Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N11397" level="3" n="6" type="chapter" type-free="capitulum">
              <p xml:id="N114FA">
                <s xml:id="N11545" xml:space="preserve">
                  <pb chead="Prime partis" file="0017" n="17"/>
                in proportione que eſt medietas duple vt conſtat:
                  <lb/>
                quia illa eſt proportio diuiſionis: et prima pars
                  <lb/>
                proportionalis impar eſt medietas totius aggre­
                  <lb/>
                gati ex omnibus imparibus: et prima par que eſt
                  <lb/>
                ſecunda eſt medietas aggregati ex omnibus pa-
                  <lb/>
                ribus: vt patet ex duabus primis partibus corre-
                  <lb/>
                larii: ergo medietas omnium imparium ſe habet
                  <lb/>
                ad medietatem omnium parium in proportione
                  <lb/>
                que eſt medietas duple: quod fuit probandum.</s>
              </p>
              <note position="left" xml:id="N1155C" xml:space="preserve">Quartū
                <lb/>
              correlar̄.</note>
              <p xml:id="N11562">
                <s xml:id="N11563" xml:space="preserve">¶ Sequitur quarto /  diuiſo corpore per partes
                  <lb/>
                proportionales proportione irrationali que eſt
                  <lb/>
                medietas triple: omnes partes impares talis di-
                  <lb/>
                uiſionis ſe habent in proportione tripla: et etiam
                  <lb/>
                omēs pares: et omnes inter quas mediant tres in
                  <lb/>
                proportione nouocupla: et aggregatum ex omni-
                  <lb/>
                bus imparibus ſe habet ad aggregatum ex omni­
                  <lb/>
                bus paribus in proportione que eſt medietas tri-
                  <lb/>
                ple. </s>
                <s xml:id="N11576" xml:space="preserve">Hoc correlarium cum precedenti ſimilem de-
                  <lb/>
                monſtrationem admittit.</s>
              </p>
              <p xml:id="N1157B">
                <s xml:id="N1157C" xml:space="preserve">Tertia concluſio: </s>
                <s xml:id="N1157F" xml:space="preserve">Ad diuidendū cor­
                  <lb/>
                pus in partes proportionales infinitis ſpeciebus
                  <lb/>
                proportionis irrationalis maioris dupla: vt pu-
                  <lb/>
                ta proportione que eſt totius diametri ad exceſſū
                  <lb/>
                quo ipſa diameter excedit coſtam et totius diame­
                  <lb/>
                tri cum medietate exceſſus quo excedit coſtam vel
                  <lb/>
                ad quarta in vel ad quintã vel ad ſextã vt ſuperiꝰ
                  <lb/>
                dictum eſt: pro prima parte proportionali capi-
                  <lb/>
                endus eſt exceſſus quo quãtitas maior excedit mi­
                  <lb/>
                norem in tali proportione: et quãtitas miuor pro
                  <lb/>
                reſiduo vt ſi velis partiri corpꝰ in partes propor­
                  <lb/>
                tionales proportione que eſt totius diametri ad
                  <lb/>
                exceſſum quo diameter excedit coſtam: capienda
                  <lb/>
                eſt coſta quadrati cuius illud corpus diuidendum
                  <lb/>
                eſt diameter pro prima parte proportionali: et ſic
                  <lb/>
                pro reſiduis maneat exceſſus que eſt quãtitas mi-
                  <lb/>
                nor talis proportionis: et pro ſecunda capien-
                  <lb/>
                da eſt coſta quadrati cuius totum aggregatum ex
                  <lb/>
                omnibus ſequentibus primam eſt diameter: et ad­
                  <lb/>
                dandam tertiam capiatur coſta quadrati cuius
                  <lb/>
                eſt diameter aggregatum ex omnibus ſequenti-
                  <lb/>
                bus primam et ſecundam. </s>
                <s xml:id="N115AC" xml:space="preserve">Et ad diuidendum ali-
                  <lb/>
                quod corpus proportione que eſt totius diametri
                  <lb/>
                ad medietatē exceſſus quo excedit coſtaꝫ, pro pri-
                  <lb/>
                ma parte ꝓportionali capiendus eſt exceſſus quo
                  <lb/>
                maior quantitas excedit minorem tali proporti-
                  <lb/>
                one. </s>
                <s xml:id="N115B9" xml:space="preserve">Conſtituendum .n. eſt totum corpus diameter
                  <lb/>
                alicuius quadrati / et tunc pro prima parte propor­
                  <lb/>
                tionali capienda eſt tanta pars illius corporis 
                  <lb/>
                pro omnibus ſequentibus non maneat niſi medie­
                  <lb/>
                tas exceſſus quo tale corpus exiſtens diameter ex­
                  <lb/>
                cedit coſtam eiuſdem quadrati: et addandam ſe-
                  <lb/>
                cundam partem proportionalem conſtituatur to­
                  <lb/>
                tum / quod ſequitur primã diameter alicuius qua-
                  <lb/>
                drati: et pro ſecūda parte capiatur tantum /  pro
                  <lb/>
                ſequentibus non maneat niſi medietas exceſſus
                  <lb/>
                quo talis diameter excedit ſuam coſtam / et ſic con­
                  <lb/>
                ſequenter. </s>
                <s xml:id="N115D2" xml:space="preserve">Patet hec concluſio eo modo quo ſe-
                  <lb/>
                cūda huius capitis. </s>
                <s xml:id="N115D7" xml:space="preserve">Hic poteris multa correlaria
                  <lb/>
                inferre ſed iam ad ea inferenda ex predictis faci-
                  <lb/>
                lem haberes aditum. </s>
                <s xml:id="N115DE" xml:space="preserve">Et hec de proportione irra-
                  <lb/>
                tionali: et de diuiſione corporum eadem irratio-
                  <lb/>
                nali proportione: de qua non eſt facile cum rotio-
                  <lb/>
                ne loqui.</s>
              </p>
            </div>
            <div xml:id="N115E7" level="3" n="7" type="chapter" type-free="capitulum">
              <head xml:id="N115EC" xml:space="preserve">Capitulum ſeptimum / in quo agi­
                <lb/>
              tur de proportione ordinum par-
                <cb chead="Capitulū ſeptimū."/>
              tium proportionalium interſcala-
                <lb/>
              riter ſe habentium.</head>
              <p xml:id="N115F6">
                <s xml:id="N115F7" xml:space="preserve">OCcurrit nonnūquam in mate-
                  <lb/>
                teria de motu locali quo ad effectū et mo-
                  <lb/>
                tu augmentationis comparatio alicuius
                  <lb/>
                ordinis aliquarum partium proportionalium in­
                  <lb/>
                terſcalariter ſe habentiū ad alium ordinem par-
                  <lb/>
                tium proportionalium: vt cum volumus compara­
                  <lb/>
                re totum ordinem partium imparium toti ordini
                  <lb/>
                partium parium: vt iam ex parte tangebatur in
                  <lb/>
                precedēti capite: ideo non abs re pro noticia huiꝰ
                  <lb/>
                pono aliquas concluſiones.</s>
              </p>
              <p xml:id="N1160C">
                <s xml:id="N1160D" xml:space="preserve">Prima cõcluſio. </s>
                <s xml:id="N11610" xml:space="preserve">Diuiſo corpore per
                  <lb/>
                partes proportionales quauis proportione: et ca­
                  <lb/>
                ptis certis ordinibus partium proportionalium
                  <lb/>
                interſcalariter ſe habentium: totum corpus ab-
                  <lb/>
                ſoluentibus: tunc illi ordines ſe habent continuo
                  <lb/>
                in proportione diuiſionis: vt ſi corpus diuidatur
                  <lb/>
                proportione dupla: et capiantur oēs partes inter
                  <lb/>
                quas mediant due pro primo ordine puta prima
                  <lb/>
                quarta, ſeptima, decima, tridecima .etc̈ / et deinde
                  <lb/>
                pro ſecundo ordine ſecunda, quinta, octaua, vn-
                  <lb/>
                decima, decima quarta, et ſic cõſequenter. </s>
                <s xml:id="N11627" xml:space="preserve">et demū
                  <lb/>
                pro tertio ordine capiantur tertia, ſexta, nona,
                  <lb/>
                duodecima, quindecima, et ſic deinceps. </s>
                <s xml:id="N1162E" xml:space="preserve">Dico / 
                  <lb/>
                primus ordo ſe habet ad ſecundū in ꝓportiõe du-
                  <lb/>
                pla: et etiam ſecundus ad tertium in proportione
                  <lb/>
                dupla. </s>
                <s xml:id="N11637" xml:space="preserve">Et eſto /  centum ordines caperes illi etiaꝫ
                  <lb/>
                in proportione dupla continuo ſe haberent. </s>
                <s xml:id="N1163C" xml:space="preserve">Pa-
                  <lb/>
                tet hoc / quoniam cuiuſlibet illorum ordinum con-
                  <lb/>
                tinuo partes correſpõdentes ſe habent in eadem
                  <lb/>
                proportione: igitur in quacū proportione ſe ha­
                  <lb/>
                bent continuo prime partes illorum ordinum in
                  <lb/>
                eadem proportione continuo ſe habent ille ordi-
                  <lb/>
                nes: ſed prime partes ſe habent in proportione di­
                  <lb/>
                uiſionis / vt conſtat: igitur et illi ordines. </s>
                <s xml:id="N1164D" xml:space="preserve">Proba-
                  <lb/>
                tur tamen cõſequētia per hanc regulam. </s>
                <s xml:id="N11652" xml:space="preserve">Quado-
                  <lb/>
                cū aliqua diuiduntur equali ꝓportione in qua-
                  <lb/>
                cū proportione ſe habent prime partes propor­
                  <lb/>
                tionales in eadem proportione ſe habent et ipſa
                  <lb/>
                tota: quoniam ſunt partes aliquote eiuſdē deno-
                  <lb/>
                minationis. </s>
                <s xml:id="N1165F" xml:space="preserve">Modo in quacū proportione ſe ha­
                  <lb/>
                bent partes aliquote eiuſdem denominationis in
                  <lb/>
                eadem ſe habent et ipſa tota quorum ſunt partes
                  <lb/>
                aliquote / vt poſtea demonſtrabitur igitur.</s>
              </p>
              <p xml:id="N11668">
                <s xml:id="N11669" xml:space="preserve">Secunda concluſio per modum do-
                  <lb/>
                cumenti poſita. </s>
                <s xml:id="N1166E" xml:space="preserve">Ad ſciendū quota pars vel quote
                  <lb/>
                partes aliquote eſt quilibet illorum ordinum vi-
                  <lb/>
                dendum eſt quot ſint ordines: et tunc cõſtituantur
                  <lb/>
                in numeris tot proportiões diuiſionis quot ſunt
                  <lb/>
                illi ordinis dempta vna: et coadunētur omnes ter­
                  <lb/>
                mini illarum proportionum: et diuidatur totū in
                  <lb/>
                tot partes aliquotas quotꝰ eſt numerus reſultãs
                  <lb/>
                et dentur primo ordini tot ex illis partibas qnotꝰ
                  <lb/>
                eſt maximus numerus in illis proportionibus: et
                  <lb/>
                ſecundo ordini tot quotus eſt ſecundus numerus:
                  <lb/>
                et ſic conſequenter. </s>
                <s xml:id="N11685" xml:space="preserve">Et ſic videbis quot partes ali-
                  <lb/>
                quotas et cuiꝰ denominationis continet primꝰ or­
                  <lb/>
                do: et ſecundus, et tertius, et ſic conſequenter. </s>
                <s xml:id="N1168C" xml:space="preserve">Exē-
                  <lb/>
                plum / vt ſi pedale fuerit diuiſum in partes propor­
                  <lb/>
                tionales proportione dupla conſtituantur tres
                  <lb/>
                ordines / vt paulo ãte exēplo expreſſimꝰ / q2 ibi tres
                  <lb/>
                ſunt ordines conſtituti: et proportio diuiſionis eſt
                  <lb/>
                dupla: conſtituas in numeris duas proportiones </s>
              </p>
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