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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IO. BAPT. BENED.
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<
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xml:space
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">Diameter verò ſphæræ ſeſquialter eſt longitudine axi Tetraedri, conſonantiæ
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diapentis. </
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<
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xml:space
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">Axis autem Tetraedri ſeſquitertius eſt longitudinis ſemidiametro ſphæ-
<
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ræ conſonantiæ diateſſaron. </
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<
s
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xml:space
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">Ita quod iſti tres termini, qui ſunt, diameter ſphæræ,
<
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axis Tetraedri, & ſemidiameter ſphæræ conſtituunt etiam valde perfectam harmo-
<
lb
/>
niam huiuſmodi numeris contentam .6. 4. 3. corpulentia verò Exaedri ad corpu-
<
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/>
lentiam Tetraedri tripla eſt, conſonantiæ iam ſupradictæ diapaſondiapente. </
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<
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xml:space
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">Si ve-
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rò de vniſono aliquid videre deſideras, conſidera æqualitatem dupli quadrati dia-
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/>
metri ipſius ſphæræ, cum omnibus baſibus Exaedri, vel potentia diametri ſphæræ
<
lb
/>
cum duabus potentijs ſimul ſumptis, quarum vna eſt lateris Tetraedri, reliqua verò
<
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lateris Exaedri, vel æqualitatem numerorum laterum Tetraedri, cum baſibus Exae
<
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dri. </
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<
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xml:space
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">Nec mihi videtur ſilentio inuoluendum eſſe, antequam vlterius progrediar no
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tabilem ſympatiam inter triangulum æquilaterum, & Tetraedron (
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type
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corpus non ſit) non ſolum ob
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harum duarum figurarum. </
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xml:space
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">(nam omnes
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aliæ alterabiles eſſe poſſunt, ijſdem lateribns exiſtentibus, cum ex quadrato rom-
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bus, vel ex pentagono ęquiangulo, pentagonum non æquiangulum & c. efficiatur)
<
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</
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<
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xml:space
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">ſed quod quemadmodum latus trianguli æquilateri ſeſquitertium potentia eſt per-
<
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pendiculari ipſum per æqualia diuidenti, ita latus Tetraedri, ſeſquialterum eſt po-
<
lb
/>
tentia axi ipſius Tetraedri, vnde cum dempta fuerit illa proportio ſeſquitertia, ex
<
lb
/>
hac ſeſquialtera relinquetur nobis proportio ſeſquioctaua, inter perpendicularem
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trianguli, & axem Tetraedri (quod etiam ſupra demonſtrauimus.) </
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">Tranſeamus nunc
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hęc, nec omittamus tamen ſympatias quaſdam inter Exaedron, Octaedron, & Tetra
<
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/>
edron, hoc eſt quod eadem proportio ſit inter corpulentias Exaedri, & Octaedri,
<
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quæinter eorum ſuperficies, nec non, vt latus Exaedri ad ſemidiametrum ſphæræ.
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">Proportio verò baſis Exaedri ad baſim Tetraedri, vtlatus Tetraedri ad perpendicu
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larem diuidentem per æqualia eius baſim.</
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<
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xml:space
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">Hactenus ſatis dictum ſit de Tetraedro, Exaedro, & Octaedro cum ſphæra. </
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dum nunc cenſeo aliquid de reliquis duobus mirabilibus corporibus, quamuis ferè
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omnia hæc ab antiquis philoſophis inuenta ſint, quorum primum eſt, quod tam ba-
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ſis Duodecaedri, quam Icoſaedri, ab vno
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circulo circunſcriptibiles ſunt, ve
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rùm, talis paſſio accidit etiam baſibus Exaedri & Octaedri. </
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<
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xml:space
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">Præterea quemadmo-
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dum in Duodecaedro, quilibet angulus ſolidus terminatur tribus angulis pentago-
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norum æquiangulorum ita in Icoſaedro, quilibet angulus ſolidus viceuerſa termi-
<
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natur quinque angulis triangulorum æquiangulorum. </
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>
<
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xml:space
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">Et tam vnum, quam alte-
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rum horum corporum, triginta lateribus continetur. </
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<
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xml:space
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">Et tot ſolidos angulos trian-
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gulares, habet Duodecaedron, quot baſes triangulares continet Icoſaedron.</
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<
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xml:space
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">Et Icoſaedron, tot ſolidos angulos
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, quot baſes
<
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pentagonas
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type
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">pẽtagonas</
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habet Duo
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decaedron. </
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<
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xml:id
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xml:space
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">Et tam vnum quam alterum habet .60. angulos ſuperficiales. </
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<
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proportio eſt omnium baſium ſimul
<
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>
Duodecaedri ad omnes baſes ſimul
<
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ſumptas ipſius Icoſaedri, quæ corpulentiæ ipſius Duodecaedri ad corpulentiam
<
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Icoſaedri (quamuis hęc paſſio accidat Exaedro cum Octaedro, vt ſpra diximus) quę
<
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quidem proportio, eadem etiam eſt, quę lateris Exaedri ad latus Icoſaedri, vt ſu-
<
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pra iam dictum fuit.</
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