Alvarus, Thomas, Liber de triplici motu, 1509

Page concordance

< >
Scan Original
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
< >
page |< < of 290 > >|
    <echo version="1.0">
      <text xml:lang="la">
        <div xml:id="N10132" level="1" n="1" type="body">
          <div xml:id="N1194D" level="2" n="2" type="other" type-free="pars">
            <div xml:id="N11E85" level="3" n="2" type="chapter" type-free="capitulum">
              <p xml:id="N1255B">
                <s xml:id="N12699" xml:space="preserve">
                  <pb chead="Secūde partis" file="0027" n="27"/>
                bus quã ſit pars aliquota denoīata a numero ſub­
                  <lb/>
                duplo ad numerum parem in quo ſunt conſtituti
                  <lb/>
                dati termini: et aggregatum ex duobus mediis
                  <lb/>
                immediatis equaliter diſtantibus ab extremis
                  <lb/>
                eſt maius quaꝫ talis pars aliquota. </s>
                <s xml:id="N126B1" xml:space="preserve">vt captis his
                  <lb/>
                terminis .12.11.9.6. aggregatum ex .12. et ſex. eſt
                  <lb/>
                minus quam medietas aggregati oīm illorū me­
                  <lb/>
                dietas denomīatur a numero binario qui eſt ſub­
                  <lb/>
                duplus ad numerū quaternariū in quo illi termi-
                  <lb/>
                ni ſunt conſtituti: et aggregatum ex .11. et .9. eſt ma­
                  <lb/>
                ius quã medietas. </s>
                <s xml:id="N126C0" xml:space="preserve">Probatur: et ſint a.b.c.d.e.f.6.
                  <lb/>
                termini continuo minores et minores maiori con­
                  <lb/>
                tinuo dnr̄ia ſeſe excedentes: et q2 illi ſunt conſtitu­
                  <lb/>
                ti in numero ſenario dico /  aggregatū ex primo
                  <lb/>
                et vltimo eſt minor pars totius ꝙ̄ pars aliquota
                  <lb/>
                eiuſdem totius denoīata a numero ſubduplo ad
                  <lb/>
                ſenarium que eſt vna tertia. / et aggregatū ex duo­
                  <lb/>
                bus intermediis īmediatis equaliter diſtantibus
                  <lb/>
                ab extremis puta c.d. eſt maius quã talis pars ali­
                  <lb/>
                quota totius puta quã tertia. </s>
                <s xml:id="N126D5" xml:space="preserve">Probat̄̄ / q2 tale ag­
                  <lb/>
                gregatū cõponitur ex tribus partialibus aggre­
                  <lb/>
                gatis adequate puta ex aggregato ex a. et f. et ex
                  <lb/>
                aggregato ex b. et e. et aggregato et c. et d. et ag-
                  <lb/>
                gregatū ex a. et f. eſt minus ſecundo aggregato et
                  <lb/>
                ſecundū minus tertio. </s>
                <s xml:id="N126E2" xml:space="preserve">igitur aggregatū ex a. et f.
                  <lb/>
                eſt minus quaꝫ tertia totius: et aggregatū ex c.d.
                  <lb/>
                maius quã tertia totius. </s>
                <s xml:id="N126E9" xml:space="preserve">Patet hec conſequentia /
                  <lb/>
                quia quando aliquid cõponitur ex tribus quoruꝫ
                  <lb/>
                quodlibet cuilibet alteri eſt inequale: maius illoꝝ
                  <lb/>
                eſt maius quã tertia: et ſic dices quando cõponitur
                  <lb/>
                ex quatuor adequate quorū quodlibet cuilibet al­
                  <lb/>
                teri eſt īequale: et ex .5. et ex .6. / et ſic deinceps vt po­
                  <lb/>
                ſtea oſtendetur. </s>
                <s xml:id="N126F8" xml:space="preserve">Iam probo minorem videlicet / 
                  <lb/>
                aggregatū ex a. et f. eſt minus ſecundo aggrega-
                  <lb/>
                to puta ex b. et e. / q2 ſi tanto exceſſu. </s>
                <s xml:id="N126FF" xml:space="preserve">et dnr̄a a exce-
                  <lb/>
                deret b. quanta e. excedit f. / tunc aggregatū ex a. et
                  <lb/>
                f. eſſet equale aggregato ex b. et e. / vt patet ex ſecū­
                  <lb/>
                da concluſione: ſed modo aggregatū ex a.f. eſt mi­
                  <lb/>
                nus quã tunc: quia a. eſt tãtum ſicut tunc et f. eſt mi­
                  <lb/>
                nus quã tunc: quia maiori dnr̄ia exceditur modo
                  <lb/>
                quã tunc ab eodē puta e. / igitur aggregatū ex a. et
                  <lb/>
                f. eſt minus quã aggregatū ex b. et e. / et eadē ratio­
                  <lb/>
                ne ꝓbabis /  aggregatū ex b. et e. eſt minus aggre­
                  <lb/>
                gato ex c. et d. / et ſic patet minor et totū correlariū /
                  <lb/>
                quoniã et ſi iſta ſit particularis demonſtratio tñ
                  <lb/>
                dat formã vniuerſaliter ꝓbandi quibuſcū ter-
                  <lb/>
                minis paribus conſtitutis. </s>
                <s xml:id="N1271A" xml:space="preserve">¶ Similia correlaria
                  <lb/>
                poteris inferre q̇buſcun termīs īpari nūero cõ­
                  <lb/>
                ſtitutis ſiue continuo maioribus et maioribus ma­
                  <lb/>
                iori continuo dnr̄a ſe excedentibus: ſiue eocontra
                  <lb/>
                etc. / que omnia predictorum auxilio facile monſtra­
                  <lb/>
                ri poſſunt.</s>
              </p>
              <note position="left" xml:id="N12743" xml:space="preserve">1. ele. ior.
                <lb/>
              3. con.
                <lb/>
              4. ꝓprie­
                <lb/>
              tas arith­
                <lb/>
              metice
                <lb/>
              medieta­
                <lb/>
              tis.</note>
              <p xml:id="N12753">
                <s xml:id="N12754" xml:space="preserve">Tertia concluſio in hac medietate
                  <lb/>
                arithmetica / quod ſub extremis continetur cum q̈­
                  <lb/>
                drato differentie. </s>
                <s xml:id="N1275B" xml:space="preserve">equale eſt quadrato medii. </s>
                <s xml:id="N1275E" xml:space="preserve">Hec
                  <lb/>
                concluſio eſt tertia decimi elementorum iordani et
                  <lb/>
                breuitatis cauſa hic non demonſtratur / quia eius
                  <lb/>
                demõſtratio prolixa eſt eo  dependet ex decima
                  <lb/>
                quarta et decima nona primi elementorum eiuſ-
                  <lb/>
                dem iordani. </s>
                <s xml:id="N1276B" xml:space="preserve">¶ Aduerte tamen pro intelli­
                  <lb/>
                gentia contextus ipſius concluſionis /  illud dici­
                  <lb/>
                tur contineri. </s>
                <s xml:id="N12772" xml:space="preserve">ſub extremis arithmetice ꝓportio-
                  <lb/>
                nalitatis quod reſultat ex ductu vnius extremi in
                  <lb/>
                alterum: vt numerus octonarius continetur ſub
                  <lb/>
                extremis huius ꝓportionalitatis .4.3.2. quia du-
                  <lb/>
                cendo .4. per .2. reſultant octo. </s>
                <s xml:id="N1277D" xml:space="preserve">Bis em̄ .4. ſūt octo
                  <cb chead="Capitulum ſecundum"/>
                </s>
                <s xml:id="N12783" xml:space="preserve">Item .32. continētur ſub extremis huius ꝓportio­
                  <lb/>
                nalitatis arithmetice .8.7.4. qm̄ ducendo .8. per .
                  <lb/>
                4. reſultant: 32. </s>
                <s xml:id="N1278A" xml:space="preserve">Quater enim octo ſunt .32.
                  <note position="right" xlink:href="note-0027-01a" xlink:label="note-0027-01" xml:id="N127D7" xml:space="preserve">quadra-
                    <lb/>
                  tū medii</note>
                </s>
                <s xml:id="N12792" xml:space="preserve">¶ Ad­
                  <lb/>
                uerte vlterius /  quadratū medii termini eſt illud
                  <lb/>
                quod reſultat ex ductu medii termini in ſeipſuꝫ:
                  <note position="right" xlink:href="note-0027-02a" xlink:label="note-0027-02" xml:id="N127DF" xml:space="preserve">q̈dratuꝫ
                    <lb/>
                  dnr̄ie.</note>
                vt
                  <lb/>
                numerus nouenarius eſt quadratum medii in hac
                  <lb/>
                arithmetica proportionalitate .4.3.2. quia reſul-
                  <lb/>
                tat ex ductu numeri ternarii in ſeipſum. </s>
                <s xml:id="N127A4" xml:space="preserve">Nam ter
                  <lb/>
                tria ſunt nouē. </s>
                <s xml:id="N127A9" xml:space="preserve">¶ Quadratū autē differentie eſt il­
                  <lb/>
                lud quod reſultat ex ductu differentie in ſeipſum:
                  <lb/>
                vt in hac arithmetica medietate .8.6.4. numerus
                  <lb/>
                quaternarius eſt quadratū dnr̄e. </s>
                <s xml:id="N127B2" xml:space="preserve">Nã differentia
                  <lb/>
                eſt numerus binarius / vt conſtat. </s>
                <s xml:id="N127B7" xml:space="preserve">Binarius enim
                  <lb/>
                ductus in ſeipſum quaternarium educit / vt cõſtat.
                  <lb/>
                </s>
                <s xml:id="N127BD" xml:space="preserve">¶ His dictis ſenſus concluſionis eſt talis. </s>
                <s xml:id="N127C0" xml:space="preserve">Nume-
                  <lb/>
                rus reſultans ex ductu vnius extremi in alterū in
                  <lb/>
                medietate arithmetica continua cum numero re-
                  <lb/>
                ſultante ex ductu differentie in ſeipſam eſt equalis
                  <lb/>
                numero qui fit ex ductu medii in ſeipſū: vt in hac
                  <lb/>
                medietate .8. que fiunt ex ductu vnius extremi in al­
                  <lb/>
                terum iuncto quaternario numero qui fit ex dictu
                  <lb/>
                differentie in ſeipſaꝫ ſunt equalia .36. que fiunt ex
                  <lb/>
                ductu ſenarii medii termini in ſeipſum.</s>
              </p>
              <note position="right" xml:id="N127E7" xml:space="preserve">4. cõclu-
                <lb/>
              ſio prīa
                <lb/>
              ꝓprietaſ
                <lb/>
              medieta­
                <lb/>
              tis geo-
                <lb/>
              metrice.</note>
              <p xml:id="N127F5">
                <s xml:id="N127F6" xml:space="preserve">Quarta concluſio in medietate geo-
                  <lb/>
                metrica q̈tuor terminis conſtituta ſi primus ad ſe­
                  <lb/>
                cundū ſicut tertius ad quartum: ita primus ad ter­
                  <lb/>
                tiū ſicut tertius ad quartū ſe habeat neceſſe eſt: vt
                  <lb/>
                quia ſicut ſe habent octo ad quatuor ita ſe habēt
                  <lb/>
                ſex. ad .tria. / conſequens eſt /  ſicut ſe habent .octo
                  <lb/>
                ad .ſex. ita quatuor ad tria. </s>
                <s xml:id="N12805" xml:space="preserve">Probatur / ſint a.b.
                  <lb/>
                c.d. quatuor termini in medietate geometrica: et
                  <lb/>
                habeat ſe a. ad .b. ſicut c. ad d. / tūc dico /  ſicut ſe hꝫ
                  <lb/>
                a. ad .c. ita b. ad d. </s>
                <s xml:id="N1280E" xml:space="preserve">Qḋ ſic ꝓbat̄̄ et ṗmo ī nūerꝪ / q2 ſi
                  <lb/>
                ſicut ſe habet a. ad b. ita .c. ad .d.b. eſt pars vel par­
                  <lb/>
                tes aliquote reſpectu a. eiuſdem denoīationis ſi-
                  <lb/>
                cut d. ipſius c. et vltra b. eſt pars aliquota vel par­
                  <lb/>
                tes aliq̊te eiuſdē denoīationis reſpectu a. ſicut d.
                  <lb/>
                reſpectu c. / ergo ſicut ſe habet a. ad c. ita b. ad d. / qḋ
                  <lb/>
                fuit probandū. </s>
                <s xml:id="N1281D" xml:space="preserve">Secunda conſequētia patet ex vn-
                  <lb/>
                decima ſuppoſitione huius capitis: et prima ptꝫ
                  <lb/>
                ex hoc / quod inferius probabitur. </s>
                <s xml:id="N12824" xml:space="preserve">Si aliqui duo
                  <lb/>
                numeri maiores habent ↄ̨ſimiles proportiones
                  <lb/>
                ad duos minores: illi minores numeri ſūt partes
                  <lb/>
                aliquote maiorū conſimilis denoīationis. </s>
                <s xml:id="N1282D" xml:space="preserve">Et ſit
                  <lb/>
                hec prima proprietas geometrice medietatis.</s>
              </p>
              <p xml:id="N12832">
                <s xml:id="N12833" xml:space="preserve">Probatur iaꝫ vniuerſaliter / ſint a.b.c.d. quatuor
                  <lb/>
                termini in hac medietate geometrica conſtituti ſi­
                  <lb/>
                ue continuo ꝓportionabiles, ſiue diſcontinue, ſi-
                  <lb/>
                ue proportione rationali, ſiue irrationali. </s>
                <s xml:id="N1283C" xml:space="preserve">et ipſi-
                  <lb/>
                us a. ad b. ſit f. proportio: et ſimiliter ipſius c. ad
                  <lb/>
                ipſum d. ſit f. proportio: et ſit a. ad .c.g. ꝓportio. </s>
                <s xml:id="N12843" xml:space="preserve">et
                  <lb/>
                tunc dico /  etiam b. ad d. eſt g. proportio. </s>
                <s xml:id="N12848" xml:space="preserve">Quod
                  <lb/>
                probatur ſic / et capio ꝓportionem g. / que eſt a. ad
                  <lb/>
                c. / et volo /  a deperdat ꝓportioneꝫ f. quam habet
                  <lb/>
                ad b. ita  in fine maneat equale ipſi b. / vt oportet
                  <lb/>
                et c. perdat eandem proportionem f. quam ex hy-
                  <lb/>
                potheſi habet ad ipſum d. ita  in fine maneat eq̈­
                  <lb/>
                le ipſi d. / et arguo ſic. </s>
                <s xml:id="N12857" xml:space="preserve">huius ꝓportionis g. que eſt a
                  <lb/>
                ad c. equalem omnino ꝓportionē deperdit termi-
                  <lb/>
                nus maior ſicut minor: quia vter f. proportioneꝫ /
                  <lb/>
                vt patet ex hypotheſi: igitur facta tali diminutio­
                  <lb/>
                ne adhuc manet inter reſiduum maioris termini et
                  <lb/>
                minoris. </s>
                <s xml:id="N12864" xml:space="preserve">eadem proportio g. / vt patet ex ſecunda
                  <lb/>
                parte decime ſuppoſitionis ſecundi capitis ſecun­
                  <lb/>
                de partis ſed reſiduū maioris termini eſt b. et reſi­
                  <lb/>
                duū mīoris d. / vt pꝫ ex hypotheſi: igit̄̄ b. ad d. ē g. ꝓ­ </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum