Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N11E85" level="3" n="2" type="chapter" type-free="capitulum">
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                <s xml:id="N12310" xml:space="preserve">
                  <pb chead="Secūde partis" file="0025" n="25"/>
                aggregato ex .b.c. / quod fuit probandum </s>
                <s xml:id="N1232C" xml:space="preserve">Sed iam
                  <lb/>
                probo /  facta tali variatione aggregatum ex .a.
                  <lb/>
                d. componitur ex duobus equalibus adequate il-
                  <lb/>
                lis duobus ex quibus adequate componitur ag-
                  <lb/>
                gregatum ex .b.c. / quia facta tali variatione .a. ef-
                  <lb/>
                ficit̄̄ eq̈le ipſi b. et d. efficit̄̄ eq̈le ipſi .c. / vt ↄ̨ſtat: igit̄̄
                  <lb/>
                facta tali variatiõe aggregatū ex a.d. ↄ̨ponit̄̄ ade­
                  <lb/>
                te ex duobus aqualibus illis duobus puta .b.c. ex
                  <lb/>
                quibus componitur adequate aggregatum ex .b.
                  <lb/>
                c. / quod fuit oſtendēdum. </s>
                <s xml:id="N12341" xml:space="preserve">Et ſic patet prima pars
                  <lb/>
                </s>
                <s xml:id="N12345" xml:space="preserve">Secūda pars probatur: et ſint a.b.c.d. quattuor
                  <lb/>
                numeri a.d. circūſtantes .b. vero et .c. intermedii et
                  <lb/>
                diſtet .a. ab .b.g. differētia et .c. excedat .d. / tunc dico /
                  <lb/>
                 ſi aggregatū ex .b.c. eſt equale aggregato ex .a.
                  <lb/>
                d.b.c. equaliter diſtant ab .a.d. </s>
                <s xml:id="N12350" xml:space="preserve">Quod ſic proba-
                  <lb/>
                tur / quia .a diſtat a.b.g. differentia: et .c.a.d. diſtat
                  <lb/>
                eadē differētia. </s>
                <s xml:id="N12357" xml:space="preserve">igitur illi intermedii equaliter di­
                  <lb/>
                ſtãt ab illis extremis. </s>
                <s xml:id="N1235C" xml:space="preserve">Probatur minor / quia ſi .c.
                  <lb/>
                non eadem differentia diſtat a.d. ſicut a. ab .b. ca-
                  <lb/>
                pio / igitur vnum terminū qui ſit .f. a quo .c. diſtet
                  <lb/>
                eadē differentia qua .a. diſtat ab .b. / et tunc ex prio­
                  <lb/>
                ri parte aggregatuꝫ ex a. et .f. eſt equale aggrega­
                  <lb/>
                to ex .b.c. et per te aggregatum ex .a.d. eſt ēt equa-
                  <lb/>
                le aggregato ex .b.c: igitur aggregatum ex .a.f. eſt
                  <lb/>
                equale aggregato ex .a.d. / patet conſequentia ꝑ il­
                  <lb/>
                laꝫ dignitatē que eidē tertio equantur inter ſe ſūt
                  <lb/>
                equalia. </s>
                <s xml:id="N12371" xml:space="preserve">et vltra aggregatum ex .a.f. eſt equale ag­
                  <lb/>
                gregato ex .a.d. / ergo ſequitur /  eodeꝫ cõmuni dē­
                  <lb/>
                pto puta a. reſidua manebunt equalia videlicet .f.
                  <lb/>
                et .d. et .c. diſtat .g. differētia qua a. diſtat ab .b. ab
                  <lb/>
                ipſo .f. / ergo .c. diſtat .g. differentia ab ipſo .d. / et ſic
                  <lb/>
                b.c. equaliter diſtant ab .a.d. numeris circunſtan-
                  <lb/>
                tibus / quod fuit probandum. </s>
                <s xml:id="N12380" xml:space="preserve">Patet tamen conſe­
                  <lb/>
                quentia / quia que ſunt equalia qualiter diſtant a
                  <lb/>
                quouis tertio
                  <note position="left" xlink:href="note-0025-01a" xlink:label="note-0025-01" xml:id="N1240D" xml:space="preserve">īueſtigat̄̄
                    <lb/>
                  itas ſe­
                    <lb/>
                  cūde con­
                    <lb/>
                  cluſionis
                    <lb/>
                  Iordanꝰ
                    <lb/>
                  .1. ele.</note>
                </s>
                <s xml:id="N1238C" xml:space="preserve">
                  <gap/>
                Hec cõcluſio in propria forma in­
                  <lb/>
                ſtantiam patitur: ſed ſic poſita eſt / quia ita poni­
                  <lb/>
                tur a iordano primo elementorum. </s>
                <s xml:id="N12394" xml:space="preserve">Nam iſti nu-
                  <lb/>
                meri .8.8. equaliter diſtãt ab his duobus .4.4. in
                  <lb/>
                iſta ſerie .4.8.8.4. / et tamen extrema coniūcta nõ
                  <lb/>
                equantur mediis. </s>
                <s xml:id="N1239D" xml:space="preserve">Item iſti duo numeri .4.1. equa­
                  <lb/>
                liter diſtant ab his duobus extremis .8.5. in iſta
                  <lb/>
                ſeries .8.4.1.5. / et tamen medii iuncti non equãtur
                  <lb/>
                extremis coniunctis / vt conſtat. </s>
                <s xml:id="N123A6" xml:space="preserve">Item illi numeri .
                  <lb/>
                4. et .4. coniuncti equantur his numeris ſimul iun­
                  <lb/>
                ctis .4. et .4. / et tamen duo intermedii non equali­
                  <lb/>
                ter diſtant a duobus extremis: quia non diſtant.
                  <lb/>
                  <note position="left" xlink:href="note-0025-02a" xlink:label="note-0025-02" xml:id="N1241D" xml:space="preserve">Senſus
                    <lb/>
                  ſecūde cõ­
                    <lb/>
                  cluſionis</note>
                </s>
                <s xml:id="N123B6" xml:space="preserve">¶ Intellige igitur concluſionē in ſenſu in quo ma­
                  <lb/>
                thematici eam intelligunt. </s>
                <s xml:id="N123BB" xml:space="preserve">puta /  ſi duo nume-
                  <lb/>
                ri equaliter diſtēt a duobus numeris extrimis ita­
                  <lb/>
                 primus excedat ſecundum eadē differentia qua
                  <lb/>
                tertius quartum: vel primus excedatur a ſecundo
                  <lb/>
                ea differentia qua tertius exceditur a quarto illi
                  <lb/>
                intermedii ſimul iuncti extremis copulatis equã-
                  <lb/>
                tur. </s>
                <s xml:id="N123CA" xml:space="preserve"> ſi intermedii ab extremis diſtãtes ſimul iū­
                  <lb/>
                cti extremis equantur ab extremis eos equidiſta­
                  <lb/>
                re neceſſe eſt.
                  <note position="left" xlink:href="note-0025-03a" xlink:label="note-0025-03" xml:id="N12427" xml:space="preserve">Primu
                    <lb/>
                  correlari­
                    <lb/>
                  um.</note>
                </s>
                <s xml:id="N123D6" xml:space="preserve">¶ Ex hac concluſione ſequitur arith­
                  <lb/>
                metice medietatis diſiūcte quattuor terminis ab­
                  <lb/>
                ſolute extrema ſimul iuncta collectis medii equa­
                  <lb/>
                ri.
                  <note position="left" xlink:href="note-0025-04a" xlink:label="note-0025-04" xml:id="N12431" xml:space="preserve">tertia ꝓ-
                    <lb/>
                  prietas
                    <lb/>
                  medieta­
                    <lb/>
                  tis arith­
                    <lb/>
                  metice.</note>
                </s>
                <s xml:id="N123E4" xml:space="preserve">Et hec eſt tertia ꝓprietas mediedatis arithme­
                  <lb/>
                tice. </s>
                <s xml:id="N123E9" xml:space="preserve">Patet hoc correlarium facile ex precedēti cõ­
                  <lb/>
                cluſione </s>
                <s xml:id="N123EE" xml:space="preserve">Nam ſi quattuor termini proportionen­
                  <lb/>
                tur arithmetice et diſiiuncte ea differētia que erit
                  <lb/>
                inter primū et ſecundum. erit inter tertium et quar­
                  <lb/>
                tū </s>
                <s xml:id="N123F7" xml:space="preserve">Quare medii equaliter diſtabunt ab extremis
                  <lb/>
                coniunctis / igitur mediis equabuntur externa col­
                  <lb/>
                lecta iuxta doctrinam concluſionis. </s>
                <s xml:id="N123FE" xml:space="preserve">Et dixi notã-
                  <cb chead="Capitulum ſecundum"/>
                ter in correlario. </s>
                <s xml:id="N12404" xml:space="preserve">quattuor terminis quia ſi ponã­
                  <lb/>
                tur plures termini non oportet illud verificari.</s>
              </p>
              <p xml:id="N1243F">
                <s xml:id="N12440" xml:space="preserve">Quare inconſiderate aliqui illam proprietatem
                  <lb/>
                abſolute ponūt. </s>
                <s xml:id="N12445" xml:space="preserve">Patet enim inſtantia in his ter­
                  <lb/>
                minis .2.5.7.11.1.4. manifeſtum eſt enim /  aggre­
                  <lb/>
                gatum ex extremis minus eſt aggregato ex inter-
                  <lb/>
                mediis. </s>
                <s xml:id="N1244E" xml:space="preserve">Imo implicat aggregatum ex extremis
                  <lb/>
                equari omnibus itermediis ſimul ſumptis cum
                  <lb/>
                ſunt plures termini quattuor: quoniam ſuper ag­
                  <lb/>
                gregatum ex extermis puta ex primo et vltimo ad­
                  <lb/>
                dequatur aggregato ex ſecūdo et penultimo. </s>
                <s xml:id="N12459" xml:space="preserve">ergo
                  <lb/>
                non aggregato ex omnibus intermediis quia il-
                  <lb/>
                lud erit maiꝰ. </s>
                <s xml:id="N12460" xml:space="preserve">Si autem velis dicere ꝓprietatē il-
                  <lb/>
                lam intelligi /  aggregatum ex ṗmo et vltimo ade­
                  <lb/>
                quatur aggregato ex ſecūdo et penultimo: et etiã
                  <lb/>
                equatur aggregato ex tertio et ante penultimo .etc̈ /
                  <lb/>
                patet hoc eſſe falſum in datis terminis. </s>
                <s xml:id="N1246B" xml:space="preserve">Nã in il-
                  <lb/>
                lis duo et .14. conſtituunt .1.6. tertius tñ et ante pe­
                  <lb/>
                nultimus puta .7. et .10. conſtituunt .1.7. / igitur.</s>
              </p>
              <note position="right" xml:id="N12472" xml:space="preserve">Secūduꝫ
                <lb/>
              correlari­
                <lb/>
              um.</note>
              <p xml:id="N1247A">
                <s xml:id="N1247B" xml:space="preserve">¶ Sequitur ſecundo /  poſitis quattuor terminis
                  <lb/>
                proportionabilibus arithmetice ſiue cõiuncte ſi-
                  <lb/>
                ue diſiuncte aggregatum ex primo et vltimo ē me­
                  <lb/>
                dietas aggregati ex omnibus ſimul et etiam ag-
                  <lb/>
                gregatum ex ſecūdo. </s>
                <s xml:id="N12486" xml:space="preserve">et tertio eſt medietas totius
                  <lb/>
                aggregati ex omnibus ſimul. </s>
                <s xml:id="N1248B" xml:space="preserve">Patet / quia illa ag­
                  <lb/>
                gregata ſunt eq̈lia ex cõcluſione et adequate com­
                  <lb/>
                ponunt aggregatū ex omnibus illis quattuor ter­
                  <lb/>
                minis: igitur vtrum illorū aggregatum eſt me-
                  <lb/>
                dietas aggregati ex omnibus illis terminis ſimĺ
                  <lb/>
                ſumptis / quod fuit probãdum.
                  <note position="right" xlink:href="note-0025-05a" xlink:label="note-0025-05" xml:id="N12535" xml:space="preserve">Tertium
                    <lb/>
                  correlari­
                    <lb/>
                  um.</note>
                </s>
                <s xml:id="N1249D" xml:space="preserve">¶ Sequitur tertio /
                  <lb/>
                 poſitis ſex terminis ſi octo.
                  <note position="right" xlink:href="note-0025-06a" xlink:label="note-0025-06" xml:id="N1253F" xml:space="preserve">Cal. ḋ 10
                    <lb/>
                  ele.</note>
                </s>
                <s xml:id="N124A7" xml:space="preserve">ſiue .10. et in quo-
                  <lb/>
                cun numero pari cõtinuo proportionabilibus
                  <lb/>
                arithmetice. </s>
                <s xml:id="N124AE" xml:space="preserve">aggregatum ex primo et vltimo et ag­
                  <lb/>
                gregatum ex ſecundo et penultimo et aggregatū
                  <lb/>
                ex tertio et ante penultimo / et ſic conſequenter eſt
                  <lb/>
                pars aliquota aggregati ex omnibus illis ter-
                  <lb/>
                minis denominata a numero ſubduplo ad nume-
                  <lb/>
                rum parem in quo conſtituuntur tales termini. </s>
                <s xml:id="N124BB" xml:space="preserve">vt
                  <lb/>
                ſi ſint ſex termini aggregatum ex primo et ſexto et
                  <lb/>
                etiam aggregatum ex ſecundo et quinto et ex ter-
                  <lb/>
                tio et quarto eſt vna tertia aggregati ex omnibus
                  <lb/>
                illis ſex terminis: et ſi fuerint octo talia aggrega­
                  <lb/>
                ta erunt quarte / q2 quarta denominatur a nume-
                  <lb/>
                ro ſubduplo ad numerum octonarium. </s>
                <s xml:id="N124CA" xml:space="preserve">Proba-
                  <lb/>
                tur hoc / et ſint ſex termini .a.b.d.c.e.f. ↄ̨tinuo arith­
                  <lb/>
                metice proportionabiles. </s>
                <s xml:id="N124D1" xml:space="preserve">et arguitur ſic / aggrega­
                  <lb/>
                tum ex a.f. eſt equale aggregato ex .b.e. / vt patet ex
                  <lb/>
                concluſione / quia illa extrema equaliter diſtãt ab
                  <lb/>
                illis mediis et eadem ratione aggregatum ex .c.d
                  <lb/>
                eſt equale aggregato ex b.e. / igitur ibi ſūt tria ag­
                  <lb/>
                gregata omnino equalia: et illa componunt ag-
                  <lb/>
                gregatum ex omnibus illis .6. adequate: igitur qḋ­
                  <lb/>
                libet illorum aggregatorum eſt vna tertia totius
                  <lb/>
                </s>
                <s xml:id="N124E3" xml:space="preserve">Et iſto modo probabis quando fuerint octo ter-
                  <lb/>
                mini / quia inuenies ibi quattuor aggregata equa­
                  <lb/>
                lia: et quando decem inuenies quin. </s>
                <s xml:id="N124EA" xml:space="preserve">Et ſic dein-
                  <lb/>
                ceps inuenies talia aggregata equalia in ſubdu­
                  <lb/>
                plo numero ad numerum terminorum: quoniam
                  <lb/>
                ſemper pro quolibet tali aggregato capis duos
                  <lb/>
                terminos / et per conſequens dualitatem illorum
                  <lb/>
                terminorum. </s>
                <s xml:id="N124F7" xml:space="preserve">Modo in quolibet numero pari in
                  <lb/>
                duplo pauciores dualitates reperiūtur quam vni­
                  <lb/>
                tates. </s>
                <s xml:id="N124FE" xml:space="preserve">Et ſic patet correlarium.
                  <note position="right" xlink:href="note-0025-07a" xlink:label="note-0025-07" xml:id="N12547" xml:space="preserve">Quartū
                    <lb/>
                  correlari­
                    <lb/>
                  um.</note>
                </s>
                <s xml:id="N12506" xml:space="preserve">¶ Sequitur quar­
                  <lb/>
                to /  ſint quattuor termini non continuo propor-
                  <lb/>
                tionabiles arithmetice continuo tamen minores
                  <lb/>
                et minores continuo ſe excedētes minori et mino- </s>
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