Alvarus, Thomas, Liber de triplici motu, 1509

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                  <pb chead="Secūde partis" file="0024" n="24"/>
                tione ſequitur /  ſi duo numeri ſe habentes in ali­
                  <lb/>
                qua proportione acquirãt ↄ̨tinuo partes aliquo­
                  <lb/>
                tas eiuſdem denominationis: ſemper manebunt
                  <lb/>
                in eadem proportione. </s>
                <s xml:id="N12144" xml:space="preserve">Patet / q2 vter illorū eq̈­
                  <lb/>
                lem proportionem acquirit. </s>
                <s xml:id="N12149" xml:space="preserve">Patet / quia ſi vter
                  <lb/>
                illorum numerorum illas partes aliquotas eiuſ-
                  <lb/>
                dem denominationis deperderet eq̈lē ꝓportionē
                  <lb/>
                deꝑderet / vt patet ex ſuppoſitione: igitur quando
                  <lb/>
                acquirit equalem acquirit.</s>
              </p>
              <p xml:id="N12154">
                <s xml:id="N12155" xml:space="preserve">Duodecima ſuppoſitio. </s>
                <s xml:id="N12158" xml:space="preserve">Si aliquid
                  <lb/>
                componitur ex duobus ſiue equalibus ſiue īequa­
                  <lb/>
                libus: et quantum deperdit vnum illorum tantuꝫ
                  <lb/>
                acquirit reliquum: compoſitum ex illis nichil ac-
                  <lb/>
                quirit vel deperdit ſed ſemper manet equale. </s>
                <s xml:id="N12163" xml:space="preserve">Et
                  <lb/>
                hanc peto quia nota eſt ex ſe.</s>
              </p>
              <note position="left" xml:id="N12168" xml:space="preserve">cal. de in­
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              duc. gra-
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              ſum et de
                <lb/>
              mo. 10.</note>
              <p xml:id="N12172">
                <s xml:id="N12173" xml:space="preserve">Prima concluſio </s>
                <s xml:id="N12176" xml:space="preserve">Omne compoſitū
                  <lb/>
                ex duobus inequalibus inter que eſt mediuꝫ eſt du­
                  <lb/>
                plum ad medium inter illa vt compoſitum ex .4. et
                  <lb/>
                2. eſt duplum ad ternarium numerum qui mediat
                  <lb/>
                inter illos </s>
                <s xml:id="N12181" xml:space="preserve">Probatur / ſint a.c. duo īequalia .a ma­
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                ius et .c. minus et ſit .b. medium inter .a.c. compoſi­
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                tum ex a.c. ſit .d. / tunc dico /  .d. eſt duplum ad .b.
                  <lb/>
                </s>
                <s xml:id="N12189" xml:space="preserve">Quod ſic probo / quia cū .b. ſit medium: equali dif­
                  <lb/>
                ferentia diſtat ab extremis ex prima ſuppoſitiõe /
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                capio igitur illam differentiã ſiue exceſſum qua .a
                  <lb/>
                excedit b. / et addo illam .c. / et manifeſtum eſt /  .a. et
                  <lb/>
                b. manēt equalia: et ſimiliter .c. et .b. quia ipſi .c. ad
                  <lb/>
                dictus eſt exceſſꝰ / quo excedebatur a.b. / igitur ag-
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                gregatum ex .a. et .c. componitur ex duobus equa­
                  <lb/>
                lidus .b. adequate. </s>
                <s xml:id="N1219A" xml:space="preserve">igitur tale aggregatum eſt du­
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                plum ad .b. et tale aggregatum eſt .d. / igitur d. eſt
                  <lb/>
                duplum ad .b. et .d. eſt in tantum quantum erat añ
                  <lb/>
                variationem .a.c. / vt patet ex vltima ſuppoſitione /
                  <lb/>
                igitut .d. ante variationem a.c. eſt duplum ad .b. /
                  <lb/>
                quod fuit probandum.
                  <note position="left" xlink:href="note-0024-01a" xlink:label="note-0024-01" xml:id="N12298" xml:space="preserve">p̄mū cor-
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                  relarium</note>
                </s>
                <s xml:id="N121AC" xml:space="preserve">¶ Ex hac concluſione ſequi­
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                tur:  mediū inter duo inequalia eſt medietas ag­
                  <lb/>
                gregati ex eis. </s>
                <s xml:id="N121B3" xml:space="preserve">Patet / quia eſt ſubdupluꝫ / ergo me­
                  <lb/>
                dietas.
                  <note position="left" xlink:href="note-0024-02a" xlink:label="note-0024-02" xml:id="N122A0" xml:space="preserve">Secūduꝫ
                    <lb/>
                  correlari­
                    <lb/>
                  um.</note>
                </s>
                <s xml:id="N121BD" xml:space="preserve">¶ Sequitur ſecūdo /  medietas aggrega­
                  <lb/>
                ti ex duobus inequalibus inter que eſt mediuꝫ: eq̈­
                  <lb/>
                liter ab vtro illorum diſtat. </s>
                <s xml:id="N121C4" xml:space="preserve">Probatur / q2 medi­
                  <lb/>
                etas illorum eſt equalis medio inter illa / vt patet
                  <lb/>
                ex precedenti correlario: ergo ſequitur /  equali-
                  <lb/>
                ter diſtat ab vtro. </s>
                <s xml:id="N121CD" xml:space="preserve">cum mediuꝫ ſit /  equaliter di­
                  <lb/>
                ſtat ab extremis / vt patet ex prima ſuppoſitione.
                  <lb/>
                  <note position="left" xlink:href="note-0024-03a" xlink:label="note-0024-03" xml:id="N122AA" xml:space="preserve">Tercium
                    <lb/>
                  correlari­
                    <lb/>
                  um.</note>
                ¶ Sequitur tertio /  omnis numerus circū ſe poſi­
                  <lb/>
                torum numerorum et equaliter ab eo diſtantium
                  <lb/>
                eſt medietas. </s>
                <s xml:id="N121DD" xml:space="preserve">Quod ſi eoruꝫ fuerit medietas illos
                  <lb/>
                ab eo eque diſtare conueniet. </s>
                <s xml:id="N121E2" xml:space="preserve">Probatur / ſint .a.c.
                  <lb/>
                duo numeri inter quos mediat .b. ſit aggregatū
                  <lb/>
                ex .a.c.d. / tunc .b. eſt medietas ipſius .d. / vt patet ex
                  <lb/>
                ṗmo correlario et ſi .b. eſt medietas aggregati .a.c.
                  <lb/>
                equaliter diſtat ab .a. et .c. / vt patet ex ſecundo cor-
                  <lb/>
                relario / ergo .a.c. equaliter diſtant .a.b.
                  <note position="left" xlink:href="note-0024-04a" xlink:label="note-0024-04" xml:id="N122B4" xml:space="preserve">Quartū
                    <lb/>
                  correlari­
                    <lb/>
                  um.</note>
                </s>
                <s xml:id="N121F4" xml:space="preserve">¶ Sequi-
                  <lb/>
                tur quarto /  cõiuncte arithmetice medietatis me­
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                diis terminus extremorum ſimul iunctorum ē me­
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                dietas: vt captis his terminis .a.bc. continuo ꝓ-
                  <lb/>
                portionabilibꝰ arithmetice .b. medius terminus
                  <lb/>
                eſt medietas aggregati ex .a.c. </s>
                <s xml:id="N12201" xml:space="preserve">Patꝫ ex primo cor­
                  <lb/>
                relario
                  <note position="left" xlink:href="note-0024-05a" xlink:label="note-0024-05" xml:id="N122BE" xml:space="preserve">prima ꝓ­
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                  prietas
                    <lb/>
                  medieta-
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                  tis arith­
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                  metice.</note>
                </s>
                <s xml:id="N1220B" xml:space="preserve">Et hec ſit prima ꝓprietas arithmetice me­
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                dietatis </s>
                <s xml:id="N12210" xml:space="preserve">Et intelligas hanc proprietatem quan-
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                do tales termini continuo proportionaabiles hac ꝓ­
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                portionalitate fuerint impares: vel quantitates
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                continue. </s>
                <s xml:id="N12219" xml:space="preserve">Alias plerū non inuenires medium in­
                  <lb/>
                ter tales terminos. </s>
                <s xml:id="N1221E" xml:space="preserve">ſicut inter .2.3.4.5
                  <note position="left" xlink:href="note-0024-06a" xlink:label="note-0024-06" xml:id="N122CC" xml:space="preserve">Quintū
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                  correlari­
                    <lb/>
                  um.</note>
                </s>
                <s xml:id="N12226" xml:space="preserve">¶ Sequitur
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                quinto /  diſpoſitis .3. terminis continuo ꝓportio­
                  <cb chead="Capitulum ſecundum"/>
                nabilibꝰ arithmetice: aggregatū ex maxīo termīo
                  <lb/>
                et mīmo ē due tertie aggregati ex illis tribꝰ termi­
                  <lb/>
                nis: et diſpoſitis .5. continuo proportionalibus
                  <lb/>
                arithmetice aggregatum ex maximo et minimo ē
                  <lb/>
                due quinte:
                  <note position="right" xlink:href="note-0024-07a" xlink:label="note-0024-07" xml:id="N122D6" xml:space="preserve">Secūda
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                  ꝓprietas
                    <lb/>
                  medietaſ
                    <lb/>
                  arithme-
                    <lb/>
                  tice.</note>
                et etiam aggregatum ex ſecūdo termi­
                  <lb/>
                no et quarto eſt due quinte: et poſitis .7. aggrega­
                  <lb/>
                tum ex maximo et minimo eſt due ſeptime ſimili-
                  <lb/>
                ter aggregatum ex ſecundo et ſexto et ex tertio et
                  <lb/>
                quinto. </s>
                <s xml:id="N12243" xml:space="preserve">et vniuerſaliter vbicū plures termini in
                  <lb/>
                numero impari arithmetice continuo proportio­
                  <lb/>
                nantur ſemper aggregatum ex quibuſcū duo-
                  <lb/>
                bus equaliter diſtantibus a medio eſt due partes
                  <lb/>
                aliquote. </s>
                <s xml:id="N1224E" xml:space="preserve">aggregati ex omnibus illis quarū par­
                  <lb/>
                tium aliquotarum vtra denominatur a numero
                  <lb/>
                impari a quo denominantur illi termini. </s>
                <s xml:id="N12255" xml:space="preserve">vt ſi ter­
                  <lb/>
                mini ſint vndeci3 denominabuntur due vndecime
                  <lb/>
                et ſi .13. due tridecime. </s>
                <s xml:id="N1225C" xml:space="preserve">Probatur hoc correlarium /
                  <lb/>
                et ſigno tres terminos .a.b.c. / et arguo ſic / aggrega­
                  <lb/>
                tum ex .a.c. eſt duplum ad .b. quia .b. eſt terminꝰ me­
                  <lb/>
                dius inter .a.c. ſed aggregatum ex a.b.c. componi­
                  <lb/>
                tur adeq̈te ex .b. et aggregato ex .a.c. duplo ad .b. /
                  <lb/>
                vt patet ex concluſione: ergo b. eſt vna tertia totiꝰ
                  <lb/>
                aggregati cum ter in illo contineatur adequate et
                  <lb/>
                per conſequens aggregatum ex .a.c. due tertie / qḋ
                  <lb/>
                fuit probandum. </s>
                <s xml:id="N1226F" xml:space="preserve">Item poſitis quin trrminis .a
                  <lb/>
                b.c.d.e. aggregatum ex .a. et .e. eſt duplum ad ter-
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                minum medium .c. et ſimiliter aggregatum ex .b. et
                  <lb/>
                d. / vt patet ex concluſioīe et totum aggregatum ex
                  <lb/>
                illis quin terminis componitur adequate ex c. et
                  <lb/>
                ex aggregato .a. et .e. et aggregato ex .b. et .d. et vtrū­
                  <lb/>
                 illorum aggregatorum eſt duplum ad .c. / vt pro­
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                batum eſt: ergo .c. eſt vna quinta totius aggrega-
                  <lb/>
                ti ex illis quin terminis: cum quīquies in illo ag­
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                gregato contineatur: et per conſequens aggrega­
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                tum ex .a. et .e. eſt due quinte: et ſimiliter aggrega-
                  <lb/>
                tum ex .b.d. cum ſit duplum ad .c </s>
                <s xml:id="N12288" xml:space="preserve">Et iſto modo pro­
                  <lb/>
                babis capiendo quotcū alios terminos īpares
                  <lb/>
                continuo arithmetice ꝓportionabiles. </s>
                <s xml:id="N1228F" xml:space="preserve">Et iſta ſit
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                ſecunda proprietas medietatis arithmetice.</s>
              </p>
              <p xml:id="N122E4">
                <s xml:id="N122E5" xml:space="preserve">Secunda concluſio </s>
                <s xml:id="N122E8" xml:space="preserve">Si duo nume-
                  <lb/>
                ri a duobus numeris circum ſe poſitis equaliṫ di­
                  <lb/>
                ſtent: illis coniunctis erunt equales. </s>
                <s xml:id="N122EF" xml:space="preserve">Quod ſi eis
                  <lb/>
                equales fuerint: ab eis equidiſtare neceſſe eſt vt ca­
                  <lb/>
                ptis his terminis .2.3.4.5. numerus quinarus et
                  <lb/>
                binarius circunſtantes quaternarium et ternariū
                  <lb/>
                equaliter ſimul iuncti equantur quaternario et ter­
                  <lb/>
                nario ſimul iunctis et quia quinarius et binariꝰ
                  <lb/>
                ſimul iuncti equales ſunt quaternario et binario
                  <lb/>
                ſimul iuncti: ideo neceſſario ab illis equaliter di-
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                ſtant. </s>
                <s xml:id="N12302" xml:space="preserve">Probatur concluſio / et ſint .a.b.c.d.a.d. cir-
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                cunſtantes reliqui vero intermedii: et diſtat .a. ab
                  <lb/>
                b.g. dnr̄a ita  .a. ſit maior numerus et eadem .g
                  <lb/>
                dnr̄ia excedat .c. ipſum .d. / tunc dico /  aggregatū
                  <lb/>
                ex .a.d. extremis numeris eſt equale aggregato ex
                  <lb/>
                b.c. intermediis a quibus alii equaliter diſtant.</s>
              </p>
              <p xml:id="N1230F">
                <s xml:id="N12310" xml:space="preserve">Quod probatur ſic / et volo /  .a. perdat .g. dnr̄iaꝫ /
                  <lb/>
                ita  fiat equale b. et .d. acquirat illam ita  fiat
                  <lb/>
                equale .c. / et arguo ſic / facta tali variatione in a.d.
                  <lb/>
                aggregatū ex .a.d. ↄ̨ponit̄̄ adeq̈te ex duobꝰ eq̈libꝰ
                  <lb/>
                aliis duobus ex quibus adequate cõponitur ag-
                  <lb/>
                gretatum ex .b.c. / igitur facta tali variatiõe in .a.
                  <lb/>
                d. aggregatum ex .a.d. eſt equale aggregato ex .b
                  <lb/>
                c. et illud aggregatum ex .a.d. facta tali variatio­
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                ne eſt equale aggregato .a.d. ante talem variatio­
                  <lb/>
                nem / vt patet ex vltima ſuppoſitione: igitur aggre­
                  <lb/>
                gatum ex .a.c. ante talem variationem eſt equale </s>
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