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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                <p>
                  <s xml:id="echoid-s5097" xml:space="preserve">
                    <pb o="421" rhead="EPISTOL AE." n="433" file="0433" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0433"/>
                  tum, ſi dempſeris ex quadrato ipſius .100000. ſemidiametro ſphęræ, </s>
                  <s xml:id="echoid-s5098" xml:space="preserve">tuncradix qua-
                    <lb/>
                  drata reſidui, erit perpendicularis à centro ſphæræ ad centrum pentagoni partium,
                    <lb/>
                  79461. cuius tertia pars, ſi multiplicata fuerit cum pentagono ſupra reperto dicti cor
                    <lb/>
                  poris producet vnam ex .12. pyramidibus componentibus dictum Duodecaedron,
                    <lb/>
                  quæ pyramis, demum, multiplicata per .12. dabit totam corpulentiam ipſius Duo
                    <lb/>
                  decaedri partium .2785354925791680.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5099" xml:space="preserve">Nunc verò ſi experiri voluerimus vtrum iſti calculi duorum corporum maiorum
                    <lb/>
                  ſint rectè ſupputati,
                    <reg norm="dicemus" type="simple">dicemꝰ</reg>
                  ſi ad
                    <reg norm="corpus" type="simple">corpꝰ</reg>
                  .12.
                    <reg norm="baſium" type="context">baſiũ</reg>
                  ,
                    <reg norm="quod" type="wordlist">qđ</reg>
                  eſt
                    <reg norm="partium" type="context">partiũ</reg>
                  .2785354925791680
                    <lb/>
                  conuenit numerus partium .2536010579470260. ipſius Icoſaedri, quid conueniet
                    <lb/>
                  lateri cubi partium .115476. & inueniemus conuenire latus ipſius Icoſaedri partium
                    <lb/>
                  105138. eo quod probatum ſit in
                    <ref id="ref-0026">.10. propoſitione .14. li. Eucl.</ref>
                  eandem
                    <reg norm="proportionem" type="context">proportionẽ</reg>
                    <lb/>
                  eſſe corpulentiæ ipſius Duodecaedri ad corpulentiam ipſius Icoſaedri, quæ lateris
                    <lb/>
                  cubi ad latus Icoſaedri.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5100" xml:space="preserve">Hæc autem corpora, ita ſibi inuicem, & cum eorum ſphæra harmonicè
                    <reg norm="conueniunt" type="context">conueniũt</reg>
                    <lb/>
                  quemadmodum antiqui philoſophi inuenerunt, vt
                    <reg norm="mirandum" type="context">mirandũ</reg>
                  non ſit, ipſos credidiſ-
                    <lb/>
                  ſe omnia quæ natura conſtant, aliquo pacto exiſtis corporibus fieri. </s>
                  <s xml:id="echoid-s5101" xml:space="preserve">Conſidera quæ-
                    <lb/>
                  ſo quomodo conueniant inuicem Tetraedron, Octaedron, & Icoſaedron, cum uniuſ-
                    <lb/>
                  cuiuſque baſes ſint triangulares æquilateræ intelli gendo ſemper hæc corpora ab ea-
                    <lb/>
                  dem ſphæra circunſcriptibilia.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5102" xml:space="preserve">Octaedron, cum Tetraedro etiam in hoc conuenit, quod latus Octaedri æquale
                    <lb/>
                  ſit ei perpendiculari, quæ diuidit baſim Tetraedri per æqualia, vtſupra demonſtra-
                    <lb/>
                  uimus.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5103" xml:space="preserve">Harmonicis etiam interua llis hæc duo corpora inuicem concordantur, cum baſis
                    <lb/>
                  Tetraedri ad baſim Octaedri ſeruet proportionem ſeſquitertiam, conſonantiæ dia-
                    <lb/>
                  teſſaron. </s>
                  <s xml:id="echoid-s5104" xml:space="preserve">Et proportio omnium ſuperficierum ſiue baſium Octaedri ſimul ſumpta-
                    <lb/>
                  rum, ad omnes baſes ipſius Tetraedri ſimul ſumptas ſit ſeſquialtera, conſonantiæ dia
                    <lb/>
                  pentis. </s>
                  <s xml:id="echoid-s5105" xml:space="preserve">Neque omittendum eſt, quod proportio Octaedriad triplum Tetraedri ſit,
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                  vt latus Octaedri ad latus Tetraedri.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5106" xml:space="preserve">Proportio verò lateris Octaedri, ad axem Tetraedri, potentia eſt ſeſquioctaua,
                    <lb/>
                  vt ſupra vidimus interuallum ſcilicet harmonicum toni maioris.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5107" xml:space="preserve">Harmonia verò Tetraedri, & Exaedri
                    <reg norm="cum" type="context">cũ</reg>
                  eorum ſphæra, talis eſt, vt proportio dia
                    <lb/>
                  metriſphæræ, potentia, tripla ſit lateri Exaedri, & ſeſquialtera lateri Tetraedri, ex
                    <lb/>
                  quo ſequitur latus Tetraedri potentia duplum exiſtere lateri Exaedri. </s>
                  <s xml:id="echoid-s5108" xml:space="preserve">Interuallum
                    <lb/>
                  enim triplum in harmonicis, componitur ex diapaſon, & diapente, & ſonat ſpeciem
                    <lb/>
                  diapentis. </s>
                  <s xml:id="echoid-s5109" xml:space="preserve">Duplum verò eſt diapaſon, ſeſquialterum autem eſt di apente, quę con-
                    <lb/>
                  ſonantiæ perfectiſſimæ ſunt.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5110" xml:space="preserve">Proportio verò diametri ſphæræ, potentia dupla eſt lat eri Octaedri, conſonantię
                    <lb/>
                  diapaſon. </s>
                  <s xml:id="echoid-s5111" xml:space="preserve">Ex quo ſequitur proportionem lateris Tetraedri ad latus Octaedri, po-
                    <lb/>
                  tentia, ſeſquitertiam eſſe, hoc eſt conſonantiæ diateſſaron, & proportionem lateris
                    <lb/>
                  Octaedri ad latus Exaedri, potentia, ſeſquialteram eſſe, ita quod quatuor iſtæ poten
                    <lb/>
                  tiæ, ideſt diametri ſphæræ, lateris Tetraedri, lateris Octaedri, & lateris Exaedri con-
                    <lb/>
                  ſtituunt harmoniam ferè perfectiſſimam, ijs terminis comprehenſam .6. 4. 3. 2. (dixi
                    <lb/>
                  ferè, quia ditonus ſupra terminum .3. vel ſemiditonus ſub termino .2. hoc loco non
                    <lb/>
                  reperitur, cuius quidem terminus eſſet .2. cum duabus quintis.)</s>
                </p>
                <p>
                  <s xml:id="echoid-s5112" xml:space="preserve">Adde quod diameter ſphæræ triplus eſt longitudine ad
                    <reg norm="perpendicularem" type="context">perpendicularẽ</reg>
                  ductam
                    <lb/>
                  à centro ſphæræ ad baſim Octaedri, quæ proportio, vt ſupra dictum eſt, dicitur dia-
                    <lb/>
                  paſondiapente, practici verò eam vocant duodecimam.</s>
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