Mercurial > hg > mpdl-xml-content

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  <metadata>
    <dcterms:identifier>ECHO:CE3XGS5P.xml</dcterms:identifier>
    <dcterms:creator identifier="GND:118872621">Cavalieri, Bonaventura</dcterms:creator>
    <dcterms:title xml:lang="it">Lo specchio ustorio, overo, Trattato delle settioni coniche : et alcuni loro mirabili effeti intorno al lume, caldo, freddo, suono, e moto ancora</dcterms:title>
    <dcterms:date xsi:type="dcterms:W3CDTF">1632</dcterms:date>
    <dcterms:language xsi:type="dcterms:ISO639-3">ita</dcterms:language>
    <dcterms:rights>CC-BY-SA</dcterms:rights>
    <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license>
    <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder>
  </metadata>
  <text xml:lang="it" type="free">
<div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="0001" n="1"/>
<pb file="0002" n="2"/>
<pb file="0003" n="3"/>
<handwritten/>
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<div xml:id="echoid-div2" type="section" level="1" n="2">
<head xml:id="echoid-head1" xml:space="preserve">LO <lb/>SPECCHIO <lb/>VSTORIO <lb/>OVERO <lb/>TRATTATO</head>
<head xml:id="echoid-head2" xml:space="preserve">Delle Settioni Coniche, <lb/>ET ALCVNI LORO MIRABILI EFFETTI <lb/>Intorno al Lu<unsure/>me, Caldo, Freddo, Suono, <lb/>e<unsure/> Moto ancor@.</head>
<head xml:id="echoid-head3" xml:space="preserve">DEDICATO <lb/>A GL’ILLVSTRISSIMI <lb/>SIGNORI SENATORI <lb/>DI BOLOGNA</head>
<head xml:id="echoid-head4" xml:space="preserve">Da F. Bonauentura Caualieri Milaneſe Gieſuato <lb/>di S. GIROLAMO <lb/>AVTORE <lb/>E Matematico Primario nell’Inclito Studio dell’iſteſſa Cittd.</head>
  <figure>
    <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0005-01"/>
  </figure>
<p>
  <s xml:id="echoid-s1" xml:space="preserve">In Bologna, preſſo Clemente Ferroni 1632. <lb/></s>
  <s xml:id="echoid-s2" xml:space="preserve">Conlicenza de’Superiori.</s>
  <s xml:id="echoid-s3" xml:space="preserve"/>
</p>
<pb file="0006" n="6"/>
<pb file="0007" n="7"/>
</div>
<div xml:id="echoid-div3" type="section" level="1" n="3">
<head xml:id="echoid-head5" xml:space="preserve">ILLVSTRISSIMI <lb/>SIGNORI</head>
<head xml:id="echoid-head6" xml:space="preserve">Padroni Colendiſsimi.</head>
  <figure>
    <image file="0007-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0007-01"/>
  </figure>
<p>
  <s xml:id="echoid-s4" xml:space="preserve">ARDEA in honor di Lu-<lb/>cina pretioſa lampada <lb/>nel mezo del famoſo <lb/>Tempio, che allo ſpun-<lb/>tar delle grandezze, e <lb/>marauiglie dell’antica <lb/>Roma, le fù dalla ſuperſtitioſa Gentilità, <lb/>come à ſingolar protettrice de’parti, con-<lb/>ſecrato, con cosi diligente, anzi ſtraordina-
<pb file="0008" n="8"/>
ria cura, da ſemplici Verginelle cuſtodita, <lb/>che ſtimando eſſe da terreſtre luce douer-<lb/>ne rimaner’offeſi gli occhi dicosì alto Nu-<lb/>me, non con altro, che con celeſte fuoco, <lb/>generato dal riuerbero di lucidiſſimo ſpec-<lb/>chio, eſpoſto al Sole, induſtrioſamente ſi <lb/>accẽdeua. </s>
  <s xml:id="echoid-s5" xml:space="preserve">Lodeuole coſtume inuero, reli-<lb/>gioſo penſiero, quãdo dalla loro vana Dei-<lb/>tà non foſſe ſtato profanato. </s>
  <s xml:id="echoid-s6" xml:space="preserve">Io perciò, che <lb/>non da i fauori di Lucina, ma (doppo la <lb/>diuina benignità) dall’aura propitia, che <lb/>dalla magnanimità ſpira de i generoſi cuo-<lb/>ri delle SS. </s>
  <s xml:id="echoid-s7" xml:space="preserve">VV. </s>
  <s xml:id="echoid-s8" xml:space="preserve">Illuſtriſs riconoſco i miei, <lb/>benche deboliſsimi parti, eſſere ſomma-<lb/>mente felicitati, de’quali, come non iſde-<lb/>gnaſte aggradire la publicatione del pri-<lb/>mo, così prendo ardire di offerire di nuo-<lb/>uo, oltre me ſteſſo, il ſecondo; </s>
  <s xml:id="echoid-s9" xml:space="preserve">vorrei pur’ <lb/>anche, non dall’eſca, efociletrarne cadu-<lb/>ca fiamma; </s>
  <s xml:id="echoid-s10" xml:space="preserve">ma, ad imitation di Eſſempio <lb/>così celebre, con queſto mio SPECCHIO <lb/>VSTORIO, ſimile in parte à quello, che
<pb file="0009" n="9"/>
veniua nel culto di Lucina adoperato, nel <lb/>voſtro nobiliſsimo Teatro, o, per dir me-<lb/>glio, nel Tempio dell’eternità delle voſtre <lb/>glorie, accendere vna celeſte, &amp; </s>
  <s xml:id="echoid-s11" xml:space="preserve">ineſtin-<lb/>guibile lampada di diuotione, entro al cui <lb/>ſplendore viuamente appariſſel’intenſiſ <lb/>ſ@mo ardore, che hò di perpetuamente ho-<lb/>norare, e ſeruire, chi mi hà così altamente <lb/>obligato. </s>
  <s xml:id="echoid-s12" xml:space="preserve">Ma per ottener queſto, pure mi <lb/>èauiſo non eſſermi d’vopoil ſalire con la <lb/>verga di Prometeo à rapirne il fuoco dal <lb/>carro del Sole, o pure, che à quella infoca-<lb/>ta ruota io lo riuolga; </s>
  <s xml:id="echoid-s13" xml:space="preserve">ma che mi baſti e-<lb/>ſporlo al ſplendidiſsimo cerchio, del qua-<lb/>le diuinamente le voſtre ſingolariſsime<unsure/> <lb/>virtù coronandoſi, fanno chiaramẽte rim-<lb/>bo<unsure/>mbar d’ogn’intorno</s>
</p>
<p style="it">
  <s xml:id="echoid-s14" xml:space="preserve">Igneus est ollis vigor, &amp; </s>
  <s xml:id="echoid-s15" xml:space="preserve">ca<unsure/>lestis origo.</s>
  <s xml:id="echoid-s16" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s17" xml:space="preserve">A queſto ſeruirà dunque principalmente <lb/>il<unsure/> preſente mio Specchio. </s>
  <s xml:id="echoid-s18" xml:space="preserve">Seruirà per pe-<lb/>g<unsure/>no di gratitudine, e per rappreſentar’in-<lb/>ſ<unsure/>ieme la diuotiſsima ſeruitù, che alle Illu-
<pb file="0010" n="10"/>
ſtriſsime Signorie loro continuamẽte pro-<lb/>feſſo. </s>
  <s xml:id="echoid-s19" xml:space="preserve">Alle quali riuerentemente inchinan-<lb/>domi, deſidero per fine il colmo d’ogni fe-<lb/>licità. </s>
  <s xml:id="echoid-s20" xml:space="preserve">Bologna il dì 19. </s>
  <s xml:id="echoid-s21" xml:space="preserve">Agoſto 1632.</s>
  <s xml:id="echoid-s22" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s23" xml:space="preserve">D. </s>
  <s xml:id="echoid-s24" xml:space="preserve">VV. </s>
  <s xml:id="echoid-s25" xml:space="preserve">SS. </s>
  <s xml:id="echoid-s26" xml:space="preserve">Illuſtriſs.</s>
  <s xml:id="echoid-s27" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s28" xml:space="preserve">Diuotiſs. </s>
  <s xml:id="echoid-s29" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s30" xml:space="preserve">obligatiſs. </s>
  <s xml:id="echoid-s31" xml:space="preserve">Setuitore</s>
</p>
<p>
  <s xml:id="echoid-s32" xml:space="preserve">F. </s>
  <s xml:id="echoid-s33" xml:space="preserve">Bonauentura Caualier</s>
</p>
<pb file="0011" n="11"/>
</div>
<div xml:id="echoid-div4" type="section" level="1" n="4">
<head xml:id="echoid-head7" xml:space="preserve">AL CORTESE <lb/>LETTORE.</head>
  <figure>
    <image file="0011-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0011-01"/>
  </figure>
<p style="it">
  <s xml:id="echoid-s34" xml:space="preserve">REstaua queſto mio S P E C C H I O <lb/>V S T O R I O ſepolto ancora per <lb/>qualche tempo, e forſi per ſempre, <lb/>non oſtante, ch’io haueſſi di già ſco-<lb/>perto tanto del ſuo lume, ch’io mi <lb/><gap/>ſſi perſuadere, mettendolo in publico (almeno per <lb/><gap/>rioſità, ſe non per altro) ch’ei non foſſe per ri-<lb/><gap/>re del tutto inuiſibile; </s>
  <s xml:id="echoid-s35" xml:space="preserve">ſe le frequẽti eſſortatio-<lb/><gap/>perſona, il cui giuditio vi@ne da me molio sti-<lb/><gap/>, ed al quale haueuo di già confidato il mio pen-<lb/><gap/>non mi haueſſero fatto romper gli argini del ti-<lb/><gap/>, e riſoluermi à metterlo in proſpettiua delia <lb/><gap/>. </s>
  <s xml:id="echoid-s36" xml:space="preserve">Non era quello però del tutto irragioneuole, <lb/><gap/>attandoſi di coſa tenuta da molti per poco <lb/><gap/>che fauoloſa, penſauo, che da queſti tali doueſſe <lb/><gap/>io parere eſſere ſenz’altro eſſame rigettato alla <lb/><gap/>come altretanio vano, quanto era da loro sti-<lb/><gap/>per ohimerico lo Specchio di Archimede, proto-<lb/><gap/>del mio. </s>
  <s xml:id="echoid-s37" xml:space="preserve">A questo però contraponendo l’auto-<lb/><gap/>d’lſtorici famoſi, che ci fanno autentica fede di
<pb file="0012" n="12"/>
quello Specchio, e le ragioni dimostratiue, che la di <lb/>lui poſſibiltà ci perſuadono; </s>
  <s xml:id="echoid-s38" xml:space="preserve">pareami aſſat credibile, <lb/>che chi non è del tutto incapace di ragione, doueſſe <lb/>pur’anco ſentirſi far qualche forza da argomenti di <lb/>verità tanto euidenti. </s>
  <s xml:id="echoid-s39" xml:space="preserve">Ma quì non ſi terminaua la <lb/>mia difficoltà; </s>
  <s xml:id="echoid-s40" xml:space="preserve">poiche non accompagnando io l’opera <lb/>fatta con la teorica, giudicauo ad alcuni douer queſto <lb/>ri@ſcir così poco accetto, che nõ preſentandogli in ma-<lb/>no lo Specchio bell’, e fatto, foſſero per riputar per co-<lb/>ſa friuola &amp; </s>
  <s xml:id="echoid-s41" xml:space="preserve">di niun momẽto, ogni diſcorſo ſpecola-<lb/>tiuo, benche indrizzato alla di lui fabrica; </s>
  <s xml:id="echoid-s42" xml:space="preserve">mentre <lb/>non ſi metteuano (come ſi ſaol dirc) le mani in paſta, <lb/>e non ſi veniua all’atto prattico, preualendo la coſa <lb/>fatta à qualunque ragione, che ſi poſſa addurre, ch’ <lb/>ella ſia fattibile. </s>
  <s xml:id="echoid-s43" xml:space="preserve">Il che veramente confeſſo, c’hauria <lb/>hauuto molta forza per diſtormi dall’impreſa @ſe <lb/>l’altrui ragioni non mi haueſſero finalmente in un <lb/>certo modo violentato à far queſta riſolutione; </s>
  <s xml:id="echoid-s44" xml:space="preserve">con <lb/>il perſuadermi, che anco il promouere nuoui penſiert, <lb/>era coſa gradita da’studioſi, e tanto più, quãto ſi la-<lb/>ſciaua altrui campo d’acquiſtarſi gran parte della <lb/>gloria nel perfettionar quella fabrica, alla cui strut-<lb/>tura la ſpecolatiua hauea ſolò gettato i fondamenti@ <lb/>&amp; </s>
  <s xml:id="echoid-s45" xml:space="preserve">che queſto era modo d’inuentar ſingolare; </s>
  <s xml:id="echoid-s46" xml:space="preserve">poiche <lb/>l’imparar Mercurio dalla Teſtuggine di formar la
<pb file="0013" n="13"/>
Lira, o dal batter vicendeuol de’martelli l’inuentar <lb/>Pitagora la Muſica, fù vn veder prima, in vn cer-<lb/>to modo, la concluſione, e da quella preconoſciuta in-<lb/>@eſtigar poſcia i ſuoi principĳ; </s>
  <s xml:id="echoid-s47" xml:space="preserve">ma il diſcorrer’il Co-<lb/>lombo, che biſognaua per le tali, e tali ragioni, che vi <lb/>foſſero l’lndie nuoue, e poi trouarle, fù vn caminare <lb/>da’principĳ alla concluſione, come per appunto par, <lb/>che accada in questo propoſito: </s>
  <s xml:id="echoid-s48" xml:space="preserve">Dicendomi inſieme, <lb/>che chi era conſapeuole delle mie molte occupationi, <lb/>m’hauria ſcuſato, accettando volontieri per hora la <lb/>parte della ſpecolatiua, per vederne poi con più com-<lb/>modità la coſa ridotta in prattica, o per opera miæ, <lb/>o d’altri, c’habbino più agio, e commodità di farlo, &amp; </s>
  <s xml:id="echoid-s49" xml:space="preserve"><lb/>eſp@@ienza nell’arte di fondere, e di luſtrare iſpecchi <lb/>di metallo, non potendoſi queſto ottenere in grado <lb/>perfetto, ſe non da chi è stato per molto tempo sù <lb/>l’eſſe@citio di fabricarli. </s>
  <s xml:id="echoid-s50" xml:space="preserve">Queſte eſſortationi adun-<lb/>que ſurno potenti à farmi riſoluere di dare in luce <lb/>queste mie poche ſpecolationi intorno al detto ſog-<lb/>getto principalmente, hauendomi inſieme non poco <lb/>fatto accelerare queſta riſolutione l’hauer’io ſmar-<lb/>vito tal parte di quelle in alcune ſcritture, come ac-<lb/>cenno parimente nel Cap. </s>
  <s xml:id="echoid-s51" xml:space="preserve">31. </s>
  <s xml:id="echoid-s52" xml:space="preserve">che mi poteua far <lb/>dubitare di non eſſer da altri preoccupato nel farle <lb/>paleſi. </s>
  <s xml:id="echoid-s53" xml:space="preserve">Et benche io ſappi finalmente, che alcuni
<pb file="0014" n="14"/>
ſcorgendo à prima vista il titolo di S P E C C H I O <lb/>V S T O R I O, diranno queſta eſſer materia, della <lb/>quale ſe ne sà hormai, quanto ſe ne p@ò ſapère, ha-<lb/>uendo trattato pure de’Specchi Vſtorĳ Vitellione, <lb/>Rogerio Bacconi, Orontio, il Cardano, il Getaldo il <lb/>Porta, il P. </s>
  <s xml:id="echoid-s54" xml:space="preserve">Gruemberger, il P. </s>
  <s xml:id="echoid-s55" xml:space="preserve">Biancano, che ne toc-<lb/>@a vn poco nella ſua Echometria, e finalmente il Ma-<lb/>gini, &amp; </s>
  <s xml:id="echoid-s56" xml:space="preserve">al@ri, che con la loro eſquiſitezza di dottrina <lb/>ci hãno inſegnato tanto, che nõ laſciano luogo di poter <lb/>dir più coſa nuoua intorno à ſimil ſoggetto. </s>
  <s xml:id="echoid-s57" xml:space="preserve">A que ſto <lb/>per ò nõ riſponderò altro, ſe nõ che ſi cõptaccino queſti <lb/>tali di veder’vn poco tutto il Trattato prima, e poi <lb/>che giudichino, ſe la coſa stà così, come dal titolo gli <lb/>pare di poter’à prima fronte congetturare: </s>
  <s xml:id="echoid-s58" xml:space="preserve">Dirò ben <lb/>queſto ſolo, che ſe conſideraranno bene in particolare <lb/>il libro del Magin@, trouaranno, ch’egli non trat@ò <lb/>coſa alcuna de’Specchi Parabolici, Iperbolici, o El. <lb/></s>
  <s xml:id="echoid-s59" xml:space="preserve">littici, ma ſolo delle apparenze dello Specchio sferico; </s>
  <s xml:id="echoid-s60" xml:space="preserve"><lb/>e maſſime per quãto s’aſpetta al rappreſentar le ima-<lb/>gini, che come facile da fabricare in comparatione di <lb/>questi altri, potè anco da lui eſſer ridotto in pratti-<lb/>ca, &amp; </s>
  <s xml:id="echoid-s61" xml:space="preserve">acquiſtarſi quella lode, che all’eminẽza del ſuo <lb/>valore giustamente viene attribuita; </s>
  <s xml:id="echoid-s62" xml:space="preserve">ma non per-<lb/>ciò dourà stimarſi ſuperfluo quest altro mio Diſcor-<lb/>ſo, trattando egli di coſa molto differente da quella,
<pb file="0015" n="15"/>
che da eſſo venne ſpiegata. </s>
  <s xml:id="echoid-s63" xml:space="preserve">Con l’occaſione poi di que-<lb/>sto Specchio Vſtorio, vẽgono aggiũte al preſente T@at <lb/>tato alcune altre ſpecolationi, in particolare circa il <lb/>Suono, &amp; </s>
  <s xml:id="echoid-s64" xml:space="preserve">vni@erſalmente intorno à qualunque co-<lb/>ſa, che per linea retta ſi diffonda, come accade al Cal-<lb/>do, e Freddo ancora, &amp; </s>
  <s xml:id="echoid-s65" xml:space="preserve">ad altre qualità, diſcorren-<lb/>doſi inſieme qualche coſa circa il Moto; </s>
  <s xml:id="echoid-s66" xml:space="preserve">queste però <lb/>ſono da me ſoggiunte più per abbondanza, che <lb/>per neceſſità di dottrina, hauendole collegate inſieme <lb/>ſotto il titolo delle Settioni Coniche, che perciò l’hò <lb/>anco voluto metter’in fronte delle pagine, più toſto, <lb/>che il titolo di Specchio Vſtorio. </s>
  <s xml:id="echoid-s67" xml:space="preserve">Accetta dunque vo-<lb/>lontieri, benigno Lettore, quanto le mie poche forze <lb/>per hora ti offeriſcono, e col gradire il buon deſiderio, <lb/>che hò di ſeruire alla publica vtilità, fà, che <lb/>la mia debolezza auanzando ſe steſſa, <lb/>talmente ſi auualori, ch’arriui à <lb/>poter fare coſe maggiori.</s>
  <s xml:id="echoid-s68" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s69" xml:space="preserve">∵</s>
</p>
  <figure>
    <image file="0015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0015-01"/>
  </figure>
<pb file="0016" n="16"/>
</div>
<div xml:id="echoid-div5" type="section" level="1" n="5">
<head xml:id="echoid-head8" style="it" xml:space="preserve">Licenza del Reuerendiſs. P. Generale.</head>
<p>
  <s xml:id="echoid-s70" xml:space="preserve">NOi F. </s>
  <s xml:id="echoid-s71" xml:space="preserve">Girolamo Longhi da Milano Generale de’Fra-<lb/>ti Gieſuati di S. </s>
  <s xml:id="echoid-s72" xml:space="preserve">Girolamo, per le preſenti noſtre, con-<lb/>cediamo facoltà al R. </s>
  <s xml:id="echoid-s73" xml:space="preserve">P. </s>
  <s xml:id="echoid-s74" xml:space="preserve">F. </s>
  <s xml:id="echoid-s75" xml:space="preserve">Bonauentura Caualiet<unsure/>i Sacer <lb/>dote Profeſſo dell’iſteſs’O dine, e Lettor publico delle Ma-<lb/>tematiche in Bologna, di potere f@t ſtampare il Libro, in-<lb/>titolato, Lo Specchio V ſtorio, oſſeruando peiò le coſe ſolite <lb/>ad oſſeruat ſi in queſto geu<unsure/>ere. </s>
  <s xml:id="echoid-s76" xml:space="preserve">Date in Milano nel Con-<lb/>uento di S. </s>
  <s xml:id="echoid-s77" xml:space="preserve">Girolamo alli 7. </s>
  <s xml:id="echoid-s78" xml:space="preserve">di Genaro 1632.</s>
  <s xml:id="echoid-s79" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s80" xml:space="preserve">F. </s>
  <s xml:id="echoid-s81" xml:space="preserve">Girolamo Long hi Generale.</s>
  <s xml:id="echoid-s82" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s83" xml:space="preserve">IO F. </s>
  <s xml:id="echoid-s84" xml:space="preserve">Coſtantino Buci dell’Or<unsure/>dine de’G@ſuati di S. </s>
  <s xml:id="echoid-s85" xml:space="preserve">Gi-<lb/>rolamo, di commandamento del Reuerendiſs. </s>
  <s xml:id="echoid-s86" xml:space="preserve">P. </s>
  <s xml:id="echoid-s87" xml:space="preserve">Gen@-<lb/>rale, hò viſto il libto del M. </s>
  <s xml:id="echoid-s88" xml:space="preserve">R. </s>
  <s xml:id="echoid-s89" xml:space="preserve">P. </s>
  <s xml:id="echoid-s90" xml:space="preserve">F. </s>
  <s xml:id="echoid-s91" xml:space="preserve">Bonauentura Caualie-<lb/>ri dell’iſteſs’Ordine, intitolato, Lo Specchio Vſtorio, ne ha-<lb/>ucndoui trouato coſa, che contr<unsure/>ar@j alla Fede, o à buoni <lb/>coſtumi, perciò giudico, che ſi poſsi ſtampare.</s>
  <s xml:id="echoid-s92" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s93" xml:space="preserve">D. </s>
  <s xml:id="echoid-s94" xml:space="preserve">Homobonus de Bonis Pœnitentiarius, pro Eminentiſs. <lb/></s>
  <s xml:id="echoid-s95" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s96" xml:space="preserve">Reuerendiſs. </s>
  <s xml:id="echoid-s97" xml:space="preserve">D. </s>
  <s xml:id="echoid-s98" xml:space="preserve">Card. </s>
  <s xml:id="echoid-s99" xml:space="preserve">Archiepiſc.</s>
  <s xml:id="echoid-s100" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s101" xml:space="preserve">Imprimatnr.</s>
  <s xml:id="echoid-s102" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s103" xml:space="preserve">Fr. </s>
  <s xml:id="echoid-s104" xml:space="preserve">Hieronymus Onuphri@s Doct. </s>
  <s xml:id="echoid-s105" xml:space="preserve">Collegiatus, Lector pu-<lb/>blicus, &amp; </s>
  <s xml:id="echoid-s106" xml:space="preserve">Sanctiſs. </s>
  <s xml:id="echoid-s107" xml:space="preserve">Inquiſitio@is Conſultor, pro Reue-<lb/>rendiſs. </s>
  <s xml:id="echoid-s108" xml:space="preserve">P. </s>
  <s xml:id="echoid-s109" xml:space="preserve">Inquiſit. </s>
  <s xml:id="echoid-s110" xml:space="preserve">Bonon.</s>
  <s xml:id="echoid-s111" xml:space="preserve"/>
</p>
<pb file="0017" n="17"/>
</div>
<div xml:id="echoid-div6" type="section" level="1" n="6">
<head xml:id="echoid-head9" xml:space="preserve">TAVOLA</head>
<head xml:id="echoid-head10" xml:space="preserve">De’Capi del preſente Trattato.</head>
<note style="it" position="right" xml:space="preserve"> <lb/>INtroduttione alla materia da trattarſi, nella quale ſi diſcor-<lb/># re, d’onde habbi bauuto origine la dottrina delle Se@tioni <lb/># Coniche. # pag. 1 <lb/>Che coſa ſia cono, e come ſi generi. Cap. 1. # 9 <lb/>Che coſa ſiano Settioni Coniche, e come nel Cono ſi produchino. <lb/># Cap. 2. # 12 <lb/>Di quante ſorti di Settioni Coniche per il ſudetto ſegamento ſi poſ <lb/># ſono nel Cono generare. Cap. 3. # 13 <lb/>Che coſa ſiano le Settioni Oppoſte, e come ſi generino. Cap. 4. # 19 <lb/>Come dalle coſe dette nel ſudetto Capitolo potiamo con ageuolezz4<unsure/> <lb/># comprendere i fondamenti de gli Horology<unsure/> Solari. Cap. 5. # 21 <lb/>D’alc<unsure/>uni termini, che ſi ade<unsure/>prane intorno alle Settioni Con@che. <lb/># Cap. 6. # 22 <lb/>D’vn principio cauato dalla Proſpettiua per le coſe ſuſſeguenti. <lb/># Cap. 7. # 25 <lb/>Come ſi adatti queſto principio an@o alli Specchi, che non ſon piani. <lb/># Cap. 8. # 27 <lb/>Delle ammirabili proprietà delle Settioni Coniche, incominciandoſi <lb/># dalla prima della Parabola. Cap. 9. # 29 <lb/>Della ſeconda proprietà della Parabola. Cap. 10. # 33 <lb/>Della terza proprietà della parabola. Cap. 11. # 36 <lb/>Della quarta proprietà della Parabola. Cap. 12. # 38 <lb/>Qual@@e quãti ſiano nell’I perbola, Elliſſi, &amp; Oppoſte Settioni, i pun-<lb/># ti, che ſi chiamano fochi di quelle. Cap. 13. # 42 <lb/>Della prima proprietà dell’Iperbola. Cap. 14. # 45 <lb/>Della ſeconda proprietà dell’Iperbola. Cap. 15. # 47 <lb/>Della terza proprietà dell’Iperbola. Cap. 16. # 49 <lb/>Della quarta proprietà dell’Iperbola. Cap. 16. # 52 <lb/></note>
<pb file="0018" n="18"/>
</div>
<div xml:id="echoid-div7" type="section" level="1" n="7">
<head xml:id="echoid-head11" xml:space="preserve">TAVOLA</head>
<note style="it" position="right" xml:space="preserve"> <lb/>Della prima proprietà dell’Elliſſi. Cap. 17. # 54 <lb/>Della ſeconda proprietà dell’Elliſſi. Cap. 18. # 55 <lb/>Della terza p@oprietà dell’Elliſſi. Cap. 19. # 55 <lb/>Della quarta proprietà dell’Elliſſi. Cap. 20. # 57 <lb/>Della proprietà, ancor lei belliſſima, della circon ferenza di circolo <lb/># intorno alle incidenti, e rifleſſe. Cap. 21. # 60 <lb/>Delle ſuperficie, che ſi poſſono generare dalle Settioni Coniche, e <lb/># come à quelle ſi accommodino le già dimoſtrate loro proprietà, <lb/># e de’loro nomi. Cap. 22. # 63 <lb/>Epil go delle ſudette proprietà delle Settioni Coniche, applicate <lb/># alle da loro generate ſuperſicie. Cap. 23. # 65 <lb/>Tauola Specolaria. # 71 <lb/>Dell’vſo della precedente Tauola Specolaria. Cap. 24. # 73 <lb/>Digreſſione intorno le R@frattioni. # 7@ <lb/>Come ſi poſſi accendere il fuoco, per il rifleſſo de’raggi Solari. <lb/># Cap. 25. # 78 <lb/>Come per rif<unsure/>leſſione ſi poſſi accender fuoco con il riuerbero della fiã <lb/># @@ma, o deicarboni acceſi. Cap. 26. # 84 <lb/>Come in due <gap/> potiamo ſeruirci delli ſudetti Specchi. Cap. <lb/># 27. # 86 <lb/>Dello Specchio Vstovio d’Archimede. Cap. 28. # 89 <lb/>Della Linea Vſtoria di Gio. Battiſta Porta, che abbrucia in infini-<lb/># to. Cap. 29. # 96 <lb/>In qual ſenſo stimi l’Autore, che la ſudetta Linea Vstoria ſi poſſ@ <lb/># ſoſtenere. Cap. 29. # 98 <lb/>Dello Specchio Vſtorio imaginato dall’Autore, e varietà di quello. <lb/># Cap. 30. # 102 <lb/>Come ſi può probabilmente congetturare, che lo Specchio di Arcbi <lb/># mede, Proclo, e del Porta, non molto diſcordi da quello, che ſi è <lb/># dichiarato nel Capo antecedente. Cap. 31. # 111 <lb/>Come con li ſudetti Specchi potiamo di notte mandare il lume lon-<lb/># tano. Cap. 32. # 125 <lb/>Come potiamo ſentir quel ſuono, che per altro non s’vdirebbe, o ſen-<lb/></note>
<pb file="0019" n="19"/>
</div>
<div xml:id="echoid-div8" type="section" level="1" n="8">
<head xml:id="echoid-head12" xml:space="preserve">DE’CAPI.</head>
<note style="it" position="right" xml:space="preserve">
# tir meglio quello, che debolmente ſi ſence. Cap. 33. # 129
Come per il contrario potiamo inuigorir’il ſzono, ſi che ſia ſentito
# più gagliardo, che non ſi ſentirebbe. Cap. 34. # 131
Come ſi poſſa fabricare vna stanza talmente, che chi starà in vn’
# angolo d<unsure/> quella, ſenta il ſuono fatto nell’altr’angolo diamc<unsure/>tral
# mente oppoſto, non ſentendo quelli, che ſaranno nel mezo. Cap.
# 35. # 132
De i V aſi Teatrali di Vitruuio. Cap.36. # 134
Delle altre ſuperficie, che dal vario mouimento, ò fluſſo delle Set
# tioni Coniche poſſono eſſer generate. Cap. 37. # 148
Della cognitione del Moto. Cap. 38. # 151
Del mouimento de’corpi graui. Cap. 39. # 153
Qual ſorte di linea deſeriuano i graui nel loro moto, ſpiccati che ſia-
# no dal proiciente. Cap. 40. # 163
Come ſi deſcriuino le Settioni Coniche. Cap. 41. # 172
De i modi particolari di deſcriuere le Settioni Coniche, che s’aſpet-
# tano all’inuention ſolida. Cap. 42. # 174
De i modi particolari di deſcriuere le Settioni Coniche, che s’oſpet-
# tano all’inuention piana vera. Cap. 43. # 179
Come ſi deſcriua la Iperbola con vn filo, primo modo della inuen-
# tion piana vera. Cap. 44. # 182
Come ſi deſoriua la Parabola con vn filo, prime modo della inuen
# tion piana vera. Cap. 45. # 184
Come ſi deſcriua la Parabola, mediante gl’iſtrumenti ſodi, compo-
# sti di regoli, ch è il ſecondo modo dell’inuention piana vera.
# Cap. 46. # 187
Come ſ@ deſcriua la Iperbola con le righe, ſecondo modo dell’inuen-
# tion piana vera. Cap. 47. # 189
Come ſi deſcriua l’Elliſſi con le righe, ſecondo modo dell’inuention
# piana vera. Cap. 48. # 191
De i mod@ particolari di deſcriuere le Settioni Coniche, appartenẽ
# ti all’inuention piana per i punti continuati. Cap. 49. # 195
Come ſi deſcriua l’Iperbola, &amp; Elliſſi per i pũti cõtinuati. C. 50. # 198
</note>
<pb file="0020" n="20"/>
</div>
<div xml:id="echoid-div9" type="section" level="1" n="9">
<head xml:id="echoid-head13" xml:space="preserve">TAVOLA DE’CAPI.</head>
<note style="it" position="right" xml:space="preserve"> <lb/>D’vn’altra maniera molto facile, &amp; eſpediente di deſcriuere per i <lb/># punti continuati la Parabola, che habbi per foco vn determina <lb/># to punto. Cap. 51. # 201 <lb/>Come dalla Parabola ſi poſſono dedurre inſinite Iperbcle, che con <lb/># mirabile analogia vann<unsure/>o mutando i lati traſuerſi, mantenendo <lb/># però ſempre l’iſteſſo lato retto. Cap. 52. # 204 <lb/>In qual maniera ſi poſſi deſcriuere l’Iperbola equilatera, il cui foco <lb/># diſti dalla ſua cima, quante noi vo<unsure/>rremo. Cap. 53. # 209 <lb/>Come ſi deſcriua l’Elliſſi, c’babbi ciaſcuno de’ſuoi focbi distãti dall’ <lb/># estremit à dell’aſſe, quanto ſi voglia. Cap. 54. # 213 <lb/>D’altre maniere ancora di dedurre le Settioni Coniche vicendeuol-<lb/># mente l’vna dall’altra, o dalla circonferenz a del cerchio. Cap. <lb/># 55. &amp; vlt. # 219 <lb/></note>
</div>
<div xml:id="echoid-div10" type="section" level="1" n="10">
<head xml:id="echoid-head14" style="it" xml:space="preserve">IL FINE.</head>
  <figure>
    <image file="0020-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0020-01"/>
  </figure>
<pb o="1" file="0021" n="21"/>
</div>
<div xml:id="echoid-div11" type="section" level="1" n="11">
<head xml:id="echoid-head15" xml:space="preserve">INTRODVTTIONE</head>
<head xml:id="echoid-head16" style="it" xml:space="preserve">Alla materia da trattarſi, <lb/>Nella quale ſi diſcorre d’onde habbi hauuto origine <lb/>la dottrina delle Settioni Coniche.</head>
<p>
  <s xml:id="echoid-s112" xml:space="preserve">FV@ da gli antichi Filoſofi, che di <lb/>ben regolare le operationi hu-<lb/>mane ſi preſero cura, in tãto pre-<lb/>gio, e ſtima tenuto ſempre il tem-<lb/>po, che non ſolo con eleganza di <lb/>parole, ma con induſtrioſe operationi ancora, <lb/>non mancorno giamai di farne capaci, quan-<lb/>to importaſſe il ben compartirlo nelle noſtre <lb/>attioni. </s>
  <s xml:id="echoid-s113" xml:space="preserve">Così ſolea dir Democrito quello eſ-<lb/>ſere vna pretioſiſſima ſpeſa. </s>
  <s xml:id="echoid-s114" xml:space="preserve">Talete lo chia-<lb/>maua ſapientiſſimo in natura. </s>
  <s xml:id="echoid-s115" xml:space="preserve">Seneca lo com-<lb/>paraua ad vn fiume rapidiſſimo. </s>
  <s xml:id="echoid-s116" xml:space="preserve">Biante di-<lb/>ceua douerſi talmente diſpenſare, come ſe aſ-<lb/>ſai, e poco doueſſimo campare. </s>
  <s xml:id="echoid-s117" xml:space="preserve">Marco Var-<lb/>rone riſolutamente affermaua nõ eſſerui per-<lb/>dita più graue di quella del Tempo; </s>
  <s xml:id="echoid-s118" xml:space="preserve">onde il <lb/>Prencipe de’Poeti di queſto irreparabil dan-<lb/>no ci volſe amm onire anch’egli nel 3. </s>
  <s xml:id="echoid-s119" xml:space="preserve">della@. <lb/></s>
  <s xml:id="echoid-s120" xml:space="preserve">Geor. </s>
  <s xml:id="echoid-s121" xml:space="preserve">dicendo.</s>
  <s xml:id="echoid-s122" xml:space="preserve"/>
</p>
<pb o="2" file="0022" n="22"/>
<p>
  <s xml:id="echoid-s123" xml:space="preserve">Et fugit interea fugit irreparabile tempus. <lb/></s>
  <s xml:id="echoid-s124" xml:space="preserve">c<unsure/> Ouidio nel 6. </s>
  <s xml:id="echoid-s125" xml:space="preserve">de’Faſti.</s>
  <s xml:id="echoid-s126" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s127" xml:space="preserve">Tempora labuntur, tacitiſq; </s>
  <s xml:id="echoid-s128" xml:space="preserve">ſeneſcimus annis, <lb/>Et fugiunt fræno non remorante dies.</s>
  <s xml:id="echoid-s129" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s130" xml:space="preserve">Queſto è il Serpe de gli Egittij, che il tutto cõ-<lb/>prende; </s>
  <s xml:id="echoid-s131" xml:space="preserve">queſto è il Baſiliſco, che ſi rende fra <lb/>gli altri ſer penti così contumace al morire, <lb/>queſt’è la Falce di Saturno, ch’ogni coſa mie-<lb/>te, ogni coſa recide. </s>
  <s xml:id="echoid-s132" xml:space="preserve">Vedẽdo adunque quan-<lb/>to egli foſſe pretioſo, ma dall’altro cãto quan-<lb/>to volubile, e fugace, e quanta poca parte ne <lb/>foſſe per toccar’à ciaſcun’huomo, s’ingegnor-<lb/>no i più ſottili di trouar modo più ſicuro, che <lb/>foſſe poſſibile di miſurare, come foſſe tant’oro, <lb/>vna coſa di sì alto pregio; </s>
  <s xml:id="echoid-s133" xml:space="preserve">e vedẽdo, che il tẽpo <lb/>era vna ſcaturigine del moto, o per dir meglio <lb/>vna miſura di quello, poiche dice pur’Ariſt. </s>
  <s xml:id="echoid-s134" xml:space="preserve">nel <lb/>4. </s>
  <s xml:id="echoid-s135" xml:space="preserve">della Fiſica al Teſ. </s>
  <s xml:id="echoid-s136" xml:space="preserve">101. </s>
  <s xml:id="echoid-s137" xml:space="preserve">Tẽpus eſt numerus mo-<lb/>tus ſecundum prius, &amp; </s>
  <s xml:id="echoid-s138" xml:space="preserve">poſterius, e perciò douerſi <lb/>quello ſcompartire, per hauerne il tempo, e <lb/>queſto potendo eſſere, e nelle coſe à noi vici-<lb/>ne, e nelle lontane: </s>
  <s xml:id="echoid-s139" xml:space="preserve">Furono alcuni, che ſi pre-<lb/>ualſero del moto vicino, cioè del cõtinuo fla<unsure/>f-<lb/>ſo dell’acqua, o della poluere, o del girar delle <lb/>ruote per via de’cõtrapeſi, o di molle gagliar-
<pb o="3" file="0023" n="23"/>
de. </s>
  <s xml:id="echoid-s140" xml:space="preserve">Così ſi trouorno gli Arenarij, gli Horolo-<lb/>gij da ruote, e le Cleſſidri, &amp; </s>
  <s xml:id="echoid-s141" xml:space="preserve">il primo, che à <lb/>Roma le faceſſe vedere fù Scipion Naſica <lb/>l’anno 594. </s>
  <s xml:id="echoid-s142" xml:space="preserve">doppo l’edificatione di Roma. <lb/></s>
  <s xml:id="echoid-s143" xml:space="preserve">Fra gli Autori poi, che ſi preualſero del moto <lb/>lontano, cioè di quel delle ſtelle fiſſe, per mi-<lb/>ſurar’il tempola notte, poterno ben ſeruirſi di <lb/>quelle, e maſſime delle ſempre apparenti nel-<lb/>la ſua regione, come noi dell’Orſa minore, o <lb/>maggiore, del Dragone, di Ceffeo, o di Caſ-<lb/>ſiopea, tralaſciãdone i Pianeti, come ſoggetti <lb/>à diuerſi accidẽti, come d’irregolarità di moti <lb/>apparenti, di ſtationi, direttioni, e retrogra-<lb/>dationi, o di parallaſſi, e ſimili, eccettuatone <lb/>però il Sole; </s>
  <s xml:id="echoid-s144" xml:space="preserve">ma per miſurar’il tempo di gior-<lb/>no, non gli reſtò altro, che il moto, e lume pur <lb/>del Sole, che nell’abiſſo del ſuo ſplẽdore i pic-<lb/>coliſſimi lumi delle ſtelle ci naſconde; </s>
  <s xml:id="echoid-s145" xml:space="preserve">egli è <lb/>però vero, che nel principio aſſai rozamente <lb/>parue, che ſi portaſſero, come accader ſuole <lb/>di tutte le nuoue inuentioni, poiche in Roma <lb/>particolarmente, Città così inſigne, nõ ſi no-<lb/>taua altro, che in dodici Tauole l’Orto, e l’Oc-<lb/>caſo del Sole, quali ſi eſponeuano publicamẽ-<lb/>te; </s>
  <s xml:id="echoid-s146" xml:space="preserve">al che doppo alcuni anni ſi aggiunſe anco-
<pb o="4" file="0024" n="24"/>
ra il momento del Mezo giorno, e queſto però <lb/>ſolamente ne i tem pi ſereni, qual ſi manifeſta-<lb/>ua al popolo per vn publico banditore, come <lb/>racconta Plinio nel lib. </s>
  <s xml:id="echoid-s147" xml:space="preserve">8. </s>
  <s xml:id="echoid-s148" xml:space="preserve">il che durò ſino alla <lb/>prima guerra contro Cartagineſi: </s>
  <s xml:id="echoid-s149" xml:space="preserve">Eſſendo fi-<lb/>nalmente poi da M. </s>
  <s xml:id="echoid-s150" xml:space="preserve">Val. </s>
  <s xml:id="echoid-s151" xml:space="preserve">Meſſaia Conſ collo-<lb/>cato appreſſo i Roſtri in vna colõna l’Horolo-<lb/>gio Solare, che moſtraua tutte le hore del gior <lb/>no compitamente, ſecondo che dice M. </s>
  <s xml:id="echoid-s152" xml:space="preserve">Var-<lb/>rone; </s>
  <s xml:id="echoid-s153" xml:space="preserve">ſimile al quale parimẽte n’hebbero vno <lb/>ancora i Lacedemonij per opera di Anaſſime-<lb/>ne Mileſio. </s>
  <s xml:id="echoid-s154" xml:space="preserve">Altri ſecondo le diuerſe ſuperfi-<lb/>cie, nelle quali diſſegnorno l’Horologio, tro-<lb/>uorno varij modelli, e gli diedero diuerſi no-<lb/>mi; </s>
  <s xml:id="echoid-s155" xml:space="preserve">così Beroſo Caldeo inuentò l’Emiciclio, <lb/>Ariſtarco Samio la Scaffa, &amp; </s>
  <s xml:id="echoid-s156" xml:space="preserve">il Diſco nel pia-<lb/>no, Eudoſſo la Rete, Scopa Siracuſano il Plin-<lb/>to, e Dioniſiodoro il Cono.</s>
  <s xml:id="echoid-s157" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s158" xml:space="preserve">Ma vaglia à dire il vero, che frà tutti que-<lb/>ſti modi quello è ſempre ſtato più ricercato, e <lb/>ſtimato, che c’inſegna deſcriuer l’Horologio <lb/>Solare in vna ſuperficie piana, poiche l’haue-<lb/>re i noſtri edificij, e le noſtre fabriche, circon-<lb/>date per il più da muraglie piane, sforzò que-<lb/>ſti ingegni ad applicarſi all’inueſtigare il mo-
<pb o="5" file="0025" n="25"/>
do di diſſegnarli ne i piani, non ſolo orizonta-<lb/>li, ma anco verticali, o inchinati in qualſiuo-<lb/>glia maniera, più che nelle altre ſuperficie. <lb/></s>
  <s xml:id="echoid-s159" xml:space="preserve">E perche nella diſſegnatione di queſti Horo-<lb/>logij vid dero, che l’eſtremità delle linee hora-<lb/>rie o foſſero dall’occaſo, o dall’orto del Sole, <lb/>o dal mezo dì, o meza notte computate, anda-<lb/>uano tutze à terminare in due linee curue, che <lb/>haueuano le ſue conueſſità cõtrapoſte, riſpon-<lb/>denti alli due Tropici in Cielo, che ſi mutaua-<lb/>no ſecondo la varia ſituatione de’piani, ne’ <lb/>quali ſi deſcriueua l’Horologio, hebbero mo-<lb/>tiuo, che queſte linee curue poteuano cadere <lb/>ſotto regola, e che non naſceuano da altro, <lb/>che dall’interſegatione del piano dell’Horolo-<lb/>gio con le ſuperficie de’duoi Coni cõtrapoſti, <lb/>che hanno per cima commune il centro del <lb/>mondo, e per baſe i Tropici, cioè, che nõ era-<lb/>no altro, che Settioni del Cono, chiamate poi <lb/>Iperbole contra poſte (quali ſaranno fra poco <lb/>dichiarate che coſa ſiano) perciò queſta fù la <lb/>cauſa primaria, e potiſſima, per la quale i Geo-<lb/>metri ſi aſſottigliorno nel cõſiderare in quan-<lb/>ti modi vn piano potea ſegare duoi Coni, oue-<lb/>ro vn ſolo, ſi che le communi Settioni del pia-
<pb o="6" file="0026" n="26"/>
no ſegãte, con le loro ſuperficie, veniſſero dif-<lb/>ferenti, e così ſi accorſero, che quando vno de’ <lb/>Tropici foſſe reſtato tutto ſopra il piano dell’ <lb/>Orizonte, come nell’eleuatione del Polo di <lb/>G. </s>
  <s xml:id="echoid-s160" xml:space="preserve">66. </s>
  <s xml:id="echoid-s161" xml:space="preserve">30. </s>
  <s xml:id="echoid-s162" xml:space="preserve">minuti in circa, e nella maggior di <lb/>queſta, non già più la predetta ſorte di Settio-<lb/>ne, ma altre veniuano à naſcere nell’horolo-<lb/>gio orizontale, quali chiamorno Parabola, &amp; </s>
  <s xml:id="echoid-s163" xml:space="preserve"><lb/>Elliſſi, e talhor’anco Circolo, perciò ſupponẽ-<lb/>do, che dalla cognition di queſte Settioni pẽ-<lb/>deſſe la fondamẽtal dottrina de gli Horologij <lb/>Solari, la maggior parte ſi diede à ſpecolare <lb/>intorno à queſte, e di quì nacque la dottrina <lb/>delle Settioni Coniche. </s>
  <s xml:id="echoid-s164" xml:space="preserve">Così Platone fù il pri-<lb/>mo, che vi applicaſſe il penſiero, che poi mi-<lb/>rabilmente ſe ne ſeruì anco per riſoluer’il Del-<lb/>fico Problema, come ottenne con l’incrocia-<lb/>mẽto di due Parabole. </s>
  <s xml:id="echoid-s165" xml:space="preserve">Euclide ne ſcriſſe quat-<lb/>tro libri, come anco Ariſteo, Menechmo, Ar-<lb/>chimede, A pollonio Pergeo; </s>
  <s xml:id="echoid-s166" xml:space="preserve">e finalmente per <lb/>queſta via la dottrina de gli Horologij Solari <lb/>s’è talmente perfettionata, che pare nõ ſi poſſi <lb/>paſſar più oltre. </s>
  <s xml:id="echoid-s167" xml:space="preserve">Altri poi più profondamente <lb/>ſpecolando, viddero, che le Settioni Coniche <lb/>haueuano che fare in altri effetti di Natura
<pb o="7" file="0027" n="27"/>
ancora, &amp; </s>
  <s xml:id="echoid-s168" xml:space="preserve">in particolare, che l’accender fuo-<lb/>co in virtù de’raggi ſolari, l’inuigorire il ſuo-<lb/>no, e ſimili effetti, originauano parimente da <lb/>quelle, la onde non è mãcato, chi habbi ſcrit-<lb/>to de’Specchi Vſtorij, come Vitellione, Oron-<lb/>tio, il Getaldo, il Porta, &amp; </s>
  <s xml:id="echoid-s169" xml:space="preserve">altri, moſtrando in <lb/>queſte coſe ancora l’eccellẽza di tal dottrina. <lb/></s>
  <s xml:id="echoid-s170" xml:space="preserve">L’intention mia dunque non è già di voler’in-<lb/>ueſtigar le più aſtruſe, e recondite proprietà <lb/>di queſte Settioni, come altri hãno fatto; </s>
  <s xml:id="echoid-s171" xml:space="preserve">o di <lb/>trattare ex profeſſo la materia de gli Horolo-<lb/>gij Solari, mettendone io il Cap. </s>
  <s xml:id="echoid-s172" xml:space="preserve">5. </s>
  <s xml:id="echoid-s173" xml:space="preserve">ſolo, per <lb/>darne vna tale, e quale cognitione, à chi non <lb/>ne ſapeſſe coſa alcuna; </s>
  <s xml:id="echoid-s174" xml:space="preserve">o, volend’io pur trat-<lb/>tare de’Specchi Vſtorij, di ſtar ſolamente sù <lb/>quello, che hanno ſcritto i ſudetti Autori, ma <lb/>di moſtrare, come potiamo probabilmẽt<unsure/>e cre-<lb/>dere, che la ſtruttura dello Specchio d’Archi-<lb/>mede tutta dipenda dalle Settioni Coniche <lb/>ben’inteſe; </s>
  <s xml:id="echoid-s175" xml:space="preserve">queſto è il mio principale intento, <lb/>al che aggiungo poi, con tale occaſione, altre <lb/>ſpecolationi naturali, maſſime intorno al ſuo-<lb/>no, che forſi non ſaranno ingrate: </s>
  <s xml:id="echoid-s176" xml:space="preserve">Ma perche <lb/>chi vuole intender queſte coſe da’fondamẽti, <lb/>hà di biſogno della cognitione d’alcune pro-
<pb o="8" file="0028" n="28" rhead="Delle Settioni Coniche."/>
prietà delle dette Settioni, perciò hò voluto <lb/>iſtruir’il Lettore, come che non haueſſe viſto <lb/>coſa veruna in queſto genere, nel che mi per-<lb/>donerãno gl’intelligenti, ſe alla loro ſofficien-<lb/>za queſto gli pareſſe ſouerchio, poiche ciò non <lb/>è fatto per loro; </s>
  <s xml:id="echoid-s177" xml:space="preserve">e perche le coſe cõmuni ven-<lb/>gono ſa pute, e ſcritte da molti, perciò mi ſcu-<lb/>ſeranno parimente, trouando quà alcune co-<lb/>ſe poſte da altri ancora, ſe ben credo le ragio-<lb/>ni ſaranno per il più differenti, poiche per ſer-<lb/>uare l’ordine della dottrina, e per non rimet-<lb/>ter’il Lettore ad altri libri, mi è parſo ben fat-<lb/>to il raccoglier quà tutto ciò, che li può biſo-<lb/>gnare in queſta materia: </s>
  <s xml:id="echoid-s178" xml:space="preserve">La dottrina fonda-<lb/>mentale adunque viene da me trattata ſino al <lb/>Cap. </s>
  <s xml:id="echoid-s179" xml:space="preserve">24. </s>
  <s xml:id="echoid-s180" xml:space="preserve">quale à chi rincreſceſſe, baſterà ve-<lb/>derne ſolamente li Cap. </s>
  <s xml:id="echoid-s181" xml:space="preserve">22. </s>
  <s xml:id="echoid-s182" xml:space="preserve">23. </s>
  <s xml:id="echoid-s183" xml:space="preserve">24. </s>
  <s xml:id="echoid-s184" xml:space="preserve">cercando <lb/>almeno d’intendere i nomi: </s>
  <s xml:id="echoid-s185" xml:space="preserve">Dal Cap. </s>
  <s xml:id="echoid-s186" xml:space="preserve">23. </s>
  <s xml:id="echoid-s187" xml:space="preserve">ſino <lb/>al 41. </s>
  <s xml:id="echoid-s188" xml:space="preserve">s’applica poi la dottrina antecedente al-<lb/>la materia; </s>
  <s xml:id="echoid-s189" xml:space="preserve">e nel rimanente s’inſegnano varie <lb/>deſcrittioni delle dette Settioni, della qual <lb/>parte baſterà vederne quel tanto, che più pia-<lb/>cerà; </s>
  <s xml:id="echoid-s190" xml:space="preserve">ma tempo è hormai di dar principio à <lb/>queſta dottrina.</s>
  <s xml:id="echoid-s191" xml:space="preserve"/>
</p>
<pb o="9" file="0029" n="29"/>
</div>
<div xml:id="echoid-div12" type="section" level="1" n="12">
<head xml:id="echoid-head17" xml:space="preserve">Che coſa ſia Cono, e come <lb/>ſi generi. Cap. I.</head>
<p>
  <s xml:id="echoid-s192" xml:space="preserve">IL Cono è quel corpo ſolido, che <lb/>da’prattici ſuole eſſer chiama-<lb/>mato, Piramide rotonda, che <lb/>fù da Euclide nell’11. </s>
  <s xml:id="echoid-s193" xml:space="preserve">lib. </s>
  <s xml:id="echoid-s194" xml:space="preserve">alla <lb/>def. </s>
  <s xml:id="echoid-s195" xml:space="preserve">18. </s>
  <s xml:id="echoid-s196" xml:space="preserve">(preſo in ſenſo men’ <lb/>vniuerſale) definito, naſcer dalla reuolutione <lb/>del triangolo rettangolo, ſtan do fermo vn de’ <lb/>lati, che ſtanno intorno all’angolo retto, ſino, <lb/>che eſſo triangolo ritorni di onde ſi partì: </s>
  <s xml:id="echoid-s197" xml:space="preserve">ma <lb/>p<unsure/>erche queſta definitione cõprende folamen-<lb/>toei Coni, che hanno l’aſſe della reuolutione <lb/>perpẽdicolare alla baſe; </s>
  <s xml:id="echoid-s198" xml:space="preserve">perciò riceueremo da <lb/>A pollonio Pergeo la definitione vniuerſale, <lb/>poſta nel principio de’ſuoi Elementi Conici, <lb/>in queſta maniera. </s>
  <s xml:id="echoid-s199" xml:space="preserve">Se da vn punto poſto fuo-<lb/>ri del piano d’vn dato circolo ſarà tirata vna <lb/>retta linea ſino alla circonferenza di eſſo cir-<lb/>colo, di quà, e di là indefinitamente prolon-<lb/>gata, quale ſi riuolga intorno alla circonfe-<lb/>renza ſino, che ritorni di onde ſi partì; </s>
  <s xml:id="echoid-s200" xml:space="preserve">la ſu-
<pb o="10" file="0030" n="30" rhead="Delle Settioni"/>
perficie deſcritta dalla detta linea, ſi chiame-<lb/>rà ſuperficie conica, e Cono ſi dirà il ſolido <lb/>rinchiuſo dalla detta ſuperficie, e dal circolo <lb/>propoſto, qual vien chiamato baſe del Cono, <lb/>e cima il ponto ſoprapreſo; </s>
  <s xml:id="echoid-s201" xml:space="preserve">aſſe poi vien det-<lb/>ta la retta linea, che congiunge eſſa cima con <lb/>il centro del circolo, che è di lui baſe, quale, <lb/>quando ſtà perpendicolarmente ſopra la baſe, <lb/>fà, che il Cono ſi chiami equicrure, e quando <lb/>ſia inchinato ſopra di quella, fà, che ſi dica Co-<lb/>no ſcaleno; </s>
  <s xml:id="echoid-s202" xml:space="preserve">di quelli s’intende la definitione <lb/>d’Euclide, e di queſti quella d’Apollonio, dẽ-<lb/>tro la quale vengono parimẽte rinchiuſi i Co-<lb/>ni d’Euclide, per eſſer queſta più vniuerſale, <lb/>e però baſterà, che noi ci appigliamo à que-<lb/>ſta, per farci capaci d’ambedue le ſorti de’Co-<lb/>ni in vn ſol colpo, il che più chiaramente s’in-<lb/>tenderà dalle quì poſte figure.</s>
  <s xml:id="echoid-s203" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div13" type="section" level="1" n="13">
<head xml:id="echoid-head18" xml:space="preserve">Eſſempio ſopra la prima Figura.</head>
<p style="it">
  <s xml:id="echoid-s204" xml:space="preserve">S Ia il triangolo, A B E, che hà l’angolo, A E B, <lb/>retto, e riuolgaſi eſſo triangolo, A B E, intor-<lb/>no all’, A E, fiſſa, ſin che ritorni di onde ſi par-<lb/>tì; </s>
  <s xml:id="echoid-s205" xml:space="preserve">la, B E, adunque deſcriuerà il circolo, B G, il
<pb o="11" file="0031" n="31" rhead="Coniche. Cap. I."/>
cui diametro è, B G, &amp; </s>
  <s xml:id="echoid-s206" xml:space="preserve">il triangolo deſcriuerà il <lb/>ſolido, A B G, che da Euclide vien chiamato Cono, <lb/>&amp; </s>
  <s xml:id="echoid-s207" xml:space="preserve">è equicrure, per eſſer l’aſſe, A E, perpendicolare <lb/>alcircolo, B G. </s>
  <s xml:id="echoid-s208" xml:space="preserve">Sia horail circola, N P, fuori del <lb/>cui piano ſia preſo il punto, C, e da eſſo tirata la, C <lb/>N, alla circonferenza del circolo, N P, &amp; </s>
  <s xml:id="echoid-s209" xml:space="preserve">indefini-<lb/>tamente prolongata, come in, D, M, e s’intenda ri-<lb/>uolgerſi la retta, D M, per la circonferenza del cir-<lb/>colo, N P, ſopra il ponto fiſſo, C, ſino che ritorni di <lb/>onde ſi parti; </s>
  <s xml:id="echoid-s210" xml:space="preserve">la ſuperficie dunque deſcritta da tal <lb/>linea, non ſolo dal ponto, C, verſola baſe, N P, ma <lb/>anco verſo la parte opposta, cioè verſo, D, vien <lb/>da Apollonio chiamata ſuperficie conica, &amp; </s>
  <s xml:id="echoid-s211" xml:space="preserve">il ſolido, <lb/>C N P, compreſo dalla ſuperficie conica verſo, N P, <lb/>e<unsure/> dal circolo, N P, vien chiamato Cono, e cima il pon-<lb/>to, C, baſe il circolo, N P, &amp; </s>
  <s xml:id="echoid-s212" xml:space="preserve">aſſe la retta, C O che <lb/>congiunge la cima, cioè il ponto, C, con il centro del <lb/>circolo, N P, che ſia, O; </s>
  <s xml:id="echoid-s213" xml:space="preserve">quale può eſſer, che ſia per-<lb/>pendicolare ſopra la baſe, come nell’altra figura è la, <lb/>A E, (poiche anco la generatione del Cono, A B G, <lb/>benche equicrure, ſi può intendere al modo d’Apol-<lb/>lonio) e può eſſer, che vi ſtia inchinata, come la, C O; <lb/></s>
  <s xml:id="echoid-s214" xml:space="preserve">nel qual caſo tal Cono ſi chiama ſc<unsure/>aleno; </s>
  <s xml:id="echoid-s215" xml:space="preserve">e questo <lb/>baſti per intendere, che coſa ſia Cono, e come ſi generi.</s>
  <s xml:id="echoid-s216" xml:space="preserve"/>
</p>
<pb o="12" file="0032" n="32" rhead="Delle Settioni"/>
</div>
<div xml:id="echoid-div14" type="section" level="1" n="14">
<head xml:id="echoid-head19" xml:space="preserve">Corollario.</head>
<p style="it">
  <s xml:id="echoid-s217" xml:space="preserve">DI quì ſi fà manifeſto, che la parte, C D, de-<lb/>ſcriue anch’eſſa ſuperficie conica, la quale <lb/>terminando nella circonferenza del circolo, <lb/>H D, di che grandezza ſi vogli, ma parallelo al cir-<lb/>colo, N P, rinchiude con eſſo ctrcolo, H D, il Cono, <lb/>H C D, dalla parte oppoſta al circolo, N P; </s>
  <s xml:id="echoid-s218" xml:space="preserve">che però <lb/>ſi può chiamare Cono inuerſo, in riſpetto del Cono, <lb/>N C P.</s>
  <s xml:id="echoid-s219" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div15" type="section" level="1" n="15">
<head xml:id="echoid-head20" style="it" xml:space="preserve">Che coſa ſi ano Settioni Coniche, e come nel Cono <lb/>ſi produchino. Cap. II.</head>
<p>
  <s xml:id="echoid-s220" xml:space="preserve">COncioſiacoſa, che il Cono poſsi <lb/>eſſer ſegato, ouer troncato da <lb/>diuerſe ſorti di ſuperficie, hora <lb/>però non intenderemo, che ſia <lb/>ſegato con altro, che con ſuper-<lb/>ficie piane. </s>
  <s xml:id="echoid-s221" xml:space="preserve">Fà dunque di meſtieri andar con-<lb/>ſiderãdo in quanti modi ſia poſſibile tagliarlo, <lb/>ſi che ne venghino fatte differenti Settioni di <lb/>ſpecie; </s>
  <s xml:id="echoid-s222" xml:space="preserve">e perche il commun ſegamento di due <lb/>ſuperficie è ſempre linea; </s>
  <s xml:id="echoid-s223" xml:space="preserve">perciò intenden do <lb/>noi, che vn piano tagli il Cono, in che modo ſi
<pb o="13" file="0033" n="33" rhead="Coniche. Cap. II."/>
voglia, taglia anco la ſuperficie di eſſo Cono, <lb/>compoſta della ſuperficie conica, e della baſe, <lb/>talhora ambedue, e talhor la conica ſola, e pe-<lb/>rò nella ſuperficie di eſſo Cono vien ſempre <lb/>generata vna linea, ch’è il commun ſegamen-<lb/>to della ſuperficie del Cono, e del piano ſegã-<lb/>te: </s>
  <s xml:id="echoid-s224" xml:space="preserve">Queſta linea adunque può con nome com-<lb/>mune dirſi Settion Conica, ſe ben’Apollonio <lb/>non ſuol chiamare ogni tal linea Settion Co-<lb/>nica, ma ſolamente alcune, come quì da baſ-<lb/>ſo s’intenderà.</s>
  <s xml:id="echoid-s225" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div16" type="section" level="1" n="16">
<head xml:id="echoid-head21" style="it" xml:space="preserve">Di quante ſorti di Settioni Coniche per il ſudetto <lb/>ſegamenio ſi poſſono nel Cono generare. <lb/>Cap. III.</head>
<p>
  <s xml:id="echoid-s226" xml:space="preserve">LE Settioni Coniche, preſe nel <lb/>ſenſo commune di ſopra dichia-<lb/>rato, non poſſono eſſere più di <lb/>cinque.</s>
  <s xml:id="echoid-s227" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s228" xml:space="preserve">La prima è, quando il piano <lb/>ſegante taglia il Cono per la lunghezza dell’ <lb/>aſſe, ò almeno per la cima; </s>
  <s xml:id="echoid-s229" xml:space="preserve">e ſempre ſe ne pro-<lb/>ducono tre linee rette, due nella ſuperficie co-<lb/>nica, &amp; </s>
  <s xml:id="echoid-s230" xml:space="preserve">vna nella baſe, le quali ſono il perime-
<pb o="14" file="0034" n="34" rhead="Delle Settioni"/>
tro d’vn triãgolo, che paſſa per l’aſſe dell’iſteſ-<lb/>ſo Cono, quando il ſegamento è per l’aſſe, co-<lb/>me dimoſtrò Apollonio nel 1. </s>
  <s xml:id="echoid-s231" xml:space="preserve">de’Conici alla <lb/>prop. </s>
  <s xml:id="echoid-s232" xml:space="preserve">3. </s>
  <s xml:id="echoid-s233" xml:space="preserve">moſtrando inſieme produrſi ſempre <lb/>triangolo, benche il piano ſegante non paſsi <lb/>per l’aſſe, ma ſolamente paſsi per la cima di <lb/>eſſo Cono.</s>
  <s xml:id="echoid-s234" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s235" xml:space="preserve">La ſeconda è, quando il piano ſegante è pa-<lb/>rallelo alla baſe; </s>
  <s xml:id="echoid-s236" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s237" xml:space="preserve">allhora ſe ne produce cir-<lb/>conferenza di circolo, per la 4. </s>
  <s xml:id="echoid-s238" xml:space="preserve">del 1. </s>
  <s xml:id="echoid-s239" xml:space="preserve">de’Co-<lb/>nici; </s>
  <s xml:id="echoid-s240" xml:space="preserve">ma queſte due non ſogliono propriamen-<lb/>te eſſer chiamate Settioni Coniche, dando ſolo <lb/>tal nome à queſte tre vltime.</s>
  <s xml:id="echoid-s241" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s242" xml:space="preserve">La terza parimente ſi fà, quando eſſendoſi <lb/>prima ſegato il Cono per l’aſſe con vn piano, e <lb/>prodottoſene il triangolo, che paſſa per l’aſſe, <lb/>ſi ſega poi con vn’altro piano, sì il Cono, co-<lb/>me il triangolo già fatto, &amp; </s>
  <s xml:id="echoid-s243" xml:space="preserve">anco la baſe di eſ-<lb/>ſo Cono, in tal maniera, che la retta linea, che <lb/>vien prodotta nella baſe, ſia perpendicolare <lb/>alla baſe del detto triangolo, e quella, che vien <lb/>diſegnata nel triangolo per l’aſſe, ſia parallela <lb/>ad vn de’lati del detto triangolo; </s>
  <s xml:id="echoid-s244" xml:space="preserve">la linea dun-<lb/>que diſegnata nella ſuperficie conica da detto <lb/>piano ſegante, da Apollonio nel 1. </s>
  <s xml:id="echoid-s245" xml:space="preserve">libro alla
<pb o="15" file="0035" n="35" rhead="Coniche. Cap. III."/>
propoſ. </s>
  <s xml:id="echoid-s246" xml:space="preserve">11. </s>
  <s xml:id="echoid-s247" xml:space="preserve">vien chiamata Parabola, ch’èla <lb/>prima delle Settioni, che da Apollonio ſon <lb/>chiamate Coniche.</s>
  <s xml:id="echoid-s248" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s249" xml:space="preserve">La quarta ſi produrrà, quãdo, ſtando le me-<lb/>deſime coſe dette per la terza, ſolo ſarà varie-<lb/>tà in queſto, che la linea diſegnata dal piano <lb/>ſegante nel triangolo per l’aſſe in vece d’eſſer <lb/>parallela, ſarà concorrente con vn de’lati di <lb/>detto triangolo fuori della cima del Cono, e <lb/>la linea diſegnata dal piano ſegante, che è la <lb/>quarta Settione, e la ſeconda appreſſo Apol-<lb/>lonio, viene da lui alla prop. </s>
  <s xml:id="echoid-s250" xml:space="preserve">11. </s>
  <s xml:id="echoid-s251" xml:space="preserve">del primo di-<lb/>mandata Iperbola.</s>
  <s xml:id="echoid-s252" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s253" xml:space="preserve">La quinta ſi hauerà finalmente, quando <lb/>ſtando le medeſime coſe dette per la Parabo-<lb/>la, e per l’lperbola, ſolo vi ſarà variatione in <lb/>queſto, che la linea diſegnata nel triãgolo per <lb/>l’aſſe dal piano ſegante, in vece d’eſſer paral-<lb/>lela, ò concorrente con vn de’lati fuori della <lb/>cima del detto triangolo, taglierà ambedue i <lb/>lati di quello (non eſſendo però il piano ſegan-<lb/>te parallelo alla baſe del Cono, ò ſubcontraria-<lb/>mente poſto, poiche ſe ne produrria circolo) <lb/>e ſarà tal Settione la linea diſegnata dal piano <lb/>ſegãte nella ſuperficie conica, chiamata Eliſsi
<pb o="16" file="0036" n="36" rhead="Delle Settioni"/>
da Apollonio nel lib. </s>
  <s xml:id="echoid-s254" xml:space="preserve">1. </s>
  <s xml:id="echoid-s255" xml:space="preserve">alla prop. </s>
  <s xml:id="echoid-s256" xml:space="preserve">13. </s>
  <s xml:id="echoid-s257" xml:space="preserve">E poi-<lb/>che non è poſſiſibile ſegar’ il Cono con vn pia-<lb/>no in altro modo, che con le ſudette conditio-<lb/>ni, come à chi più attentamente lo conſidera-<lb/>rà, ſi farà manifeſto; </s>
  <s xml:id="echoid-s258" xml:space="preserve">perciò ſtabiliremo con <lb/>Apollonio, che cinque, largamente parlando, <lb/>ouero tre ſolamente, ſtrettamente prenden-<lb/>dole, ponno eſſer le Settioni Coniche, cioè Pa-<lb/>rabola, Iperbola, &amp; </s>
  <s xml:id="echoid-s259" xml:space="preserve">Eliſſi; </s>
  <s xml:id="echoid-s260" xml:space="preserve">le quali fà di meſtie-<lb/>ri con qualche diligenza and are eſſaminando, <lb/>per le mirabili proprietà, che in ſe racchiudo-<lb/>no: </s>
  <s xml:id="echoid-s261" xml:space="preserve">parendomi bene di accennare, che tal vol-<lb/>ta ſi chiamano con queſti nomi gli ſpatij ſotto <lb/>queſte curue, e ſotto rette linee compreſi, il <lb/>che però dal modo di parlare facilmente s’in-<lb/>tenderà.</s>
  <s xml:id="echoid-s262" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s263" xml:space="preserve">Eſſempio ſopra la ſeconda figura.</s>
  <s xml:id="echoid-s264" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s265" xml:space="preserve">LE ſudette coſe ſi poſſono ageuolmẽte compren <lb/>dere nelle quì addotte figure; </s>
  <s xml:id="echoid-s266" xml:space="preserve">e ſe bene tutte <lb/>le dette Settioni ſi generano in tutti i Coni, <lb/>come da altri è ſtato dimoſtrato; </s>
  <s xml:id="echoid-s267" xml:space="preserve">nondimeno, per più <lb/>chiarezza nelli eſſempĳ ci ſeruiremo de i Coni Equi-<lb/>cruri. </s>
  <s xml:id="echoid-s268" xml:space="preserve">Siano dunque tre Coni, A B C, e benche non
<pb o="17" file="0037" n="37" rhead="Coniche. Cap. III."/>
vi ſia diſegnato l’aße, s’intenda però per quello di-<lb/>ſteſo vn piano, che produchi i tre lati, A B, B C, <lb/>C A, &amp; </s>
  <s xml:id="echoid-s269" xml:space="preserve">il triãgolo, A B C; </s>
  <s xml:id="echoid-s270" xml:space="preserve">ſarà dunque la prima <lb/>Settione linea retta, cioè, A B, A C; </s>
  <s xml:id="echoid-s271" xml:space="preserve">poinella pr@-<lb/>ma ſigura di queste tre ſi dia vn taglio al Cono, A B <lb/>C, con vn piano parallelo alla baſe, B C, che produ-<lb/>chi il circolo, D E, ſarà la di lui circonferenza la ſe-<lb/>conda Settione: </s>
  <s xml:id="echoid-s272" xml:space="preserve">sĳ poinell’isteſſa figura vn’ altro <lb/>piano, che ſeghi la baſe nella retta, R V, perpendico-<lb/>lare alla, B C, baſe del triangolo, A B C, e l’iſteſſo <lb/>ſeghi il triangolo, A B C, nell a retta, O X, parallela <lb/>allato A C, e ſeghi la ſuperficie conica nella linea, R <lb/>O V, queſta ſarà la Parabola. </s>
  <s xml:id="echoid-s273" xml:space="preserve">Maſe nella ſeconda <lb/>figura (fatte le isteſſe coſe) X O, concorrerà con il <lb/>lato, C A, prodotto oltre la cima, come nel ponto, K, <lb/>ſarà la linea, R O V, ſegnata nella ſuperficie conica, <lb/>chiamata Iperbola: </s>
  <s xml:id="echoid-s274" xml:space="preserve">Ma ſe finalmente la, O X, come <lb/>nella terza figura (fait’ il medeſimo) ſegarà am-<lb/>bedue i lati del triangolo, A B C, eßendo pure la, Z <lb/>Y, commu<unsure/>n ſegamẽto del piano, O X B C, perpen-<lb/>dicolare alla baſe, B C, prolongata pur che detto pia-<lb/>no ſegante non ſia parallelo alla baſe, B C, ne ſubcõ-<lb/>trariamente poſto (che ſaria, quando ſuppoſto eſſere <lb/>il Cono, A B C, ſcaleno, l’ãgolo, A X O, foſſe eguale <lb/>all’angolo, A B C, e A O X, all’, A C B,) la linea
<pb o="18" file="0038" n="38" rhead="Delle Settioni"/>
ſegnata nella ſuperficie conica, cioè, O R X V, vien <lb/>chiamata Eliſſi, che è l’vltima delle dette Settioni <lb/>Coniche.</s>
  <s xml:id="echoid-s275" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s276" xml:space="preserve">Queste coſe le potiamo apprendere mecanicamen-<lb/>te, con facilità, e guſto, in vn bicchiero di forma co-<lb/>nica, mediãte qualche liquore, come acqua, con l’im-<lb/>mergolo in diuerſi modi dentro di quella; </s>
  <s xml:id="echoid-s277" xml:space="preserve">poiche ſe <lb/>l’immergeremo in tal modo dentro l’acqua, che la ſu-<lb/>perficie di quella paſsi preciſamente per il fondo del <lb/>bicchiero, che è la cima del Cono, e ſeghi l’orlo del <lb/>bicchiero in qualſiuoglia modo, ſe ne produrrà l’am-<lb/>bito del triangolo, che è la prima Settion Conica, qual <lb/>paſſerà per l’aſſe, quando la ſuperficie dell’acqua paſ-<lb/>ſerà per l’aſſe; </s>
  <s xml:id="echoid-s278" xml:space="preserve">ma quando la ſuperficie dell’acqua <lb/>ſarà parallela al piano dell’orlo del bicchiero, ſe ne <lb/>produrrà la circonferenza di circolo, ſeconda Settion <lb/>Conica, &amp; </s>
  <s xml:id="echoid-s279" xml:space="preserve">immergendolo, come richiede la produttio-<lb/>ne delle altre tre Settioni, le vedremo chiaramente <lb/>nel ſegamento della ſuperficie dell’acqua con <lb/>la ſuperficie del bicchiero; </s>
  <s xml:id="echoid-s280" xml:space="preserve">quali coſe pe-<lb/>rò, come faciliſſime, basterà bre-<lb/>@emente ha@erle accen-<lb/>nate.</s>
  <s xml:id="echoid-s281" xml:space="preserve"/>
</p>
<pb o="19" file="0039" n="39" rhead="Coniche. Cap. IV."/>
</div>
<div xml:id="echoid-div17" type="section" level="1" n="17">
<head xml:id="echoid-head22" style="it" xml:space="preserve">Che coſa ſiano le Settioni Opposte, e come <lb/>ſi generino. Cap. IV.</head>
<p>
  <s xml:id="echoid-s282" xml:space="preserve">COn l’occaſione di queſte Settio-<lb/>ni Coniche, nõ poſſo far di me-<lb/>no di non dir qualche coſa del-<lb/>le Settioni Oppoſte, per l’vtili-<lb/>tà, che n’apporta la loro cogni-<lb/>tione principalmente per gli ho@ologij Solari.</s>
  <s xml:id="echoid-s283" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s284" xml:space="preserve">Siano dunque nella terza figura i due Co-<lb/>ni, B A D, N A M, fra loro inuerſi, la cui cõ-<lb/>mune cima il ponto, A, e le baſi ſiano i circoli, <lb/>B D, N M, ſia poi dato vn taglio à queſti due <lb/>Coni cõ vn piano, che paſsi per la lor cõmune <lb/>cima, A E, per l’aſſe, che produca in eſsi i triã-<lb/>goli, B A D, A N M, dipoi ſia data alli medeſi-<lb/>mi vn’altro taglio cõ vn piano, che ſeghi il tri-<lb/>angolo, A N M, nella retta, V G, che prodotta <lb/>concorra con il lato, N A, nel punto, F, e ſia <lb/>la, T H, commun ſegamento della baſe, N M, <lb/>e del piano ſegãte, perpendicolare alla, N M, <lb/>baſe del triangolo, A N M; </s>
  <s xml:id="echoid-s285" xml:space="preserve">ſarà dunque la li-<lb/>nea, T G H, diſegnata dal piano ſegante nel-<lb/>la ſuperficie conica, A N M, vn’Iperbola; <lb/></s>
  <s xml:id="echoid-s286" xml:space="preserve">prolonghi ſi hora la, V F, verſo, B D, ſino che
<pb o="20" file="0040" n="40" rhead="Delle Settioni"/>
l’incontri, come in, I, &amp; </s>
  <s xml:id="echoid-s287" xml:space="preserve">intendaſi, che il det-<lb/>to piano ſegãte nella ſuperficie conica, A B D, <lb/>ſegni la linea, E F R, e nella baſe la retta, E R, <lb/>ſarà anco, E R, perpendicolare à, B D, baſe <lb/>del triangolo, B A D, per eſſere, B D, paralle-<lb/>la ad, N M, &amp;</s>
  <s xml:id="echoid-s288" xml:space="preserve">, E R, à, T H, per la 10. </s>
  <s xml:id="echoid-s289" xml:space="preserve">dell’ 11. <lb/></s>
  <s xml:id="echoid-s290" xml:space="preserve">de gli Elem. </s>
  <s xml:id="echoid-s291" xml:space="preserve">E perche la, I F, ſegnata dal me-<lb/>deſimo piano ſegante il triangolo, B A D, <lb/>prodotta con@orre con il lato, B A, ſteſo pur <lb/>oltre la cima, A, eſſendo il cõcorſo in, G, per-<lb/>ciò anco la linea, E F R, è vn’Iperbola, adun-<lb/>que con vn ſol piano habbiamo prodotton el-<lb/>le ſuperficie coniche d’ambedue i Coni, B A <lb/>D, A M N, due Iperbole; </s>
  <s xml:id="echoid-s292" xml:space="preserve">queſte dunque da <lb/>Apollonio nel lib. </s>
  <s xml:id="echoid-s293" xml:space="preserve">1. </s>
  <s xml:id="echoid-s294" xml:space="preserve">alla propoſ. </s>
  <s xml:id="echoid-s295" xml:space="preserve">14. </s>
  <s xml:id="echoid-s296" xml:space="preserve">ſon chia-<lb/>mate Settioni Oppoſte, e ſono della quinta <lb/>ſpecie delle Settioni Coniche, ò per <lb/>dir meglio, della terza <lb/>ſpecie, conforme ad <lb/>Apollonio.</s>
  <s xml:id="echoid-s297" xml:space="preserve"/>
</p>
  <figure>
    <image file="0040-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0040-01"/>
  </figure>
<pb o="21" file="0041" n="41" rhead="Coniche. Cap. V."/>
</div>
<div xml:id="echoid-div18" type="section" level="1" n="18">
<head xml:id="echoid-head23" style="it" xml:space="preserve">Come dalle coſe dette ne<unsure/> ſudetto Capitolo potiamo <lb/>con ageuolezza comprendere i fondamenti <lb/>de gli Horologĳ Solari, <lb/>Cap. V.</head>
<p>
  <s xml:id="echoid-s298" xml:space="preserve">Clò ſarà facile, ſe nella medeſima <lb/>terza figura intenderemo, A, eſ-<lb/>ſere il centro del mondo, NM, <lb/>BD, gli due tropici, &amp;</s>
  <s xml:id="echoid-s299" xml:space="preserve">, BM, <lb/>che riuolgendoſi intorno le cir-<lb/>conferenze de’tropici, deſcriua le ſuperficie <lb/>coniche, BAD, NAM, le quali ſiano ſegate <lb/>dal piano diſtante dal centro del mondo, per <lb/>la retta, AC, che ſarà il piano dell’Horolo-<lb/>gio, sì come, AC, loſtile, il qual piano dell’ <lb/>Horologio verrà, ſegando le dette ſuperficie <lb/>coniche, à produrre le due Iperbole, ouero <lb/>Oppoſte Settioni, TGH, EFR; </s>
  <s xml:id="echoid-s300" xml:space="preserve">e sì come <lb/>il raggio del Sole poſto in B, che paſſa per la <lb/>cima dello ſtile, ch’è il põto, A, ſe non incon-<lb/>traſſe il piano dell’Horologio, ſcorreria al põ-<lb/>ti, M, così i raggi dello ſteſſo Sole, poſto ne <lb/>gl’altri punti dell’hore del Tropico, BD, paſ-<lb/>ſati oltre il ponto, A, ſcorreriano ſino al luo-<lb/>go oppoſto nell’altro tropico, NM, ſe non in-
<pb o="22" file="0042" n="42" rhead="Delle Settioni"/>
contraſſero il piano dell’Horologio, che gli <lb/>trattiene nella linea, ouero Iperbola, TGM, <lb/>in luidiſegnata, nella quale perciò vengono <lb/>à terminar le ombre: </s>
  <s xml:id="echoid-s301" xml:space="preserve">laſcio gli altri particola-<lb/>ri, che ſi potrebbero dire, poiche hora non in-<lb/>traprendo di trattare di tal materia, baſti ſolo <lb/>hauer’accẽnato così in vniuerſale, come hab <lb/>bino che fare queſte Settioni Oppoſte con gli <lb/>Horologij Solari, e come queſta dottrina alle <lb/>coſe celeſti ancora mirabilmente ſi adatti.</s>
  <s xml:id="echoid-s302" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div19" type="section" level="1" n="19">
<head xml:id="echoid-head24" style="it" xml:space="preserve">D’alcunitermini, che ſi adoprano intorno alle <lb/>Settions<unsure/> Coniche. Cap. VI.</head>
<p>
  <s xml:id="echoid-s303" xml:space="preserve">SE noi riſguardaremo la ſudetta <lb/>ſecõda figura, cõforme alle de-<lb/>finitioni d’Apollonio, trouare-<lb/>mo chiamarſi la retta, OX, in <lb/>tutte tre le Settioni, diametro, <lb/>il ponto, O, cima, &amp;</s>
  <s xml:id="echoid-s304" xml:space="preserve">, RV, or-<lb/>dinata mente applicata al diametro, OX, co-<lb/>me an co vien detta qualſiuoglia altra paral-<lb/>lela alla, RV, che termini in eſſa Settione; </s>
  <s xml:id="echoid-s305" xml:space="preserve">e <lb/>ſela, OX, taglia ad angoli retti, queſte tali li-<lb/>nee ordi natamẽte ad e<unsure/>ſſa applicate, acquiſta
<pb o="23" file="0043" n="43" rhead="Coniche. Cap. VI."/>
il nome di aſſe di tal Settione, ma ſe le taglia <lb/>ad angoli nõ retti, gli reſta ſolo il nome di dia-<lb/>metro, diuidendole però ò ſia aſſe, ò ſolamen-<lb/>te diametro, ſempre in parti vguali. </s>
  <s xml:id="echoid-s306" xml:space="preserve">Se pren-<lb/>deremo poi il quadrato della metà di qualſiuo-<lb/>glia delle ſudette, ordinatamente applicate <lb/>à detto diametro, trouaremo eſſer ſempre <lb/>eguale al parallelogramo rettangolo, largo <lb/>quant’è la parte troncata dal diametro della <lb/>conſiderata Settione, &amp; </s>
  <s xml:id="echoid-s307" xml:space="preserve">adiacente ad vn’altra <lb/>linea, che ſi chiama lato retto, occupandola <lb/>tutta, ſe la Settione è Parabola, ò più di tutta, <lb/>ſe è Iperbola, ò mãco di tutta, ſe è Eliſsi; </s>
  <s xml:id="echoid-s308" xml:space="preserve">e per <lb/>ſa pere quanto ſia l’ecceſſo, ò il mancamento, <lb/>ſi preuale Apollonio d’vn’altra linea, chiama-<lb/>ta lato tranſuerſo, ſi che nell’Iperbola eccede <lb/>il detto parallelogramo, e nello Eliſsi manca <lb/>d’vn parallelogramo ſimile al contenuto ſot-<lb/>to il lato retto, e traſuerſo; </s>
  <s xml:id="echoid-s309" xml:space="preserve">il qual lato traſ-<lb/>uerſo nel ſecondo caſo è figura delle tre già <lb/>dette, ela, OK, è nel terzo, e la, OX, che è <lb/>tutt’vno col diametro dell’Eliſsi; </s>
  <s xml:id="echoid-s310" xml:space="preserve">la Parabola <lb/>poi hà ſolamente il lato retto.</s>
  <s xml:id="echoid-s311" xml:space="preserve"/>
</p>
<pb o="24" file="0044" n="44" rhead="Delle Settioni"/>
</div>
<div xml:id="echoid-div20" type="section" level="1" n="20">
<head xml:id="echoid-head25" xml:space="preserve">Eſſempio ſopra la quarta Figura.</head>
<p style="it">
  <s xml:id="echoid-s312" xml:space="preserve">IN tutte tre le Settioni Coniche, quì eſpoſte, A <lb/>E, è diametro, AH, lato retto, e preſo doue <lb/>ſi voglia nell’, AE ilponto, S, e da quello or-<lb/>dinatamente applicata la, SR, il quadrato di, SR, <lb/>è vguale al rettangolo, ZA, largo quant’è la parte, <lb/>AS, tagliata via dal diametro, AE, per laretta, <lb/>SR, adeguatamente adiacente ad, AH, lato retto; <lb/></s>
  <s xml:id="echoid-s313" xml:space="preserve">e queſto nella Parabola, il che almeno ſi cerchi d’intẽ-<lb/>dere, per capir’il recto; </s>
  <s xml:id="echoid-s314" xml:space="preserve">poiche il ſaper queſto circa le <lb/>altre Settioni, non fà più che tanto di biſogno, per <lb/>quello, che ſi hà da dire, e perciò ſe ad alcuno le coſe <lb/>diqueſto Capitolo pareſſero alquanto oſcure, intenda <lb/>quecto, e tralaſci il recto, che vien quì da me poſto, <lb/>per il cõpimento, che richiede la dottrina: </s>
  <s xml:id="echoid-s315" xml:space="preserve">nell’Iper-<lb/>bola poi, AV, è lato trãſuerſo, come, AH, lato ret-<lb/>to, &amp; </s>
  <s xml:id="echoid-s316" xml:space="preserve">il quadrato di, SR, è vguale al rettangolo, <lb/>AZ, compreſo da, SA, &amp;</s>
  <s xml:id="echoid-s317" xml:space="preserve">, AO, ouero, SZ, ecce-<lb/>dente del parallelogramo, HZ, ſimile al parallelo-<lb/>gramo ſotto i duoi lati, VA, AH, per la 24. </s>
  <s xml:id="echoid-s318" xml:space="preserve">del 6. </s>
  <s xml:id="echoid-s319" xml:space="preserve"><lb/>de gli Elem. </s>
  <s xml:id="echoid-s320" xml:space="preserve">Finalmente nell’Eliſsi il lato tranſ. </s>
  <s xml:id="echoid-s321" xml:space="preserve"><lb/>uerſo è, AE, come il retto è, AH, &amp; </s>
  <s xml:id="echoid-s322" xml:space="preserve">il quadrato di, <lb/>SR, è vguale al rettangolo, AZ, deficiente del pa-<lb/>rallelogramo, HZ, ſimile al parallelogramo conte
<pb o="25" file="0045" n="45" rhead="Coniche. Cap. VI."/>
nuto ſotto, HA, AE, lato retto, e traſuerſo, per la <lb/>24. </s>
  <s xml:id="echoid-s323" xml:space="preserve">del 6. </s>
  <s xml:id="echoid-s324" xml:space="preserve">Quali coſe però, come hò detto, à chi pa-<lb/>reſſero difficili, le tralaſci, ſaluando ſolo in mente, co-<lb/>mei quadrati delle meze ordinatamente applicate al <lb/>diametro della Parabola ſono eguali alli rettangoli <lb/>ſotto le portioni del diametro da quelle tagliate via <lb/>verſo la cima, e ſotto il lato retto, sì come ſi è detto <lb/>eſſere per eſſempio nella Parabola il quadrato, SR, <lb/>eguale alrettangolo ſotto, SA, &amp;</s>
  <s xml:id="echoid-s325" xml:space="preserve">, AH, lato retto, <lb/>come anco il quadrato, EC, ſarà eguale al rettan-<lb/>golo ſotto, EA, e l’iſteſſo lato retto, AH.</s>
  <s xml:id="echoid-s326" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div21" type="section" level="1" n="21">
<head xml:id="echoid-head26" style="it" xml:space="preserve">D’vn principio cauato dalla Proſpettiua per le coſe <lb/>ſuſſeguenti. Cap. VII.</head>
<p>
  <s xml:id="echoid-s327" xml:space="preserve">INanzi, che noi dichiariamo al-<lb/>tro, fà di meſtieri ridurre à me-<lb/>moria quel principio cauato <lb/>dalla Proſpettiua, ch’è la baſe, <lb/>e fondamento della dottrina <lb/>delle rifleſſioni, e ciò per intelligenza delle <lb/>coſe ſuſſeguenti. </s>
  <s xml:id="echoid-s328" xml:space="preserve">Prouano adunque i Pro-<lb/>ſpettiui, che quando vna linea radioſa incon-<lb/>tra la ſuperficie d’vno ſpecchio piano, ſempre <lb/>dal punto dell’incidenza ſi riflette ad angoli
<pb o="26" file="0046" n="46" rhead="Delle Settioni"/>
vguali, d’onde però ſi raccoglie, che quando <lb/>la incidente ſarà perpẽdicolare ſopra la ſuper-<lb/>ficie dello ſpecchio, la rifleſſa tirata dal me-<lb/>deſimo punto di detta ſuperficie ſarà ancor <lb/>lei perpendicolare ſopra l’iſteſſa; </s>
  <s xml:id="echoid-s329" xml:space="preserve">e però la <lb/>rifl@ſſa ritornerà per la ſtrada della inciden-<lb/>te; </s>
  <s xml:id="echoid-s330" xml:space="preserve">ma quando la incidente incontrerà tal ſu-<lb/>perficie ad angolo acuto, la rifle<unsure/>ſſa anderà <lb/>dall’altra banda ad angolo pur’acuto, &amp; </s>
  <s xml:id="echoid-s331" xml:space="preserve">egua-<lb/>le all’ angolo fatto dall’ incidente, quale <lb/>ſuol’eſſer chiamato angolo dell’incidenza, e <lb/>quell’altro angolo di rifleſſione; </s>
  <s xml:id="echoid-s332" xml:space="preserve">prouano adũ-<lb/>que, che queſti due angoli ſono ſempre egua-<lb/>li, ſupponendo queſto principio, come eui-<lb/>dente, cioè, che la Natura opera ſempre per <lb/>la più breue ſtrada, ſe non è impedita; </s>
  <s xml:id="echoid-s333" xml:space="preserve">della <lb/>qual coſa non ſtarò adducendo quà la dimo-<lb/>ſtratione, potendoſi vedere in Euclide, Vitel-<lb/>lione, &amp; </s>
  <s xml:id="echoid-s334" xml:space="preserve">Alazeno, e noi la prenderemo, come <lb/>dimoſtrata. </s>
  <s xml:id="echoid-s335" xml:space="preserve">Il Keplero però nella ſua Aſtro-<lb/>nomia Ottica, moſtrando di non reſtar ſodiſ-<lb/>fatto delle ragioni de’ſudetti Autori, cerca <lb/>dimoſtrar queſto, trahendolo dalla natura del <lb/>moto, come ſi può vedere al cap. </s>
  <s xml:id="echoid-s336" xml:space="preserve">1. </s>
  <s xml:id="echoid-s337" xml:space="preserve">alla prop. <lb/></s>
  <s xml:id="echoid-s338" xml:space="preserve">19. </s>
  <s xml:id="echoid-s339" xml:space="preserve">Da queſto poi cauano i ſudetti Autori,
<pb o="27" file="0047" n="47" rhead="Coniche. Cap. VII."/>
che la ſuperficie, nella quale è poſta la linea <lb/>incidente e rifleſſa, ſega ſempre perpendico-<lb/>larmente la ſuperficie dello Specchio.</s>
  <s xml:id="echoid-s340" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s341" xml:space="preserve">Eſſempio ſopra la quinta Figura.</s>
  <s xml:id="echoid-s342" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s343" xml:space="preserve">SIa lo Specchio piano, A C, e dal punto, E, pre-<lb/>ſo fuori del piano di queſto Specchio, ſia tirata <lb/>al punto, B, la retta E B, che ſi r@fletta in, H, <lb/>e per le, E B H, ſi diſtenda vn piano, che ſeghi il <lb/>piano dello Specchio nella retta, A C, ſi chiama dũ <lb/>que, E B, incidente, B, punto d’incidenza, ò di ri-<lb/>fleſſione, B H, rifleſſa, l’angolo, E B A, angolo a’in-<lb/>cidenza, l’ang@lo, H B C, di rifle<unsure/>ſſione, quali pro-<lb/>uano eſſer ſempre vguali, &amp; </s>
  <s xml:id="echoid-s344" xml:space="preserve">il piano, E A B C H, <lb/>nel quale giacciono la incidente, E B, e rifleſſa, B H, <lb/>che ſi chiama ſuperficie rifleſſiua, ò di rifle<unsure/>ſſione, <lb/>moſtrano ſempre eſſer perpendicolare al piano dello <lb/>Specchio, A C.</s>
  <s xml:id="echoid-s345" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div22" type="section" level="1" n="22">
<head xml:id="echoid-head27" style="it" xml:space="preserve">Come ſi adatti questo principio anco alli Specchi, <lb/>che non ſono piani. Cap. VIII.</head>
<p>
  <s xml:id="echoid-s346" xml:space="preserve">SIa dunque nella 6. </s>
  <s xml:id="echoid-s347" xml:space="preserve">fig. </s>
  <s xml:id="echoid-s348" xml:space="preserve">10 ſpecchio cõ-<lb/>cauo, M B N, e conueſſo, R B V, e <lb/>fuori di quello ſia il pũto, E, dal qua-
<pb o="28" file="0048" n="48" rhead="Delle Settioni"/>
le ſi tiri la, E B, incidẽte in ambedue gli Spec-<lb/>chi nel commun punto, B, per eſſempio, che <lb/>ſi rifletta in, H, prouano adunque i Proſpetti-<lb/>ui queſte due, E B, B H, far pure angoli vgua-<lb/>liſopra la retta linea, che paſſando per il pun-<lb/>to, B, tocca lo Specchio pure in quel punto; <lb/></s>
  <s xml:id="echoid-s349" xml:space="preserve">la qual tangente ſia la retta, A C, che tocchi <lb/>ambedue le ſuperficie di queſti Specchi; </s>
  <s xml:id="echoid-s350" xml:space="preserve">fan-<lb/>no dunque la incidẽte, e la rifleſſa ne gli Spec-<lb/>chi, che non ſon piani, angoli vguali ſopra la <lb/>tangente eſſe ſuperficie nel pũto dell’inciden-<lb/>za, e così vi ſi accommoda il ſopradetto prin-<lb/>cipio.</s>
  <s xml:id="echoid-s351" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div23" type="section" level="1" n="23">
<head xml:id="echoid-head28" xml:space="preserve">Corollario.</head>
<p style="it">
  <s xml:id="echoid-s352" xml:space="preserve">POiche il Keplero nel luogo ſopracitato prouò il <lb/>ſudetto principio, cauando la ragione dalla <lb/>natura del moto, che hà da fare co’l ſuono, co’l <lb/>caldo, e co’l freddo; </s>
  <s xml:id="echoid-s353" xml:space="preserve">perciò prenderemo il ſudetto <lb/>principio nõ ſolo in materia del lume, ma dal moto, &amp; </s>
  <s xml:id="echoid-s354" xml:space="preserve"><lb/>anco dal ſuono (come hà fatto il P. </s>
  <s xml:id="echoid-s355" xml:space="preserve">Biancano Geſuita <lb/>nella ſua Echometria) del caldo, del freddo, &amp; </s>
  <s xml:id="echoid-s356" xml:space="preserve">in <lb/>ſomma d’ogni coſa, il cui moto ſia per retta linea, per <lb/>dire il tutto in vna ſol parola; </s>
  <s xml:id="echoid-s357" xml:space="preserve">accommodando le ſe-<lb/>guenti dimo strationi in aſtratto alle linee, per accom-
<pb o="29" file="0049" n="49" rhead="Coniche. Cap. VIII."/>
modarle poi alle linee lucide, ſonore, calde, fredde, <lb/>ò di qualunque ſorte poi ſi ſiano.</s>
  <s xml:id="echoid-s358" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div24" type="section" level="1" n="24">
<head xml:id="echoid-head29" style="it" xml:space="preserve">Delle ammirabili proprietà delle Settioni Coniche, <lb/>incomincian doſi dalla prima Parabola. <lb/>Cap. IX.</head>
<p>
  <s xml:id="echoid-s359" xml:space="preserve">QVali, e quante ſiano le proprietà <lb/>delle Settioni Coniche, come <lb/>anco delle altre ſorti di linee, <lb/>ò figure, ſtimo veramente, che <lb/>ſia difficiliſſimo poterlo ſape-<lb/>re; </s>
  <s xml:id="echoid-s360" xml:space="preserve">ma l’hauer cognitione d’alcune principa-<lb/>li, &amp; </s>
  <s xml:id="echoid-s361" xml:space="preserve">in riſpetto delle altre, quaſi fondamen-<lb/>tali, ciò credo poterſi da noi con la ſcorta del-<lb/>la buona Geometria, non men facil, che per-<lb/>fettamente ottenere, perciò andaremo ne’ſe-<lb/>guenti Capitoli eſſaminandone alcune, quali <lb/>ſtimo eſſere delle più nobili, principali, e fon-<lb/>damentali, che per hauer’ anco del maraui-<lb/>glioſo, e potere inſieme arrecare di molte vti-<lb/>lità ridotte alla materia, ſpero debbano da <lb/>ciaſcuno volõtieri eſſer lette, e con molto gu-<lb/>ſto forſe eſſere inteſe, delle quali alcune fur-<lb/>no dimoſtrate da altri, &amp; </s>
  <s xml:id="echoid-s362" xml:space="preserve">alcune non ancora,
<pb o="30" file="0050" n="50" rhead="Delle Settioni"/>
per quanto io mi ſappi, toccate; </s>
  <s xml:id="echoid-s363" xml:space="preserve">perciò dare-<lb/>mo principio dalla Parabola, e ſua prima pro-<lb/>prietà.</s>
  <s xml:id="echoid-s364" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s365" xml:space="preserve">Sia dunque la Parabola, A Q Y, nella 7. </s>
  <s xml:id="echoid-s366" xml:space="preserve">fig. <lb/></s>
  <s xml:id="echoid-s367" xml:space="preserve">il cui aſſe ſia, A V, al quale ſia ordinatamente <lb/>applicata, Q Y, alla quale l’aſſe, A V, ſarà per-<lb/>pendicolare; </s>
  <s xml:id="echoid-s368" xml:space="preserve">ſi prendano poi dẽtro, Q Y, punti <lb/>come ſi voglia, per eſſempio, G, X, da’ quali <lb/>verſo la Parabola, Q A Y, ſiano tirate le, G M, <lb/>X Z, parallele all’ aſſe, A V, ſino che incontri-<lb/>no la Parabola, come ne i punti, M, Z, da’qua-<lb/>li, come da’ punti d’incidenza s’intendan ri-<lb/>fletterſi le, M I, Z I, hà dunque tal Settione <lb/>queſto di ammirabile, che tutte le rifleſse vã-<lb/>no à ferire in vn determinato punto dell’aſse, <lb/>che ſia, @, che taglia via dall’aſse verſo la cima <lb/>della Parabola vn pezzo di linea, come, I A, <lb/>che è ſempre la quarta parte del lato retto di <lb/>eſsa, qual ſia, A T: </s>
  <s xml:id="echoid-s369" xml:space="preserve">queſto già è ſtato dimoſtra-<lb/>to da altri, come da Vitellione nella ſua Proſ-<lb/>pettiua da Orontio Fineo, e da Marin Ghe-<lb/>taldo nel lib. </s>
  <s xml:id="echoid-s370" xml:space="preserve">dello Specchio vſtorio, tuttauia <lb/>non voglio tralaſciare di addurne quà la di-<lb/>moſtratione, per eſser degna d’eſsere inteſa.</s>
  <s xml:id="echoid-s371" xml:space="preserve"/>
</p>
<pb o="31" file="0051" n="51" rhead="Coniche. Cap. IX."/>
</div>
<div xml:id="echoid-div25" type="section" level="1" n="25">
<head xml:id="echoid-head30" style="it" xml:space="preserve">Dimoſtratione.</head>
<p>
  <s xml:id="echoid-s372" xml:space="preserve">PEri punti, M, Z, tirando le tangenti la <lb/>Parabola nei medeſimi punti, che pro-<lb/>dotte incontrino l’aſse, V A, prolonga-<lb/>ta, come ne i punti, N, O, ſi tirino dalli me-<lb/>deſimi punti, M, Z, le, M C, Z P, ordinata-<lb/>mente applicate ad, A V, (che ſono la metà <lb/>delle intiere applicate) che ſeghino l’aſse, A <lb/>V, nei punti, C, P, pongaſi poi, che la rifleſsa <lb/>dal pũto, Z, habbia incontrato l’aſse nel pun-<lb/>to, I<unsure/>, dico, che la, I A, è vn quarto di, A T, la-<lb/>to retto della preſente Parabola; </s>
  <s xml:id="echoid-s373" xml:space="preserve">imperoche <lb/>per eſser’, O S, tangente la Parabola nel pũto <lb/>Z, dell’incidenza ſarà l’angolo, X Z S, dell’in-<lb/>cidẽza eguale all’angolo, O Z I, della rifleſſio-<lb/>ne, per la dottrina dichiarata nell’ant. </s>
  <s xml:id="echoid-s374" xml:space="preserve">cap. </s>
  <s xml:id="echoid-s375" xml:space="preserve">ma <lb/>l’angolo, X Z S, è anco vguale all’angolo, S O <lb/>V, interiore delle parallele, X Z, V O, adũque <lb/>i duoi angoli, Z O I, I Z O, ſarãno vguali, &amp; </s>
  <s xml:id="echoid-s376" xml:space="preserve">an-<lb/>co i lati, O I, I Z, ſaran pur’vguali, il che ſi cõ-<lb/>ſerni, con queſt’altra coſa ancora, cioè, che la <lb/>parte, P A, è vguale all’, A O, per la 35. </s>
  <s xml:id="echoid-s377" xml:space="preserve">del <lb/>1. </s>
  <s xml:id="echoid-s378" xml:space="preserve">de’Conici: </s>
  <s xml:id="echoid-s379" xml:space="preserve">Per eſser poi, A P, diuiſa nel <lb/>punto, I, quattro rettangoli, P A I, (ouero il
<pb o="32" file="0052" n="52" rhead="Delle Settioni"/>
rettãgolo ſotto, P A, e ſotto la quadrupla d’, A <lb/>I, con il quadrato d’, I P,) ſarãno vguali al qua-<lb/>drato di, P A I, diſteſa, ouero al quadrato d’, O <lb/>I, per l’8. </s>
  <s xml:id="echoid-s380" xml:space="preserve">del 2. </s>
  <s xml:id="echoid-s381" xml:space="preserve">(cambiando la parte, P A, in, <lb/>A O, che gli è vguale, per la 35. </s>
  <s xml:id="echoid-s382" xml:space="preserve">del 1. </s>
  <s xml:id="echoid-s383" xml:space="preserve">de i <lb/>Conici) ouero al quadrato d’, I Z, ch’è vgua-<lb/>le à I O, cioè alli duoi quadrati, I P, P Z, ele-<lb/>uato il quadrato, I P, commune, reſterà il qua-<lb/>drato, P Z, eguale al rettangolo ſotto, P A, e <lb/>ſotto la quadrupla d’, A I, ma il medeſimo qua <lb/>drato di, P Z, per eſser meza ordinatamente <lb/>applicata all’aſse, A V, (per le coſe dette al <lb/>Cap. </s>
  <s xml:id="echoid-s384" xml:space="preserve">6@) è vguale al rettangolo ſotto, P A, e <lb/>ſotto il lato retto, A T, adunque il rettangolo <lb/>ſotto, P A, e la quadrupla d’, A I, è vguale al <lb/>rettangolo ſotto la medeſima, P A, e ſotto, A <lb/>T, adunque la quadrupla d’, A I, è vguale all’, <lb/>A T, adunque, A I, è vn quarto d’, A T, la-<lb/>to retto della Parabola, Q A Y; </s>
  <s xml:id="echoid-s385" xml:space="preserve">Nell’iſteſso <lb/>modo prouaremo, che la rifleſſa, M I, ſega <lb/>l’aſſe, A V, in vn punto, che recide verſo, A, <lb/>vn quarto d’, A T, adũque non può incontrar <lb/>l’aſſe ſe non nel punto, I, il che di tutte l’altre <lb/>rifleſſe nel medeſimo modo prouaremo, adun-<lb/>que tutte le dette rifleſſe concorrono nel ſol
<pb o="33" file="0053" n="53" rhead="Coniche. Cap. IX."/>
punto, I, diſtante dalla cima, A, per vn quar-<lb/>to di, A T, lato retto di eſſa Parabola, Q A Y, <lb/>il qual punto, I, di quì inanzi chiamaremo fo-<lb/>co della Parabola. </s>
  <s xml:id="echoid-s386" xml:space="preserve">Ciò anco baſterà, che in-<lb/>tendino quelli, che non poteſſero capire la ſo-<lb/>pradetta Dimoſtratione; </s>
  <s xml:id="echoid-s387" xml:space="preserve">e queſta ſi è regiſtra-<lb/>ta per prima proprietà fra le ammirabili, che <lb/>hà la Parabola.</s>
  <s xml:id="echoid-s388" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div26" type="section" level="1" n="26">
<head xml:id="echoid-head31" xml:space="preserve">Corollario.</head>
<p style="it">
  <s xml:id="echoid-s389" xml:space="preserve">SI raccogli e poi di quà, che ſe nell’isteſſa figura <lb/>prendere mo per incide nti le, I M, I Z, che ſi <lb/>pariono dal foco, I, le ſue ri<unsure/>fleſſe ſaranno le, <lb/>M G, Z X parallele all’aſſe, A V, raccogliẽ do queſt’ <lb/>altra coſa, che è il conuerſo della proprietà, che ſi è <lb/>dimoſtrata, cioè, che le linee, che, partendoſi dal foco <lb/>della parabola la vanno ad incontra<unsure/>re, ſi riflettono <lb/>dai punti dell’incidenza parallele all’aſſe della me-<lb/>deſima Parabola per di dentro.</s>
  <s xml:id="echoid-s390" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div27" type="section" level="1" n="27">
<head xml:id="echoid-head32" style="it" xml:space="preserve">Della ſeconda proprietà dalla Parabola. Cap. X.</head>
<p>
  <s xml:id="echoid-s391" xml:space="preserve">LA ſeconda proprietà marauiglioſa <lb/>di queſta Settione è, che partendoſi <lb/>le incidenti dalla medeſima retta li-
<pb o="34" file="0054" n="54" rhead="Delle Settioni"/>
nea ordinatamente applicata all’aſſe, cami-<lb/>nando parallele all’aſſe ſino, che incontrino la <lb/>Parabola, e riflettendoſi finalmente nell’aſſe <lb/>al ſudetto foco; </s>
  <s xml:id="echoid-s392" xml:space="preserve">la compoſta di qualſiuoglia <lb/>incidente, e ſua rifleſſa, è vguale alla compo-<lb/>ſta di qualſiuoglia altra incidẽte, e ſua rifleſſa.</s>
  <s xml:id="echoid-s393" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div28" type="section" level="1" n="28">
<head xml:id="echoid-head33" xml:space="preserve">Eſſempio.</head>
<p style="it">
  <s xml:id="echoid-s394" xml:space="preserve">NElla medeſima 7. </s>
  <s xml:id="echoid-s395" xml:space="preserve">fig. </s>
  <s xml:id="echoid-s396" xml:space="preserve">intenderemo facilmen-<lb/>te questo, prendendo per ordinat amente <lb/>applicata all’aſſe, A V, la, Q Y, dalla <lb/>quale ſi partono le incidenti, G M, X Z, parallele <lb/>all’aſſe, A V, che incontrano la Parabola ne i pun-<lb/>ti, M, Z, da<unsure/> quali punti d’incidenza partendoſi le <lb/>rifleſſe, concorrono nel punto<unsure/>, I, foco di eſſa Para-<lb/>bola. </s>
  <s xml:id="echoid-s397" xml:space="preserve">Dico dunque, che la composta di, G M, M I, <lb/>non ſolo è vguale alla compoſta di, X Z, Z I, ouero <lb/>alla compoſta di, V A, A I, ma anco alla composta <lb/>di qualſiuoglia tale incidente, che principia da i pun-<lb/>ti della retta, Q Y, e ſua rifleſſa; </s>
  <s xml:id="echoid-s398" xml:space="preserve">la dimo ſtr atione <lb/>della qual coſa non hò ancor visto in alcun’Autore, <lb/>ſolo vien’accennata tal proprietà dal Keplero nell’ <lb/>Aſtronomia Ottica, al Cap. </s>
  <s xml:id="echoid-s399" xml:space="preserve">4. </s>
  <s xml:id="echoid-s400" xml:space="preserve">nella Preparation 4. <lb/></s>
  <s xml:id="echoid-s401" xml:space="preserve">De Refractionum menſura, mentre inſegna vn
<pb o="35" file="0055" n="55" rhead="Coniche. Cap. X."/>
modo di deſcriuer la Parabola con vn filo (il che io <lb/>ancora accennarò quì da baſſo) non ne adducendo <lb/>però alcuna dimostratione, che perciò mi par bene <lb/>metterla in queſto luogo.</s>
  <s xml:id="echoid-s402" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div29" type="section" level="1" n="29">
<head xml:id="echoid-head34" style="it" xml:space="preserve">Dimostratione.</head>
<p>
  <s xml:id="echoid-s403" xml:space="preserve">FAcciſi queſto ſopra la 7. </s>
  <s xml:id="echoid-s404" xml:space="preserve">fig. </s>
  <s xml:id="echoid-s405" xml:space="preserve">ſopradetta, <lb/>nella quale già habbiamo prouato, che, <lb/>Z I, è vguale all’, I O, ouero alle, P A I, <lb/>(cambiando O A, in, A P, à lei vguale, come <lb/>ſi diſſe di ſopra) e però aggiungẽdoli le vgua-<lb/>li, X Z, V P, ſaranno le, X Z I, eguali alle, V A <lb/>I; </s>
  <s xml:id="echoid-s406" xml:space="preserve">così, M I, è vguale all’, I N, per l’iſteſſa ra-<lb/>gione, che, Z I, fù prouata eguale ad, I O, &amp;</s>
  <s xml:id="echoid-s407" xml:space="preserve">, <lb/>I N, è vguale all’, I A C, dunque, I M, è vgua-<lb/>le à, I A C, &amp; </s>
  <s xml:id="echoid-s408" xml:space="preserve">aggiunteli le vguali, M G, C V, <lb/>ſaranno le, G M I, eguali alle, V A I, &amp; </s>
  <s xml:id="echoid-s409" xml:space="preserve">in con-<lb/>ſeguenza anco vguali alle, X Z I, il <lb/>che nell’iſteſſo modo di qual-<lb/>ſiuoglia altre, parimente <lb/>ſi dimoſtrarà: <lb/></s>
  <s xml:id="echoid-s410" xml:space="preserve">qual ſi prenda per ſeconda <lb/>proprietà.</s>
  <s xml:id="echoid-s411" xml:space="preserve"/>
</p>
<pb o="36" file="0056" n="56" rhead="Delle Settioni"/>
</div>
<div xml:id="echoid-div30" type="section" level="1" n="30">
<head xml:id="echoid-head35" style="it" xml:space="preserve">Della terza proprietà della Parabola. <lb/>Cap. XI.</head>
<p>
  <s xml:id="echoid-s412" xml:space="preserve">SIa la Parabola, B A C, nell’8. </s>
  <s xml:id="echoid-s413" xml:space="preserve">fig. <lb/></s>
  <s xml:id="echoid-s414" xml:space="preserve">il cui aſſe, O A, indiffinitamen-<lb/>te prolongato verſo, A, come <lb/>in, X, e ſia foco di detta Parabo-<lb/>la il punto, M, e da che parte ſi <lb/>voglia fuori di eſſa incontrino la ſuperficie pa-<lb/>rabolica per eſſempio le rette linee, T I, F K, <lb/>nei punti, I, M, le quali ſiano ſempre per drit-<lb/>to al foco, M, hà dunque la Parabola queſt’al-<lb/>tra mirabile proprietà, che dalli detti punti <lb/>d’incidenza ſi partono le rifleſſe dalla parabo-<lb/>la per di fuori ſempre parallele all’aſſe, cioè <lb/>all’, A O, le quali rifleſſe ſiano le, I V, K Y, pro-<lb/>dotte come ſi voglia in, V, Y, queſta proprietà <lb/>ancora non hò viſto in altri, ſe ben facilmente <lb/>ſi dimoſtra, come hora s’intenderà.</s>
  <s xml:id="echoid-s415" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div31" type="section" level="1" n="31">
<head xml:id="echoid-head36" style="it" xml:space="preserve">Dimoſtratione.</head>
<p>
  <s xml:id="echoid-s416" xml:space="preserve">SIa la, BC, ordinatamente applicata all’aſ<unsure/>-<lb/>ſe, A O, e ſi prolonghino, V I, Y K, ſino <lb/>che incontrino, B C, come in, D, E, &amp; </s>
  <s xml:id="echoid-s417" xml:space="preserve">le,
<pb o="37" file="0057" n="57" rhead="Coniche. Cap. XI."/>
T I, F K, ſi prolonghino parimente dentro la. <lb/></s>
  <s xml:id="echoid-s418" xml:space="preserve">Parabola, ſino che incontrino il foco, M, al <lb/>quale ſtan per dritto, come ſi ſuppone, dipoi <lb/>per i punti, I, K, ſi tirino le tangenti, S R, ℞ L, <lb/>che tocchino la Parabola ne gl’iſteſſi punti, I, <lb/>K, le quali prolongate ſeghino, X A, ne i pun-<lb/>ti, R, L; </s>
  <s xml:id="echoid-s419" xml:space="preserve">perche dunque, D I, ſi parte dalla or-<lb/>dinatamente applicata, B C, &amp; </s>
  <s xml:id="echoid-s420" xml:space="preserve">incontra la. </s>
  <s xml:id="echoid-s421" xml:space="preserve"><lb/>Parabola in, I, la ſua rifl@ſſa andarà dal pun-<lb/>to, I, ad, M, per il Cap. </s>
  <s xml:id="echoid-s422" xml:space="preserve">9. </s>
  <s xml:id="echoid-s423" xml:space="preserve">adunque la, I M, è <lb/>tal rifleſſa, adunque fanno angoli vguali ſopra <lb/>la tangente, S R, peril Cap. </s>
  <s xml:id="echoid-s424" xml:space="preserve">8. </s>
  <s xml:id="echoid-s425" xml:space="preserve">cioèl’angolo, <lb/>SID, ſarà vguale all’angolo, R I M, ma, S I D, <lb/>è vguale all’, V I R, &amp; </s>
  <s xml:id="echoid-s426" xml:space="preserve">R I M, all’angolo, T I S, <lb/>perche ſono alla cima, adũque l’angolo, T I S, <lb/>dell’incidenza di, T I, ſarà eguale all’angolo, <lb/>V I R, adunque, V I R, ſarà l’angolo della ri-<lb/>fleſſione, &amp; </s>
  <s xml:id="echoid-s427" xml:space="preserve">I V, ſarà la rifleſſa di, T I, la quale <lb/>camina per di fuori parallela all’aſſe, O A. </s>
  <s xml:id="echoid-s428" xml:space="preserve"><lb/>Nell’iſteſſo modo dimoſtraremo eſſere, K Y, <lb/>la rifleſſa di, F K, la quale, K Y, è pure paral-<lb/>lela all’aſſe, O A, el’iſteſſo prouaremo ditut-<lb/>te l’altre, adunque è vero, che le rifleſſe delle <lb/>dette incidenti, che di fuori incõtrano la Pa-<lb/>rabola, e tutte coſpirano nel punto, M, foco
<pb o="38" file="0058" n="58" rhead="Delle Settioni"/>
di detta Parabola per di fuori ſi partono dai <lb/>punti dell’incidenza tutte parallele all’aſſe, <lb/>il che ſi douea dimoſtrare.</s>
  <s xml:id="echoid-s429" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div32" type="section" level="1" n="32">
<head xml:id="echoid-head37" xml:space="preserve">Corollario.</head>
<p style="it">
  <s xml:id="echoid-s430" xml:space="preserve">DI quì ſi caua, che ſe prenderemo le parallele <lb/>all’aſſe, A O, che ſono, V I, Y Z, per inci-<lb/>denti, che le ſue rifleſſe ſaranno, I T, K F, <lb/>che stando per dritto al foco, M, da quello ſi dilonga-<lb/>no; </s>
  <s xml:id="echoid-s431" xml:space="preserve">raccogliendo queſt’altra coſa, che è il conuer ſo del-<lb/>la ſudetta proprietà, cioè, che i raggì paralleli all’aſſe <lb/>della Parabola, che l’incontrano per di fuori, ſi riflet-<lb/>tono dai punti dell’incidenza pur per di fuori, stan-<lb/>do dette rifleſſe ſempre per dritto al foco dieſſa Para-<lb/>bola, dal quale ſi vanno allontanando.</s>
  <s xml:id="echoid-s432" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div33" type="section" level="1" n="33">
<head xml:id="echoid-head38" style="it" xml:space="preserve">Della quarta proprietà della Parabo<unsure/>la. <lb/>Cap. XII.</head>
<p>
  <s xml:id="echoid-s433" xml:space="preserve">SIa nella nona figura la Parabola, <lb/>A C E, aſſe, A D, al quale ſiano <lb/>ordinatamence a pplicate, C E, <lb/>B F, che ſeghino l’aſſe ne i pun-<lb/>ti, D, M, hà dunque queſt’al-<lb/>tra proprietà, che mettiam o per quarta, che
<pb o="39" file="0059" n="59" rhead="Coniche. Cap. XII."/>
il quadrato di, C E, al quadrato di, B F, e co-<lb/>me, D A, ad, A M, come ſono anco i quadra-<lb/>ti delle metà, D E, M F, ouero, C D, B M, <lb/>quali ſono ſempre nella proportione, che han-<lb/>no le parti dell’aſſe interpoſte fra la cima del-<lb/>la Parabola, e quelle applicate, la cui dimo-<lb/>ſtratione per eſſer bella, facile, e breue, non <lb/>tralaſcierò di quì regiſtrarla.</s>
  <s xml:id="echoid-s434" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div34" type="section" level="1" n="34">
<head xml:id="echoid-head39" style="it" xml:space="preserve">Dimostratìone.</head>
<p>
  <s xml:id="echoid-s435" xml:space="preserve">SIa dunque della detta Parabola lato ret-<lb/>to, A N, adũque per le coſe dette al Cap. <lb/></s>
  <s xml:id="echoid-s436" xml:space="preserve">6. </s>
  <s xml:id="echoid-s437" xml:space="preserve">il quadrato di, D E, èvguale al rettã-<lb/>golo ſotto, D A, A N, e cosìil quadrato di, M <lb/>F, eguale al rettãgolo ſotto, M A, A N, adun-<lb/>que i duoi quadrati, D E, M F, hanno l’iſteſſa <lb/>proportione, che hanno i rettangoli, D A N, <lb/>M A N; </s>
  <s xml:id="echoid-s438" xml:space="preserve">ma queſti per hauer l’altezza cõmu-<lb/>ne, A N, ſono come le baſi, D A, A M, cioè <lb/>come queſti aſſi, adunque anco i quadrati, D <lb/>E, M F, ſarãno come, D A, A M, e così ſaran-<lb/>no i quadrati, C E, B F, che ſono quadrupli <lb/>de’quadrati, D E, M F, per eſſer’ilati doppij, <lb/>cioè per eſſer la, C E, doppia di, D E, e la, B
<pb o="40" file="0060" n="60" rhead="Delle Settioni"/>
F, doppia di, M F, e queſta Dimoſtratione è <lb/>d’Apollon. </s>
  <s xml:id="echoid-s439" xml:space="preserve">poſta alla 20. </s>
  <s xml:id="echoid-s440" xml:space="preserve">del Primo de’Conici.</s>
  <s xml:id="echoid-s441" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div35" type="section" level="1" n="35">
<head xml:id="echoid-head40" style="it" xml:space="preserve">Altra Dimoſtratione ſopra la decima Figura.</head>
<p>
  <s xml:id="echoid-s442" xml:space="preserve">IN altro modo dimoſtro io queſta proprie-<lb/>tà, ſenza hauer biſogno del lato retto: </s>
  <s xml:id="echoid-s443" xml:space="preserve">ſia <lb/>dũque nella 10. </s>
  <s xml:id="echoid-s444" xml:space="preserve">fig. </s>
  <s xml:id="echoid-s445" xml:space="preserve">il Cono, A B C, ſega-<lb/>to prima da vn piano ꝑl’aſſe, c’habbia prodot-<lb/>to il triãgolo, A B C, dipoi ſia ſegato cõ vn’al-<lb/>tro piano, che faccila Parabola, R O V, il cui <lb/>diametro ſia, O X, &amp; </s>
  <s xml:id="echoid-s446" xml:space="preserve">il cõmun ſegamento del <lb/>detto piano, e della baſe del Cono, che è, B C, <lb/>ſia, R V, quale ſarà perpendicolare à, B C, &amp;</s>
  <s xml:id="echoid-s447" xml:space="preserve">, <lb/>O X, parallela ad, A C, per le coſe dette al <lb/>Cap. </s>
  <s xml:id="echoid-s448" xml:space="preserve">3. </s>
  <s xml:id="echoid-s449" xml:space="preserve">ſia poi nel diametro, O X, preſo doue <lb/>ſi voglia vn punto, come, S, per il quale nel <lb/>piano della Parabola ſi tirila, M N, paralle-<lb/>la ad, R V, e per l’iſteſſo punto nel piano del <lb/>triangolo, A B C, ſi tiri la, I H, ch@ prodotta, <lb/>ſeghi i lati del triangolo ne i punti, I, H, come <lb/>la, M N, ſeghi la Parabola nei punti, M, N, <lb/>ſarà dun que il piano, nel qual ſon poſtele, I H, <lb/>M N, parallelo alla baſe, B C, per la 15. </s>
  <s xml:id="echoid-s450" xml:space="preserve">dell’ <lb/>11. </s>
  <s xml:id="echoid-s451" xml:space="preserve">delli El@m. </s>
  <s xml:id="echoid-s452" xml:space="preserve">adunque la Settion di queſto
<pb o="41" file="0061" n="61" rhead="Coniche. Cap. XII."/>
piano con la ſuperficie Conica ſarà circonfe-<lb/>rẽza di circolo, adunque i quattro punti, I, M, <lb/>H, N, ſono in tal circonferenza, e per eſſere, <lb/>M N, I H, parallele alle, R V, B C, contengo-<lb/>no angolo retto, per la 10. </s>
  <s xml:id="echoid-s453" xml:space="preserve">dell’ 11. </s>
  <s xml:id="echoid-s454" xml:space="preserve">come fan-<lb/>no le, RV, BC, adunque caſcãdo la, M S, per-<lb/>pendicolarmente ſopra, I D<unsure/>, che è diametro <lb/>del generato circolo, ſarà il quadrato, M S, v-<lb/>guale al rettangolo, I S H, come il quadrato, <lb/>R X, al rettangolo, B X C, ma i rettangoli, B <lb/>X C, I S H, ſono come le, B X, I S, per eſſere le <lb/>loro altezze, X C, S H, vguali (e ciò, perche, <lb/>S C, è parallelogramo) cioè ſono come le, X O, <lb/>O S, per i triangoli, O B X, O I S, che ſono ſi-<lb/>mili, adũque i quadrati, R X, M S, ouero i qua-<lb/>drati, R V, M N, ſaranno nella proportione <lb/>delle, X O, O S, come ſi propoſe di dimoſtrare.</s>
  <s xml:id="echoid-s455" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s456" xml:space="preserve">Non rincreſca al Lettore l’intender prima <lb/>queſte coſe così in aſtratto, e cerchi d’appren-<lb/>derle, che ſentirà poi maggior guſto, quando <lb/>le vedrà applicate, e conoſcerà euidente<unsure/>men-<lb/>te l’vtilità, che poſſono apportare queſte Set-<lb/>tioni Coniche, e così ſegua d’intendere il re-<lb/>ſto intorno alle rimanenti.</s>
  <s xml:id="echoid-s457" xml:space="preserve"/>
</p>
<pb o="42" file="0062" n="62" rhead="Delle Settioni"/>
</div>
<div xml:id="echoid-div36" type="section" level="1" n="36">
<head xml:id="echoid-head41" style="it" xml:space="preserve">Quali, e quanti ſiano nell’Iperbola, Eliſſi, &amp; Op-<lb/>poste Settioni i punti, che ſi chiamano <lb/>foshi di quelle. Cap. XiII.</head>
<p>
  <s xml:id="echoid-s458" xml:space="preserve">SIa nella vndecima figura l’Iper-<lb/>bola ſola, C B D, il cui diame-<lb/>tro, E B, lat<unsure/>o traſuerſo, B A, la-<lb/>to retto, B N, e prẽdaſi la quar-<lb/>ta parte di, A B, che termini in, <lb/>B, che ſia, Z B, facendo il rettãgolo, Z N, di-<lb/>poi adattiſi alla retta, A B, lato traſuerſo vn <lb/>rettangolo eguale à, Z N, cl<unsure/>i’ecceda d’vna fi-<lb/>gurà quadrata, come c’inſegna la 29. </s>
  <s xml:id="echoid-s459" xml:space="preserve">del 6. <lb/></s>
  <s xml:id="echoid-s460" xml:space="preserve">libro de gli Elementi, e ſia ciò fatto dalla par-<lb/>te, B, eſſendo l’ecceſſo il quadrato di, B O, ſia <lb/>poi fatto l’iſteſſo dalla parte, A, e ſia pur l’ec-<lb/>ceſſo il quadrato d’, A I<unsure/>, per eſſer dũque fat-<lb/>ta l’vna, el altra applicatione all’iſteſſa linea, <lb/>ſarãno detti ecceſſi, cioè detti quadrati eguali, <lb/>e per<unsure/>ò eguali anco i lor lati, cioè le<unsure/>, B O, A I; </s>
  <s xml:id="echoid-s461" xml:space="preserve"><lb/>ſiano hora le Oppoſte Settioni, delle quali v-<lb/>na ſia l’Iperbol<unsure/>a, C B D, l’altra, G A H, lato <lb/>tr<unsure/>aſuerſo, B A, lato retto dall’Iperbola, C B D, <lb/>eſſo, B N, e della, G A H, eſſo, A M, quai lati <lb/>retti ſaranno eguali, per la 14. </s>
  <s xml:id="echoid-s462" xml:space="preserve">del 1. </s>
  <s xml:id="echoid-s463" xml:space="preserve">de’Co-
<pb o="43" file="0063" n="63" rhead="Coniche. Cap. XIII."/>
nici; </s>
  <s xml:id="echoid-s464" xml:space="preserve">dipoi applichiamo pure al lato traſuerſo, <lb/>A B, vn parallelogramo rettãgolo eguale alla <lb/>quarta parte del rettangolo ſotto, A B, B N, <lb/>come à dire al rettangolo, Z N, (fatto pur, <lb/>Z B, vn quarto di, A B, e poi compito l’iſteſſo <lb/>rettangolo, che fù già fatto) e ciò in tal mo-<lb/>do, che l’ecceſſo venghi vna volta verſo, B, &amp; </s>
  <s xml:id="echoid-s465" xml:space="preserve"><lb/>vn’altra verſo, A, è dunque manifeſto, che ci <lb/>verranno i medeſimi ecceſſi di prima, e le me-<lb/>deſime linee, B O, A I, il che pur ſaria, ſe in ve-<lb/>ce del rettangolo, Z N, ci preualeſſimo d’vn <lb/>quarto del rettangolo ſotto, B A, A M, poiche <lb/>i rettangoli ſotto, A B N, B A M, ſono eguali; <lb/></s>
  <s xml:id="echoid-s466" xml:space="preserve">i punti adũque per tale ſtrada trouati, ſi chia-<lb/>mino fochi dell’Iperbola, C B D, cioè il pun-<lb/>to, O, &amp; </s>
  <s xml:id="echoid-s467" xml:space="preserve">I, e parimẽte fochi delle Settioni Op-<lb/>poſte, C B D, G A H; </s>
  <s xml:id="echoid-s468" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s469" xml:space="preserve">il foco, O, ſi chiami <lb/>foco interiore, &amp;</s>
  <s xml:id="echoid-s470" xml:space="preserve">, I, foco eſteriore dell’Iper-<lb/>bola, C B D, sì come, I, foco interiore della <lb/>Iperbola, G A H, &amp;</s>
  <s xml:id="echoid-s471" xml:space="preserve">, O, foco eſteriore della <lb/>medeſima. </s>
  <s xml:id="echoid-s472" xml:space="preserve">Sia poil’Eliſsi, T X V Y, il cui dia-<lb/>metro, ouer lato traſuerſo, V T, lato retto, T <lb/>P, applichiſi poi alla retta, T V, di quà, e dilà <lb/>vn parallelogramo rettangolo eguale à vn <lb/>quarto del rettãgolo ſotto i due lati, V T, T P,
<pb o="44" file="0064" n="64" rhead="Delle Settioni"/>
deficiente d’vna figura quadrata, come c’in-<lb/>ſegna la 28. </s>
  <s xml:id="echoid-s473" xml:space="preserve">del 6. </s>
  <s xml:id="echoid-s474" xml:space="preserve">de gli Elem.</s>
  <s xml:id="echoid-s475" xml:space="preserve"><unsure/> che ſiano i ret-<lb/>tangoli, V R T, T S V, e gli ecceſſi i quadrati <lb/>di, T R, S V, quali ſaranno pur’eguali, per eſ-<lb/>ſer fatta l’applicatione all’iſteſſa linea, e però <lb/>le rette, T R, S V, ſaranno eguali: </s>
  <s xml:id="echoid-s476" xml:space="preserve">I punti a-<lb/>dunque, R, S ſi chiamino tutti due fochi del-<lb/>lo Eliſſi, T X V Y; </s>
  <s xml:id="echoid-s477" xml:space="preserve">ond’è manifeſto, che in que-<lb/>ſta, e nelle ſopradette Settioni, vi ſono due fo-<lb/>chi, ambedue di dẽtro nello Eliſſi, e nelle Op-<lb/>poſte Settioni, ma in relatione d’vna ſola Iper <lb/>bola vno interiore, l’altro eſteriore, come ſi è <lb/>detto, ſappiamo anco in queſte Settioni, qua-<lb/>li ſi chiamino fochi, e quanti ſiano, il che ſi <lb/>propoſe da dichiarare: </s>
  <s xml:id="echoid-s478" xml:space="preserve">Apollonio però chia-<lb/>ma queſti non fochi, ma pũti fatti dalla <lb/>comparatione, ò applicatione del <lb/>ſudetto rettangolo al <lb/>lato traſuerſo, <lb/>come ſi può vedere nel libro 3. <lb/></s>
  <s xml:id="echoid-s479" xml:space="preserve">de’Conici alla propoſi-<lb/>tione 45.</s>
  <s xml:id="echoid-s480" xml:space="preserve"/>
</p>
  <figure>
    <image file="0064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0064-01"/>
  </figure>
<pb o="45" file="0065" n="65" rhead="Coniche. Cap. XIV."/>
</div>
<div xml:id="echoid-div37" type="section" level="1" n="37">
<head xml:id="echoid-head42" style="it" xml:space="preserve">Della prima proprietà dell’Iperbola. <lb/>Cap. XIV.</head>
<p>
  <s xml:id="echoid-s481" xml:space="preserve">LA proprietà dell’Iperbola, che <lb/>metteremo per prima, veramẽ-<lb/>te marauiglioſa lei ancora, è, <lb/>che tutte le linee rette, che <lb/>per di dẽtro incontrano l’Iper-<lb/>bola, le quali ſe fuori di quella ſi prolongaſſe-<lb/>ro, andrebbono tutte à ferir nel di lei foco e-<lb/>ſteriore, dalli punti dell’incidenza ſi rifletto-<lb/>no nel foco interiore; </s>
  <s xml:id="echoid-s482" xml:space="preserve">la qual coſa parimente <lb/>nõ hò ancor viſto in altri dimoſtrata, e ſi pro-<lb/>uarà in queſto modo.</s>
  <s xml:id="echoid-s483" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div38" type="section" level="1" n="38">
<head xml:id="echoid-head43" style="it" xml:space="preserve">Dimoſtratione ſopra la àuodecima figura.</head>
<p>
  <s xml:id="echoid-s484" xml:space="preserve">SIa l’Iperbola, A G F, A M, diametro, A <lb/>B, lato traſuerſo, nel qual prolõgato ver-<lb/>ſo, C, ſi troui il foco eſteriore, C, &amp; </s>
  <s xml:id="echoid-s485" xml:space="preserve">il fo-<lb/>co interiore ſia, E; </s>
  <s xml:id="echoid-s486" xml:space="preserve">dipoi ſiano che linee rette <lb/>ſi vogliano, K D, Y P, che per di dẽtro incon-<lb/>trino l’Iperbola ne i punti, D, P, le quali ſt<unsure/>ia-<lb/>no per dritto al punto, C, foco eſteriore, di-<lb/>co, che ſi rifletteranno d<unsure/>a’punti, D, P, d’inci-
<pb o="46" file="0066" n="66" rhead="Delle Settioni"/>
denza al foco int eriore, E. </s>
  <s xml:id="echoid-s487" xml:space="preserve">Cõgiunghinſi dun-<lb/>que, D E, P E, e per i punti, D, P, paſſino le <lb/>rette linee, R O, ℞ Z, che tocchino in quei <lb/>punti la Iperbola, G A F; </s>
  <s xml:id="echoid-s488" xml:space="preserve">è dunque manifeſto <lb/>per la 48. </s>
  <s xml:id="echoid-s489" xml:space="preserve">del 3. </s>
  <s xml:id="echoid-s490" xml:space="preserve">de’Conici, che l’angolo, C D <lb/>O, è vguale all’, O D E, ma, C D O, è vguale <lb/>all’, R D K, che gli è alla cima, adunque l’an-<lb/>golo, R D K, s’adegua all’angolo, O D E, ma, <lb/>R D K, è l’angolo della incidenza della retta, <lb/>K D, adunque, O D E, è l’angolo della rifleſ-<lb/>ſione, &amp;</s>
  <s xml:id="echoid-s491" xml:space="preserve">, D E, ſua rifleſſa, che termina n<unsure/>el pun-<lb/>to, E; </s>
  <s xml:id="echoid-s492" xml:space="preserve">nell’iſteſſo modo prouaremo, che, P E, <lb/>è la rifleſſa della, Y P, che và pure à terminare <lb/>nel foco, E, e così d’ogn’altra; </s>
  <s xml:id="echoid-s493" xml:space="preserve">adunque cia-<lb/>ſcheduna di queſte incidenti hà la ſua rifleſſa, <lb/>che và à terminare nel foco, E, interiore, il che <lb/>biſognaua prouare.</s>
  <s xml:id="echoid-s494" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div39" type="section" level="1" n="39">
<head xml:id="echoid-head44" xml:space="preserve">Corollario.</head>
<p style="it">
  <s xml:id="echoid-s495" xml:space="preserve">DI quì ſi raccoglie queſt’altra coſa, cioè, che <lb/>le rette linee, che vanno ad incontrare la <lb/>Iperbola per di dentro, partendo ſi dal fo-<lb/>co interiore, come, E, hanno le ſ@e rifleſſe, che ſi <lb/>partono dalli punti dell’incidenza per di dentro, al.</s>
  <s xml:id="echoid-s496" xml:space="preserve">
<pb o="47" file="0067" n="67" rhead="Coniche. Cap. XIV."/>
lontanandoſi da quelli, le quali stanno ſempre per <lb/>dritto al foco eſteriore, come al foco, C, dal qual pa-<lb/>rimente ſi vanno allontanando; </s>
  <s xml:id="echoid-s497" xml:space="preserve">il che ſi farà chiaro, <lb/>prendendo nella ſopr apoſta figura le, E D, E P, per <lb/>incidenti, poiche verranno ad eſſere le loro r@fleſſe le, <lb/>D K, P Y, che per di dentro ſi allontanano da i pun-<lb/>ti, D, P, dell’incidenza, e dal foco eſteriore, C, al <lb/>quale stanno ſempre per dritto, e ciò, perche ſi è pro-<lb/>uato, che le due, E D, D K, fanno angoli vguali ſo-<lb/>pra la tangenìe, O R, come le, E P, P Y, ſopra la <lb/>tangente, ℞ Z.</s>
  <s xml:id="echoid-s498" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div40" type="section" level="1" n="40">
<head xml:id="echoid-head45" style="it" xml:space="preserve">Della ſeconda proprietà dell’Iperbola. <lb/>Cap. X V.</head>
<p>
  <s xml:id="echoid-s499" xml:space="preserve">RIpigliſi la figura pur’adoperata <lb/>nell’antecedente Capit. </s>
  <s xml:id="echoid-s500" xml:space="preserve">e pre-<lb/>ſo qualſiuoglia pũto nell’lper-<lb/>bola, come, G, fatto centro, C, <lb/>con la diſtanza, C G, ſi deſcri-<lb/>ui l’arco, G F, che ſeghi, C K, in, K, C M, in, <lb/>M, C Y, in, Y, e l’lperbola in, F, queſta è dũ-<lb/>que la proprietà marauiglioſa, che ſi regiſtra <lb/>per ſeconda, cioè, che la Compoſta dell’in-<lb/>cidente, K D, e rifleſſa, D E, è vguale non.</s>
  <s xml:id="echoid-s501" xml:space="preserve">
<pb o="48" file="0068" n="68" rhead="Delle Settioni"/>
ſolo alla compoſta dell’incidente, Y P, e ri-<lb/>fleſſa, P E, ouero alla compoſta dell’inciden-<lb/>te, M A, e rifleſſa, A E, ma è vguale à qualſi-<lb/>uoglia altra cõpoſta d’vna tale incidẽte, e ſua <lb/>rifleſſa; </s>
  <s xml:id="echoid-s502" xml:space="preserve">che perciò tali compoſte vengono tut-<lb/>te ad eſſere eguali fra di loro; </s>
  <s xml:id="echoid-s503" xml:space="preserve">queſto parimẽ-<lb/>te non hò viſto da altri dimoſtrato, ſe ben fa-<lb/>cilmente, ſuppoſta vna propoſit. </s>
  <s xml:id="echoid-s504" xml:space="preserve">d’Apollonio, <lb/>in queſto modo ſi prouarà.</s>
  <s xml:id="echoid-s505" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div41" type="section" level="1" n="41">
<head xml:id="echoid-head46" style="it" xml:space="preserve">Dimostratione.</head>
<p>
  <s xml:id="echoid-s506" xml:space="preserve">PErche dũque proua Apollonio alla pro-<lb/>poſit. </s>
  <s xml:id="echoid-s507" xml:space="preserve">51. </s>
  <s xml:id="echoid-s508" xml:space="preserve">del 3. </s>
  <s xml:id="echoid-s509" xml:space="preserve">de’Conici, che la linea <lb/>retta tratta dal foco eſteriore dell’Iper-<lb/>bola al punto del toccamento fatto da vna li-<lb/>nea ſopra l’Iperbola, ſupera la retta linea ti-<lb/>rata dall’iſteſſa Iperbola, della quantità <lb/>del lato traſuerſo, ouero aſſe, come lui lo chia-<lb/>ma; </s>
  <s xml:id="echoid-s510" xml:space="preserve">perciò, C D, tratta dal foco eſteriore, C, <lb/>al punto, D, punto di toccamento della retta, <lb/>R O, ſuperarà, D E, tratta dall’iſteſſo punto <lb/>di toccamẽto al foco interiore, E, della quan-<lb/>tità di, A B, adunque, C D, ſarà eguale alle
<pb o="49" file="0069" n="69" rhead="Coniche. Cap. XV."/>
due, A B, D E, aggiunta commune, D K, ſarà <lb/>la, C K, eguale alle tre, K D, D E, A B; </s>
  <s xml:id="echoid-s511" xml:space="preserve">e per-<lb/>che, C K, è vguale à, C M, perciò, C M, anco-<lb/>ra ſarà vguale alle tre, K D, D E, A B, tolta <lb/>via la commune parte, B A, reſtarãno le, K D, <lb/>D E, eguali alle, M A, B C, ma, B C, ma, B C, è vguale <lb/>ad, A E, adunque le, K D, D E, ſaranno eguali <lb/>alle, M A, A E; </s>
  <s xml:id="echoid-s512" xml:space="preserve">così prouaremo ancora, che <lb/>le, Y P, P E, ſono eguali alle, M A, A E, la on-<lb/>de la compoſta, K D E, ſarà eguale alla com-<lb/>poſta, Y P E, &amp; </s>
  <s xml:id="echoid-s513" xml:space="preserve">ad ogn’altra ſimile, il che ſi do-<lb/>uea dimoſtrare. </s>
  <s xml:id="echoid-s514" xml:space="preserve">Si tenga poi in memoria la <lb/>propoſit. </s>
  <s xml:id="echoid-s515" xml:space="preserve">d’Apollonio ſopracitata, cioè, che, <lb/>C D, ſupera, D E, della quantità del lato traſ-<lb/>uerſo, A B, per ſeruirſene à ſuo tempo.</s>
  <s xml:id="echoid-s516" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div42" type="section" level="1" n="42">
<head xml:id="echoid-head47" style="it" xml:space="preserve">Della terza propriet à dell’Iperbola. <lb/>Cap. XVI.</head>
<p>
  <s xml:id="echoid-s517" xml:space="preserve">LA proprietà dell’Iperbola, che <lb/>io metto al terzo luogoè, che <lb/>tutte le linee rette, che per di <lb/>fuori anderanno ad incontrare <lb/>l’Iperbola, ſtando per dritto al <lb/>foco interiore, doue tutte concorrerebbono,
<pb o="50" file="0070" n="70" rhead="Delle Settioni"/>
ſe dentro eſſa ſi prolongaſſero, haurãno le ſue <lb/>rifleſſe, che partẽdoſi da i punti dell’inciden-<lb/>za prolongate, anderanno tutte à concorrere <lb/>nel foco eſteriore della medeſima lperbola, <lb/>della qual coſa la dimoſtratione, come le al-<lb/>tre ſudette, credo che ſia nuoua, e perciò quà <lb/>non manch erò diſoggiungerla.</s>
  <s xml:id="echoid-s518" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div43" type="section" level="1" n="43">
<head xml:id="echoid-head48" style="it" xml:space="preserve">Dimostratione.</head>
<p>
  <s xml:id="echoid-s519" xml:space="preserve">PReuagliamoci pure della figura antece-<lb/>dẽte, nella quale s’intendano le due ret-<lb/>te, N D, Q P, incontrar, venendo di <lb/>fuori, l’Iperbola ne i pũti, D, P, da’quali, di-<lb/>co, che partendoſi le loro rifleſſe, anderanno <lb/>ad vnirſi nel foco, C, mentre le incidenti ſtia-<lb/>no per dritto al foco, E: </s>
  <s xml:id="echoid-s520" xml:space="preserve">Intendanſi dunque <lb/>le, N, D, Q P, prodotte ſino al foco, E, e ti-<lb/>rate le, C D, C P, s’intendano pure indefini-<lb/>tamente prodotte dentro l’Iperbola, come in, <lb/>K, Y; </s>
  <s xml:id="echoid-s521" xml:space="preserve">è manifeſto adunque per il Cap. </s>
  <s xml:id="echoid-s522" xml:space="preserve">14. </s>
  <s xml:id="echoid-s523" xml:space="preserve">che <lb/>le K D, D E, ſaranno incidente, erifleſſa, efa-<lb/>ranno angoli eguali ſopra la tangente l’Iper-<lb/>bola nel punto, D, qual ſia pur la, R O; </s>
  <s xml:id="echoid-s524" xml:space="preserve">e per-<lb/>che queſti angoli ſono alla cima con gl’ango-
<pb o="51" file="0071" n="71" rhead="Coniche. Cap. XV."/>
li, C D O, N D R, perciò anco queſti, che ſo-<lb/>no fatti dalle due, N D, D C, ſopra l’iſteſſa <lb/>tangente nel punto, D, dell’incidenza, ſaran-<lb/>no eguali, adunque eſſendo incidente, N D, <lb/>farà, D C, ſua rifleſſa, che vien da, D, e tcrmi-<lb/>na nel foco eſteriore, C. </s>
  <s xml:id="echoid-s525" xml:space="preserve">Nell’iſteſſo modo in-<lb/>teſa la tangente, ℞ Z, toccare parimente l’I-<lb/>perbola in, P, moſtraremo, P C, eſſer la rifleſ-<lb/>ſa dell’incidente, Q P, che viene da, P, punto <lb/>dell’incidenza, e termina in, C, foco eſterio-<lb/>ra, il che dell’altre parimẽte ſi prouerà; </s>
  <s xml:id="echoid-s526" xml:space="preserve">è dun-<lb/>que vera queſta proprietà, cioè, che ſe le ret-<lb/>te linee ſtando per dritto al foco interiore del-<lb/>l’Iperbola, s’incontrerãno in quella, le rifleſſe <lb/>delle medeſime anderanno da que’punti d’in-<lb/>cidenza à concorrer tutte nel foco eſteriore, <lb/>il che biſognaua prouare.</s>
  <s xml:id="echoid-s527" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div44" type="section" level="1" n="44">
<head xml:id="echoid-head49" xml:space="preserve">Corollario.</head>
<p style="it">
  <s xml:id="echoid-s528" xml:space="preserve">DA queſto ſi raccoglie, che ſe per il contra-<lb/>rt<unsure/>o prenderemo le, C D, C P, per inciden-<lb/>ti, le loro ri<unsure/>fleſſe ſaranno le D N, P Q, <lb/>che ſtanno per dritto al foco interiore, E, dal q@ale, <lb/>e dall’lperbola ſi vanno dilongando, cauã to in ſom-
<pb o="52" file="0072" n="72" rhead="Delle Settioni"/>
ma quest’altra eoſa, cioè, che le rette linee, che dal <lb/>foco eſteriore vanno ad mcontrare l’Iperbola, ban-<lb/>ne le loro ri<unsure/>fleſſe, che partendoſi da i punti dell’in-<lb/>cidenza per di fuori, ſtãno ſempre per dritto al foco <lb/>interiore, che è il conuerſo della ſudetta proprietà.</s>
  <s xml:id="echoid-s529" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div45" type="section" level="1" n="45">
<head xml:id="echoid-head50" style="it" xml:space="preserve">Della quarta proprietà della Iperbola. <lb/>Cap. XVI.</head>
<p>
  <s xml:id="echoid-s530" xml:space="preserve">SIa nella 13. </s>
  <s xml:id="echoid-s531" xml:space="preserve">figura l’Iperbola, <lb/>A C E, diametro, D A, lato traſ-<lb/>uerſo, H A, dimoſtra Apollonio <lb/>alla 21. </s>
  <s xml:id="echoid-s532" xml:space="preserve">del primo de’Conici, <lb/>che ſe tiraremo le ordinatamẽ-<lb/>te applicate al diametro, come le, C E, B F, i <lb/>quadrati di quelle ſaranno come in rettango-<lb/>li, H D A, H I A, la qual dimoſtratione non <lb/>ſtò à repeter quà, per eſſer breue; </s>
  <s xml:id="echoid-s533" xml:space="preserve">ma non poſ-<lb/>ſo già mancare diaddurre la preſente, <lb/>che non hà biſogno del lato ret-<lb/>to, come quella d’Apol-<lb/>lonio.</s>
  <s xml:id="echoid-s534" xml:space="preserve"/>
</p>
  <figure>
    <image file="0072-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0072-01"/>
  </figure>
<pb o="53" file="0073" n="73" rhead="Coniche. Cap. XVI."/>
<p style="it">
  <s xml:id="echoid-s535" xml:space="preserve">Dimo stratione ſopra la 14. </s>
  <s xml:id="echoid-s536" xml:space="preserve">Figura.</s>
  <s xml:id="echoid-s537" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s538" xml:space="preserve">NElla ſudetta ſigura perciò ſia preſo vn <lb/>punto, come ſi voglia nel diametro, <lb/>O X, come, I, e per quello ſi tiri, D H, <lb/>parallela à, B C, &amp; </s>
  <s xml:id="echoid-s539" xml:space="preserve">N M, ad, R V, che termini <lb/>nell’Iperbola nei punti, N, M: </s>
  <s xml:id="echoid-s540" xml:space="preserve">Prouaremo a-<lb/>dunque, che’l quadrato, N I, è vguale al rettã-<lb/>golo, D I H, &amp; </s>
  <s xml:id="echoid-s541" xml:space="preserve">R X, al rettãgolo, B X C, come <lb/>ſi fece nel Cap. </s>
  <s xml:id="echoid-s542" xml:space="preserve">12. </s>
  <s xml:id="echoid-s543" xml:space="preserve">moſtrando il quadrato, M <lb/>S, eſſer’eguale al rettangolo, I S H, &amp; </s>
  <s xml:id="echoid-s544" xml:space="preserve">il qua-<lb/>drato, R X, al rettangolo, B X C. </s>
  <s xml:id="echoid-s545" xml:space="preserve">Più oltre il <lb/>rettãgolo, B X C, al rettangolo, D I H, hà per <lb/>la 13. </s>
  <s xml:id="echoid-s546" xml:space="preserve">del 6. </s>
  <s xml:id="echoid-s547" xml:space="preserve">la proportione cõpoſta di, B X, à <lb/>D I, cioè di, O X, ad, O I, (pereſſer’, O D I, O B <lb/>X, triangoli ſimili) e di quella, che hà, X C, <lb/>ad, I H, cioè, X K, à, K I, per eſſer, K I H, K <lb/>X C, triangoli ſimili, le quali due proportioni <lb/>di, X O, ad, O I, edi, X K, à, K I, compon-<lb/>gono la proportione del rettangolo, K X O, <lb/>alrettangolo, K I O, adunque il rettangolo, <lb/>B X C, al rettangolo, D I H, ſarà come il ret-<lb/>tangolo, K X O, à K I O, e così ſarà ancora il <lb/>quadrato, R X, al quadrato, N I, ouero il qua-<lb/>drato, R V, alquadrato, N M, il che biſogna-<lb/>ua dimoſtrare.</s>
  <s xml:id="echoid-s548" xml:space="preserve"/>
</p>
<pb o="54" file="0074" n="74" rhead="Delle Settioni"/>
</div>
<div xml:id="echoid-div46" type="section" level="1" n="46">
<head xml:id="echoid-head51" style="it" xml:space="preserve">Della prima proprietà dell’Eliſſi. <lb/>Cap. XVII.</head>
<p>
  <s xml:id="echoid-s549" xml:space="preserve">SIa nella decimaquinta figura la <lb/>Eliſſi, A C B D, aſſe, A B, fochi, <lb/>H, E, la proprietà regiſtrata per <lb/>prima è dunque queſta, cioè, <lb/>che tutte le rette linee tratte <lb/>dall’vno de’fochi, come da, H, ſino all’Eliſſe, <lb/>A C B D, hanno le rifleſſe loro, che partendoſi <lb/>dalli punti dell’incidenza, vanno tutte à con-<lb/>corre re nel rimanente foco.</s>
  <s xml:id="echoid-s550" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div47" type="section" level="1" n="47">
<head xml:id="echoid-head52" style="it" xml:space="preserve">Dimostratione.</head>
<p>
  <s xml:id="echoid-s551" xml:space="preserve">SIa vna di quelle incidenti la, H C, e ſi <lb/>congiunghino i punti, C, E, e ſia la ret-<lb/>ta, N M, che tocchi l’Eliſsi nel punto, C, <lb/>le due adunque, H C, C E, fanno angoli egua-<lb/>li ſopra la tangente, M N, per la 48. </s>
  <s xml:id="echoid-s552" xml:space="preserve">del 3. </s>
  <s xml:id="echoid-s553" xml:space="preserve">de’ <lb/>Conici, adunque eſſendo, H C, incidente ſa-<lb/>rà, C E, ſua rifleſſa, sì come ſe ſupporremo, <lb/>E C, per incidente, ſarà, C H, ſua rifleſſa, che <lb/>và à terminare nel rimanente foco, H, è dun-<lb/>que manifeſta queſta proprietà.</s>
  <s xml:id="echoid-s554" xml:space="preserve"/>
</p>
<pb o="55" file="0075" n="75" rhead="Coniche. Cap. XVIII."/>
</div>
<div xml:id="echoid-div48" type="section" level="1" n="48">
<head xml:id="echoid-head53" style="it" xml:space="preserve">Della ſeconda proprietà dell’Eliſſi. <lb/>Cap. XVIII.</head>
<p>
  <s xml:id="echoid-s555" xml:space="preserve">LA ſeconda proprietà dell’Eliſſi <lb/>è queſta, che preſe le inciden-<lb/>ti, erifleſſe, come ſi dice nell’ <lb/>antecedente Capitolo, la com-<lb/>poſta di qualſiuoglia incidente, <lb/>e ſua rifleſſa è vguale all’aſſe, &amp; </s>
  <s xml:id="echoid-s556" xml:space="preserve">in conſeguen-<lb/>za eguale alla compoſta di qualſiuoglia altra <lb/>incidente, erifleſſa, preſa come ſopra; </s>
  <s xml:id="echoid-s557" xml:space="preserve">come <lb/>per eſſempio, la compoſta di, H C, C E, nell’ <lb/>antecedẽte figura è vguale all’aſſe, A B, e però <lb/>ſarà vguale alla cõpoſta di qual ſi voglia inci-<lb/>dẽte, e ſua rifleſſa; </s>
  <s xml:id="echoid-s558" xml:space="preserve">q̃ſta è dimoſtrata da Apoll. <lb/></s>
  <s xml:id="echoid-s559" xml:space="preserve">nel lib. </s>
  <s xml:id="echoid-s560" xml:space="preserve">3. </s>
  <s xml:id="echoid-s561" xml:space="preserve">alla propoſ. </s>
  <s xml:id="echoid-s562" xml:space="preserve">52. </s>
  <s xml:id="echoid-s563" xml:space="preserve">petò non ne addur-<lb/>rò la dimoſtratione, potẽdoſi quella in lui ve-<lb/>dere, e queſto per non ripetere tutto ciò, che <lb/>da altri è ſtato dimoſtrato, ꝑ maggior breuità.</s>
  <s xml:id="echoid-s564" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div49" type="section" level="1" n="49">
<head xml:id="echoid-head54" style="it" xml:space="preserve">Della terza proprietà dell’Eliſsi. Cap. X. X.</head>
<p>
  <s xml:id="echoid-s565" xml:space="preserve">GVardiſi la figura decimaquinta, e s’in-<lb/>tenda, che per di fuori vna linea ret-<lb/>ta incontri l’Eliſſi, ſtando per dritto
<pb o="56" file="0076" n="76" rhead="Delle Settioni"/>
ad vno de’fochi di eſſa, come la, G C, indriz-<lb/>zata verſo il foco, H, dico, che la ſua rifleſſa <lb/>ſtarà per dritto all’altro foco, E, pur per di fuo <lb/>ri allontanzndoſi da quello. </s>
  <s xml:id="echoid-s566" xml:space="preserve">Dal punto dũque <lb/>dell’incidenza, C, ſi tiri la retta, C F, che ſtia <lb/>per dritto al foco, E: </s>
  <s xml:id="echoid-s567" xml:space="preserve">Dico, che queſta è la ri-<lb/>fleſſa di, G C; </s>
  <s xml:id="echoid-s568" xml:space="preserve">la onde hauremo queſta pro-<lb/>prietà nello Eliſſi, meſſa per terza, che le ret-<lb/>te linee, le quali ſtando per dri@to all’vn de’ <lb/>fochi, incontrano per di fuori l’Eliſſi, tutte <lb/>hanno le rifleſſe, che partendoſi da@ punti del-<lb/>l’incidenza, ſtaran per dritto all’altro foco, dal <lb/>quale s’anderanno diſcoſtando.</s>
  <s xml:id="echoid-s569" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div50" type="section" level="1" n="50">
<head xml:id="echoid-head55" style="it" xml:space="preserve">Dimostrationt<unsure/>.</head>
<p>
  <s xml:id="echoid-s570" xml:space="preserve">SI prolunghino dunque le, G C, F C, ſino <lb/>che incõtrino i fochi, H, E, ſaranno dun-<lb/>que (inteſaui pur la tangente, M N,) <lb/>gli angoli, M C E, N C H, vguali, e però quel-<lb/>li, che gli ſtanno alla cima, che ſono, G C M, <lb/>F C N, ſaranno parimente vguali, adunque <lb/>eſſendo, G C, incidente, C F, che ſtà per drit-<lb/>to al foco, E, e da quello ſi dilonga, ſarà ſua <lb/>rifleſſa, il che di tutte le altre nell’iſteſſo mo-
<pb o="57" file="0077" n="77" rhead="Coniche. Cap. XIX."/>
do ſi dimoſtrerà; </s>
  <s xml:id="echoid-s571" xml:space="preserve">è dunque veratal proprietà, <lb/>come anco ſe prenderemo, F C, per inciden-<lb/>te, verrà ad eſſere, C G, parimente ſua rifleſ-<lb/>ſione per dritto al foco, H, il che era biſogno <lb/>di dimoſtrare.</s>
  <s xml:id="echoid-s572" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div51" type="section" level="1" n="51">
<head xml:id="echoid-head56" style="it" xml:space="preserve">Della quarta proprietà<unsure/> dell’Eliſſi. <lb/>Cap. XX.</head>
<p>
  <s xml:id="echoid-s573" xml:space="preserve">SIa nella decimaſeſta figura l’E-<lb/>liſſi, A C D E, il cui diametro, <lb/>A D, &amp; </s>
  <s xml:id="echoid-s574" xml:space="preserve">à quello ordinatamen-<lb/>te applicate le, C E, B F, mo-<lb/>ſtra Apollonio, che il quadra-<lb/>to, M E, al quadrato, I F, è come il rettango-<lb/>lo, D M A, alrettangolo, D I A, e così, che i <lb/>quadrati di tutte le ordinatamente applicate <lb/>al diametro dell’Eliſſi ſono, come i rettango-<lb/>li ſotto le parti del diametro fatte da quelle <lb/>ordinatamente applicate, il che anco conuie-<lb/>ne alli quadrati, C E, B F, cioè delle intiere <lb/>applicate; </s>
  <s xml:id="echoid-s575" xml:space="preserve">la dimoſtratione di queſto addot-<lb/>ta da Apollonio alla propoſ. </s>
  <s xml:id="echoid-s576" xml:space="preserve">21. </s>
  <s xml:id="echoid-s577" xml:space="preserve">del primo de’ <lb/>Conici, non la metterò, potendoſi in lui vede-<lb/>re; </s>
  <s xml:id="echoid-s578" xml:space="preserve">non tralaſcierò però la preſente, come
<pb o="58" file="0078" n="78" rhead="Delle Settioni"/>
quella, che non hà biſogno dellato retto, co-<lb/>me l’altra d’Apollonio.</s>
  <s xml:id="echoid-s579" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div52" type="section" level="1" n="52">
<head xml:id="echoid-head57" style="it" xml:space="preserve">Dimoſtratione.</head>
<p>
  <s xml:id="echoid-s580" xml:space="preserve">SIa il Cono, H K L, ſegato da vn piano per <lb/>l’aſſe, che facci il triangolo, H K L, e poi <lb/>ſegato da vn’altro piano, che tagliãdo la <lb/>ſuperficie Conica, produchi l’Eliſſi, T R S V, <lb/>il qual prodotto ſeghi<unsure/> la baſe del Cono, K L, <lb/>prodotta, nella retta, P ℞, che ſarà perpendi-<lb/>colare à, K L, che s’incontri con lei in, P; </s>
  <s xml:id="echoid-s581" xml:space="preserve">ſi <lb/>prendino poi nel diametro dell’Eliſſi (che ſia, <lb/>T S,) che punti ſi voglino, come, I, O, peri <lb/>quali ſi tirino le, N M, R V, parallele à P ℞, <lb/>che prodotte incontrino l’Eliſſi nei punti, N, <lb/>M, R, V, quali ſaranno ordinatamente appli-<lb/>cate al diametro, T S, e per gli ſteſsi punti ſi ti-<lb/>rino le A X, Y Z, che ſeghino i lati, H K, H L, <lb/>nei punti, A, X, Y, Z: </s>
  <s xml:id="echoid-s582" xml:space="preserve">Perche dunque, N M, è <lb/>parallela à, P ℞, &amp; </s>
  <s xml:id="echoid-s583" xml:space="preserve">A X, à, K P, sì come le, K P, <lb/>P ℞, contengono angolo retto, così ſarãno ad <lb/>angolo retto le, N M, A X, e così prouaremo <lb/>eſſere ad angolo retto le, V R, Y Z, &amp; </s>
  <s xml:id="echoid-s584" xml:space="preserve">i quat-
<pb o="59" file="0079" n="79" rhead="Coniche. Cap. XX."/>
tro punti, M, A, N, X, eſſer in vna circonſe-<lb/>renza di circolo, come anco i quattro, V, Y, <lb/>R, Z, e perciò concluderemo, come nel Cap. <lb/></s>
  <s xml:id="echoid-s585" xml:space="preserve">12. </s>
  <s xml:id="echoid-s586" xml:space="preserve">e 16. </s>
  <s xml:id="echoid-s587" xml:space="preserve">eſſere il quadrato, N I, eguale alret-<lb/>tangolo, A I X, &amp; </s>
  <s xml:id="echoid-s588" xml:space="preserve">il quadrato, R O, al rettan-<lb/>golo, Y O Z, ma il rettãgolo, Y O Z, al rettan-<lb/>golo, A I X, hà la proportione cõpoſta di quel-<lb/>la, che hà, Y O, ad, A I, (cioè, O T, à, T I, per <lb/>la ſimilitudine de’triãgoli, T A I, T Y O,) e di <lb/>quella, che hà, O Z, ad, I X, cioè (per i ſimili <lb/>triangoli, S Z O, S X I,) di quella, che hà, O S, <lb/>ad, S I, ma le due proportioni di, O T, à T I, e <lb/>di, O S, ad, S I, cõpongono la proportione del <lb/>rettãgolo, S O T, al rettãgolo, S I T, adunque <lb/>il rettangolo, Y O Z, al rettangolo, A I X, cioè <lb/>il quadrato, R O, al quadrato, N I, ouero il <lb/>quadrato, R V, al quadrato, N M, ſarà come il <lb/>rettangolo, S O T, al rettangolo, S I T, il che <lb/>di tutte l’altre ſi dimoſtrerà; </s>
  <s xml:id="echoid-s589" xml:space="preserve">ſi è dunque pro-<lb/>uato à ſufficienza eſſer vera queſta proprie-<lb/>tà, che fù ſtabilita per quarta.</s>
  <s xml:id="echoid-s590" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s591" xml:space="preserve">Sono veramẽte moltiſſime le proprietà del-<lb/>le ſudette Settioni Coniche, e de gli ſpatij poi <lb/>ſotto quelle, &amp; </s>
  <s xml:id="echoid-s592" xml:space="preserve">altre rette compreſe, curioſe in <lb/>vero, e marauiglioſe, come altri hanno dimo-
<pb o="60" file="0080" n="80" rhead="Delle Settioni"/>
ſtrato, e come comprenderà, chi vedrà l’Ope-<lb/>ra mia da ſtamparſi intorno alla miſura de’pia-<lb/>ni, e ſolidi, nella quale mi ſono sforzato dida-<lb/>re intiera cognitione di tutte le figure pia-<lb/>ne, e regolate, che ordinariamente da’Geo-<lb/>metri ſogliono conſiderarſi, &amp; </s>
  <s xml:id="echoid-s593" xml:space="preserve">anco d’alcune <lb/>ſtraordinarie, quanto alla proportione, che <lb/>hanno fra loro, e così anco de’ſolidi, ma quì <lb/>hò ſolamente voluto regiſtrar quelle, ch’eſ-<lb/>ſendo loro ancora belle, e marauiglioſe, ſi ſo-<lb/>no ſcontrate eſſer parimente al @nio propoſi-<lb/>to, come più à baſſo intenderemo; </s>
  <s xml:id="echoid-s594" xml:space="preserve">fra tan-<lb/>to, chi non capiſſe le dimoſtrationi, le laſci, e <lb/>cerchi almeno di ſapere, che coſa ſi pretende <lb/>di dimoſtrare (come mi ſon’ingegnato d@ſpie-<lb/>garlo inanzi la dimoſtratione) che arriuarà <lb/>nulladimeno anco alla cognition di quelle <lb/>coſe, alle quali tal dottrina vien preordinata.</s>
  <s xml:id="echoid-s595" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div53" type="section" level="1" n="53">
<head xml:id="echoid-head58" style="it" xml:space="preserve">Della proprietà, ancor lei belliſſima, della cir-<lb/>conferenza dicircolo intorno alle inci-<lb/>denti, er@fleſſe. Cap. XXI.</head>
<p>
  <s xml:id="echoid-s596" xml:space="preserve">LA proprietà di ſopra accennata è que-<lb/>ſta, che da al@@@è ſtata dimoſtrata, <lb/>cioè, che hauendo noi vn ſemicirco-
<pb o="61" file="0081" n="81" rhead="Coniche. Cap. XXI."/>
lo, tutte le rette linee, che eſſendo parallele <lb/>al dilei aſſe, incõtrano la circonferenza, han-<lb/>no le ſue ri<unsure/>fleſſe, che cõcorrono tutte non già <lb/>in vn ſol punto, ma sì bene in diuerſi punti del <lb/>diametro, cominciando da quello, che taglia <lb/>dall’aſſe verſo la cima vn quarto del diame-<lb/>tro, e da quello verſo la cima diſcoſtandoſi per <lb/>di fuori in infinito, auuertendo, che le rifleſ-<lb/>ſe, che vengono dalla circonferenza, che ſot-<lb/>tende il lato dell’Eſſagono, tutte concorrono <lb/>dentro il circolo, e quelle, che vengono dal <lb/>compimento del ſudetto arco, tutte concor-<lb/>ronodi fuori, equella, che incontra detta cir-<lb/>conferenza nel punto della ſeparatione di <lb/>queſti due archi hà la ſua rifleſſa, che concor-<lb/>re preciſamente nella cima del detto ſemicir-<lb/>colo.</s>
  <s xml:id="echoid-s597" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div54" type="section" level="1" n="54">
<head xml:id="echoid-head59" xml:space="preserve">Eſſempio ſopra la 17. figura.</head>
<p style="it">
  <s xml:id="echoid-s598" xml:space="preserve">SIa il ſe<unsure/>micircolo, A C H, aſſe, A D, prolonga-<lb/>to verſo A, ind ffinitan@ẽte, e ſi tiri la B N, <lb/>che tagli dalla circonferẽza, C A H, gl@ar@ht, <lb/>B A, A N, ſoſtendentt il lato dell’Eſſagono, ſiano <lb/>poitre rette linee, E O, F N, G M, parall@le all’ <lb/>aſſe, D A, delle quali, E O, incontri la circonferen-
<pb o="62" file="0082" n="82" rhead="Delle Settioni"/>
zanell’arco, A N, G M, nell’arco, N H, &amp; </s>
  <s xml:id="echoid-s599" xml:space="preserve">F N, <lb/>nel punto, N, che ſepara i detti archi, A N, N H; <lb/></s>
  <s xml:id="echoid-s600" xml:space="preserve">vien dunque prouato, che la rifleſſa di, E O, batte <lb/>dentro, come in, I, e la r@fleſſa di G M, batte di fuo-<lb/>ri, come in, R, e quella ai, F N, batte preciſamente <lb/>in, A, e tutte generalmente battono, cominc@ãdo dal <lb/>mezo di, A D, ouer quarto del diametro, che ſia Z, e <lb/>dilungand ſi da quello in infinito: </s>
  <s xml:id="echoid-s601" xml:space="preserve">E adunque queſta <lb/>la ſopradetta proprietà, dalla quale ſi può compren-<lb/>dere, che veramente volendoſi ſeruir di questa per <lb/>vnir le linee radioſe, ella non pare molto à propoſito, <lb/>non gli raccogliendo tutti in vn punto, come la Pa-<lb/>rabola, tutt auia, poiche la parte intorno è proſſima al <lb/>punto, A, gli raccoglie tanto vicini, per eſſer’iui i<unsure/> <lb/>toccamenti delle tangentinon così diradati come vi-<lb/>cino à i punti, B, N, perciò potendo i raggi ſolari per <lb/>eſſempio per tal’auut@inamento operare, come ſi de-<lb/>ſidera, come accẽ<unsure/>dere il fuoco, aggiuntala facilità di <lb/>dare alla materia più d’ogn’altra la curuità sferica, <lb/>per la vnigene@@à delle parti: </s>
  <s xml:id="echoid-s602" xml:space="preserve">quindi auuiene, che <lb/>i fabricatori de’Specc<unsure/>hi ſi ſiano preualſi di queſta fi-<lb/>gura, enon dell’altre, la qual prattica, e dottrina è <lb/>stata con facilità ſpiegata dal Magini nel ſuo Libro <lb/>dello Specchio Sferico, che perciò quà non ne dirò al-<lb/>tro, rimettendo il Lettore à quello, che ne bà ſcritto
<pb o="63" file="0083" n="83" rhead="Coniche. Cap. XXI."/>
luis e queſto basti quanto alle Settioni Coniche, ſem-<lb/>plicemente conſiderate.</s>
  <s xml:id="echoid-s603" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div55" type="section" level="1" n="55">
<head xml:id="echoid-head60" style="it" xml:space="preserve">Delle Superficie, che ſi poſſone<unsure/> generare dalle Set-<lb/>tioni Coniche, e come à quelle s’accomodino<unsure/> le<unsure/> <lb/>già dimo strate loro proprietà, e de’lor <lb/>nomi. Cap. XXII.</head>
<p>
  <s xml:id="echoid-s604" xml:space="preserve">COncioſiacoſa, che il moto, ò fluſ-<lb/>ſo delle linee generi ſuperficie, <lb/>non è dubbio alcuno, che mo-<lb/>uendoſi le Settioni Coniche in <lb/>qualunque modo ſi voglia, con <lb/><gap/>cõtinuo fluſſo generarãno ſuperficie; </s>
  <s xml:id="echoid-s605" xml:space="preserve">ma <lb/>perche i mouimenti poſſono farſi in varij mo-<lb/>di, hora però le conſideraremo ſolamente mo-<lb/>uerſi, riuolgendoſi intorno al ſuo aſſe ſino, che <lb/>ritornino di onde ſi partirono; </s>
  <s xml:id="echoid-s606" xml:space="preserve">nel qual modo <lb/>generano ſuperficie, che diuerſamẽte, confor-<lb/>me alli altri Autori, ſi douran nominare, ſecõ-<lb/>do la varietà delle dette Settioni Coniche; </s>
  <s xml:id="echoid-s607" xml:space="preserve">ſe <lb/>adunque la riuoluta è circonferenza di circo-<lb/>lo, ſi chiamerà la generata ſuperficie, confor-<lb/>me al ſolito, ſuperficie sferica; </s>
  <s xml:id="echoid-s608" xml:space="preserve">ma ſe quella <lb/>ſarà Parabolica, ſu<unsure/>perficie Parabolica; </s>
  <s xml:id="echoid-s609" xml:space="preserve">ſe Iper-
<pb o="64" file="0084" n="84" rhead="Delle Settioni"/>
bola, Iperbolica: </s>
  <s xml:id="echoid-s610" xml:space="preserve">e ſe Eliſſi, pur’Elittica chia-<lb/>mandola concaua, ſe ci preualeremo dilei, co-<lb/>me concaua, ouero conueſſa, ſe ci preualere-<lb/>mo dilei, come di conueſſa; </s>
  <s xml:id="echoid-s611" xml:space="preserve">e perche le dette <lb/>Settioni Coniche nel riuolgerſi nel modo ſu-<lb/>detto, generano le dette ſuper<unsure/>ficie, conſti<unsure/>tu-<lb/>endoſi in tutti i luoghi di quelle, perciò gli <lb/>vengono inſieme à communicare, ciaſcuna <lb/>alla ſua, le loro proprietà; </s>
  <s xml:id="echoid-s612" xml:space="preserve">sì che dũque quel-<lb/>lo, che ſi è detto quanto alle linee incidenti, e <lb/>rifleſſe per le ſemplici Settioni Coniche, s’in-<lb/>tenderà ancora per le da loro generate ſu@<gap/> <lb/>ficie, à’quali pure inſieme inſieme ſi de<gap/> <lb/>intẽdere trasferiti i nomi d’aſſe, efochi, <gap/> <lb/>ſiano communi alle Settioni Coniche genera@ <lb/>ti, &amp; </s>
  <s xml:id="echoid-s613" xml:space="preserve">alle generate ſuperficie: </s>
  <s xml:id="echoid-s614" xml:space="preserve">Intẽdendo poi <lb/>ancora, che i corpi ſolidi rinchiuſi dalle dette <lb/>ſuperficie, ſole, come dalla Sferica, &amp; </s>
  <s xml:id="echoid-s615" xml:space="preserve">Elitti-<lb/>ca, ouero ancor compreſi dalli piani, che gli <lb/>ſegano, troncando il loro aſſe, come nella Pa-<lb/>rabolica, &amp; </s>
  <s xml:id="echoid-s616" xml:space="preserve">Iperbolica, hãno altri nomi, chia-<lb/>mandoſi il compreſo dalla Sferica, conforme <lb/>al ſolito, Sfera, il compreſo dall’Elittica, Sfe-<lb/>roide; </s>
  <s xml:id="echoid-s617" xml:space="preserve">dalla Parabolica, e piano ſegante, Co-<lb/>noide Parabolico, e dall’Iperbolica, e piano
<pb o="65" file="0085" n="85" rhead="Coniche. Cap. XXII."/>
pur ſegante, Conoide Iperbolico, e che le det-<lb/>te ſuperficie ſi chiamano anco ſuperficie di <lb/>queſti corpi, come la Elittica, ſi chiama an-<lb/>cor ſuperficie dello Sferoide, la Parabolica, <lb/>ſuperficie del Conoide Parabolico, e così le <lb/>altre; </s>
  <s xml:id="echoid-s618" xml:space="preserve">nomi, che ſono in vſo appreſſo d’Archi-<lb/>mede, come ſi può vedere nel Libro, De Co-<lb/>@oid@bus, &amp; </s>
  <s xml:id="echoid-s619" xml:space="preserve">Sphæroidibus, dell’iſteſſo.</s>
  <s xml:id="echoid-s620" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div56" type="section" level="1" n="56">
<head xml:id="echoid-head61" style="it" xml:space="preserve">Epilogo delle ſudette proprietà delle Settioni <lb/>Coniche, applicate alle da loro generate <lb/>ſuperficie. Cap. XXIII.</head>
<p>
  <s xml:id="echoid-s621" xml:space="preserve">AVanti però, che ſi venga à que-<lb/>ſto, nõ rincreſca al Lettore ſta-<lb/>bilirſi in mẽte prima queſti no-<lb/>mi, per maggior chiarezza, <lb/>breuità, e più facile intelligen-<lb/>za; </s>
  <s xml:id="echoid-s622" xml:space="preserve">linee rette adunque, ouer raggi luminoſi, <lb/>ò linee ſonore, calde, fredde, &amp;</s>
  <s xml:id="echoid-s623" xml:space="preserve">c. </s>
  <s xml:id="echoid-s624" xml:space="preserve">ſaranno dz<unsure/> <lb/>noi chiamate conuergenti, quando indiffini-<lb/>tamente prolongate, anderãno tutte ad vnirſi <lb/>in vn dato punto: </s>
  <s xml:id="echoid-s625" xml:space="preserve">l’iſteſſe chiamaremo diuer-<lb/>genti, quando tutte ſi partiranno da vn dato <lb/>punto commune; </s>
  <s xml:id="echoid-s626" xml:space="preserve">parallele poi ſi chiamerãno,
<pb o="66" file="0086" n="86" rhead="Delle Settioni"/>
conforme al ſolito, cioè, quando ſaranno tali, <lb/>auuertendo d’intender ſempre la conuergen-<lb/>za, ò diuergenza dal ſolo dato punto, mentre <lb/>non ſi aggiũghi altro, come per eſſempio, s’io <lb/>voleſſi, che i raggi conuergenti ad vn punto, <lb/>foſſero conuergenti ad vn’altro punto, prima <lb/>li chiamerò conuergenti, poi conuergẽti ad al-<lb/>tro pũto, il che s’intenda ancora circa la diuer <lb/>genza; </s>
  <s xml:id="echoid-s627" xml:space="preserve">e quando la conuergenza, ò diuergẽza <lb/>non ſia preciſamente in vn punto, ma ben vi ſi <lb/>auuicini, allhora gli chiamaremo conuergenti, <lb/>ò diuergenti proſſimamente ad vn punto, ò da <lb/>vn punto; </s>
  <s xml:id="echoid-s628" xml:space="preserve">e quando diremo di voler fare i rag-<lb/>gi paralleli, che ſiano conuergenti, non inten-<lb/>deremo già, che ſiano inſieme paralleli, e cõ-<lb/>uergenti, che ſaria implicanza, ma che eſſendo <lb/>paralleli ſino all’incidẽza, doppo quella diuen-<lb/>tino poi dell’altra natura, cioè conuergenti, ò <lb/>diuergenti, come occorrerà: </s>
  <s xml:id="echoid-s629" xml:space="preserve">E queſto hò vo-<lb/>luto auuertire, per douermi ſeruire di queſta <lb/>fraſe, cioè di fare i raggi, che ſono d’vna natu-<lb/>ra diuentar d’vn’altra natura, e ciò mediante <lb/>le ſudette ſuperficie, come hora s’intenderà.</s>
  <s xml:id="echoid-s630" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s631" xml:space="preserve">In virtù adunque delle coſe dell’antece-<lb/>dente Capit. </s>
  <s xml:id="echoid-s632" xml:space="preserve">trasferendo la prima proprietà
<pb o="67" file="0087" n="87" rhead="Coniche. Cap. XXII."/>
della Parabola, dimoſtrata al Cap. </s>
  <s xml:id="echoid-s633" xml:space="preserve">9. </s>
  <s xml:id="echoid-s634" xml:space="preserve">alla ſu-<lb/>perficie Parabolica, diremo, che queſta, rice-<lb/>uendo nella ſua concauità le rette linee paral-<lb/>lele all’aſſe, riflettẽdole poſcia tutte al ſuo fo-<lb/>co, le fà à quello conuergenti, ſi che (per dir <lb/>breuemente, come inanzi ſi vſerà) ella fà le <lb/>parallele conuergenti; </s>
  <s xml:id="echoid-s635" xml:space="preserve">e dal Coroll. </s>
  <s xml:id="echoid-s636" xml:space="preserve">ſi racco-<lb/>glie, ch’ella farà le diuergenti parallele. </s>
  <s xml:id="echoid-s637" xml:space="preserve">Dal <lb/>Capit. </s>
  <s xml:id="echoid-s638" xml:space="preserve">11. </s>
  <s xml:id="echoid-s639" xml:space="preserve">ſi deduce, che l’iſteſſa farà con la <lb/>ſua conueſſità parallele quelle, che ſaranno <lb/>conuergenti al ſuo foco; </s>
  <s xml:id="echoid-s640" xml:space="preserve">e per il Corol. </s>
  <s xml:id="echoid-s641" xml:space="preserve">farà <lb/>diuergenti dal ſuo foco quelle, che ſarãno pa-<lb/>rallele. </s>
  <s xml:id="echoid-s642" xml:space="preserve">Dal Cap. </s>
  <s xml:id="echoid-s643" xml:space="preserve">14. </s>
  <s xml:id="echoid-s644" xml:space="preserve">ſi hà, che la ſuperficie <lb/>concaua Iperbolica fà cõuergenti al foco ſuo <lb/>interiore quelle, che dentro di lei incontran-<lb/>dola ſono cõuergenti nel foco eſteriore; </s>
  <s xml:id="echoid-s645" xml:space="preserve">e dal <lb/>Corol. </s>
  <s xml:id="echoid-s646" xml:space="preserve">ſi hà, che l’iſteſſa fà diuergenti dal ſuo <lb/>foco eſteriore le diuergenti dall’interiore. <lb/></s>
  <s xml:id="echoid-s647" xml:space="preserve">Dal 16. </s>
  <s xml:id="echoid-s648" xml:space="preserve">Cap. </s>
  <s xml:id="echoid-s649" xml:space="preserve">cauiamo, che la ſuperficie con-<lb/>ueſſa Iperbolica fà le conuergenti per di fuo-<lb/>rial ſuo foco interiore eſſere, co’l rifletterle, <lb/>conuergenti nel foco eſteriore; </s>
  <s xml:id="echoid-s650" xml:space="preserve">e dal Corol. </s>
  <s xml:id="echoid-s651" xml:space="preserve"><lb/>che l’iſteſſa fà le diuergenti dal foco eſterio-<lb/>re, co’l rifletterli, eſſer diuergenti dal foco in-<lb/>teriore. </s>
  <s xml:id="echoid-s652" xml:space="preserve">Dal Cap. </s>
  <s xml:id="echoid-s653" xml:space="preserve">17. </s>
  <s xml:id="echoid-s654" xml:space="preserve">habbiamo, che la ſu-
<pb o="68" file="0088" n="88" rhead="Delle Settioni"/>
perficie cõcaua Elittica, fà le diuergẽti da l’vn <lb/>de’ſuoi fochi, con il rifletterle, eſſer conuer-<lb/>genti all’altro foco. </s>
  <s xml:id="echoid-s655" xml:space="preserve">Dal Cap. </s>
  <s xml:id="echoid-s656" xml:space="preserve">19. </s>
  <s xml:id="echoid-s657" xml:space="preserve">ſi caua, <lb/>che la ſuperficie Elittica conueſſa fà le con-<lb/>uergenti all’@vn de’ſuoi fochi per di fuora via, <lb/>con il rifletterle, eſſer diuergenti pur di fuora <lb/>via dall’altro foco. </s>
  <s xml:id="echoid-s658" xml:space="preserve">Dal Cap. </s>
  <s xml:id="echoid-s659" xml:space="preserve">21. </s>
  <s xml:id="echoid-s660" xml:space="preserve">finalmente <lb/>noi habiamo, che la ſuperficie concaua Sferi-<lb/>ca farà le parallele proſſimamente conuergẽ-<lb/>ti, ouero farà le proſſimamẽte diuergenti pa-<lb/>rallele, ogni volta, che la portion di ſuperficie <lb/>Sferica, che ſarà preſa, non molto ſi allarghi <lb/>dalla cima di eſſa, che è il fondo dello Spec-<lb/>chio Sferico vſitato, poiche queſta, come hab-<lb/>biamo detto, proſſimamente vnirà quelle ri-<lb/>fleſſe in vn punto, che alla ſimilitudine de gli <lb/>altri poſſi chiamar foco di eſſo circolo, che è <lb/>alla metà del ſemidiametro: </s>
  <s xml:id="echoid-s661" xml:space="preserve">Poteaſi poi mo-<lb/>ſtrare alla ſimilitndine della Parabola al Cap. <lb/></s>
  <s xml:id="echoid-s662" xml:space="preserve">11 che la ſuperficie Sferica conueſſa farà le <lb/>conuergenti proſſimamente in tal punto, con <lb/>il rifletterle, eſſer parallele, caminando l’vne, <lb/>e l’altre di fuori, e farà parimente le paralle-<lb/>le diuergenti proſſimamente da quel punto, <lb/>il che però hò tralaſciato, potendoſi facilmen-
<pb o="69" file="0089" n="89" rhead="Coniche. Cap. XXIII."/>
te capire ſopra la Dimoſtratione della terz@ <lb/>proprietà della Parabola al Cap. </s>
  <s xml:id="echoid-s663" xml:space="preserve">11. </s>
  <s xml:id="echoid-s664" xml:space="preserve">preua-<lb/>lendoſi della figura di quello, che è l’ottaua, <lb/>come che, B A C, foſſe la circonferenza di cir-<lb/>colo, M, il ſuo foco, &amp; </s>
  <s xml:id="echoid-s665" xml:space="preserve">incidenti, e rifleſſe le <lb/>medeſime iui poſte, accomodandoui la Dimo-<lb/>ſtratione iui addotta, che nell’iſteſſo modo à <lb/>queſta ancora potrà ſeruire.</s>
  <s xml:id="echoid-s666" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div57" type="section" level="1" n="57">
<head xml:id="echoid-head62" xml:space="preserve">Corollario.</head>
<p style="it">
  <s xml:id="echoid-s667" xml:space="preserve">DAlle ſudette coſe ſommariamente è manife-<lb/>ſto, che noi potiamo con le ſudette ſuperfi-<lb/>cie far le parallele conuergenti, le conuer-<lb/>genti parallele, le parallele diuergenti, le <lb/>conuergenti diuergenti, le conuergenti conuergenti <lb/>ad altro punto, le diuergenti parallele, le diuergenti <lb/>conuergenti, le diuergenti diuergenti da altro pun. <lb/></s>
  <s xml:id="echoid-s668" xml:space="preserve">to; </s>
  <s xml:id="echoid-s669" xml:space="preserve">nelle quali è compre ſa tutta la varietà, che poſ-<lb/>ſon fare quanto all’equidistanza, conuergenza, e di-<lb/>uergenza. </s>
  <s xml:id="echoid-s670" xml:space="preserve">E però bò formate<unsure/> la preſente T auola, <lb/>per poter vedere qual ſuperficie ci ſia di biſogno, per <lb/>far fare alle linee quello, che per ſe steſſe non fa-<lb/>rebbono; </s>
  <s xml:id="echoid-s671" xml:space="preserve">che perciò, conſiderate le mols<unsure/>e vtilità, <lb/>ch’ella può apportare in materia principalmente de
<pb o="70" file="0090" n="90" rhead="Delle Settioni"/>
gli Specchi, miè parſo di chiamarla, Ta@ola Specola-<lb/>ria, intendendo però ſotto nome di Specchi non ſola-<lb/>mente quelli, che ſono atti à rappreſentar le <lb/>imagini, ma quelle ſuperficie ancora, <lb/>dalle quali ſi poſſono r flettere le <lb/>linee ſonore, calde, fred-<lb/>de, &amp; </s>
  <s xml:id="echoid-s672" xml:space="preserve">c.</s>
  <s xml:id="echoid-s673" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s674" xml:space="preserve">L’vſo della quale immediatamente ſarà <lb/>doppo eſſa Tauela à baſtanza <lb/>dicbiarato. <lb/></s>
  <s xml:id="echoid-s675" xml:space="preserve">∴</s>
</p>
  <figure>
    <image file="0090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0090-01"/>
  </figure>
<pb o="71" file="0091" n="91" rhead="Coniche. Cap. XXIII."/>
</div>
<div xml:id="echoid-div58" type="section" level="1" n="58">
<head xml:id="echoid-head63" style="it" xml:space="preserve">TAVOLA SPECOLARIA. <lb/>Potiamo per via della rifleſſione con la ſuperficie ſcritta nell’area <lb/>di questa Tauola fare <lb/>L E</head>
<note position="right" xml:space="preserve"> <lb/># Parallele. # Conuergenti<unsure/>. # Diuergenti. <lb/>Paralle \\ le. # Cioè con la ſu \\ peificie piana. # Con la conucſla \\ Parabolica, e \\ proſsimamẽte \\ con la Sferica. # Con la concaua \\ Parabolica, e \\ proſsima<unsure/>mẽte \\ con la Sferica. <lb/>Cõuer \\ genti. # Con la cõcaua \\ Parabolica, e \\ p@oſsimamẽte \\ con la Sferica. # Con la cõcaua, \\ e cõu<unsure/>eſſa Iper \\ bolica, e con \\ la piana. # Con la concaua \\ Elittica. <lb/>Diuer \\ genti # Con la cõueſſa \\ Parabol@ca, e \\ proſsimamẽte \\ con la Sferica. # Con la conueſſa \\ Elittica. # Con la cõcaua, \\ e cõueſſa Iper- \\ bolica, e con \\ la piana. <lb/>Cõuer.<unsure/> \\ gẽti ad \\ altro \\ pũto di \\ dentro # Non ſe gli con- \\ ul<unsure/>ene, per non \\ eſſer lot<unsure/>o con \\ uergenti. # Con la cõcaua, \\ e cõueſſa Iper- \\ bolica. # Con la concaua \\ Elittica. <lb/>Cõuer- \\ gẽti ad \\ altro \\ pũto di \\ fuori. # Non ſe gli con- \\ uiene, per non \\ eſſer loro con \\ uergenti. # Con la cõueſſa \\ Iperbolica. # Con la concaua \\ Elittica. <lb/>Diuer<unsure/> \\ gẽti da \\ altre<unsure/> \\ pũto di \\ dentro # Non ſe gli con- \\ uiene, per non \\ eſſer loro di- \\ uergenti. # Con la conueſſa \\ Elittica. # Con la conueſſa \\ Ipeibolica. <lb/>Diuer<unsure/> \\ gẽti da \\ altro \\ pũto di \\ ſuori. # Non ſe gli con- \\ uiene, per non \\ eſſer loro di- \\ uergenti. # Con la conueſſa \\ Elittica. # Con la concaua \\ Ipeibolica. <lb/></note>
<pb o="72" file="0092" n="92" rhead="Delle Settioni"/>
</div>
<div xml:id="echoid-div59" type="section" level="1" n="59">
<head xml:id="echoid-head64" style="it" xml:space="preserve">Dell’vſo della precedente Tauola Specolaria. <lb/>Cap. XXIV.</head>
<p>
  <s xml:id="echoid-s676" xml:space="preserve">Q Vando noi haueremo vna mol-<lb/>tiplicità di linee rette, che ſia-<lb/>no tutte d’vna medeſima natu-<lb/>ra, ò qualirà delle tre ſudette, <lb/>cioè, ò parallele, ò conuergen-<lb/>ti, ò diuergenti; </s>
  <s xml:id="echoid-s677" xml:space="preserve">e vorremo cõmutarle di qua-<lb/>lità, conforme alla diuerſità, che apporta l’e-<lb/>quidiſtanza, conuergenza, e diuergenza, en-<lb/>traremo nella ſoprapoſta Tauoletta, trouãdo <lb/>in fronte di quella la natura, ò qualità, della <lb/>quale ſono le linee da commutarſi, e lateral-<lb/>mente la natura, ò qualità, nella quale vo-<lb/>gliamo commutarle; </s>
  <s xml:id="echoid-s678" xml:space="preserve">che dirimpetto à quel-<lb/>le nell’area di eſſa Tauola, comprenderemo <lb/>qual ſuperficie ſia atta à fare tal’effetto: </s>
  <s xml:id="echoid-s679" xml:space="preserve">co-<lb/>me per eſſempio, ſe haueſſimo vna moltipli-<lb/>cità di parallele, e le voleſſimo fare conuer-<lb/>genti, trouareſſimo nella fronte della Tauola <lb/>le parallele, e lateralmente le conuergenti, <lb/>raccogliendo dirimpetto à quella la ſuperfi-<lb/>cie concaua Parabolica, ch’è atta à far tal’ef-<lb/>fetto, come ſi è inteſo nel Cap. </s>
  <s xml:id="echoid-s680" xml:space="preserve">9. </s>
  <s xml:id="echoid-s681" xml:space="preserve">c 23. </s>
  <s xml:id="echoid-s682" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s683" xml:space="preserve">an-
<pb o="73" file="0093" n="93" rhead="Coniche. Cap. XXIV."/>
co la Sferica, che può farle conuergenti proſ-<lb/>fimamente in vn punto, ſe tali le deſideraſſi-<lb/>mo, come ſi è detto nel fine del Cap. </s>
  <s xml:id="echoid-s684" xml:space="preserve">23. </s>
  <s xml:id="echoid-s685" xml:space="preserve">la <lb/>quale hò meſſo ancora ne gli altri luoghi, do-<lb/>ue hò veduto, che può fare proſſimamente <lb/>ſ<unsure/>e<unsure/>ffetto della ſuperficie, che appreſſo lei vien <lb/>notata nell’iſteſſa caſella. </s>
  <s xml:id="echoid-s686" xml:space="preserve">Hò poi ancor po-<lb/>ſto la ſuperficie piana, doue ella può operare, <lb/>per non laſciar vuote le caſelle; </s>
  <s xml:id="echoid-s687" xml:space="preserve">come dirim-<lb/>petto à parallele in fronte, e à parallele late-<lb/>ralmente, hò meſſo la Piana, perche riceuen-<lb/>do le parallele, le riflette parallele (come anco <lb/>ſe ſon conuergenti, le ribatte conuergenti, e <lb/>ſe diuergenti, pur le riflette diuergenti) nel <lb/>qual caſo non mutano natura, ma ſolamente <lb/>ſito, poiche doue prima caminauano verſo la <lb/>ſuperficie piana, doppo l’incidenza da lei ſi <lb/>diſcoſtano; </s>
  <s xml:id="echoid-s688" xml:space="preserve">e perciò dirimpetto a’cõuergen-<lb/>ti in fronte, e conuergenti nel lato, &amp; </s>
  <s xml:id="echoid-s689" xml:space="preserve">à diuer-<lb/>genti in fronte, e nel lato, hò meſſo ancor la <lb/>piana, come quella, che non gli fà mutar na-<lb/>tura, ma ſolamente ſito. </s>
  <s xml:id="echoid-s690" xml:space="preserve">Pongaſi di più, che <lb/>noi vogliamo fare le conuergenti eſſer con-<lb/>uergenti ad altro punto di fuori, trouaremo <lb/>dunque nell’area la ſuperficie conueſſa Iper-
<pb o="74" file="0094" n="94" rhead="Delle Settioni"/>
bolica, che fà queſt’effetto, come ſi è viſto nel <lb/>Capit. </s>
  <s xml:id="echoid-s691" xml:space="preserve">16. </s>
  <s xml:id="echoid-s692" xml:space="preserve">e 23. </s>
  <s xml:id="echoid-s693" xml:space="preserve">intendendo quello (ad altro <lb/>punto di fuori, ò di dentro) cioè di fuori, ad vn <lb/>punto poſto fuori delle conuergenti, ò diuer-<lb/>genti, ſe concorreſſero nel punto, à cui ſtãno <lb/>per dritto; </s>
  <s xml:id="echoid-s694" xml:space="preserve">e di dentro, quãdo quel punto ſteſ-<lb/>ſe dẽtro di quelle; </s>
  <s xml:id="echoid-s695" xml:space="preserve">e perche il dire di farle con-<lb/>uergenti ad altro punto, ò diuergenti da altro <lb/>punto, par che ſupponga, che già le linee, che <lb/>noi habbiamo, ſiano conuergẽti, ò diuergen-<lb/>ti, cioè che concorrino, ò che ſi allarghino da <lb/>qualche punto, perciò, non conuenendo que-<lb/>ſto alle parallele, l’hò anco poſto ſotto di <lb/>loro, dirimpetto à quelle caſelle laterali, <lb/>doue ſi fà mentione di far le linee conuergen-<lb/>ti ad altro punto, ò diuergenti da altro punto <lb/>di dentro, ò di fuori, dicendo con ragione non <lb/>conuenirli queſto, per non eſſer conuergenti, <lb/>ò diuergenti. </s>
  <s xml:id="echoid-s696" xml:space="preserve">Di più dalla ſudetta Tauola <lb/>per il contrario potiamo ſapere l’effetto d’v-<lb/>na data ſuperficie delle ſopranominate, cer-<lb/>candola nell’area, poiche dirimpetto à lei in <lb/>fronte trouaremo la natura, che vien tramu-<lb/>tata da lei nella natura, ò qualità, che gli <lb/>ſtà dir impetto lateralmente, e quante volte
<pb o="75" file="0095" n="95" rhead="Coniche. Cap. XXIV."/>
incõtraremo tal ſuperficie nell’area, tãte pro-<lb/>prietà hauerà; </s>
  <s xml:id="echoid-s697" xml:space="preserve">come per eſſempio, cerco nell’ <lb/>area la ſuperficie conueſſa Elittica, volẽdo ſa-<lb/>pere quali, e quante proprietà habbia, trouo <lb/>dunque vna volta dirimpetto à lei in fronte <lb/>le conuergenti, elateralmente diuergenti, di-<lb/>rò dunque, ch’ella fà le conuergẽti diuergen-<lb/>ti; </s>
  <s xml:id="echoid-s698" xml:space="preserve">e ſimilmente, che fà le conuergenti diuer-<lb/>genti da altro punto di dentro, e di fuori, che <lb/>ſon’altre due (ſe ben queſte ſono più toſto di-<lb/>ſtintioni della proprietà generale di far le cõ-<lb/>uergenti diuergenti, poſta nel Capit. </s>
  <s xml:id="echoid-s699" xml:space="preserve">19. </s>
  <s xml:id="echoid-s700" xml:space="preserve">e <lb/>23. </s>
  <s xml:id="echoid-s701" xml:space="preserve">che veramente diſtinte proprietà) in al-<lb/>tre però le trouaremo conforme, che ſi ſo-<lb/>no diſtinte ne i ſoprapoſti capi, come guar-<lb/>dãdo in detta Tauola ſi comprẽderà e queſto <lb/>baſti quanto all’intelligenza della coſtruttio-<lb/>ne, &amp; </s>
  <s xml:id="echoid-s702" xml:space="preserve">vſo della ſoprapoſta Tauola Specolaria.</s>
  <s xml:id="echoid-s703" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div60" type="section" level="1" n="60">
<head xml:id="echoid-head65" xml:space="preserve">Digreſsione intor no le Refrattioni.</head>
<p style="it">
  <s xml:id="echoid-s704" xml:space="preserve">CHi poteſſe veramente formar la Tau<unsure/>ola del-<lb/>le Superfic@e, che per Refrattione produco-<lb/>no i ſudetti effetti, faria coſa digrandiſſi-<lb/>mo momento nella Proſpettiua, e di gran conſeg@e@-
<pb o="76" file="0096" n="96" rhead="Delle Settioni"/>
za, ma ſin’hora non ſi troua, chi habbia potuto pre-<lb/>ciſamente arriuarui, peril mancamento diregola v-<lb/>niuerſale, qual’è nelle r fl ſſioni, che l’angolo della <lb/>incidenza ſia eguale à quello della rifleſſione, poiche <lb/>non ſi sà come paſſi nella Refrattione, intendendo noi <lb/>in quella ſolamente, che nell’entrare ne i diafani più <lb/>denſi le inciden@i ſi accostano alla perpendicolare, <lb/>che dal punto dell’incidenza vien tirateſopra la ſu-<lb/>perficie deldiafano, ò ſopra la tangente in quel pun-<lb/>to, e che entranao ne i diafani più rari, da quella ſi <lb/>diſcoſtano, facendoſi maggiore, e minor’angolo dire-<lb/>frattione, quanto è maggiore, ò minore l’angolo del-<lb/>l’incidenza, ma con che regola ſi vadano diminuen-<lb/>do gli angoli della Refrattione in vn diafano, ouero <lb/>accreſcendo in relatione de gli angoli dell’incidenza, <lb/>ciò ſin’hora non ſi è con modo ſiouro, e d<unsure/>imo stratiua-<lb/>mente, per quanto io ſappi, potuto prouare; </s>
  <s xml:id="echoid-s705" xml:space="preserve">tengono <lb/>alcuni, che la Parabola cristallina vniſcale parallele <lb/>in vn punto: </s>
  <s xml:id="echoid-s706" xml:space="preserve">Il Kepleronell’Astronomia Ottica ſti-<lb/>ma, che ſia vn’Iperbola, come la Mecanica gli dimo-<lb/>stra, ſe ben dice vederla vn poco più acuta della <lb/>Iperbola nella cima, com’egli accenna al Cap. </s>
  <s xml:id="echoid-s707" xml:space="preserve">4. </s>
  <s xml:id="echoid-s708" xml:space="preserve">trat-<lb/>tando della miſura delle Refrattioni, ſaria dun-<lb/>que d<unsure/>a stimarſi molto vna ſimil Tauola per le re-<lb/>frattioni, e poiche ſin’hora non vi ſi è potuto arri-
<pb o="77" file="0097" n="97" rhead="Coniche. Cap. XXIV."/>
uare, ſi ſono però alcuni sforzati almeno proſſima-<lb/>mente di ottener questo, e così hanno moſtrato farlo <lb/>le Sfere criſtalline, le lenti conueſſe di portioni di Sfe-<lb/>ra, come anco alcuni de gli altri effetti ſopranotati <lb/>eſſer fatti per refrattione dalle lenti concaue, conueſ-<lb/>ſe, e miſte, appigliandoſi in particolare alla figura <lb/>Sferica per la facilità di produrla in materia, ſi come <lb/>l’altre ſono difficiliſſime da farſi, come chi ſi metterà <lb/>all’eſperienza comprenderà facilmente; </s>
  <s xml:id="echoid-s709" xml:space="preserve">di queſte len-<lb/>ti adunque hauendone à lungo trattato il Keplero <lb/>nella ſua Diottrica, non dirò altro, potendo il Let-<lb/>tore in quello vedere in buona parte ciò, che ſi può <lb/>dire in materia di refrattione intorno à queſte lenti: <lb/></s>
  <s xml:id="echoid-s710" xml:space="preserve">E poſciache ſiamo arriuati cõ la ſpecolatione intorno <lb/>alla R fleſſione, e ſuperficie rifleſſiue, à queltermine, <lb/>e perfetta cognitione, che mi è parſa di biſogno per <lb/>intelligenza delle coſe ſeguenti, veniamo hora all’ <lb/>applicatione aiciò, che è stato da noi con<unsure/>ſiderato, <lb/>rida<unsure/>cendolo alla prattica, acciò viuamente appa-<lb/>riſcal’vtilità, che poſſono apportare queste <lb/>Settioni Coniche, e da lor generate ſu-<lb/>perficie, intorno alle coſe di Natura, nel prin-<lb/>cipio di questo Trattato da me <lb/>accennate.</s>
  <s xml:id="echoid-s711" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s712" xml:space="preserve">∴</s>
</p>
<pb o="78" file="0098" n="98" rhead="Delle Settioni"/>
</div>
<div xml:id="echoid-div61" type="section" level="1" n="61">
<head xml:id="echoid-head66" style="it" xml:space="preserve">Come ſi poſsi accendere il fuoco per il ri<unsure/>fleſſo de’ <lb/>raggi Solari. Cap. XXV.</head>
<p>
  <s xml:id="echoid-s713" xml:space="preserve">BEnche i raggi Solari vigoroſi eſ-<lb/>chino dal centro del Sole (dico <lb/>vigoroſi principalmente quan-<lb/>to alla virtù calorifica) come i <lb/>Proſpettiui aſſeriſcono (ſpargẽ-<lb/>doſi però ancora da ogni punto Solare ad o-<lb/>gni poſitione raggi luminoſi, ſe ben quanto <lb/>alla virtù calorifica, non così efficaci) bẽche, <lb/>dico, quelli eſchino da vn ſol pũto, e però ſia-<lb/>no diuergenti, tuttauia in tanta lontananza ſi <lb/>reputano quelli, che ſi riceuono nella ſuperfi-<lb/>cie d’vno Specchio, come paralleli, e però per <lb/>volerli raccoglier’in vn ſol pũto, nel quale ſarà <lb/>vnita tutta la virtù calorifica, e perciò ſi cauſa-<lb/>rà l’incẽdio, che è vn voler far le parallele con-<lb/>uergenti, trouaremo in fronte della Tauola <lb/>Specolaria parallele, e lateralmente conuer-<lb/>genti, e nell’area vedremo eſſer’atta à queſto <lb/>ſeruitio la cõcaua Parabolica, e proſſimamen-<lb/>te farlo la concaua Sferica, e però con queſte <lb/>haurem l’intẽto noſtro, facendoſi l’incẽdio ne’ <lb/>loro fochi, come s’è detto diſopra. </s>
  <s xml:id="echoid-s714" xml:space="preserve">Tuttauia il
<pb o="79" file="0099" n="99" rhead="Coniche. Cap. XXV."/>
Padre Gruemberger Geſuita, Matematico ce-<lb/>leberrimo, nel ſuo Libretto dello Specchio <lb/>Elittico, volendo pur preualerſi de’medeſimi <lb/>raggi, quali realmente ſono, cioè diuergenti <lb/>dal centro del Sole, vi adopera la ſuperficie <lb/>concaua Elittica, imaginandoſi, che il cẽtro <lb/>del Sole ſia vno de’fochi di tal ſuperficie Elit-<lb/>tica, e l’altro foco ſia il pũto della combuſtio-<lb/>ne, deſcriuendo l’Eliſsi, e ſuperficie Elittica, <lb/>alla quale conuengono detti fochi, ſecondo <lb/>le diuerſe diſtãze del Sole nel diſcendere dal-<lb/>la maſsima alla minima lontananza della ter-<lb/>ra: </s>
  <s xml:id="echoid-s715" xml:space="preserve">la qual diligẽza però in queſto negotio po-<lb/>tria forſi parer altrui ſuperflua, per la ſomma <lb/>difficultà di mettere in prattica tal’operatio-<lb/>ne, douẽdo deſcriuerſi vna portione di sì grãd’ <lb/>Eliſſe, e ſe pure lo vogliamo fare ſtretto ſtret-<lb/>to, perche da ambedue i capi v’appariſca no-<lb/>tabile curuatura, può eſſer, che rieſchi, ma che <lb/>ſia diſtinto dalla Parabola, non credo, che con <lb/>gl’inſtrumenti adoperati da noi ſi poſſa fare, <lb/>il che diuerrà più chiaro, adducendo la ragio-<lb/>ne, perchei raggi Solari ſi reputino come pa-<lb/>ralleli.</s>
  <s xml:id="echoid-s716" xml:space="preserve"/>
</p>
<pb o="80" file="0100" n="100" rhead="Delle Settioni"/>
</div>
<div xml:id="echoid-div62" type="section" level="1" n="62">
<head xml:id="echoid-head67" style="it" xml:space="preserve">Dimostratione ſopra la 18. Figura.</head>
<p>
  <s xml:id="echoid-s717" xml:space="preserve">INtendaſi, che, A, ſia il centro del Sole, &amp;</s>
  <s xml:id="echoid-s718" xml:space="preserve">, <lb/>A C, A D, due raggi, che da quello ſi par <lb/>tino, che veramente ſono diuergenti, C <lb/>D, la larghezza dello ſpecchio, &amp;</s>
  <s xml:id="echoid-s719" xml:space="preserve">, B E, paral-<lb/>lela à, C D, &amp;</s>
  <s xml:id="echoid-s720" xml:space="preserve">, B C, E D, l’vna, e l’altra della <lb/>lũghezza d’vn miglio, e lo Specchio, C D, lar-<lb/>go dieci braccia, eſſendo il cẽtro del Sole nel-<lb/>la maſſima vicinanza alla terra, ſarà dunque, <lb/>C A, 1101. </s>
  <s xml:id="echoid-s721" xml:space="preserve">ſemidiametri terreni, che commu-<lb/>nemente ſi ſtiman’eſſere di 3436. </s>
  <s xml:id="echoid-s722" xml:space="preserve">miglia, cioè <lb/>ſarà, C A, miglia 3783036. </s>
  <s xml:id="echoid-s723" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s724" xml:space="preserve">A B, vn miglio <lb/>manco, adunque, C A, ad, A B, ſarà come, <lb/>3783036. </s>
  <s xml:id="echoid-s725" xml:space="preserve">à, 1383035. </s>
  <s xml:id="echoid-s726" xml:space="preserve">e per eſſere, A B E, <lb/>A C D, triangoli ſimili, ſarà, C D, à, B E, co-<lb/>me, C A, ad, A B, e per la conuerſione della <lb/>proportione ſarà, C D, all’ecceſſo di, C D, ſo-<lb/>pra, B E, come, A C, à C B, cioè come, 378-<lb/>3036. </s>
  <s xml:id="echoid-s727" xml:space="preserve">à, 1. </s>
  <s xml:id="echoid-s728" xml:space="preserve">adunque inteſa, C D, larghezza di <lb/>10. </s>
  <s xml:id="echoid-s729" xml:space="preserve">braccia diuiſa in 3783036. </s>
  <s xml:id="echoid-s730" xml:space="preserve">di queſte ne <lb/>manca vna ſola vnità à, B E, e tanto mancano <lb/>le, B C, E D, dall’eſſer parallele, e perciò vn <lb/>cotale ſuario farà l’Eliſſi deſcritta con eſſat-<lb/>tiſſima diligẽza ſopra li detti duoi fochi, dalla
<pb o="81" file="0101" n="101" rhead="Coniche. Cap. XXV."/>
Parabola, di cui ſarà foco vn de’fochi della <lb/>detta Eliſsi; </s>
  <s xml:id="echoid-s731" xml:space="preserve">veggaſi hora ſe l’arte può diſcer-<lb/>nere vn 3783036. </s>
  <s xml:id="echoid-s732" xml:space="preserve">eſimo di 10. </s>
  <s xml:id="echoid-s733" xml:space="preserve">braccia, ouero <lb/>vn 378303. </s>
  <s xml:id="echoid-s734" xml:space="preserve">eſimo d’vn brac. </s>
  <s xml:id="echoid-s735" xml:space="preserve">che chiaramẽte <lb/>ſi conoſcerà ſe ſia di biſogno d’vſar così eſqui-<lb/>ſita |diligẽza ꝑ preualerſi dello Specchio Elit-<lb/>tico, che diuerſo poi anco ſi deue fabricare ꝑ <lb/>le diuerſe diſtãze del Sole dal centro della ter-<lb/>ra, ouero ſe ſia meglio preualerſi dello Spec-<lb/>chio Parabolico, che nõ hà biſogno d’eſſer va-<lb/>riato, per la varietà di tali diſtãze, ne anco ſe <lb/>il Sole foſſe doue ſon le ſtelle fiſſe, e quelle foſ-<lb/>ſero diſtanti da noi tanto, che l’orbe del Sole <lb/>foſſe inſenſibile in comparatione delle ſtelle <lb/>fiſſe, come ſtimò Ariſtarco, &amp; </s>
  <s xml:id="echoid-s736" xml:space="preserve">i ſuoi ſeguaci; <lb/></s>
  <s xml:id="echoid-s737" xml:space="preserve">poiche ſe pur voleſſimo cõcepire per vna cer-<lb/>ta analogia vn’altro foco nella Parabola, non <lb/>come Parabola, ma come vn’acutiſſima Eliſſi, <lb/>quello ſi deue intendere infinitamente diſtã-<lb/>te dalla ſua cima, dal quale però ne vẽgonole <lb/>linee parallele, e facẽdoſi queſto da vn’immẽ-<lb/>ſa diſtãza, qual’è quella del Sole anco nella ſua <lb/>maſſima vicinanza al centro della terra, la re-<lb/>putiamo in comparatione de’noſtri Specchi, <lb/>come diſtanza infinita, e però potiamo ſtar
<pb o="82" file="0102" n="102" rhead="Delle Settioni"/>
nello Specchio Parabolico, poiche volendo <lb/>aſſalire la fabrica dello Specchio Elittico, da-<lb/>remo al ſicuro nel Parabolico, quando anco <lb/>con ogni eſſattezza poſſibile a’nof<unsure/>tri iſtro-<lb/>menti pretendiamo d’hauerlo fatto Elittico; <lb/></s>
  <s xml:id="echoid-s738" xml:space="preserve">ſi è poi ſnppoſto il Sole nella maſſima vici-<lb/>nanza à terra, e lo Specchio di diametro di <lb/>dieci braccia, &amp; </s>
  <s xml:id="echoid-s739" xml:space="preserve">i raggi ſolari nella diſtan-<lb/>za d’vn miglio, perche l’argomento ſtringa <lb/>più fortemente per le diſtanze maggiori del <lb/>Sole, per le minori larghezze de’Specchi di <lb/>quel, che ſian 10. </s>
  <s xml:id="echoid-s740" xml:space="preserve">braccia, alla quale l’arte <lb/>noſtra forſi non può arriuare, ſe nõ con gran-<lb/>diſsima difficoltà, e per i raggi preſi in minor <lb/>diſtanza, che d’vn miglio, che ſempre più, e <lb/>più ſi vanno alla equidiſtanza auicinando.</s>
  <s xml:id="echoid-s741" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s742" xml:space="preserve">Sò però, che lo ſuario de’raggi diuergenti, <lb/>dai paralleli, ch’andariano nello Specchio ad <lb/>incontrare i me deſimi punti, ſi deue conſide-<lb/>rare quanto all’angolo, che viene contenuto <lb/>dalla parallela, e diuergente, che concorrono <lb/>nell’iſteſſo punto, ſe ben ſi è miſurato con la <lb/>retta linea, che reſtaria intrapreſa tra eſſa <lb/>diuergente, e parallela, dalla quale ſi diſ-<lb/>coſta, ciò hò però fatto per maggor chiarez-
<pb o="83" file="0103" n="103" rhead="Coniche. Cap. XXV."/>
za, non vi eſſendo molta differenza per inten-<lb/>dere quanto ſi pretende: </s>
  <s xml:id="echoid-s743" xml:space="preserve">tuttauia per non <lb/>tralaſciare adietro coſa, che poſſa far chiaro il <lb/>noſtro concetto, non mancarò di dichiararla <lb/>parimente in queſto modo; </s>
  <s xml:id="echoid-s744" xml:space="preserve">Sia pure nella 19. <lb/></s>
  <s xml:id="echoid-s745" xml:space="preserve">figura, A, cẽtro del Sole, dal quale nello Spec-<lb/>chio, C O D, largo dieci braccia, diſcendano <lb/>li duoi raggi, A C, A D, eſſendo lo Specchio, <lb/>C O D, talmente ſituato verſo il Sole, che il <lb/>ſuo aſſe, che ſia, O N, prolongato, concorri <lb/>nel centro di quell@; </s>
  <s xml:id="echoid-s746" xml:space="preserve">incontrino adunque <lb/>detti raggi la ſuperficie dello Specchio in, C, <lb/>D, e ſia il Sole viciniſsimo à terra, e nel punto, <lb/>C, cõcorra il raggio, P C, parallelo all’aſſe, A <lb/>O, e nel piano, P C A, ſia tirata la, H F, tãgen-<lb/>te lo Specchio in, C; </s>
  <s xml:id="echoid-s747" xml:space="preserve">gionta dũque, C D, dico, <lb/>che lo ſuario della diuergẽte, A C, dalla paral-<lb/>lela, P C, è l’angolo, P C A, che è acutiſſimo, <lb/>come ſarà manifeſto, ſe nel triãgolo, C N A, <lb/>trouaremo l’angolo, C A N, per le Tauole de’ <lb/>Seni, imperoche, come, A C, 3783036. </s>
  <s xml:id="echoid-s748" xml:space="preserve">miglia, <lb/>cioè braccia 11349108000. </s>
  <s xml:id="echoid-s749" xml:space="preserve">à, C N, che è la <lb/>metà di, C D, cioè br. </s>
  <s xml:id="echoid-s750" xml:space="preserve">5. </s>
  <s xml:id="echoid-s751" xml:space="preserve">così è 10000000000. </s>
  <s xml:id="echoid-s752" xml:space="preserve"><lb/>à 4. </s>
  <s xml:id="echoid-s753" xml:space="preserve">cioè à, C N, ſeno di, C A N, qual ſarà cir-<lb/>ca 20. </s>
  <s xml:id="echoid-s754" xml:space="preserve">ſcrupuli quinti, cioè inſenſibile à noi, e
<pb o="84" file="0104" n="104" rhead="Delle Settioni"/>
però anco, P C A, ſarà inſenſibile, che è l’an-<lb/>golo di tale ſuario. </s>
  <s xml:id="echoid-s755" xml:space="preserve">Stimo però, che il ſudetto <lb/>Padre, come perſona di valore, conoſciute <lb/>queſte difficoltà, habbi tuttauia per eſſercitio <lb/>de’ſpecolatiui, eletto più lo Specchio Elitti-<lb/>co, che il Parabolico, trattãdo più to ſto quan-<lb/>to alla Teorica, e matematicamẽte, che quan-<lb/>to alla Prattica, e fiſicamente, poiche ſpeco-<lb/>latiuamente s’intende bene, che douria eſſere <lb/>vn’Eliſſi, ma in Prattioa, operan do anco dili-<lb/>gentiſſimamente, ci verrà fatta la Parabola, <lb/>alla conſtruttion della quale trouandoni noi <lb/>pur ſomma difficoltà, ci contentiamo poi an-<lb/>co della Sferica.</s>
  <s xml:id="echoid-s756" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div63" type="section" level="1" n="63">
<head xml:id="echoid-head68" style="it" xml:space="preserve">Come per rifleſſione ſi poſſi accender fuoco con il ri-<lb/>uerbero della fiamma, ò de i carboni acceſi. <lb/>Cap. XXVI.</head>
<p>
  <s xml:id="echoid-s757" xml:space="preserve">SI potrà parimente eccitar l’in-<lb/>cendio al riuerbero della fiam-<lb/>ma, ò de’carboni acceſi, oppo-<lb/>nendogli lo Specchio concauo <lb/>Parabolico, Sferico, &amp; </s>
  <s xml:id="echoid-s758" xml:space="preserve">anco <lb/>Iperbolico; </s>
  <s xml:id="echoid-s759" xml:space="preserve">e ciò non manca di ragione, poi-
<pb o="85" file="0105" n="105" rhead="Coniche. Cap. XXVI."/>
che dalla fiamma, ouero da vn’aggregato di <lb/>carboni acceſi ſi partono infinite linee à tutte <lb/>le poſitioni, che non eſſendo impedite, cami-<lb/>nano ſin doue ſi eſtende la loro attiuità, den-<lb/>tro la quale vi ſono anco le parallele, che per-<lb/>ciò ſi vniranno in vn punto, cauſando iui l’in-<lb/>cendio, e ciò quãdo vi ſi opponga lo Specchio <lb/>Parabolico, &amp; </s>
  <s xml:id="echoid-s760" xml:space="preserve">anco proſſimamente il conca-<lb/>uo Sferico, e ſimilmente l’lperbolico, poiche <lb/>dentro quegl’infiniti raggi vi ſono ancora i <lb/>cõuergenti alfoco eſteriore dell’Iperbola, che <lb/>ſi vnitãno perciò nel di lei foco interiore, do-<lb/>ue ecciteranno l’incendio; </s>
  <s xml:id="echoid-s761" xml:space="preserve">è ben vero, che gli <lb/>altri raggi, che à queſti paralleli, e cõuergen-<lb/>tinel detto punto ſi auuicinano, aiuteranno <lb/>loro ancora detto incendio, benche non ſi vni-<lb/>ſcano tutti inſieme, &amp; </s>
  <s xml:id="echoid-s762" xml:space="preserve">iui cauſarãno calor grã-<lb/>de; </s>
  <s xml:id="echoid-s763" xml:space="preserve">eſperienza di queſto hò fatto io, che con <lb/>vno Specchio sferico di piõbo ancor mal po-<lb/>lito, hò acceſo il fuoco nella materia arida al <lb/>fuoco di carboni; </s>
  <s xml:id="echoid-s764" xml:space="preserve">e di più l’hò fatto con la ſu-<lb/>perficie Parabolica, cioè con vn Cãnone Para-<lb/>bolico, che hauea il ſuo foco vicino alla cima, <lb/>eſſendo eſſo Specchio Parabolico trõcato pur <lb/>nella cima, qual’era di ſtagno, e mal polito,
<pb o="86" file="0106" n="106" rhead="Delle Settioni"/>
tal che opponendolo al fuoco, ò alla fiamma <lb/>di ben poca legna, nella diſtanza di tre brac-<lb/>cia, ponendo la mano lì, dou’era la parte trõ-<lb/>cata, &amp; </s>
  <s xml:id="echoid-s765" xml:space="preserve">il foco della Parabola, non vi ſi potea <lb/>ſoſtenere, anzi vi s’acceſe fuoco; </s>
  <s xml:id="echoid-s766" xml:space="preserve">la qual coſa <lb/>potria alcuno applicare al riſcaldamẽto delle <lb/>ſtanze, ò alle diſtillationi; </s>
  <s xml:id="echoid-s767" xml:space="preserve">baſtami però d’ha-<lb/>uere al curioſo Lettore accennato queſto, la-<lb/>ſciãdo poi alla ſua induſtria il cercare il reſto.</s>
  <s xml:id="echoid-s768" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div64" type="section" level="1" n="64">
<head xml:id="echoid-head69" style="it" xml:space="preserve">Come in due maniere potiamo ſeruirci delli ſudetti <lb/>Specchi. Cap. XXVII.</head>
<p>
  <s xml:id="echoid-s769" xml:space="preserve">POtendoci noi ſeruire della ſuper-<lb/>ficie Sferica, Parabolica, &amp; </s>
  <s xml:id="echoid-s770" xml:space="preserve">Iper-<lb/>bolica intiera, ò d’vna parte ſo-<lb/>la, conforme che quella può eſ-<lb/>ſer diuerſa, diuerſamente ancora <lb/>chiamaremo lo Specchio, dãdoci queſte il mo-<lb/>do d’accendere il fuoco in che ſito vogliamo; <lb/></s>
  <s xml:id="echoid-s771" xml:space="preserve">Se adunque prenderemo di queſta ſuperficie <lb/>quella parte, ch’è intorno alla cima, queſta <lb/>abbrucierà tra’l corpo focoſo, e lo Specchio; </s>
  <s xml:id="echoid-s772" xml:space="preserve"><lb/>ma ſe vogliamo, che l’incendio ſia di dietro <lb/>dello Specchio, biſognerà pigliare vna parte
<pb o="87" file="0107" n="107" rhead="Coniche. Cap. XXVII."/>
diquella, diſcoſta dalla cima tanto, che laſci <lb/>fuori di ſe il foco di tal ſuperficie verſo la ci-<lb/>ma, come per eſſempio; </s>
  <s xml:id="echoid-s773" xml:space="preserve">Sia nella 20. </s>
  <s xml:id="echoid-s774" xml:space="preserve">figura v-<lb/>na tal ſuperficie la, CAF, cioè, verbi gra-<lb/>tia, Parabolica, il cui aſſe ſia, AI, e foco, O; <lb/></s>
  <s xml:id="echoid-s775" xml:space="preserve">tagliando adunque tal ſuperficie con vn pia-<lb/>no, al quale, A I, ſia perpendicolare, che diui-<lb/>da detta ſuperficie nelle due, B A G, B C F G, <lb/>è manifeſto, che la parte, B A G, intorno la ci-<lb/>ma, A, abbruciarà tra lei, e’l fuoco, ò Sole nel <lb/>punto, O, e la parte, B C F G, abbruciarà di <lb/>dietro nell’iſteſſo foco, O, qual però chiama-<lb/>remo Cãnone Parabolico, e quando vorremo, <lb/>che abbruci lontano, intendendo prodotta <lb/>la ſuperficie parabolica, per eſſempio in, D <lb/>E, e l’aſſe in, H, prendendo il Cãnone para-<lb/>bolico, C D E F, quello pure abbruciarà nel <lb/>punto, O, più lon tano dallo Specchio, che <lb/>nõ facea il Cãnone, B C F G, e così potremo <lb/>abbruciare infinitamente lontano, prolongã-<lb/>do ſempre detta ſupeificie Parabolica, e prẽ-<lb/>dendola in quella diſtanza, che ci biſogna. </s>
  <s xml:id="echoid-s776" xml:space="preserve"><lb/>Potiamo poi anco di queſto Cãnone, ò Spec-<lb/>chio prendere vn ſol pezzo, come, R E Z S, <lb/>che non ſolo abbrucierà di dietro da lui nel
<pb o="88" file="0108" n="108" rhead="Delle Settioni"/>
punto, O, ma anco da vna parte, qual ſi potrà <lb/>chiamar Fruſto della ſuperf<unsure/>icie Parabolica; </s>
  <s xml:id="echoid-s777" xml:space="preserve">e <lb/>l’iſteſſo s’intenda detto per la Sferica, poſcia-<lb/>che nella 17. </s>
  <s xml:id="echoid-s778" xml:space="preserve">figura la faſcia Sferica, BCH <lb/>N, e la minor di quella, abbrucierà dietro di <lb/>lei, come lo Specchio Sferico, B A N, dinãzi; <lb/></s>
  <s xml:id="echoid-s779" xml:space="preserve">e l’iſteſſo s’incenda per l’Iperbolica; </s>
  <s xml:id="echoid-s780" xml:space="preserve">potremo <lb/>dunque con queſte cauſar l’incendio da che <lb/>parte, e lontano, quanto noi vorremo. </s>
  <s xml:id="echoid-s781" xml:space="preserve">Non <lb/>poſſo poi tralaſciar di dire, come la sfera, ò <lb/>lente chriſtallina eſpoſta al fuoco de’carbo-<lb/>ni, ò alla fiamma, non cauſa l’incendio, co-<lb/>me li Specchi, la onde pare, che ſi poteſſe rac-<lb/>cogliere, che la rifleſſione foſſe più potẽte del-<lb/>la refrattione, tuttauia ciò non determi-<lb/>no in virtù di queſto, poiche ſa-<lb/>ria di biſogno eſſaminar pri-<lb/>ma molt’altre <lb/>coſe, che per breuità <lb/>tralaſcio.</s>
  <s xml:id="echoid-s782" xml:space="preserve"/>
</p>
  <figure>
    <image file="0108-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0108-01"/>
  </figure>
<pb o="89" file="0109" n="109" rhead="Coniche Cap. XXVIII."/>
</div>
<div xml:id="echoid-div65" type="section" level="1" n="65">
<head xml:id="echoid-head70" style="it" xml:space="preserve">Dello Specchio Vctorio d’Archimede. <lb/>Cap. XXVIII.</head>
<p>
  <s xml:id="echoid-s783" xml:space="preserve">LEggeſi nell’antiche hiſtorie del-<lb/>le guerre de’Romani, ch’eſsẽ-<lb/>do aſſediata Siracuſa, così per <lb/>terra, come per mare, da Ap-<lb/>pio, e Marco Marcello, con ap-<lb/>parato grandiſſimo da guerra, vi ſi oppoſe <lb/>talmente il valore, e l’induſtria d’Archime-<lb/>de, che per lui ſolo parea ſi ſoſteneſſe l’im-<lb/>petuoſo aſſalto d’vn’eſſercito sì potente, e <lb/>già d’altre Illuſtri Città vittorioſo; </s>
  <s xml:id="echoid-s784" xml:space="preserve">onde <lb/>Tito Liuio nella Dec. </s>
  <s xml:id="echoid-s785" xml:space="preserve">1. </s>
  <s xml:id="echoid-s786" xml:space="preserve">al Cap. </s>
  <s xml:id="echoid-s787" xml:space="preserve">24. </s>
  <s xml:id="echoid-s788" xml:space="preserve">fù sfor-<lb/>zato à dire; </s>
  <s xml:id="echoid-s789" xml:space="preserve">Et habuiſſet tanto impetu cæpta res <lb/>fortunam, niſi vnus bomo Syracuſis ea tempeſtate <lb/>fuiſſet; </s>
  <s xml:id="echoid-s790" xml:space="preserve">e veramente ſi può credere facilmen-<lb/>te, che, s’egli haueſſe hauuto la fortuna <lb/>alquanto più propitia, haurebbe alla Patria <lb/>ſaluata la libertà, ed à ſe medeſimo la vita. <lb/></s>
  <s xml:id="echoid-s791" xml:space="preserve">Imperoche in gratia del ſuo Re hauea fa-<lb/>bricato così ſtupende machine per i biſo-<lb/>gni da guerra, che i preparamenti fatti con <lb/>immenſa ſpeſa, e con molti ſtenti da i ne-<lb/>mici, per l’oppugnatione, erano da lui con
<pb o="90" file="0110" n="110" rhead="Delle Settioni"/>
tale artificio deluſi, e reſi del tutto inutili, <lb/>che parea più toſto ſcherzaſſe, che combat-<lb/>teſſe da douero con nemici così potenti. <lb/></s>
  <s xml:id="echoid-s792" xml:space="preserve">Perciò il medeſimo parlando di sì eminente <lb/>ingegno, ſoggiunſe nell’iſteſſo luogo. </s>
  <s xml:id="echoid-s793" xml:space="preserve">Archi-<lb/>medes is erat, vnicus ſpectator Cæli, ſyderumque, <lb/>mirabilior tamen inventor, ac machinator bellico-<lb/>rum tormentorum, operumq; </s>
  <s xml:id="echoid-s794" xml:space="preserve">quibus ea, quæ bo-<lb/>ctes ingenti mole agerent, ipſe perleui momento lu-<lb/>dificaretur. </s>
  <s xml:id="echoid-s795" xml:space="preserve">Furno le machine diuerſe, con le <lb/>quali così di vicino, come di lontano ſcaglia-<lb/>ua pietre di molta grandezza; </s>
  <s xml:id="echoid-s796" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s797" xml:space="preserve">era di gran <lb/>marauiglia veder con vna mano di ferro, le-<lb/>gata ad vna forte catena, prender le naui per <lb/>la prora, e drizzatele ſopra la poppa, laſciarle <lb/>poſcia con ineuitabil naufragio, precipitoſa-<lb/>mente cadere. </s>
  <s xml:id="echoid-s798" xml:space="preserve">Altri parimente furono gli <lb/>ordegni, con che valoroſiſſimamente facea <lb/>à’nemici reſiſtenza, come raccontano Polibio <lb/>nellib. </s>
  <s xml:id="echoid-s799" xml:space="preserve">8. </s>
  <s xml:id="echoid-s800" xml:space="preserve">Plutarco nella vita di M. </s>
  <s xml:id="echoid-s801" xml:space="preserve">Marcello, <lb/>Dione, &amp; </s>
  <s xml:id="echoid-s802" xml:space="preserve">altri Hiſtorici famoſi, che tutti con-<lb/>cordemente eſſaltano l’ingegno d’Archime-<lb/>de, come coſa ſopra humana, e quaſi diuina: </s>
  <s xml:id="echoid-s803" xml:space="preserve"><lb/>Ma fra tutte le marauiglioſe inuentioni di sì <lb/>grand’huomo, non vi è, per mio credere, co-
<pb o="91" file="0111" n="111" rhead="Coniche. Cap. XXVIII."/>
ſa, che habbi arrecato maggior ſtupore, ne <lb/>che ſia ſtata tenuta in maggior pregio, ò che <lb/>habbi dato più da ſpecolare à i curioſi, di <lb/>quel famoſo Specchio, con il quale, eſſendoſi <lb/>ritirate le naui, quant’è vn tiro d’arco, per nõ <lb/>ſentire i duriſſimi colpi delle pietre, che con-<lb/>tinuamente erano ſcagliate dalle mura, in vir-<lb/>tù de’raggi ſolari, vniti inſieme, vſcendo im-<lb/>petuoſamente, à guiſa di fulmine, il fuoco dal <lb/>medeſimo Specchio, causò vn’incendio così <lb/>formidabile, che la maggior parte delle naui <lb/>fù ridotta in cenere. </s>
  <s xml:id="echoid-s804" xml:space="preserve">Così riferiſce Galeno <lb/>πει' χράσεων lib. </s>
  <s xml:id="echoid-s805" xml:space="preserve">3. </s>
  <s xml:id="echoid-s806" xml:space="preserve">dicendo: </s>
  <s xml:id="echoid-s807" xml:space="preserve">Hoc vtiq; </s>
  <s xml:id="echoid-s808" xml:space="preserve">modo a-<lb/>iunt, puto, Archimedem per comburentia ſpecula <lb/>hoſtium triremes incendiſſe. </s>
  <s xml:id="echoid-s809" xml:space="preserve">Succenditur verò faci-<lb/>lè à comburente ſpeculo, &amp; </s>
  <s xml:id="echoid-s810" xml:space="preserve">lana, &amp; </s>
  <s xml:id="echoid-s811" xml:space="preserve">ctuppa, &amp; </s>
  <s xml:id="echoid-s812" xml:space="preserve">el-<lb/>lychtnium, &amp; </s>
  <s xml:id="echoid-s813" xml:space="preserve">ferula, &amp; </s>
  <s xml:id="echoid-s814" xml:space="preserve">quidquid deniq; </s>
  <s xml:id="echoid-s815" xml:space="preserve">ſimiliter <lb/>eſt aridũ, &amp; </s>
  <s xml:id="echoid-s816" xml:space="preserve">rarũ Il medeſimo racconta Zonara <lb/>Greco, Autore antichiſſimo, nel 3. </s>
  <s xml:id="echoid-s817" xml:space="preserve">Tomo del-<lb/>le ſue Hiſtorie, hauer fatto Proclo ſotto Co-<lb/>ſtantinopoli, con grandiſſimo danno dell’ar-<lb/>mata nemica, c’hauea aſſediato quella Città.</s>
  <s xml:id="echoid-s818" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s819" xml:space="preserve">Queſte coſe adunque, benche da grauiſ-<lb/>ſimi Scrittori riferite, furono tuttauia da mol-<lb/>ti tenute per fauole più toſto, che per verità,
<pb o="92" file="0112" n="112" rhead="Delle Settioni"/>
parendoli molto improbabile poterſi cauſare <lb/>incendio per via di Specchi in così gran diſtã-<lb/>za, in quanta s’intende, che fece Archimede <lb/>dalle mura di Siracuſa. </s>
  <s xml:id="echoid-s820" xml:space="preserve">Altri per il contrario, <lb/>non volendo metter dubbio nelle relationi di <lb/>così illuſtri, e ſegnalati Scrittori, crederno be-<lb/>ne tal coſa poter’eſſer ſtata, ma nel penetrare <lb/>il vero modo, hanno incontrato di molte diffi-<lb/>coltà, ſi nell’inueſtigar la ſorma di quello Spec <lb/>chio, ſi anco nel ridurſi à metterlo in prattica: <lb/></s>
  <s xml:id="echoid-s821" xml:space="preserve">Imperoche ſentendo mentouare, che quello <lb/>foſſe di forma Parabolica, ſi ſono meſſi con o-<lb/>gni induſtria à cõſiderare le proprietà di que-<lb/>ſta forma di Specchio, inſegnando varij modi <lb/>per diſegnare la Parabola, acciò fattane la <lb/>ſagma, ſe ne poteſſe poi formare lo Specchio <lb/>Parabolico, come ſi può vedere in Vitellione, <lb/>Marin Ghetaldo, Orõtio, Cardano, Gio. </s>
  <s xml:id="echoid-s822" xml:space="preserve">Bat-<lb/>tiſta Porta, &amp; </s>
  <s xml:id="echoid-s823" xml:space="preserve">altri valenti Matematici, coſpi-<lb/>rando forſi tutti nel marauiglioſo Specchio di <lb/>Archimede; </s>
  <s xml:id="echoid-s824" xml:space="preserve">ma per quanto ſi ſiano affaticati <lb/>queſti ingegni, non pare, che ci habbino da-<lb/>to vna chiara cognitione della ſtruttura di <lb/>quello, poſciache ci hãno ſolamente inſegna-<lb/>to cauſarſi l’incendio in vn ſol punto, mercè
<pb o="93" file="0113" n="113" rhead="Coniche. Cap. XXVIII."/>
dello Specchio Parabolico, cioè nel concorſo <lb/>de’raggi ſolari, più, e men lontano, ſecondo <lb/>che eſſo ſarà più, e men cauo; </s>
  <s xml:id="echoid-s825" xml:space="preserve">la onde hanno <lb/>ſ<unsure/>timato molti, che lo Specchio d’Archimede <lb/>abbruciaſſe così lontano, perche foſſe di for-<lb/>ma Parabolica, e con tal proportione fabri-<lb/>cato, che faceſſe il concorſo de’raggi in tan-<lb/>ta diſtanza, in quanta le ſudette naui ſi erano <lb/>ritirate. </s>
  <s xml:id="echoid-s826" xml:space="preserve">Ma chi non vede quanto ciò dal ve-<lb/>riſimile ſi diſcoſti? </s>
  <s xml:id="echoid-s827" xml:space="preserve">poiche vno Specchio Pa-<lb/>rabolico, che habbi il foco lontano da lui ſolo <lb/>trenta piedi, come ben dice il Porta nella ſua <lb/>Magia naturale, e tanto poco differente dal <lb/>piano (s’ei non foſſe di ſmiſurata grandezza) <lb/>che l’arte noſtra non lo può diſtinguere, <lb/>e perciò ne men fabricare; </s>
  <s xml:id="echoid-s828" xml:space="preserve">come dunque ſi <lb/>può credere, che lo Specchio d’Archimede <lb/>foſſe tale, che il concorſo de’raggi ſi faceſſe <lb/>lontano quanto vn tiro d’arco? </s>
  <s xml:id="echoid-s829" xml:space="preserve">Di più vno <lb/>Specchio tale cauſa l’incendio in vna deter-<lb/>minata diſtanza, e ſolo da vna bãda, cioè ver-<lb/>ſo il Sole; </s>
  <s xml:id="echoid-s830" xml:space="preserve">doue che l’abbruciamẽto delle Na-<lb/>ui portaua diuerſe diſtanze, e forſe diuerſi ſiti <lb/>ancora, e però in conſeguenza, ò biſognaua <lb/>hauerne più d’vno, ò mouere il medeſimo, per
<pb o="94" file="0114" n="114" rhead="Delle Settioni"/>
aggiuſtarſi alle diſtanze, il che pare, che da <lb/>vna muraglia d’vna Rocca non così ageuol-<lb/>mente ſi poteſſe fare: </s>
  <s xml:id="echoid-s831" xml:space="preserve">E finalmente il ſottiliſ-<lb/>ſimo ingegno d’Archimede mi dà à credere, <lb/>ch’egli penetraſſe più à dentro di quello, che <lb/>l’vniuerſale intende à prima viſta.</s>
  <s xml:id="echoid-s832" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s833" xml:space="preserve">Altri hanno penſato, che qualche materia <lb/>de’Specchi ſtrauagãte, à noi incognita, ado-<lb/>peraſſe, che haueſſe virtù di cauſar l’incendio <lb/>tanto lõtano; </s>
  <s xml:id="echoid-s834" xml:space="preserve">ma io non credo ne quello det-<lb/>to di ſopra, per le già addotte ragioni, ne me-<lb/>no queſto, mentre alla materia non s’accom-<lb/>pagni la figura, poiche la rifleſſione ricerca <lb/>l’vno, e l’altro; </s>
  <s xml:id="echoid-s835" xml:space="preserve">cioè per parte della materia, <lb/>ſomma lucidezza, e politura, che ſuol venire <lb/>particolarmẽte dalla durezza dell’iſteſſa ma-<lb/>teria, per parte poi della figura richiede, per-<lb/>che s’vniſchino tutti i raggi in vn punto, che <lb/>ſia Parabolica.</s>
  <s xml:id="echoid-s836" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s837" xml:space="preserve">Non ſono però mancati ancora di quelli, <lb/>che hanno ſtimato, che Archimede non per <lb/>via di rifleſſione, ò refrattione abbruciaſſe <lb/>l’Armata, ma con alcune pietre, chiamate da’ <lb/>Greci πυριγης, ſcagliandole nelle naui, all’v-<lb/>ſanza delle palle di fuochi artificiati, vi ſuſci-
<pb o="95" file="0115" n="115" rhead="Coniche. Cap. XXVIII."/>
taſſe l’incendio; </s>
  <s xml:id="echoid-s838" xml:space="preserve">E di queſta opinione pare, <lb/>che ſia Tomaſo Linacro, Interprete de i libri <lb/>di Galeno, De Temperaturis; </s>
  <s xml:id="echoid-s839" xml:space="preserve">che cõmentan-<lb/>do le ſudette parole, e volendo ſpiegare la <lb/>voce Greca, πυριε, l’interpreta, come che <lb/>ſignifichi le ſudette pietre di fuoco. </s>
  <s xml:id="echoid-s840" xml:space="preserve">Ma ciò <lb/>hà parimente dell’improbabile, poiche, come <lb/>pur’iui ſoggiunge Galeno, dette pietre non <lb/>s’infuocano, ſe non s’infrangono; </s>
  <s xml:id="echoid-s841" xml:space="preserve">ma chi po-<lb/>teua in fiangere, e ſpoluerizzare, per dir così, <lb/>dette pietre nelle naui de’Romani? </s>
  <s xml:id="echoid-s842" xml:space="preserve">ouero, <lb/>come di già ſpezzate, poteuano ſcagliare il <lb/>fuoco dalle mura ſino alle dette naui? </s>
  <s xml:id="echoid-s843" xml:space="preserve">come <lb/>argomẽta Dauid Riualto nel fine de’ſuoi cõ-<lb/>menti ſopra Archimede nello Scholio De Spe-<lb/>culis Vſtorijs Archimedis. </s>
  <s xml:id="echoid-s844" xml:space="preserve">Ma quel, che più im-<lb/>porta, ancor che ſi poteſſero lanciare nelle <lb/>naui coſe, che per la percoſſa, toccando le <lb/>medeſime, ſi accendeſſero, come hò ſentito da <lb/>huomini prattici hoggidì vſarſi in guerra con <lb/>certe palle artificioſe, econ vaſi di fuochi ſtra-<lb/>uaganti, tuttauia quello, che in queſto nego-<lb/>tio chiariſce il tutto è, che la relatione de gli <lb/>Scrittori, e maſſime di Galeno, in queſto lu<unsure/>o-<lb/>go ci vien ſignificando, che Archimede ado-
<pb o="96" file="0116" n="116" rhead="Delle Settioni"/>
peraſſe gli Specchi, &amp; </s>
  <s xml:id="echoid-s845" xml:space="preserve">i raggi del Sole, poiche <lb/>perappunto dà l’eſſempio d’vna caſa, che nel-<lb/>la Miſia, parte dell’Aſia, ſi abbruciò per il ca-<lb/>lor del Sole, eſſendoſi attaccato fuoco nel tet-<lb/>to, mediante la reſina, &amp; </s>
  <s xml:id="echoid-s846" xml:space="preserve">il letame de’Colom-<lb/>bi, ſoggiungendo poi l’incendio delle naui di <lb/>Marco Marcello, il che dimoſtra hauer’egli <lb/>creduto, che quello pur fuſſe cauſato da i rag-<lb/>gi del Sole. </s>
  <s xml:id="echoid-s847" xml:space="preserve">In queſta ſtrauaganza di penſie-<lb/>ri adunque ſono dati quelli, che pur’hanno <lb/>voluto dar credenza alle relationi di così ce-<lb/>lebri, e così illuſtri Scrittori. </s>
  <s xml:id="echoid-s848" xml:space="preserve">Inãzi però, ch’io <lb/>ſpieghi qual ſia il mio penſiero intorno à que-<lb/>ſto, fà di meſtieri dir qualche coſa intorno al-<lb/>la Linea Vſtoria di Gio. </s>
  <s xml:id="echoid-s849" xml:space="preserve">Battiſta Porta, deſcrit <lb/>ta da lui nella ſua Magia naturale nel lib. </s>
  <s xml:id="echoid-s850" xml:space="preserve">17. <lb/></s>
  <s xml:id="echoid-s851" xml:space="preserve">al Cap. </s>
  <s xml:id="echoid-s852" xml:space="preserve">17.</s>
  <s xml:id="echoid-s853" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div66" type="section" level="1" n="66">
<head xml:id="echoid-head71" style="it" xml:space="preserve">Della Linea Vſtoria di Gio. Battiſta Porta, che ab-<lb/>brucia in infinito. Cap. XXIX.</head>
<p>
  <s xml:id="echoid-s854" xml:space="preserve">IL Porta dunque nel ſudetto luogo, dopò <lb/>hauer riferito ciò, che mediãte gli Spec-<lb/>chi Vſtorij fecero Archimede à Siracuſa, <lb/>e Proclo à Coſtãtinopoli, parla d’vn ſuo Spec-
<pb o="97" file="0117" n="117" rhead="Coniche. Cap. XXIX."/>
chio, come di coſa molto diuerſa da quelli, <lb/>tanto, per ſuo dir, marauiglioſo, che non cre-<lb/>de l’ingegno humano poter paſſar più oltre, <lb/>reputand<unsure/>o quelli de’ſudetti Autori molto à <lb/>queſto inferiori, com’egli dimoſtra, mentre <lb/>nell’iſteſſo luogo ſoggiunge; </s>
  <s xml:id="echoid-s855" xml:space="preserve">Sed longè cæteris <lb/>præſtãtiorem modum trademus, à nemine equidem, <lb/>quod ſciam, traditum, antiquorum omnium, &amp; </s>
  <s xml:id="echoid-s856" xml:space="preserve">re-<lb/>centiorum inuentionem ſuperantem, nec putò huma-<lb/>num ingenium maiora excogitare poſſe. </s>
  <s xml:id="echoid-s857" xml:space="preserve">Hoc Specu-<lb/>lum non ad decem, viginti, centum, aut mille paſ-<lb/>ſus comburit, vel ad determinatam diſtantiam, ſed <lb/>in infinitum, nec in cono accendit, vbi radĳ coeunt, <lb/>ſed à Speculi centro Vſtoria Linea procedit, cui@ſuis <lb/>longitudinis, quæ obuia omnia comburit. </s>
  <s xml:id="echoid-s858" xml:space="preserve">Præterea <lb/>accendit retro, ante, &amp; </s>
  <s xml:id="echoid-s859" xml:space="preserve">ex omniparte. </s>
  <s xml:id="echoid-s860" xml:space="preserve">Vero è, che <lb/>venendo poi à ſpiegare il ſuo penſiero, in ve-<lb/>ce di manifeſtarcelo, cuopre il ſecreto con <lb/>parole à bello ſtudio traſportate, e ci laſcia ſi-<lb/>tibondi della vera cognitione d’vn tanto ar-<lb/>tificio. </s>
  <s xml:id="echoid-s861" xml:space="preserve">Ma perche preuale in me più d’ogni <lb/>altra coſa il deſiderio di giouare al publico, <lb/>perciò ſpiegarò con parole più chiare, che ſia <lb/>poſſibile, quanto mièſouuenuto nello ſpeco-<lb/>lare intorno à queſto mirabil Problema, che
<pb o="98" file="0118" n="118" rhead="Delle Settioni"/>
il Porta ci propone di fare cõil ſuo Specchio, <lb/>e vederemo inſieme ſe ſia veriſimile, che Ar-<lb/>chimede, Proclo, &amp; </s>
  <s xml:id="echoid-s862" xml:space="preserve">il medeſimo Porta ſi ac-<lb/>cordino nell’inuentione, come io da molti ſe-<lb/>gni vado congetturando.</s>
  <s xml:id="echoid-s863" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div67" type="section" level="1" n="67">
<head xml:id="echoid-head72" style="it" xml:space="preserve">In qual ſenſo ſtimi l’Autore che la ſudetta Linea <lb/>Vstoria ſi poſſa ſoſtenere. Cap. XXIX.</head>
<p>
  <s xml:id="echoid-s864" xml:space="preserve">CHe ſia poſſibile vnire molte li-<lb/>nee radioſe fra diloro parallele <lb/>in vn punto, cioè farle di paral-<lb/>lele conuergenti, ciò è manife-<lb/>ſto perle coſe dette di ſopra, e <lb/>nella Tauola Specolaria trouiamo farci que-<lb/>ſto ſeruitio la ſuperficie concaua parabolica; <lb/></s>
  <s xml:id="echoid-s865" xml:space="preserve">ma con quale artificio ſi poſſino ſtringere mol <lb/>te linee radioſe in vna ſola (sì come ſi condu-<lb/>cono ad vn pũto per via dello Specchio Para-<lb/>bolico) la quale perciò habbi forza d’abbru-<lb/>ciare tutto quello, che incontra, non ſolo, di-<lb/>co, di non poterlo penetrare, ma parermi aſ-<lb/>ſolutamente impoſſibile, imperoche ò queſto <lb/>ſi farà per rifleſſione, ò per refrattione; </s>
  <s xml:id="echoid-s866" xml:space="preserve">facciſi <lb/>pure in qualunque de i due modi, è neceſſario
<pb o="99" file="0119" n="119" rhead="Coniche. Cap XXIX."/>
prima condurre i raggi paralleli in vn punto, <lb/>il che ſappiamo di già fare cõ lo Specchio Pa-<lb/>rabolico, e che iui poi ritrouino qualche cor-<lb/>po, che rifletti, ò rifranga i medeſimi raggi, <lb/>facendoli tutti caminare per vna linea ſola, <lb/>quale veramente haurebbe le conditioni, che <lb/>il Porta ci promette: </s>
  <s xml:id="echoid-s867" xml:space="preserve">quindi per il contrario, <lb/>quella Linea preſa come incidente, haureb-<lb/>be non vna rifleſſa, ò rifratta, ma diuerſe, il <lb/>che, come beniſſimo dice il Keplero nella ſua <lb/>Diottrica, alla pag. </s>
  <s xml:id="echoid-s868" xml:space="preserve">21. </s>
  <s xml:id="echoid-s869" xml:space="preserve">è contro le leggi del-<lb/>la Proſpettiua, poiche ſecondo quella, eguali <lb/>angoli d’incidenza cauſano eguali angoli di <lb/>rifleſſione, ò rifrattione, e perciò alla mede-<lb/>ſima incidente non poſſono corriſpondere dal <lb/>punto della incidẽza diuerſe rifleſſe, ò rifrat-<lb/>te, ma ſi bene vna ſola, adunque per il cõtra-<lb/>rio vna moltiplicità di raggi paralleli ſi potrà <lb/>bene far concorrere in vn punto, ma che quel-<lb/>li ſi poſſino ſtringere in vna Linea ſola, ciò re-<lb/>puto con il Keplero aſſolutamente impoſſi-<lb/>bile; </s>
  <s xml:id="echoid-s870" xml:space="preserve">Inteſa dũque la Linea Vſtoria del Porta, <lb/>à queſta maniera pare à me coſa molto impro <lb/>babile; </s>
  <s xml:id="echoid-s871" xml:space="preserve">e ſe lo Specchio d’Archimede haueſ-<lb/>ſe hauuto à fare vna tale operatione, credo
<pb o="100" file="0120" n="120" rhead="Delle Settioni"/>
quanto à me, che vano ſaria ſtato lo sforzo del <lb/>medeſimo, per abbrnciar le naui di M. </s>
  <s xml:id="echoid-s872" xml:space="preserve">Mar-<lb/>cello, come pure intendiamo, ch’egli fece.</s>
  <s xml:id="echoid-s873" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s874" xml:space="preserve">Reſta dunque, ch’andiamo vn poco conſi-<lb/>derando, ſe pur’è poſſibile in qualche modo <lb/>fare quel, che promette il Porta, allontanan-<lb/>doſi poco ancora dal ſuo ſenſo. </s>
  <s xml:id="echoid-s875" xml:space="preserve">Dico adunque <lb/>ſe per linea intẽderemo non ſtrettiſſimamen-<lb/>te quello, che importa il nome di linea, ma <lb/>vn poco più alla groſſa, cioè che voglia dir tã-<lb/>to quanto vn Cilindro, ò Cannoncino di lu-<lb/>me, di che ſottigliezza vogliamo, indiffinita-<lb/>mente prolongato, che veramcnte in queſto <lb/>ſenſo parmi eſſer poſſibile il farlo, ſi che quei <lb/>raggi, che caminarãno paralleli per ſpatio di <lb/>vn braccio, ſi poſſino ſtringere in vn Cãnon-<lb/>cino di lume groſſo vn’oncia, e manco anco-<lb/>ra, al qual potiamo dar nome di linea, alla ſi-<lb/>militudine delle linee abuſate da noi in ma-<lb/>teria, che hanno tuttauia qualche groſſezza; <lb/></s>
  <s xml:id="echoid-s876" xml:space="preserve">nel qual ſenſo credo, che’l Porta ſi poſſa ſo-<lb/>ſtenere, e che ſia poſſibile fare (con modera-<lb/>tione però) quanto egli propone, cioè abbru-<lb/>ciare dinanzi, e di dietro, anzi da ogni parte <lb/>dello Specchio, doue egli non arrechi impedi-
<pb o="101" file="0121" n="121" rhead="Coniche Cap. XXIX."/>
mento, e ciò non in vn punto, ò nella coinci-<lb/>denza de’raggi, ma in ogniluogo, doue ſi e-<lb/>ſtenda quel Cannoncino di lume, che di ſua <lb/>natura cauſarebbe l’incendio anco in ogni di-<lb/>ſtãza, ſe i medeſimi raggi non ſi andaſſero cõ-<lb/>tinuamente debilitando. </s>
  <s xml:id="echoid-s877" xml:space="preserve">Stima veramente <lb/>il Keplero, che la combuſtione ſi cauſi per il <lb/>ſegamento de’raggi luminoſi concorrenti in <lb/>vn punto, non sò però s’egli intenda, che ſo-<lb/>lo in queſto modo, e non altrimenti ſi cauſi <lb/>l’accenſione in virtù de’medeſimi raggi, com-<lb/>unque egli creda, à me pare probabiliſſimo, <lb/>che quelli non vniti in vn punto, ma anco in <lb/>anguſto ſpatio coſtretti, poſſino generar fuo-<lb/>co, poiche gli Specchi sferici cauſano l’incen-<lb/>dio, e pur ſappiamo, che non vniſcono in vn <lb/>ſol pũto ſe non quelli, che vengono rifleſſi dal <lb/>medeſimo cerchio parallelo alla bocca dello <lb/>Specchio, che pur ſon pochi; </s>
  <s xml:id="echoid-s878" xml:space="preserve">e la mano, che <lb/>riceue i raggi del Sole rifleſsi dallo Specchio, <lb/>non nel punto del concorſo, ma alquanto da <lb/>quello diſcoſto, ſente ben tanto calore, che <lb/>tenendouela più d’vn poco, ci accorgia-<lb/>mo, che iui è forza digenerar fuoco; </s>
  <s xml:id="echoid-s879" xml:space="preserve">perciò <lb/>non credo vi ſarà dubbio, che quel Cannonci-
<pb o="102" file="0122" n="122" rhead="Delle Settoni"/>
no di lume, eletto di conueneuol groſſezza, <lb/>non ſia per abbruciare per qualche anco nota-<lb/>bil diſtanza. </s>
  <s xml:id="echoid-s880" xml:space="preserve">Inteſa dunque in queſto modo <lb/>la Linea Vſtoria del Porta, parmi poterſi ſoſte-<lb/>nere, anzi hauer molta probabilità, che con-<lb/>cordi con l’inuentione e di Proclo, &amp; </s>
  <s xml:id="echoid-s881" xml:space="preserve">anco di <lb/>Archimede, come da’ſeguenti Capi ſi potrà <lb/>meglio comprendere.</s>
  <s xml:id="echoid-s882" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div68" type="section" level="1" n="68">
<head xml:id="echoid-head73" style="it" xml:space="preserve">Dello Specchio Vſtorio imaginato dall’Autore, <lb/>e varietà di quello. Cap. XXX.</head>
<p>
  <s xml:id="echoid-s883" xml:space="preserve">PEr dimoſtrare la probabilità del-<lb/>la Linea Vſtoria già detta, e de <lb/>gli Specchi Vſtorij di Proclo, e <lb/>d’Archimede, è finalmente ne-<lb/>ceſſario, ch’io ſpieghi ciò, che <lb/>ſpecolando mi è ſouuenuto, manifeſtãdo l’eſ-<lb/>ſemplare di queſto mio Specchio, all’eſplica-<lb/>tione del quale è principalmente ordinato il <lb/>preſente Trattato.</s>
  <s xml:id="echoid-s884" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s885" xml:space="preserve">Ma prima dirò pur’anch’io col medeſimo <lb/>Porta; </s>
  <s xml:id="echoid-s886" xml:space="preserve">Sed profectò indignũ facinus duco ignarę ple <lb/>bi propalare. </s>
  <s xml:id="echoid-s887" xml:space="preserve">Prode at ergo in lucẽ, vt ſumma Dei <lb/>immenſa bonitas laudetur, veneretur. </s>
  <s xml:id="echoid-s888" xml:space="preserve">Poiche nõ
<pb o="103" file="0123" n="123" rhead="Coniche. Cap. XXX."/>
ſarà poco con que<unsure/>ſto debol lume, ch io porgo <lb/>per l’intelligenza d’vn sì nobil ſoggetto, po-<lb/>ter, venendo alla prattica, eſſequire quanto <lb/>dalla ſpecolatiua hauremo imparato.</s>
  <s xml:id="echoid-s889" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s890" xml:space="preserve">Per far dunque gli effetti, ch’egli ci pro-<lb/>poſe, ſtimo douerſi lo Specchio di tal ſorte fa-<lb/>bricare, cioè, che vniſca i raggi ſolari in vn <lb/>punto, e perciò, per mio credere, egli dourà <lb/>eſſer Parabolico, conforme alla dottrina di ſo-<lb/>pra inſegnata, douendo fare i raggi paralleli <lb/>cõuergenti; </s>
  <s xml:id="echoid-s891" xml:space="preserve">quãto poi all’vnire eſſi raggi vici-<lb/>no, ò lontano, io al contrario de gl’altri pen-<lb/>ſo, che ſarà meglio, che l’vnione, ò foco di t<unsure/>a-<lb/>le Spechio non ſia dal medeſimo molto lonta-<lb/>no, ſi che non veniamo à dare nella difficoltà <lb/>di quelli, che cercano di mandar tal concorſo <lb/>lontano, che perciò ſi riducono à lauorare <lb/>vno Specchio inſenſibilmente dal piano diffe-<lb/>rente. </s>
  <s xml:id="echoid-s892" xml:space="preserve">Fatto queſto Specchio Parabolico, ſe al-<lb/>tro non vi ſi aggiongeſſe, non è dubbio, che <lb/>non accenderebbe fuoco più lontano di quel-<lb/>lo, che ſia il punto del concorſo; </s>
  <s xml:id="echoid-s893" xml:space="preserve">per hauer <lb/>dunque queſta operatione in altre diſtanze <lb/>ancora, è neceſſario portar più oltre quella for <lb/>za, che hanno i raggi ſolari inanzi, ò doppo,
<pb o="104" file="0124" n="124" rhead="Delle Settioni"/>
vicino al concorſo, cioè ò fare i raggi conuer-<lb/>genti paralleli, e ciò inanzi, ouero i raggi di-<lb/>uergenti pur paralleli, e ciò doppo il concor-<lb/>ſo; </s>
  <s xml:id="echoid-s894" xml:space="preserve">ricorrendo adunque alla Tauola Specola-<lb/>ria, trouaremo inãzi al concorſo douerſi ado-<lb/>perare la conueſſa Parabolica, e dopo la con-<lb/>caua pur Parabolica, auuertendo, che è neceſ-<lb/>ſario ſiano inſieme vniti il foco dello Specchio <lb/>grande, &amp; </s>
  <s xml:id="echoid-s895" xml:space="preserve">il foco dello Specchio piccolo, al-<lb/>trimẽte non dourà riuſcire l’operatione, ſtret-<lb/>tamente parlando: </s>
  <s xml:id="echoid-s896" xml:space="preserve">Vniti dunque, che ſiano <lb/>queſti duoi fochi, lo Specchio piccolo conueſ-<lb/>ſo, vibrarà quei raggi, ch’egli riceuerà nella <lb/>ſua cõueſſita (ch’erano cõuergenti al foco del <lb/>lo Specchio grãde, ch’è vnito col foco del pic-<lb/>colo) paralleli all’aſſe del piccolo, che perciò, <lb/>per ragion d’vnione dourãno per notabil ſpa-<lb/>tio conſeruare la medeſima forza, c’hebbero <lb/>nel dipartirſi dal piccolo, benche in queſta <lb/>ſeconda rifleſſione venghino alquanto à in-<lb/>debolirſi. </s>
  <s xml:id="echoid-s897" xml:space="preserve">Se dunque riuoltaremo l’aſſe dello <lb/>Specchietto verſo quel luogo, doue ſi vorrà <lb/>accender fuoco, quel Cãnoncino di lume, che <lb/>vſcirà dallo Specchietto piccolo, attaccarà <lb/>iui fuoco, anzi à guiſa di trapano, dourà tra-
<pb o="105" file="0125" n="125" rhead="Coniche. Cap. XXIX."/>
forare quelle materie combuſtibili, ch’egli in-<lb/>contrarà; </s>
  <s xml:id="echoid-s898" xml:space="preserve">auuertaſi però, che la conuerſione <lb/>dello Specchietto dourà ſempre farſi intorno <lb/>al ſuo foco, come centro. </s>
  <s xml:id="echoid-s899" xml:space="preserve">Ma per maggior di-<lb/>chiaratione, eccone l’eſſempio, con l’eſſem-<lb/>plare dello Specchio.</s>
  <s xml:id="echoid-s900" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s901" xml:space="preserve">Eſſempio ſopra la vigeſimaprima figura.</s>
  <s xml:id="echoid-s902" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s903" xml:space="preserve">Sla lo Specchio Parabolico, T X B, il cui foco il <lb/>punto, I, talmente indrizzato verſo il Sole, <lb/>A M Q, che l’aſſe di quello prolongato, come, <lb/>X M, vadi à ferire nel cẽtro del Sole, dal quale vẽ-<lb/>gano nello Specchio, T X B, iraggi, A T, M X, Q B<unsure/>, <lb/>paralleli, ſotto i quali intenderemo tutti gli altri, che <lb/>caſcano nell’iſteſſo Specchio. </s>
  <s xml:id="echoid-s904" xml:space="preserve">E’dunque manifeſto, che <lb/>queſti raggi, che formano vn Cannone di lume largo, <lb/>quanto è lo Specchio, T X B, diuentaranno doppo l’in-<lb/>cidenza conuergẽti al foco, I; </s>
  <s xml:id="echoid-s905" xml:space="preserve">Sia hora fatto vn’altro <lb/>Specchietto Parabolico c<unsure/>õueſſo, D R G, il cui foco, I, <lb/>ſia pochiſſimo diſcoſto dalla cima R, e ſiano talmente <lb/>il grande, &amp; </s>
  <s xml:id="echoid-s906" xml:space="preserve">il piccolo inſieme collocati, che il foco <lb/>dell’vno, e dell’altro ſia vnito nel punto, I, eſſen-<lb/>do l’aſſe del piccolo, F I R; </s>
  <s xml:id="echoid-s907" xml:space="preserve">pongaſi poi, che ſi vo-<lb/>glia accendere il fuoco verſo, P; </s>
  <s xml:id="echoid-s908" xml:space="preserve">eſſendo adunque
<pb o="106" file="0126" n="126" rhead="Delle Settioni"/>
lo Specchietto talmente accommodato che ſia conuer-<lb/>tibile intorno al punto I, lo riuolgeremo tanto, che <lb/>ſtia per dritto al punto, P, come, F R P: </s>
  <s xml:id="echoid-s909" xml:space="preserve">Perche dũ-<lb/>que ſi è moſtrato nel Cap. </s>
  <s xml:id="echoid-s910" xml:space="preserve">11. </s>
  <s xml:id="echoid-s911" xml:space="preserve">che la ſuperficie con-<lb/>ueſſa Parabol@ca riceuendo le conuergenti al ſuo foco, <lb/>le riflette parallele all’aſſe, perciò lo Specch@etto, D <lb/>R G, riceuendo per fianco i raggi, che dalio Specchio <lb/>grande ſi partono conuergenti al foco, I, inanzi, che <lb/>vi arriuino, ma ben vicino à quello, li rifletterà in <lb/>angusto ſpatio paralleli all’aſſe F R, cioè tutt@ri-<lb/>stretti nel Cannoncino, ò Cilindretto luminoſo, P @ <lb/>Z, che anderà pure à ferire al punio, P, &amp; </s>
  <s xml:id="echoid-s912" xml:space="preserve">iuicauſa-<lb/>rà l’incendio nella materia di facil combu stione in <lb/>quella diſtanza, che l’indebolimento de’raggi cauſa-<lb/>to per le due r@fleſſioni ci permetterà, el’eſperienza <lb/>c’inſegnarà; </s>
  <s xml:id="echoid-s913" xml:space="preserve">è adunque chiaro, che questo artificio <lb/>non è altro, che stringer quei raggi, che caminaua-<lb/>no paralleli, per eſſempio, nell’ampiezza d’vn <lb/>braccio alla ſottigliezza d’vn dito, pur facendoli ca-<lb/>minar paralleli, creſcendo la forza de’raggi, ſecondo <lb/>la reciproca proportione de i quadrati delle groſſezze <lb/>de’Cilindri; </s>
  <s xml:id="echoid-s914" xml:space="preserve">cioè, ſe il grande ſarà decuplo del pic-<lb/>colo in groſſezza, il piccolo haurà forza d@ riſcaldare <lb/>cẽto volte più efficace della forza del grande, douẽdo <lb/>l’iſteſſa quãtità de’raggi operar’in vno ſpatio ſubcẽ-
<pb o="107" file="0127" n="127" rhead="Coniche. Cap. XXX."/>
tuplo à quello, nel quale bauria da operare il grãde; <lb/></s>
  <s xml:id="echoid-s915" xml:space="preserve">questa proportione però ſi verificarebbe preciſamen-<lb/>te, ſe nelle due rifleſſioni non ſi ſcapitaſſe niente, co-<lb/>me pur’accade, che perciò biſogna leuarne la tara, che <lb/>importano dette rifleſſioni, che così ſapremo quanta <lb/>debba eſſer la forza del Cilindro piccolo doppo le due <lb/>r@fleſſioni generato.</s>
  <s xml:id="echoid-s916" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s917" xml:space="preserve">In altro modo poi ſi potrà ottener l’iſteſſo con <lb/>lo Specchio troncato, ò Cannone Parabolico, che <lb/>h<gap/>l ſuo foco di dietro, collocando iui lo Specchietto <lb/>nel modo ſopradetto, come ſi vede nella figura 22.</s>
  <s xml:id="echoid-s918" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s919" xml:space="preserve">Di più, ſe in vece del cõueſſo adopraremo il con-<lb/>cauo dello Specchietto Parabolico, riceuendo i raggi <lb/>diuergenti doppo il concorſo nell’vna, el’altra figu-<lb/>ra, otteneremo il medeſimo, e di tutto questo ne hab. <lb/></s>
  <s xml:id="echoid-s920" xml:space="preserve">biamo ſicura dimoſtratione.</s>
  <s xml:id="echoid-s921" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s922" xml:space="preserve">Ma ſe vogliamo credere, che l’vnione de’raggi <lb/>fatta non preciſamente, ma proſsimamente in vn <lb/>punto, equiuaglia, quanto al far quello, che cerchia-<lb/>mo, all’vnione fatta preciſamente in vn punto, po-<lb/>tremo in vece di Specchietii Paraboli<unsure/>ci, ſeruirci delli <lb/>Sferici, ouero adoperare le lenti, poiche la lente caua, <lb/>ò traguardo, farà l’offitio dello Specchietto conueſſo, e <lb/>la lente conueſſa del concauo, quella dourà riceuere i <lb/>raggi conuergẽti, e queſta diuergenti, imparando dal
<pb o="108" file="0128" n="128" rhead="Delle Settioni"/>
Keplero nella ſua Diottrica, quanto ſi a lontano il fo-<lb/>co loro dalle medeſime lenti, poiche quello dourà sta-<lb/>re vnito col foco dello Specchio grande.</s>
  <s xml:id="echoid-s923" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s924" xml:space="preserve">Maperche la ſudetta Diottrica non ſarà forſi così <lb/>alle mani di ciaſcuno, perciò mi è parſo bene metter <lb/>quà quel poco, che vi bò trouato poter’eſſere à no-<lb/>ſtro propoſito, in materia dell’vnire, ò diſunire i <lb/>raggi per via di queste lenti, cioè.</s>
  <s xml:id="echoid-s925" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s926" xml:space="preserve">Che la lente conueſſa da vna ſola banda, ma di <lb/>portione minore di G. </s>
  <s xml:id="echoid-s927" xml:space="preserve">30. </s>
  <s xml:id="echoid-s928" xml:space="preserve">oppoſta perpendicolarmen-<lb/>te à i raggi paralleli, con il conueſſo verſo loro, gli v-<lb/>niſce proſſimamente in vn punto, lontano dal con-<lb/>ueſſo tre ſemidiametri di eſſa conueſſità in circa, ſe <lb/>però non ſi rifrangeſſero anco nella baſe. </s>
  <s xml:id="echoid-s929" xml:space="preserve">Prop. </s>
  <s xml:id="echoid-s930" xml:space="preserve">34.</s>
  <s xml:id="echoid-s931" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s932" xml:space="preserve">Che la medeſima riuolta al contrario, gli vniſſe <lb/>lontano dal cõueßo per due ſemidiametri di eßa con-<lb/>ueſſità in circa. </s>
  <s xml:id="echoid-s933" xml:space="preserve">Prop. </s>
  <s xml:id="echoid-s934" xml:space="preserve">35.</s>
  <s xml:id="echoid-s935" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s936" xml:space="preserve">Che la lente conueſſa d’ambedue le bãde della con-<lb/>ueſſità dell’iſteſſo cerchio, eſpoſta perpendicolarmen-<lb/>te à i raggi paralleli, gli vniſſe lontano dal conueſſo <lb/>(che riſguarda eſſi paralleli) vn ſemidiametro della <lb/>medeſima conueſſità in circa. </s>
  <s xml:id="echoid-s937" xml:space="preserve">Prop. </s>
  <s xml:id="echoid-s938" xml:space="preserve">39.</s>
  <s xml:id="echoid-s939" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s940" xml:space="preserve">E però le medeſime lenti faranno per il contrario <lb/>i raggi diuergenti dal punto, nel quale ſi è detto farſi <lb/>il concorſo, paralleli, douendoſi queſte adoprare dop-
<pb o="109" file="0129" n="129" rhead="Coniche. Cap. XXX."/>
po l’vnione fatta dallo Specchio grande.</s>
  <s xml:id="echoid-s941" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s942" xml:space="preserve">Potiamo ancora mecanicamente trouare il punto <lb/>del loro concorſo, mediante i raggi del Sole, oſſeruan-<lb/>do doue gli raccogliono; </s>
  <s xml:id="echoid-s943" xml:space="preserve">ouero in vna carnera ſerra-<lb/>ta, che habbi ſolo vn pertugio nella finestra, doue eſ-<lb/>ſa lente ſi deue collocare, oſſeruando, quanto lontano <lb/>da quella caſca la distintiſſima pittura de gli ogge@@i <lb/>di fuori ſopra la carta, postali dirimpetto, che iui è il <lb/>p@nto del concorſo.</s>
  <s xml:id="echoid-s944" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s945" xml:space="preserve">Quanto poi alle lenti caue, c’inſegna ſolamẽte, che <lb/>fanno le parallele, ouero diuergenti pur diuergẽti nel-<lb/>le Prop. </s>
  <s xml:id="echoid-s946" xml:space="preserve">90. </s>
  <s xml:id="echoid-s947" xml:space="preserve">91. </s>
  <s xml:id="echoid-s948" xml:space="preserve">92. </s>
  <s xml:id="echoid-s949" xml:space="preserve">93. </s>
  <s xml:id="echoid-s950" xml:space="preserve">94. </s>
  <s xml:id="echoid-s951" xml:space="preserve">la onde per il @õirario fa-<lb/>ranno le conuergenti parallele, ò conuergenti ad altro <lb/>punto, e perciò ſi dourãno adoprare inanzi al concorſo <lb/>fatto dallo Specchio grande, imparãdo dall’eſperien-<lb/>za il ſuo vero luogo.</s>
  <s xml:id="echoid-s952" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s953" xml:space="preserve">Potremo finalmente, in vece dello Specchio con-<lb/>cauo grande, adoperar la lente cõueſſa, combinata con <lb/>li Specchietti, ò con le lenti, che n’hauremo il medeſi-<lb/>mo: </s>
  <s xml:id="echoid-s954" xml:space="preserve">E questo fù auuertito dal ſottiliſſimo ingegno <lb/>del Keplero nella medeſima Diottrica alla Prop. </s>
  <s xml:id="echoid-s955" xml:space="preserve">106. <lb/></s>
  <s xml:id="echoid-s956" xml:space="preserve">come ci manifesta dicendo. </s>
  <s xml:id="echoid-s957" xml:space="preserve">Quod I. </s>
  <s xml:id="echoid-s958" xml:space="preserve">Baptiſta pro-<lb/>fitetur radios Solis primum colligere, poſt col-<lb/>lectos in infinitum mittere, &amp; </s>
  <s xml:id="echoid-s959" xml:space="preserve">ſic comburere, <lb/>etſi de Speculis loquitur, videtur tamen de
<pb o="110" file="0130" n="130" rhead="Delle Settioni"/>
perſpicillis intelligi debere, quia de induſtria <lb/>occultauit ſententiam. </s>
  <s xml:id="echoid-s960" xml:space="preserve">Doue ſoggiunge poi la <lb/>combinatione della lente caua, e conueſſa per fare tal’ <lb/>effetto, credendo queſto conuenire alle lenti più toſto, <lb/>che alli Specchi; </s>
  <s xml:id="echoid-s961" xml:space="preserve">ma noi ſappiamo di già per dimoſtra-<lb/>tione, che lo deuono fare i Specchi Parabolici, che <lb/>quanto alle lenti ci è per anco naſcosto qual figura <lb/>debbano hauere per vnire in vn punto, ò d ſunir da <lb/>quello; </s>
  <s xml:id="echoid-s962" xml:space="preserve">non dubito però,<unsure/> che ſe il Keplero, come d’in-<lb/>gegno perſpicace, haueſſe fatto rifleſſione alle proprie. <lb/></s>
  <s xml:id="echoid-s963" xml:space="preserve">tà de’Specchi Parabolici, nõ haueſſe creduto it mede-<lb/>ſimo de’Specchi, ch’egli mostrò di creder delle lenti.</s>
  <s xml:id="echoid-s964" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s965" xml:space="preserve">Non tacerò anco, che ſe in vece di Specchietti Pa-<lb/>rabolici gli adopraremo @perbolici, talmente fabrica-<lb/>ti, che habbino il foco esteriore tanto lontano, quan-<lb/>ta ſarà la maggior distanza, nella quale vorremo ab-<lb/>bruciare, che me deſimamente douremo hauer l’intẽ-<lb/>to noſtro, e ciò forſi ſarà d’aiuto all’Operario, men-<lb/>tre egli non vien riſtretto alla forma Parabolica, ma <lb/>ſe gli allarga il campo dalla moltiplicità dell’Iperbo-<lb/>lette, che in vece di Cilindretti vibraranno Con@ lu-<lb/>minoſi, che potranno hauer le baſi non più larghe di <lb/>quelle, c’haurebbono detti Cilindretti. </s>
  <s xml:id="echoid-s966" xml:space="preserve">La groſſezza <lb/>poi del metallo, del quale ſi formarãno gli Specchiet-<lb/>ti, ò del chriſtallo, del quale ſaranno fabricate le len-
<pb o="111" file="0131" n="131" rhead="Coniche. Cap. XXX."/>
ti, farà, che reſtino per qualche t@mpo contumaci alla <lb/>forza de’raggi, che da loro venendo vibrati verſo <lb/>materie ài facil combuſtione, accenderanno in quelle <lb/>il fuoco, inanzi che li Specchietti, o l@ lenti patiſca-<lb/>no, come altri potrebbe temere, ſe ben questi ancora <lb/>not abilmente ſi riſcalderanno.</s>
  <s xml:id="echoid-s967" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s968" xml:space="preserve">Come ſi può probabilmentecongetturare, che lo Spec-<lb/>chio a’Archimede, Proclo, e del Por@a, non molto <lb/>diſoordi da quello, che ſi è dichiarato nel Capo an-<lb/>tecedente. </s>
  <s xml:id="echoid-s969" xml:space="preserve"># Cap. </s>
  <s xml:id="echoid-s970" xml:space="preserve">XXXI.</s>
  <s xml:id="echoid-s971" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s972" xml:space="preserve">VEggaſihora, c’habbiamo inteſo <lb/>queſte coſe, s’egli hà del veri-<lb/>ſimile, che l’inuentione accen-<lb/>nata nell ant. </s>
  <s xml:id="echoid-s973" xml:space="preserve">Cap. </s>
  <s xml:id="echoid-s974" xml:space="preserve">s’accordi con <lb/>quelle de’ſudetti Autori; </s>
  <s xml:id="echoid-s975" xml:space="preserve">io per <lb/>me lo tẽgo per probabiliſſimo, prima per ſen-<lb/>tir noi quaſi tutti gl’I ſtorici concordemente <lb/>aſſerire lo Specchio d’Archimede eſſere ſtar<unsure/>o <lb/>Parabolico, come ancoil Porta accenna del <lb/>ſuo, eſi può ragioneuolmente credere anco di <lb/>Proclo (ſe bene non ardirei negare, che in ve-<lb/>ce de’Parabolici non ſi foſſero ſeruiti forſi an-<lb/>co de i Sferici, ma haueſſero dato ad intende-
<pb o="112" file="0132" n="132" rhead="Delle Settioni"/>
re quelli eſſer Parabolici, perche altri non co-<lb/>sì facilmente trouaſſe l’artificio, per la diffi-<lb/>coltà di fare il Parabolico) &amp; </s>
  <s xml:id="echoid-s976" xml:space="preserve">eſſendo o Para-<lb/>bolico, o Sferico, per hauer noi inteſo eſſer’im-<lb/>poſſibile con quelli abbruciar’in tanta diſtan-<lb/>za, ſenza l’aggiunta di qualche altra coſa, la <lb/>quale o deue operare per rifleſſione, o per ri-<lb/>frattione; </s>
  <s xml:id="echoid-s977" xml:space="preserve">ſe per rifleſſione, ſarà ſtato Spec-<lb/>chio; </s>
  <s xml:id="echoid-s978" xml:space="preserve">ſe per rifrattione, dourà ſtimarſi, che foſ-<lb/>e lente. </s>
  <s xml:id="echoid-s979" xml:space="preserve">Secondo, mi vien ciò molto cõfermato <lb/>pal modo di ſpiegar l’operatione dell’attaccar <lb/>fuoco; </s>
  <s xml:id="echoid-s980" xml:space="preserve">poiche di Proclo dice pur Zonara di ſo-<lb/>pra citato, come riferiſce il Porta; </s>
  <s xml:id="echoid-s981" xml:space="preserve">Nam ſpecula <lb/>ex ære fabricaſſe Vstoria fertur Proclus, in quæ cum <lb/>Solares rad<unsure/>ĳ impegiſſent, @gnem inde fulminis inſtar <lb/>erumpentem, claſſiarios, ipſaſq́ combuſiſſe, doue <lb/>quell’ignem fulminis in ſtar erumpentem, mi par, <lb/>che ci rappreſenti quel Cilindretto Vſtorio, <lb/>che di ſopra ſi è dichiarato. </s>
  <s xml:id="echoid-s982" xml:space="preserve">E di Archimede <lb/>l’ifteſſo Zonara nel Tomo 2. </s>
  <s xml:id="echoid-s983" xml:space="preserve">parla pur, dicen-<lb/>do; </s>
  <s xml:id="echoid-s984" xml:space="preserve">Speculo enim quodam verſus Solem ſuſpenſo, <lb/>radios @xcepit, aereque ob denſitatem, &amp; </s>
  <s xml:id="echoid-s985" xml:space="preserve">leuitatem <lb/>Spe<unsure/>culi ex ĳs radĳs incenſo, efficit, vt m<unsure/>gens flam-<lb/>ma recta in naues illata omnes eas cremaret: </s>
  <s xml:id="echoid-s986" xml:space="preserve">che <lb/>pur cõferma la vibratione del ſudetto Cãnon-
<pb o="113" file="0133" n="133" rhead="Coniche. Cap. XXXI."/>
cino dilume. </s>
  <s xml:id="echoid-s987" xml:space="preserve">Il medeſimo ſi può credere del-<lb/>la Linea Vſtoria del Porta, la qual dice vſcir <lb/>dal cẽtro dello Specchio, eſſer di qualſiuoglia <lb/>lunghezza, &amp; </s>
  <s xml:id="echoid-s988" xml:space="preserve">abbruciare da che parte ſi vo-<lb/>glia tutto ciò, che incontra, il che molto ſi ac-<lb/>corda cõ il già detto Cilindretto Vſtorio; </s>
  <s xml:id="echoid-s989" xml:space="preserve">anzi <lb/>ſe bene eſſo Porta, traſponẽdo le parole, ci hà <lb/>naſcoſto il loro ſenſo, ſi raccoglie però à pez-<lb/>zo à pezzo, ch’egli parla d’vn’artificio, che <lb/>conſta d’vna coſa grande, e piccola, cioè di <lb/>Specchi Parabolici, ò Sferici, poiche dice: </s>
  <s xml:id="echoid-s990" xml:space="preserve">Sed <lb/>exeuntem radium ex Speculi ſuperficie Parabolica, <lb/>&amp; </s>
  <s xml:id="echoid-s991" xml:space="preserve">c. </s>
  <s xml:id="echoid-s992" xml:space="preserve">e poi; </s>
  <s xml:id="echoid-s993" xml:space="preserve">Nec refert Parabola ſit, aut Sphærica. <lb/></s>
  <s xml:id="echoid-s994" xml:space="preserve">e più à baſſo; </s>
  <s xml:id="echoid-s995" xml:space="preserve">Fenestra perforetur obliquè, vt re-<lb/>cipiat Speculum Parabolicum: </s>
  <s xml:id="echoid-s996" xml:space="preserve">più oltre poi; </s>
  <s xml:id="echoid-s997" xml:space="preserve">At <lb/>ſi parua magnæ in proportione non reſpondet, ſcitò <lb/>te nil operaſſe, magna ſit circa baſim, parua circa ver-<lb/>ticem, primæ æquidistans. </s>
  <s xml:id="echoid-s998" xml:space="preserve">e finalmente periſpie-<lb/>gar la ſeconda forma del medeſimo Specchio, <lb/>che ſi può vedere nella figura 22. </s>
  <s xml:id="echoid-s999" xml:space="preserve">ſoggiunge <lb/>nel fine: </s>
  <s xml:id="echoid-s1000" xml:space="preserve">Sicordi fuerit, vt accenſio anterius fiat, <lb/>ex ſectione, quæ circa baſim eſt, conficiatur torques, <lb/>in cuius medĳ puncto accommodetur artificium, vt <lb/>regreſſus radius in anterius prodat: </s>
  <s xml:id="echoid-s1001" xml:space="preserve">veggaſi ſe ſi <lb/>può dire più chiaro; </s>
  <s xml:id="echoid-s1002" xml:space="preserve">il che vien confermato
<pb o="114" file="0134" n="134" rhead="Delle Settioni"/>
da quello, ch’egli ſoggiũge parimente nel Ca-<lb/>pit. </s>
  <s xml:id="echoid-s1003" xml:space="preserve">19. </s>
  <s xml:id="echoid-s1004" xml:space="preserve">al Tit. </s>
  <s xml:id="echoid-s1005" xml:space="preserve">Refractione longiſſimè ignem accen-<lb/>dere: </s>
  <s xml:id="echoid-s1006" xml:space="preserve">poiche dice; </s>
  <s xml:id="echoid-s1007" xml:space="preserve">Conficit eodem modo lineas <lb/>tranſuersè inciſas parallelas, dixit Almeon; </s>
  <s xml:id="echoid-s1008" xml:space="preserve">cioè, <lb/>che ſi deuono fare le conuergẽti, ò diuergenti <lb/>(ſignificate per quel tranſuersè inciſas) paral-<lb/>lele, che ſi fà con gl’artificij detti di ſopra; </s>
  <s xml:id="echoid-s1009" xml:space="preserve">e <lb/>più à baſſo poi; </s>
  <s xml:id="echoid-s1010" xml:space="preserve">Videbis ignem per occultum, &amp; </s>
  <s xml:id="echoid-s1011" xml:space="preserve">a@ <lb/>pertum radium incidẽtem in ſuperficiem rectã, &amp; </s>
  <s xml:id="echoid-s1012" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s1013" xml:space="preserve">E finalmente quanto allo Specchio d’Archi-<lb/>mede, ch’egli adoperandolo o Parabolico, ò <lb/>Sferico, vi accompagnaſſe qualche altra coſa, <lb/>che o per rifleſsione, ò per rifrattione vibra-<lb/>ua quei raggi, che veniuano in lei raccolti, ce <lb/>lo manifeſtano le parole di Zetzes, Autore an-<lb/>tichiſsimo, che volendoci eſſo Specchio deſ-<lb/>criuere, ne parla in tal maniera, come ſi può <lb/>vedere nell’Archimede commentato da Da-<lb/>uid Riualto, che da quello le hà tradotte, e <lb/>poſte nel fine de’ſuoi Commenti, nello Scho-<lb/>lio, poſto al Titolo De Speculis Vstorĳs Arcbi-<lb/>medis, citato di ſopra nel Cap. </s>
  <s xml:id="echoid-s1014" xml:space="preserve">28.</s>
  <s xml:id="echoid-s1015" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1016" xml:space="preserve">Cum autem Marcellus remouiſſet illas ad iactum <lb/>arcus:</s>
  <s xml:id="echoid-s1017" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1018" xml:space="preserve">Hexagonum aliquod Speculum fabricauit ſenex:</s>
  <s xml:id="echoid-s1019" xml:space="preserve"/>
</p>
<pb o="115" file="0135" n="135" rhead="Coniche. Cap. XXXI."/>
<p style="it">
  <s xml:id="echoid-s1020" xml:space="preserve">A diſtantia autem commenſurati Speculi</s>
</p>
<p style="it">
  <s xml:id="echoid-s1021" xml:space="preserve">Parua talia ſpecilla cum poſuiſſet, quadrupla <lb/>angulis:</s>
  <s xml:id="echoid-s1022" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1023" xml:space="preserve">Quæ mouebantur laminis, &amp; </s>
  <s xml:id="echoid-s1024" xml:space="preserve">quibuſdam ſcul. <lb/></s>
  <s xml:id="echoid-s1025" xml:space="preserve">pturis,</s>
</p>
<p style="it">
  <s xml:id="echoid-s1026" xml:space="preserve">Medium illud poſuit radiorum Solis,</s>
</p>
<p style="it">
  <s xml:id="echoid-s1027" xml:space="preserve">Australis, &amp; </s>
  <s xml:id="echoid-s1028" xml:space="preserve">Æ stiualis, &amp; </s>
  <s xml:id="echoid-s1029" xml:space="preserve">Hyemalis:</s>
  <s xml:id="echoid-s1030" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1031" xml:space="preserve">Refractis deinceps in hoc radĳs,</s>
</p>
<p style="it">
  <s xml:id="echoid-s1032" xml:space="preserve">Exarſio ſublata e st formidabilis ignita nauibus.</s>
  <s xml:id="echoid-s1033" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1034" xml:space="preserve">Et has in cinerẽ redegit longitudine arcus iactus.</s>
  <s xml:id="echoid-s1035" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div69" type="section" level="1" n="69">
<head xml:id="echoid-head74" style="it" xml:space="preserve">Hist. 35. Chil. 2.</head>
<p>
  <s xml:id="echoid-s1036" xml:space="preserve">Io per me confeſſo non hauer trouato Autore, <lb/>che paſsi più inanzi di queſto nello ſpiegare <lb/>la forma del ſudetto Specchio d’Archimede; <lb/></s>
  <s xml:id="echoid-s1037" xml:space="preserve">dalle quali parole credo non oſcuramente ſi <lb/>poſsi comprẽdere, che oltre lo Specchio o Pa-<lb/>rabolico, ò Sferico che ſi foſſe, vi adopraſ-<lb/>ſe ancora o Specchietti, ò pur lenti; </s>
  <s xml:id="echoid-s1038" xml:space="preserve">quel ter-<lb/>mine poi di Hexagonuw aliquod Speculum ci ma-<lb/>nifeſta, che queſto Scrittore nõ ſapeua di che <lb/>ſorte foſſe tale Specchio, ma hauendo forſi <lb/>ſentito dire, che lo Specchio cõcauo Sferico <lb/>abbrucia dinanzi per quanto di quà, e di là ſi <lb/>eſtẽde il lato dell Eſſagono, come ſi è ſpiega-<lb/>to nel Cap. </s>
  <s xml:id="echoid-s1039" xml:space="preserve">21. </s>
  <s xml:id="echoid-s1040" xml:space="preserve">ſi riſolſe à ſcriuere, ch’era vn
<pb o="116" file="0136" n="136" rhead="Delle Settioni"/>
qualche Specchio Eſſagono; </s>
  <s xml:id="echoid-s1041" xml:space="preserve">il rimanente poi, <lb/>che ſoggiunge de gli Specchietti, ò vetri col-<lb/>locati in diſtãza proportionata allo Specchio <lb/>(poiche crederei diceſſe più toſto; </s>
  <s xml:id="echoid-s1042" xml:space="preserve">A diſtantia <lb/>autem commenſurata ſpeculo; </s>
  <s xml:id="echoid-s1043" xml:space="preserve">ma che foſſero cor-<lb/>rottele parole) moſsi con laminette, ò ſcultu-<lb/>re, ci moſtra la volubilità di quelli per attac-<lb/>car fuoco in diuerſe bande, hauendone più <lb/>d’vno, poſcia che molto ſi riſcaldano, e perciò <lb/>deuonſi mutare; </s>
  <s xml:id="echoid-s1044" xml:space="preserve">quel quadrupla angulis, poi non <lb/>ſa prei, che ſi voleſſe dire, ſe non forſi, che la <lb/>larghezza della baſe de’Specchietti Parabo-<lb/>lici (s’erano tali) foſſe quadrupla della pro-<lb/>fondità di quelli, il che ſaria ſtato, quando il <lb/>foco loro foſſe ſtato in sù la bocca dello Spec-<lb/>chietto, poiche la ordinatamente applicata <lb/>all’aſſe della Parabola, la qual paſſa per il di <lb/>lei foco, è eguale al lato retto di quella, e per-<lb/>ciò è quadrupla della parte dell’aſſe troncata <lb/>via da lei verſo la cima. </s>
  <s xml:id="echoid-s1045" xml:space="preserve">Quella parola, &amp; </s>
  <s xml:id="echoid-s1046" xml:space="preserve"><lb/>hyemalis, moſtra l’efficacia di quello Spec-<lb/>chio, che anco d’Inuerno facea tale operatio-<lb/>ne. </s>
  <s xml:id="echoid-s1047" xml:space="preserve">Refractis deinceps in hoc radĳs; </s>
  <s xml:id="echoid-s1048" xml:space="preserve">Se queſte <lb/>poi vſciſſero da perſona intelligente de’ter-<lb/>mini di Proſpettiua, e non da ſemplice Iſtori-
<pb o="117" file="0137" n="137" rhead="Coniche. Cap. XXXI."/>
co, biſognaria credere, che nõ foſſe vno Spec-<lb/>chio, ma vna lente quello, ch’adoperò Archi-<lb/>mede; </s>
  <s xml:id="echoid-s1049" xml:space="preserve">ma perche queſto Autore forſi non <lb/>hebbe intentione di ſpiegar’altro, che il rom-<lb/>pimento de’raggi, che ſi fà tanto per rifleſſio-<lb/>ne, come per rifrattione, perciò crederemo <lb/>ragioneuolmente, che quel termine, Refractis, <lb/>non più la rifrattione, che la rifleſſione ci poſſi <lb/>ſignificare. </s>
  <s xml:id="echoid-s1050" xml:space="preserve">Il dir poi Exarſio ſublata est for-<lb/>midabilis ignita nauibus. </s>
  <s xml:id="echoid-s1051" xml:space="preserve">Moſtra probabilmen-<lb/>te, che foſſe attaccato fuoco in più d’vna na-<lb/>ue, per douer cagionare vn’incendio così for-<lb/>midabile, il che ſi rincontra con quel, che ſi è <lb/>detto di ſopra; </s>
  <s xml:id="echoid-s1052" xml:space="preserve">e finalmente ci dichiara que-<lb/>ſto Autore la diſtanza, nella quale ſeguì que-<lb/>ſto incendio, mentre dice; </s>
  <s xml:id="echoid-s1053" xml:space="preserve">Et has in cinerem re-<lb/>degit longitudine arcus iactus; </s>
  <s xml:id="echoid-s1054" xml:space="preserve">cioè nella diſtan-<lb/>za d’vn tiro d’arco abbruciò le naui, e le riduſ-<lb/>ſe in cenere. </s>
  <s xml:id="echoid-s1055" xml:space="preserve">Queſta è tutta l’importanza del <lb/>negotio, accendere il fuoco in tanta diſtanza, <lb/>poiche l’abbruciar d’appreſſo, anzi per lo ſpa-<lb/>zio anco di 4. </s>
  <s xml:id="echoid-s1056" xml:space="preserve">braccia, è coſa facile, e notiſ. <lb/></s>
  <s xml:id="echoid-s1057" xml:space="preserve">ſima à tutti, in comparatione di quello; </s>
  <s xml:id="echoid-s1058" xml:space="preserve">ma <lb/>cauſar l’incẽdio lontano quãto è vn tiro d’ar-<lb/>co, Hocopus, hic labor. </s>
  <s xml:id="echoid-s1059" xml:space="preserve">Hò inteſo da perſone pra-
<pb o="118" file="0138" n="138" rhead="Delle Settioni"/>
tiche, quando ſi tira con l’arco à ſegno, cioè <lb/>di punto in bianco, poter’andare lo ſtrale per <lb/>inſino à ducento paſſi, m<unsure/>a di tiro eleuato, <lb/>com’à dire, circag. </s>
  <s xml:id="echoid-s1060" xml:space="preserve">45. </s>
  <s xml:id="echoid-s1061" xml:space="preserve">ch’è il maſſimo, poterſi <lb/>tirar lontano ſino à quattrocento paſſi; </s>
  <s xml:id="echoid-s1062" xml:space="preserve">che ſe <lb/>s’intendeſſe il ſudetto Autore d’vn’arco non <lb/>caricato à mano, ma con leue, martinelli, e ſi-<lb/>mili ordigni, crederei ſi poteſſe con queſti ar-<lb/>riuare alla diſtanza di mezo miglio, e più an-<lb/>cora. </s>
  <s xml:id="echoid-s1063" xml:space="preserve">E che parli d’vn tal’arco il detto Auto-<lb/>re, hà del probabiliſſimo, poiche racconta Po-<lb/>libio, che hauendo i Romani prouato la tem-<lb/>peſta delle pietre, ch’erano ſcagliate dalle <lb/>mura, ritrouandoſi aſſai lontani da quelle, ſti-<lb/>mò Marco Marcello, che ad Archimede foſſe <lb/>biſogno di tanta diſtanza, e che à tanto ſpatio <lb/>foſſero caricate le frombole, le baleſtri, e ſimi-<lb/>li ſtromenti da tirar pietre; </s>
  <s xml:id="echoid-s1064" xml:space="preserve">la onde coman-<lb/>dò, ſi accoſtaſſe alle mura, per veder d’in-<lb/>gannar’Archimede, edi rendere inutili quel-<lb/>le machine, dalle quali ſentiua nella propria <lb/>Armata tant’offe@a; </s>
  <s xml:id="echoid-s1065" xml:space="preserve">ma egli fù l’ingannato, <lb/>poiche ritrouò non minor’apparato da vicino, <lb/>che da lontano, come racconta il medeſimo <lb/>Polibio dicendo; </s>
  <s xml:id="echoid-s1066" xml:space="preserve">Ad extremum M. </s>
  <s xml:id="echoid-s1067" xml:space="preserve">Marcellus
<pb o="119" file="0139" n="139" rhead="Coniche. Cap. XXXI."/>
his difficultatibus circumuentus, clam ſilentio noctis <lb/>naues propius admouere eſt coactus, quæ poſtquam <lb/>intra tali iactum terræ appropin quaſſent, alium rur-<lb/>ſus apparatum aduerſus eos, qui è nauibus dimica-<lb/>bant, idem vir perstruxerat. </s>
  <s xml:id="echoid-s1068" xml:space="preserve">Murum crebris ca-<lb/>uis ad bumanæ ſtaiuræ modum: </s>
  <s xml:id="echoid-s1069" xml:space="preserve">ſed quæ extrinſecus <lb/>palmares eſſent aperuit. </s>
  <s xml:id="echoid-s1070" xml:space="preserve">Ibi Sagittarĳs, ac Scorpiũ-<lb/>culis ab interiore muri parte appoſitis, per istos pe-<lb/>tens hostem, inutiles nauium Romanorum epibatas <lb/>reddebat, ex quo eueniebat, vt inimicos &amp; </s>
  <s xml:id="echoid-s1071" xml:space="preserve">procul <lb/>poſitos, &amp; </s>
  <s xml:id="echoid-s1072" xml:space="preserve">in proximo ſtantes, non ſolum quicquam <lb/>eorum exequi vetaret, quæ propoſuerant, ſedetiam <lb/>plurimos illorum occiderer<unsure/>. </s>
  <s xml:id="echoid-s1073" xml:space="preserve">E così và ſeguitando <lb/>di raccontare le machine, e gli artificij, con <lb/>che egli diffendendoſi offendeua le nemiche <lb/>naui. </s>
  <s xml:id="echoid-s1074" xml:space="preserve">Hora egli hà del credibile, che veden-<lb/>do M. </s>
  <s xml:id="echoid-s1075" xml:space="preserve">Marcello non poterſi diffendere, ne con <lb/>lo ſtare tanto lontano, quanto era dianzi che <lb/>s’accoſtaſſe alle mura, ne con lo ſtar tanto vi-<lb/>cino, che intra teli iactum terræ appropinquaſſet; <lb/></s>
  <s xml:id="echoid-s1076" xml:space="preserve">egli faceſſe vna ritirata tale, che ne viarriuaſ-<lb/>ſero i Sagittarij, ne men le frõbole, ò baleſtre, <lb/>e che per vltimo rimedio adoperaſſe lo Spec-<lb/>chio, e le abbruciaſſe; </s>
  <s xml:id="echoid-s1077" xml:space="preserve">long<unsure/>itudine arcus iactus; </s>
  <s xml:id="echoid-s1078" xml:space="preserve"><lb/>per la lunghezza d’vn tiro d’arco, non carica-
<pb o="120" file="0140" n="140" rhead="Delle Settioni"/>
to con la ſemplice mano, ma ſi bene con leue, <lb/>ò altri ordigni; </s>
  <s xml:id="echoid-s1079" xml:space="preserve">poiche alla prima già ſi troua-<lb/>ua con le naui fuor del tiro delle ſaette, come <lb/>dice Polibio, e pazzia par che ſarebbe ſtato il <lb/>fermarſi di nuouo, doue poco dianzi hauea di <lb/>già prouato il continuo tempeſtare delle pie-<lb/>tre, ch’erano di groſſezza tale ſcagliate dalle <lb/>mura, che tal’vna era di 250. </s>
  <s xml:id="echoid-s1080" xml:space="preserve">libre: </s>
  <s xml:id="echoid-s1081" xml:space="preserve">Emi ri-<lb/>cordo hauer ſentito dire à vn Siracuſano, che <lb/>le naui per vltimo ſi ritiraſſero in vn luogo, <lb/>che ſi chiama Bocca di porto, che ſtà verſo <lb/>Settentrione in riſpetto di Siracuſa, doue mi <lb/>dicea eſſer fama, che Archimede abbruciaſſe <lb/>in gran parte le dette naui, nella diſtanza par <lb/>(ſe mal non mi ricordo) che affermaſſe di più <lb/>di mezo miglio, ſe ben poi per tradimento di <lb/>Merico Prefetto di Acradina, che gli aperſe <lb/>vna porta, fù poi preſa la Città. </s>
  <s xml:id="echoid-s1082" xml:space="preserve">Hora l’hauer-<lb/>le abbruciate verſo Settentrione, dalla qual <lb/>banda non potea eſſer drittamẽte il Sole, con-<lb/>ferma maggiormente non poter quello Spec-<lb/>chio eſſer ſtato d’altra ſorte, che di quella, che <lb/>di ſopra ſi è dichiarato. </s>
  <s xml:id="echoid-s1083" xml:space="preserve">Aggiungaſi ancora, <lb/>ch’eſſo diceua hauer viſto vn’antichiſsimo di-<lb/>ſegno di tale Specchio, e che ſi ricordaua, che
<pb o="121" file="0141" n="141" rhead="Coniche. Cap. XXXI."/>
era notabilmente cauo, il che pur conferma <lb/>ciò, che ſi è detto di ſopra.</s>
  <s xml:id="echoid-s1084" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1085" xml:space="preserve">Non hò voluto veramente tacere quel-<lb/>lo, che mi è venuto in mente, ò che hò potuto <lb/>intẽdere da altri, per rẽder credibile vn ſimile <lb/>artificio, e queſto hò fatto per inanimar’al-<lb/>tri ad affaticaruiſi intorno, ne penſo ſia poco <lb/>prima l’hauer’inteſo per dimoſtratione, che <lb/>operãdo conforme all’eſſemplare, debba ſe-<lb/>guirne l’effetto; </s>
  <s xml:id="echoid-s1086" xml:space="preserve">poi il vedere da tanti rincon-<lb/>tri, quanto queſto penſiero mirabilmente <lb/>concordi con quello, che ſi troua ſcritto de <lb/>gli Specchi d’Archimede, di Proclo, e del <lb/>Porta; </s>
  <s xml:id="echoid-s1087" xml:space="preserve">e finalmente l’hauer noi relatione di <lb/>Scrittori inſigni, e di prima claſſe, ciò eſſer <lb/>ſtato fatto, ci deue ben far credere, che ſia <lb/>coſa fattibile; </s>
  <s xml:id="echoid-s1088" xml:space="preserve">ma l’eſſer ſtato fatto da pochi, <lb/>che ſia veramente coſa molto difficile, ma <lb/>non impoſſibile da farſi. </s>
  <s xml:id="echoid-s1089" xml:space="preserve">Non hò, dico, volu-<lb/>to naſcondere ſotto il ſilentio, ò mettere in <lb/>cifra il mio ſenſo in coſa, che può apportare <lb/>tanto guſto, marauiglia, &amp; </s>
  <s xml:id="echoid-s1090" xml:space="preserve">vtilità à ſtudioſi <lb/>de’ſegreti di Natura, eſſendo l’huomo, come <lb/>diceua Platone, nato non per ſe ſteſſo ſolo, ma <lb/>per giouare à gli altri ancora, hauendo ſpie-
<pb o="122" file="0142" n="142" rhead="Delle Settioni"/>
gato nel miglior modo, che hò ſaputo, ciò, che <lb/>la ſpecolatiua mi hà ſomminiſtrato in coſa tã-<lb/>to recondita, e tanto curioſa, acciò quelli, che <lb/>hanno prattica nel lauorare i Specchi, com-<lb/>modità di ſpendere, e di tempo più, che nõ hò <lb/>io, eſſendo occupato in altra ſorte di ſtudij, <lb/>che non mi permettono il poterli applicare, <lb/>quãto ſaria di biſogno, adopr<unsure/>ãdo ui l’ingegno, <lb/>e la mano, cauino dal ſepolcro dell’oblio vn’ <lb/>inuention sì rara, che per tanti anni è ſtata na-<lb/>ſcoſta anco à’più ſottili inueſtigatori delle o-<lb/>pere marauiglioſe di Natura: </s>
  <s xml:id="echoid-s1091" xml:space="preserve">Come pur’an-<lb/>cora reſtiamo incapaci, cred’io, ſin’hora della <lb/>colũba di Archita, che volaua, delle lucerne, <lb/>che ardeuano perpetuamente ne’ſepolcri del <lb/>capo fatto da Alberto Magno, che parlaua, <lb/>e di ſimili altri ſecreti; </s>
  <s xml:id="echoid-s1092" xml:space="preserve">riconoſcendo noi nel <lb/>noſtro Problema molto vantaggio ſopra di <lb/>quelli, poiche non ſolo ſappiamo, che è ſtato <lb/>fatto, ma intendiamo anco ſpecolatiuamen-<lb/>te il modo, con che ſi dourebbe fare, che tan-<lb/>to non ne ſappiamo forſi di quelli. </s>
  <s xml:id="echoid-s1093" xml:space="preserve">Laſcierò <lb/>ben poi ad altri penſare la maniera ſi di fabri-<lb/>car detti Specchi, come anco di ſituarli con-<lb/>forme all’eſſemplare, tralaſciando molte coſe,
<pb o="123" file="0143" n="143" rhead="Coniche. Cap. XXXI."/>
che potrei dire, per quanto s’aſpetta à facili-<lb/>tar l’operatione, acciò gli altri habbino d’af-<lb/>faticarſi loro ancora, e per non eſſer giuſta-<lb/>mente ripreſo d’hauer publicato affatto vn <lb/>tal ſecreto, del quale non ſi sà ſe altri n’habbi <lb/>mai voluto ſcriuere, ſe nõ il Porta, con parole <lb/>oſcure, &amp; </s>
  <s xml:id="echoid-s1094" xml:space="preserve">enimmatiche, &amp; </s>
  <s xml:id="echoid-s1095" xml:space="preserve">il Keplero pur’ in <lb/>poche parole attribuendolo alle lenti, come <lb/>ſi è detto di ſopra; </s>
  <s xml:id="echoid-s1096" xml:space="preserve">e ſe ben ſi crede da alcuni, <lb/>che Archimede ſcriueſſe ancora lui vn’ Ope-<lb/>ra de’ Specchi, commentata da vn tal Goga-<lb/>ua, ciò però è tenuto per coſa molto dubbio-<lb/>ſa; </s>
  <s xml:id="echoid-s1097" xml:space="preserve">La difficoltà dunque, che vi reſta per met-<lb/>terlo in prattica, ſpero che mi liberarà da que <lb/>ſta cenſura; </s>
  <s xml:id="echoid-s1098" xml:space="preserve">oltre che l’hauer’io ſmarrito alcu-<lb/>ne ſcritture, nelle quali per mia memoria ha-<lb/>ueuo deſcritto le ſudette coſe, per non eſ-<lb/>ſer da altri preoccupato, mi hà fatto fare in <lb/>parte queſta riſolutione. </s>
  <s xml:id="echoid-s1099" xml:space="preserve">Vi agginngerò an-<lb/>cora l’eſſortationi dell’Illuſtriſs. </s>
  <s xml:id="echoid-s1100" xml:space="preserve">Sig. </s>
  <s xml:id="echoid-s1101" xml:space="preserve">Ceſare <lb/>Marſili, non meno ornato di quelle parti no-<lb/>biliſſime, che à Caualiero ſi conuengono, che <lb/>verſatiſſimo nelle diſcipline Matematiche, al <lb/>quale parendo, che tal penſiero non ha-<lb/>ueſſe così dell’ ordinario, ſtimò eſſer coſa
<pb o="124" file="0144" n="144" rhead="Delle Settioni"/>
molto conueneuole, ch’io preſentaſſi qu@ſta <lb/>nuoua inuentione à queſto Illuſtriſs. </s>
  <s xml:id="echoid-s1102" xml:space="preserve">Senato <lb/>di Bologna, publicandola ſi per la ſopradetta <lb/>ragione, ſi anco, perche ſi porgeſſe materia à’ <lb/>ſtudioli d’eſſercitar l’ingegno, e la mano, per <lb/>arriuare alla perfettione di così curioſo tro-<lb/>uato; </s>
  <s xml:id="echoid-s1103" xml:space="preserve">riferiſcano quelli dunque principal-<lb/>mente al generoſo ſpirito di queſto Signore <lb/>l’hauer’io paleſato ciò, che penſai di tener na-<lb/>ſcoſto (hauendo pur’anco, ſecondo il ſuo pen-<lb/>ſiero, poſto il libretto delle figure in vltimo, <lb/>cõforme, ch’anch’egli ſtampãdo, mi diſſe di vo <lb/>ler fare) e gradiſchino l’affetto mio, e da tanti <lb/>cõtraſegni di verità inanimati, faccino della <lb/>benignità della Natura iſperienza, che ap-<lb/>plaudẽdo à’noſtri sforzi, accompagnarà forſi <lb/>queſto con gli altri fauori, de’quali, come mi-<lb/>niſtra della Diuina Prouidenza, ſi è com-<lb/>piacciuta arricchir ſingolarmente <lb/>fra gli altri queſto noſtro <lb/>ſecolo.</s>
  <s xml:id="echoid-s1104" xml:space="preserve"/>
</p>
  <figure>
    <image file="0144-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0144-01"/>
  </figure>
<pb o="125" file="0145" n="145" rhead="Coniche. Cap. XXXI."/>
</div>
<div xml:id="echoid-div70" type="section" level="1" n="70">
<head xml:id="echoid-head75" style="it" xml:space="preserve">Come con lt ſudetti Specchi potiamo di notte manda-<lb/>re i<unsure/>l lume lontano. Cap. XXXII.</head>
<p>
  <s xml:id="echoid-s1105" xml:space="preserve">EManifeſto, che ſe noi collocare-<lb/>mo nel foco della Parabola la <lb/>fiamma d’vna candela, che tut-<lb/>te le linee radioſe, ò luminoſe <lb/>ſi rifletteranno dalla Paraboli-<lb/>ca ſuperficie parallele, che pe-<lb/>rò il lume an darà aſſai lontano, e tutto lo ſpa-<lb/>tio rinchiuſo dẽtro eſſe parallele verrà ad eſſe-<lb/>re illuminato, l’iſteſſo farà lo Specchio Sferico; <lb/></s>
  <s xml:id="echoid-s1106" xml:space="preserve">l’Iperbolico poi le ribatterà ſem@re diuergen-<lb/>ti, mentre ſia la fiamma nel ſuo foco interio-<lb/>re, e ciò farà con la concaua, e con la conueſ-<lb/>ſa, mentre quella ſia nel foco eſteriore; </s>
  <s xml:id="echoid-s1107" xml:space="preserve">l’E-<lb/>littico poi le ribatterà ad vn punto, cioè all’ <lb/>altro foco, doue il lume ſarà viuaciſſimo, ma <lb/>in poco ſpatio; </s>
  <s xml:id="echoid-s1108" xml:space="preserve">con queſti adunque potremo <lb/>illuminare vna ſtanza, ò gran ſala con pochi <lb/>lumi, ſoſpendendo intorno à quella alc uni di <lb/>queſti ſopranominati Specchi, non vi comprẽ-<lb/>dendo però l’Elittico, poiche vniſce ſolo in <lb/>vn punto.</s>
  <s xml:id="echoid-s1109" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1110" xml:space="preserve">Sò beniſſimo queſte coſe eſſer ſtate da altri
<pb o="126" file="0146" n="146" rhead="Delle Settioni"/>
ancora accẽnate, ma io le hò quì di nuouo po-<lb/>ſte, ſi per applicarle alle altre Settioni Coni-<lb/>che, ſi anco per farle naſcere con diletto del-<lb/>la ſua ragion fondamẽtale. </s>
  <s xml:id="echoid-s1111" xml:space="preserve">Laſcio di trattar’ <lb/>intorno alle imagini, come le poſſino moſtrar’ <lb/>inuerſe, grandi, piccole, torbide, chiare, <lb/>e pendule nell’aria, hauẽdone trattato il Ma-<lb/>gini nel ſuo Libretto dello Specchio Sferico; <lb/></s>
  <s xml:id="echoid-s1112" xml:space="preserve">e l’applicar’i medeſimi ſintomi à’ſudetti Spec-<lb/>chi ricercaria maggior lunghezza di quella, <lb/>ch’io pretendo in queſto mio Trattato.</s>
  <s xml:id="echoid-s1113" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1114" xml:space="preserve">Potrei anco dire, come l’effetto del Can-<lb/>nocchiale ſi haurebbe forſi anco dalla combi-<lb/>natione di queſti Specchi, ò de’Specchi con le <lb/>lenti, ſe ben la facilità del produrre la figura <lb/>Sferica farà, che ci preuagliamo più toſto di <lb/>queſta, che dell’altre; </s>
  <s xml:id="echoid-s1115" xml:space="preserve">Concioſiacoſa adunque, <lb/>che lo Specchio cõcauo facci l’operatione del-<lb/>la lente conueſſa, e lo Specchio conueſſo della <lb/>lente caua, è manifeſto, che ſe combinare-<lb/>mo lo Specchio concauo con il conueſſo, o-<lb/>uero con la lente caua, douremo hauer l’ef-<lb/>fetto del Cãnocchiale, e tale forſi fù lo Spec-<lb/>chio di Tolomeo, la onde con tale occaſione <lb/>non mancherò di dire, come hauendo più
<pb o="127" file="0147" n="147" rhead="Coniche. Cap. XXXII."/>
volte ſentito cercar da alcuni il modo di fa-<lb/>re vn paro d’occhiali, che faceſſero l’effetto <lb/>del Cannocchiale, io penſai, che ciò in tal <lb/>modo ſi poteſſe fare, cioè, che ſi collocaſſe vn <lb/>traguardo da vna bãda, e dall’altra vno Spec-<lb/>chietto cauo, poiche mettendoci noi queſto <lb/>paro d’occhiali, con il contra porui vno Spec-<lb/>chio piano auuicinato, ò allontanato, quanto <lb/>cõporta il veder diſtin tamente l’oggetto den-<lb/>tro lo Specchietto cauo (ſcorgendoſi però l’v-<lb/>no, e l’altro nello Specchio piano, antepoſto <lb/>alla noſtra faccia) ſi ottenerà l’effetto del Cã-<lb/>nocchiale, egli è però vero, che douẽdo ſta-<lb/>re queſti allo ſcoperto, faranno il medeſimo, <lb/>che il vetro cauo, ò conueſſo, adoperati fuor <lb/>della canna, anzi per farſi vna rifleſſione <lb/>di più, cioè dallo Specchio piano, verremo <lb/>anco perciò à ſcapitar più nell’operatione; <lb/></s>
  <s xml:id="echoid-s1116" xml:space="preserve">ciò però con queſta occaſione hò voluto ac-<lb/>cẽnare, come per vna bizzarria, per dar qual-<lb/>che ſodisfattione à’curioſi, che voglion cer-<lb/>car miglior pane, che di farina, poiche all’ec-<lb/>cellenza del Cannocchiale, non arriuaranno <lb/>mai, per mio credere, ne i Specchi combina-<lb/>ti inſieme, ne accompagnati con le lenti, co-
<pb o="128" file="0148" n="148" rhead="Delle Settioni"/>
me, chine vorrà far proua, credo ſi potrà aſ-<lb/>ſicurare. </s>
  <s xml:id="echoid-s1117" xml:space="preserve">Hora dunque baſterà quello, che <lb/>ſi è detto di ſopra intorno al lume, e calore, <lb/>potendo noi nell’iſteſſo tempo intendere le <lb/>medeſime coſe anco per il freddo, che dilatã-<lb/>doſi dal corpo freddo ad ogni poſitione per li-<lb/>nea retta, e perciò nell’infinite linee, che ſi <lb/>partono dal corpo freddo, come dalla neue, <lb/>eſſendoui dentro le parallele, che ſono vnite <lb/>dallo Specchio Parabolico, e le diuergenti, <lb/>che ſono vnite dall’Elittico, e le conuergen-<lb/>ti vnite dall’Iperbolico, perciò con opporre <lb/>alcun di queſti Specchi ad vna maſſa di neue, <lb/>ò di ghiaccio, ſentiremo nel loro foco eſſere il <lb/>freddo fatto molto gagliardo, ma per quefto <lb/>effetto ſarà più atto l’Iperbolico di tutti, come <lb/>quello, che raccoglierà maggior quantità di <lb/>linee fredde; </s>
  <s xml:id="echoid-s1118" xml:space="preserve">e queſto baſti ancora circa il <lb/>freddo, potendoſi forſi in vn certo modo cre-<lb/>der, che tale effetto accadeſſe anco in-<lb/>torno à gli odori, prouando noi di-<lb/>latarſi pur quelli dalli corpi <lb/>odoriferi verſo ogni <lb/>banda.</s>
  <s xml:id="echoid-s1119" xml:space="preserve"/>
</p>
<pb o="129" file="0149" n="149" rhead="Coniche. Cap. XXXIII."/>
</div>
<div xml:id="echoid-div71" type="section" level="1" n="71">
<head xml:id="echoid-head76" style="it" xml:space="preserve">Come potiamo ſentir quel ſuono, che per altro uon <lb/>s vdirebbe, ò ſentir meglio quello, che de-<lb/>bolmente ſi ſente. Cap. XXXIII.</head>
<p>
  <s xml:id="echoid-s1120" xml:space="preserve">IN due maniere noi potiamo ot-<lb/>tener queſto, ma prima fà di <lb/>meſtieri conſiderare ſe il ſuono, <lb/>che ſi hà da ſentire, è vn ſolo, e <lb/>vicino, ouero ſe è vn ſolo, e lon-<lb/>tano aſſai, ouero ſe ſono più ſuoni inſieme, <lb/>come ſariano i ragionamenti fatti in vna <lb/>piazza da varie adunanze d’huomini, che diſ-<lb/>ſcorreſſero, ouer’il mormorio d’vn fiume, ò ſi-<lb/>mili altri ſuoni. </s>
  <s xml:id="echoid-s1121" xml:space="preserve">Se adun<unsure/> que il ſuono ſarà vn <lb/>ſolo, e vicino, non è dubbio alcuno, che ſarà <lb/>d’ogn’altro più atto lo Specchio Elittico, met-<lb/>tendo l’orecchio in vn de’fochi, e nell’altro <lb/>ſtandoui il corpo ſonante; </s>
  <s xml:id="echoid-s1122" xml:space="preserve">ma quando ſara vn <lb/>ſolo, e molto lontano, allhora ſarà atto à que-<lb/>ſto ſeruitio anco il Parabolico, mettendo l’o-<lb/>recchio nel di lui foco; </s>
  <s xml:id="echoid-s1123" xml:space="preserve">e finalmente, quando <lb/>foſſer più ſuoni inſieme, allhora più atto di tut <lb/>ti ſarà l’Iperbolico, come quello, che racco-<lb/>glierà più linee ſonore; </s>
  <s xml:id="echoid-s1124" xml:space="preserve">potiamo poi in due <lb/>modi ottener queſto, cioè o preualendoſi
<pb o="130" file="0150" n="150" rhead="Delle Settioni"/>
dello Specchio Elittico, o Parabolico, o Iper-<lb/>bolico, o del Cannone, ma ſtimo più atto di <lb/>tutti il Gannone, perche l’orecchio ſi può ac-<lb/>commodare dietro di quello ſenza impedire <lb/>le linee ſonore, doue nello Specchio douen-<lb/>do l’orecchio ſtarli dinanzi, può apportar-<lb/>ui qualche impedimento; </s>
  <s xml:id="echoid-s1125" xml:space="preserve">auuertendo però, <lb/>che queſti iſtromenti vogliono eſſer grãdi per <lb/>le linee ſonore più, che per le lucide, poiche <lb/>il ſuono non ſoggiace così à queſte leggi, co-<lb/>me il lume, propagandoſi quello anco per li-<lb/>nea fleſſuoſa, cagionandoſi egli dalla pulſio-<lb/>ne nell’organo dell’vdito, fatta dall’aria tre-<lb/>mante di più, e men veloci tremori, che fan-<lb/>no l’alto, e’l baſſo, il graue, e l’acuto nel ſuo-<lb/>no, il qual tremore comincia dal corpo ſonan-<lb/>te, e ad ogni poſitione ſi và continuamen-<lb/>te diffondendo per dritta linea, <lb/>quando non troui oſtacolo, <lb/>ma per dritta linea, e <lb/>per fleſſuoſa, <lb/>quando ritroui impedimento.</s>
  <s xml:id="echoid-s1126" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div72" type="section" level="1" n="72">
<head xml:id="echoid-head77" xml:space="preserve">∴</head>
<pb o="131" file="0151" n="151" rhead="Coniche. Cap. XXXIV."/>
</div>
<div xml:id="echoid-div73" type="section" level="1" n="73">
<head xml:id="echoid-head78" style="it" xml:space="preserve">Come per il contrario potiamo inuigorire il ſuono, <lb/>ſi che ſia ſentito più gagliardo, che non ſi <lb/>ſentirebbe. Cap. XXXIV.</head>
<p>
  <s xml:id="echoid-s1127" xml:space="preserve">QVeſto pur ſi potrà fare con i <lb/>medeſimi Specchi, &amp; </s>
  <s xml:id="echoid-s1128" xml:space="preserve">in par-<lb/>ticolare con il Cannone Elit-<lb/>tico, fatto con la debita pro-<lb/>portione, e miſura, ſi che ſia-<lb/>no ſuoi fochi il punto della voce, &amp; </s>
  <s xml:id="echoid-s1129" xml:space="preserve">il punto <lb/>dell’vdito; </s>
  <s xml:id="echoid-s1130" xml:space="preserve">poiche formando la voce in vn di <lb/>quei fochi, ſi ſentirà (per vnirſi molte linee <lb/>ſonore in vn ſol pũto) gagliarda più, che ſen-<lb/>za il detto Cannone ſi ſentirebbe; </s>
  <s xml:id="echoid-s1131" xml:space="preserve">Il Cannon <lb/>Parabolico poi le manderà parallele, e l’Iper-<lb/>bolico diuergenti; </s>
  <s xml:id="echoid-s1132" xml:space="preserve">intendẽdo però per Spec-<lb/>chi nel ſuono quelli, c’hauranno la ſuperficie <lb/>in qualche modo liſcia, ſeben non rappreſen-<lb/>taſſero le imagini; </s>
  <s xml:id="echoid-s1133" xml:space="preserve">e chi faceſſe vna tromba <lb/>Elittica, Parabolica, ò Iperbolica, che <lb/>haueſſe il foco, doue ſe li dà la vo-<lb/>ce, forſi faria meglio delle <lb/>vſitate.</s>
  <s xml:id="echoid-s1134" xml:space="preserve"/>
</p>
<pb o="132" file="0152" n="152" rhead="Delle Settioni"/>
</div>
<div xml:id="echoid-div74" type="section" level="1" n="74">
<head xml:id="echoid-head79" style="it" xml:space="preserve">Come ſi poſſa fabricare vna stanza talmente, che <lb/>chi ſtarà in vn’angolo di quella, ſenta il ſuo-<lb/>no fatto nell’altro angolo diametral-<lb/>mente oppoſto, non ſentendo quel-<lb/>li, che ſaranno nel mezo. <lb/>Cap. XXXV.</head>
<p>
  <s xml:id="echoid-s1135" xml:space="preserve">NOn ſolo lo Specchio, e Canno-<lb/>ne Elittico, Parabolico, ò Iper-<lb/>bolico faranno i ſudetti effetti, <lb/>ma ancora qualſiuoglia pezzo <lb/>della ſuperficie di quelli, e pe-<lb/>rò ſe noi fabricaremo vna ſtanza contal’arte, <lb/>che il volto ſia vn pezzo, ò fruſto di ſuperficie <lb/>Elittica in tal modo diſegnata, chei due fochi <lb/>di quella venghino ad eſſer ne gli angoli op-<lb/>poſti di detta ſtanza, prouaremo, che ſtando <lb/>in vn di quegli angoli con l’orecchio in vn de <lb/>i detti fochi, ſentiremo ciò, che dirà vn’altro <lb/>nell’altr’angolo dall’altro foco baſſamente, ſi <lb/>che non ſia inteſo da quelli, che ſaranno in <lb/>mezo; </s>
  <s xml:id="echoid-s1136" xml:space="preserve">sò, che ſcorrendo la voce ſopra d’vna <lb/>terſa ſuperficie, ſenza interrompimento alcu-<lb/>no, ſuol farſi ſentire più gagliarda dell’ordi-<lb/>nario, come ſi ſente longo vn fiume, che ſia
<pb o="133" file="0153" n="153" rhead="Coniche. Cap. XXXV."/>
placido, ouero vn muro, che ſia bẽ pulito, ò da <lb/>angolo à angolo, come nella ſala del Sereniſs. <lb/></s>
  <s xml:id="echoid-s1137" xml:space="preserve">Duca di Mantoua, tuttauia sò anco, cheſe à <lb/>queſtos’aggiõgerà, che ſia tal ſuperficie Elitti <lb/>ca, fatta nel modo di ſopra, che farà quel mi-<lb/>glior’effetto, che ſia poſſibil fare; </s>
  <s xml:id="echoid-s1138" xml:space="preserve">ſarà poi be-<lb/>ne, che il reſto della ſtanza ſia ben pulito, eli-<lb/>ſcio, e ſia di ſuperficie Elittica per vn verſo, e <lb/>dritta per l’altro (acciò, che’l muro ſtia à piõ-<lb/>bo, conforme all’ordinario)e che non vi ſiano <lb/>cornici, ouero cordoni, che così ſi darà quel <lb/>maggior’aiuto alla voce, ò ſuono, che ſia poſ-<lb/>ſibil darui all’aperta; </s>
  <s xml:id="echoid-s1139" xml:space="preserve">dico all’aperta, poiche <lb/>per canali rinchiuſi sò molto bene poterſi par-<lb/>lar di lontano, ma in quelli non vi è artificio, <lb/>per conto di rifleſſione, ma ſemplicemente <lb/>mantengono la voce gagliarda, per la ſuper-<lb/>ficie terſa del canale, e per il tremito dell’aria, <lb/>che ſenza patire turbamẽto per ſtrada, incor-<lb/>rotto peruiene all’orecchio, e di quì ſi può rac <lb/>cogliere, che all’a perta eſſendo vna cauità di <lb/>muro, ò di mõti di ſuperficie Elittica, faremo <lb/>ſentire vn’Echo perfettiſſima, ſe ſtando nell’ <lb/>vn de’fochi di quella, l’vditore ſarà nell’altro <lb/>foco, poiche ſentirà la voce primaria, e poi
<pb o="134" file="0154" n="154" rhead="Delle Settioni"/>
la rifleſſa ingagliardita. </s>
  <s xml:id="echoid-s1140" xml:space="preserve">Di quì può naſcer, che <lb/>la ſeluaggia Ninfa Echo ſia da’Poeti ſtata fa-<lb/>u<unsure/>oleggiata per habitatrice de’caui ſpechi, <lb/>forſi perche da queſti più perfettamente (co-<lb/>me da ſuperficie, che all’Elittica ſi vanno ac-<lb/>coſtando) che da ſuperficie piane ci riſponda, <lb/>benche ancor da queſte ſi formi l’Echo, co-<lb/>me nella ſua Echomatria hà dimoſtrato il P. <lb/></s>
  <s xml:id="echoid-s1141" xml:space="preserve">Biancano Geſuita.</s>
  <s xml:id="echoid-s1142" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div75" type="section" level="1" n="75">
<head xml:id="echoid-head80" style="it" xml:space="preserve">De i Vaſi Teatrali di Vitruuio. <lb/>Cap. XXXVI.</head>
<p>
  <s xml:id="echoid-s1143" xml:space="preserve">POtiamo ancora dalle coſe dette <lb/>di ſopra, comprendere in parte <lb/>la ragione del formare i Teatri <lb/>circolari, cioè, perche gli vdi-<lb/>tori nõ ſolo ſentano la voce pri-<lb/>maria, che dalla Scena, come da centro per <lb/>l’ampiezza del Teatro ſi diffonde, ma anco la <lb/>ſecondaria, cioè la rifleſſa dalla rotondità del <lb/>medeſimo Teatro: </s>
  <s xml:id="echoid-s1144" xml:space="preserve">Anzi per render’eſſa voce <lb/>ſonora, &amp; </s>
  <s xml:id="echoid-s1145" xml:space="preserve">armonica all’vdito, ſoleuano gli an-<lb/>tichi collocar certi vaſi dentro le ſedie, ſopra <lb/>certe celle incauate nel muro, crederei io,
<pb o="135" file="0155" n="155" rhead="Coniche. Cap. XXXV."/>
à guiſa de i Nicchi, ne’quali ſi ſoglion mette-<lb/>re le Statue, in tal maniera però, che ſteſſero <lb/>ſoſpeſi con certi cunei, ſenza toccare il muro, <lb/>con la bocca riuolta in giù, ſino al numero di <lb/>tredici, nel medeſimo corſo, e ne’Teatri pic-<lb/>coli, ma di trẽtaſette in tre corſi, e ciò ne’Tea-<lb/>tri grandi, accommodati ſecondo il genere <lb/>armonico chromatico, e diatonico, formati <lb/>con tal proportione fra di loro, che toccati, <lb/>riſonaſſero il diateſſeron, e’l diapente per or-<lb/>dine al diſdiapaſon, cioè la quarta, e la quin-<lb/>ta per ordine alla quintadecima, dẽtro la qua-<lb/>le rinchiudeuano gli Antichi tutte le conſo-<lb/>nanze, benche Vitruuio arriui ſino alle diciot-<lb/>to; </s>
  <s xml:id="echoid-s1146" xml:space="preserve">le quali coſe egli ci manifeſta nel libro 5. <lb/></s>
  <s xml:id="echoid-s1147" xml:space="preserve">al Cap. </s>
  <s xml:id="echoid-s1148" xml:space="preserve">5. </s>
  <s xml:id="echoid-s1149" xml:space="preserve">della ſua Architettura, mentre di-<lb/>ce; </s>
  <s xml:id="echoid-s1150" xml:space="preserve">Itaex his indagationibus mathematicis ratio-<lb/>nibus fiunt vaſa ærea pro ratione magnitudinis <lb/>Theatri, eaq; </s>
  <s xml:id="echoid-s1151" xml:space="preserve">it a fabricentur, vt cum tanguntur, ſo-<lb/>nitum facere poſſint inter ſe diateſſeron, diapente <lb/>ex ordine ad diſdiapaſon. </s>
  <s xml:id="echoid-s1152" xml:space="preserve">Postea inter ſedes Thea-<lb/>tri conſtitutis cellis, ratione muſica ibi collocentur, <lb/>it a vti nullum parietem tangãt, circaq; </s>
  <s xml:id="echoid-s1153" xml:space="preserve">habeant lo-<lb/>cum vacuum, &amp; </s>
  <s xml:id="echoid-s1154" xml:space="preserve">à ſummo capite ſpatium, ponan-<lb/>turq; </s>
  <s xml:id="echoid-s1155" xml:space="preserve">inuerſa, &amp; </s>
  <s xml:id="echoid-s1156" xml:space="preserve">habeant in parte, quæ ſpectat ad
<pb o="136" file="0156" n="156" rhead="Delle Settioni"/>
Scenam, ſuppoſitos cuneos, ne minus altos ſemipe-<lb/>de, contraq; </s>
  <s xml:id="echoid-s1157" xml:space="preserve">eas cellas relinquantur aperturæ infe-<lb/>riorum graduum cubilibus longæ pedes duos, altæ <lb/>ſemipedem. </s>
  <s xml:id="echoid-s1158" xml:space="preserve">E così da queſti vaſi riflettendoſi <lb/>la voce con molta ſonoritſa4;</s>
  <s xml:id="echoid-s1159" xml:space="preserve">, &amp; </s>
  <s xml:id="echoid-s1160" xml:space="preserve">armonia, arri-<lb/>uaua alle orecchie de gli vditori, com’egli di-<lb/>ce doppo hauer ſpiegato le conſonanze, che <lb/>deuono formar tra di loro detti vaſi, ſoggiõ-<lb/>gendo: </s>
  <s xml:id="echoid-s1161" xml:space="preserve">It a hac ratiocinatione vox ab Scena, vti <lb/>à centro profuſa ſe circumagẽs, tactuq; </s>
  <s xml:id="echoid-s1162" xml:space="preserve">feriens ſingu-<lb/>lorum vaſorum caua @x@it auerit auctam clarita-<lb/>tem &amp; </s>
  <s xml:id="echoid-s1163" xml:space="preserve">concentu conuenientem ſibi conſonantiam. <lb/></s>
  <s xml:id="echoid-s1164" xml:space="preserve">Queſti vaſi, benche foſſero tredici, non ren-<lb/>deuano però tutti ſuoni differenti, ma eſſen-<lb/>do diſpoſti in ſemicircolo, erano vniſoni quel-<lb/>li, che diſtauano vgualmente da quel di me-<lb/>zo: </s>
  <s xml:id="echoid-s1165" xml:space="preserve">Ma per maggior chiarezza ſupponiamo <lb/>per detti tredici vaſi le tredici lettere maiuſ-<lb/>cole, poſte indritto, ſe ben’i Vaſi vanno diſpo-<lb/>ſti in giro, conforme alla rotondità del Tea <lb/>tro; </s>
  <s xml:id="echoid-s1166" xml:space="preserve">intenderemo dunque, che i Vaſi AA. </s>
  <s xml:id="echoid-s1167" xml:space="preserve">fra <lb/>di loro, e così BB. </s>
  <s xml:id="echoid-s1168" xml:space="preserve">CC. </s>
  <s xml:id="echoid-s1169" xml:space="preserve">DD. </s>
  <s xml:id="echoid-s1170" xml:space="preserve">EE. </s>
  <s xml:id="echoid-s1171" xml:space="preserve">FF. </s>
  <s xml:id="echoid-s1172" xml:space="preserve">ſiano <lb/>A B C D E F G F E D C B A <lb/>vniſoni, e perciò d’egual grandezzapur fra di <lb/>loro, hora tra queſti i vaſi eſtremi AA. </s>
  <s xml:id="echoid-s1173" xml:space="preserve">dourã-
<pb o="137" file="0157" n="157" rhead="Coniche. Cap. XXXV."/>
no, dice, riſonare il Nete hyperboleon, per <lb/>eſſempio, nel genere armonico, cioè eſſere <lb/>acutiſſimi; </s>
  <s xml:id="echoid-s1174" xml:space="preserve">Poi i ſecondi BB. </s>
  <s xml:id="echoid-s1175" xml:space="preserve">douranno far’il <lb/>Nete diezeugmenon, cioè eſſer più baſsi del-<lb/>li AA. </s>
  <s xml:id="echoid-s1176" xml:space="preserve">per vn Diateſſeron, cioè per vna quar-<lb/>ta, e perciò i corpi di queſti BB. </s>
  <s xml:id="echoid-s1177" xml:space="preserve">douran’eſſer <lb/>ſeſquiterzi delli AA. </s>
  <s xml:id="echoid-s1178" xml:space="preserve">poiche il Diateſſeron <lb/>conſiſte nella proportion ſeſquiterza. </s>
  <s xml:id="echoid-s1179" xml:space="preserve">Parimẽ <lb/>te gli CC. </s>
  <s xml:id="echoid-s1180" xml:space="preserve">ſarãno più baſsi de ſecondi BB. </s>
  <s xml:id="echoid-s1181" xml:space="preserve">vna <lb/>quarta, &amp; </s>
  <s xml:id="echoid-s1182" xml:space="preserve">à quelli ſimilmente in ſeſquiter-<lb/>za proportione, per formar pur queſti CC. <lb/></s>
  <s xml:id="echoid-s1183" xml:space="preserve">con li BB. </s>
  <s xml:id="echoid-s1184" xml:space="preserve">il Diateſſeron. </s>
  <s xml:id="echoid-s1185" xml:space="preserve">I quarti DD. </s>
  <s xml:id="echoid-s1186" xml:space="preserve">ſa-<lb/>ranno poi più baſſi delli CC. </s>
  <s xml:id="echoid-s1187" xml:space="preserve">per vn tuono, <lb/>e perciò à i CC. </s>
  <s xml:id="echoid-s1188" xml:space="preserve">douranno hauer proportio-<lb/>ne ſeſquiottaua. </s>
  <s xml:id="echoid-s1189" xml:space="preserve">I quinti EE. </s>
  <s xml:id="echoid-s1190" xml:space="preserve">ſaranno più <lb/>baſſi delli DD. </s>
  <s xml:id="echoid-s1191" xml:space="preserve">vna quarta; </s>
  <s xml:id="echoid-s1192" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1193" xml:space="preserve">i ſeſti FF. </s>
  <s xml:id="echoid-s1194" xml:space="preserve">pur <lb/>vna quarta più baſſi delli EE. </s>
  <s xml:id="echoid-s1195" xml:space="preserve">E ſinalmente il <lb/>medio G. </s>
  <s xml:id="echoid-s1196" xml:space="preserve">fra tutti baſſiſſimo lõtano dalli FF. </s>
  <s xml:id="echoid-s1197" xml:space="preserve"><lb/>medeſimamente per vna quarta. </s>
  <s xml:id="echoid-s1198" xml:space="preserve">Queſte ſo-<lb/>no lec<unsure/>onſonanze, che per il detto di Vitruuio <lb/>par che debbino far que<unsure/>ſti vaſi, quando ſian <lb/>tocchi dalla voce, o da altra coſa, che gli per-<lb/>cuota; </s>
  <s xml:id="echoid-s1199" xml:space="preserve">ſe ben pare, che ſi poteſſero in altro <lb/>modo ancora talmente ordinare, che fareb-<lb/>bono forſi anco miglior conſonanza, queſto
<pb o="138" file="0158" n="158" rhead="Delle Settioni"/>
però laſcierò conſiderare alli Muſici prattici, <lb/>che facilmente comprenderanno, qual ſia la <lb/>migli r concordanza, che poſſino hauer fra <lb/>di<unsure/> loro queſti vaſi.</s>
  <s xml:id="echoid-s1200" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1201" xml:space="preserve">Ma benche s’intendano queſte conſonan-<lb/>ze, e che Vitruuio c’inſegni, che queſti vaſi <lb/>deuono ſtar, doue ſono le celle, com’egli dice <lb/>nelluogo ſopracitato, con l’altre circonſtan-<lb/>ze, nondimeno pare, che la noſtra curioſità <lb/>non troui piena ſodisfattione nella dichiara-<lb/>tione, ch’egli fà intorno à queſti vaſi, poiche <lb/>ne c’inſegna, che forma debbano hauere, ne <lb/>men le celle, ne con qual proportione, riſpet-<lb/>to alle grandezze de’Teatri, ſi habbino da fa-<lb/>bricare, e con qual regola ſituarſi, dicendo <lb/>ſolo; </s>
  <s xml:id="echoid-s1202" xml:space="preserve">F@unt vaſa ærea pro ratione magnitudinis <lb/>Theatri; </s>
  <s xml:id="echoid-s1203" xml:space="preserve">moſtrando, che deuono eſſer caui, <lb/>per quelle parole; </s>
  <s xml:id="echoid-s1204" xml:space="preserve">Tactuq́; </s>
  <s xml:id="echoid-s1205" xml:space="preserve">feriens ſingulorum <lb/>vaſorum caua; </s>
  <s xml:id="echoid-s1206" xml:space="preserve">non paſſando più oltre ne lui, <lb/>ne i ſuoi commentatori, almeno quelli, che <lb/>hò potuto vedere; </s>
  <s xml:id="echoid-s1207" xml:space="preserve">Non mancherò perciò di <lb/>dire anco intorno à queſto il mio pẽſiero, non <lb/>perche io ſtimi d’indouinare il modo de gli <lb/>antichi; </s>
  <s xml:id="echoid-s1208" xml:space="preserve">ma perche mi pare probabilmen-<lb/>te, che nella maniera, che penſo io, poteſſe
<pb o="139" file="0159" n="159" rhead="Coniche. Cap. XXXV."/>
farſi vna coſa ſimile à queſta, che Vitru-<lb/>uio accenna, e per eccitare gli ſtudioſi ad ap-<lb/>plicare il penſiero à queſt’altro Problema, de-<lb/>gno della curioſità di quelli, che vann<unsure/>o cer-<lb/>cando coſe nuoue, e che non ceſſan mai d’af <lb/>faticarſi per diſcoprire i teſori, che ſotto le <lb/>ruine delle antiche diſcipline ſtanno ſepolti. <lb/></s>
  <s xml:id="echoid-s1209" xml:space="preserve">Io dunque ſtimarei, che quei vaſi doueſſero <lb/>hauere vna delle tre forme di ſopra dichia-<lb/>rate, cioè o Parabolica, o @perbolica, o pure <lb/>Elittica, come quelle, che habbiamo di già <lb/>viſto eſſere attiſſime per vnire ad vn punto le <lb/>conuergenti, diuergenti, e parallele, o diſu-<lb/>nire da quello, vſcendo la voce, come da vn <lb/>pũto della bocc<unsure/>a del recitãte, e diffondendoſi <lb/>in giro in cõſeguenza linee diuergẽti. </s>
  <s xml:id="echoid-s1210" xml:space="preserve">Fra quel <lb/>le tre poi crederei più toſto cõuenirli la forma <lb/>Iperbolica, che le altre due; </s>
  <s xml:id="echoid-s1211" xml:space="preserve">e che le celle do-<lb/>ueſſero eſſere di cõcauità Elittica, per il l<unsure/>on-<lb/>go, e per il largo, fabricando così la cella, co-<lb/>meil vaſo con tal proportione, in riſpetto del <lb/>Teatro, che de i due fochi della cella Elitti-<lb/>ca (cioè, che è vn pezzo della ſuper<unsure/>ficie del-<lb/>lo Sferoide) vno foſſe in quel luogo, doue ſo <lb/>gliono ſtare irecitanti, o cantori, l’altro vici-
<pb o="140" file="0160" n="160" rhead="Delle Settioni"/>
no alla cella dinanzi à quella, e dirimpetto al-<lb/>la fronte di eſſa, acciò nel medeſimo punto <lb/>fuſſe anco il foco interiore del vaſo Iperbo-<lb/>lico riuolto con la bocca in giù, eſſendo que-<lb/>ſta collocatione ſim@le à quella dello Spec-<lb/>chio grande Parabolico, e dello Specchietto, <lb/>inſegnata nel Cap. </s>
  <s xml:id="echoid-s1212" xml:space="preserve">20. </s>
  <s xml:id="echoid-s1213" xml:space="preserve">e quaſi vn ſimile arti-<lb/>ficio. </s>
  <s xml:id="echoid-s1214" xml:space="preserve">Hora che la figura, e diſpoſitione de i <lb/>detti vaſi, e delle celle debba farci ſortir l’ef-<lb/>fetto, che pretendiamo, adeſſo ſi farà manife-<lb/>ſto. </s>
  <s xml:id="echoid-s1215" xml:space="preserve">La voce adunque, ch’vſcirà dalla Seena, <lb/>e ſi diffonderà per l’am piezza<unsure/> del Teatro, ca-<lb/>minerà per linee diuergenti dal foco ſtatuito <lb/>nel ſudetto luogo della Scena, &amp; </s>
  <s xml:id="echoid-s1216" xml:space="preserve">arriuarà nel-<lb/>le celle di ſuperficie Elittica, adunque le det-<lb/>te celle rifletteranno tutte le linee ſonore, ch’ <lb/>eſſe riceueranno, facẽdole diuentare conuer-<lb/>genti all’altro foco, che già ſi è conſtituito in-<lb/>nanzi à loro, dirim petto alla fronte, ma quel-<lb/>lo ſarà anco foco interiore del vaſo Iperboli-<lb/>co, che iui ſarà collocato, adunque le linee <lb/>ſonore, che in virtù della cella ſi ſaranno riu-<lb/>nite nel foco interiore del ſudetto vaſo, tra-<lb/>paſſandolo, &amp; </s>
  <s xml:id="echoid-s1217" xml:space="preserve">andando à ferire nella conca-<lb/>uità Iperbolica del vaſo, come diuergenti, ſi
<pb o="141" file="0161" n="161" rhead="Coniche. Cap. XXXV."/>
rifletteranno da quella pur diuergenti, poi-<lb/>che nel Corollario del Cap. </s>
  <s xml:id="echoid-s1218" xml:space="preserve">14. </s>
  <s xml:id="echoid-s1219" xml:space="preserve">ſi è inſegnato, <lb/>che le rette linee, che, partendoſi dal foco in-<lb/>teriore dell’Iperbola, la vanno ad incontrare, <lb/>ſi riflettono poſcia diuergẽti dal foco eſterio-<lb/>re, così dilatandoſi ſempre più, potrà arriuar <lb/>la voce per rifleſſo à tutti gli vditori, il che <lb/>non ſeguirebbe, quando il vaſo foſſe di forma <lb/>Parabolica, che rifletterebbe dette linee ſo-<lb/>nore parallele, per il Corol. </s>
  <s xml:id="echoid-s1220" xml:space="preserve">del Capit. </s>
  <s xml:id="echoid-s1221" xml:space="preserve">9. </s>
  <s xml:id="echoid-s1222" xml:space="preserve">o di <lb/>forma Elittica, che le riunirebbe in vn punto <lb/>ſolo, per il Cap. </s>
  <s xml:id="echoid-s1223" xml:space="preserve">17 che perciò parmi più del-<lb/>le altre conuenirli la figura Iperbolica; </s>
  <s xml:id="echoid-s1224" xml:space="preserve">eſſen-<lb/>do adunque tredici vaſi ſituati, come ſi è det-<lb/>to, ſi faranno tredici rifleſſioni della medeſima <lb/>voce per ciaſcun’orecchio, &amp; </s>
  <s xml:id="echoid-s1225" xml:space="preserve">in ſomma tan-<lb/>te, quanti ſaranno i vaſi, con quell’armonia, <lb/>che dalla proportion di quelli riſultarà.</s>
  <s xml:id="echoid-s1226" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1227" xml:space="preserve">Queſta è vna delle coſe, che mi ſon venu-<lb/>te in mente circa i detti vaſi, intorno a’quali <lb/>altri ſpecolando, incontraranno forſi modi <lb/>più facili, che alla debolezza del mio inge-<lb/>gno non poſſono così<unsure/> facilmente ſouuenire, <lb/>a quali, ſpero nondimeno, non ſia per eſſere <lb/>ne anco diſcaro il poter cõ gli altri eſſaminare
<pb o="142" file="0162" n="162" rhead="Delle Settioni"/>
ancora queſto mio penſiero. </s>
  <s xml:id="echoid-s1228" xml:space="preserve">Ch’egli vera-<lb/>mente confronti co’l modo de gli antichi, ac-<lb/>cennato da Vitruuio, non ardirei d’affermar-<lb/>lo, e ma<unsure/>ſſime, che vi ſono alcune parole nel ſu-<lb/>detto Capitolo, che più toſto moſtrano la co-<lb/>ſa ſtare altrimente da quel, che ſiè dichiara-<lb/>to; </s>
  <s xml:id="echoid-s1229" xml:space="preserve">poiche ſe bene conſideraremo il nome di <lb/>cella, non pare, che ſignifichi, come vn Nic-<lb/>chio da Statue; </s>
  <s xml:id="echoid-s1230" xml:space="preserve">ma più toſto vna coſa ſerrata, <lb/>o ripoſto, come appunto ſi dice, cella vinaria, <lb/>la cãtina, cella penuaria, la ſaluarobba, e cel-<lb/>le le caſelle delle Api, che pur hanno il recin-<lb/>to attorno; </s>
  <s xml:id="echoid-s1231" xml:space="preserve">così ſtando sù la proprietà della <lb/>voce, cella, dobbiamo ſtimar più toſto, che <lb/>ſignifichi vna coſa ſerrata, come vna conſer-<lb/>ua della voce rifleſſa dal vaſo ſoprapoſtoli, a-<lb/>perta però nella parte ſuperiore, &amp; </s>
  <s xml:id="echoid-s1232" xml:space="preserve">inferiore; <lb/></s>
  <s xml:id="echoid-s1233" xml:space="preserve">il che, ſe è vero, hauremo da credere, che, ſpic-<lb/>candoſi la voce dalla bocca del recitãte, non <lb/>s’intẽda Vitruuio, che debba immediatamen-<lb/>te andar’à ferir nella cella, come diceuo io, <lb/>ma sì ben nel vaſo ſoprapoſtoli, e da quello po <lb/>ſcia s’incauerni, per dir così, nella cella, vſcẽ-<lb/>do dalla parte inferiore, che deue eſſer’a perta <lb/>à queſto e<unsure/>ffetto, per arriuar per il rifleſſo del-
<pb o="143" file="0163" n="163" rhead="Coniche. Cap. XXXV."/>
l’onde dell’aria all’orecchie de gli vditori, che <lb/>è aſſai diffe<unsure/>rente dal ſudetto penſiero; </s>
  <s xml:id="echoid-s1234" xml:space="preserve">quello <lb/>nõdimeno ſi accetti, come modo nuouamen-<lb/>te imagina to, e propoſto, perche da altt<unsure/>i con <lb/>il diſcorſo, e con l’eſperienza poſſi eſſere eſſa-<lb/>minato, ne per queſto tralaſciamo ancora di <lb/>põderar le parole di Vitruuio, per poterne pe-<lb/>netrar’il vero ſenſo. </s>
  <s xml:id="echoid-s1235" xml:space="preserve">Quelle veramente par, <lb/>che ci vengano ſignifican do, che le celle ſiano <lb/>luoghi ſerrati d’intorno, il che non ſolo vien <lb/>confermato dal nome di cella, come già ſi è <lb/>detto; </s>
  <s xml:id="echoid-s1236" xml:space="preserve">ma da quelle parole ancora; </s>
  <s xml:id="echoid-s1237" xml:space="preserve">Circaque <lb/>habeant locum vacuũ, &amp; </s>
  <s xml:id="echoid-s1238" xml:space="preserve">à ſummo capite fpati<unsure/>um; <lb/></s>
  <s xml:id="echoid-s1239" xml:space="preserve">parlando dei vaſi, che ſe foſſe la cella vn Nic-<lb/>chio, non parerebbono molto à propoſito, e <lb/>da quelle altre, &amp; </s>
  <s xml:id="echoid-s1240" xml:space="preserve">habeant in parte, quæ ſpectat <lb/>ad Scenã, ſuppoſitos cuneos; </s>
  <s xml:id="echoid-s1241" xml:space="preserve">biſognando perciò, <lb/>che vi ſia, chi ſoſtenti detti cunei, cioè forſi il <lb/>recinto della cella; </s>
  <s xml:id="echoid-s1242" xml:space="preserve">e da quelle altre; </s>
  <s xml:id="echoid-s1243" xml:space="preserve">contraq; </s>
  <s xml:id="echoid-s1244" xml:space="preserve"><lb/>eas cellas relinquantur apertur æ inferiorũ graduum <lb/>cubilibus; </s>
  <s xml:id="echoid-s1245" xml:space="preserve">acciò la voce poſſi vſcire dalle cel-<lb/>le, aggiunge Daniel Barbaro; </s>
  <s xml:id="echoid-s1246" xml:space="preserve">la onde ſarà <lb/>l’ingreſſo dalla parte ſuperiore della cella per <lb/>il rifleſſo fatto primieramente dal va@o, il che <lb/>pare, che confermino quelle altte parole; </s>
  <s xml:id="echoid-s1247" xml:space="preserve">Ita
<pb o="144" file="0164" n="164" rhead="Delle Settioni"/>
hac ratiocinatione vox ab Scena; </s>
  <s xml:id="echoid-s1248" xml:space="preserve">Vti à centro pro-<lb/>fuſa ſe circumagens, tactuq; </s>
  <s xml:id="echoid-s1249" xml:space="preserve">feriens ſingulorũ va-<lb/>ſorum caua, excitauerit auctam claritatem, &amp; </s>
  <s xml:id="echoid-s1250" xml:space="preserve">c. </s>
  <s xml:id="echoid-s1251" xml:space="preserve">do-<lb/>ue par, che voglia intender Vitruuio, che la <lb/>voce vſcita dalla Scena debba andar di poſta <lb/>à ferir ne i vaſi, da’quali ſi rifle<unsure/>tta nelle <lb/>celle, edi lì poi ſi ſparga per il Teatr<unsure/>o; </s>
  <s xml:id="echoid-s1252" xml:space="preserve">per la <lb/>qual coſa crederei pur’anco, che i vaſi doueſ-<lb/>ſero eſſer’Iperbolici, ma che haueſſero il foco <lb/>interiore, nõ vicino, ma nella Scena, dal qua-<lb/>le riceuendo le linee ſonore, come diuergen-<lb/>ti, le riſtetteſſero nella cella pur diuergenti, <lb/>che perciò ſtimarei, che i vaſi non doueſſero <lb/>ſtar ſopra le celle, come tanti capelletti total-<lb/>mente inuerſi, ma riuolti con la bocca parte <lb/>verſo la Scena, parte verſo la cella, la quale <lb/>non pare, c’habbi poi da far’altro, che di con-<lb/>ſeruar la voce à guiſa di canale ſerrato, ſpar-<lb/>gendola per l Auditorio dalla parte da baſſo; <lb/></s>
  <s xml:id="echoid-s1253" xml:space="preserve">Vna tale cõſtitutione adunque par, che ſi poſ-<lb/>ſi dedurre dalle parole del medeſimo Vitru-<lb/>uio.</s>
  <s xml:id="echoid-s1254" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1255" xml:space="preserve">Non riputarei però ne anco improbabile, <lb/>cheivaſi, in vece di ſopraſtare alle celle à gui-<lb/>ſa di capelletti, foſſero dentro le medeſime
<pb o="145" file="0165" n="165" rhead="Coniche. Cap. XXXVI."/>
celle, hauendo però lo ſpatio aperto di ſopra, <lb/>come il medeſimo Vitruuio accẽna con quel-<lb/>le parole; </s>
  <s xml:id="echoid-s1256" xml:space="preserve">Circaq; </s>
  <s xml:id="echoid-s1257" xml:space="preserve">habeant locum vacuum, &amp; </s>
  <s xml:id="echoid-s1258" xml:space="preserve">à <lb/>ſummo capite ſpatium; </s>
  <s xml:id="echoid-s1259" xml:space="preserve">che danno inditio que-<lb/>ſti vaſi douer’eſſer dentro le celle, poiche ſe <lb/>foſſero fuori, haurebbono ſenz’altro lo ſpatio <lb/>di ſopra, ſenza che biſognaſſe dirlo, e maſſi-<lb/>me, che tale ſpatio par, che vi biſogni per l’in-<lb/>greſſo della voce, come il Barbaro eſpone, a-<lb/>dunque il vaſo ſarà dentro, e non fuori della <lb/>cella: </s>
  <s xml:id="echoid-s1260" xml:space="preserve">il che quando ſia vero, intenderemo, <lb/>che la voce vadi à ferir nel vaſo, non per linea <lb/>retta, ma ſi ben fleſſuoſa, per la dilatatione <lb/>delle onde dell’aria, come per la caduta del <lb/>ſaſſo ſegue delle onde dell’acqua, dalla cui <lb/>concauità ſi rifletta poi nella cella, e di nuo-<lb/>uo nel vaſo, facendoſi vna reciprocatione di <lb/>voce per queſti rifleſsi, &amp; </s>
  <s xml:id="echoid-s1261" xml:space="preserve">vn’Echo quaſi infi-<lb/>nita, vſcendo tuttauia per le aperture dinan-<lb/>zi, che ſideuono fare pur nelle celle, ſecondo <lb/>Vitruuio, e ſpargendoſi per l’Auditorio, eſ-<lb/>ſendo eſſe celle à guiſa di coperti da forni, o <lb/>di Teſtudini, come accennano ancora Agoſtin <lb/>Gallo, &amp; </s>
  <s xml:id="echoid-s1262" xml:space="preserve">Aloiſi Pirouano, Commentatori di <lb/>Vitruuio; </s>
  <s xml:id="echoid-s1263" xml:space="preserve">conforme à queſto penſiero adun-
<pb o="146" file="0166" n="166" rhead="Delle Settioni"/>
que nella figura 23. </s>
  <s xml:id="echoid-s1264" xml:space="preserve">hò fatto vn poco d’vno <lb/>abbozzo, per iſpiegare queſto concetto; </s>
  <s xml:id="echoid-s1265" xml:space="preserve">In-<lb/>tenderemo adũque eſſer la cella, O A R, fat-<lb/>ta in volta, &amp; </s>
  <s xml:id="echoid-s1266" xml:space="preserve">aperta di ſopra, doue è la lette-<lb/>ra, A, come anco dinanzi verſo l’Auditorio, <lb/>ſecõdo il foro, F P, il vaſo poi ſarà, K, diſtan-<lb/>te dalla baſe, O R, almeno per vn mezo pie-<lb/>de, ſoſtentato dalli cunei, S A, I A; </s>
  <s xml:id="echoid-s1267" xml:space="preserve">entrarà <lb/>la voce adunque perilforo, A, e di lì nel vaſo, <lb/>K, dal quale ſi rifletterà nella cella, O A R, <lb/>vſcendo per il foro, F P, all’Auditorio: </s>
  <s xml:id="echoid-s1268" xml:space="preserve">Se altri <lb/>ſtimaſſe poi, che il vaſo ſteſſe al cõtrario, cioè <lb/>con la boccain sù, potrà far l’eſperienza del-<lb/>l’vna, e l’altra poſitura, e vedere qual rieſca <lb/>meglio: </s>
  <s xml:id="echoid-s1269" xml:space="preserve">Perla circonferenza poi, C B D, ſ@ã <lb/>no diſpoſte le tredici celle, tutte ſimili alla cel-<lb/>la, O A R, eſſendo ſerrati i vani tra l’vna, e <lb/>l’altra, e quelle ſeparate con le pareti, acciò <lb/>tutte inſieme venghino à formare, come vn <lb/>grãde ſcalino, ſouraſtamte à gli altri, ne’quali <lb/>ſogliono ſedere gli Vditori: </s>
  <s xml:id="echoid-s1270" xml:space="preserve">Chi poi non tra-<lb/>mezaſſe le celle, ma, C B D, ſoſſe, come vn <lb/>canale ſemicircolare, dentro il quale entraſſe <lb/>la voce per le bocche, C, D, vſcendo incor-<lb/>rotta per le aperture dinanzi, ſimili ad, F P,
<pb o="147" file="0167" n="167" rhead="Coniche. Cap. XXXVI."/>
ma rinforzata dentro il canale, C B D, per i <lb/>rifleſſi fatti da i vaſi, diſpoſti come ſopra, cre-<lb/>doanco, che non faria mal’effetto.</s>
  <s xml:id="echoid-s1271" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1272" xml:space="preserve">Ciaſcuno tuttauia s’accoſti à quella opi-<lb/>nione, che gli parerà più probabile, ch’io non <lb/>determino perciò coſa veruna: </s>
  <s xml:id="echoid-s1273" xml:space="preserve">Quanto alla <lb/>ſigura de i vaſi poidirò ancor queſto, ch’io hò <lb/>viſto appreſſo i ſudetti Commentatori, che gli <lb/>moſtrano, come tante campanelle, il che pur <lb/>conferma la figura Iperbolica già detta di ſo-<lb/>pra (qualunque ſia la loro conſtitutione) alla <lb/>quale pur le campane par che ſi vadano acco-<lb/>ſtando. </s>
  <s xml:id="echoid-s1274" xml:space="preserve">Queſte figure però non s’hanno da <lb/>intendere così rigoroſamente, che non ſia le-<lb/>cito lo ſua riare alquanto, baſtando pur, che <lb/>à quelle ci accoſtiamo; </s>
  <s xml:id="echoid-s1275" xml:space="preserve">poiche dice il medeſi-<lb/>mo Vitruuio nel ſudetto Capitolo, che molti <lb/>ſeruen doſi non di vaſi di metallo, ma di terra, <lb/>come di vrne, ouer’olle, fecero ne’Teatri pur’ <lb/>anco vtiliſſimi effetti. </s>
  <s xml:id="echoid-s1276" xml:space="preserve">Haueuano poi gli An-<lb/>tichi altra ſorte d@ vaſi, come dice Herone, <lb/>per cauſare il tuonone’medeſimi Teatri, alla <lb/>cognition de’quali, come anco de’ſopradetti <lb/>vaſi, accoppiando la buona dottrina con l’e-<lb/>ſperienza, non dubito, che non potiamo ar-
<pb o="148" file="0168" n="168" rhead="Delle Settioni"/>
riuare doppo qualche fatica noi ancora.</s>
  <s xml:id="echoid-s1277" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div76" type="section" level="1" n="76">
<head xml:id="echoid-head81" style="it" xml:space="preserve">Delle altre ſuperficie, che dal vario mouimento, <lb/>ò fluſſo delle Settioni Coniche poſſono eſſer <lb/>generate. Cap. XXXVII.</head>
<p>
  <s xml:id="echoid-s1278" xml:space="preserve">POſciache noi habbiamo conſide-<lb/>rato le ſuperficie generate dalle <lb/>Settioni Coniche, per il riuol-<lb/>gimento intorno al ſuo aſſe, vi <lb/>reſtano da vedere quelle, che <lb/>poſſono eſſer prodotte per il vario mouimen-<lb/>to delle iſteſſe Settioni, e concioſiacoſa che <lb/>infiniti ſiano i moti, che poſſon fare, infinite <lb/>ſaranno anco le ſuperficie da lor generabi-<lb/>li, tuttauia di quei mouimenti due ſoli an-<lb/>cora ne conſiderarcmo, e due delle dette ſu-<lb/>perficie, quali così ſpiegaremo ſopra la 24. <lb/></s>
  <s xml:id="echoid-s1279" xml:space="preserve">figura.</s>
  <s xml:id="echoid-s1280" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1281" xml:space="preserve">Siano dunque le due Parabole, B A C, V T <lb/>Y, per eſſempio, ſuoi aſſi, A M, T X, &amp; </s>
  <s xml:id="echoid-s1282" xml:space="preserve">à quel-<lb/>li ordinatamente applicate, B C, V Y, e dal <lb/>punto, C, ſia tirata la, C H, parallela all’aſſe, <lb/>A M, e ſia, I, foco della Parabola, B A C, qua-<lb/>le ſi riuolga intorno ad, H C, fiſſa, ſino che ri-
<pb o="149" file="0169" n="169" rhead="Coniche. Cap. XXXVII."/>
torni di onde ſi partì, tal che deſcriua la ſuper-<lb/>ficie, B A C D E, &amp; </s>
  <s xml:id="echoid-s1283" xml:space="preserve">il punto, I, la circonferen-<lb/>za, I O, &amp;</s>
  <s xml:id="echoid-s1284" xml:space="preserve">, B C, il circolo, B E; </s>
  <s xml:id="echoid-s1285" xml:space="preserve">è dunque ma-<lb/>nifeſto, che i fochi della Parabola, B A C, con-<lb/>ſtituita in diuerſi ſiti in tal reuolutione, ſaran-<lb/>no tutti nella circonferenza, I O, ſi che eſpo-<lb/>ſta vna tal ſuperficie verſo il Sole, talmente, <lb/>che, H C, ſia per dritto al centro di quello, el-<lb/>la abbruſcierà non in vn punto, ma nella cir-<lb/>conferenza, I O; </s>
  <s xml:id="echoid-s1286" xml:space="preserve">e per il contrario molti lumi <lb/>poſti nella circonferenza, I O, rifletteranno il <lb/>ſuo ſplendore per linee parallele, e per linee <lb/>diuergenti, quando, B A C, foſſe vn’Iperbola, <lb/>o per conuergenti, quando foſſe vna portion <lb/>d’Eliſſi, terminanti pure in circonferẽza di cir-<lb/>colo (intendendo però hora conuergenti non <lb/>ad vn punto tutte, ma alla circonferenza d’vn <lb/>circolo, come anco diuergenti) e tutto quello <lb/>in ſomma, che ſi è detto dell’vnire, o diſunire <lb/>le linee radioſe, o ſonore da vn punto, quà s’@n-<lb/>tenderà quanto ad vnirle in vna circonferen-<lb/>za di circolo, o diſunirle da quella; </s>
  <s xml:id="echoid-s1287" xml:space="preserve">ſupponen-<lb/>do per la Parabola, B A C, poſta per eſſempio, <lb/>e l’Iperbola, e l’Eliſſi, o ſua portione.</s>
  <s xml:id="echoid-s1288" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1289" xml:space="preserve">Sia poi, V T Y, Parabola, Iperbola, o Eliſſi,
<pb o="150" file="0170" n="170" rhead="Delle Settioni"/>
o pa@te di quelle, X T, aſſe, &amp;</s>
  <s xml:id="echoid-s1290" xml:space="preserve">, V Y, ordinata-<lb/>mente applicata à, T X, e foco il punto, S, e ſi <lb/>moua, ſolleuandoſi talmente, che ſtia ſempre <lb/>parallela al piano, V T Y, e la, V Y, deſcriua <lb/>vn parallel@gramo rettangolo, P Y, perpendi-<lb/>colal<unsure/>e ſopra il piano, V T Y, &amp; </s>
  <s xml:id="echoid-s1291" xml:space="preserve">il foco, S, la ret-<lb/>ta, S R, terminata nel piano della ſettione, P <lb/>O Q, quieſcente@ nel fin del moto, il cui aſſe <lb/>ſia, O @; </s>
  <s xml:id="echoid-s1292" xml:space="preserve">ſe adunque ſegaremo la ſuperficie <lb/>generata dalla ſettione, P O Q, con vn piano <lb/>paralleloa à, V T X, ſe ne farà vna iſteſſa ſettio-<lb/>ne, che hauerà il foco ne<unsure/>lla retta, R S, &amp; </s>
  <s xml:id="echoid-s1293" xml:space="preserve">in <lb/>ſomma tutci<unsure/> i fochi delle ſettioni intermedie <lb/>ſaranno nella retta, R S, nella quale la ſuper-<lb/>ficie, P O Q Y T V, abbruſcierà, eſſendo con <lb/>ll<unsure/> concauo eſpoſta al Sole; </s>
  <s xml:id="echoid-s1294" xml:space="preserve">ſecondo la drittura <lb/>delli aſſi delle medeſime Settioni. </s>
  <s xml:id="echoid-s1295" xml:space="preserve">Tutto quel-<lb/>lo adunque, che ſi è detto, quanto all’vnire le <lb/>linee radioſe, o ſonore ad vn punto, o diſunir-<lb/>le da quello, s’intenderà ancora poterſi fare <lb/>con queſte ſuperficie, quanto all’vnirle, ò di-<lb/>ſunirle da vna linea retta; </s>
  <s xml:id="echoid-s1296" xml:space="preserve">E però ſe, R S, foſ-<lb/>ſe vna corda ſonante, nõ è dubbio alcuno, che <lb/>il ſuono per ragion di rifl@ſſione, eſſe@do la ſu-<lb/>perficie, P O Q Y T V, Elittica, ſi rifletterà in
<pb o="151" file="0171" n="171" rhead="Coniche. Cap. XXXVII."/>
vna retta linea, ma, eſſendo Parabolica, per ſu-<lb/>perficie parallele, &amp;</s>
  <s xml:id="echoid-s1297" xml:space="preserve">, eſſendo Iperbolica, per <lb/>ſuperficie diuergenti, intendendo pero la di-<lb/>uergenza, e conuergenza non in vn punto, ma <lb/>in vna retta linea, dalle quali coſe ſi manife-<lb/>ſta; </s>
  <s xml:id="echoid-s1298" xml:space="preserve">che ſi può inuigorire in diue@ſi modi il ſuo-<lb/>no de gl’@ſtrumenti da corde, il che però baſti <lb/>di hauer’al curioſo Lettore accennato.</s>
  <s xml:id="echoid-s1299" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div77" type="section" level="1" n="77">
<head xml:id="echoid-head82" style="it" xml:space="preserve">Della cognitione del Moto. <lb/>Cap. XXXVIII.</head>
<p>
  <s xml:id="echoid-s1300" xml:space="preserve">DI quanta importanza ſia la co-<lb/>gnitione del moto nelle coſe na-<lb/>turali, e di che momẽto, per ben <lb/>filoſofare, credo eſſer manifeſtiſ-<lb/>ſimo à ciaſcuno, che in queſto <lb/>gran Teatro di Natura habbi talhora fiſſato <lb/>lo ſguardo nelle di lei marauiglioſe bellezze. <lb/></s>
  <s xml:id="echoid-s1301" xml:space="preserve">I mouimenti de’Cieli, le traſmutationi, che ſi <lb/>veggon fare continuamente intorno à queſto <lb/>globo terr@ſtre, eccitorno la curioſità à con-<lb/>templarli, come ſingolari artificij di Natura, <lb/>con ch’ella così mirabili effetti ci rappreſen-<lb/>ta, e sforzorno le menti humane à giudicare
<pb o="152" file="0172" n="172" rhead="Delle Settioni"/>
altro per appunto non eſſer Natura (come il <lb/>Prencipe de’Peripatetici nel 2. </s>
  <s xml:id="echoid-s1302" xml:space="preserve">della Fiſica <lb/>c’inſegna) che vn principio di moto, e di quie-<lb/>te; </s>
  <s xml:id="echoid-s1303" xml:space="preserve">di onde poi con gran ragione ſi deduſſe <lb/>quella communiſſima, e veriſſima ſentenza. <lb/></s>
  <s xml:id="echoid-s1304" xml:space="preserve">Ignorato motu ignoratur Natura. </s>
  <s xml:id="echoid-s1305" xml:space="preserve">Altiſſima dot-<lb/>trina in vero, intorno alla quale degnamente <lb/>ſi ſono affaticati i più ſublimi ingegni, che ſia-<lb/>no ſtati al Mondo, parendoli come à dire, che <lb/>chi haueſſe vn’eſſatta cognition di quello, po-<lb/>ſcia diuentaſſe attiſſimo in vn certo modo all’ <lb/>intendere, ed à penetrare tutti gli effetti di <lb/>Natura. </s>
  <s xml:id="echoid-s1306" xml:space="preserve">Ma quanto vi aggiunga la cognitio-<lb/>ne delle ſcienze Matematiche, giudicate da <lb/>quelle famoſiſſime Scuole de’Pitagorici, e de’ <lb/>Platonici, ſommamente neceſſarie per inten-<lb/>der le coſe Fiſiche, ſpero in breue ſarà mani-<lb/>feſto, per la nuoua dottrina del moto promeſ-<lb/>ſaci dall’eſquiſitiſſimo Saggiatore della Natu-<lb/>ra, dico dal Sig. </s>
  <s xml:id="echoid-s1307" xml:space="preserve">Galileo Galilei, ne’ſuoi Dia-<lb/>logi, proteſtando io hauer’hauuto e motiuo, <lb/>e lume ancora in parte intorno à quel poco, <lb/>ch’io dirò del moto in queſto mio Trattato, <lb/>per quanto alle Settioni Coniche ſi aſpetta, <lb/>da i ſottiliſſimi diſcorſi di quello, e del Reue-
<pb o="153" file="0173" n="173" rhead="Coniche. Cap. XX XVIII."/>
rẽdiſs. </s>
  <s xml:id="echoid-s1308" xml:space="preserve">P. </s>
  <s xml:id="echoid-s1309" xml:space="preserve">Abbate D. </s>
  <s xml:id="echoid-s1310" xml:space="preserve">Benedetto Caſtelli Mona-<lb/>co Caſſinenſe, Matem. </s>
  <s xml:id="echoid-s1311" xml:space="preserve">di N. </s>
  <s xml:id="echoid-s1312" xml:space="preserve">S. </s>
  <s xml:id="echoid-s1313" xml:space="preserve">e molto inten-<lb/>dente di queſte materie, ambid ue miei Mae-<lb/>ſtri. </s>
  <s xml:id="echoid-s1314" xml:space="preserve">Rimetto dunque il Lettore in ciò, ch’io <lb/>ſupporrò al dottiſs. </s>
  <s xml:id="echoid-s1315" xml:space="preserve">libro, che da sì grand’in-<lb/>gegno in breue dourà porſi in luce, e ſi cõten-<lb/>terà di queſto poco, ch’io dirò, per manifeſta-<lb/>re, che coſa habbino che fare le Settioni Co-<lb/>niche con così alto, e così nobil ſoggetto.</s>
  <s xml:id="echoid-s1316" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div78" type="section" level="1" n="78">
<head xml:id="echoid-head83" style="it" xml:space="preserve">Del mouimento de’corpi graui. <lb/>Cap. XX XIX.</head>
<p>
  <s xml:id="echoid-s1317" xml:space="preserve">BEnche intorno à’corpi graui di-<lb/>uerſiſſime coſe ſi poteſſero con-<lb/>ſiderare, tutte belle, e tutte cu-<lb/>rioſe, hora però non cercaremo <lb/>altro, ſe non che ſorte di linea <lb/>ſia quella, per la quale ſi moue eſſo graue, mer-<lb/>eè prima dell’interna grauità, poi del proiciẽ-<lb/>te, e finalmente dell’vno, e dell’altro accop-<lb/>piati inſieme, per vedere, ſe vi haueſſero che <lb/>fare le Settioni Coniche, e quali ſiano, quan-<lb/>do ciò ſia vero.</s>
  <s xml:id="echoid-s1318" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1319" xml:space="preserve">Dico adunque, ſe noi conſideraremo il mo-
<pb o="154" file="0174" n="174" rhead="Delle Settioni"/>
to del graue fatto per la ſola interna grauità, <lb/>in qualunque modo poi ella ſi operi, che quel-<lb/>lo ſarà ſempre indrizzato yerſo il centro vni-<lb/>uerſale delle coſe graui, cioè verſo il cẽtro del-<lb/>la terra, &amp; </s>
  <s xml:id="echoid-s1320" xml:space="preserve">vniuerſalmente conſpirare tutti i <lb/>graui à queſto centro, poiche ſi veggono in <lb/>tutti i luoghi della ſuperficie terreſtre ſcen-<lb/>dere, non impediti, à per pẽdicolo ſopra l’Ori-<lb/>zonte, &amp; </s>
  <s xml:id="echoid-s1321" xml:space="preserve">è manifeſto, che le linee rette per-<lb/>pendicolari alla ſuperficie della sfera prolon-<lb/>gate, vanno tutte à ferire nel centro di quella, <lb/>che poi la terra ſia sferica è manifeſtiſſimo, ſi <lb/>per via delli Eccliſſi, come anco d’altri acci-<lb/>denti, che euidentemẽtẽ queſto ci dimoſtrano.</s>
  <s xml:id="echoid-s1322" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1323" xml:space="preserve">Dico più oltre, che conſiderato il mobile, <lb/>che da vn proiciẽte viene ſpinto verſo alcuna <lb/>parte, ſe non haueſſe altra virtù motrice, che <lb/>lo cacciaſſe verſo vn’altra banda, andarebbe <lb/>nel luogo ſegnato dal proiciente per dritta li-<lb/>nea, mercè della virtù impreſſali pur per drit-<lb/>ta linea, dalla quale drittura non è ragioneuo-<lb/>le, che il mobile ſi diſcoſti, mentre non vi è al-<lb/>tra virtù motrice, che ne lo rimoua, e ciò quã-<lb/>do fra li duoi termini non ſia impedimẽto; </s>
  <s xml:id="echoid-s1324" xml:space="preserve">co-<lb/>me, per eſſempio, vna palla d’Artiglieria vſci-
<pb o="155" file="0175" n="175" rhead="Coniche. Cap. XXXIX."/>
ta dalla bocca del pezzo, ſe non haue ſſe altro, <lb/>che la virtù impreſſali dal fuoco, andarebbe à <lb/>dare di punto in bianco nel ſegno poſto à drit-<lb/>tura della canna, ma perche vi è vn’altro mo-<lb/>tore, che è l’interna grauità di eſſa palla, quin-<lb/>di auuiene, che da tal drittura ſia quella sfor-<lb/>zata deuiare, accoſtandoſi al centro della <lb/>terra.</s>
  <s xml:id="echoid-s1325" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1326" xml:space="preserve">Dico ancora, che quel proietto non ſolo an-<lb/>darebbe per dritta linea nel ſegno oppoſto, <lb/>ma che in tempi eguali paſſarebbe pur ſpatij <lb/>eguali della medeſima linea, mentre quel<unsure/> mo-<lb/>bile foſſe à tal moto indifferent@; </s>
  <s xml:id="echoid-s1327" xml:space="preserve">e mentre an-<lb/>cora il mezo non li faceſſe qualche reſiſtenza, <lb/>poiche non ci ſarebbe cauſa di ritardarſi, ne di <lb/>accelerarſi: </s>
  <s xml:id="echoid-s1328" xml:space="preserve">ſi che il graue, mercè della inter-<lb/>na grauità, non anderà ſe non verſo il centro <lb/>della terra, ma quello, mercè della virtù im-<lb/>preſſali, potrà incaminarſi verſo ogni banda; <lb/></s>
  <s xml:id="echoid-s1329" xml:space="preserve">eſſendo due adunque nel proietto le virtù mo-<lb/>trici, vna la grauità, l’altra la virtù impreſſa, <lb/>ciaſcuna di loro ſeparatamente farebbe ben <lb/>caminare il mobile per linea retta, come ſi è <lb/>detto, ma accoppiate inſieme non lo faranno <lb/>andare per linea retta, ſe nõ in queſti due caſi,
<pb o="156" file="0176" n="176" rhead="Delle Settioni"/>
nel primo, quãdo dalla virtù impreſſa ſia ſpin-<lb/>to il graue per la perpẽdicolare all’Orizonte; <lb/></s>
  <s xml:id="echoid-s1330" xml:space="preserve">il ſecondo, quando non ſolo la virtù impreſſa, <lb/>ma anco la grauità moua il graue vniforme-<lb/>mente, perche gli accoſtamenti fatti in tempi <lb/>eguali al centro della terra, partendoſi da vna <lb/>retta linea, ſariano ſempre eguali, come anco <lb/>li ſpatij decorſi ne’medeſimi tempi dell’iſteſſa <lb/>linea, per la quale viene ſpinto eſſo graue; </s>
  <s xml:id="echoid-s1331" xml:space="preserve">e <lb/>perciò il mobile ſarebbe ſempre nella medeſi-<lb/>ma linea retta: </s>
  <s xml:id="echoid-s1332" xml:space="preserve">Ma quando vno de’duoi non <lb/>foſſe vniforme, allhora nõ caminarebbe il mo-<lb/>bile ſpinto dalla grauità, e dalla virtù impreſ-<lb/>ſa, altrimente per linea retta, ma ſi bene per <lb/>vna curua, la cui qualità, e conditione dipen-<lb/>derebbe dalla detta vniformità, e diſſormità <lb/>di moto accoppiate inſieme. </s>
  <s xml:id="echoid-s1333" xml:space="preserve">Hora nel graue, <lb/>che, ſpiccandoſi dal proiciente, viene indriz-<lb/>zato verſo qual ſi ſia parte, per eſſempio, moſ-<lb/>ſo per vna linea eleuata ſopra l’Orizonte, vi è <lb/>bene la grauità, che opera, ma quella non fà <lb/>altro, che ritirare il mobile dalla drittura del-<lb/>la ſudetta linea eleuata, non hauendo che far <lb/>niente con l’altro moto, ſe non per quanto vie-<lb/>ne il graue allontanato dal centro della ter-
<pb o="157" file="0177" n="177" rhead="Coniche. Cap. XXXIX"/>
ra, aſtraendo adunque nel graue la inclinatio-<lb/>ne al centro di quella, come anco ad altro luo-<lb/>go, egli reſta indifferente al moto conferitoli <lb/>dal proiciente, e perciò ſe non vi foſſe l’@mpe-<lb/>dimento dell’ambiente, quello ſarebbe vni-<lb/>forme: </s>
  <s xml:id="echoid-s1334" xml:space="preserve">ragioneuolmẽte adunque ſi potrà ſup <lb/>porre, che i graui ſpinti dal proiciente verſo <lb/>qualunque parte, mercè della virtù impreſſa, <lb/>caminino vniformemente, non hauendo riſ-<lb/>guardo all’impedimento dell’aria, che per eſ-<lb/>ſer tenuiſſima, e fluidiſſima, per qualche nota-<lb/>bile ſpatio, può eſſer, che gli permetta la ſu-<lb/>detta vniformità.</s>
  <s xml:id="echoid-s1335" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1336" xml:space="preserve">Reſta hora, chefacciamo rifleſſione all’ac-<lb/>coſtamento del graue, fatto alcentro della <lb/>terra, mercè dell’interna grauità, che vien <lb/>detto moto naturale, &amp; </s>
  <s xml:id="echoid-s1337" xml:space="preserve">al diſcoſtamento da <lb/>quello, per l’impulſo cõferitoli, che ſi chia-<lb/>ma moto violento; </s>
  <s xml:id="echoid-s1338" xml:space="preserve">cheil graue, che ſi patte <lb/>dal la quiete, e ſi moue al centro, ſi vada ſem-<lb/>pre velocitãdo, quãto più ſi accoſta al centro, <lb/>o per dir meglio, quanto più ſi allontana dal <lb/>ſuo principio, e che il violento, o dal centro ſi <lb/>vada ſempre ritardando, ciò è ſta@o ſa puto da <lb/>tutti i Filoſofi ancora, ma con qual proportio-
<pb o="158" file="0178" n="178" rhead="Delle Settioni"/>
ne s’acceleri il moto naturale, e ſi ritardi il <lb/>violento, ce lo inſegna nuouamente, e ſingo-<lb/>larmente il Sig. </s>
  <s xml:id="echoid-s1339" xml:space="preserve">Galileo ne’ſuoi Dialogi alla <lb/>pag. </s>
  <s xml:id="echoid-s1340" xml:space="preserve">217. </s>
  <s xml:id="echoid-s1341" xml:space="preserve">dicendo eſſer l’incremento della ve-<lb/>locità, ſecondo il progreſſo de’numeri diſpa-<lb/>ri continuati dall’vnità, &amp; </s>
  <s xml:id="echoid-s1342" xml:space="preserve">il decremento, ſe-<lb/>condo la medeſim<unsure/>a ſerie, numerando al con-<lb/>trario; </s>
  <s xml:id="echoid-s1343" xml:space="preserve">cioè, Se, per eſſempio, vn mobile an-<lb/>dando verſo il cen@ro in vn<unsure/>a battuta di polſo, <lb/>farà vn braccio di ſpatio, nella ſeconda ne fa-<lb/>rà 3. </s>
  <s xml:id="echoid-s1344" xml:space="preserve">nella terza 5. </s>
  <s xml:id="echoid-s1345" xml:space="preserve">nella quarta 7. </s>
  <s xml:id="echoid-s1346" xml:space="preserve">nella quin-<lb/>ta 9. </s>
  <s xml:id="echoid-s1347" xml:space="preserve">e così di man’in mano; </s>
  <s xml:id="echoid-s1348" xml:space="preserve">ma ſe per il con-<lb/>trario il mobile andaſſe all’in sù, facendo in <lb/>vna battuta di polſo braccia 9. </s>
  <s xml:id="echoid-s1349" xml:space="preserve">nella ſeconda <lb/>ne faria braccia 7. </s>
  <s xml:id="echoid-s1350" xml:space="preserve">nella terza 5. </s>
  <s xml:id="echoid-s1351" xml:space="preserve">nella quarta <lb/>3. </s>
  <s xml:id="echoid-s1352" xml:space="preserve">e nella quinta 1. </s>
  <s xml:id="echoid-s1353" xml:space="preserve">riducendoſi al nullo grado <lb/>nel punto della rifleſſione, che è fine del moto <lb/>violento, e principio del naturale. </s>
  <s xml:id="echoid-s1354" xml:space="preserve">A queſta <lb/>medeſima concluſione mi ſono ancor’io sfor-<lb/>zato di arriuare per altra via, doppo hauerla <lb/>ſentita dal ſudetto Sig. </s>
  <s xml:id="echoid-s1355" xml:space="preserve">Galileo, conſiderando <lb/>in vn cerchio i gradi delle velocità, che, dalla <lb/>quiete incominciando, vanno creſcendo ſino <lb/>al maſſimo nel medeſimo cerchio, rappreſen-<lb/>tandomi il centro il nullo grado di velocità,
<pb o="159" file="0179" n="179" rhead="Coniche. Cap. XXXIX."/>
o vogliamo dire la quiete, e le circonferenze, <lb/>che ſi poſſono deſcriuere intorno al medeſimo <lb/>centro, i gradi delle diuerſe velocità, quali ſe <lb/>li vogliamo prender tutti, conuiene, che noi <lb/>intendiamo diſſegnati tutti i cerchi poſſibili à <lb/>deſcriuerſi ſopra quel centro, che facendo la <lb/>ſomma delle loro circonferenze, potremo di-<lb/>re di ſapere la vera quantità di tutti i gradi di <lb/>velocità, che intermediano tra la quiete, &amp; </s>
  <s xml:id="echoid-s1356" xml:space="preserve">il <lb/>maſſimo grado in quel cerchio: </s>
  <s xml:id="echoid-s1357" xml:space="preserve">Hora, perche <lb/>queſto pare coſa impoſſibile, cioè il ſommare <lb/>infinite circonferenze, io mi preuaglio dell’a-<lb/>rea dell’@ſteſſo cerchio, e ne cauo le proportio-<lb/>ni delle aggregate velocità, incominciãdo dal <lb/>centro, o dalla quiete, e procedendo ſino alla <lb/>circonferenza eſtrema, cioè ſino al maſſimo; <lb/></s>
  <s xml:id="echoid-s1358" xml:space="preserve">hauendo dimoſtrato io nella mia Geometria, <lb/>che qual proportione hanno i cerchi frà loro, <lb/>tale anco l’hãno tutte le circonferẽze, deſcrit-<lb/>tibili ſopra il cẽtro dell’vno, à tutte le circon-<lb/>ferenze, deſcrittibili ſopra il cen@ro dell’altro, <lb/>perciò ſe nel noſtro cerchio, nel quale voglio <lb/>miſurare le aggregate velocità, con la diſtan-<lb/>za di vn terzo del ſemidiametro, per eſſempio, <lb/>deſcriuerò vn cerchio, la cui circonferenza mi
<pb o="160" file="0180" n="180" rhead="Delle Settioni"/>
rappreſenti vn tal grado di velocità; </s>
  <s xml:id="echoid-s1359" xml:space="preserve">ſaprò, <lb/>che qual proportione hà il cerchio grande al <lb/>piccolo, tale ancora l haueranno tu<unsure/>tte le cir-<lb/>conferenze concentr<unsure/>iche del cerchio grande <lb/>à tutte le circonferenze concentriche del pic-<lb/>colo, cioè tutti i gradi di velocità acquiſtati <lb/>nel trapaſſare dalla quiete al grado maſſimo, <lb/>à tutti i gradi acquiſtati paſſando dall’iſteſſa <lb/>quiete al grado intermedio, che habbiamo <lb/>preſo, ma i cerchi ſono tra loro, come i qua-<lb/>drati de’ſemidiametri, adunque anco dette <lb/>velocità creſcerãno ſecondo l’incremento de’ <lb/>quadrati de’ſemidiametri, ma con qual pro-<lb/>portione creſce la velocità nel mobile, creſco-<lb/>no anco li ſpatij decorſi dall’iſteſſo mobile<unsure/>, co-<lb/>me è ragioneuole, poiche chi acquiſta altre-<lb/>tanta velocità, quanta ſi<unsure/> ritroua hauere, gua-<lb/>dagna ancora forza di trapaſſare altretanto <lb/>ſpatio, quanto faceua, e così nell’altre propor-<lb/>tioni; </s>
  <s xml:id="echoid-s1360" xml:space="preserve">adunque li ſpatij decorſi dal mobile, nel <lb/>quale ſi vanno aggregando le velocità, ſaran-<lb/>no, come i quadrati de’ſemidiametri de’cer-<lb/>chi, ne’quali ſi poſſono conſiderare dette ve-<lb/>locità, cioè come i quadrati de’rempi, quali <lb/>intenderemo nel ſemidiametro del dato cer-
<pb o="161" file="0181" n="181" rhead="Coniche. Cap. XXXIX."/>
chio; </s>
  <s xml:id="echoid-s1361" xml:space="preserve">ſe quello adunque ſi ſupponeſſe diuiſo <lb/>in cinque parti eguali, poſto, che il quadrato <lb/>d’vna di queſte parti foſſe 1. </s>
  <s xml:id="echoid-s1362" xml:space="preserve">il quadrato di due <lb/>ſarebbe 4. </s>
  <s xml:id="echoid-s1363" xml:space="preserve">di tre 9. </s>
  <s xml:id="echoid-s1364" xml:space="preserve">di quattro 16. </s>
  <s xml:id="echoid-s1365" xml:space="preserve">di cinque <lb/>25. </s>
  <s xml:id="echoid-s1366" xml:space="preserve">e tal proportione haurebbono i cinque <lb/>cerchi deſcritti ſopra queſti cinque ſemidia-<lb/>metri, e perciò, ſottrahendo ciaſcuno antece-<lb/>dente dal ſuo conſeguente, reſtarebbono que-<lb/>ſti numeri 1. </s>
  <s xml:id="echoid-s1367" xml:space="preserve">3. </s>
  <s xml:id="echoid-s1368" xml:space="preserve">5. </s>
  <s xml:id="echoid-s1369" xml:space="preserve">7. </s>
  <s xml:id="echoid-s1370" xml:space="preserve">9. </s>
  <s xml:id="echoid-s1371" xml:space="preserve">che moſtrarebbono la <lb/>progreſſione del minimo cerchio, e delli ſe-<lb/>guenti reſidui, o armille, che ci rappreſenta-<lb/>no i gradi acquiſtati dal mobile cõtinuamen-<lb/>te ne’ſudetti tempi eguali. </s>
  <s xml:id="echoid-s1372" xml:space="preserve">Perche dunque i <lb/>graui partendoſi dalla quiete, vanno ad ogni <lb/>momento acquiſtando nuouo grado di velo-<lb/>cità (hauendo il motore aſſiſtente, che ſempre <lb/>opera, cioè la grauità) quale non perdono, per <lb/>non ripugnarli, ne eſſergli tolto dall’ambien-<lb/>te, almeno, che ſe n’accorga per qualche no-<lb/>tabil ſpatio (ciò dico, poiche ad vna grandiſ-<lb/>ſima velocità finalmente l’ambiente reſiſte <lb/>notabilmente, non comportando egli di eſſer <lb/>moſſo con tanta furia, del che il volo de gli vc-<lb/>celli ce ne può nell’aria, &amp; </s>
  <s xml:id="echoid-s1373" xml:space="preserve">il nuotar nell’ac-<lb/>que, in parte aſſicurare) perciò i ſpatij ſcoi<unsure/>ſi
<pb o="162" file="0182" n="182" rhead="Delle Settioni"/>
da quelli in tempi eguali creſceranno, cõfor-<lb/>me l’incremento de’numeri diſpari continua-<lb/>ti dall’vnità; </s>
  <s xml:id="echoid-s1374" xml:space="preserve">e gl’i<unsure/>ſteſſi graui douendo trapaſ-<lb/>ſare da vn dato grado di velocità alla quiete, <lb/>come nel moto violento terrãno l’ordine con-<lb/>trario della medeſima ſerie de’numeri diſpari. <lb/></s>
  <s xml:id="echoid-s1375" xml:space="preserve">Queſte coſe però ſiano da me dette, come per <lb/>vn paſſaggio, che perciò non mi ſono ſpiegato <lb/>con figura, ne con quella chiarezza, che biſo-<lb/>gnarebbe, poiche rimetto il Lettore à quello, <lb/>che la ſottigliezza del Sig. </s>
  <s xml:id="echoid-s1376" xml:space="preserve">Galileo c’inſegna-<lb/>rà nell’Opera del moto, che ci promette ne’ <lb/>ſuoi Dialogi. </s>
  <s xml:id="echoid-s1377" xml:space="preserve">Intendiamo adunque la condi-<lb/>tione del moto nel graue, ſi per ragion dell’ <lb/>impulſo, ſi anco dell’interna grauità, le quali <lb/>coſe ſuppoſte, trapaſſaremo hora à cercare, <lb/>qual ſorte di linea ſia quella, che deſcriue il <lb/>graue, ſpinto da qualche forza, non per <lb/>la perpendicolare all’Orizonte, <lb/>ma per qualſiuoglia altra <lb/>banda.</s>
  <s xml:id="echoid-s1378" xml:space="preserve"/>
</p>
  <figure>
    <image file="0182-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0182-01"/>
  </figure>
<pb o="163" file="0183" n="183" rhead="Coniche. Cap. XL."/>
</div>
<div xml:id="echoid-div79" type="section" level="1" n="79">
<head xml:id="echoid-head84" style="it" xml:space="preserve">Qual ſorte dilinea deſcriuano i graui nelloro moto, <lb/>ſpiccati che ſiano dal proiciente. <lb/>Cap. XL.</head>
<p>
  <s xml:id="echoid-s1379" xml:space="preserve">SE noi conſideraremo il graue, <lb/>che hà da mouerſi ſeparatamẽ-<lb/>te dal proiciente, quando ca-<lb/>mina di conſerua con quello, <lb/>come à dire; </s>
  <s xml:id="echoid-s1380" xml:space="preserve">la pietra, che ſi <lb/>moue per vn poco di ſpatio in <lb/>compagnia della mano, o della froinbola, o al-<lb/>tro iſtrumento, non è dubbio alcuno, che è <lb/>sforzata à far quella ſtrada, che farà ancora il <lb/>ſuo motore; </s>
  <s xml:id="echoid-s1381" xml:space="preserve">come, per eſſempio, dourà anda-<lb/>re in giro con la mano, ſino ch’egli da quella ſi <lb/>diſſepari, ò in qualſiuoglia altro modo, ch’eſſa <lb/>mano cami<unsure/>ni; </s>
  <s xml:id="echoid-s1382" xml:space="preserve">ma ſeparato che ſia, non hà più <lb/>obligo di accompagnar la mano: </s>
  <s xml:id="echoid-s1383" xml:space="preserve">l’impulſo poi <lb/>conferito dal proiciente nel punto della ſepa-<lb/>ratione è ſempre per linea retta, cioè per quel-<lb/>la, che è à drittura del moto, che viene ad eſ-<lb/>ſer la tangente di quella curua, per la quale ſi <lb/>è fatto il moto, tangente dico nel punto della <lb/>ſeparatione, come pariinente c’inſegna il Sig. <lb/></s>
  <s xml:id="echoid-s1384" xml:space="preserve">Galileo ne’ſuoi Dialogi alla pag. </s>
  <s xml:id="echoid-s1385" xml:space="preserve">186. </s>
  <s xml:id="echoid-s1386" xml:space="preserve">quan-
<pb o="164" file="0184" n="184" rhead="Delle Settioni"/>
do il moto della mano ſia per linea cur<unsure/>ua, oue-<lb/>ro quando ſia fatto per linea retta è pure vn <lb/>pezzo dell’iſteſſa linea prolongata: </s>
  <s xml:id="echoid-s1387" xml:space="preserve">Per que-<lb/>ſta linea retta adunque andarebbe il proietto <lb/>ogni volta, che la grauità non lo ritiraſſe da <lb/>quella continuamente verſo il cẽtro della ter-<lb/>ra, vero è, che quando l’impulſo foſſe per la <lb/>perpendicolare all’Orizonte, anco la grauità <lb/>tirarebbe eſſo graue per l’iſteſſa linea, e così il <lb/>moto del proietto in queſto caſo ſarebbe pur <lb/>linea retta, come ſi è detto nel Cap. </s>
  <s xml:id="echoid-s1388" xml:space="preserve">antecedẽ-<lb/>te; </s>
  <s xml:id="echoid-s1389" xml:space="preserve">ma quando l’impulſo non ſia fatto per la <lb/>detta perpendicolare, ma per qualſiuoglia al-<lb/>tra linea retta, ne ſeguirà, che detto graue ſia, <lb/>mercè dell’interna grauità, ritirato verſo il <lb/>centro, &amp; </s>
  <s xml:id="echoid-s1390" xml:space="preserve">in conſeguenza tolto fuori di quella <lb/>drittura, talmente, che in tempi eguali non lo <lb/>abbaſſarà per ſpatij eguali da quella linea <lb/>dritta, ma sì bene per ſpatij diſeguali, che cre-<lb/>ſceranno, come ſi è detto, ſecondo l’incremen-<lb/>to de’numeri diſpari continuati dall’vnità.</s>
  <s xml:id="echoid-s1391" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1392" xml:space="preserve">Dico adunque, che i graui ſpinti dal proi-<lb/>ciente à qualſiuoglia banda, fuorche per la <lb/>perpendicolare all’Orizonte, ſeparati che ſia-<lb/>no da quello, &amp; </s>
  <s xml:id="echoid-s1393" xml:space="preserve">eſcluſo l’impedimẽto dell’am-
<pb o="165" file="0185" n="185" rhead="Coniche. Cap. XL."/>
biente, deſcriuono vna linea curua, inſenſi-<lb/>bilmente differente dalla Parabola.</s>
  <s xml:id="echoid-s1394" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div80" type="section" level="1" n="80">
<head xml:id="echoid-head85" style="it" xml:space="preserve">Dimoſtratione ſopra la figura 25.</head>
<p>
  <s xml:id="echoid-s1395" xml:space="preserve">SVppongaſi per, TO, la drittura della cã-<lb/>na d’vn pezzo d’Artiglieria, la cui bocca <lb/>ſia nel punto, O, e queſta s’intenda ſtare <lb/>à liuello, ouero eleuata, ò inclinata, come ſi <lb/>voglia, benche la figura moſtri il tiro à liuello; <lb/></s>
  <s xml:id="echoid-s1396" xml:space="preserve">la Palla dunque cacciata dalla forza del fuo-<lb/>co per la linea dritta, TO, arriuata alla boc-<lb/>ca, O, doue non hà più il ſoſtegno della cãna, <lb/>non è dubbio alcuno, che ſe non haueſſe la <lb/>grauità, che la tira cõtinuamente verſo il cen-<lb/>tro, caminarebbe per, O Q, vniformemente, <lb/>non conſiderato l’impedimento dell’ambien-<lb/>te; </s>
  <s xml:id="echoid-s1397" xml:space="preserve">ſia dunque il tempo, nel quale ſcorrereb-<lb/>be la, O Q, diuiſo in quattro parti eguali, co-<lb/>me à dire, in quattro battute dimuſica; </s>
  <s xml:id="echoid-s1398" xml:space="preserve">ſega-<lb/>ta dunque parimẽte, O Q, nelle quattro par-<lb/>ti eguali, O H, H M, M R, R Q, ciaſcuna di <lb/>eſſe parti ſaria trapaſſata dalla palla in vna <lb/>battuta, ma perche la grauità la tira verſo il <lb/>centro, pongaſi, che nel tempo, che ella ſcor-
<pb o="166" file="0186" n="186" rhead="Delle Settioni"/>
rerebbe per, O H, quella ſia potente à farla <lb/>abbaſſare verſo il centro, quant’è la, O B, qua-<lb/>le ſia patte della, O X, tirata à piombo dal pũ-<lb/>to, O, ſotto la, T Q; </s>
  <s xml:id="echoid-s1399" xml:space="preserve">ſimilmente per il punto, <lb/>H, paſſi la, H V, parallela à, O X, come anco <lb/>peripunti, M, R, Q, le, M℞, R Π, Q Y, pa-<lb/>rallele alla iſteſſa, O X, e nel paſſaggio della <lb/>palla per, H M, ſia l’abbaſſamento, quant’è, <lb/>B L, per, M R, quant’<unsure/>è, L P, e per, R Q, quãt’ <lb/>è, P X, e per i punti, B, L, P, X, ſi tirino le pa-<lb/>rallele alla, T Q, cioè, S F, E K, Z C, A Y, co-<lb/>me anco da i punti, T, D, G, N, preſi nella, T <lb/>O, ſuppoſta eguale ad, O Q, e che la diuida-<lb/>no pure in quattro parti vguali, ſi tirino le, T <lb/>A, D Δ, G, N I, parallele ad, O X, e ſiano <lb/>queſte con le, H V, M ℞, R Π, Q Y, ſegate dal-<lb/>le, A Y, Z C, E K, S F, ne i punti, A, Δ, Γ, I; </s>
  <s xml:id="echoid-s1400" xml:space="preserve">V, <lb/>℞, Π, Y; </s>
  <s xml:id="echoid-s1401" xml:space="preserve">Z, C; </s>
  <s xml:id="echoid-s1402" xml:space="preserve">E, K; </s>
  <s xml:id="echoid-s1403" xml:space="preserve">S, F: </s>
  <s xml:id="echoid-s1404" xml:space="preserve">Sarà dunque la palla <lb/>nel tempo, c’haurebbe ſcorſa la, O H, abbaſ-<lb/>ſataſi per la quãtità di, O B, ouero, H F, egua-<lb/>li, come lati oppoſti del parallelogramo, O F, <lb/>cioè in vece di eſſere nel punto, H, ſarà in, F, <lb/>così nel tempo della ſcorſa per, H M, ſarà ab-<lb/>baſſata in, K, per, M R, in, C, e per, R Q, in, <lb/>Y, eſſendo queſti abbaſſamẽti eguali alle, O B,
<pb o="167" file="0187" n="187" rhead="Coniche. Cap. XL."/>
O L, O P, O X, cioè accreſcendoſi ſecondo la <lb/>quantità delle, O B, B L, L P, P X, ma queſte <lb/>ſi aumẽtano ſecondo la ſerie de’numeri diſpa-<lb/>ri continuati dall’vnità; </s>
  <s xml:id="echoid-s1405" xml:space="preserve">adunque poſta, O B; <lb/></s>
  <s xml:id="echoid-s1406" xml:space="preserve">1. </s>
  <s xml:id="echoid-s1407" xml:space="preserve">B L, ſarà 3. </s>
  <s xml:id="echoid-s1408" xml:space="preserve">L P, 5. </s>
  <s xml:id="echoid-s1409" xml:space="preserve">P X, 7. </s>
  <s xml:id="echoid-s1410" xml:space="preserve">ouero, O L, ſarà <lb/>4. </s>
  <s xml:id="echoid-s1411" xml:space="preserve">O P, 9. </s>
  <s xml:id="echoid-s1412" xml:space="preserve">O X, 16. </s>
  <s xml:id="echoid-s1413" xml:space="preserve">ma così anco procedono i <lb/>quadrati delle, O H, O M, O R, O Q, ouero <lb/>delle, B F, L K, P C, X Y, che à quelle s’aggua-<lb/>gliano, come lati oppoſti de’parallelogrami, <lb/>O F, O K, O C, O Y, adunque eſſendo il qua-<lb/>drato di, B F, 1. </s>
  <s xml:id="echoid-s1414" xml:space="preserve">ſarà quello di, L K, 4. </s>
  <s xml:id="echoid-s1415" xml:space="preserve">di, P C, <lb/>9. </s>
  <s xml:id="echoid-s1416" xml:space="preserve">e di, X Y, 16. </s>
  <s xml:id="echoid-s1417" xml:space="preserve">e perciò ſarà il quadrato di, <lb/>X Y, al quadrato, P C, come, X O, ad, O P, &amp; </s>
  <s xml:id="echoid-s1418" xml:space="preserve"><lb/>il quadrato, P C, al quadrato, L K, come, P O, <lb/>ad, O L, e finalmente il quadrato, L K, al qua-<lb/>drato, B F, come, L O, ad, O B; </s>
  <s xml:id="echoid-s1419" xml:space="preserve">ma ſe noi de-<lb/>ſcriueremo la Semiparabola, O Y, ouero la <lb/>Parabola, che ſia, A O Y, qual paſſi peril pun-<lb/>to, O, ſua cima, e per li pũti, A, Y, anco i qua-<lb/>drati deli<unsure/>e intrapreſe frà la, O X, e la Parabo-<lb/>la, A O Y, ſaranno nella medeſima proportio-<lb/>ne, nella quale ſono le, O X, O P, O L, O B, <lb/>poiche queſta è la quarta proprietà della Pa-<lb/>rabola, dimoſtrata nel Cap. </s>
  <s xml:id="echoid-s1420" xml:space="preserve">12. </s>
  <s xml:id="echoid-s1421" xml:space="preserve">adunque i lati <lb/>di quei quadrati ſaranno congruenti à i lati,
<pb o="168" file="0188" n="188" rhead="Delle Settioni"/>
P C, L K, B F, (&amp; </s>
  <s xml:id="echoid-s1422" xml:space="preserve">à i lati, P Z, L E, B S, appli-<lb/>cãdo la dimoſtratione da queſt’altra banda) e <lb/>però i punti, F, K, C, ſono nella Parabola, A <lb/>O Y, come anco li, S, E, Z, cioè la palla ne i <lb/>punti, O, F, K, C, Y, ſarà ſempre nella Para-<lb/>bola, A O Y, eſſendo cima di quella il punto, <lb/>O, doue ſi ſpicca dal proiciẽte; </s>
  <s xml:id="echoid-s1423" xml:space="preserve">e l’@ſt@ſſo pro-<lb/>uaremo di tutti gli altri punti, ne’quali ella ſi <lb/>può ritrouare, ſubdiuidendo la, O H, con le <lb/>rimanenti in quante parti vguali ci ſarà biſo-<lb/>gno, &amp; </s>
  <s xml:id="echoid-s1424" xml:space="preserve">applicandoui l’iſteſſa dimoſtratione; <lb/></s>
  <s xml:id="echoid-s1425" xml:space="preserve">adunque egli è vero, quanto ſi è propoſto di <lb/>prouare.</s>
  <s xml:id="echoid-s1426" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1427" xml:space="preserve">Ma perche ſi vegga anco in figura il tiro e-<lb/>leuato, ouero abbaſſato, ſi è deſcritta la tãgen-<lb/>te, Φ Ω, nel punto, E; </s>
  <s xml:id="echoid-s1428" xml:space="preserve">ſe adunque il graue foſ-<lb/>ſe ſpinto per la retta, E Φ, ouero per la, E Ω, <lb/>eſſendo la ſeparatione nel punto, E, ſi proua-<lb/>ria nell’iſteſſo modo, che la interna grauità ri-<lb/>trahendolo continuamente dalla retta, E Φ, lo <lb/>mantenerebbe ſempre nella Parabola, E O Y, <lb/>ouero diſcoſtandolo da, E Ω, lo terrebbe nel-<lb/>la curua, A E, parte della Parabola, A O Y, e <lb/>s’intenderia in tal caſo il punto, E, per cima, <lb/>E Γ<unsure/>, per diametro, douendoſi tirare le ordina-
<pb o="169" file="0189" n="189" rhead="Coniche. Cap. XL."/>
tamente applicate all’, E Γ, parallele alla tan-<lb/>gente, Ω Φ, adattandouiſi la dimoſtratione <lb/>nell’iſteſſa maniera.</s>
  <s xml:id="echoid-s1429" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1430" xml:space="preserve">Egli è però vero, che eſſendo le parallele <lb/>ad, O X, ſtrade, per le quali s’intende ſcen-<lb/>dere il graue, finalmente egli andarebbe, ſe <lb/>non trouaſſe l’impedimento della terra, à col-<lb/>locarſi nel centro di quella; </s>
  <s xml:id="echoid-s1431" xml:space="preserve">la onde realmente <lb/>non ſono parallele, ma per le diſtanze de’no-<lb/>ſtri titi le abuſiamo, come parallele, eſſendo il <lb/>loro ſtringimẽto in ſi poco ſpatio, come inſen-<lb/>ſibile, e perciò ſi è detto, che deſcriuono vna <lb/>linea curua, inſenſibilmente differente dalla <lb/>Parabola. </s>
  <s xml:id="echoid-s1432" xml:space="preserve">Di maggior’importanza è bene <lb/>l’impedimento dell’aria, quãdo il tiro ſia lon-<lb/>ghiſſimo, che contraſta e con l’impulſo, e con <lb/>la grauità, ma in poco ſpatio, pẽſo, che non ſia <lb/>di molta cõſideratione, &amp; </s>
  <s xml:id="echoid-s1433" xml:space="preserve">il venir’all’eſſame di <lb/>queſto contraſto non è coſa ſi facile, ne che in <lb/>poche parole ſi poteſſe, credo, ſpiegare, perciò <lb/>ci contentaremo di queſto poco, per intender <lb/>le varie cõditioni, e nobiltà delle Settioni Co-<lb/>niche, hauendole anco il Keplero in ſopremo <lb/>grado nobilitate, mentre ci hà fatto vedere <lb/>con manifeſte ragioni ne’Cõmentarij di Mar-
<pb o="170" file="0190" n="190" rhead="Delle Settioni"/>
te, e nell’Epitome Copernicano, che le circo-<lb/>lationi de’ Pianeti intorno al Sole non ſono al-<lb/>trimente circolari, ma elittiche. </s>
  <s xml:id="echoid-s1434" xml:space="preserve">Ci baſterà <lb/>queſto adunque, cauãdo dalla ſudetta dottri-<lb/>na per noſtra vtilit<unsure/>à l’infraſcritto Corollario.</s>
  <s xml:id="echoid-s1435" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div81" type="section" level="1" n="81">
<head xml:id="echoid-head86" xml:space="preserve">Corollario.</head>
<p style="it">
  <s xml:id="echoid-s1436" xml:space="preserve">DI quì è manifeſto, che data la diſtanza del <lb/>tiro, come, O H, &amp; </s>
  <s xml:id="echoid-s1437" xml:space="preserve">eſperimentato il ca-<lb/>lare, che vien fatto in tal diſtanza, che <lb/>ſia, H F, alia medeſima eleuatione ſapremo gli abbaſ-<lb/>ſamenti, che ſi faranno dal ſegno in qualſiuoglia di-<lb/>stanza: </s>
  <s xml:id="echoid-s1438" xml:space="preserve">Come, per eſſempio, nel tiro, O M, doppio <lb/>dell’eſperimentato, O H, la ſceſa ſarà, M K, qua-<lb/>drupla di, H F, nella diſtanza, O R, ſarà, R C, <lb/>nell’ O Q, Q Y, &amp; </s>
  <s xml:id="echoid-s1439" xml:space="preserve">vniuerſalmente qual propor-<lb/>tione haur a il quadrato di, O H, tiro già prouato, al <lb/>quadrato di qual’altra distanza ſi voglia, tale l’ha-<lb/>urà la diſceſa, H F, à quella, che ſi farà in tal di-<lb/>stanza, alla medeſima eleuatione; </s>
  <s xml:id="echoid-s1440" xml:space="preserve">dal che impariamo <lb/>ancora, che i proietti non poſſono mai caminare per <lb/>dritta linea, ſe non moſſi per la perpẽdicolare all’Ori-<lb/>zonte, benche tal volta per la poca distanza questo <lb/>rieſchi inſenſibile.</s>
  <s xml:id="echoid-s1441" xml:space="preserve"/>
</p>
<pb o="171" file="0191" n="191" rhead="Coniche. Cap. XL."/>
<p style="it">
  <s xml:id="echoid-s1442" xml:space="preserve">Hora basteranno le ſudette coſe intarno alle vti-<lb/>lità, che potiamo cauare da queſte Settioni Coniche, <lb/>bauendone fatto come vna ricercata, e toccato leg-<lb/>giermente diuerſe materie, alle quali eſſendo applica-<lb/>te, fanno moſtra della ſua nobiltà, acciò da queſto po-<lb/>co argomentiamo, quali, e quanti deuino eſſer le loro <lb/>prerogatiue in queſto gran cãpo della Natura, e quan-<lb/>to à ſi gran Maeſtra deuano riuſcire artificioſe. </s>
  <s xml:id="echoid-s1443" xml:space="preserve">E ſe <lb/>noi, che ſolo ne vediamo la ſcorza, ſcopriamo nondi-<lb/>meno effetti così merauiglioſi, quali dobbiamo crede-<lb/>re ſian quelli, che con la ſua ſagaciſſima industria ne <lb/>deue ſaper ritrarre eſſa Natura, guidata dalla Sa-<lb/>pienza diuina, che nel profondo delle ſue più recondi-<lb/>te proprietà, &amp; </s>
  <s xml:id="echoid-s1444" xml:space="preserve">eccellenze le comprende? </s>
  <s xml:id="echoid-s1445" xml:space="preserve">E chi <lb/>meglio vuole intender questo, facci vn poco ri-<lb/>fl@ſſione à quello, che noi ſappiamo di Mecanica, poi <lb/>guardi alla struttura del corpo humano, che vedrà <lb/>nell’hauer preparato tanti organi, e tanti stromenti <lb/>da eſſercitar moti diuerſiſſimi, ſenza che l’vn l’altro <lb/>impediſchi, con ſi marauiglioſo artificio, quanto ella <lb/>ci auanzinell’intender la maniera del mouer peſi, co-<lb/>sì nel ſaper di Proſpettiua nell’occhio, del Suono nel-<lb/>l’orecchio, riuſcendo non meno ammirabile nelle coſe <lb/>piccoliſſime, che nelle grandiſſime. </s>
  <s xml:id="echoid-s1446" xml:space="preserve">Perciò ragioneuol-<lb/>mente ſtimaremo, che ella in mille, e mille effetti, tut-
<pb o="172" file="0192" n="192" rhead="Delle Settioni"/>
ti marauiglioſi, parimente ſi preuaglia di queſte Set-<lb/>tioni Coniche, mentre, non oſtante il noſtro poco ſa-<lb/>pere, ci rieſcono nulladimeno tanto douitioſe, e fecon-<lb/>de, quanto habbiamo di già potuto comprendere. </s>
  <s xml:id="echoid-s1447" xml:space="preserve">Re-<lb/>sta hora, che vediamo, come le medeſime ſi poſſin de-<lb/>ſcriuere.</s>
  <s xml:id="echoid-s1448" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div82" type="section" level="1" n="82">
<head xml:id="echoid-head87" style="it" xml:space="preserve">Come ſi deſcriuino le Settioni Coniche. <lb/>Cap. XLI.</head>
<p>
  <s xml:id="echoid-s1449" xml:space="preserve">BEnche apporti molto diletto <lb/>l’intendere le proprietà, e vir-<lb/>tù delle Settioni Coniche, ſi co-<lb/>me dalli antecedenti Capi hab-<lb/>biamo potuto almeno ſuperfi-<lb/>cialmente comprendere; </s>
  <s xml:id="echoid-s1450" xml:space="preserve">tuttauia non ci po-<lb/>triano arrecare le vtilità da noi accennate, ſe <lb/>anco non ſapeſſimo deſcriuerle, e farle in ma-<lb/>teria, per ridurle all’atto prattico, al che per <lb/>compimento di tal dottrina ſuppliranno li ſuſ-<lb/>ſeguenti Capitoli. </s>
  <s xml:id="echoid-s1451" xml:space="preserve">E concioſiacoſa che molti <lb/>habbino inſegnati diuerſi modi di deſcriuerle, <lb/>non addurrò però quà, ſe nõ quelli, che ſaran-<lb/>no ſtimati più facili, e più belli, che in parte <lb/>ancora, per quanto hò potuto in altri Auttori
<pb o="173" file="0193" n="193" rhead="Coniche. Cap. XLI."/>
comprendere, farãno nuoui; </s>
  <s xml:id="echoid-s1452" xml:space="preserve">e perche s’int<unsure/>en-<lb/>dano meglio, farà bene prima vedere così in <lb/>vniuerſale, come queſti modi particolari, che <lb/>ſono molti, ſi riduchino à’ſuoi modi generali: <lb/></s>
  <s xml:id="echoid-s1453" xml:space="preserve">Se adunque gli anderemo eſſaminando, troua-<lb/>remo quelli ridurſi prima, e principalmente à <lb/>due, il pr<unsure/>imo modo ſarà, quando noi cauaremo <lb/>tali Settioni dal Cono, quale potiamo chiama-<lb/>re, inuention ſolida; </s>
  <s xml:id="echoid-s1454" xml:space="preserve">il ſecondo, quãdo che noi <lb/>con qualche iſtrumento fondato ſopra alcuna <lb/>loro proprietà, le deſcriueremo nella ſuperfi-<lb/>cie piana, che potiamo chiamare inuention <lb/>piana: </s>
  <s xml:id="echoid-s1455" xml:space="preserve">Queſta poi, o ſi fà preciſamente deſcri-<lb/>uendo veramente dette Settioni, o ſi fà perap-<lb/>proſſimatione alla vera, cioè per punti conti-<lb/>nuati, per i quali poi tirando vna linea, che <lb/>appreſs’à poco ſi vadi accommodãdo alla fleſ-<lb/>ſuoſità di quei diſſegnati punti, ſi deſcriue, ſe <lb/>non preciſamente, almeno proſſimamente tal <lb/>Settione, ſi che inſenſibilmente ſia dalla vera <lb/>differente; </s>
  <s xml:id="echoid-s1456" xml:space="preserve">Queſti modi particolari adunque <lb/>ſi riducono primieramente à due modi gene-<lb/>rali, cioè all’inuention ſolida, &amp; </s>
  <s xml:id="echoid-s1457" xml:space="preserve">all’inuention <lb/>piana, e queſto ſecondo à due altri, cioè all’in-<lb/>uention piana vera, e all’inuention piana per
<pb o="174" file="0194" n="194" rhead="Delle Settioni"/>
punti continuati, che ſono in tutto tre modi <lb/>generali, che abbracciano tutti i modi parti-<lb/>colari da me quì poſti, ò da altri Auttori inſe-<lb/>gnati: </s>
  <s xml:id="echoid-s1458" xml:space="preserve">Hora veniamo à i modi particolari, che <lb/>ſi contengono ſotto queſti generali, quali ſe <lb/>non tutti, almeno in parte, conforme à quel, <lb/>che ſi è detto di ſopra, ſaranno quì da me re-<lb/>giſtrati.</s>
  <s xml:id="echoid-s1459" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div83" type="section" level="1" n="83">
<head xml:id="echoid-head88" style="it" xml:space="preserve">De i modi particolari di deſcriuere le Settioni Coni-<lb/>che, che s’aſpettano all’inuention ſolida. <lb/>Cap. XLII.</head>
<p>
  <s xml:id="echoid-s1460" xml:space="preserve">IL primo ſarà, quando noi fare-<lb/>mo fabricare al torno vn Cono <lb/>di legno, o di ſtagno, o d’altra <lb/>materia, che habbi conſiſtenza, <lb/>e che ſi poſſi non difficilmẽte ta-<lb/>gliare, e poi lo ſegaremo nel modo, che richie-<lb/>dela generatione della deſiderata Settione: <lb/></s>
  <s xml:id="echoid-s1461" xml:space="preserve">Come, pereſſempio, viſta la 2. </s>
  <s xml:id="echoid-s1462" xml:space="preserve">figura, ſe vorre-<lb/>mola Parabola, gli daremo il taglio, come ve-<lb/>diamo nel primo Cono, cioè in tal maniera, <lb/>che diſſegnato il triangolo, A B C, che paſſa <lb/>per l’aſſe, la commune ſettione del piano ſe-
<pb o="175" file="0195" n="195" rhead="Coniche. Cap. XLII."/>
gante, che è per produr nella ſuperficie del <lb/>Cono la Parabola, e di eſſo triangolo, ſia paral-<lb/>lela all’vn de’lati di eſſo triangolo, come ad, <lb/>A C; </s>
  <s xml:id="echoid-s1463" xml:space="preserve">ſia dunque fatto queſto taglio, ſi che ne <lb/>ſia venuta la linea, R O V, queſta dunque ſa-<lb/>rà Parabola, ſeruendoci per traſportarla poi in <lb/>piano del tronco, O B R V; </s>
  <s xml:id="echoid-s1464" xml:space="preserve">Nell’iſteſſo modo <lb/>faremo l’Iperbola, tagliando laſuperficie del <lb/>Cono al modo, che ſi vede nel 2. </s>
  <s xml:id="echoid-s1465" xml:space="preserve">Cono dell’ <lb/>iſteſſa figura; </s>
  <s xml:id="echoid-s1466" xml:space="preserve">e l’Eliſſi, nel modo, che ci mo-<lb/>ſtra il 3. </s>
  <s xml:id="echoid-s1467" xml:space="preserve">Cono. </s>
  <s xml:id="echoid-s1468" xml:space="preserve">Si poſſono poi delineare nella <lb/>ſuperficie del Cono, o con l’immergerlo in <lb/>qualche liquore, che tinga, facendo queſto, <lb/>conforme, che ſi diſſe nel Cap. </s>
  <s xml:id="echoid-s1469" xml:space="preserve">3. </s>
  <s xml:id="echoid-s1470" xml:space="preserve">quanto all’ <lb/>immerſion del bicchiero di forma conica nell’ <lb/>acqua, poiche l’eſtremo margine della tintura <lb/>ci moſtrerà, doue habbiamo à fare il taglio; <lb/></s>
  <s xml:id="echoid-s1471" xml:space="preserve">ouero ci preualeremo del lume del Sole, e d’vn <lb/>filo dritto, che col centro del Sole ſtia poſto in <lb/>quel piano, che è atto, con tagliar la ſuperfi-<lb/>cie conica à produr tal ſettione (ſarà poi atto, <lb/>quando la ſeghi con le conditioni dichiarate <lb/>nel Cap. </s>
  <s xml:id="echoid-s1472" xml:space="preserve">3.) </s>
  <s xml:id="echoid-s1473" xml:space="preserve">imperoche l’ombra di tal filo de-<lb/>lineata ſopra la ſuperficie conica, ſarà la deſi-<lb/>derata Settione, e ci moſtrerà, doue ſi haurà
<pb o="176" file="0196" n="196" rhead="Delle Settioni"/>
da far’il taglio. </s>
  <s xml:id="echoid-s1474" xml:space="preserve">Si può ancora, per non hauere <lb/>à far queſto ſegamento, acco mmodare vn’aſ-<lb/>ſicella ſottile frà due pezzi, per eſſempio, di le-<lb/>gno, attaccandola à quelli con pironcini, ela-<lb/>uorando poi al torno ogni coſa inſieme, che <lb/>ſe detta aſſicella ſarà poſta frà quei pezzi, e <lb/>quelli sù’l torno, conforme, che richiede la <lb/>produttione di tal Settione, ci verrà lauorata <lb/>l’aſſicella, conforme al noſtro biſogno, ſenz’ <lb/>hauere à far’il ſudetto taglio; </s>
  <s xml:id="echoid-s1475" xml:space="preserve">Si può ottener <lb/>queſto ancora in altri modi, ma baſtino per <lb/>maggior breuità li già dichiarati.</s>
  <s xml:id="echoid-s1476" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1477" xml:space="preserve">II ſecondo modo particolare ſarà, quando <lb/>non faremo fare il ſudetto Cono, ma perla ſu-<lb/>perficie conica ci preualeremo d’vno ſtile, che <lb/>col girar’intorno ad vn punto fiſſo, che s’in-<lb/>tẽde per cima del Cono imaginario, riuolgen-<lb/>doſi ancora per la circonferenza d’vn circolo, <lb/>diſſegnarà in vna ſuperficie piana, ſituata ſo-<lb/>pra il ſoggetto piano (in tal’eleuatione, che <lb/>poſſi, ſegando l’imaginato Cono, produrla) <lb/>la deſiderata Settione, e ciò con l’hauer liber-<lb/>tà non ſolo di girar’intorno alla cima del Co-<lb/>no, ma anco d’alzarſi, &amp; </s>
  <s xml:id="echoid-s1478" xml:space="preserve">abbaſſarſi ſopra l’in-<lb/>chinato piano, paſſando ſempre per la ſudetta
<pb o="177" file="0197" n="197" rhead="Coniche. Cap. XLII."/>
cima; </s>
  <s xml:id="echoid-s1479" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1480" xml:space="preserve">in queſto modo hà inuentato, e fabri-<lb/>cato l’iſtrumento da diſſegnare le Settioni <lb/>Coniche il Sig. </s>
  <s xml:id="echoid-s1481" xml:space="preserve">Alfonſo da Isè, huomo molto <lb/>verſato nelle Matematiche, &amp; </s>
  <s xml:id="echoid-s1482" xml:space="preserve">attiſſimo alle <lb/>operationi, il quale lo migliora poi in tal <lb/>maniera, che lo fà atto al deſcriuere qualſi-<lb/>uoglia Settione, che ſenſibilmẽte ſi diſtingua <lb/>dalla linea retta, in quel modo appunto, che <lb/>il Sig. </s>
  <s xml:id="echoid-s1483" xml:space="preserve">Guid’Vbaldo dal Monte diſſegna le <lb/>portioni di circonferenze de’circoli, benche <lb/>ſiano di ſemidiametro di grandezza notabile, <lb/>preualendoſi in queſto iſtrumento in vece <lb/>di eſſa circonferenza, per la quale douria <lb/>ſcorrere lo ſtile, ò lato del Cono, che deue <lb/>diſſegnar la Settione, preualendoſi, dico, d’v-<lb/>na ſquadra zoppa, ò mobile, aperta talmente, <lb/>che facci l’angolo della circonferenza, da lei <lb/>ſupplita; </s>
  <s xml:id="echoid-s1484" xml:space="preserve">nel congiungimento de i lati della <lb/>quale ſtà fiſſo il lato del Cono, à quell’inclina-<lb/>tione, che fà biſogno, tutto intiero, ouero vn <lb/>pezzo ſolo, ſecondo che vogliamo, dentro il <lb/>quale è la detta ſquadra, ſtà accomodato nel <lb/>debito ſito il piano, nel qual ſi hà da diſſegna-<lb/>re la Settione, che ſi vuole, sù per il quale ſcor-<lb/>re la punta d’vn’altro lato mobile del Cono,
<pb o="178" file="0198" n="198" rhead="Delle Settioni"/>
ma però ſempre aggiacente al lato fiſſo di eſſo <lb/>Cono, dalla cui pũta vien diſſegnata la Settio-<lb/>ne ſopra l’inchinato piano; </s>
  <s xml:id="echoid-s1485" xml:space="preserve">non lo ſpiego con <lb/>figura, parendomi, per non eſſer coſa mia, à <lb/>baſtanza hauerlo dichiarato così in aſtratto.</s>
  <s xml:id="echoid-s1486" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1487" xml:space="preserve">Ilterzo, &amp; </s>
  <s xml:id="echoid-s1488" xml:space="preserve">vltimo modo ſarà (benche ſia <lb/>ſimile à queſto ſecondo) quando in vece d’ac-<lb/>comodare il piano, in cui ſi hà da diſſegnare <lb/>la determinata Settione, al Cono, &amp; </s>
  <s xml:id="echoid-s1489" xml:space="preserve">al ſogget-<lb/>to piano, noi accomodaremo il Cono, &amp; </s>
  <s xml:id="echoid-s1490" xml:space="preserve">il <lb/>ſoggetto piano à quello, intẽdendo vn’imagi-<lb/>nario Cono talmente piegato ſopra il ſogget-<lb/>to piano (qual ſaria quello d’vna tauola) che <lb/>vẽga detto ſoggetto piano à eſſer ſituato tal-<lb/>mẽte in riſpetto di quel Cono, che ſia atto, ſe-<lb/>gando tal ſuperficie, à produrre tal Settione, <lb/>qual ſi deſidera (ſarà poi atto, ſe hauerà le con-<lb/>ditioni, in riſpetto del Cono, dichiarate nel <lb/>Cap. </s>
  <s xml:id="echoid-s1491" xml:space="preserve">3.) </s>
  <s xml:id="echoid-s1492" xml:space="preserve">e di queſto Cono imaginario non vi <lb/>è altro, che ſia reale, ſe nõ vno ſtile, che ſi moue <lb/>di due moti, cioè per la circõferenza d’vn circo <lb/>lo, ſopra vn pũto fiſſo, che s’intẽde per la cima <lb/>del detto Cono, e sù, e giù per la detta cima, <lb/>poiche eſſendo diſuguali le rette linee tirate <lb/>dalla cima del Cono à ciaſcuna delle dette
<pb o="179" file="0199" n="199" rhead="Coniche. Cap. XLII."/>
Settioni, fuor che al circolo non ſubcontra-<lb/>riamente generato, è di biſogno, che ſi ritiri <lb/>in sù, e in giù, paſſando ſempre per la cima, s’e-<lb/>gli hà da ſtar con la punta continuamente nel <lb/>ſoggetto piano, nel quale con tal modo diſſe-<lb/>gnarà la deſiderata Settione; </s>
  <s xml:id="echoid-s1493" xml:space="preserve">Vn tale iſtro-<lb/>mento poi hò viſto appreſſo li Molto RR. </s>
  <s xml:id="echoid-s1494" xml:space="preserve">PP. <lb/></s>
  <s xml:id="echoid-s1495" xml:space="preserve">Geſuiti, qual mi dicono eſſere inuentione, e <lb/>fabrica del P. </s>
  <s xml:id="echoid-s1496" xml:space="preserve">Scheiner dell’iſteſſa Cõpagnia.</s>
  <s xml:id="echoid-s1497" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1498" xml:space="preserve">Vi poſſono eſſere forſi altri modi particola-<lb/>ri ancora, attinenti all’inuention ſolida, ma <lb/>per mio giudicio credo, che ſaranno pochiſſi-<lb/>mo differenti dalli ſudetti; </s>
  <s xml:id="echoid-s1499" xml:space="preserve">e perciò baſteran-<lb/>no queſti per eſplicatione de’modi particola-<lb/>ri di queſta inuention ſolida. </s>
  <s xml:id="echoid-s1500" xml:space="preserve">Trapaſſaremo <lb/>dunque à gli altri, che s’aſpettano all’inuen-<lb/>tion piana, e prima à quelli, che appartengono <lb/>all’inuention piana vera.</s>
  <s xml:id="echoid-s1501" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div84" type="section" level="1" n="84">
<head xml:id="echoid-head89" style="it" xml:space="preserve">De’modi particolari di deſcriuere le Settioni <lb/>Coniche, che s’aſpettano all’inuention <lb/>piana vera. Cap. XLIII.</head>
<p>
  <s xml:id="echoid-s1502" xml:space="preserve">QVeſti modi gli potiamo diſtinguere in <lb/>due mẽbri principali, il primo de’qua-<lb/>li ſarà il modo di deſeriuerle con vn
<pb o="180" file="0200" n="200" rhead="Delle Settioni"/>
filo, il ſecondo poi ſarà difarle con iſtrumen-<lb/>ti ſodi, come con righe dilegno, ò di metallo: <lb/></s>
  <s xml:id="echoid-s1503" xml:space="preserve">Quanto al primo, ridurrò ſolamente à memo-<lb/>ria il modo di deſcriuere l’Eliſſi con vn filo, v-<lb/>ſato da tutti i prattici, cominciãdo da queſto, <lb/>per eſſer più trito, e noto à ciaſcheduno. </s>
  <s xml:id="echoid-s1504" xml:space="preserve">Sia <lb/>dunque nella 26. </s>
  <s xml:id="echoid-s1505" xml:space="preserve">figura vn filo di che lõghez-<lb/>za ſi voglia, e preſi in vn piano, come ſi voglia, <lb/>due punti diſtãti, per minor’interuallo, che nõ <lb/>è il filo, quali ſiano, O, E, ſi metta l’vn de’ca-<lb/>pi del filo in vno, comein, O, e l’altro capo <lb/>nel rimanente, cioè in, E, e ſia il filo, O C E, <lb/>dentro il quale ſia poſta la punta dello ſtile, <lb/>N C, che ſia, C, che poſſi ſcorrere liberamen-<lb/>te sù, e giù, per il filo, e per i punti, O, E, ſi <lb/>tirila retta, O E, di quà, e di là indiffinita-<lb/>mente prolongata; </s>
  <s xml:id="echoid-s1506" xml:space="preserve">ſcorra poi la punta, C, ſino <lb/>c’habbia dato vna volta attorno i due punti, <lb/>O, E, tenendo ſempre teſe le due parti del <lb/>filo, O C E, da lui ſeparate, ſi che ſiano drit-<lb/>te (nel che è difettoſa queſt’operatione, per <lb/>non ci poter noi mai di queſto aſſicurare, cioè, <lb/>che queſta tenſione ſia fatta con tal tempera-<lb/>mento, che queſte parti teſe hora non ſiano <lb/>più longhe, &amp; </s>
  <s xml:id="echoid-s1507" xml:space="preserve">hora più corte) &amp; </s>
  <s xml:id="echoid-s1508" xml:space="preserve">habbi deſcrit-
<pb o="181" file="0201" n="201" rhead="Coniche. Cap. XLIII."/>
to la linea, A C B F, è manifeſto per la ſecon-<lb/>da proprietà dell’Eliſſi, che fù dichiarata al <lb/>Cap. </s>
  <s xml:id="echoid-s1509" xml:space="preserve">18. </s>
  <s xml:id="echoid-s1510" xml:space="preserve">che queſta ſarà Eliſſi, e ſuoi fochi ſa-<lb/>rannoi punti, O, E, diametro primo, A B, e <lb/>ſecondo quella retta linea, che ſega, A B, nel <lb/>mezo perpendicolarmente terminata nell’E-<lb/>liſſi, A M B F, qual ſia, M F, che paſſi per, R, e <lb/>che diuida, A B, in parti vguali, qual poi ſi chia <lb/>ma centro dell’Eliſſi; </s>
  <s xml:id="echoid-s1511" xml:space="preserve">sì come, M F, ſi dice an-<lb/>co diametro minore, o maggiore, ſecondo che <lb/>farà minore, o maggiore del diametro, A B; <lb/></s>
  <s xml:id="echoid-s1512" xml:space="preserve">Di quì ſi fà manifeſto, che ſe noi voleſſimo far <lb/>l’Eliſſi, i fochi della quale haueſſero vna da-<lb/>ta diſtãza, come, O E, &amp; </s>
  <s xml:id="echoid-s1513" xml:space="preserve">anco vna data diſtan-<lb/>za da gli eſtremi, A B, egualmente lontani da, <lb/>O, E, nell’iſteſſa drittura, che prendendo vn <lb/>filo, e mettendo vno de’ſuoi eſtremi in, O, e <lb/>facendolo paſſare d’attorno alla punta dello <lb/>ftile, collocata in, A, e da quello traendolo ſi-<lb/>no ad, E, &amp; </s>
  <s xml:id="echoid-s1514" xml:space="preserve">iui religandolo, ſi che le parti, O <lb/>A, E A, raccomandate alla punta dello ſtile, <lb/>ſteſſero dritte, e facendola reuolutione, e de-<lb/>ſcrittione, come ſopra, ſi produrria la deſiata <lb/>Eliſſi, qual pur ſia la, A M B F, di doue ci ver-<lb/>rà determinato il ſecondo diametro, M F; </s>
  <s xml:id="echoid-s1515" xml:space="preserve">Ma
<pb o="182" file="0202" n="202" rhead="Delle Settioni"/>
ſe per il contrario voleſſimo farla d’vna deter-<lb/>minata longhezza, e larghezza, allhora non <lb/>poſſono ſupporſi i fochi, O, E, ma vengono <lb/>à determinarſi con tal ſuppoſitione; </s>
  <s xml:id="echoid-s1516" xml:space="preserve">e ciò ba-<lb/>ſti intorno al deſcriuere l’Eliſsi con vn filo.</s>
  <s xml:id="echoid-s1517" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div85" type="section" level="1" n="85">
<head xml:id="echoid-head90" style="it" xml:space="preserve">Come ſi deſcriua la Iperbola con vn filo, primo <lb/>modo della inuention piana vera. <lb/>Cap. XLIV.</head>
<p>
  <s xml:id="echoid-s1518" xml:space="preserve">LAſcio in vltimo la Parabola, poi-<lb/>che doppo le altre s’è imparato <lb/>à deſcriuerla con vn filo. </s>
  <s xml:id="echoid-s1519" xml:space="preserve">Siano <lb/>dunque nella vigeſimaſettima <lb/>figura dati due fochi della Iper <lb/>bola, da deſcriuerſi, A, E, volendo, che’l foco <lb/>interiore, che ſia, E, diſti dalla cima dell’Iper-<lb/>bola, per la data retta linea, E D, che prodot-<lb/>ta paſſi per l’altro foco, A, e ſia, A C, tolta e-<lb/>guale à, D E; </s>
  <s xml:id="echoid-s1520" xml:space="preserve">ſarà dunque, D C, lato trauerſo <lb/>dital’Iperbola, prendaſi hora vn filo, come, A <lb/>Z G H Z E, i cui capi ſi leghino alli punti, E, A, <lb/>poi ſia lo ſtile, F Z, la cui pũta poſta in, D, ten-<lb/>ghi il filo teſo, ſi che vna parte ſi ſtenda ſopra, <lb/>D A, l’altra ſopra, D E, &amp; </s>
  <s xml:id="echoid-s1521" xml:space="preserve">il reſto raddoppia-
<pb o="183" file="0203" n="203" rhead="Coniche. Cap. XLIV."/>
to paſſi per vn piccol foro, viciniſſimo alla ci-<lb/>ma dello ſtile, e tenendo quello in mano, e con <lb/>l’altra mano il raddoppiato filo bẽ teſo, ſi pre-<lb/>ma lo ſtile ſopra il piano, in cui ſi vuole diſſe-<lb/>gnare la lperbola, venendo da, D, verſo, G, in <lb/>tal maniera, che ſempre eſchino fuori dal pic-<lb/>ciol foro dello ſtile parti eguali di filo, co-<lb/>me ſi vede, eſſendo ſituatoin, Z, che così la <lb/>diſegnata linea ſarà Iperbola, qual ſia, D H, e <lb/>ciò perche la, A Z, ſupera la, Z E, in tutti i ſi-<lb/>ti, della quantità dellato traſuerſo, ò aſſe, D <lb/>C, facendoſi ſempre eguali addittioni, come <lb/>vuole la p. </s>
  <s xml:id="echoid-s1522" xml:space="preserve">51. </s>
  <s xml:id="echoid-s1523" xml:space="preserve">del 3. </s>
  <s xml:id="echoid-s1524" xml:space="preserve">di Apollonio. </s>
  <s xml:id="echoid-s1525" xml:space="preserve">Potiamo <lb/>ancora, prodotta, A Z, in, X, intendere, che, <lb/>A X, ſia vna riga, alla quale ſtia ſempre aggia-<lb/>cente vna parte del ſilo, che hora s’intenda eſ-<lb/>ſere, E Z X, cioè la parte, Z X, poiche mouẽ-<lb/>doſi detta riga intorno al centro, A, e lo ſtile <lb/>ſcorrendo per la longhezza di lei, doue il filo, <lb/>E Z X, legatoin, X, lo neceſſitarà verrà à de-<lb/>ſcriuere l’Iperbola, H D L, facẽdo nell’vno, e <lb/>nell’altro modo, anco dalla parte, L, eq̃ſta ſe-<lb/>cõda operatione ſi caua dalla ſeconda proprie <lb/>tà dell’Iperbola di ſopra dimoſtrata al C. </s>
  <s xml:id="echoid-s1526" xml:space="preserve">15.</s>
  <s xml:id="echoid-s1527" xml:space="preserve"/>
</p>
<pb o="184" file="0204" n="204" rhead="Delle Settioni"/>
</div>
<div xml:id="echoid-div86" type="section" level="1" n="86">
<head xml:id="echoid-head91" style="it" xml:space="preserve">Come ſi deſcriua la Parabola con vn filo; primo <lb/>modo della inuention piana vera. <lb/>Cap. XLV.</head>
<p>
  <s xml:id="echoid-s1528" xml:space="preserve">SIa nella 28. </s>
  <s xml:id="echoid-s1529" xml:space="preserve">figura il punto P, e <lb/>ſi habbi da deſcriuere con vn <lb/>filo vna Parabola, della quale <lb/>il detto punto, P, ſia foco, che <lb/>diſti dalla di lei cima, per la da-<lb/>ta retta, P A, che ſarà parte dell’aſſe di tal Pa-<lb/>rabola, qual ſia indiffinitamente prolongata <lb/>verſo, P, come in, C, per il qual punto, C, ſia <lb/>tirata la, B D, ad angolo retto ſopra, A C, pro-<lb/>longata indiffinitamente, come in, B D; </s>
  <s xml:id="echoid-s1530" xml:space="preserve">ſia <lb/>dunque per il punto, A, tirata la, H G, paral-<lb/>lelaà, B D, che perciò ſarà perpendicolare ad, <lb/>A C, indiffinitamente pur prodotta, come in, <lb/>H, G; </s>
  <s xml:id="echoid-s1531" xml:space="preserve">habbiſi poi vna ſquadra di legno, ò di <lb/>metallo, che ſia, M F E, il cui lato, M N, ſcor-<lb/>ra ſopra la retta, H G, &amp; </s>
  <s xml:id="echoid-s1532" xml:space="preserve">ſia poi vn filo legato <lb/>in, P, preciſamente longo quanto è la, P A C, <lb/>e nel principio del moto ſia il punto, N, dell’ <lb/>angolo della ſquadra collocato in, A, come <lb/>anco lo ſtile, R O, che hebbi la punta, O, in, <lb/>A, e l’altro capo del filo ſtia legato in, E, nel
<pb o="185" file="0205" n="205" rhead="Coniche. Cap. XLV."/>
qual ſito le due parti del filo ſeparate dalla <lb/>punta, O, dello ſtile, R O, ſtaranno diſteſe <lb/>ſopra le, A P, A C; </s>
  <s xml:id="echoid-s1533" xml:space="preserve">ſi moua poi la ſquadra, M <lb/>N E, verſo, G, mantenendo ſempre il lato, M <lb/>N, nella retta, H G, e nell’iſteſſo tempo ſi mo-<lb/>ua lo ſtile longo il lato, N E, mantenẽdo ſem-<lb/>pre il filo adherente al lato, N E, che così con <lb/>la ſua punta deſcriuerà la ſemiparabola, A D, <lb/>poſto, che termini in, D, e nell’iſteſſo modo ri-<lb/>uoltata la ſquadra, ſi deſcriua la ſemiparabola, <lb/>A B, che termini in, B, che così haueremo fat-<lb/>ta la Parabola, B A D, il cui aſſe ſarà, A C; </s>
  <s xml:id="echoid-s1534" xml:space="preserve">e <lb/>foco il punto, P, e cima il punto, A, &amp; </s>
  <s xml:id="echoid-s1535" xml:space="preserve">è mani-<lb/>feſto, che, B A D, ſarà Parabola, poiche eſſen-<lb/>do il filo ſempre il medeſimo, vengono ad eſ-<lb/>ſer’eguali le incidenti parallele all’aſſe, A C, <lb/>e rifleſſe al punto, P, tolta inſieme ciaſcuna <lb/>incidente, e ſua rifleſſa, eguali dico à qualſiuo-<lb/>glia incidente, e ſua rifleſſa, che è la ſeconda <lb/>proprietà della Parabola dimoſtrata al Cap. <lb/></s>
  <s xml:id="echoid-s1536" xml:space="preserve">10. </s>
  <s xml:id="echoid-s1537" xml:space="preserve">Ecco dunque compitamente deſcritte le <lb/>tre Settioni Coniche di Apollonio, non ſolo <lb/>l’Eliſſi con il filo, ma la Iperbola, e finalmente <lb/>anco la Parabola, della quale appunto dice il <lb/>Keplero nell’Aſtronomia Ottica al Capit. </s>
  <s xml:id="echoid-s1538" xml:space="preserve">4.</s>
  <s xml:id="echoid-s1539" xml:space="preserve">
<pb o="186" file="0206" n="206" rhead="Delle Settioni"/>
doppo hauer’accennato la deſcrittion dell’E-<lb/>liſſi, &amp; </s>
  <s xml:id="echoid-s1540" xml:space="preserve">Iperbola fatta col filo (Diu dolui, non <lb/>poſſe, ſic eti<unsure/>am Parabolem deſcribi. </s>
  <s xml:id="echoid-s1541" xml:space="preserve">Tande<unsure/>m analo-<lb/>gia monſtrauit, &amp; </s>
  <s xml:id="echoid-s1542" xml:space="preserve">Geometrica comprobat, non mul-<lb/>tò operoſius &amp; </s>
  <s xml:id="echoid-s1543" xml:space="preserve">hanc deſignare) doue non hauen-<lb/>do egli poſto la dimoſtratione, fece ch’io ap-<lb/>plicandoui incontraſſi queſta ragione, che quì <lb/>con le altre hò voluto regiſtrare.</s>
  <s xml:id="echoid-s1544" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1545" xml:space="preserve">Le Settioni oppoſte, poi paſſano ſotto il ca-<lb/>po dell’Iperbola, potendoſi con vn filo diſſe-<lb/>gnare, come l’Iperbola, facendone vna, e poi <lb/>l’altra, con l’iſteſſa diſtanza de’fochi. </s>
  <s xml:id="echoid-s1546" xml:space="preserve">E ben-<lb/>che finalmente io ſappi, che non potiamo così <lb/>aggiuſtatamente operare col filo, che ſiamo <lb/>ſicuri d’hauer diſſegnate le vere Settioni, nõ-<lb/>dimeno le hò meſſe ſotto il capo dell’inuẽtion <lb/>piana vera, poiche ſi deue intendere l’opera-<lb/>tione fatta con vn filo, che non patiſchi queſta <lb/>imperfettione, che del reſto ella è poi vera in-<lb/>uẽtione di tal Settione, non hauẽdo poi quel-<lb/>lo, che preſcriue tal’operatione, obligo di mo-<lb/>ſtrare, che ſi poſſi, o nõ ſi poſſi trouar’vn tal fi-<lb/>lo, o veramẽte dicaſi, ch’ella è inuẽtion vera, <lb/>ſe il filo nõ patirà tal’imperfettione, e nõ ve-<lb/>ra, ſe pur ſarà di tale imperfettione, ma ſolo,
<pb o="187" file="0207" n="207" rhead="Coniche. Cap. XLV."/>
che ſi auuicina alla vera, e s’intendi queſta <lb/>ſotto l’altro capo.</s>
  <s xml:id="echoid-s1547" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div87" type="section" level="1" n="87">
<head xml:id="echoid-head92" style="it" xml:space="preserve">Come ſi deſcriua la Parabola, mediante gl’iſtrumen-<lb/>tiſodi, composti diregoli, ch’è i<unsure/>lſecondo mo-<lb/>do dell’inuention piana vera. <lb/>Cap. XLVI.</head>
<p>
  <s xml:id="echoid-s1548" xml:space="preserve">QVeſto noi lo cõſeguiremo me-<lb/>diante due ſole ſquadre, acco-<lb/>modate inſieme, &amp; </s>
  <s xml:id="echoid-s1549" xml:space="preserve">adoperate <lb/>nel modo, che ſi dirà. </s>
  <s xml:id="echoid-s1550" xml:space="preserve">Sia <lb/>dunque da deſc<unsure/>riuerſi la Pa-<lb/>rabola, il cui foco diſti dalla <lb/>cima per la retta, A P, nella 29. </s>
  <s xml:id="echoid-s1551" xml:space="preserve">figura, per <lb/>eſſer dunque queſta la quarta parte del lato <lb/>retto di tal Parabola, ſapremo pur’il lato ret-<lb/>to di eſſa Parabola, qual ſia la, A E, poſta ad <lb/>angolo retto ſopra, A P, qual prodotta indif-<lb/>finitamente ver<unsure/>ſo, P, come in, M, intendiamo <lb/>douer’eſſer’aſſe della Parabola, che ſi hà da <lb/>deſcriuerſi, ſiano poi fabricate due ſquadre <lb/>di legno, ò di metallo, che ſiano, N L M, A I <lb/>K, e talmente poſte, che il lato d’vna di quel-<lb/>le, come, L M, ſi facci ſempre ſcorrere sù per
<pb o="188" file="0208" n="208" rhead="Delle Settioni"/>
la retta, A M, che perciò, N L, ſtarà ſempre <lb/>ad angolo retto ſopra, A M, dipoi ſi prenda, <lb/>L K, nella, L M, terminata in, K, che ſia egua-<lb/>le al lato retto, A E, &amp; </s>
  <s xml:id="echoid-s1552" xml:space="preserve">iui mettaſi vn pironci-<lb/>no, ò altra coſa, che coſtringa la gamba, I K, <lb/>della ſquadra, A I K, à paſſar sẽpre per il pun-<lb/>to, K, il quale ſi potrà mettere o di quà, o di <lb/>là o nel mezo della gamba, I K, facendoui vn <lb/>canaletto; </s>
  <s xml:id="echoid-s1553" xml:space="preserve">vn’altro parimente ſe ne metta nel <lb/>punto, A, che coſtringa il lato, A I, paſſar sẽ-<lb/>pre per il punto, A, e nel punto, I, ſi metta lo <lb/>ſtile, R I, che habbi la punta in, I, quale s’in-<lb/>tenda mouerſi sù, e giù per il lato, N, L, ſtan-<lb/>doli ſempre aggiacente, mẽtre anco ilati, I A, <lb/>I K, ſcorrerãno peri punti, A, K, e s’intenda <lb/>principiarſi la deſcrittione dal punto, A, nel <lb/>qual principio i tre punti, I, L, A, ſaranno vn <lb/>ſolo, poi mouẽdoſi la ſquadra, N L M, ſi che per <lb/>eſſempio ſi ſia coſtituita, doue hora ſtà, inten-<lb/>deremo, cheil pũto, I, ſia ſcorſo da, L, in, I, de-<lb/>ſcriuẽdo la curua, A I, mẽtre i lati, A I, I K, ſarã-<lb/>no ſcorſi per i pũti, A, K, mantenẽdoſi ſempre <lb/>aggiacenti à quelli, e così ſeguitaremo in tal <lb/>modo à deſcriuere la curua, I B; </s>
  <s xml:id="echoid-s1554" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1555" xml:space="preserve">è manifeſto, <lb/>chela, A I B, ſarà linea vera Parabolica, poiche
<pb o="189" file="0209" n="209" rhead="Coniche. Cap. XLVI."/>
nel triangolo rettangolo per eſſempio, A I K, <lb/>il quadrato della perpendicolare, I L, è vgua-<lb/>le al rettangolo ſotto, A L, &amp;</s>
  <s xml:id="echoid-s1556" xml:space="preserve">, L K, ouero, A E, <lb/>lato retto, al quale, L K, ſi tolſe vguale, cioè <lb/>il quadrato di, I L, ordinatamente applicata <lb/>all’aſſe, A M, è vguale al rettangolo ſotto, L <lb/>A, parte dell’aſſe tra lei, e la cima, A, e ſotto <lb/>il lato retto, A E, e così prouaremo accadere <lb/>in tutti gli altri ſiti delle due ſquadre, A I K, <lb/>N L M, adunque A B, è Parabola, la cui cima <lb/>è il punto, A, foco, P, &amp; </s>
  <s xml:id="echoid-s1557" xml:space="preserve">aſſe, A M, ſupponen-<lb/>do però d’hauer deſcritto l’altra parte, A C, il <lb/>che faremo nell’i<unsure/>ſteſſo modo: </s>
  <s xml:id="echoid-s1558" xml:space="preserve">Queſta maniera <lb/>poi ſi caua dalla quarta proprietà della Para-<lb/>bola, dimoſtrata al Cap. </s>
  <s xml:id="echoid-s1559" xml:space="preserve">12. </s>
  <s xml:id="echoid-s1560" xml:space="preserve">e la fig. </s>
  <s xml:id="echoid-s1561" xml:space="preserve">N I K A, <lb/>credo forſe ſia il Greco {λα}μβδα d’lſidoro Mile-<lb/>ſio, da lui inuentato per deſcriuer la Parabola.</s>
  <s xml:id="echoid-s1562" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div88" type="section" level="1" n="88">
<head xml:id="echoid-head93" style="it" xml:space="preserve">Come ſi deſcriua la Iperbola con le righe, ſecondo mo-<lb/>do dell’inuention piana vera. Cap. XLVII.</head>
<p>
  <s xml:id="echoid-s1563" xml:space="preserve">SIa da deſcriuerſi la Ipeibola, il cui lato <lb/>retto nella trigeſima figura ſia, C B, <lb/>e lato tr auerſo, B A, perpendicolare <lb/>à, B C, prodotto verſo, B, indiffinitamente, <lb/>come in, Z, e c osì, A C, verſo, C, indiffinita-
<pb o="190" file="0210" n="210" rhead="Delle Settioni"/>
mente, come in, X, intendaſi poi la retta, D <lb/>E, mobile in sù, e in giù ſempre perpendico-<lb/>larmente à, B Z, e ſia la, D F, che contenga <lb/>vn mezo retto con, D E, ſtando ſempre il pũ-<lb/>to, D, nell’interſegatione delle due, D E, A <lb/>X, quale ſegarà, B Z, in diuerſi punti, eſſendo <lb/>coſtituito l’iſtromento in diuerſi ſiti, s’intẽda <lb/>però hora in vn determinato ſito, e ſeghi la, D <lb/>F, eſſa, B Z, in, F, ſia poi vna ſquadra, B E F, <lb/>come nella Parabola fù la, A I K, il cui punto, <lb/>E, dell’angolo retto (che s’intenda per la pun-<lb/>ta d’vno ſtile) ſcorra sù, e giù per la retta, G <lb/>E, ſtando ſempre in quella, e fra tantoi lati di <lb/>lei paſſino ſempre per i punti, B, F, è manife-<lb/>ſto, che principiandoſi i<unsure/>l moto dal punto, B, <lb/>doue ſaranno vniti i tre punti, G, B, E, ſi par-<lb/>tirà da, B, il punto, G, ſcorrẽdo ſopra la, A Z, <lb/>e la, E D, conducendo ſeco la retta, D F, che <lb/>parimente porterà il punto, F, sù per la retta, <lb/>B Z, e fra tanto il punto, E, ſcorrerà sù per, G <lb/>E, da, G, in, E, perſeuerãdo i due lati, E B, E F, <lb/>di paſſar ſempre per i punti, B F; </s>
  <s xml:id="echoid-s1564" xml:space="preserve">habbia dun-<lb/>que la punta dello ſtile, E, in tal moto deſcrit-<lb/>ta la curua, B E, dico, che queſta ſarà Iperbo-<lb/>lica, poiche il quadrato, G E, è vguale al ret-
<pb o="191" file="0211" n="211" rhead="Coniche. Cap. XLVII."/>
tangolo, B G F, cioè, B G D, che eccede il ret-<lb/>tãgolo ſotto, C B, lato retto, e ſotto, B G, par-<lb/>te dell’aſſe fra’l pũto, B, e<unsure/>l’ordinatamente ap-<lb/>plicata, G E, d’vn rettangolo ſimile al rettan-<lb/>golo contenuto ſotto il lato retto, C B, e traſ-<lb/>uerſo, B A, e così moſtraremo accadere ne gli <lb/>altri ſiti dell’iſtrumento nell’iſteſſo modo; </s>
  <s xml:id="echoid-s1565" xml:space="preserve">a-<lb/>dunque, B E, è Iperbolica, e così anco deſcri-<lb/>ueremo quel, che manca dalla parte verſo, D, <lb/>la onde ſi haurà l’intiera Iperbola, il cui dia-<lb/>metro ſarà, B Z, cima il punto, B, lato retto, <lb/>C B, e traſuerſo, B A, già ſuppoſti, il che bi-<lb/>ſognaua fare: </s>
  <s xml:id="echoid-s1566" xml:space="preserve">In vece poi delle rette linee da <lb/>noi diſegnate per minor briga, e confuſione, <lb/>intenderemo zante righe congionte inſieme, <lb/>come richiede la ſtabilità dell’iſtrumento, e la <lb/>libertà del mouerſi delle parti di quello, e que-<lb/>ſta maniera ſi caua dal Cap. </s>
  <s xml:id="echoid-s1567" xml:space="preserve">16.</s>
  <s xml:id="echoid-s1568" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div89" type="section" level="1" n="89">
<head xml:id="echoid-head94" style="it" xml:space="preserve">Come ſi deſcriua l’Eliſsi con le righe, ſecondo modo <lb/>dell’inuention piana vera. Cap. XLVIII.</head>
<p>
  <s xml:id="echoid-s1569" xml:space="preserve">SIa finalmente da deſcriuerſi l’Eliſs<unsure/>i, <lb/>dato nella 31. </s>
  <s xml:id="echoid-s1570" xml:space="preserve">figura il lato retto, C <lb/>B, e trauerſo, B A, che ſtiano ad an-
<pb o="192" file="0212" n="212" rhead="Delle Settioni"/>
golo retto: </s>
  <s xml:id="echoid-s1571" xml:space="preserve">eſſendo pur dũque, D E, che ſcor-<lb/>ra sù, e giù per la, B A, perpendicolarmente <lb/>à quella, giũta, A C, quella porti sù, e giù per <lb/>C A, il punto, D, con la retta, D F, che ſtia <lb/>ad angolo ſemiretto ſopra, D E, ſegando la, <lb/>B A, come in, F, ſia poi la ſquadra, B E F, il <lb/>cui punto, E, dell’angolo retto ſcorra sù per, <lb/>G E, e fra tanto i lati, E B, E F, paſſino ſem-<lb/>pre per i pũti, B, F, e ſi principij il moto in, B, <lb/>ouero in, A, e ſia l’iſtrumento vna volta nel ſi-<lb/>to, che ſi vede, e per il punto, E, s’intenda la <lb/>punta d’vno ſtile, che deſcriua la curua, B E <lb/>A; </s>
  <s xml:id="echoid-s1572" xml:space="preserve">dico, che queſta ſarà Eliſſi, poiche il qua-<lb/>drato di, G E, è vguale al rettangolo, B G F, <lb/>cioè, B G D, per eſſer, G F, G D, eguali, che <lb/>riſguardano gli angoli ſemiretti, G D F, G F <lb/>D, ma il rettangolo, B G D, i<unsure/>nãca dal rettan-<lb/>golo, C B G, ſotto tutto il lato retto, ela tron-<lb/>cata via dell’aſſe per la, G E, che è, G B, man-<lb/>ca, dico, d’vn rettangolo ſimile al rettangolo <lb/>ſotto ambedue i lati, C B, retto, &amp;</s>
  <s xml:id="echoid-s1573" xml:space="preserve">, B A, traſ-<lb/>uerſo; </s>
  <s xml:id="echoid-s1574" xml:space="preserve">adunque il punto, E, è nello Eliſsi, di <lb/>cui ſon lati, C B, B A, così prouaremo eſſerui <lb/>gli altri punti della curua, B E A; </s>
  <s xml:id="echoid-s1575" xml:space="preserve">adunque <lb/>queſta è Eliſſi, o ſemieliſſi; </s>
  <s xml:id="echoid-s1576" xml:space="preserve">nell’iſteſſo modo
<pb o="193" file="0213" n="213" rhead="Coniche. Cap. XLVIII."/>
deſcriueremo l’altra parte, adoprando l’i<unsure/>ſtru-<lb/>mento, che dourà eſſer compoſto di righe, in <lb/>vece di linee, dall’altra bãda, e così haueremo <lb/>l’intiero Eliſsi, di cui ſaran lato retto, C B, e <lb/>traſuerſo, B A, comeſi preteſe di fare: </s>
  <s xml:id="echoid-s1577" xml:space="preserve">Ma va-<lb/>glia à dire il vero, che per iſtrumenti di righe <lb/>non credo ſi poſsi migliorare di quello, che fù <lb/>inuentato dal Sig. </s>
  <s xml:id="echoid-s1578" xml:space="preserve">Guid’Vbaldo dal Monte, <lb/>huomo veramente intendentiſſimo delle Ma-<lb/>tematiche, ch’accoppiò inſieme il natiuo ſplẽ-<lb/>dore con il bel lume di sì alte dottrine, il qua-<lb/>le iſtrumento fù da lui dichiarato, &amp; </s>
  <s xml:id="echoid-s1579" xml:space="preserve">inſegna-<lb/>tane la fabrica, nel fine dell’Opera ſua de’Pia-<lb/>nisferij, come ciaſcun’à ſuo commodo potrà <lb/>vedere.</s>
  <s xml:id="echoid-s1580" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1581" xml:space="preserve">Le Settioni oppoſte poi ſi deſcriuerãno co-<lb/>me due Iperbole, che hanno commune illato <lb/>traſuerſo, &amp; </s>
  <s xml:id="echoid-s1582" xml:space="preserve">eguali i lati retti, per la 14. </s>
  <s xml:id="echoid-s1583" xml:space="preserve">del <lb/>primo de’Conici.</s>
  <s xml:id="echoid-s1584" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1585" xml:space="preserve">Vna ſimil maniera di deſcriuere dette Set-<lb/>tioni Coniche con le righe, mi fù moſtrata <lb/>parecchi anni ſono dal Sig. </s>
  <s xml:id="echoid-s1586" xml:space="preserve">Mutio Oddi da <lb/>Vrbino, hora Ingegnero della Sereniſsima <lb/>Republica di Lucca, perſona conſumata ne’ <lb/>ſtudi di Matematica, e molto intelligente si<unsure/>
<pb o="194" file="0214" n="214" rhead="Delle Settioni"/>
della Teorica, come della Prattica ancora. <lb/></s>
  <s xml:id="echoid-s1587" xml:space="preserve">Egli è però vero, ch’eſſendomi ſuanito, per la <lb/>longhezza del tempo, dalla memoria, quale <lb/>veramente foſſe il modo, ricordandomi ſolo, <lb/>che v’entrauano le ſquadre, con occaſione di <lb/>hauere à inſegnare la loro deſcrittione per via <lb/>di righe, mi meſsi à penſarui, e mi ſouuenne <lb/>queſta maniera, che hò ſpiegato di ſopra, qua-<lb/>le, quando s’abbatti con il modo del ſudetto <lb/>Autore (il che ſarà manifeſto dal libro, che il <lb/>medeſimo mi accenna voler ſtã pare in breue, <lb/>con il detto modo) dourà darſi la lode al ſuo <lb/>primo inuentore. </s>
  <s xml:id="echoid-s1588" xml:space="preserve">E que ſto hò voluto dire, <lb/>non mi parẽdo ben fatto il veſtirmi delle pẽ-<lb/>ne d’altri; </s>
  <s xml:id="echoid-s1589" xml:space="preserve">che perciò, ſe ben hò raccolto quà <lb/>alcuni di queſti modi, che ſono d’altri Autori, <lb/>acciò chi leggerà queſto mio Trattato, ne <lb/>habbia di dinerſe ſorti, per appigliarſi à qual <lb/>più li piacerà; </s>
  <s xml:id="echoid-s1590" xml:space="preserve">nõ tralaſcio tuttauia di nomi-<lb/>nare, come mi pare il douere, ilor proprij Au-<lb/>tori. </s>
  <s xml:id="echoid-s1591" xml:space="preserve">Mi ſcriue poi il medeſimo vltimamente, <lb/>ch’anch’egli deſcriue tutte tre le Settioni con <lb/>il filo, come pur hò inſegnato di ſopra, con <lb/>accompagnarui la propria ragione, che per la <lb/>Parabola, e l’Iperbola non hò ancora viſto ap-
<pb o="195" file="0215" n="215" rhead="Coniche. Cap. XLVIII."/>
preſſo di altri.</s>
  <s xml:id="echoid-s1592" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1593" xml:space="preserve">Diqueſti modi poi ci contentaremo quãto <lb/>all’inuẽtion piana vera, per non regiſtrar quà <lb/>tutto quello, che han detto gli altri, laſcian-<lb/>do all’induſtria dell’arteſice la coſtruttione <lb/>de’ſudetti iſtrumenti, acciò rieſchino più age-<lb/>uoli, e più facili da maneggiare, per non vo-<lb/>ler con troppo pregiudicio della breuità, an-<lb/>dar ſminuzzando ogni minima coſa, che dall’<unsure/> <lb/>induſtrioſo Operario può, vſandoui qualche <lb/>poco di diligenza, con facilità eſſer condotta <lb/>à perfettione, e però di queſti modi ſia detto <lb/>à baſtanza, facendo paſſaggio à quelli della <lb/>inuention piana, fatta per continuati punti.</s>
  <s xml:id="echoid-s1594" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div90" type="section" level="1" n="90">
<head xml:id="echoid-head95" style="it" xml:space="preserve">Dei modi partico<unsure/>lari di deſcriuere le Settioni Coni-<lb/>che, appartenenti all’Inuention piana per i <lb/>punti continuati. Cap. XLIX.</head>
<p>
  <s xml:id="echoid-s1595" xml:space="preserve">POtrei addur quà le varie inuen-<lb/>tioni di diuerſi Autori, come di <lb/>Orontio Fineo, di Marin Getal-<lb/>do, e d’altri, che ſi ſono ingegna-<lb/>ti di deſcriuerle per i punti con-<lb/>tinuati, ma perche non vorrei ecceder’in lon-
<pb o="196" file="0216" n="216" rhead="Delle Settioni"/>
ghezza; </s>
  <s xml:id="echoid-s1596" xml:space="preserve">perciò ſpiegarò ſolo quella vniuer-<lb/>ſal ragione, ſopra la quale ſtanno fondati que-<lb/>ſti vltimi modi di diſſegnare le dette Settioni <lb/>per continuati punti, con aggiunta di qualche <lb/>coſa del mio. </s>
  <s xml:id="echoid-s1597" xml:space="preserve">Dico adunque quaſi tutti quei <lb/>modi, o almeno i principali eſſer fondati ſo-<lb/>pra le tre vltime proprietà di dette Settioni <lb/>Coniche, che da me ſono ſtate ſpiegate ne i <lb/>Capitoli 12. </s>
  <s xml:id="echoid-s1598" xml:space="preserve">16. </s>
  <s xml:id="echoid-s1599" xml:space="preserve">20. </s>
  <s xml:id="echoid-s1600" xml:space="preserve">vediamo perciò prima <lb/>queſto intorno la Parabola. </s>
  <s xml:id="echoid-s1601" xml:space="preserve">Sia dunque dato <lb/>il lato retto, A Z, nella trigeſimaſeconda ſi-<lb/>gura, prolongato indiffinitamente verſo, A, <lb/>come in, X; </s>
  <s xml:id="echoid-s1602" xml:space="preserve">volendo adunque deſcriuere vna <lb/>Parabola, il cui aſſe ſia, X A, e lato retto, A Z, <lb/>tiraremo dalla eſtremità la, X G, ad angolo <lb/>retto (ſe ben verrà deſcritta anco, che non ſia <lb/>ad angolo retto, il che però ſuppõ@o per mag-<lb/>gior chiarezza) e poi prenderemo molti pun-<lb/>ti in, AX, più ſpeſsi, che ſia poſſibile, però quà <lb/>per eſſempio non notaremo, ſe non li tre pũti, <lb/>M, L, H, da’quali tiraremo dalla parte mede-<lb/>ſima le, H B, L D, M F, parallele ad, X G, in-<lb/>diffinitamente prolõgate, dipoi deſcriueremo <lb/>dall’altra parte i ſemicircoli, Z C H, Z O L, <lb/>Z I M, Z V X, tirando da, A, la retta, A V, per-
<pb o="197" file="0217" n="217" rhead="Coniche. Cap. XLIX."/>
pendicolare ſopra, A X, indiſfinitamente <lb/>prodotta, che ſeghi le circonferenze de’ſudet-<lb/>ti ſemicircoli ne i punti, C, O, I, V, prendere-<lb/>mo poi in, H B, la, H B, eguale ad, AC, <lb/>in, L D, la, L D, eguale ad, A O, in M F, la, <lb/>M F, eguale ad, A I, e finalmente la, X G, e-<lb/>guale ad, A V; </s>
  <s xml:id="echoid-s1603" xml:space="preserve">dico adunque, chei punti, B, <lb/>D, F, G, ſarãno nella Parabola, il cui lato ret-<lb/>to è, A Z, poiche il quadrato, X G, cioè, V A, <lb/>è vguale al rettangolo ſotto, X A, compreſa <lb/>tra, X G, &amp; </s>
  <s xml:id="echoid-s1604" xml:space="preserve">il punto eſtremo della retta, X A, <lb/>e ſotto il lato retto, A Z, per eſſere, Z V X, ſe-<lb/>micircolo, &amp;</s>
  <s xml:id="echoid-s1605" xml:space="preserve">, A V, perpendicolare ſopra il <lb/>diametro, Z X, e così il quadrato, M F, è vgua-<lb/>le al rettangolo, M A Z, &amp; </s>
  <s xml:id="echoid-s1606" xml:space="preserve">il quadrato, L D, <lb/>al rettangolo, L A Z, &amp; </s>
  <s xml:id="echoid-s1607" xml:space="preserve">il quadrato, H B, al <lb/>rettangolo, H A Z, e però i punti, G, F, D, B, <lb/>ſaranno nella Parabola, il cui lato retto ſarà, <lb/>A Z; </s>
  <s xml:id="echoid-s1608" xml:space="preserve">trouando dunque tali punti, che ſian vi-<lb/>cini, e facendo paſſare vna curua per quelli, <lb/>decorſa dalla punta d’vno ſtile, che paſſi per <lb/>i medeſimi punti, verrà proſſimamẽte deſcrit-<lb/>ta da quello la ſemiparabola da queſta banda, <lb/>e nell’iſteſſo modo deſcriueremo la rimanen-<lb/>te dall’altra, &amp; </s>
  <s xml:id="echoid-s1609" xml:space="preserve">hauremo l’intiera Parabola,
<pb o="198" file="0218" n="218" rhead="Delle Settioni"/>
deſcritta per i punti continuati, il cui lato ret-<lb/>to ſarà, A Z, che prima ſi propoſe, e queſto mo-<lb/>do è cauato dalla 4. </s>
  <s xml:id="echoid-s1610" xml:space="preserve">proprietà della Parabola <lb/>al Capit. </s>
  <s xml:id="echoid-s1611" xml:space="preserve">12.</s>
  <s xml:id="echoid-s1612" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div91" type="section" level="1" n="91">
<head xml:id="echoid-head96" style="it" xml:space="preserve">Come ſi deſcriua l’Iperbola, &amp; Eliſsi per ì <lb/>punticontinuati. Cap. L.</head>
<p>
  <s xml:id="echoid-s1613" xml:space="preserve">SIa nella trigeſimaterza figura, <lb/>B A, lato traſuerſo d’vn’Iper-<lb/>bola, &amp; </s>
  <s xml:id="echoid-s1614" xml:space="preserve">Eliſſi, da deſcriuerſi, <lb/>come ſopra, &amp;</s>
  <s xml:id="echoid-s1615" xml:space="preserve">, A F, à quello <lb/>perpendicolare ſia lor cõmun <lb/>lato retto, e giunti i punti, B, <lb/>F, ſia la, B, F, come anco la, B A, indiffinita-<lb/>mente prodotta verſo, F, A, come in, D, C; <lb/></s>
  <s xml:id="echoid-s1616" xml:space="preserve">per far queſto dunque, prẽderemo molti pun-<lb/>ti, e ſpeſſi nelle, B A, A C, ma noi per eſſempio <lb/>ne notaremo due ſoli, e ſupporremo di voler <lb/>fare, che il punto, A, non ſolo ſia cima dell’I-<lb/>perbola, ma anco della Eliſſi da deſcriuerſi. </s>
  <s xml:id="echoid-s1617" xml:space="preserve"><lb/>Siano dunque li due punti preſi in, B A, eſſi, <lb/>M, N, &amp; </s>
  <s xml:id="echoid-s1618" xml:space="preserve">in, A C, eſſi, R, C, per i quali ſi pro-<lb/>longhino indiffinitamente parallele ad, F A, <lb/>di quà, e di là le, H ℞, G &amp;</s>
  <s xml:id="echoid-s1619" xml:space="preserve">, E X, D Y, che ſe-
<pb o="199" file="0219" n="219" rhead="Coniche. Cap. L."/>
ghino la, B D, ne i punti, H, G, E, D, ſi pren-<lb/>da poi in, H ℞, la, M ℞, eguale ad, M A, in, <lb/>G &amp;</s>
  <s xml:id="echoid-s1620" xml:space="preserve">, la, N &amp;</s>
  <s xml:id="echoid-s1621" xml:space="preserve">, eguale ad, N A, in, E X, la, R X, <lb/>eguale ad, R A, &amp; </s>
  <s xml:id="echoid-s1622" xml:space="preserve">in, D Y, la, C Y, eguale ad, <lb/>A C, e ſopra le, H ℞, G &amp;</s>
  <s xml:id="echoid-s1623" xml:space="preserve">, E X, D Y, s’inten-<lb/>dino deſcritti ſemicircoli, che ſeghino la, B C, <lb/>ne i pũti, K, Z, T, V, cioè il ſemicircolo ſopra, <lb/>H ℞, ſeghi la, B A, in, K, quel ſopra, G &amp;</s>
  <s xml:id="echoid-s1624" xml:space="preserve">, la <lb/>iſteſſa, B A, in, Z, quel ſopra, E X, la, A C, in, <lb/>T, e finalmente quel ſopra, D Y, la, A C, pur <lb/>in, V, e prendaſi la, M K, in, M ℞, cioè, M O, <lb/>eguale ad, M K, che termini in, B A, e così, <lb/>N P, eguale ad, N Z; </s>
  <s xml:id="echoid-s1625" xml:space="preserve">R Q, eguale ad, R T, &amp; </s>
  <s xml:id="echoid-s1626" xml:space="preserve"><lb/>C S, eguale à, C V; </s>
  <s xml:id="echoid-s1627" xml:space="preserve">dico dunque, che i punti, <lb/>O, P, ſono nell Eliſſi, di cui è lato retto, F A, <lb/>e traſuerſo, A B, &amp;</s>
  <s xml:id="echoid-s1628" xml:space="preserve">, Q, S, nell’Iperbola, che <lb/>hà i medeſimi lati retto, e traſuerſo, imperoche <lb/>il quadrato, M K, cioè, M, O, è vguale al ret-<lb/>tangolo, H M ℞, cioè, H M A, per eſſer, M ℞, <lb/>eguale ad, M A, come, M O, ad, M K, cioè è <lb/>eguale al rettangolo ſotto, M A, &amp; </s>
  <s xml:id="echoid-s1629" xml:space="preserve">H M, defi-<lb/>ciente dal rettangolo ſotto, M A, A F, di vn <lb/>rettangolo ſimile al contenuto ſotto, B A, A F; <lb/></s>
  <s xml:id="echoid-s1630" xml:space="preserve">adunque per la quarta proprietà, ſarà il pun-<lb/>to, O, nell’Eliſſi, di cui ſon lati, F A, A B, così
<pb o="200" file="0220" n="220" rhead="Delle Settioni"/>
prouaremo eſſerui il punto, P, &amp; </s>
  <s xml:id="echoid-s1631" xml:space="preserve">ogn’altro <lb/>punto in tal modo trouato; </s>
  <s xml:id="echoid-s1632" xml:space="preserve">per quelli adũque <lb/>diſſegnata la curua, come ſopra, che ſia, B O <lb/>P A, diremo queſta eſſer proſſimamente ſemi-<lb/>eliſſi, di cui ſon lati, F A, A B, così faremo la <lb/>rimanente dall’altra parte, &amp; </s>
  <s xml:id="echoid-s1633" xml:space="preserve">hauremo deſcrit <lb/>to l’Eliſſi per i punti continuati, di cui ſaran-<lb/>no lati, F A, retto, &amp;</s>
  <s xml:id="echoid-s1634" xml:space="preserve">, A B, traſuerſo.</s>
  <s xml:id="echoid-s1635" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1636" xml:space="preserve">Nell’altra figura poi, il quadrato, R Q, per <lb/>eſſer’eguale al quadrato, T R, ſarà anco egua-<lb/>le al rettangolo ſotto, E R X, cioè, E R A, ec-<lb/>cedente il rettangolo, F A R, di vn rettango-<lb/>lo ſimile al contenuto ſotto, B A, lato traſuer-<lb/>ſo, &amp;</s>
  <s xml:id="echoid-s1637" xml:space="preserve">, A F, lato retto; </s>
  <s xml:id="echoid-s1638" xml:space="preserve">adunque il punto, Q, ſa-<lb/>rà nell’Iperbola, di cui ſono lato retto, F A, e <lb/>traſuerſo, A B; </s>
  <s xml:id="echoid-s1639" xml:space="preserve">così moſtraremo eſſerui il pun-<lb/>to, S, &amp; </s>
  <s xml:id="echoid-s1640" xml:space="preserve">ogn’altro in tal modo ritrouato, de-<lb/>ſcriuendo adunque, come ſopra, la curua, A Q <lb/>S, ſarà queſta ſemiIperbola, e nell’iſteſſo mo-<lb/>do, fatta dall’altra parte la rimanente, haure-<lb/>mo l’intiera Iperbola deſcritta per i punti con-<lb/>tinuati, di cui ſarãno lato retto, F A, e traſuer-<lb/>ſo, A B, &amp; </s>
  <s xml:id="echoid-s1641" xml:space="preserve">il punto, A, cõmune cima dell’Iper-<lb/>bola, &amp; </s>
  <s xml:id="echoid-s1642" xml:space="preserve">Eliſſi, e l’iſteſſo ſi farà, quando le, B A, <lb/>A C, non foſſero aſſi, ma ſolo diametri: </s>
  <s xml:id="echoid-s1643" xml:space="preserve">Nella
<pb o="201" file="0221" n="221" rhead="Coniche. Cap. L."/>
medeſima maniera poi ſi potranno deſcriuere <lb/>le oppoſte Settioni, come due Iperbole, ſe-<lb/>condo quello, che ſi è detto anco di ſopra.</s>
  <s xml:id="echoid-s1644" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div92" type="section" level="1" n="92">
<head xml:id="echoid-head97" style="it" xml:space="preserve">D’vn’altra maniera molto facile, &amp; eſpediente, di <lb/>deſcriuere per i punti continuati la Parabola, <lb/>che habbi per foco vn determinato <lb/>punto. Cap. L I.</head>
<p>
  <s xml:id="echoid-s1645" xml:space="preserve">SIa nella figura 34. </s>
  <s xml:id="echoid-s1646" xml:space="preserve">la retta, A K, <lb/>indiffinitamente prolongata, <lb/>quale vogliamo conſtituir per <lb/>aſſe della Parabola da deſcri-<lb/>uerſi, &amp; </s>
  <s xml:id="echoid-s1647" xml:space="preserve">in quella ſi prendano, <lb/>come ſi voglia due punti, B, A, cioè, B, che <lb/>debba eſſer foco, &amp;</s>
  <s xml:id="echoid-s1648" xml:space="preserve">, A, cima della ſudetta Pa-<lb/>rabola; </s>
  <s xml:id="echoid-s1649" xml:space="preserve">pigliſi poi, B C, eguale à, B A, e ſopra <lb/>il centro, C, con la diſtanza, C A, ſi deſcriui il <lb/>circolo, A O F Z, che ſeghi, A K, in, F, e per, <lb/>S, ſi tiri la, N F G, perpendicolare ad, A K, <lb/>nella quale ſi prendano le, N F, F G, eguali <lb/>ad, F A; </s>
  <s xml:id="echoid-s1650" xml:space="preserve">Dico, cheipunti, N, G, ſono nella <lb/>Parabola, il cuifoco è il punto, B, ouero il cui <lb/>lato retto è, F A, quadrupla di, A B: </s>
  <s xml:id="echoid-s1651" xml:space="preserve">Poiche <lb/>il quadrato, N F, ouero, F G, è vguale al qua-
<pb o="202" file="0222" n="222" rhead="Delle Settioni"/>
drato, F A, cioè al rettangolo ſotto, F A, el’i-<lb/>ſteſſa, F A, lato retto, e però i punti, N, G, ſa-<lb/>ranno in tal Parabola. </s>
  <s xml:id="echoid-s1652" xml:space="preserve">Prendiſi hora nella, F <lb/>A, doue ſi voglia il pũto, E, per il quale ſi pro-<lb/>duchi di quà, e di là indiffinitamẽte la, M H, <lb/>parallela ad, N G, che ſeghi la circonferenza, <lb/>A O F Z, nei punti, O, Z, e la diſtanza, A O, <lb/>ouero, A Z, tirate le, A O, A Z, ſi traſporti sù <lb/>la, M H, terminandola di quà, e di là in, E, e <lb/>ne i punti, M, H; </s>
  <s xml:id="echoid-s1653" xml:space="preserve">Dico, che queſti ſaranno <lb/>nella detta Parabola; </s>
  <s xml:id="echoid-s1654" xml:space="preserve">poiche il quadrato, M <lb/>E, ouero, O A, che gli è vguale, è parimente <lb/>vguale à i quadrati, O E, E A, ma il quadra-<lb/>to, O E, è vguale al rettãgolo, F E A, che con <lb/>il quadrato, E A, fà il rettangolo, F A E; </s>
  <s xml:id="echoid-s1655" xml:space="preserve">adũ-<lb/>que il quadrato, M E, è vguale al rettangolo <lb/>ſotto, E A, parte troncata da eſſa verſo, A, e <lb/>ſotto, A F, lato retto; </s>
  <s xml:id="echoid-s1656" xml:space="preserve">adũque per le coſe det-<lb/>te al Cap. </s>
  <s xml:id="echoid-s1657" xml:space="preserve">6. </s>
  <s xml:id="echoid-s1658" xml:space="preserve">ſarà il punto, M, in tal Parabola; </s>
  <s xml:id="echoid-s1659" xml:space="preserve">e <lb/>nell’iſteſſo modo prouaremo eſſerui il pũto, H.</s>
  <s xml:id="echoid-s1660" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1661" xml:space="preserve">Sia hora preſo in, A F, prodotta oltre il pũ-<lb/>to, F, eſſo, K, per il quale ſi tiri la, L K Q, pa-<lb/>rallela ad, N G, e ſi facci ſopra, A K, vn ſemi-<lb/>circolo, che ſeghi con la ſua circonferenza la, <lb/>N F, in, T; </s>
  <s xml:id="echoid-s1662" xml:space="preserve">Tolta dunque la diſtanza, A T,
<pb o="203" file="0223" n="223" rhead="Coniche. Cap. LI."/>
la traſportaremo ſopra, L, Q, terminandola <lb/>di quà, e di là à i punti, L, Q, e communemen-<lb/>te nel punto, K, prouando noi i punti, L, Q, <lb/>eſſere nella detta Parabola, poiche il quadra-<lb/>to, L K, cioè, T A, per eſſere, K T A, ſemicir-<lb/>colo, è vguale al rettãgolo, K A F, ſotto, K A, <lb/>troncata da, L K, &amp; </s>
  <s xml:id="echoid-s1663" xml:space="preserve">A F, lato retto; </s>
  <s xml:id="echoid-s1664" xml:space="preserve">adunque <lb/>il pũto, L, è in tal Parabola, come anco ſi pro-<lb/>uarà del punto, Q; </s>
  <s xml:id="echoid-s1665" xml:space="preserve">In tal modo adunque preſi <lb/>molti, e ſpeſsi punti nella, A K, e per quelli, di <lb/>quà, e di là dalla, A K, prodotte indiffinita-<lb/>mente rette linee, perpendicolari ad, A K, e <lb/>tolte le diſtanze da i punti, doue dette paral-<lb/>lele ſegano la circonferenza, A O F Z, ſino al <lb/>punto, A, ouero da i punti ſegnati al modo ſu-<lb/>detto nella retta, N G, trouaremo i punti vici-<lb/>niſſimi, per i quali tirata, come ſi è detto, vna <lb/>linea curua, ſi diſſegnarà la Parabola, il cui la-<lb/>to retto ſarà, F A, e ſuo foco il punto, B, coſa <lb/>veramente degna d’eſſer ſaputa; </s>
  <s xml:id="echoid-s1666" xml:space="preserve">ſia dunque <lb/>tal Parabola la, L A Q, nella deſcrittione del-<lb/>la quale, continuata ſotto il punto, F, cõuie-<lb/>ne auuertire, che i ſegamenti, fatti nella ret-<lb/>ta, N G, per occaſione de i ſemicircoli, da de-<lb/>ſcriuerſi, come è, A T K, tal volta ſarãno den-
<pb o="204" file="0224" n="224" rhead="Delle Settioni"/>
tro, N G, ò che batteranno in, N, G, e tal <lb/>volta ſi faranno oltre i punti, N, G, nella me-<lb/>deſima, N G, di quà, e di là indiffinitamente <lb/>prolongata.</s>
  <s xml:id="echoid-s1667" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div93" type="section" level="1" n="93">
<head xml:id="echoid-head98" style="it" xml:space="preserve">Come dalla Parabola ſi poſſono dedurre infinite <lb/>Iperbole, che con mirabile analogia vanno mutan-<lb/>do i lati traſuerſi, mantenendo però ſempre l’iſteſ-<lb/>ſo lato retto. Cap. LII.</head>
<p>
  <s xml:id="echoid-s1668" xml:space="preserve">SIa nella fig. </s>
  <s xml:id="echoid-s1669" xml:space="preserve">35. </s>
  <s xml:id="echoid-s1670" xml:space="preserve">il circolo, D P G, <lb/>diametro, D G, e centro, F, dal <lb/>quale ſuppongaſi hauer noi de-<lb/>dotta la Parabola, D Q H, (mi <lb/>ſia lecito chiamare queſte Set-<lb/>tioni, come che foſſero intiere) nel modo im-<lb/>parato dal Cap.</s>
  <s xml:id="echoid-s1671" xml:space="preserve">ant.</s>
  <s xml:id="echoid-s1672" xml:space="preserve">e ſe ne deuino cauar le ſo-<lb/>pradette Iperbole. </s>
  <s xml:id="echoid-s1673" xml:space="preserve">Tiraremo adunque dentro <lb/>la Parabola, D Q H, già fatta, quante ſi voglia <lb/>linee perpendicolari all’aſſe, D G, che perciò <lb/>ſarãno parallele fra di loro, come, per eſſem-<lb/>piola, G Y, dalla eſtremità del diametro, D G, <lb/>&amp;</s>
  <s xml:id="echoid-s1674" xml:space="preserve">, O X, ambedue indiſſinitamẽte prolongate <lb/>in, Y, X; </s>
  <s xml:id="echoid-s1675" xml:space="preserve">preſa dunque la diſtãza, D Q, e tiaſ-<lb/>feritala ſopra, O X, cominciãdo dal punto, O,
<pb o="205" file="0225" n="225" rhead="Coniche. Cap. LII."/>
ſi che termini in, R, ſimilmente tolta la, D H <lb/>e ſteſala da, G, ſopra, G I; </s>
  <s xml:id="echoid-s1676" xml:space="preserve">Dico, che li punti, <lb/>R, I, ſarãno in vn’lperbola; </s>
  <s xml:id="echoid-s1677" xml:space="preserve">e ſe di nuouo pren-<lb/>deremo, O S, eguale à, D R, e, G M, eguale à, <lb/>D I, ſaranno pur li punti, S, M, in vna nuoua <lb/>Iperbola; </s>
  <s xml:id="echoid-s1678" xml:space="preserve">ſimilmente prendendo, O T, egua-<lb/>le à, D S, e, G N, à, D M, ſaranno i punti, T, <lb/>N, in vn’altra Iperbola, e così procedẽdo con <lb/>l’iſteſſo modo, potremo deſcriuere infinite di <lb/>queſte Iperbole, tutte generate in vn certo <lb/>modo dalla Parabola, ciaſcuna però median-<lb/>tile Iperbole antecedenti, ſino che s’arriui al-<lb/>la Parabola, che riconoſce poi per ſuo genito-<lb/>re il cerchio: </s>
  <s xml:id="echoid-s1679" xml:space="preserve">Mache ciò ſia vero, ſi prouarà in <lb/>queſto modo.</s>
  <s xml:id="echoid-s1680" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1681" xml:space="preserve">Pongaſi, D A, eguale, e per dritto à, D G, <lb/>come anco, D V, ad angolo retto ſopra, G D, <lb/>&amp; </s>
  <s xml:id="echoid-s1682" xml:space="preserve">eguale pure all’iſteſſa, e s’intendino diſſe-<lb/>gnate le curue, D R I, D S M, D T N, median-<lb/>te li molti punti, che potremo trouare ſimili <lb/>alli, R, S, T; </s>
  <s xml:id="echoid-s1683" xml:space="preserve">I, M, N; </s>
  <s xml:id="echoid-s1684" xml:space="preserve">Perche dunque il qua-<lb/>drato, D H, è vguale alli quadrati, D G, G H, <lb/>come anco il quadrato, D Q, s’adegua alli <lb/>duoi quadrati, D O, O Q, e di queſti il qua-<lb/>drato, O Q, è eguale al rettangolo, G D O,
<pb o="206" file="0226" n="226" rhead="Delle Settioni"/>
ouero, A D O, &amp; </s>
  <s xml:id="echoid-s1685" xml:space="preserve">il quadrato, G H, è eguale <lb/>al quadrato, G D, ouero al rettangolo, A D <lb/>G, perciò il quadrato, D H, ouero, G I, ſarà <lb/>eguale al rettangolo, A D G, con il quadra-<lb/>to, D G, cioè al rettangolo, A G D, &amp; </s>
  <s xml:id="echoid-s1686" xml:space="preserve">il qua-<lb/>drato, D Q, ouero, O R, ſarà eguale al ret-<lb/>tangolo, A D O, con il quadrato, D O, cioè al <lb/>rettangolo, A O D, adunque il quadrato, G I, <lb/>al quadrato, O R, ſarà, come il rettangolo, A <lb/>G D, al rettãgolo, A O D, adunque per il Ca-<lb/>pit. </s>
  <s xml:id="echoid-s1687" xml:space="preserve">16. </s>
  <s xml:id="echoid-s1688" xml:space="preserve">ouero per la p. </s>
  <s xml:id="echoid-s1689" xml:space="preserve">21. </s>
  <s xml:id="echoid-s1690" xml:space="preserve">del P. </s>
  <s xml:id="echoid-s1691" xml:space="preserve">de’Conici, <lb/>D R I, ſarà Iperbola, e ſuo lato traſuerſo, A D, <lb/>e poſcia che per l’iſteſſa, come è il rettangolo, <lb/>A G D, al quadrato, G I, così è il lato traſuer-<lb/>ſo al retto, ſi come quelli s’è prouato, che ſono <lb/>eguali, così ſaranno eguali queſti ancora, adũ-<lb/>que, D V, che è vguale ali’, A D, ſarà lato ret-<lb/>to della Iperbola, D R I, quale perciò potre-<lb/>mo chiamare Iperbola equilatera.</s>
  <s xml:id="echoid-s1692" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1693" xml:space="preserve">Perche poi il quadrato, D I, è vguale alli <lb/>quadrati, I G, ouero, D H, &amp;</s>
  <s xml:id="echoid-s1694" xml:space="preserve">, G D, cioè à tre <lb/>quadrati di, G D, ſarà il quadrato, D I, ouero, <lb/>G M, eguale al rettangolo ſotto la tripla di, <lb/>G D, e ſotto, G D, e perciò diuiſo in partie-<lb/>guali, A D, lato traſuerſo in, B, ſarà il quadra-
<pb o="207" file="0227" n="227" rhead="Coniche. Cap. LII."/>
to, G M, doppio del rettangolo ſotto, B G D, <lb/>e come il quadrato, G M, al rettãgolo, B G D, <lb/>così ſarà, V D, à, D B, e perciò, B D, ſarà lato <lb/>traſuerſo dell’Iperbola, D S M, eſſendo poi il <lb/>quadrato, O S, ouero, D R, eguale alli qua-<lb/>drati, R O, O D, cioè al rettangolo, A O D, <lb/>con il quadrato, O D, cioè (rolta, A Z, eguale <lb/>à, D O,) al rettãgolo, Z O D, il medeſimo qua-<lb/>drato, O S, ſarà il doppio del rettangolo, B O <lb/>D, onde à quello ſarà come, V D, à D B, e per-<lb/>ciò il quadrato, G M, al quadrato, O S, ſarà <lb/>come il rettangolo, B G D, al rettangolo, B O <lb/>D, e perciò, D S M, è Iperbola, per la p. </s>
  <s xml:id="echoid-s1695" xml:space="preserve">21. </s>
  <s xml:id="echoid-s1696" xml:space="preserve">del <lb/>P. </s>
  <s xml:id="echoid-s1697" xml:space="preserve">de’Conici, il cui lato retto, V D, è doppio <lb/>del traſuerſo, D B.</s>
  <s xml:id="echoid-s1698" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1699" xml:space="preserve">Similmente il quadrato, D M, ouero, G N, <lb/>ſupera il quadrato, G M, di vn quadrato, D G, <lb/>(come faranno pure gli altri nella, G Y, delle <lb/>Iperbole ſuſſeguenti) ma il quadrato, G M, è <lb/>eguale à tre quadrati di, G D, adunque, G N, <lb/>ſarà eguale à quattro quadrati di, G D, cioè <lb/>eguale al rettangolo ſotto la quadrupla di, G <lb/>D, e ſotto, G D, cioè (fatta, C D, vn terzo di, <lb/>A D,) ſarà triplo del rettangolo, C G D, &amp; </s>
  <s xml:id="echoid-s1700" xml:space="preserve">à <lb/>quello haurà l’iſteſſa proportione, che la, V D,
<pb o="208" file="0228" n="228" rhead="Delle Settioni"/>
à, C D; </s>
  <s xml:id="echoid-s1701" xml:space="preserve">Parimente il quadrato, D S, ouero, T <lb/>O, ſupera il quadrato, S O, cioè il rettangolo, <lb/>Z O D, di vn quadrato, D O, e perciò ſarà e-<lb/>guale al rettãgolo, Z O D, con il quadrato di, <lb/>O D, cioè (aggiunta, Z ℞, eguale ad, A Z,) ſa-<lb/>rà eguale al rettangolo, ℞ O D, cioè triplo del <lb/>rettãgolo, C O D, per eſſer, C O, vn terzo di, <lb/>O ℞, e però il quadrato, T O, al rettangolo, <lb/>C O D, ſarà pure come, V D, à, D C, &amp; </s>
  <s xml:id="echoid-s1702" xml:space="preserve">il qua-<lb/>drato, G N, al quadrato, O T, ſarà come il ret-<lb/>tangolo, C G D, al rettangolo, C O D, e però <lb/>anco, D T N, ſarà vn’Iperbola, il cui lato traſ-<lb/>uerſo è, C D, del quale il lato retto, V D, vie-<lb/>ne ad eſſer triplo: </s>
  <s xml:id="echoid-s1703" xml:space="preserve">Così prouaremo le altre ſuſ-<lb/>ſeguenti, che nell’iſteſſo modo ſi poſſon gene-<lb/>rare, eſſer pure Iperbole, che haurãno ſempre <lb/>il medeſimo lato retto, V D, ma mutaranno il <lb/>traſuerſo; </s>
  <s xml:id="echoid-s1704" xml:space="preserve">cioè nella Iperbola equilatera, ouer <lb/>prima il lato retto ſarà eguale al traſuerſo, <lb/>nella ſeconda il retto ſarà doppio del traſ-<lb/>uerſo, nella terza ſarà triplo, nella quarta qua-<lb/>druplo, e così ſeguirà la proportione del lato <lb/>retto altraſuer ſo in infinito, ſecondo la ſerie <lb/>naturale de’numeri continuati dall’vnità.</s>
  <s xml:id="echoid-s1705" xml:space="preserve"/>
</p>
<pb o="209" file="0229" n="229" rhead="Coniche. Cap. LIII."/>
</div>
<div xml:id="echoid-div94" type="section" level="1" n="94">
<head xml:id="echoid-head99" style="it" xml:space="preserve">In qual maniera ſi poſſi deſcriuere l’Iperbola <lb/>equilatera, il cuifoco diſti dalla ſua ci-<lb/>ma quanto noi vorremo. <lb/>Cap. LIII.</head>
<p>
  <s xml:id="echoid-s1706" xml:space="preserve">GVardiſi pure la medeſima figura <lb/>35. </s>
  <s xml:id="echoid-s1707" xml:space="preserve">nella quale ſia, E D, vn <lb/>quarto del diametro, D G; </s>
  <s xml:id="echoid-s1708" xml:space="preserve">è <lb/>dunque manifeſto per il Capi-<lb/>tolo 21. </s>
  <s xml:id="echoid-s1709" xml:space="preserve">che il punto, E, ſarà <lb/>foco della circonferenza, D P G; </s>
  <s xml:id="echoid-s1710" xml:space="preserve">e perche, <lb/>D G, è anco lato retto della Parabola, D Q <lb/>H, &amp; </s>
  <s xml:id="echoid-s1711" xml:space="preserve">è, D E, vn quarto di quello, perciò <lb/>il punto, E, per il Capitolo 9. </s>
  <s xml:id="echoid-s1712" xml:space="preserve">ſarà pur’an-<lb/>co foco della Parabola, D Q H: </s>
  <s xml:id="echoid-s1713" xml:space="preserve">Pongaſi ho-<lb/>ra, che habbiamo da deſcriuere vn’Iperbola <lb/>equilatera, il cui foco diſti dalla cima, D, per <lb/>la retra, E D; </s>
  <s xml:id="echoid-s1714" xml:space="preserve">Prima dunque io dico, che il <lb/>punto, E, non è foco dell’Iperbola equila-<lb/>tera, D R I, poiche douendoſi, per ritrouar-<lb/>lo, adattare all’, A D, vn rettangolo eccedẽ-<lb/>te d’vna figura quadrata, eguale alla quarta <lb/>parte del rettangolo ſotto, A D, D V, ouero <lb/>del quadrato, A D, cioè eguale al rettãgolo, <lb/>A D E, è manifeſto, che, D E, non può eſſe-
<pb o="210" file="0230" n="230" rhead="Delle Settioni"/>
re l’ecceſſo fatto per la ſudetta applicatione, <lb/>poiche verrebbe il rettangolo, A D E, ad eſ-<lb/>ſere eguale al rettangolo, A E D, che è aſſur-<lb/>do; </s>
  <s xml:id="echoid-s1715" xml:space="preserve">adunque il punto, F, non può eſſer foco <lb/>dell’Iperbola equilatera, D R I, ma caſcherà <lb/>tra i punti, E, D, cone in, Φ, poiche eſſendo <lb/>il rettangolo, A D E, eguale al rettangolo, <lb/>A Φ D, e maggiore, A Φ, di, A D, biſogna re-<lb/>ciprocamente, che anco, D E, ſia maggiore <lb/>di, Φ D, quanto poi ſi allontani dal punto, D, <lb/>lo trouaremo in queſto modo. </s>
  <s xml:id="echoid-s1716" xml:space="preserve">Tagliſi, D V, <lb/>in, Π, in parti eguali, e ſi tiri la, B Π Dico, che, <lb/>B Π, è vguale à, B Φ, poiche eſſendo il rettã-<lb/>golo, A Φ D, eguale à vn quarto del quadra-<lb/>to, A D, cioè al quadrato, B D, ne ſeguirà, <lb/>che il rettangolo, A Φ D, con il qua dr. </s>
  <s xml:id="echoid-s1717" xml:space="preserve">D B, <lb/>cioè che il quadrato, B Φ, per la p. </s>
  <s xml:id="echoid-s1718" xml:space="preserve">6. </s>
  <s xml:id="echoid-s1719" xml:space="preserve">del Se-<lb/>condo de gli Elem. </s>
  <s xml:id="echoid-s1720" xml:space="preserve">ſia doppio del quadrato, <lb/>B D, ma anco il quadrato, B Π, è doppio del <lb/>quadrato, B D; </s>
  <s xml:id="echoid-s1721" xml:space="preserve">adunque il quadrato, B Π, è <lb/>vguale al quadrato, B Φ, eſſendo perciò, Φ B, <lb/>incommenſurabile à, D B; </s>
  <s xml:id="echoid-s1722" xml:space="preserve">ſappiamo dunque <lb/>quanto il foco, Φ, dell’Iperbola equilatera, <lb/>D R I, ſi allontani dalla ſua cima, D; </s>
  <s xml:id="echoid-s1723" xml:space="preserve">sì come <lb/>ſi prouarà in tutte le Iperbole equilatere, di-
<pb o="211" file="0231" n="231" rhead="Coniche. Cap. LIII."/>
uiſo per mezo il lato traſuerſo, il lor foco eſſer <lb/>diſtante da quel punto di mezo, cioè dal cen-<lb/>tro dell’Iperbola, per la quantità d’vna linea <lb/>retta, che viene ad eſſer diametro del quadra-<lb/>to, che ſi può formare ſopra eſſa metà del lato <lb/>traſuerſo, com’è la, B Φ, hauendo perciò in <lb/>tutte le Iperbole equilatere le diſtanze da i <lb/>fochi à i centri dell’Iperbole alle metà dei lo-<lb/>ro lati trauerſi l’iſteſſa proportione, cioè <lb/>quella, che hà il diametro alla coſta dell’iſteſ-<lb/>ſo quadrato.</s>
  <s xml:id="echoid-s1724" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1725" xml:space="preserve">Inteſe queſte coſe, per deſcriuer l’Iperbo-<lb/>la equilatera, il cui foco ſia il pun to, E, baſte-<lb/>rà trouare la quantità del diametro di quel <lb/>cerchio, dal quale cauando l’Iperbola equi-<lb/>latera nel modo, che dal cerchio, D P G, s’è <lb/>dedotta la, D R I, ſi potrà facilmente ritro-<lb/>uare. </s>
  <s xml:id="echoid-s1726" xml:space="preserve">Eſſendo adunque la diſtanza propoſta <lb/>dal foco alla cima dell’Iperbola eſſa, E D, ſe <lb/>noi prendeſſimo, G D, quadrupla di, E D, e <lb/>deſcritto il cerchio, D P G, cauaſſimo l’Iper-<lb/>bola equilatera, D R I, queſta hauerebbe il <lb/>foco nel pũto, Φ, e perciò non ſarebbe à pro-<lb/>poſiro, come ſi è moſtrato di ſopra, per trouar <lb/>quella adunque, che hà per foco il punto, E,
<pb o="212" file="0232" n="232" rhead="Delle Settioni"/>
farò come, Φ D, (ſuppoſto che il punto, Φ, ſia <lb/>ritrouato, come ſopra) à, D B, metà del lato <lb/>traſuerſo dell’Iperbola equilatera, il cui foco <lb/>è il punto, Φ, ouero vniuerſalmente, ſenza ha-<lb/>uere à riguardare ad altra Iperbola, farò co-<lb/>me l’ecceſſo del diametro di qual ſi voglia <lb/>quadrato alla ſua coſta, così la data diſtanza, <lb/>E D, ad vna quarta linea, che ſarà la metà del <lb/>lato traſuerſo, o retto, della noſtra Iperbola <lb/>equilatera, che haurà per foco il punto, E, e <lb/>l’iſteſſa ſarà ſemidiametro del cerchio da de-<lb/>ſcriue@ſi, dal quale cauando l’Iperbola equi-<lb/>latera nel modo, che dal cerchio, D P G, s’è <lb/>dedotta la, D R I, queſta haurà per ſuo foco <lb/>(interiore intendo ſempre) il punto, E, poi-<lb/>che la diſtanza di, E, foco dal centro della <lb/>detta Iperbola alla metà del lato traſuerſo ha-<lb/>urà l’iſteſſa proportione, che hà il diametro <lb/>alla coſta, e perciò conforme à quel, che ſi è <lb/>detto di ſopra, il punto, E, ſarà pur foco della <lb/>detta Iperbola equilatera. </s>
  <s xml:id="echoid-s1727" xml:space="preserve">Sin’hora dunque <lb/>ſappiamo deſcriuere il cerchio, la Parabola, <lb/>e l’Iperbola equilatera, mediante la traslatio-<lb/>ne delle dette linee verticali, che habbino il <lb/>lor foco diſtante dalla cima, quanto à noi pia-
<pb o="213" file="0233" n="233" rhead="Coniche. Cap. LIII."/>
cerà: </s>
  <s xml:id="echoid-s1728" xml:space="preserve">Vi reſta l’Eliſſi, la cui deſcrittione hò <lb/>con anſietà cercato ſe ſi poteua fare in vna ſi-<lb/>mil maniera, ma hauẽdo viſto il Keplero nel-<lb/>le Tauole Rodulfine, inſegnare vn modo, che <lb/>hà molta affinità con il già accẽnato di ſopra, <lb/>m’è parſo bene, per non differir più con nuo-<lb/>ue ſpecolationi il fin della ſtampa di queſto <lb/>mio Trattato, accõpagnarlo con gli altri in-<lb/>ſegnati di ſopra, aggiungendoui anco, per ſo-<lb/>disfattione de’ſtudioſi, la ſua dimoſtratione, <lb/>poiche quella non ſi hà nelle dette Tauole, <lb/>mettendo egli ſolo la ſemplice prattica, per <lb/>ſeruirſene nelle coſe celeſti.</s>
  <s xml:id="echoid-s1729" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div95" type="section" level="1" n="95">
<head xml:id="echoid-head100" style="it" xml:space="preserve">Come ſi deſcriua l’Eliſsi, che habbiciaſcun de’ſuoi <lb/>fochi distanti dall’eſtremità dell’aſſe quanto <lb/>ſi voglia. Cap. LIV.</head>
<p>
  <s xml:id="echoid-s1730" xml:space="preserve">SIa nella figura 36. </s>
  <s xml:id="echoid-s1731" xml:space="preserve">la retta, A H, <lb/>che deua eſſere diametro mag-<lb/>giore d’vno Eliſsi, nel quale ſi <lb/>prendano per fochi i punti, C, <lb/>G, egualmente diſtanri da gli <lb/>eſtremi dell’aſſe, A, H; </s>
  <s xml:id="echoid-s1732" xml:space="preserve">per deſcriuere adun-<lb/>quela pr opoſta Eliſſi, prima diuiſa, A H, per
<pb o="214" file="0234" n="234" rhead="Delle Settioni"/>
mezo in, E, ſopra’l centro, E, con li ſemidia-<lb/>metri, E A, E C, faremo li duoi ſemicircoli, A <lb/>L H, C R G, preſi poi nella circonferenza, A <lb/>L H, quanti punti, e douunque vorremo, per <lb/>eſſempio, M, L, K, da quelli tiraremo al cen-<lb/>tro, E, le, M E, L E, K E, notando i punti, O, <lb/>R, S, doue ſegano la circonferenza dell’inte-<lb/>riore ſemicircolo, da queſti poi, come anco <lb/>dalli punti, M, L, K, ſopra il diametro, A H, <lb/>caſchino le perpendicolari, M B, L E, nel cen-<lb/>tro, K I, O D, R E, S F, fatto poi centro com-<lb/>mune, G, con l’interuallo, H D, ſi deſcriua <lb/>vn pezzetto d’arco, che ſeghi, M B, in, N, e <lb/>così con l’interuallo, H E, trasferendolo in, <lb/>G P, ſi noti nella, L E, il punto, P, e con, H F, <lb/>traſportato in, G Q, ſi ſegninella, K I, il pũ-<lb/>to, Q; </s>
  <s xml:id="echoid-s1733" xml:space="preserve">Dico adunque, che i punti, N, P, Q, <lb/>ſtanno nell’Eliſsi, i cui fochi ſono, C, G; </s>
  <s xml:id="echoid-s1734" xml:space="preserve">Per <lb/>prouar queſto adũque, ſi tirino le, M G, O H; <lb/></s>
  <s xml:id="echoid-s1735" xml:space="preserve">Hora perche ne’triangoli, M E G, O E H, <lb/>M E, è vguale ad, E H, &amp;</s>
  <s xml:id="echoid-s1736" xml:space="preserve">, E G, ad, E O, e <lb/>l’angolo, O E G, commune, ſarà, per la 4. </s>
  <s xml:id="echoid-s1737" xml:space="preserve">del <lb/>primo de gli Elem. </s>
  <s xml:id="echoid-s1738" xml:space="preserve">la baſe, M G, eguale alla <lb/>baſe, O H, e perciò anco i loro quadrati ſarã-<lb/>no eguali. </s>
  <s xml:id="echoid-s1739" xml:space="preserve">E perche anco i quad. </s>
  <s xml:id="echoid-s1740" xml:space="preserve">delle H D,</s>
</p>
<pb o="215" file="0235" n="235" rhead="Coniche. Cap. LIV."/>
<p>
  <s xml:id="echoid-s1741" xml:space="preserve">G N, per la coſtruttione eguali, ſon pure e-<lb/>guali, perciò la differenza tra li duoi quad. </s>
  <s xml:id="echoid-s1742" xml:space="preserve">O <lb/>H, H D, cioè il quad. </s>
  <s xml:id="echoid-s1743" xml:space="preserve">O D, ſarà eguale alla <lb/>differenza tra li duoi quad. </s>
  <s xml:id="echoid-s1744" xml:space="preserve">M G, G N, cioè <lb/>all’ecceſſo de’quad. </s>
  <s xml:id="echoid-s1745" xml:space="preserve">M B, B G, ſopra li quad. <lb/></s>
  <s xml:id="echoid-s1746" xml:space="preserve">N B, B G, cioè all’ecceſſo del quad. </s>
  <s xml:id="echoid-s1747" xml:space="preserve">M B, ſo-<lb/>pra il quadr. </s>
  <s xml:id="echoid-s1748" xml:space="preserve">N B, nell’iſteſſo modo poi pro-<lb/>uaremo eſſer’il quad. </s>
  <s xml:id="echoid-s1749" xml:space="preserve">S F, eguale all’ecceſſo <lb/>del quad. </s>
  <s xml:id="echoid-s1750" xml:space="preserve">K I<unsure/>, ſopra il quad. </s>
  <s xml:id="echoid-s1751" xml:space="preserve">I<unsure/> Q, sì come anco <lb/>il quad. </s>
  <s xml:id="echoid-s1752" xml:space="preserve">E G, è vguale all’ecceſſo del quad. </s>
  <s xml:id="echoid-s1753" xml:space="preserve">G <lb/>P, ouero, L E, ſopra il quad. </s>
  <s xml:id="echoid-s1754" xml:space="preserve">P E; </s>
  <s xml:id="echoid-s1755" xml:space="preserve">Perche poi <lb/>i triangoli, M B E, O D E, ſono ſimili, perciò <lb/>il quad. </s>
  <s xml:id="echoid-s1756" xml:space="preserve">O D, al quad. </s>
  <s xml:id="echoid-s1757" xml:space="preserve">O E, cioè l’ecceſſo del <lb/>quad. </s>
  <s xml:id="echoid-s1758" xml:space="preserve">M B, ſopra il quad. </s>
  <s xml:id="echoid-s1759" xml:space="preserve">B N, all’ecceſſo del <lb/>quadr. </s>
  <s xml:id="echoid-s1760" xml:space="preserve">L E, ſopra il quadr. </s>
  <s xml:id="echoid-s1761" xml:space="preserve">P E, ſarà comeil <lb/>quad. </s>
  <s xml:id="echoid-s1762" xml:space="preserve">M B, al quad. </s>
  <s xml:id="echoid-s1763" xml:space="preserve">M E, ouero al quad. </s>
  <s xml:id="echoid-s1764" xml:space="preserve">L E, <lb/>adunque il quad. </s>
  <s xml:id="echoid-s1765" xml:space="preserve">M B, al quad. </s>
  <s xml:id="echoid-s1766" xml:space="preserve">L E, ſarà an-<lb/>cora come il quadr. </s>
  <s xml:id="echoid-s1767" xml:space="preserve">N B, al quadr. </s>
  <s xml:id="echoid-s1768" xml:space="preserve">P E, ma il <lb/>quad. </s>
  <s xml:id="echoid-s1769" xml:space="preserve">M B, al quad. </s>
  <s xml:id="echoid-s1770" xml:space="preserve">L E, è come il rettangolo, <lb/>A B H, al rettangolo, A E H, adũque il quad. </s>
  <s xml:id="echoid-s1771" xml:space="preserve"><lb/>N B, al quad. </s>
  <s xml:id="echoid-s1772" xml:space="preserve">P E, è come il rettang. </s>
  <s xml:id="echoid-s1773" xml:space="preserve">A B H, al <lb/>rettãg A E H, ma queſta è la 4. </s>
  <s xml:id="echoid-s1774" xml:space="preserve">proprietà dell’ <lb/>Eliſſi, dimoſtrata al Cap. </s>
  <s xml:id="echoid-s1775" xml:space="preserve">20. </s>
  <s xml:id="echoid-s1776" xml:space="preserve">adũque i punti, <lb/>N, P, ſono nella Eliſſi medeſima, cioè in quel-<lb/>la, i cui fochi ſono i punti, C, G: </s>
  <s xml:id="echoid-s1777" xml:space="preserve">ſimilmente</s>
</p>
<pb o="216" file="0236" n="236" rhead="Delle Settioni"/>
<p>
  <s xml:id="echoid-s1778" xml:space="preserve">ſupponendoſi prouato, che il quadr. </s>
  <s xml:id="echoid-s1779" xml:space="preserve">S F, èv-<lb/>guale all’ecceſſo del quad. </s>
  <s xml:id="echoid-s1780" xml:space="preserve">K I, ſopra il quad. <lb/></s>
  <s xml:id="echoid-s1781" xml:space="preserve">I Q, &amp; </s>
  <s xml:id="echoid-s1782" xml:space="preserve">eſſendo il quad. </s>
  <s xml:id="echoid-s1783" xml:space="preserve">I K, al quad K E, come <lb/>il quad. </s>
  <s xml:id="echoid-s1784" xml:space="preserve">F S, al quad. </s>
  <s xml:id="echoid-s1785" xml:space="preserve">S E, perciò il medeſimo <lb/>quad. </s>
  <s xml:id="echoid-s1786" xml:space="preserve">I K, al quad K E, ouero, L E, ſarà come <lb/>l’ecceſſo del quad. </s>
  <s xml:id="echoid-s1787" xml:space="preserve">K I, ſopra’l quad. </s>
  <s xml:id="echoid-s1788" xml:space="preserve">I Q, al qua <lb/>dr. </s>
  <s xml:id="echoid-s1789" xml:space="preserve">S E, cioè all’ecceſſo del quad. </s>
  <s xml:id="echoid-s1790" xml:space="preserve">L E, ſopra il <lb/>quad. </s>
  <s xml:id="echoid-s1791" xml:space="preserve">E P, e perciò come il quad. </s>
  <s xml:id="echoid-s1792" xml:space="preserve">K I, al quad. </s>
  <s xml:id="echoid-s1793" xml:space="preserve"><lb/>L E, cioè come il rettãgolo, H I A, al rettãgo-<lb/>lo, H E A, così ſarà il quad. </s>
  <s xml:id="echoid-s1794" xml:space="preserve">Q I, al quad. </s>
  <s xml:id="echoid-s1795" xml:space="preserve">P E, <lb/>adunque il punto, Q, è nell’Eliſſi, nel quale <lb/>èil punto, P, cioè in quello, che hà per fochi <lb/>i punti, C, G, così dunque prendendo ſpeſſi <lb/>punti nella circonferenza, A L H, deſcriuere-<lb/>mo facilmente la parte, A P H, e con l’iſteſſa <lb/>maniera l’altra metà, sì come deuono parimẽ-<lb/>te nella fig. </s>
  <s xml:id="echoid-s1796" xml:space="preserve">35. </s>
  <s xml:id="echoid-s1797" xml:space="preserve">farſi l’altre metà della Parabo-<lb/>la, e delle I perbole ſuſſeguenti, col medeſimo <lb/>modo iui dichiarato. </s>
  <s xml:id="echoid-s1798" xml:space="preserve">I<unsure/>I<unsure/> Keplero poi preualen-<lb/>doſi dell’Eliſſi nelle coſe celeſti, ſuppone, che <lb/>G, ſia il Sole, A, l’Affelio, H, Perielio, A P H, <lb/>l’orbita del Pianeta, G A, G N, G P, G Q, G <lb/>H, Gl’interualli del Pianeta, poſto in queſti <lb/>luoghi, dal Sole, l’angolo, M E A, ouero l’arco <lb/>M A, l’anomalia dell’eccentrico, l angolo, M</s>
</p>
<pb o="217" file="0237" n="237" rhead="Coniche. Cap. LIV."/>
<p>
  <s xml:id="echoid-s1799" xml:space="preserve">G A, l’anomalia coequata, ma nel circolo, &amp;</s>
  <s xml:id="echoid-s1800" xml:space="preserve">, <lb/>N G A, l’anomalia coequata vera, l’area, N G <lb/>A, l’anomalia media, il triangolo, N G E, l’e-<lb/>quation fiſica, moſtrãdo, che il Pianeta, moſ-<lb/>ſo dalla virtù ſolare, ſia sforzato deſcriuere <lb/>l’Eliſſi, i cui fochi ſono, C, G, in vn de’quali, <lb/>cioèin, G, ſtà collocato il Sole; </s>
  <s xml:id="echoid-s1801" xml:space="preserve">tanto egli hà <lb/>nobilitato queſte Settioni Coniche.</s>
  <s xml:id="echoid-s1802" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div96" type="section" level="1" n="96">
<head xml:id="echoid-head101" xml:space="preserve">Corollario.</head>
<p style="it">
  <s xml:id="echoid-s1803" xml:space="preserve">EDunque manifeſto da queſto Capa, e dalli tre <lb/>antecedenti, che, proposta qualunque diſtan-<lb/>za del foco dalla cima della data Settione, <lb/>n<unsure/>oi ſapremo deſcriuere qualſiuoglia di quelle, alla <lb/>quale conuerrà il proposto foco. </s>
  <s xml:id="echoid-s1804" xml:space="preserve">Intendendo inſie-<lb/>me, che quando ſi fabricaſſe vno Specchio Sferico <lb/>poch ſſimo cauo, ouero vna lente pochiſſimo colma, <lb/>questi<unsure/> non ſare<unsure/>bbono molto differenti dalla curuità <lb/>Parabolica, &amp; </s>
  <s xml:id="echoid-s1805" xml:space="preserve">Iperbolica; </s>
  <s xml:id="echoid-s1806" xml:space="preserve">poiche nella figura 3 5. </s>
  <s xml:id="echoid-s1807" xml:space="preserve">il <lb/>quadr<unsure/>ato, OP, iarghezza della metà delio Specchio <lb/>Sferico, D P, è ſuperato dal quad di, O Q, largbez-<lb/>za di mezo lo Specchio Parabolico, DQ, del quadr. <lb/></s>
  <s xml:id="echoid-s1808" xml:space="preserve">DO, così il quad. </s>
  <s xml:id="echoid-s1809" xml:space="preserve">medeſimo di, OP, è ſuperato dal <lb/>quad. </s>
  <s xml:id="echoid-s1810" xml:space="preserve">OR, larghezza di mezo lo Specchio Iperboli-
<pb o="218" file="0238" n="238" rhead="Delle Settioni"/>
co, D R, di duoiquadr. </s>
  <s xml:id="echoid-s1811" xml:space="preserve">di, O D, ma, O D, è la pro-<lb/>fondiità dello Specchio Sferico, D P, adunque quan-<lb/>do questo foſſe pochiſſimo cauo, ſarebbe, D O, pic-<lb/>coliſſima, &amp; </s>
  <s xml:id="echoid-s1812" xml:space="preserve">in conſeguenza ne vn tal quadrato di <lb/>più, ne due aggiunti al quadr. </s>
  <s xml:id="echoid-s1813" xml:space="preserve">O P, fariano creſcer <lb/>molto eſſa larghezza, O P, adunque gli<unsure/> Specchi Sfe-<lb/>rici poco caui, e le lenti, le quali ſiano poco colme, ſa-<lb/>ranno quaſi inſieme e Paraboliche, &amp; </s>
  <s xml:id="echoid-s1814" xml:space="preserve">Iperboliche, e <lb/>perciò accoſtandoſegli tanto, faranno ancor gli effet-<lb/>ti a quelli propinquiſſimi, il che inſieme potrà, credo, <lb/>ſeruire per iſgannar’alcuni, che ſtimano, che vn par <lb/>d’occhiali Parabolici, o Iperbolici, foſſero per far l’ef-<lb/>fetto del Canocchiale, poiche ſe così foſſe, accoſtandoſi <lb/>tanto vicino le lenti Sferiche, e pochiſſimo colme, al-<lb/>la detta curuità, ce ne dariano pur qualche ſegno, il <lb/>che non ſi vede, mentre non ſi accompagnino con il <lb/>traguardo. </s>
  <s xml:id="echoid-s1815" xml:space="preserve">Potrà inſieme ancora la dottrina di que-<lb/>ſto Corollario dar ſodisfattione à quelli, che ſtimaſſe-<lb/>ro la ſtrada diſſegnata dal proietto eſſer circolare, poi-<lb/>che eſſendo quel cerchio notabilments grande, &amp; </s>
  <s xml:id="echoid-s1816" xml:space="preserve">il <lb/>viaggio del graue poca parte dell’intiera circonfe-<lb/>renza, può eſſer, ch<unsure/>e talhora rieſcbi pure pochiſsimo <lb/>differente dalla Parabola.</s>
  <s xml:id="echoid-s1817" xml:space="preserve"/>
</p>
<pb o="219" file="0239" n="239" rhead="Coniche. Cap. LV."/>
</div>
<div xml:id="echoid-div97" type="section" level="1" n="97">
<head xml:id="echoid-head102" style="it" xml:space="preserve">Di altre maniere ancora di dedurre le Settioni Coni-<lb/>che vicen<unsure/>deuolmente l’vna dall’altra, o dal-<lb/>la circonferenza del cerchio. <lb/>Cap. LV. &amp; vlt.</head>
<p>
  <s xml:id="echoid-s1818" xml:space="preserve">POſſono anco le Settioni Coniche <lb/>dedurſi l’vna dall’altra, o dalla <lb/>circonferenza di cerchio in que-<lb/>ſto modo, cioè, per eſſempio, ſe <lb/>noi nella fig. </s>
  <s xml:id="echoid-s1819" xml:space="preserve">36. </s>
  <s xml:id="echoid-s1820" xml:space="preserve">ſegaremo pro-<lb/>portionalmente tutte le ordinatamẽte appli-<lb/>cate all’aſſe, e per i ſegamenti tiraremo vna <lb/>curua, come, A N P Q H, quella verrà Elit-<lb/>tica, e l’iſteſſo ſarebbe, ſe prolongaſſimo le <lb/>medeſime, oltre à i punti, M, L, K, tutte nel-<lb/>la medeſima proportione: </s>
  <s xml:id="echoid-s1821" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1822" xml:space="preserve">vniuerſalmente <lb/>creſcendo, e ſcemando nell’iſteſſa proportio-<lb/>ne le ordinatamente applicate al diametro di <lb/>qualſiuoglia Settione, la curua, che paſſerà <lb/>per i pũti eſtremi di quelle ſarà ancora lei Set-<lb/>tion Conica dell’iſteſſa ſorte, cioè Parabolica <lb/>la dedotta dalla Parabola, Iperbolica dalla <lb/>Iperbola, ma dal cerchio verrà fatta la Elitti-<lb/>ca. </s>
  <s xml:id="echoid-s1823" xml:space="preserve">Similmente ſe nella fig. </s>
  <s xml:id="echoid-s1824" xml:space="preserve">35. </s>
  <s xml:id="echoid-s1825" xml:space="preserve">noi prendeſ-<lb/>ſimo le ordinatamente applicate alla, D G, e
<pb o="220" file="0240" n="240" rhead="Delle Settioni"/>
le trasferiſſimo nell’iſteſſa drittura, applican-<lb/>dole ordinatamente ad vn’altra linea retta, <lb/>interpoſta fra le oppoſte tangenti, allhora ſe <lb/>quella gli foſſe perpendicolare, tirando la li-<lb/>nea curua per l’eſtremità delle applicate, <lb/>quella ſarebbe circonferenza di cerchio, ma <lb/>ſe caſcaſſe obliquamente fra le dette tangen-<lb/>ti, ſaria Elittica; </s>
  <s xml:id="echoid-s1826" xml:space="preserve">così facendo tale traslatio-<lb/>ne delle ordinatamente applicate al diame-<lb/>tro della Parabola, ne verrà pur Parabola; </s>
  <s xml:id="echoid-s1827" xml:space="preserve">e <lb/>dall’Iperbola, Iperbola, come dall’Eliſſi ſe <lb/>ne dedurrà pur’Eliſſi.</s>
  <s xml:id="echoid-s1828" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1829" xml:space="preserve">Queſti modi, con quelli, che hò ſpiegato <lb/>ne’Capit. </s>
  <s xml:id="echoid-s1830" xml:space="preserve">51. </s>
  <s xml:id="echoid-s1831" xml:space="preserve">52. </s>
  <s xml:id="echoid-s1832" xml:space="preserve">53. </s>
  <s xml:id="echoid-s1833" xml:space="preserve">per deſcriuer le dette <lb/>Settioni Coniche, credo debbano eſſer viſti <lb/>con qualche guſto da gli ſtudioſi, ſupponẽdo <lb/>chenon ſiano, maſſime quelli, così noti à tutti, <lb/>ſe bene in queſto mi potrei forſi ingannare, <lb/>poiche hauendoli io per qualche tempo ſti-<lb/>mati, come coſa di mia inuentione particola-<lb/>re; </s>
  <s xml:id="echoid-s1834" xml:space="preserve">doppo ch’io hò viſto il Libro di Bartolo-<lb/>meo Souero Friburgenſe, già Profeſſore delle <lb/>Matematiche nello Studio di Padoua, intito-<lb/>lato; </s>
  <s xml:id="echoid-s1835" xml:space="preserve">Curui, ac recta Proportio promota. </s>
  <s xml:id="echoid-s1836" xml:space="preserve">mi ſono <lb/>accorto d’eſſerm’incontrato con lui nelle me-
<pb o="221" file="0241" n="241" rhead="Coniche. Cap. LV."/>
deme ſpecolationi, benche quãto alle ragioni <lb/>ſiamo in parte differenti, come ſi può vedere. <lb/></s>
  <s xml:id="echoid-s1837" xml:space="preserve">Hora, perche qualche critico non haueſſe da <lb/>cenſurarmi, ch’io mi foſſi vſurpato l’inuẽtione <lb/>di queſt’huomo, da me ſtimato per molto in-<lb/>gegnoſo, &amp; </s>
  <s xml:id="echoid-s1838" xml:space="preserve">eſperte nelle Matematiche, mi ba-<lb/>ſterà la teſtimonianza dell’Illuſtriſs. </s>
  <s xml:id="echoid-s1839" xml:space="preserve">Sig. </s>
  <s xml:id="echoid-s1840" xml:space="preserve">Ceſa-<lb/>re Marſili, che dell’anno 1629. </s>
  <s xml:id="echoid-s1841" xml:space="preserve">vidde parte <lb/>di queſto Trattato, doue haueuo dichiarato il <lb/>modo di deſcriuer la Parabola, e l’lperbola <lb/>nella maniera ſpiegata ne i ſudetti tre Capi-<lb/>toli, cioè con dedurle dal cerchio; </s>
  <s xml:id="echoid-s1842" xml:space="preserve">come anco <lb/>ne potrà far fede il Sig. </s>
  <s xml:id="echoid-s1843" xml:space="preserve">Alfonſo da Isè nomi-<lb/>nato diiſopra, che fù in parte cauſa, ch’io mi <lb/>ci applicaſſi; </s>
  <s xml:id="echoid-s1844" xml:space="preserve">eſſendo poi il Souero ſtampato <lb/>doppo, cioè dell’anno 1630. </s>
  <s xml:id="echoid-s1845" xml:space="preserve">Ne deue arreca-<lb/>re marauiglia, che due s’incontrino ne’mede-<lb/>ſimi penſieri, parendo anzi, che la Natura ſia <lb/>inolto ſollecita nel produrr’à queſto fine huo-<lb/>mini dell’iſteſſo genio, per addottrinar, anco <lb/>contra ſua voglia, il genere humano, acciò <lb/>quello, che per la negligenza di vno reſtareb-<lb/>be ſepolto, per diligenza dell’altro venga à <lb/>porſi in luce, potendo perciò accadere, che <lb/>più d’vno ancora dia nel medeſimo ſegno.</s>
  <s xml:id="echoid-s1846" xml:space="preserve"/>
</p>
<pb o="222" file="0242" n="242" rhead="Delle Settioni"/>
<p>
  <s xml:id="echoid-s1847" xml:space="preserve">Haueuo finalmente penſiero di aggiunge-<lb/>re altre coſe, &amp; </s>
  <s xml:id="echoid-s1848" xml:space="preserve">in particolare d’inſegnare la <lb/>maniera di trouare vicendeuolmente i diame-<lb/>tri, lati, e centri, date le Settioni Coniche già <lb/>deſcritte, con altre coſe intorno alle tangen-<lb/>ti, gli aſimptoti, e le incidẽti; </s>
  <s xml:id="echoid-s1849" xml:space="preserve">ma perche que-<lb/>ſto hauria cagionato maggior lunghezza del <lb/>douere, e tedio, à chi aborriſcc dalle continue <lb/>dimoſtrationi, e figure, perciò me ne ſono vo-<lb/>luto aſtenere, maſſime, che per gl’intelligenti <lb/>è troppo il volergli ſminuzzar’ogni coſa, e per <lb/>chi non hà prattica in ſimili materie il molto, <lb/>che ſi può dire, ne anco è baſtante per fargli <lb/>capaci, perciò rimetterò, chi haueſſe biſo-<lb/>gno d’alcuni di queſti Problemi, à gli Elemẽ-<lb/>ti Conici di Apollonio Pergeo, ouero al lib. </s>
  <s xml:id="echoid-s1850" xml:space="preserve">3. <lb/></s>
  <s xml:id="echoid-s1851" xml:space="preserve">delle Linee horarie dell’Abbate Maurolico, <lb/>che con molta facilità, e breuità n’inſegna le <lb/>ſue Regole; </s>
  <s xml:id="echoid-s1852" xml:space="preserve">e facendo fine à queſto mio Trat-<lb/>tato, pregherò chiunque ne riceuerà qualche <lb/>frutto, che vogli meco rẽderne gratie alla be-<lb/>nignità dell’altiſſimo Iddio, datore d’ogni be-<lb/>ne, dalla cui infinita liberalità riconoſcendo <lb/>noi, come pretioſiſſime gioic, la vita, e l’inge-<lb/>gno, e come denati datici in contanti, dobbia-
<pb o="223" file="0243" n="243" rhead="Coniche. Cap. LV."/>
mo non ſolo à quella con ragione il tutto rife-<lb/>rire, ma anco affaticarci continuamente per <lb/>pagargliene almeno in parte l’vſura, poiche <lb/>è pur veriſſima quella ſentenza, cioè, che</s>
</p>
</div>
<div xml:id="echoid-div98" type="section" level="1" n="98">
<head xml:id="echoid-head103" style="it" xml:space="preserve">Deus nobis vſuram vitæ dedit, &amp; ingenĳ tamquam <lb/>pecuniæ, nulla praſtituta die.</head>
  <figure>
    <image file="0243-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0243-01"/>
  </figure>
<pb file="0243a" n="244"/>
  <figure>
    <image file="0243a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0243a-01"/>
  </figure>
<pb o="224" file="0244" n="245"/>
</div>
<div xml:id="echoid-div99" type="section" level="1" n="99">
<head xml:id="echoid-head104" xml:space="preserve">Errc<unsure/>ri ſcorſi per inauuettenz@ nello ſtampate.</head>
<p style="it">
  <s xml:id="echoid-s1853" xml:space="preserve">Si è fatto, per certo accidente, il più delle volte, Eliſſi, &amp; </s>
  <s xml:id="echoid-s1854" xml:space="preserve">in gene <lb/>re maſcoline, douendoſi fare, Elliſſi, &amp; </s>
  <s xml:id="echoid-s1855" xml:space="preserve">in genere feminino.</s>
  <s xml:id="echoid-s1856" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1857" xml:space="preserve">Similmente ſi è poſto due volte il numero del Cap. </s>
  <s xml:id="echoid-s1858" xml:space="preserve">16. </s>
  <s xml:id="echoid-s1859" xml:space="preserve">c Cap. </s>
  <s xml:id="echoid-s1860" xml:space="preserve">29.</s>
  <s xml:id="echoid-s1861" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s1862" xml:space="preserve">Gli altri errori d’ortografia, maſſime delle uirgole, punti, e mezi <lb/>punti, ſi laſciano alla diſcrettione del benigno Lettore, eſſendoſi <lb/>corretti gli altri più notabili, meglio, che ſi è potuto con la pẽna.</s>
  <s xml:id="echoid-s1863" xml:space="preserve"/>
</p>
<p style="it">
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</echo>
author	Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date	Wed, 29 Nov 2017 16:55:37 +0100
parents	22d6a63640c6
children