Benedetti, Giovanni Battista de
,
Io. Baptistae Benedicti ... Diversarvm specvlationvm mathematicarum, et physicarum liber : quarum seriem sequens pagina indicabit ; [annotated and critiqued by Guidobaldo Del Monte]
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EPISTOL AE.
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tum, ſi dempſeris ex quadrato ipſius .100000. ſemidiametro ſphęræ, </
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drata reſidui, erit perpendicularis à centro ſphæræ ad centrum pentagoni partium,
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79461. cuius tertia pars, ſi multiplicata fuerit cum pentagono ſupra reperto dicti cor
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poris producet vnam ex .12. pyramidibus componentibus dictum Duodecaedron,
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quæ pyramis, demum, multiplicata per .12. dabit totam corpulentiam ipſius Duo
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decaedri partium .2785354925791680.</
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<
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xml:space
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ſint rectè ſupputati,
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ſi ad
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.12.
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,
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eſt
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.2785354925791680
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conuenit numerus partium .2536010579470260. ipſius Icoſaedri, quid conueniet
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lateri cubi partium .115476. & inueniemus conuenire latus ipſius Icoſaedri partium
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105138. eo quod probatum ſit in
<
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">.10. propoſitione .14. li. Eucl.</
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eandem
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eſſe corpulentiæ ipſius Duodecaedri ad corpulentiam ipſius Icoſaedri, quæ lateris
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cubi ad latus Icoſaedri.</
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<
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<
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xml:space
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">Hæc autem corpora, ita ſibi inuicem, & cum eorum ſphæra harmonicè
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quemadmodum antiqui philoſophi inuenerunt, vt
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non ſit, ipſos credidiſ-
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ſe omnia quæ natura conſtant, aliquo pacto exiſtis corporibus fieri. </
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<
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ſo quomodo conueniant inuicem Tetraedron, Octaedron, & Icoſaedron, cum uniuſ-
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cuiuſque baſes ſint triangulares æquilateræ intelli gendo ſemper hæc corpora ab ea-
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dem ſphæra circunſcriptibilia.</
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<
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<
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">Octaedron, cum Tetraedro etiam in hoc conuenit, quod latus Octaedri æquale
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ſit ei perpendiculari, quæ diuidit baſim Tetraedri per æqualia, vtſupra demonſtra-
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uimus.</
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<
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xml:space
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">Harmonicis etiam interua llis hæc duo corpora inuicem concordantur, cum baſis
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Tetraedri ad baſim Octaedri ſeruet proportionem ſeſquitertiam, conſonantiæ dia-
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teſſaron. </
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<
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xml:space
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">Et proportio omnium ſuperficierum ſiue baſium Octaedri ſimul ſumpta-
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rum, ad omnes baſes ipſius Tetraedri ſimul ſumptas ſit ſeſquialtera, conſonantiæ dia
<
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pentis. </
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<
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xml:space
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">Neque omittendum eſt, quod proportio Octaedriad triplum Tetraedri ſit,
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vt latus Octaedri ad latus Tetraedri.</
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</
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<
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">Proportio verò lateris Octaedri, ad axem Tetraedri, potentia eſt ſeſquioctaua,
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vt ſupra vidimus interuallum ſcilicet harmonicum toni maioris.</
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<
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xml:space
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eorum ſphæra, talis eſt, vt proportio dia
<
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metriſphæræ, potentia, tripla ſit lateri Exaedri, & ſeſquialtera lateri Tetraedri, ex
<
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quo ſequitur latus Tetraedri potentia duplum exiſtere lateri Exaedri. </
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enim triplum in harmonicis, componitur ex diapaſon, & diapente, & ſonat ſpeciem
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diapentis. </
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ſonantiæ perfectiſſimæ ſunt.</
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<
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diapaſon. </
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tentia, ſeſquitertiam eſſe, hoc eſt conſonantiæ diateſſaron, & proportionem lateris
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Octaedri ad latus Exaedri, potentia, ſeſquialteram eſſe, ita quod quatuor iſtæ poten
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tiæ, ideſt diametri ſphæræ, lateris Tetraedri, lateris Octaedri, & lateris Exaedri con-
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ſtituunt harmoniam ferè perfectiſſimam, ijs terminis comprehenſam .6. 4. 3. 2. (dixi
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ferè, quia ditonus ſupra terminum .3. vel ſemiditonus ſub termino .2. hoc loco non
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reperitur, cuius quidem terminus eſſet .2. cum duabus quintis.)</
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<
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ductam
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à centro ſphæræ ad baſim Octaedri, quæ proportio, vt ſupra dictum eſt, dicitur dia-
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paſondiapente, practici verò eam vocant duodecimam.</
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