Vitruvius
,
De architectura libri decem
,
1567
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171 - 180
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514.01.113
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113
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TERTIVS.
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Vtriuſque generis proprium eſt, ut ſecundum collationem , comparationemq́; </
s
>
<
s
xml:id
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echoid-s7574
"
xml:space
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">res æquales, uel inæquales´ mutuo
<
lb
/>
dicantur, nam res omnis alteri comparata uel æqualis illi eſſe cernitur, uel inæqualis, nam uel tanta est, quan-
<
lb
/>
ta eſt alia, uel maior, uel minor. </
s
>
<
s
xml:id
="
echoid-s7575
"
xml:space
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preserve
">Proportio igitur eſt ex earum numero, quæ per ſe nec ſunt, nec intelliguntur ,
<
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ſed id, quod ſunt ad aliud referuntur. </
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>
<
s
xml:id
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xml:space
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">& </
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>
<
s
xml:id
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echoid-s7577
"
xml:space
="
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">ex proportionibus aliæ inter res æquales, aliæ inter inæquales cadent.
<
lb
/>
</
s
>
<
s
xml:id
="
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"
xml:space
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preserve
">Vnde proportio est determinata rerum duarum, quæ ſub eodem genere continentur, habitudo. </
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>
<
s
xml:id
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echoid-s7579
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xml:space
="
preserve
">Determinata in-
<
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quam non ad nos relata, neque in ſe certa , ſed huiuſmodi , ut alia eſſe non poßit. </
s
>
<
s
xml:id
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echoid-s7580
"
xml:space
="
preserve
">Sub eodem autem genere,
<
lb
/>
quia nemo lineam maiorem, uel minorem, uel æqualem ſuperficiei dixerit, ſed alteri lineæ, nec ſuperficiem cor-
<
lb
/>
pori, ſed alteri plano comparabit. </
s
>
<
s
xml:id
="
echoid-s7581
"
xml:space
="
preserve
">cum ergo proportio in quantitate ſit , duplexq́; </
s
>
<
s
xml:id
="
echoid-s7582
"
xml:space
="
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">ſit quantitatis genus ( ut di-
<
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/>
xi ) certe tam in menſuris, quàm in numeris, ac etiam in his rebus, quæ ex menſuris, & </
s
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<
s
xml:id
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xml:space
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">numeris componun-
<
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<
note
position
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left
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xlink:label
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note-514.01.113-01
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xlink:href
="
note-514.01.113-01a
"
xml:space
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">10</
note
>
tur, inuenietur, Quæ igitur ad menſuras proportio pertinet, Geometrarum propria eſt; </
s
>
<
s
xml:id
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echoid-s7584
"
xml:space
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">quæ ad numeros, eorum
<
lb
/>
qui numer andi ſcientiam profitentur; </
s
>
<
s
xml:id
="
echoid-s7585
"
xml:space
="
preserve
">quæ ad utrumque genus Muſicis accommodatur, harmonicaq́; </
s
>
<
s
xml:id
="
echoid-s7586
"
xml:space
="
preserve
">nuncupa-
<
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tur. </
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>
<
s
xml:id
="
echoid-s7587
"
xml:space
="
preserve
">Nos modo de ſimplicibus illis dicemus , ſed libro quinto de permixta illa ex utriſque generibus , cum de
<
lb
/>
muſica ratione tractabimus copioſe diſſeremus. </
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>
<
s
xml:id
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xml:space
="
preserve
">Magnitudines, & </
s
>
<
s
xml:id
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echoid-s7589
"
xml:space
="
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">numeri inter ſe comparati, aut pares,
<
lb
/>
æqualesq́; </
s
>
<
s
xml:id
="
echoid-s7590
"
xml:space
="
preserve
">ſunt, aut ſe ſe mutuo ſuccedunt . </
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>
<
s
xml:id
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xml:space
="
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">Pares, æqualesq́; </
s
>
<
s
xml:id
="
echoid-s7592
"
xml:space
="
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">intelligo, cum nihil amplius uni ſit, quàm alteri.
<
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/>
</
s
>
<
s
xml:id
="
echoid-s7593
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xml:space
="
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">impares, & </
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>
<
s
xml:id
="
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"
xml:space
="
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">ſe ſe eccedentes, cum quid maius, aut minus in eis eſſe dignoſcitur. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Aequalium, pariumq́; </
s
>
<
s
xml:id
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echoid-s7596
"
xml:space
="
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">nullum
<
lb
/>
aliud genus eſt, nam cum res inter ſe æquabili ratione conferuntur , unius formæ comparatio inter eas cadit. </
s
>
<
s
xml:id
="
echoid-s7597
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xml:space
="
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">
<
lb
/>
Inæqualium plura ſunt genera, ac totidem, quotres una altera maior eſſe potest . </
s
>
<
s
xml:id
="
echoid-s7598
"
xml:space
="
preserve
">maiorem intelligo contine-
<
lb
/>
re minorem integram, uel amplius aliquid. </
s
>
<
s
xml:id
="
echoid-s7599
"
xml:space
="
preserve
">Inæqualitatis igitur proportio, aut ſimplex est, aut compoſita. </
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>
<
s
xml:id
="
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"
xml:space
="
preserve
">ſim
<
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/>
plex es
<
unsure
/>
t, cum quod maius est, minus continet uel pluribus uicibus. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">ſi pluribus uicibus ita , ut ex inte-
<
lb
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<
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left
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xlink:label
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note-514.01.113-02
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xlink:href
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note-514.01.113-02a
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xml:space
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">20</
note
>
gro nihil ſuperſit, tunc proportio ea conficitur, quæ multiplex nominatur. </
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<
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xml:id
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xml:space
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">Ita quatuor continet duo bis . </
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>
<
s
xml:id
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xml:space
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">Sex
<
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/>
continet duo ter. </
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>
<
s
xml:id
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xml:space
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">Octo duo quater. </
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>
<
s
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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">nihil amplius. </
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>
<
s
xml:id
="
echoid-s7607
"
xml:space
="
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">ſi una uice quod plus eſt continet minus, uel unã inſuper præ-
<
lb
/>
ter integram partem eius continet, uel plures. </
s
>
<
s
xml:id
="
echoid-s7608
"
xml:space
="
preserve
">ſi unam, uel illa est dimidium, uel tertia pars, uel quarta, & </
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>
<
s
xml:id
="
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"
xml:space
="
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">ita
<
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/>
in infinitum. </
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>
<
s
xml:id
="
echoid-s7610
"
xml:space
="
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">Atque hæc proportio ſuperparticularis nominatur. </
s
>
<
s
xml:id
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"
xml:space
="
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">ſeſquialtera ſi dimidium, ſeſquitertia, ſi ter-
<
lb
/>
tiam partem, ſeſquiquarta ſi quartam continet, & </
s
>
<
s
xml:id
="
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"
xml:space
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">ita in infinitum. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">Si vero maius continet minus integre , & </
s
>
<
s
xml:id
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"
xml:space
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">
<
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/>
aliquas eius partes, tunc ſuperpartiens nominatur illarum comparatio . </
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>
<
s
xml:id
="
echoid-s7615
"
xml:space
="
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">aliquas autem partes intelligo, non
<
lb
/>
ut aiunt numerantes , ſed ex illis conſtantes. </
s
>
<
s
xml:id
="
echoid-s7616
"
xml:space
="
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">Pars enim numerans dicitur ea, quæ integram rem certis uicibus
<
lb
/>
deſumpta conſtituit. </
s
>
<
s
xml:id
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echoid-s7617
"
xml:space
="
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">Cuiuſmodi duo pars est numerans ſenarium, nam ter duo conſtituunt ſex . </
s
>
<
s
xml:id
="
echoid-s7618
"
xml:space
="
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">quatuor item
<
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numerat. </
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>
<
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xml:space
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">12. </
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>
<
s
xml:id
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xml:space
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">nam ter quatuor duodecim efficiunt . </
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>
<
s
xml:id
="
echoid-s7621
"
xml:space
="
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">huiuſmodi partes quotas etiam appellare ſolent . </
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>
<
s
xml:id
="
echoid-s7622
"
xml:space
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">ſuper-
<
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partiens igitur ratio eſt quotiens maius, minus ſemel continet , & </
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<
s
xml:id
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xml:space
="
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">partem ipſius minoris non quotam, & </
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>
<
s
xml:id
="
echoid-s7624
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xml:space
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">nu-
<
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<
note
position
="
left
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xlink:label
="
note-514.01.113-03
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xlink:href
="
note-514.01.113-03a
"
xml:space
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">30</
note
>
merantem, ſed ex illis quotis, & </
s
>
<
s
xml:id
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xml:space
="
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">numerantibus prouenientem. </
s
>
<
s
xml:id
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xml:space
="
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">ſic quinque continet tres ſemel , & </
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>
<
s
xml:id
="
echoid-s7627
"
xml:space
="
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">duas eius
<
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/>
partes, unde appellatur proportio bipartiens tertias, ſeptem continet quatuor, & </
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>
<
s
xml:id
="
echoid-s7628
"
xml:space
="
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">tres quartas, & </
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>
<
s
xml:id
="
echoid-s7629
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xml:space
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">appellatur
<
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/>
proportio tripartiens quartas, & </
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>
<
s
xml:id
="
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xml:space
="
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">nouem continet quinque, & </
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>
<
s
xml:id
="
echoid-s7631
"
xml:space
="
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">quatuor eius quintas, & </
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>
<
s
xml:id
="
echoid-s7632
"
xml:space
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">appellatur proportio
<
lb
/>
quadripartiens quintas, atque ita in infinitum. </
s
>
<
s
xml:id
="
echoid-s7633
"
xml:space
="
preserve
">Simplices igitur comparationes hæ ſunt, multiplex, ſupraparti-
<
lb
/>
cularis, ſuprapartiens. </
s
>
<
s
xml:id
="
echoid-s7634
"
xml:space
="
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">Compoſitæ autem vim, & </
s
>
<
s
xml:id
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echoid-s7635
"
xml:space
="
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">naturam ſimplicium tenent , nam vel multiplex eſt, & </
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>
<
s
xml:id
="
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xml:space
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">ſu-
<
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/>
praparticularis, vel multiplex, & </
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>
<
s
xml:id
="
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"
xml:space
="
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">ſuprapartiens. </
s
>
<
s
xml:id
="
echoid-s7638
"
xml:space
="
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">quatenus multiplex maius continebit rem integram pluri-
<
lb
/>
bus uicibus; </
s
>
<
s
xml:id
="
echoid-s7639
"
xml:space
="
preserve
">quatenus ſupra particularis, continebit amplius, aliquam eius partem, quotam ſcilicet , & </
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>
<
s
xml:id
="
echoid-s7640
"
xml:space
="
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">nu-
<
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/>
merantem; </
s
>
<
s
xml:id
="
echoid-s7641
"
xml:space
="
preserve
">quatenus ſupraparticularis continebit amplius partem non quotam , & </
s
>
<
s
xml:id
="
echoid-s7642
"
xml:space
="
preserve
">numerantem, ſed ex quo-
<
lb
/>
tis, & </
s
>
<
s
xml:id
="
echoid-s7643
"
xml:space
="
preserve
">numerantibus prouenientem. </
s
>
<
s
xml:id
="
echoid-s7644
"
xml:space
="
preserve
">multiplex ſupraparticularis eſt inter decem, & </
s
>
<
s
xml:id
="
echoid-s7645
"
xml:space
="
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">quatuor, nam decem con-
<
lb
/>
tinet quatuor bis, unde multiplicis nomen ſortitur , & </
s
>
<
s
xml:id
="
echoid-s7646
"
xml:space
="
preserve
">dimidium , vnde ſeſquialtera nuncupatur, quare du-
<
lb
/>
plex ſeſquialtera dicetur. </
s
>
<
s
xml:id
="
echoid-s7647
"
xml:space
="
preserve
">multiplex ſuprapartiens est inter octo, & </
s
>
<
s
xml:id
="
echoid-s7648
"
xml:space
="
preserve
">tria, nam octo continet tria bis, unde mul-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.113-04
"
xlink:href
="
note-514.01.113-04a
"
xml:space
="
preserve
">40</
note
>
tiplicis ratio naſcitur, & </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">duo, unde ſuprapartiens nominatur, atque ita inter octo & </
s
>
<
s
xml:id
="
echoid-s7650
"
xml:space
="
preserve
">tria erit proportio dupla
<
lb
/>
bipartiens tertias . </
s
>
<
s
xml:id
="
echoid-s7651
"
xml:space
="
preserve
">Oriuntur autem proportiones e numeris certo quodam ordine diſpoſitis , & </
s
>
<
s
xml:id
="
echoid-s7652
"
xml:space
="
preserve
">collocatis. </
s
>
<
s
xml:id
="
echoid-s7653
"
xml:space
="
preserve
">ac
<
lb
/>
primum ſupraparticulares rationes ita fient. </
s
>
<
s
xml:id
="
echoid-s7654
"
xml:space
="
preserve
">Diſpone a ternario numeros ſe ſe ut natura fert, conſequentes. </
s
>
<
s
xml:id
="
echoid-s7655
"
xml:space
="
preserve
">at-
<
lb
/>
que eo in primo , & </
s
>
<
s
xml:id
="
echoid-s7656
"
xml:space
="
preserve
">ſuperiori ordine collocabis , ſubſcribes eoſdem a binario natura ſe conſequentes quem-
<
lb
/>
admodum uides.</
s
>
<
s
xml:id
="
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"
xml:space
="
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"/>
</
p
>
<
p
style
="
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">
<
s
xml:id
="
echoid-s7658
"
xml:space
="
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">3.</
s
>
<
s
xml:id
="
echoid-s7659
"
xml:space
="
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">#4.</
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>
<
s
xml:id
="
echoid-s7660
"
xml:space
="
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">#5.</
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>
<
s
xml:id
="
echoid-s7661
"
xml:space
="
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">#6.</
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>
<
s
xml:id
="
echoid-s7662
"
xml:space
="
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">#7.</
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>
<
s
xml:id
="
echoid-s7663
"
xml:space
="
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">#8.</
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>
<
s
xml:id
="
echoid-s7664
"
xml:space
="
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">#9.</
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>
<
s
xml:id
="
echoid-s7665
"
xml:space
="
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">#10.</
s
>
<
s
xml:id
="
echoid-s7666
"
xml:space
="
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">#11.</
s
>
<
s
xml:id
="
echoid-s7667
"
xml:space
="
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">#12.</
s
>
<
s
xml:id
="
echoid-s7668
"
xml:space
="
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">#hi ſunt numeri a quibus incipit comparatio.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s7669
"
xml:space
="
preserve
">2.</
s
>
<
s
xml:id
="
echoid-s7670
"
xml:space
="
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">#3.</
s
>
<
s
xml:id
="
echoid-s7671
"
xml:space
="
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">#4.</
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>
<
s
xml:id
="
echoid-s7672
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xml:space
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<
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<
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xml:space
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">#7.</
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<
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echoid-s7675
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xml:space
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">#8.</
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<
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xml:id
="
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"
xml:space
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">#9.</
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>
<
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xml:id
="
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"
xml:space
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">#10.</
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>
<
s
xml:id
="
echoid-s7678
"
xml:space
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">#11.</
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>
<
s
xml:id
="
echoid-s7679
"
xml:space
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">#hi ſunt numeri in quos deſinit comparatio , & </
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>
<
s
xml:id
="
echoid-s7680
"
xml:space
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">ad
<
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/>
quos ſuperiores ſe habent in ſupraparticulari proportione, nam. </
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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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">ad. </
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>
<
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xml:id
="
echoid-s7683
"
xml:space
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">2. </
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<
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xml:id
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echoid-s7684
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xml:space
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">est ſeſquialtera. </
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>
<
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xml:id
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echoid-s7685
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xml:space
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">4. </
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<
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xml:id
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xml:space
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">ad. </
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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:space
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ſeſquitertia. </
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<
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xml:id
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xml:space
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">5. </
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>
<
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xml:id
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xml:space
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">ad. </
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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:id
="
echoid-s7692
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xml:space
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">ſeſquiquarta, & </
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>
<
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xml:id
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echoid-s7693
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xml:space
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">ita in infinitum. </
s
>
<
s
xml:id
="
echoid-s7694
"
xml:space
="
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">Superpartiens oritur imparibus numeris ſua ſe-
<
lb
/>
rie ſuprapoſitis, modo a quinario incipias, & </
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>
<
s
xml:id
="
echoid-s7695
"
xml:space
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">a numeris naturali ordine, ſi conſequentibus a ternario in
<
lb
/>
<
note
position
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left
"
xlink:label
="
note-514.01.113-05
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xlink:href
="
note-514.01.113-05a
"
xml:space
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">50</
note
>
hunc modum.</
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>
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xml:space
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"/>
</
p
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<
p
style
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">
<
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xml:id
="
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xml:space
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<
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xml:id
="
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"
xml:space
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">#7.</
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>
<
s
xml:id
="
echoid-s7699
"
xml:space
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">#9.</
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>
<
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xml:id
="
echoid-s7700
"
xml:space
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">#11.</
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>
<
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xml:id
="
echoid-s7701
"
xml:space
="
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">#13.</
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>
<
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xml:id
="
echoid-s7702
"
xml:space
="
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">#15.</
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>
<
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xml:id
="
echoid-s7703
"
xml:space
="
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">#17.</
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>
<
s
xml:id
="
echoid-s7704
"
xml:space
="
preserve
">#19.</
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>
<
s
xml:id
="
echoid-s7705
"
xml:space
="
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">#21.</
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>
<
s
xml:id
="
echoid-s7706
"
xml:space
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">#numeri a quibus incipit comparatio.
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lb
/>
</
s
>
<
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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
="
echoid-s7708
"
xml:space
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">#4.</
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<
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xml:id
="
echoid-s7709
"
xml:space
="
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">#5.</
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>
<
s
xml:id
="
echoid-s7710
"
xml:space
="
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">#6.</
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>
<
s
xml:id
="
echoid-s7711
"
xml:space
="
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">#7.</
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>
<
s
xml:id
="
echoid-s7712
"
xml:space
="
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">#8.</
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>
<
s
xml:id
="
echoid-s7713
"
xml:space
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">#9.</
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>
<
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xml:id
="
echoid-s7714
"
xml:space
="
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">#10.</
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>
<
s
xml:id
="
echoid-s7715
"
xml:space
="
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">#11.</
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>
<
s
xml:id
="
echoid-s7716
"
xml:space
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">#numeri in quos deſinit comparatio. </
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>
<
s
xml:id
="
echoid-s7717
"
xml:space
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">atque ita. </
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>
<
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xml:id
="
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"
xml:space
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">5. </
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>
<
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<
lb
/>
ad 3. </
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>
<
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xml:id
="
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"
xml:space
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">bipartiens tertias. </
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>
<
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xml:id
="
echoid-s7721
"
xml:space
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">7. </
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>
<
s
xml:id
="
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"
xml:space
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">ad 4. </
s
>
<
s
xml:id
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"
xml:space
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">tripartiens quartas. </
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>
<
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xml:id
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echoid-s7724
"
xml:space
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">9. </
s
>
<
s
xml:id
="
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xml:space
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">ad. </
s
>
<
s
xml:id
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echoid-s7726
"
xml:space
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">5, quadripartiens quintas, & </
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>
<
s
xml:id
="
echoid-s7727
"
xml:space
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">ſic de
<
lb
/>
ſingulis. </
s
>
<
s
xml:id
="
echoid-s7728
"
xml:space
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preserve
">quod ſi ſuperiorem imparium numerorum ordinem a quinario orſus adſcribes, inferius autem a bina-
<
lb
/>
rio natur ali ordine præcedentes numeros ſubſcribes , differentias multiplicis ſupraparticularis , ſub ratione
<
lb
/>
dupla, content as conficies.</
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>
<
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xml:id
="
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"
xml:space
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"/>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s7730
"
xml:space
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">5.</
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>
<
s
xml:id
="
echoid-s7731
"
xml:space
="
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">#7.</
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>
<
s
xml:id
="
echoid-s7732
"
xml:space
="
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">#9.</
s
>
<
s
xml:id
="
echoid-s7733
"
xml:space
="
preserve
">#11.</
s
>
<
s
xml:id
="
echoid-s7734
"
xml:space
="
preserve
">#13.</
s
>
<
s
xml:id
="
echoid-s7735
"
xml:space
="
preserve
">#15.</
s
>
<
s
xml:id
="
echoid-s7736
"
xml:space
="
preserve
">#17.</
s
>
<
s
xml:id
="
echoid-s7737
"
xml:space
="
preserve
">#19.</
s
>
<
s
xml:id
="
echoid-s7738
"
xml:space
="
preserve
">#21.</
s
>
<
s
xml:id
="
echoid-s7739
"
xml:space
="
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"/>
</
p
>
<
p
style
="
it
">
<
s
xml:id
="
echoid-s7740
"
xml:space
="
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">2.</
s
>
<
s
xml:id
="
echoid-s7741
"
xml:space
="
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">#3.</
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>
<
s
xml:id
="
echoid-s7742
"
xml:space
="
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">#4.</
s
>
<
s
xml:id
="
echoid-s7743
"
xml:space
="
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">#5.</
s
>
<
s
xml:id
="
echoid-s7744
"
xml:space
="
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">#6.</
s
>
<
s
xml:id
="
echoid-s7745
"
xml:space
="
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">#7.</
s
>
<
s
xml:id
="
echoid-s7746
"
xml:space
="
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">#8.</
s
>
<
s
xml:id
="
echoid-s7747
"
xml:space
="
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">#9.</
s
>
<
s
xml:id
="
echoid-s7748
"
xml:space
="
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">#10. </
s
>
<
s
xml:id
="
echoid-s7749
"
xml:space
="
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">Siaſeptenario incœperis per impares procedendo,
<
lb
/>
& </
s
>
<
s
xml:id
="
echoid-s7750
"
xml:space
="
preserve
">eandem in inferiori ſerie ſucceſſionem a binario ſeruaueris, multiplicis ſuperparticularis formas , ſub ratio-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.113-06
"
xlink:href
="
note-514.01.113-06a
"
xml:space
="
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">60</
note
>
ne tripla constitues.</
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>
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s
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"
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"/>
</
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</
div
>
</
div
>
</
text
>
</
echo
>