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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                <pb o="422" rhead="IO. BAPT. BENED." n="434" file="0434" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0434"/>
                <p>
                  <s xml:id="echoid-s5113" xml:space="preserve">Diameter verò ſphæræ ſeſquialter eſt longitudine axi Tetraedri, conſonantiæ
                    <lb/>
                  diapentis. </s>
                  <s xml:id="echoid-s5114" xml:space="preserve">Axis autem Tetraedri ſeſquitertius eſt longitudinis ſemidiametro ſphæ-
                    <lb/>
                  ræ conſonantiæ diateſſaron. </s>
                  <s xml:id="echoid-s5115" xml:space="preserve">Ita quod iſti tres termini, qui ſunt, diameter ſphæræ,
                    <lb/>
                  axis Tetraedri, & ſemidiameter ſphæræ conſtituunt etiam valde perfectam harmo-
                    <lb/>
                  niam huiuſmodi numeris contentam .6. 4. 3. corpulentia verò Exaedri ad corpu-
                    <lb/>
                  lentiam Tetraedri tripla eſt, conſonantiæ iam ſupradictæ diapaſondiapente. </s>
                  <s xml:id="echoid-s5116" xml:space="preserve">Si ve-
                    <lb/>
                  rò de vniſono aliquid videre deſideras, conſidera æqualitatem dupli quadrati dia-
                    <lb/>
                  metri ipſius ſphæræ, cum omnibus baſibus Exaedri, vel potentia diametri ſphæræ
                    <lb/>
                  cum duabus potentijs ſimul ſumptis, quarum vna eſt lateris Tetraedri, reliqua verò
                    <lb/>
                  lateris Exaedri, vel æqualitatem numerorum laterum Tetraedri, cum baſibus Exae
                    <lb/>
                  dri. </s>
                  <s xml:id="echoid-s5117" xml:space="preserve">Nec mihi videtur ſilentio inuoluendum eſſe, antequam vlterius progrediar no­
                    <lb/>
                  tabilem ſympatiam inter triangulum æquilaterum, & Tetraedron (
                    <reg norm="quanuis" type="context">quãuis</reg>
                    <reg norm="triangulum" type="context">triangulũ</reg>
                    <lb/>
                  corpus non ſit) non ſolum ob
                    <reg norm="inalterabilitatem" type="context">inalterabilitatẽ</reg>
                  harum duarum figurarum. </s>
                  <s xml:id="echoid-s5118" xml:space="preserve">(nam omnes
                    <lb/>
                  aliæ alterabiles eſſe poſſunt, ijſdem lateribns exiſtentibus, cum ex quadrato rom-
                    <lb/>
                  bus, vel ex pentagono ęquiangulo, pentagonum non æquiangulum & c. efficiatur)
                    <lb/>
                  </s>
                  <s xml:id="echoid-s5119" xml:space="preserve">ſed quod quemadmodum latus trianguli æquilateri ſeſquitertium potentia eſt per-
                    <lb/>
                  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
                    <lb/>
                  trianguli, & axem Tetraedri (quod etiam ſupra demonſtrauimus.) </s>
                  <s xml:id="echoid-s5120" xml:space="preserve">Tranſeamus nunc
                    <lb/>
                  hęc, nec omittamus tamen ſympatias quaſdam inter Exaedron, Octaedron, & Tetra
                    <lb/>
                  edron, hoc eſt quod eadem proportio ſit inter corpulentias Exaedri, & Octaedri,
                    <lb/>
                  quæinter eorum ſuperficies, nec non, vt latus Exaedri ad ſemidiametrum ſphæræ.
                    <lb/>
                  </s>
                  <s xml:id="echoid-s5121" xml:space="preserve">Proportio verò baſis Exaedri ad baſim Tetraedri, vtlatus Tetraedri ad perpendicu
                    <lb/>
                  larem diuidentem per æqualia eius baſim.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5122" xml:space="preserve">Hactenus ſatis dictum ſit de Tetraedro, Exaedro, & Octaedro cum ſphæra. </s>
                  <s xml:id="echoid-s5123" xml:space="preserve">
                    <reg norm="Dicem" type="context">Dicẽ</reg>
                    <lb/>
                  dum nunc cenſeo aliquid de reliquis duobus mirabilibus corporibus, quamuis ferè
                    <lb/>
                  omnia hæc ab antiquis philoſophis inuenta ſint, quorum primum eſt, quod tam ba-
                    <lb/>
                  ſis Duodecaedri, quam Icoſaedri, ab vno
                    <reg norm="eodemque" type="simple">eodemq́;</reg>
                  circulo circunſcriptibiles ſunt, ve
                    <lb/>
                  rùm, talis paſſio accidit etiam baſibus Exaedri & Octaedri. </s>
                  <s xml:id="echoid-s5124" xml:space="preserve">Præterea quemadmo-
                    <lb/>
                  dum in Duodecaedro, quilibet angulus ſolidus terminatur tribus angulis pentago-
                    <lb/>
                  norum æquiangulorum ita in Icoſaedro, quilibet angulus ſolidus viceuerſa termi-
                    <lb/>
                  natur quinque angulis triangulorum æquiangulorum. </s>
                  <s xml:id="echoid-s5125" xml:space="preserve">Et tam vnum, quam alte-
                    <lb/>
                  rum horum corporum, triginta lateribus continetur. </s>
                  <s xml:id="echoid-s5126" xml:space="preserve">Et tot ſolidos angulos trian-
                    <lb/>
                  gulares, habet Duodecaedron, quot baſes triangulares continet Icoſaedron.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5127" xml:space="preserve">Et Icoſaedron, tot ſolidos angulos
                    <reg norm="pentagonos" type="context">pẽtagonos</reg>
                  , quot baſes
                    <reg norm="pentagonas" type="context">pẽtagonas</reg>
                  habet Duo
                    <lb/>
                  decaedron. </s>
                  <s xml:id="echoid-s5128" xml:space="preserve">Et tam vnum quam alterum habet .60. angulos ſuperficiales. </s>
                  <s xml:id="echoid-s5129" xml:space="preserve">
                    <reg norm="Eademque" type="context simple">Eadẽq́;</reg>
                    <lb/>
                  proportio eſt omnium baſium ſimul
                    <reg norm="ſumptarum" type="context">ſumptarũ</reg>
                  Duodecaedri ad omnes baſes ſimul
                    <lb/>
                  ſumptas ipſius Icoſaedri, quæ corpulentiæ ipſius Duodecaedri ad corpulentiam
                    <lb/>
                  Icoſaedri (quamuis hęc paſſio accidat Exaedro cum Octaedro, vt ſpra diximus) quę
                    <lb/>
                  quidem proportio, eadem etiam eſt, quę lateris Exaedri ad latus Icoſaedri, vt ſu-
                    <lb/>
                  pra iam dictum fuit.</s>
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