Vitruvius
,
De architectura libri decem
,
1567
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514.01.115
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TERTIVS.
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tatis ad alteram, ſed unius proportionis ad alteram, uerbi gratia, ſi dixeris proportionem illam, quæ inter qua-
<
lb
/>
tuor cadit, & </
s
>
<
s
xml:id
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xml:space
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">duo, ſimilem eſſe illi proportioni , quæ cadit inter octo , & </
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>
<
s
xml:id
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xml:space
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">quatuor, nam utraque dupla eſt,
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lb
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ideo duplæ omnes, triplæ item, & </
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>
<
s
xml:id
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"
xml:space
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">quadruplæ, reliquæq́; </
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>
<
s
xml:id
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"
xml:space
="
preserve
">ſiue ſint unius generis, cuiuſmodi ſunt , quæ cadunt
<
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inter lineam, & </
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<
s
xml:id
="
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xml:space
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">lineam, inter planum, & </
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<
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xml:id
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xml:space
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">planum, inter corpus, & </
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>
<
s
xml:id
="
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"
xml:space
="
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">corpus, ſiue ſint generum diuerſorum, qua
<
lb
/>
les ſunt quæ cadunt inter lineam, & </
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>
<
s
xml:id
="
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xml:space
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">planum, inter planum, & </
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>
<
s
xml:id
="
echoid-s7849
"
xml:space
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">corpus; </
s
>
<
s
xml:id
="
echoid-s7850
"
xml:space
="
preserve
">proportionales ſunt, hoc eſt poſſunt
<
lb
/>
proportionum ſimilitudine inter ſe conferri, & </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">conſequenter ſimiles habentur, & </
s
>
<
s
xml:id
="
echoid-s7852
"
xml:space
="
preserve
">ubi eſt proportionum col-
<
lb
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latio, quam proportionalitatem uocant, ibi neceſſario eſt proportio. </
s
>
<
s
xml:id
="
echoid-s7853
"
xml:space
="
preserve
">quoniam proportionalitas, ut ita dicam,
<
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/>
nil aliud eſt, quàm proportionum reſponſus , ſed none contrario, nam inter quatuor, & </
s
>
<
s
xml:id
="
echoid-s7854
"
xml:space
="
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">duo proportio cadit,
<
lb
/>
ſed non proportionalitas. </
s
>
<
s
xml:id
="
echoid-s7855
"
xml:space
="
preserve
">In his igitur proportionum comparationibus, omne artis ſecretum ponitur. </
s
>
<
s
xml:id
="
echoid-s7856
"
xml:space
="
preserve
">In com-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.115-01
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xlink:href
="
note-514.01.115-01a
"
xml:space
="
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">10</
note
>
ponendis igitur proportionibus, illud obſeruandum est, an ambæ rationes ſint ſimiles, an diſſimiles, hoc est an
<
lb
/>
ſint ambæ ex his, quæ a maiori in minus deſinunt, an ex his, quæ a minori in maius terminantur . </
s
>
<
s
xml:id
="
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"
xml:space
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">An altera
<
lb
/>
unius, altera ſit alterius generis, nam hoc plurimum intereſt. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">ut ex regulis dignoſcemus. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">In componendis
<
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/>
etiam proportionibus duo ſe ſe offerunt conſideranda, primum est denominatio compoſitæ proportionis. </
s
>
<
s
xml:id
="
echoid-s7860
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xml:space
="
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">Al-
<
lb
/>
terum est numerorum collectio ſub eadem producta ratione conſtitutorum. </
s
>
<
s
xml:id
="
echoid-s7861
"
xml:space
="
preserve
">Primum abſoluitur hoc modo in
<
lb
/>
his, quæ ſunt ſimilium generum, ac in multiplicibus. </
s
>
<
s
xml:id
="
echoid-s7862
"
xml:space
="
preserve
">Ducas unius in alterius denominatorem, orietur compo-
<
lb
/>
ſitæ rationis denominatio. </
s
>
<
s
xml:id
="
echoid-s7863
"
xml:space
="
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">Verbigratia. </
s
>
<
s
xml:id
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xml:space
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">inter 12. </
s
>
<
s
xml:id
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"
xml:space
="
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">& </
s
>
<
s
xml:id
="
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xml:space
="
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">quatuor tripla eſt proportio, inter quatuor, & </
s
>
<
s
xml:id
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"
xml:space
="
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">octo
<
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/>
dupla. </
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>
<
s
xml:id
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"
xml:space
="
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">duc tria in duo, fiet ſex. </
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>
<
s
xml:id
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xml:space
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">ex tripla igitur, & </
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>
<
s
xml:id
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xml:space
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">dupla fiet ſextupla proportio. </
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>
<
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xml:id
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xml:space
="
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">Secundum abſoluitur, & </
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<
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xml:id
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xml:space
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<
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/>
confirmatur his numeris , duc 12. </
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>
<
s
xml:id
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xml:space
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">in octo fiet 96. </
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>
<
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xml:id
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xml:space
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">& </
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>
<
s
xml:id
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xml:space
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">4. </
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<
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xml:space
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">in. </
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>
<
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xml:space
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">4. </
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>
<
s
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"
xml:space
="
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">fient 16. </
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>
<
s
xml:id
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"
xml:space
="
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">atque ſi 96. </
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>
<
s
xml:id
="
echoid-s7880
"
xml:space
="
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">ad ſex-
<
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/>
decim comparaueris, cernes ſextuplam oriri proportionem, quã ex proępoſitis denominationibus collectã cernis.
<
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/>
</
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<
s
xml:id
="
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<
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position
="
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xlink:label
="
note-514.01.115-02
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xlink:href
="
note-514.01.115-02a
"
xml:space
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">20</
note
>
<
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it
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position
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xlink:label
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note-514.01.115-03
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xlink:href
="
note-514.01.115-03a
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/>
Tripla # 12--4
<
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/>
Dupla # 8--4
<
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/>
Sextupla # 96--16
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</
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>
Similis ratio in ſupraparticulari proportione obſeruatur. </
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<
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xml:space
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">Eſto 6. </
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<
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">ad 4. </
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<
s
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xml:space
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">ſeſ-
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quialtera, &</
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>
<
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xml:space
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">. 8. </
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<
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<
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xml:space
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">6. </
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<
s
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xml:space
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">ſeſquitertia. </
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>
<
s
xml:id
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xml:space
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">duc unum ſemis denominatorem ſeſqui-
<
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alteræ in unum & </
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>
<
s
xml:id
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xml:space
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">tertiam denominatorem ſeſquitertiæ. </
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>
<
s
xml:id
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xml:space
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">fient 2. </
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>
<
s
xml:id
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echoid-s7892
"
xml:space
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">a quibus pro-
<
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/>
portio dupla nominatur. </
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>
<
s
xml:id
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xml:space
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">Ex ſeſquialtera igitur, & </
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>
<
s
xml:id
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xml:space
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">ſeſquitertia oritur dupla, quod
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& </
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>
<
s
xml:id
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xml:space
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">in numeris cernes hoc modo. </
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<
s
xml:id
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xml:space
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">Duc 6 in 8 fient 48. </
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<
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xml:id
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xml:space
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">& </
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<
s
xml:id
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xml:space
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">4 in 6 fient 24. </
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>
<
s
xml:id
="
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xml:space
="
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">quo facto cernes duplam
<
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inter eos numeros cadere comparationem. </
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>
<
s
xml:id
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xml:space
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">Exemplum quoque in ſuprapartientibus ponam. </
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>
<
s
xml:id
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echoid-s7901
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xml:space
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">eſto bipartiens
<
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tertias ratio, qualis eſt 5 ad 3 addenda trip artienti quartas, qualis eſt 7 ad 4. </
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>
<
s
xml:id
="
echoid-s7902
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xml:space
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">denominator bipartiẽs
<
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tertias eſt 1. </
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>
<
s
xml:id
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xml:space
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">& </
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>
<
s
xml:id
="
echoid-s7904
"
xml:space
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">duæ tertiæ. </
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>
<
s
xml:id
="
echoid-s7905
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xml:space
="
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">& </
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>
<
s
xml:id
="
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"
xml:space
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">tripartiens quartas eſt unum & </
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>
<
s
xml:id
="
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xml:space
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">tres quartæ. </
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>
<
s
xml:id
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echoid-s7908
"
xml:space
="
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">ducito inuicem huiuſmodi de-
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nomin. </
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<
s
xml:id
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xml:space
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">itores, fient 2 & </
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>
<
s
xml:id
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xml:space
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">undecim duodecimæ, a quibus dupla undecupartiens duodecimas nominatur. </
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>
<
s
xml:id
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xml:space
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">quod
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& </
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>
<
s
xml:id
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echoid-s7912
"
xml:space
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">numeri hinc inde prouenientes oſtendunt. </
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>
<
s
xml:id
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xml:space
="
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">ducendo 5 in 7. </
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>
<
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xml:id
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xml:space
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">qui ſunt antecedentes, & </
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>
<
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xml:id
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xml:space
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">3. </
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<
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xml:id
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xml:space
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">in 4. </
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>
<
s
xml:id
="
echoid-s7917
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xml:space
="
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">qui
<
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/>
<
note
position
="
left
"
xlink:label
="
note-514.01.115-04
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xlink:href
="
note-514.01.115-04a
"
xml:space
="
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">30</
note
>
ſunt ſubſequentes, nam ex illis fient 35. </
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>
<
s
xml:id
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xml:space
="
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">ex his 12. </
s
>
<
s
xml:id
="
echoid-s7919
"
xml:space
="
preserve
">inter quos ſupradicta proportio cadit.</
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>
<
s
xml:id
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xml:space
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"/>
</
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<
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position
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<
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/>
Bipartiens tertias # 5--3
<
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/>
Tripartiens quartas # 7--4
<
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/>
Dupla undecupartiens duodecimas. # 35--12
<
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/>
</
note
>
<
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style
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<
s
xml:id
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xml:space
="
preserve
">Quod ſi diuerſorum generum proportiones interceßerint, ita
<
lb
/>
quod in una fiat comparatio maioris ad minus, in altera mi-
<
lb
/>
noris ad maius: </
s
>
<
s
xml:id
="
echoid-s7922
"
xml:space
="
preserve
">Tunc id obſeruandum eſt, vt partitione uta-
<
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/>
mur. </
s
>
<
s
xml:id
="
echoid-s7923
"
xml:space
="
preserve
">ita enim compoſitio duarum emerget . </
s
>
<
s
xml:id
="
echoid-s7924
"
xml:space
="
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">partitur enim
<
lb
/>
maior per minorem hunc in modum. </
s
>
<
s
xml:id
="
echoid-s7925
"
xml:space
="
preserve
">ſubdupla ratio a binario nomen capit, quemadmodum & </
s
>
<
s
xml:id
="
echoid-s7926
"
xml:space
="
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">dupla; </
s
>
<
s
xml:id
="
echoid-s7927
"
xml:space
="
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">ſeſquial-
<
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/>
tera autem ab 1 & </
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>
<
s
xml:id
="
echoid-s7928
"
xml:space
="
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">duodecima, unum igitur & </
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>
<
s
xml:id
="
echoid-s7929
"
xml:space
="
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">dimidium minus es
<
unsure
/>
t quàm duo. </
s
>
<
s
xml:id
="
echoid-s7930
"
xml:space
="
preserve
">partire igitur duo per unum
<
lb
/>
& </
s
>
<
s
xml:id
="
echoid-s7931
"
xml:space
="
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">dimidium. </
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>
<
s
xml:id
="
echoid-s7932
"
xml:space
="
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">edetur 1. </
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>
<
s
xml:id
="
echoid-s7933
"
xml:space
="
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">& </
s
>
<
s
xml:id
="
echoid-s7934
"
xml:space
="
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">tertia. </
s
>
<
s
xml:id
="
echoid-s7935
"
xml:space
="
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">Ex prædictis ergo rationibus, ratio emerget ſubſeſquitertia , nam diui-
<
lb
/>
denda est ea, quæ eſt minoris inæqualitatis, & </
s
>
<
s
xml:id
="
echoid-s7936
"
xml:space
="
preserve
">quæ prouenit inde ratio diuidendam rationem ſequi ſolet, & </
s
>
<
s
xml:id
="
echoid-s7937
"
xml:space
="
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">
<
lb
/>
experire hoc per numeros 2 4. </
s
>
<
s
xml:id
="
echoid-s7938
"
xml:space
="
preserve
">inter quos eſt ſubdupla proportio, & </
s
>
<
s
xml:id
="
echoid-s7939
"
xml:space
="
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">6. </
s
>
<
s
xml:id
="
echoid-s7940
"
xml:space
="
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">& </
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>
<
s
xml:id
="
echoid-s7941
"
xml:space
="
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">4. </
s
>
<
s
xml:id
="
echoid-s7942
"
xml:space
="
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">inter quos eſt ſeſquialte-
<
lb
/>
ra proportio. </
s
>
<
s
xml:id
="
echoid-s7943
"
xml:space
="
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">duc 2. </
s
>
<
s
xml:id
="
echoid-s7944
"
xml:space
="
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">in 6. </
s
>
<
s
xml:id
="
echoid-s7945
"
xml:space
="
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">fient 12. </
s
>
<
s
xml:id
="
echoid-s7946
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s7947
"
xml:space
="
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">4 in 4. </
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>
<
s
xml:id
="
echoid-s7948
"
xml:space
="
preserve
">fient 16. </
s
>
<
s
xml:id
="
echoid-s7949
"
xml:space
="
preserve
">comparato 12 ad 16. </
s
>
<
s
xml:id
="
echoid-s7950
"
xml:space
="
preserve
">uidebis ſubſeſ-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.115-06
"
xlink:href
="
note-514.01.115-06a
"
xml:space
="
preserve
">40</
note
>
quitertiam proporportione procreari.</
s
>
<
s
xml:id
="
echoid-s7951
"
xml:space
="
preserve
"/>
</
p
>
<
note
style
="
it
"
position
="
right
"
xml:space
="
preserve
">
<
lb
/>
ſubdupla. # 2--4
<
lb
/>
ſeſquialtera. # 6--4
<
lb
/>
ſubſeſquitertia. # 12--16
<
lb
/>
</
note
>
<
p
style
="
it
">
<
s
xml:id
="
echoid-s7952
"
xml:space
="
preserve
">Exemplum hoc ſatis eße potest pro omnibus diuerſorum generum comparatio-
<
lb
/>
nibus. </
s
>
<
s
xml:id
="
echoid-s7953
"
xml:space
="
preserve
">Cæterum & </
s
>
<
s
xml:id
="
echoid-s7954
"
xml:space
="
preserve
">illud innoteſcet, qua nam ratione plures quàm duæ rationes
<
lb
/>
inuicem componantur, nam quæ ex duabus prioribus compoſitis effecta fue-
<
lb
/>
rit, ea cum tertia eodem modo componenda es
<
unsure
/>
t, quo ſupra diximus. </
s
>
<
s
xml:id
="
echoid-s7955
"
xml:space
="
preserve
">Ad me-
<
lb
/>
moriam uero reuocanda eſt fractorum, & </
s
>
<
s
xml:id
="
echoid-s7956
"
xml:space
="
preserve
">integrorum ductio, partitio, & </
s
>
<
s
xml:id
="
echoid-s7957
"
xml:space
="
preserve
">collectio, ut facile exerceri poſſi-
<
lb
/>
mus in hoc genere uniuerſo. </
s
>
<
s
xml:id
="
echoid-s7958
"
xml:space
="
preserve
">Ex prædictis illud colligere poßumus, quod cum ſimiles proportiones componuntur.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s7959
"
xml:space
="
preserve
">verbigratia. </
s
>
<
s
xml:id
="
echoid-s7960
"
xml:space
="
preserve
">maioris inæqualitatis ratio, ſimiliter minoris inæqualitatis, & </
s
>
<
s
xml:id
="
echoid-s7961
"
xml:space
="
preserve
">utraq; </
s
>
<
s
xml:id
="
echoid-s7962
"
xml:space
="
preserve
">maior generatur, quod ex
<
lb
/>
ſuperioribus exemplis innotuit. </
s
>
<
s
xml:id
="
echoid-s7963
"
xml:space
="
preserve
">Ex duabus quoque minoris inæqualitatis rationibus, ratio prouenit minoris
<
lb
/>
inæqualitatis, & </
s
>
<
s
xml:id
="
echoid-s7964
"
xml:space
="
preserve
">utraque minor erit. </
s
>
<
s
xml:id
="
echoid-s7965
"
xml:space
="
preserve
">Sed ex una maioris, & </
s
>
<
s
xml:id
="
echoid-s7966
"
xml:space
="
preserve
">altera minoris inæqualitatis comparatione, ra-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.115-08
"
xlink:href
="
note-514.01.115-08a
"
xml:space
="
preserve
">50</
note
>
tio huiuſmodi gignitur, cuiuſmodi ea eſt, quæ ab ampliori numero nomen capit. </
s
>
<
s
xml:id
="
echoid-s7967
"
xml:space
="
preserve
">Sola uero æqualitatis ratio
<
lb
/>
in ſe ipſam ducta, rationem procreat æqualitatis. </
s
>
<
s
xml:id
="
echoid-s7968
"
xml:space
="
preserve
">Atque hæc de componendis proportionibus dicta ſint. </
s
>
<
s
xml:id
="
echoid-s7969
"
xml:space
="
preserve
">Nunc
<
lb
/>
quemadmodum ratio a ratione ſubtrahatur, & </
s
>
<
s
xml:id
="
echoid-s7970
"
xml:space
="
preserve
">quæ reliqua ſit dignoſcatur, dicendum. </
s
>
<
s
xml:id
="
echoid-s7971
"
xml:space
="
preserve
">Si prius obſeruabimus
<
lb
/>
id partitione quadam effici, & </
s
>
<
s
xml:id
="
echoid-s7972
"
xml:space
="
preserve
">nunquam maiorem a minori, ſed minorem tantum a maiori demi poſſe. </
s
>
<
s
xml:id
="
echoid-s7973
"
xml:space
="
preserve
">quis
<
lb
/>
enim maius a minori demet? </
s
>
<
s
xml:id
="
echoid-s7974
"
xml:space
="
preserve
">cum nunquam maius in minori reperiri poſſit? </
s
>
<
s
xml:id
="
echoid-s7975
"
xml:space
="
preserve
">Eſficitur demptio, ac ſubtractio
<
lb
/>
duobus modis, primo ſi partiaris maioris rationis denominatorem numerum per denominatorem minoris, id
<
lb
/>
quod reliquum erit, procreatus proportionis deominator erit. </
s
>
<
s
xml:id
="
echoid-s7976
"
xml:space
="
preserve
">Secundo in numeris experitur, qui ex ſupra
<
lb
/>
datis rationibus proueniunt, verbigratia. </
s
>
<
s
xml:id
="
echoid-s7977
"
xml:space
="
preserve
">Constituantur numeri rationis maioris, quæ & </
s
>
<
s
xml:id
="
echoid-s7978
"
xml:space
="
preserve
">est ipſa diuidenda,
<
lb
/>
& </
s
>
<
s
xml:id
="
echoid-s7979
"
xml:space
="
preserve
">ponatur ſupra numeros ipſius minoris, per quam ratio maior partiri debet, inde ducatur numerus antece-
<
lb
/>
dens diuidendæ rationis, quæ & </
s
>
<
s
xml:id
="
echoid-s7980
"
xml:space
="
preserve
">maior est, per numerum conſequentem minoris, & </
s
>
<
s
xml:id
="
echoid-s7981
"
xml:space
="
preserve
">diuidentis, certe orietur
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.115-09
"
xlink:href
="
note-514.01.115-09a
"
xml:space
="
preserve
">60</
note
>
</
s
>
</
p
>
</
div
>
</
div
>
</
text
>
</
echo
>
