Vitruvius
,
De architectura libri decem
,
1567
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514.01.114
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114
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LIBER
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<
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xml:space
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">7#9#11#13#15#17#19#21</
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<
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<
s
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xml:space
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<
s
xml:id
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xml:space
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">#Atque ita ſubſequentes rationes additione conficies. </
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>
<
s
xml:id
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echoid-s7755
"
xml:space
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preserve
">Sed diſcri-
<
lb
/>
mina ſuperpartientis muliiplicis rationis ſub ratione dupla efficies, ſi ab octonario per tres unitates diſcedenles
<
lb
/>
numeros ordinatim ſuprapones , & </
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>
<
s
xml:id
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xml:space
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">a ternario ſeriem naturæ ſeruaueris in inferiori loco, hoc modo.</
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<
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xml:id
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xml:space
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</
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<
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<
s
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xml:space
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">8#11#14#17#20#23#26#29</
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>
</
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<
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<
s
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xml:space
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">3#4#5#6#7#8#9#10.</
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>
<
s
xml:id
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echoid-s7760
"
xml:space
="
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">#quod ſi multiplicem ſub ratione tripla, ac ſuprapartiente na-
<
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ſci videre cupis, ſubſcribe ſeriem numerorum ab undecim, ſe ſe quatuor unitatibus excedentium, & </
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>
<
s
xml:id
="
echoid-s7761
"
xml:space
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">natur alem
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ordinem a ternario ſubſcribe, hoc modo.</
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>
<
s
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echoid-s7762
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xml:space
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</
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<
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<
s
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xml:space
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">11#15#19#23#27#31#35#39</
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>
</
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>
<
p
>
<
s
xml:id
="
echoid-s7764
"
xml:space
="
preserve
">3#4#5#6#7#8#9#10.</
s
>
<
s
xml:id
="
echoid-s7765
"
xml:space
="
preserve
">#Atque hæc ſatis pro exemplo ſint poſita , Cætera per quàm
<
lb
/>
facilia studioſis eſſe poſſunt, & </
s
>
<
s
xml:id
="
echoid-s7766
"
xml:space
="
preserve
">quia comparationum termini immutarri poßunt, ita ut a minori incipere , & </
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<
s
xml:id
="
echoid-s7767
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xml:space
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">
<
lb
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<
note
position
="
left
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xlink:label
="
note-514.01.114-01
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xlink:href
="
note-514.01.114-01a
"
xml:space
="
preserve
">10</
note
>
in maius terminare poßumus: </
s
>
<
s
xml:id
="
echoid-s7768
"
xml:space
="
preserve
">ideo non ignorandum eandem eße comparationem mutatis terminis , eo ſolo di-
<
lb
/>
ſcrimine, ut ſi comparamus quod maius eſt, ad minus , ut 4. </
s
>
<
s
xml:id
="
echoid-s7769
"
xml:space
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">ad. </
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>
<
s
xml:id
="
echoid-s7770
"
xml:space
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">2. </
s
>
<
s
xml:id
="
echoid-s7771
"
xml:space
="
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">tunc utimur ea particula, ſupra.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s7772
"
xml:space
="
preserve
">ut cum dicimus ſupraparticularis. </
s
>
<
s
xml:id
="
echoid-s7773
"
xml:space
="
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">ſuprapartiens. </
s
>
<
s
xml:id
="
echoid-s7774
"
xml:space
="
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">cum uero comparamus quod minus eſt ad maius, ut 2. </
s
>
<
s
xml:id
="
echoid-s7775
"
xml:space
="
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">ad
<
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/>
4. </
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>
<
s
xml:id
="
echoid-s7776
"
xml:space
="
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">tunc utimur ea particula, ſub. </
s
>
<
s
xml:id
="
echoid-s7777
"
xml:space
="
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">atque in ſubmultiplicem, ſubparticularem, ſubpartientem nominamus. </
s
>
<
s
xml:id
="
echoid-s7778
"
xml:space
="
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">
<
lb
/>
Totidem igitur differentiæ, formæq́; </
s
>
<
s
xml:id
="
echoid-s7779
"
xml:space
="
preserve
">comparationum inueniuntur , cum minus ad maius comparabitur, quod
<
lb
/>
ſunt eæ, quibus maius minori confertur. </
s
>
<
s
xml:id
="
echoid-s7780
"
xml:space
="
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">Cæterum utſciamus & </
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>
<
s
xml:id
="
echoid-s7781
"
xml:space
="
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">uno obtutu cernamus a quibus numeris pro-
<
lb
/>
portiones nomina ſua ſortiantur, hanc regulam obſeruabimus . </
s
>
<
s
xml:id
="
echoid-s7782
"
xml:space
="
preserve
">Erit enim hoc utile in ædificijs , cæterisq́; </
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>
<
s
xml:id
="
echoid-s7783
"
xml:space
="
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">re-
<
lb
/>
bus arte, & </
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>
<
s
xml:id
="
echoid-s7784
"
xml:space
="
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">ratione perfectis, ad dignoſcendum quæ nam proportio maior , quæ minor, quæ amplior, quæue
<
lb
/>
ſtrictior habeatur, magni enim refert, qua quis utatur proportione. </
s
>
<
s
xml:id
="
echoid-s7785
"
xml:space
="
preserve
">Cum alia ædes magnificæ, alia priuatorũ
<
lb
/>
<
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position
="
left
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xlink:label
="
note-514.01.114-02
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xlink:href
="
note-514.01.114-02a
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xml:space
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">20</
note
>
domus, alia templa proportione perficiantur. </
s
>
<
s
xml:id
="
echoid-s7786
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xml:space
="
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">Quare hoc diligentiſſime obſeruandum Architectis cenſemus.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s7787
"
xml:space
="
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">Denominatio proportionum partitione dignoſcitur. </
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>
<
s
xml:id
="
echoid-s7788
"
xml:space
="
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">Partitio eſt experientia quædam, qua ſcitur , quot uicibus
<
lb
/>
numerus minor , in maiori inſit, & </
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>
<
s
xml:id
="
echoid-s7789
"
xml:space
="
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">quid inde ſuperſit. </
s
>
<
s
xml:id
="
echoid-s7790
"
xml:space
="
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">ex partitione igitur, & </
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>
<
s
xml:id
="
echoid-s7791
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xml:space
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">reliquo nomen proportio quæq; </
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>
<
s
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="
echoid-s7792
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<
lb
/>
ſuum capit, Verbi gratia. </
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>
<
s
xml:id
="
echoid-s7793
"
xml:space
="
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">Quæro quæ nam proportio ſit inter octo, & </
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>
<
s
xml:id
="
echoid-s7794
"
xml:space
="
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">quatuor, ſtatim partior octo per qua-
<
lb
/>
tuor, id eſt experior quot uicibus quatuor constituant octo, & </
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>
<
s
xml:id
="
echoid-s7795
"
xml:space
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">inuenio duabus uicibus quatuor in octo ineſſe,
<
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/>
& </
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>
<
s
xml:id
="
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xml:space
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">nihil inde ſupereſſe, ab hoc igitur numero, duo, proportionis nomen deſumo, aioq́; </
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>
<
s
xml:id
="
echoid-s7797
"
xml:space
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">inter octo , & </
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>
<
s
xml:id
="
echoid-s7798
"
xml:space
="
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">quatuor
<
lb
/>
duplam cadere proportionem. </
s
>
<
s
xml:id
="
echoid-s7799
"
xml:space
="
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">Quæro iter um quæ proportio inter quinque, & </
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>
<
s
xml:id
="
echoid-s7800
"
xml:space
="
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">ſexdecim intercidat , partior
<
lb
/>
ſexdecim per quinarium, inuenio quinarium ter in ſexdecim, & </
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>
<
s
xml:id
="
echoid-s7801
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xml:space
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">amplius unum eſſe. </
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>
<
s
xml:id
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echoid-s7802
"
xml:space
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">Triplam igitur a ternis ui-
<
lb
/>
cibus proportionem illam nomino, & </
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<
s
xml:id
="
echoid-s7803
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xml:space
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">quia ſupereſt unitas, quæ pars quinta eſt quinarij, ideo addo ſeſqui quin-
<
lb
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tam, & </
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>
<
s
xml:id
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echoid-s7804
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xml:space
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">triplam eam ſeſquiquintam nomino proportionem. </
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<
s
xml:id
="
echoid-s7805
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xml:space
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">Similis ratio in reliquis reperitur. </
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<
s
xml:id
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echoid-s7806
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xml:space
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">Quod cum
<
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/>
<
note
position
="
left
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xlink:label
="
note-514.01.114-03
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xlink:href
="
note-514.01.114-03a
"
xml:space
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">30</
note
>
ita ſit, facile reperiemus, quæ nam æqualis, quæ maior, aut minor proportio intelligatur. </
s
>
<
s
xml:id
="
echoid-s7807
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xml:space
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">nam æquales & </
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>
<
s
xml:id
="
echoid-s7808
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xml:space
="
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">ſi-
<
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/>
miles ſunt, quæ eaſdem babent nominationes. </
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>
<
s
xml:id
="
echoid-s7809
"
xml:space
="
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">Inæquales, & </
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>
<
s
xml:id
="
echoid-s7810
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xml:space
="
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">dißimiles, quarum non eſt ſimilis denominatio.
<
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/>
</
s
>
<
s
xml:id
="
echoid-s7811
"
xml:space
="
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">primum id oſtendam in multiplicibus, deinde in reliquis. </
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>
<
s
xml:id
="
echoid-s7812
"
xml:space
="
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">Ex multiplicibus illæ ſunt maiores , quarum nomina-
<
lb
/>
tio a maiori numero prouenit. </
s
>
<
s
xml:id
="
echoid-s7813
"
xml:space
="
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">Quadrupla igitur maior est proportio, quàm dupla, quoniam illa a quater-
<
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/>
nario, hæc a binario ſumitur. </
s
>
<
s
xml:id
="
echoid-s7814
"
xml:space
="
preserve
">In ſuperparticularibus ratio contraria eſt, nam illa maior eſt, quæ a minori nu-
<
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mero conſtituitur, & </
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>
<
s
xml:id
="
echoid-s7815
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xml:space
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">hoc eſt ratione minutorũ, quas fractiones nominant, ne de uerbis contendamus. </
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>
<
s
xml:id
="
echoid-s7816
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xml:space
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">nam quo
<
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/>
plures ſunt fractiones, eo minor eſt proportio, ideo ſeſquitertia minor ſeſquialtera est, nam ſeſquitertia a terna-
<
lb
/>
rio, ſeſquialtera a dimidio nominatur, maior igitur pars eſt dimidium, quàm tertia pars, atque ita de ſingulis,
<
lb
/>
ſed in ſuprapartientibus intelligendum hoc est, quod ſupra non attuli. </
s
>
<
s
xml:id
="
echoid-s7817
"
xml:space
="
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">Dixi ſuprapartientem proportionem
<
lb
/>
fieri cum id, quod amplius, & </
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>
<
s
xml:id
="
echoid-s7818
"
xml:space
="
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">maius eſt continet id, quod minus eſt, & </
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>
<
s
xml:id
="
echoid-s7819
"
xml:space
="
preserve
">plures eius partes , ſed id hoc modo ca-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.114-04
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xlink:href
="
note-514.01.114-04a
"
xml:space
="
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">40</
note
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piendum est, ut tam de numero partium, quàm de ipſa denominatione, & </
s
>
<
s
xml:id
="
echoid-s7820
"
xml:space
="
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">de utroque ſimul intelligatur. </
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>
<
s
xml:id
="
echoid-s7821
"
xml:space
="
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">De nu-
<
lb
/>
mero quidem partium quando plus continet, quod minus eſt ſemel, & </
s
>
<
s
xml:id
="
echoid-s7822
"
xml:space
="
preserve
">duas uerbi gratia eius partes ſic ſuper-
<
lb
/>
bipartiens nominatur. </
s
>
<
s
xml:id
="
echoid-s7823
"
xml:space
="
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">ſi tres ſupratripartiens, atque ita in infinitum. </
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>
<
s
xml:id
="
echoid-s7824
"
xml:space
="
preserve
">De denominatione uero partium, cum
<
lb
/>
plus continet minus ſemel, & </
s
>
<
s
xml:id
="
echoid-s7825
"
xml:space
="
preserve
">partes eius, quæ ſunt uel terliæ, uel quartæ, uel huiuſmodi partes totius, unde
<
lb
/>
proportio dicitur ſuprapartiens tertias, uel quartas, uel huiuſmodi. </
s
>
<
s
xml:id
="
echoid-s7826
"
xml:space
="
preserve
">De utroque ſimul, ut cum dicimus pro-
<
lb
/>
portionem ſuprabipartientem tertias, uel quartas, atque ita in infinitum. </
s
>
<
s
xml:id
="
echoid-s7827
"
xml:space
="
preserve
">Sed in his maior ea proportio eſt,
<
lb
/>
quæ a maiori numero nomen capit, in his dico, quæ ſecundum priorem modum, quo numerus partium expri-
<
lb
/>
mitur, continetur. </
s
>
<
s
xml:id
="
echoid-s7828
"
xml:space
="
preserve
">nam ubi exprimuntur quales ſint minoris partes, ibi eadem ratio, ut ſi dicas proportionem
<
lb
/>
ſupra octo partientem undecimas eſſe maiorem proportioni, ſupr atripartientem undecimas, quoniam hæc a
<
lb
/>
numero minori, qui ternarius eſt, illa ab octonario, qui maior eſt denominatus, cum ſecunda denominatio , ut-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.114-05
"
xlink:href
="
note-514.01.114-05a
"
xml:space
="
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">50</
note
>
pote ab undecima deducta, ſit eadem in utroque, atque it a facillime comparamus eam cognitionem, qua ma-
<
lb
/>
iores, & </
s
>
<
s
xml:id
="
echoid-s7829
"
xml:space
="
preserve
">minores proportiones habemus, unde alia, ut di
<
unsure
/>
xi, in templis, alia in publicis , alia in priuatis ædifi-
<
lb
/>
cijs utemur. </
s
>
<
s
xml:id
="
echoid-s7830
"
xml:space
="
preserve
">Similis quoque ratio in atrijs, tablinis, œcis , exedris, reliquiſque conclauibus habetur. </
s
>
<
s
xml:id
="
echoid-s7831
"
xml:space
="
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">nam quæq;
<
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/>
</
s
>
<
s
xml:id
="
echoid-s7832
"
xml:space
="
preserve
">babitatio ſuis gaudet proportionibus. </
s
>
<
s
xml:id
="
echoid-s7833
"
xml:space
="
preserve
">Cæterum quoniam ſæpe locorum anguſtiæ, proportiones mutare nos co-
<
lb
/>
gunt, ſæpe etiam ſtudio alios ædificiorum proſpectus, alias formas oſtendi uolumus, ideo non ignorandum eſt, qua
<
lb
/>
nam via proportiones inter ſe permutentur, ſubtrahantur, partiantur, componantur, & </
s
>
<
s
xml:id
="
echoid-s7834
"
xml:space
="
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">amplientur. </
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>
<
s
xml:id
="
echoid-s7835
"
xml:space
="
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">Sic enim,
<
lb
/>
& </
s
>
<
s
xml:id
="
echoid-s7836
"
xml:space
="
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">medium inter extrema, & </
s
>
<
s
xml:id
="
echoid-s7837
"
xml:space
="
preserve
">extrema ex medijs inueniemus, tam in rebus, quas efficiemus , quàm in muſi-
<
lb
/>
cis conſonantijs, de quibus quinto libro dicemus. </
s
>
<
s
xml:id
="
echoid-s7838
"
xml:space
="
preserve
">Et in reliquis diſciplinis, quarum cognitione homines dele-
<
lb
/>
ctantur. </
s
>
<
s
xml:id
="
echoid-s7839
"
xml:space
="
preserve
">Quemadmodum igitur eſt quædam comparatio rerum ſub uno genere comprehenſarum, quæ propor-
<
lb
/>
tio nominatur, ita & </
s
>
<
s
xml:id
="
echoid-s7840
"
xml:space
="
preserve
">quædam proportionum inter ſe comparatio, quæ reſpectus eſt quidam non unius quanti-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-514.01.114-06
"
xlink:href
="
note-514.01.114-06a
"
xml:space
="
preserve
">60</
note
>
</
s
>
</
p
>
</
div
>
</
div
>
</
text
>
</
echo
>