Alvarus, Thomas
,
Liber de triplici motu
,
1509
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capitulum
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Prime partis
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0007
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tas ter ſumpta: adequate conſtituit ternarium
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et quater ſumpta: quaternarium. </
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<
s
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N10398
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xml:space
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">et dualitas eſt
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pars aliquota numeri octonarii. </
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<
s
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N1039D
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xml:space
="
preserve
">quoniam duali
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tas quater ſumpta adequate numerū octonariuꝫ
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conſtituit. </
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>
<
s
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N103A4
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xml:space
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">¶ Ex quo patet / dualitas non eſt ꝑrs
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aliquota numeri ſeptenarii quoniam non aliquo
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ties ſumpta: reddit illud totum adequate. </
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>
<
s
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N103AB
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xml:space
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">¶ Pro
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portio autem irrationalis: eſt illa que nõ immedi
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ate ab aliquo numero denominatur. </
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>
<
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N103B2
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xml:space
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">Alio modo
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proportio irrationalis: eſt duarum quantitatum
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ita ſe habentiū: nulla pars aliquota vnius eſt
<
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ꝑs aliq̊ta alteriꝰ vt ꝓportio q̄ ē īter diametrū et co
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ſtã ſui q̈drati. </
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<
s
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N103BD
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xml:space
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">nã diameṫ excedit coſtã et nõ aliq̊ties
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nec ꝑ aliquã ꝑtem aliquotã. </
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>
<
s
xml:id
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N103C2
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xml:space
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">vel per aliq̈s ꝑtes ali
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quotas. </
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<
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xml:space
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">vt inferius probabitur in capitulo de ꝓ-
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portione irrationali.
<
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xml:id
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xml:space
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">Diuiſio
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ꝓportio
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nū rõna-
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lium.</
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</
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<
s
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="
N103D1
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xml:space
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">¶ Proportionum auteꝫ ra-
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tionalium .5. ſunt ſpecies tres ſimplices: et due cõ
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poſite. </
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>
<
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xml:space
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">¶ Simplices ſunt iſte. </
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>
<
s
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N103DB
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xml:space
="
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">multiplex: ſuperpar
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ticularis: et ſuprapartiēs. </
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>
<
s
xml:id
="
N103E0
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xml:space
="
preserve
">¶ Compoſite vero ſunt
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multiplex. </
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<
s
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N103E5
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xml:space
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ſuprapartiens </
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<
s
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N103EA
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xml:space
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preserve
">¶ Unde proportio multiplex: eſt ꝓ
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portio qua maius continet minus aliquoties ta-
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tū vt dupla, tripla .4. enim continent .2. bis. / et .6.
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continent .2. ter tantum </
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<
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xml:space
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ros eſt ꝓportio multiplex. </
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perparticularis. </
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<
s
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">eſt proportio qua maius cõtinet
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minus ſemel tãtū: et aliquam partem eius aliquo
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tã adeq̈te. </
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>
<
s
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="
N10404
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xml:space
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">vt ꝓportio ſex ad .4. nã .6. cõtinet .4. ſe-
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mel tm̄ et medietatē q̄ eſt pars aliquota ipſoꝝ .4.
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</
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<
s
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xml:space
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">¶ Proportio autem ſuprapartiēs: eſt proportio
<
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/>
qua maius continet minus ſemel tantū: et aliquot
<
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/>
partes eius aliquotas: que ſimul non faciunt ali
<
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/>
quam eius partem aliquotam. </
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>
<
s
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="
N10413
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xml:space
="
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">vt ꝓportio que eſt
<
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inter .7. et .5. </
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<
s
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N10418
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xml:space
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">Nam .7. continent .5. ſemel tantum: et
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duas partes eius aliquotas: puta duas vnitates
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</
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>
<
s
xml:id
="
N1041E
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xml:space
="
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">¶ Sed proportio multiplex ſuperparticularis eſt
<
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illa qua maius continet minus aliquotiens: et
<
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/>
cum hoc aliquam eius partem aliquotam tantuꝫ
<
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/>
vt proportio que eſt inter nouem et .4. </
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>
<
s
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="
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xml:space
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">Nã .9. con-
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tinent .4. bis. / et vnam partem numeri quaternarii
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puta vnitatem. </
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>
<
s
xml:id
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N1042E
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xml:space
="
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">¶ Proportio autem multiplex ſu
<
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prapartiens: eſt illa qua maius continent minus
<
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/>
aliquotiens et aliquot partes eiꝰ aliquotas: que
<
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non faciunt vnam eius partem aliquotam vt pro
<
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portio que eſt inter .11. et .4. </
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>
<
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">Nã .11. continent .4. bis /
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et tres partes aliquotas ipſorum .4. et ille nõ fa-
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ciunt aliquam partem aliquotam ipſorum .4.</
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>
</
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cia quī
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numeri ꝓ
<
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portiõis
<
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rõaĺ ma
<
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/>
ioris ine
<
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q̈litatis.</
note
>
<
p
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N10474
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<
s
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="
N10475
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xml:space
="
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">¶ Harum autem proportionum: ſiue ſpecierum ꝓ
<
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portionum ſufficientia: talis ratione haberi põt
<
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vt adducit Albertus de ſaxonia ī ſuo tractatu de
<
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proportionibus poſt alios mathematicos. </
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>
<
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="
N1047E
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="
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">Qm̄
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oīs numerus: ſiue quantitas ad aliam quantitatē
<
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habens rationalem proportiouem: aut excedit
<
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eam: aut exceditur ab illa. </
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>
<
s
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="
N10487
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xml:space
="
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">Si excedit eam: aut
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continet ipſam aliquoties. </
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>
<
s
xml:id
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xml:space
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">aut ſemel tantū: et ali
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quid vltra. </
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<
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">aut pluries et aliquid vltra. </
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<
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">Si primū /
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tunc erit proportio multiplex </
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<
s
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">Si ſecūdū / aut illud
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aliquid vltra eſt vna pars eius aliquota adequa-
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te: aut ē plures partes aliquote que nõ faciūt vnã
<
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/>
partem aliquotam. </
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>
<
s
xml:id
="
N104A2
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xml:space
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">Si primum: ſic eſt ꝓportio ſu
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perparticularis. </
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<
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xml:space
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">Si ſecundum / eſt proportio ſuꝑ-
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partiens. </
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<
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">Si vero maior quantitas continet mi-
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norē pluries. </
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<
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<
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">vel illud quod vltra
<
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continet eſt pars aliquota adequate aut: plures
<
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partes aliquote: que non faciunt vnã. </
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>
<
s
xml:id
="
N104BB
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xml:space
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">Si primum /
<
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ſic eſt proportio multiplex ſuperparticulares. </
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<
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xml:space
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">Si
<
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Capitulum ſecundum
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ſecundum ſic eſt proportio multiplex ſupraparti-
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ens. </
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<
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N104C8
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xml:space
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">Et quia quantitas maior habens proportio
<
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nē rationalem ad quantitatem minorē nõ poteſt
<
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pluribus modis ad illam referri
<
gap
/>
ſiue compara-
<
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ri. </
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>
<
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">quam his quin modis conſequens eſt / non
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poſſunt eſſe plures ſpecies proportionis rationa
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lis his .5. </
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<
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N104DA
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">Quãdoquidem eodem modo venari po
<
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teſt minoris inequalitatis proportionum ſuffici
<
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entia. </
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<
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="
N104E1
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xml:space
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">Sola enim ratione: proportio maioris ine
<
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qualitatis: et minoris differunt) </
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>
<
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N104E6
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xml:space
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<
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autem poſterius dicetur.</
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</
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</
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>
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<
head
xml:id
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N104F0
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xml:space
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preserve
">Cpitulum ſecundum / in quo agitur de ſpe
<
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ciebus horum quin generum proportionū
<
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et de ipſarum generatione.</
head
>
<
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<
s
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="
N104F8
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xml:space
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">OMnis proportio ſiue omne ge
<
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nus proportiõis: infinitas habet ſpecies
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</
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<
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="
N104FE
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xml:space
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">Unde genus multiplicis: habet infinitas
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ſpecies denominatas a naturali ſerie numerorū
<
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puta duplã denominatã a binario triplã a terna
<
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rio: milleculpam a millenario: centuplam a cen-
<
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tenario. </
s
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<
s
xml:id
="
N10509
"
xml:space
="
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">et ſic in infinitū. </
s
>
<
s
xml:id
="
N1050C
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xml:space
="
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">¶ Proportio em̄ dupla:
<
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eſt illa qua maius continet minus: bis adequate
<
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vt .4. cum .2. et tripla qua maius continet minus:
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ter adequate. </
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>
<
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="
N10515
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xml:space
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">et quadrupla quater adequate. </
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>
<
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N10518
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xml:space
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">et ſic
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in infinitum. </
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<
s
xml:id
="
N1051D
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xml:space
="
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">¶ Generãtur autem omnes ꝓportio
<
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nes duple que infinite ſunt iſto modo. </
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>
<
s
xml:id
="
N10522
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xml:space
="
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">Diſpona-
<
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tur / primo ſeries naturalis numeroꝝ in vna linea
<
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et in alia linea inferiori diſponantur omnes nu-
<
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meri excedentes ſe binario: incipiendo a binario
<
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in infinitum. </
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>
<
s
xml:id
="
N1052D
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xml:space
="
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">Et iſto modo cõparando primum ſu-
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perioris linie primo inferioris: et ſecundū ſecūdo
<
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/>
et tertiū tertio.
<
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position
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right
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xlink:href
="
note-0007-02a
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xlink:label
="
note-0007-02
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xml:id
="
N10547
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xml:space
="
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">gñatio ꝓ
<
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portõnū
<
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duplarū</
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>
</
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<
s
xml:id
="
N10539
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xml:space
="
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">et ſic in infinitum inuenientur infi-
<
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nite ꝓportiõis duple. </
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>
<
s
xml:id
="
N1053E
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xml:space
="
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">in preſenti figura clare hoc
<
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poteris conſpicere.</
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>
</
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<
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<
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<
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xml:space
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</
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>
</
xhtml:table
>
<
p
xml:id
="
N10555
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<
s
xml:id
="
N10556
"
xml:space
="
preserve
">Per naturalem ſeriē numerorum: intelligas ordi
<
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/>
ne numerorū incipiēdo ab vnitate nullū numeruꝫ
<
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/>
omittendo. </
s
>
<
s
xml:id
="
N1055D
"
xml:space
="
preserve
">vt .1.2.3.4. etc̈. </
s
>
<
s
xml:id
="
N10560
"
xml:space
="
preserve
">¶ Sed infinite ꝓportio-
<
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nes triple: iſto modo generantur </
s
>
<
s
xml:id
="
N10565
"
xml:space
="
preserve
">Diſponatur / oēs
<
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/>
nūeri ſcḋm ſeriē naturalē nūerorū incipiendo ab
<
lb
/>
vnitate ī vna linea et ī linea īferiori diſponãt̄̄ oēs
<
lb
/>
nūeri excedētes ſe ṫnario. </
s
>
<
s
xml:id
="
N1056E
"
xml:space
="
preserve
">et tūc cõparãdo ṗmū īfe
<
lb
/>
rioris ordinis prīo ſuperioris et ſecūdū ſecūdo et
<
lb
/>
tertiū tertio:
<
note
position
="
right
"
xlink:href
="
note-0007-03a
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xlink:label
="
note-0007-03
"
xml:id
="
N1057E
"
xml:space
="
preserve
">gñatio ꝓ
<
lb
/>
portõnū
<
lb
/>
triplarū</
note
>
habebunt̄̄ infinite ꝓportiões triple.</
s
>
</
p
>
<
xhtml:table
xml:id
="
N10588
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<
xhtml:tr
xml:id
="
N10589
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<
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xml:id
="
N1058A
"
xml:space
="
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"/>
</
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>
</
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>
<
note
position
="
right
"
xml:id
="
N1058C
"
xml:space
="
preserve
">gñatio ꝓ
<
lb
/>
portõnū
<
lb
/>
q̈drupla
<
lb
/>
rum:</
note
>
<
p
xml:id
="
N10596
">
<
s
xml:id
="
N10597
"
xml:space
="
preserve
">Si vero velis gñare oēs ꝓportiões quadruplas:
<
lb
/>
capias nūeros excedentes ſe q̈ternario. </
s
>
<
s
xml:id
="
N1059C
"
xml:space
="
preserve
">incipiēdo
<
lb
/>
a nūero q̈ternario cū ſerie naturali nūeroꝝ.
<
note
position
="
right
"
xlink:href
="
note-0007-04a
"
xlink:label
="
note-0007-04
"
xml:id
="
N105BE
"
xml:space
="
preserve
">Gñatio
<
lb
/>
quītupla
<
lb
/>
rum.</
note
>
</
s
>
<
s
xml:id
="
N105A6
"
xml:space
="
preserve
">¶ Si
<
lb
/>
aūt quītuplã: capias oēs excedētes ſe q̇nario
<
note
position
="
right
"
xlink:href
="
note-0007-05a
"
xlink:label
="
note-0007-05
"
xml:id
="
N105C8
"
xml:space
="
preserve
">Gñatio
<
lb
/>
ſextupla
<
lb
/>
rum.</
note
>
</
s
>
<
s
xml:id
="
N105B0
"
xml:space
="
preserve
">¶ Si
<
lb
/>
ſextuplã ſenario. </
s
>
<
s
xml:id
="
N105B5
"
xml:space
="
preserve
">et ſic in infinitū vt facile eſt vide-
<
lb
/>
re in figuris ſequentibus.</
s
>
</
p
>
<
xhtml:table
xml:id
="
N105D2
">
<
xhtml:tr
xml:id
="
N105D3
">
<
xhtml:td
xml:id
="
N105D4
"
xml:space
="
preserve
"/>
</
xhtml:tr
>
</
xhtml:table
>
<
p
xml:id
="
N105D6
">
<
s
xml:id
="
N105D7
"
xml:space
="
preserve
">¶ Suꝑparticularis autē ꝓportio etiam infinitas
<
lb
/>
habet ſpecies denoīatas a partibus aliquotis: et
<
lb
/>
vnitate. </
s
>
<
s
xml:id
="
N105DE
"
xml:space
="
preserve
">puta a medietate: a tertia quarta quinta /
<
lb
/>
et ſic in infinitū. </
s
>
<
s
xml:id
="
N105E3
"
xml:space
="
preserve
">Et ideo prima ſpecies eiꝰ et maxīa
<
lb
/>
dicitur ſexquialtera. ſecūda vero ſexquitertia. ſex </
s
>
</
p
>
</
div
>
</
div
>
</
div
>
</
text
>
</
echo
>