendo ꝓportione ſexquial- </s>
              </p>
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