Vitruvius
,
De architectura libri decem
,
1567
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514.01.304
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304
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LIBER
"/>
<
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<
s
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xml:space
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">Iucundas, & </
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>
<
s
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="
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xml:space
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">gratas bominibus Architæ, & </
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>
<
s
xml:id
="
echoid-s20299
"
xml:space
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">Eratosthenis inuentiones fuiſſe, dicit Vitr. </
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>
<
s
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xml:space
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">Sed Cum ambo
<
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eandem quæs̃tionem tractarent, nitereturq́; </
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>
<
s
xml:id
="
echoid-s20301
"
xml:space
="
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">alter, atque alter ratione diuerſa eam ſoluere, ambo ſuſpecti fue-
<
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runt, non quia res eadem diuerſis modis inueniri non poßit, ſed quia, quineſcirent, cum uiderent Architam
<
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/>
ratione una, & </
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>
<
s
xml:id
="
echoid-s20302
"
xml:space
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">Eratoſthenem. </
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>
<
s
xml:id
="
echoid-s20303
"
xml:space
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">alia uti, ob eorum contentionem ſuſpicabantur, cogitantes ſtudioſe inter-ſe
<
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contendere. </
s
>
<
s
xml:id
="
echoid-s20304
"
xml:space
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preserve
">Quod perinde eſt, ac ſi quis turris altitudinem quadrante,alius radio, alius astrolabio, alius ſpe-
<
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culo, alius alio instrumento metiretur, cum vulgus neſciat eandem eſſe omnium horum inſtrumentorum ra-
<
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tionem, & </
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>
<
s
xml:id
="
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"
xml:space
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">viam, quæ ab angulorum menſura diſceernitur, ſuſpicatur menſorum contertionem, atque æmu-
<
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/>
lationem uerum inuoluere, ac confundere inſtrumentorum diuerſitate. </
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>
<
s
xml:id
="
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"
xml:space
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">Simile euenit ob Erathoſthenis, & </
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<
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<
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Architæ contentiones. </
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>
<
s
xml:id
="
echoid-s20308
"
xml:space
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">Propoſita quæſtio erat, qua ratione cubus duplicari oporteret. </
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>
<
s
xml:id
="
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"
xml:space
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">Cubus eſt, (ut dixi li-
<
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note-514.01.304-01
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">10</
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bro quinto ) ſolidum corpus,planis ſex, & </
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>
<
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">lateribus conſtans æquaiibus Metiemur cubum, ſi duxerimus
<
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aliquod unum ex ſuis lateribus in ſeipſum, & </
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>
<
s
xml:id
="
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xml:space
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">eam ſummam quoque duxerimus per idem latus. </
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>
<
s
xml:id
="
echoid-s20312
"
xml:space
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">Detur cubus
<
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cuius latera ſingula octo conſtet pedibus, duc octo in ſe,reddentur 64. </
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>
<
s
xml:id
="
echoid-s20313
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xml:space
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">ducito rurſtus 64. </
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<
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xml:id
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xml:space
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">per octo,redden-
<
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tur 5 12. </
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>
<
s
xml:id
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xml:space
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">tot igitur quadratis perdibus conſtabit cubus. </
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>
<
s
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xml:space
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">Cum igitur cubusiam 512. </
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>
<
s
xml:id
="
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xml:space
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">pedibus comſtare cogni-
<
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tum ſit. </
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>
<
s
xml:id
="
echoid-s20318
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xml:space
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">iuxta propoſitam quæstionem duplicationdus eſt. </
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>
<
s
xml:id
="
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xml:space
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">Quod ut commode ſaciamus, neceſſario eſt ſcire, qua
<
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ratione duabus propoſitis rectis. </
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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-s20321
"
xml:space
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">inæqualibus lmeis. </
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>
<
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xml:id
="
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"
xml:space
="
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">duas medias inueniamus, quæ inter ſe & </
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>
<
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xml:id
="
echoid-s20323
"
xml:space
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">cum primis
<
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/>
continua proportione reſpondeant. </
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>
<
s
xml:id
="
echoid-s20324
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xml:space
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">Vt igitur inutniretur ea mediarum linearum comparatio, modi undecim
<
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ab antiquis reperti ſunt. </
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>
<
s
xml:id
="
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xml:space
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">Alij euidentibud matbematicorum rationibus uſi ſunt, alij. </
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>
<
s
xml:id
="
echoid-s20326
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xml:space
="
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">& </
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>
<
s
xml:id
="
echoid-s20327
"
xml:space
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">præter demonſtra-
<
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tiones inſtrument a ad id aptainuenere. </
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>
<
s
xml:id
="
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xml:space
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">Archimedes organo uſus. </
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>
<
s
xml:id
="
echoid-s20329
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xml:space
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">quod meſolabium, ideſt medicapium, ſi licet
<
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/>
ita dicere, numcupauit. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">nam eo mediæ inter duas datas comparabiles inuenimus. </
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>
<
s
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xml:space
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">Plato quoque alio inſtrumẽto
<
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note-514.01.304-02a
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xml:space
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">20</
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>
uſus eſt, quod pari ratione dmeſolabium nocare poſſumus. </
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>
<
s
xml:id
="
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xml:space
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">Architas demoſtrationes attulit lineis a dimidio
<
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Cylindro ſumptis, quas in opus redigere numquam potuit. </
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>
<
s
xml:id
="
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xml:space
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">Ego rationes, & </
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>
<
s
xml:id
="
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xml:space
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">inſtrumenta explicabo, oſten-
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damq́ quomodo ad duplicationem cubi, mediarum linearum inuentio nos ducat. </
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>
<
s
xml:id
="
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xml:space
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">Quætionis primum afferam
<
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occaſionem, e qua ſumma utilitas erit maniſeſta, quam habituri ſunt Architecti in egregijs eorum operibus
<
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deformandis. </
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>
<
s
xml:id
="
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xml:space
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">Legitur ergo Epiſtola Eratosthenis ad Potlemæum Regem in hunc modumperſcripta.</
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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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">PTOLEM AEO REGI ERATOSTHENES. S.</
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>
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<
s
xml:id
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xml:space
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">Dicitur ex antiquis tragædiarum compoſitoribus unum introducere Minoa Glauco ſepulchrum excitare
<
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uolentem, cumq; </
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>
<
s
xml:id
="
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"
xml:space
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">dictum fuiſſet illud quaqua uerſus eſſe pedes centum: </
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>
<
s
xml:id
="
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xml:space
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">Dixit paruam fore arcam pro Regio
<
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ſepulchro, duplicetur igitur, & </
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>
<
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xml:id
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xml:space
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">cubus non mutetur. </
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>
<
s
xml:id
="
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"
xml:space
="
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">certe qui unum quodque latus duplicare uoluerit, non
<
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/>
<
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note-514.01.304-03
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">30</
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>
erit erroris expers. </
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>
<
s
xml:id
="
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xml:space
="
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">nam lateribus duplicatis planum quodlibet quadruplum efficietur, ipſum uero ſolidum
<
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octuplum. </
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>
<
s
xml:id
="
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xml:space
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">Quæſitum igitur est a geometris, qua ratione ſolido in eadem figura permanens duplum efficeretur.
<
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</
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>
<
s
xml:id
="
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xml:space
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">Quæſtio hæc cubi duplicatio nominata eſt. </
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>
<
s
xml:id
="
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"
xml:space
="
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">nam propoſito cubo quærebant, qua uia alterum illi duplum effice-
<
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rent. </
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>
<
s
xml:id
="
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xml:space
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">Ambigentibus, & </
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>
<
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xml:id
="
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xml:space
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">labor antibus cæteris, prius extitit Hippocrates, qui indicauit id fieri poſſe,ſi con-
<
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ſtitutis duabus lineis, quarum maior minoris eſſet dupla, duæ. </
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>
<
s
xml:id
="
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"
xml:space
="
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">in continua proportione inuenirentur. </
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>
<
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="
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<
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Quare ea res dubia in maiorem difficultatem uerſa es̃t. </
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>
<
s
xml:id
="
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xml:space
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">Aliquanto poſt Delij morbo laborantes, cum oraculo
<
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Apollinis iuberentur aram ipſius duplicare, neque qua id fieri poſſet ratione ſatis uiderent,in eandem dubietatẽ
<
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incidere, & </
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>
<
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xml:id
="
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xml:space
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">obiurgante Platone eos geometras, qui erant in Academia, ab ijs quæſitũ est, ut inuenirent quod
<
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propoſitũfuerat. </
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>
<
s
xml:id
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xml:space
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">ij cum lahori ſeſe dediſſent,& </
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>
<
s
xml:id
="
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xml:space
="
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">conantes inuenter duas medias proportione reſpondentes dua-
<
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bus propoſitis lineis: </
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>
<
s
xml:id
="
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xml:space
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">dicitur Architam Tarentinum eas inueniſſe bemicylindrorum ratione. </
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>
<
s
xml:id
="
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xml:space
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">Eudo xus uero fle-
<
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<
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="
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="
note-514.01.304-04
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xlink:href
="
note-514.01.304-04a
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xml:space
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">40</
note
>
xis quibuſdam lineis, cæterum uterque ſane approbatam harum rerum rationem inuenire, ſed neuter eas ad
<
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uſum potuit accommodare, & </
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>
<
s
xml:id
="
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xml:space
="
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">manibus experiri, excepto uno Menechmo, qui tamen parum fecit, & </
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>
<
s
xml:id
="
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"
xml:space
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">id parũ
<
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maxima cum difficultate.</
s
>
<
s
xml:id
="
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"
xml:space
="
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">Sed nos excogitauimus per organd facilem inuentionem,qua nõ tantum duas medias
<
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/>
proportionabiles duabus datis, ſed quotquot propoſitum fuerit, utinueniamus, & </
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>
<
s
xml:id
="
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"
xml:space
="
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">eo inuento, poterimus de-
<
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/>
mum ad cubum reducere propoſitum ſolidum lineis æquè diſtantibus contentum, aut etiam ex una aliam
<
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/>
figuram formare, quæ aut æqualis, aut maior ſit ſeruata ſimilitudine. </
s
>
<
s
xml:id
="
echoid-s20361
"
xml:space
="
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">Quoniam nullidubium eſt, quin huiuſ-
<
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modi inſtrumento duplicari poßint aræ, ædificiaq́, & </
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>
<
s
xml:id
="
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"
xml:space
="
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">ad cubum referri liquidonum, &</
s
>
<
s
xml:id
="
echoid-s20363
"
xml:space
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">ſiccorum menſuræ, ut
<
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modiorum, & </
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>
<
s
xml:id
="
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xml:space
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">ſimilium, quarum menſurarum larteribus uaſorum capacitas dignoſcitur.</
s
>
<
s
xml:id
="
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"
xml:space
="
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">& </
s
>
<
s
xml:id
="
echoid-s20366
"
xml:space
="
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">ut ſummatim dicã,
<
lb
/>
quæſtionis buius cognitio utilies est uolentibus duplicare,aut maiorareddere organa, e quibus tela, ſaxa, aut
<
lb
/>
ferreæ pilæ mittuntur. </
s
>
<
s
xml:id
="
echoid-s20367
"
xml:space
="
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">nam neceſſe eſt omnia in latum, & </
s
>
<
s
xml:id
="
echoid-s20368
"
xml:space
="
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">in longum creſcere proportione quadam, ſiue fo-
<
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ramina ſint, ſiue nerui, & </
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>
<
s
xml:id
="
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"
xml:space
="
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">immiſſa alia, aut quicquid opus fuerit, ſi totum proportione quadam, ſiue fo-
<
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/>
<
note
position
="
left
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xlink:label
="
note-514.01.304-05
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xlink:href
="
note-514.01.304-05a
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xml:space
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">50</
note
>
fieri non poteſt ſine medij inuentione. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">Demonſtrationem igitur, & </
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>
<
s
xml:id
="
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xml:space
="
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">apparaiumprædicti organi tibi ſubſcripſi,
<
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ac primum demonſtrationem.</
s
>
<
s
xml:id
="
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"
xml:space
="
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">#Proponantur duæ rectæ, & </
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>
<
s
xml:id
="
echoid-s20373
"
xml:space
="
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">inæquales lineæ, quarum altera ſit a b, al-
<
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/>
tera c, d, quæramus inter duas medias comparabiles ratione continua, ita quidem ut quemadmodum ſe ha
<
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/>
bet prima ad ſecundam, ita ſe habeat ſecunda ad tertiam, & </
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>
<
s
xml:id
="
echoid-s20374
"
xml:space
="
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">tertia ad quartam. </
s
>
<
s
xml:id
="
echoid-s20375
"
xml:space
="
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">Detur utranque ex pro-
<
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/>
poſitis lineis ad perpendiculum cadere in rectam, quæ ſit b d, & </
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>
<
s
xml:id
="
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"
xml:space
="
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">maior ſit a b, monor uero c d, ab. </
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>
<
s
xml:id
="
echoid-s20377
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xml:space
="
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">a,
<
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/>
autem ad c, ducatur linea, quæ ulterius producta cadat in lineam b d, ibierit e. </
s
>
<
s
xml:id
="
echoid-s20378
"
xml:space
="
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">Ducatur etiam ab a,
<
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/>
ſigno in lineam b d, linea, quæſit a f, & </
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>
<
s
xml:id
="
echoid-s20379
"
xml:space
="
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">ab f, rurſum ducatur linea a b, æquè distans, quæ ſitf g.
<
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/>
</
s
>
<
s
xml:id
="
echoid-s20380
"
xml:space
="
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">quæ lineam a c, in puncto, g, tangat. </
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>
<
s
xml:id
="
echoid-s20381
"
xml:space
="
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">Similiter ducatur linea a ſigno g, æquidiſtans lineæ a f, & </
s
>
<
s
xml:id
="
echoid-s20382
"
xml:space
="
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">
<
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/>
fit illa g h, quæ lineam b d, tangat, in ſigno b, a quo ſigno erigatur linea æquidistans ipſi a b, & </
s
>
<
s
xml:id
="
echoid-s20383
"
xml:space
="
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">ſit
<
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/>
<
note
position
="
left
"
xlink:label
="
note-514.01.304-06
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xlink:href
="
note-514.01.304-06a
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xml:space
="
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">60</
note
>
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