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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:08:12 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <archimedes> <info> <author>Gallaccini, Teofilo</author> <title>Perigonia, o vero degli angoli (Ms. L. IV. 5 della Biblioteca degli Intronati di Siena, cc. 1r. - 86r.)</title> <date>ca. 1590-1598</date> <place>Siena</place> <translator>A cura die Annalisa Simi</translator> <lang>it</lang> <cvs_file>galla_perig_it_1590.xml</cvs_file> <echodir>/permanent/echo/biblioteca_siena/galla_perig_it_1590</echodir> </info> <text> <body> <chap> <pb pagenum="folio 1r"></pb> <p type="head"> <s>PERIGONIA, O VERO <lb></lb>DEGLI ANGOLI. </s> </p> <p type="head"> <s>Di Teofilo Gallaccini </s> </p> <p type="main"> <s>Quantunque secondo l’opinione d’alcuni le Scienze Matematiche sieno di maniera dichiarate, che si mostrano non haver bisogno di più chiarezza; con tutto ciò, avendo soggette alla speculatione dello ‘ntelletto nostro son capaci d’altre considerationi; onde non sarà maraviglia se le medesime cose della Geometria secondo diversa consideratione benchè determinate pel diverso modo di considerare si facciano diverse. </s> <s>Onde l’Angolo essendo una particella di tutto l’obietto della Geometria, considerato da noi in varie maniere e secondo la diversa applicatione e secondo gli effetti diversi, diverrà diverso, dico (favellando filosoficamente) non nell’esser reale, ma nel formale; ed essendo diverso ci darà occasione di considerare di lui diverse positioni le quali si notaranno qui appresso.</s> </p> <p type="main"> <s>Se l’angolo sia cosa reale o intelligibile o immaginaria. </s> <s>Questa si considerarà. Nel 1° cap. </s> </p> <p type="main"> <s>L’angolo di quante maniere sia. Nel 2° cap. </s> </p> <p type="main"> <s>Che cosa sia l’angolo ed in che cosa sia collocata l’essenza sua. Nel 3° cap. </s> </p> <p type="main"> <s>Che per la varia division del cerchio si ritrovano tutte le specie degli angoli. Nel 4° cap. </s> </p> <p type="main"> <s>D’altri tagliamenti del cerchio onde risultano diverse maniere d’angoli. Nel 5° cap. </s> </p> <p type="main"> <s>Se ogn’angolo sia divisibile. Nel 6° cap.</s> </p> <p type="main"> <s>Se si dà l’angolo indeterminato sì come si dà il determinato. Nel 7° cap. </s> </p> <p type="main"> <s>Se l’angolo si riduca al tutto alla pianezza e ugualità della linea retta o vero alla curvità della circolare. Nel 8° cap. </s> </p> <p type="main"> <s>Per qual ragione alcune volte il cerchio sia detto tutto angolo. Nel 9° cap. </s> </p> <p type="main"> <s>L’angolo a che serva nell’Universo. Nel 10° cap. </s> </p> <p type="main"> <s>Di quale utilità e di che uso sia l’angolo nella Geometria. Nel cap. 11° </s> </p> <pb pagenum="folios 1v-2r"></pb> <p type="main"> <s></s> </p> <p type="main"> <s>Nell’Astronomia. Nel cap. 12° </s> </p> <p type="main"> <s>Nella Prospettiva. Nel cap. 13° </s> </p> <p type="main"> <s>Se ‘l colore è obietto proprio del vedere. Nel cap. 14° </s> </p> <p type="main"> <s>L’uso degli angoli nella prospettiva appartenente agli specchi e a’ reflettimenti de’ raggi del Sole. Nel cap. 15° </s> </p> <p type="main"> <s>Nella Prospettiva scenografica. Nel cap. 16° </s> </p> <p type="main"> <s>Nelle Meccaniche. Nel cap. 17° </s> </p> <p type="main"> <s>Nell’Architettura ornata e militare. Nel cap. 18° </s> </p> <p type="main"> <s>Nell’Arte militare. Nel cap. 19° </s> </p> <p type="main"> <s>Nell’Agricoltura. Nel cap. 20° </s> </p> <p type="main"> <s>Nella Navigatoria. Nel cap. 21° </s> </p> <p type="main"> <s>Nel Disegno, in quanto abbrevia la Pittura, la Scoltura e la Plastica. Nel cap. 22° </s> </p> <p type="main"> <s>Nell’Arti fabrili. Nel cap. 23° </s> </p> <p type="main"> <s>Se alcuna volta l’angolo si può chiamar tagliamento di linea, di figura o di corpo. Nel cap. 24° </s> </p> <p type="main"> <s>Se gli angoli solidi e piani sieno i medesimi. Nel cap. 25° </s> </p> <p type="main"> <s>Tutte queste cose prenderemo a considerare per dimostrar altrui in qual cosa consista la natura, la proprietà e la vista dell’Angolo, il quale via più prontamente d’ogni altra cosa geometrica ci si è offerto perciò che, essendo due gli accidenti della linea, principale elemento della geometria, uno che è l’esser infinita, e potersi dividere e terminare: e l’altro il cagionar col mezzo del toccamento d’altra linea l’angolo, ed esendo più noto quest’effetto che altro accidente, questo solo ci dà occasione di contemplare tutto quello che appartiene alla sua natura. </s> <s>Onde con l’aiuto del divino favore daremo cominciamento.</s> </p> <p type="main"> <s>//</s> </p> </chap> <chap> <p type="head"> <s>Se l’angolo sia cosa reale o intelligibile o immaginaria</s> </p> <p type="head"> <s>Cap. 1</s> </p> <p type="main"> <s>In due maniere al parer mio par che l’angolo si possa considerare, cioè o in quanto al fondamento e alla materia in cui si truova (che non è inconveniente far questa consideratione nella Geometria; perciochè la Matematica considera le cose in quanto sono in materia, benchè di essa niente si curi!) overo in quanto a se stesso, cioè come è fatto di cose intelligibili, ed in quanto la sua forma è più tosto intelligibile che sensibile; che non è altro che un dire che degli Angoli altri sono materiali, altri intelligibili, i quali sono puramente matematici. </s> <s>E per cominciar da questo membro della divisione con ordine contraposto dirò che sì come dalla materia sensibile si ritrahe la forma sensibile (almeno per accidente se non per sé) così per una certa proportione dalla potenza della materia intelligibile si cava la forma intellligibile. </s> <s>Perciochè se le Matematiche hanno la materia intelligibile, che in quanto materia non è senza potenza overo habilità: e questa non dee esser superflua; dunque necessariamente bisogna che da essa si tragga qualche forma, la quale non può esser non proportionata alla materia; però se la materia è intelligibile, la forma sarà intelligibile. </s> <s>La materia dell’angolo senza dubbio alcuno è ‘l punto e le linee, che non sono obbietti del senso per loro stessi, ma dello ‘ntelletto, e perciò si appellano intelligibili; onde ragionevolmente la materia dell’Angolo è intelligibile, dalla quale è inconveniente farsi astrattione. </s> <s>La forma di esso, che si toglie dal grembo di questa materia è lo stesso contatto delle linee in uno stesso punto, overo la stessa inclinatione e congiognimento in un punto comune. </s> <s>La quale è intelligibile perciochè solamente si apprende <pb pagenum="folios 2v-3r"></pb>dallo intelletto e perché prende la qualità dalla natura della materia onde si leva. </s> <s>Oltre acciò se quella forma si appella materiale; che si stacca dal seno della materia e quella immateriale che da essa non si ritrahe; per qual cagione quella forma che si cava dalla potenza della materia intelligibile non si potrà dire intelligibile? Adunque, per tutte queste ragioni concludiamo l’Angolo esser cosa intelligibile. </s> <s>Ma si cerchi hora se è vero che sia cosa reale, cioè in quanto al fondamento ed alla materia da cui si regge. </s> <s>Chiara cosa è che l’Angolo, fra tutta l’università delle cose, non si può truovar se non in due generi, cioè o nel naturale, o nell’artificiale; perciochè se riguardiamo le cose naturali, o l’artificiali vi scorgeremo ogni maniera d’angoli. </s> <s>Perciochè le cose artificiali, o sieno opere overo stromenti, quasi tutte son terminate con figure, che hanno qualche specie d’angolo, escettuate però tutte quelle che hanno la figura sferica e ritonda. </s> <s>Se altri volesse addurre gli essempij di tutte si allongarebbe troppo il ragionamento, anzi sarebbe superfluo, essendo tutto ciò notissimo a ciascuno. </s> <s>Questo solo dirò degli Angoli artificiali, che essendo essi o a squadra o a soprasquadra, o a sottosquadra, sì come si ritrahe da Leon Battista Alberti nell’Architettura, non posson non esser reali; perciochè vengono così appellati da uno stromento reale, e naturale, che gli produce, il quale è la squadra, che materialmente mostra l’angolo retto. </s> <s>E le cose naturali essendo per lo più corpi terminati da qualche ragion di figura, oltre a quelli che son di figura in tutto d’Angoli nuda, sono in gran parte angolari; ciò sono o pietre o gemme o piante overo isole. </s> <s>Fra l’isole due sono nell’Italia nel lago dell’Anguillara, che si mostrano hora di figura triangolare hora di ritonda; ma di quadrata <lb></lb>//<lb></lb>non mai, sì come si può vedere appo Giorgio Agricola nel 2° lib. </s> <s>Della natura delle cose, che dalla terra scorreno. </s> <s>Le gemme si truovano di figure angolari, come l’Androdomante, il Basalte Miseno, il Cristallo, il Pangonio, il Diamante ed altre come si può vedere appo Giorgio Agricola nel primo lib. </s> <s>Della natura delle cose fossili. </s> <s>Così ancho le pietre si trovano angolari, come sono quelle presso a Bolseno in una grotta dove mostrano quasi una muraglia simile a quelle degli antichi Romani, d’opera reticolata. </s> <s>Si truovano ancora alcune pietre sciolte di grandezza alquanto meno di una nicciuola e di colore simile al ferro di figura esagona formate a ponta di diamante da ogni banda in guisa tale che mostrano due piramidi esagone. </s> <s>Nella margassita si truovano alcune volte noccioletti di forma cubica; e così ancho fra queste cose si veggano altre figure angolari. </s> <s>Le piante anchora si mostrano la maggior parte angolari; perciochè il Cipero, il Gionco odorato, l’Assaro, la Cicerbita, l’Aragallide, la Centaurea minore, la Baccara, il Marzobbio, l’Hormino, la Bettonica, il Sinfito, il Climeno, l’Otiopide, il Cirsio, il Bunio, la Lappa minore ed altre anchora hanno diversamente i fusti angolari. </s> <s>Oltre acciò i fiori, le foglie e ‘semi sono in diverse maniere angolari, come ‘l seme del Sezeli Massiliense è quadrato, sì come si può ritrarre da Dioscoride. </s> <s>Di modo che quindi possiamo concludere che per cagion delle cose naturali, ed artificiali, nelle quali vediamo le specie degli Angoli, l’Angolo habbia qualche realità. Oltre a questo quelle cose sono reali, che si ritruovano fuor dello ‘ntelletto humano; onde l’Angolo essendo nelle cose artificiali o nelle naturali si truovarà insieme con esse fuor dello ‘nteletto e perciò potrà dirsi reale. </s> <s>Ansi, queste fuor dello ‘ntelletto <pb pagenum="folios 3v-4r"></pb>non possono esser se non corporee terminate da qualche maniera di figura: overo appartenenti a corpi in quanto son termine di essi, o delle superficie loro; onde necessariamente bisogna che habbiano Angoli e sotto essi apparischino. </s> <s>Perciochè (sì come il Padre Ignatio Danti nella Prospettiva d’Euclide, sopra la 8a suppositione di esse) tutte le cose visibili vedersi sotto qualche angolo, poiché la figura compresa da’ raggi visuali è un conio la cui ponta si ferma nel centro dell’occhio, nella quale i medesimi raggi formano angoli diversi secondo la diversità delle cose vedute. </s> <s>Onde sì come l’atto del vedere de’ nostri occhij è cosa reale, così ‘l conio formato da’ raggi visuali sarà reale; perciochè il vedere stesso non si forma se non col mezzo del detto conio, la cui base è l’obbietto e ‘l cui termine è ‘l centro dell’humor cristallino dell’occhio, amendue cose reali; di maniera che essendo reale il conio non può non esser reale l’angolo da esso formato. </s> <s>Non starò hora a cercar se la visione si faccia per ricevimento di specie, come vogliono i Peripatetici: o per mandar fuore i raggi visuali, come piace a’ Platonici. perciochè questo luogo non è proportionato a tali questioni; onde le lassarò trattare a’ Filosofi, mentre delle cose dell’anima discorrono. </s> <s>Mi bastarà bene haverne accennato in quanto a quella parte che matematicamente si può considerare per pruovar la realità dell’Angolo. </s> <s>Da queste cose anchora si può ritrarre che l’Angolo sia cosa reale; che si ritruova fra i dieci generi delle cose detti da’ Filosofi “Predicamenti”. Onde Proclo sopra Euclide mostra che de’ Filosofi antichi altri posero l’angolo nel predicamento della relatione altri nella quantità, ed altri nella qualità; in qual <lb></lb>//<lb></lb>unque sia di questi generi, e cagione che inferiamo che sia cosa reale. </s> <s>Ma vediamo hora se l’Angolo si possa dir cosa immaginaria. </s> <s>Le cose immaginarie sono di due ragioni; perciochè o sono puramente immaginarie o impuramente. </s> <s>Le impuramente immaginarie son quelle che parte son reali e parte immaginate. </s> <s>La parte reale si è ‘l primo concetto che è fondamento e origine della immaginatione, cioè del secondo concetto: overo quella simiglianza che ha la immaginatione della cosa con la cosa reale, o con quella parte che è reale. </s> <s>Perciochè (come dice Temistio nel 3° dell’Anima, cap. 13°) l’immaginatione è un certo vestigio del senso; perciochè si forma dalle cose sensate. </s> <s>E però la cosa immaginaria, non puramente tale è un certo vestigio della cosa sensata. </s> <s>Ma la parte immaginaria si è quella cosa che è similitudine del secondo concetto separato in tutto dalla cosa reale. </s> <s>Le puramente immaginarie son quelle cose, che l’Immaginativa per se stessa si è fabbricate senza haverne alcuna simiglianza nelle cose sensate, e per dirla più apertamente sono le stesse fintioni e le stesse chimere e capricci che fuor della Immaginativa non hanno essere. </s> <s>Però Alesandro Afrodiseo sopra ‘l primo libro dell’Anima, nel cap. 2°, divise le immaginationi in vere e false. </s> <s>Quasi dir volesse che le pure immaginationi fosser false e le non pure vere. </s> <s>Le pure immaginationi, per Gio. </s> <s>Grammatico, nel 3° dell’Anima, parte 16a, son quelle che fingono le forme: e le non pure quelle che ricevono le forme, cioè le specie e simiglianze dalle cose sensate e reali. </s> <s>Così le cose immaginarie, che sono effetti dell’immaginatione, saranno o vere o false, o finte o reali, o pura fintione, o simiglianza di cosa sensata e reale. </s> <s>Hora per applicar tutto questo discorso al proposito nostro diremo <pb pagenum="folios 4v-5r"></pb>che l’angolo considerato, in quanto si fonda in cosa reale, essendo uno effetto delle linee ritruovate in materia, non dee dirsi puramente cosa immaginaria, ma non puramente, cioè in quanto alla dependenza che è schiettamente sensata e reale: ed in quanto alla simiglianza d’una cosa reale. </s> <s>E così si può dire, che l’Angolo sia cosa immaginaria, ma nel secondo modo non si conviene affermare; perciochè sarebbe al tutto cosa falsa, e le Matematiche non sarebbero fondate in principij veri e certi; ma in falsi ed incerti. </s> <s>Che tanto è dire l’Angolo esser puramente immaginario quanto falso e incerto: e ponendosi falso e incerto, le linee e ‘l punto che concorrono alla sua produttione sarebbero al tutto falsi ed incerti, onde seguirebbe la Geometria esser collocata in cosa non vera e non haver certezza alcuna. </s> <s>La qual cosa fu ingegnosamente avvertita da Niccolò Tartaglia nel primo lib. della sua Geometria, nel cap. primo. </s> <s>Adunque o non bisogna dire che l’angolo sia cosa immaginaria o se pur si dice più tosto si affermi che sia cosa non puramente immaginaria, che altramente; perciochè in questa maniera è più ragionevole.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Di quante maniere sia l’angolo</s> </p> <p type="head"> <s>Cap. 2</s> </p> <p type="main"> <s>Fra le cose naturali ed artificiali due generi sono che abbracciano la diversità degli Angoli. </s> <s>Il primo contiene tutte le superficie piane, il secondo tutti i corpi solidi, perciochè sì come al producimento degli Angoli nelle superficie piane concorreno le linee così alla costitution di essi ne’ solidi concorreno insieme con le linee le superficie anchora; perciochè sì come nel primo modo le linee son fra loro inclinate, così nel secondo sono le superficie, come si può trarre dalla quinta definitione dell’undecimo libro di Euclide. </s> <s>E questa divisione si ritrahe da Proclo. </s> <s>Ma ripigliando la di <lb></lb>//<lb></lb>visione, si dee ripartire tutto ‘l genere degli Angoli, che si ritruovano nelle superficie piane; secondo la qualità delle figure che le terminano; perciochè altre sono di figura rettilinea, altre di curvilinea, e altre di mista; di modo che gli Angoli delle rettilinee si dicano rettilinei, que’ delle curvilinee curvilinei, que’ delle miste misti: e come le due prime specie di figure sono simplici, così gli angoli loro sono simplici: e come la terza specie delle figure è mista, così gli angoli son misti. </s> <s>Così anchora considerando di nuovo il 2° genere si vede tutta la diversità degli Angoli de’ corpi solidi nascer dal vario producimento loro. </s> <s>Perciochè o si producano dal tagliamento fatto da diversi cerchi, o dal diametro, o dall’asse, come avviene nella sfera, ansi in ogni corpo sferico: o dal percotimento, che fanno i raggi del Sole nello specchio concavo, o convesso, mentre si ripiegano: o dal cadimento, che fanno le cose sopra ‘l piano, le quali balsando si torrono ‘l movimento, come le palle da giuocare battute sopra ‘l piano: o dal concorso delle superficie del corpo solido ugualmente e da figure uguali e della medesima specie terminato e da linee terminanti uguali, come avvien ne’ corpi regolari, overo da superficie variamente figurate e disuguali e di specie diversa terminato disugualmente insieme con linee disuguali, come accade ne’ corpi irregolari: o da linee, che habbiano diversa positione, come se altre sieno nel piano, ed altre sopra ‘l piano cadenti sopra esse, in maniera che in uno stesso tempo le tocchi tutte in guisa tale che d’ogni ‘ntorno produca gli angoli uguali, come avviene negli spatij e ne’ vani delle stanze: o da due superficie piane circolari opposte, e da una curva che col mezzo di due circonferen <pb pagenum="folios 5v-6r"></pb>ze si congiogne con essa ad angoli retti, come avviene nel cilindro o nella colonna: o da due superficie una tonda e conessa ristretta in un punto ed una piana e circolare, che è la base, che seco si congiogne per una circonferenza, come si vede nel conio: overo dal raccoglimento di cinque superficie, che per mezzo di linee disuguali si legano insieme delle quali la minore che è base è opposta all’Angolo della cima, ed è di figura quadrata; e questo avviene alla piramide. </s> <s>Di maniera che ne’ solidi quante sono le maniere del producimento degli angoli, tanti sono gli angoli; in quanto che ‘l diverso modo gli dà qualche diversità, come si vede chiaro da chi bene osserva el componimento di tutti i corpi solidi. </s> <s>Ma finalmente tutte le specie degli angoli solidi si riducano a due generi, cioè all’angolo composto ed all’angolo simplice: al composto: al composto che si mostra in due foggie, nella prima cioè come è composto d’angoli di diversa specie, come di rettilineo e misto: nella seconda come è composto d’angoli della medesima specie, cioè di rettilinei: al simplice in quanto che si riduce all’angolo minore che è elemento degli altri angoli e questo ne’ solidi è composto di più angoli minimi, ed è simplice rispetto agli altri angoli massimi de’ solidi, poiché questi son composti di quelli, come si vede nell’acutezza delle piramidi. </s> <s>E per abbracciar insieme tutti gli angoli e piani e solidi, bisogna insieme con Euclide ridurgli a tre specie, ciò sono retto, ottuso ed acuto.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Che cosa sia l’angolo ed in che sia collocata la sua essenza</s> </p> <p type="head"> <s>Cap. 3</s> </p> <p type="main"> <s>E’ cosa convenevole che, avanti che ricerchiamo la definition dell’angolo, vediamo il significato del suo <lb></lb>//<lb></lb>nome. </s> <s>E adunque l’angolo detto da’ Greci “gonia”; onde le cose angolari grecamente si appellano “goniadis”; onde Varrone, nel primo libro della Lingua latina dice che l’angolo però si appella gonia; perciochè in esso sia un luogo angustissimo; che non significa altro che un certo restrignimento, che risulta da due linee, che si terminano in un ponto commune; poiché non per altro da Proclo è detto una simiglianza di ristregnimento. </s> <s>Oltre acciò l’angolo non denota altro, che un certo gombito, o cantone, grecamente detto anchon, prodotto dal piegamento e congiognimento di due righe, sì come si può vedere appo Vitruvio nel lib. 3° cap. 3°, dove favellando delle colonne scannellate, dice la scannellatura farsi in guisa, che postavi dentro la squadra, e girata intorno, tocchi con le sue piegate righe il concavo e l’estremità del canale. </s> <s>Il piegamento adunque che fa la squadra formando angolo retto si dice anchon grecamente; perciochè le righe che la compongono separate dalla squadra hanno altro nome; che si appellano regole; ma nella squadra, perché congiognendosi formano un piegamento a modo di gombito che non è altro che ‘l luogo, dove si congiongano l’ossa del braccio, che si chiama cubito, overo gombito. </s> <s>Però non più regole, ma anchone si chiamano; ansi i piegamenti di queste righe in fra loro scambievoli sono chiamati anchones da Guglielmo Filandro, nell’ Annotationi sopra Vitruvio nel 3° lib. 3° cap., sì come anchora da Vitru. nel 8° lib. cap. . . si esplica. </s> <s>Perciochè quivi egli dice que’ piegamenti, che fanno angoli retti si chiamano anchones, sì come vediamo nella squadra, etc. </s> <s>E molto meglio Daniel <pb pagenum="folios 6v-7r"></pb>Barbaro l’esplicò; perciochè egli esponendo ‘l medesimo luogo, disse: anchones sono le braccia della squadra etc. </s> <s>Dalla simiglianza forse di questo piegamento, Ancona, città della Marca, è stata così nominata; per cagion della figura, che rappresenta il suo posto. </s> <s>Così anchora è nominata una parte del Nilo, la quale è navigabile finalmente agone, overo agonia è un luogo, dove si facevano i giuochi degli antichi, nel quale, nella parte principale, era di figura non angolare; che perciò (secondo che è piaciuto ad alcuni) era chiamato agone, cioè privo d’angoli. </s> <s>Oltre acciò, se si riguarda bene l’angolo, non è altro che quell’accostamento di due cose vicine, le quali, da diverse bande tirate, si strengano per far la forma dell’angolo: e vien dalla parola englis (sì come dice Festo) che vuol dir “esser” vicino ciò che si ha da ristregnere; che egli dice “englis”, cioè che presso si accosti. </s> <s>Ma lassiamo da parte queste cose le quali in qualche parte si allontanano dall’essenza dell’angolo; che in questo luogo intendiamo cercare di definirlo con una definitione, o almeno con una descrittion commune. </s> <s>Dico adunque cercar l’essenza dell’angolo non vi esser altro mezzo che l’osservation del suo producimento. </s> <s>Noi vediamo formarsi l’angolo con la inclinatione delle linee ed eseguirsi diversamente, cioè o dirittamente overo obliquamente; obliquamente quando da tali inclinationi risultano angoli disuguali: dirittamente quando ne risultano angoli uguali. </s> <s>Dirittamente dico, non quando una linea è per diritto dell’altra, o si continua con l’altra; <lb></lb>//<lb></lb>ma quando sopra una linea cade un’altra perpendicolare. </s> <s>Obliquamente, quando una linea si piega o più o meno sopra la linea piana; onde più o meno si accosta alla linea piana. </s> <s>Supposta questa cognitione, facilmente potremo addurre una deffinition commune a tutti gli angoli ed è:</s> </p> <p type="main"> <s>“L’angolo è un concorso e contatto di linee in un punto commune; che per la diritta e per l’obliqua inclinatione di esse, riceve diversità di specie e di grandezze”.</s> </p> <p type="main"> <s>E’ un concorso di linee perciochè tutte le linee, che fanno angolo concorreno insieme, cioè convengono e si congiogano in un punto, il che non avviene alle parallele, come è manifesto per la trentacinquesima def. del primo d’ Euclide; che se accadesse, non sarebbero più parallele, come si vede nel 5° postulato e nella dimostration del 18° Teorema del primo. </s> <s>Perciochè esse non fanno angolo; onde avviene che non chiudino spatio da banda alcuna. </s> <s>E’ un contatto; perciochè nel concorrere insieme le linee si toccano in uno stesso ponto, che le congiogne, e continua: e questo concorso e contatto è commune a tutte le maniere degli angoli, o rettilinei o curvilinei, o misti. </s> <s>Si dice farsi in un ponto; perciochè le linee che producono e chiudono l’angolo, finiscono in un ponto commune, che è termine, e congiognimento delle linee, che si toccano insieme. </s> <s>Onde quantunque per loro stesse sieno contigue, con tutto ciò, col mezzo del punto divengon continue; sì come insegna Aristotile nel lib. … della Filosofia naturale part. … Ma per intender questo si dee notare, che altro è continuarsi simplicemente, che è lo stesso che allongarsi infinitamente, o indeterminatamente, come si vede nella dimostratione del Teor. 9° del primo d’Euclide <pb pagenum="folios 7v-8r"></pb>e ‘n quella del Teor. 13° e del Prob. 10, ed altro è continuarsi col mezzo del ponto, quasi che esso sia ‘l collegamento e ‘l chiodo, che congiogne due linee disgionte. </s> <s>Nel primo modo s’interpon sì bene ‘l ponto nel continuamento della linea, ma continuata si perde; perciochè non vi si cagiona l’angolo, o però non bisogna che vi rimanga ‘l ponto. </s> <s>Nel secondo in maniera s’interpone che vi rimane; perché mentre è mezzo e legamento delle linee contigue, è anchora termine commune di esse, e terminamento dell’angolo. </s> <s>Nella stessa guisa si può formar la definitione commune dell’angolo solido, ritrahendola dall’osservatione del suo nascimento. </s> <s>Ma solamente vi sarà diversità in quanto che ‘l concorso e ‘l contatto si fa non di linee sole, ma di linee e di superficie insieme: non terminante in un ponto solo, ma ancho in una linea commune, che riceve diversità di specie e di grandezze e per la diversità delle inclinationi di esse. </s> <s>A questo proposito non mi par disdicevole, per maggior dichiaratione dell’angolo solido, notar la definitione formata da Bonaiuto Lorini nella sua Fortificatione; perciochè egli nella quarta definitione dice: l’angolo si addomanda quella parte, dove due linee si congiongano insieme, cioè AB. BC., che si congiongano in B. dove formano in tal parte l’angolo. </s> <s>Dice quella parte perché intende dell’angolo materiale e solido; perciochè, nella fortificatione, come in tutta l’architettura, non solamente i ponti, le linee e le superficie e ‘corpi si considerano come applicati alla materia sensibile e trattabile, così ancho gli angoli che in tali cose consistono. </s> </p> <p type="main"> <s>Quindi adunque non solamente è manifesto qual sia l’essenza dell’angolo, ma anchora in qual cosa consista, cioè nella materia intelligibile, che sono le linee e le superficie, volendo noi non trapassare i confini della pura Geometria: e nella forma parimente intelligibile, che è l’accostamento e contatto di due linee condotto e terminato in un ponto: overo di tre o più superficie, terminato in un ponto ed in una o più linee, come si vede ne’ cinque corpi regolari.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Che per la divisione del cerchio si ritruovano tutte le specie degli angoli</s> </p> <p type="head"> <s>Cap. 4</s> </p> <p type="main"> <s>Per intender meglio questa propositione, fa di mestiero supporre, come cosa nota e pruovata per la <lb></lb>//<lb></lb>Geometria la verità d’un’altra propositione così fatta. </s> <s>Ogni angolo potersi formare con la scambievole intersegation delle linee rette o curve. </s> <s>E posta già questa notitia ed essendo vero che ogni figura angolare si forma dalla varia division del cerchio e non possendosi formar figura alcuna, senza la varia costitution degli angoli, sarà anchora vero che per la diversa division del cerchio si truovino tutte le specie degli angoli; onde avendo noi primieramente cognitione e certezza della verità del primo supposto, veniamo facilmente in cognitione e certezza della verità del secondo. </s> <s>E per confermation del primo si dee considerare, che le figure o regolari o irregolari son di due maniere, altre al tutto dependenti da altra figura simplice, uniforme e perfetta: altra non dependente da alcuna; ma da essa scaturiscono tutte. </s> <s>Quelle di questa fatta sono tutte le figure circolari; perciochè ‘l cerchio è quasi materia di tutte le figure rettilinee; poiché dal diverso tagliamento di lui si traggano tutte le specie di figure rettilinee. </s> <s>Ma quelle dell’altra maniera sono tutte le figure rettilinee che dal detto tagliamento si formano, e perciò dependono dalla figura uniforme, simplice e perfetta, che è la circolare, la quale non ha dependenza da altra figura; che perciò si è detta simplice, ma da essa dependono tutte; poiché da lei si traggano, come dal seno della materia, col mezzo del taglio diversamente fatto. </s> <s>Adunque tagliato variamente ‘l cerchio si formano diverse foggie di rettilinee figure, e così per conseguenza diverse maniere d’angoli. </s> <s>Ma per intender meglio tutta la natura del detto tagliamento, si dee osservare, che in diversi modi dalla pratica geometrica si può eseguire; perciochè o si fa dalla metà del diametro o dal diametro intero, o dalla diversa division del diametro o dalla corda o dalla linea detta saetta, che (secondo <pb pagenum="folios 8v-9r"></pb>Daniel Barbaro, nel primo lib. di Vitru. cap. 5) è quella linea, che dal mezzo della corda con angoli uguali ascende alla circonferenza dell’arco: o dalla intersegation fatta da altri cerchi. </s> <s>Consideriamo adunque ciascuna specie di questi tagliamenti del cerchio, acciò con più facilità veniamo a ritrarne il nascimento di qualunque maniera d’angolo, e primeramente con ordine ripiegato, si consideri come dalla intersegation de’ centri si producano gli angoli. </s> <s>Questo si dimostra da Euclide nel primo Problema del primo degli Elementi, là dove dalla intersegation di due cerchi trahe la formation del triangolo equilatero; onde risultano tre angoli acuti. </s> <s>Nel qual luogo il Commandino, con la medesima intersegation de’ cerchi, ma fatta diversamente, formando i triangoli equicruri e scaleni produce gli angoli acuti e gli ottusi: e noi anchora variando ‘l modo della intersegatione nella sposition del medesimo Probl. formando altro triangolo ritroviamo l’angolo retto, come si vede qui appresso. </s> <s>Nel qual modo si suppon la pratica insegnata da Euclide e si allonga da ogni banda la linea retta AB. e posto ‘l centro D. e lo intervallo DE. si descriva ‘l cerchio EFG. così posto ‘l centro E. e lo in ED si descriva ‘l cerchio DFH. Di maniera che amendue si taglino nel punto F. Quindi l’arco FD. si divida in tre parti uguali una delle quali si trasporti due volte nell’arco FH., cioè accanto al taglio F. che saranno nei tagli FKL e divisa la parte KL. per mezzo, vi si faccia ‘l taglio I. che sarà per diritto sopra ‘l centro E. Di poi si congiongano i tagli ID. IE. vedremo cagionarsi ‘l triangolo rettangolo IED. <lb></lb>//<lb></lb>con l’angolo retto IED. costituito dalla perpendicolare IE. per la decima def. del primo d’Euclide. </s> <s>Alla qual cosa si sottoscrive Amonio sopra Porfirio dicendo, che gli angoli retti non posson muoversi se non al centro. </s> <s>Perciò che quivi consiste naturalmente la quiete; onde da Aristotile nelle Meccaniche, tale angolo è detto angolo della quiete. </s> <s>E ‘l punto E. dove detto angolo si termina non è altro che ‘l centro del cerchio DFH. e ciò avviene per la intersegation de’ cerchi minori e maggiori col mezzo de’ quali si truova ‘l punto onde dee partirsi la perpendicolare e ‘l punto dove dee terminare sopra una linea retta piana già data la quale è DBAC. e che ciò sia vero, con la stessa operatione del su detto primo Prob. facendo centro il segno D. e intervallo la linea DH. si faccia la portion del cerchio LM. e facendo centro ‘l segno H e intervallo HD. si faccia la portione del cerchio KN. la quale tagliarà la LM. nel segno O. necessariamente la perpendicolare che da esso si muovarà cadrà nel centro; perciochè ‘l taglio O. e ‘l centro E. son diritto fra loro, e si fanno termini della linea perpendicolare. </s> <s>Oltre acciò, pel secondo Postulato di Euclide, allongata la perpendicolare IE. fine al P. che finisca nella circonferenza del cerchio DFH. e allongata la linea piana DE. fine in H. che termini nella circonferenza del medesimo cerchio necessariamente il detto cerchio verrà tagliato in quattro quarte uguali, perciochè le linee DH. IP. sono diametri che lo dividono prima in due e poi in quattro parti uguali; che la DH. il taglia per metà e la IP. divide la detta metà in due parti uguali, e questo avviene infallibilmente perciochè i diametri che dividono due volte ‘l cerchio si tagliano fra loro in due parti uguali; perciochè passano pel centro; che se non passassen pel centro non potrebbero dividersi insieme in due parti uguali come si dimostra da Euclide nella quarta prop. del 3°, onde non si taglierebbe ‘l cerchio in quattro quarte uguali; determinate da quattro <pb pagenum="folios 9v-10r"></pb>tagliamenti ugualmente lontani fra loro, cioè DIHP. i quali congionti con quattro linee si costituiscan quadrato e per conseguenza quattro angoli retti; perciochè ciascuno è nel suo mezzo cerchio, come si può pruovare con la trentunesima del 3° e ciascun lato corrisponde a ciascuno arco della quarta del cerchio; che dalla quarta parte del cerchio si trahe la quarta del quadrato; che tal proportione ha un lato del quadrato con la parte dello spatio a lui sottoposta a tutto ‘l quadrato, quale ha una quarta del cerchio a tutto ‘l cerchio. </s> <s>Si costituiranno dico da quattro tagliamenti del cerchio quattro angoli retti che non possano non essere uguali fra loro sì perché questa è proprietà degli angoli retti; sì perché ne’ detti tagliamenti si costituisce lo spatio parallelogrammo, che ha gli angoli e lati opposti uguali fra loro, come si vede nella prop. trentaquattresima del primo. </s> <s>Onde si potrebbe ritrar un corollario, cioè:</s> </p> <p type="main"> <s>“Che dentro un cerchio, ritruovate quattro quarte uguali e stabilito un triangolo rettangolo si costituisca un quadrato e truovato pel suo tagliamento un angolo retto, facilmente si possa haver cognition di quattro retti.”</s> </p> <p type="main"> <s>Ma questo, oltre che è noto per le cose dette, non si dimostrarà per fuggir la longhezza. </s> <s>Si è detto che dalla saetta si divide ‘l cerchio onde si cagionano gli angoli. </s> <s>Perciò che questa linea che è perpendicolare in quanto all’effetto, ma è detta metaforicamente saetta, sorgendo dal mezzo della corda, non può non tagliare il cerchio; e questo accadendo diversamente secondo la diversa positione della corda, ancho ‘l cerchio diversamente vien diviso; onde quantunque da questa linea che è retta sia tagliato il cerchio ad angoli retti sferali, od almeno da essa, e dalla circonferenza sieno costituiti angoli misti; con tutto ciò considerata la dispositione e l’habilità de’ tagliamenti fatti nel cerchio, vedremo espressamente sorger <lb></lb>//<lb></lb>diverse specie d’angoli, secondo la diversità delle figure rettilinee che per essi tagli si formano, come vedrà con esperienza qualunque è esercitato nelle pratiche del Compasso nel formar le figure moltilatere, con la duplication de’ lati ritruovata dal taglio fatto per modo di saetta con la forza dell’arcuatione, come in altro luogo da noi s’è ragionato alla longa, e come si vede in questa figura. </s> <s>La qual cosa si vede espressa nella trentunesima del 3°, là dove si mostra l’angolo minore del retto esser nella maggior portione del cerchio, e l’angolo maggiore esser nella minore e quindi nasce la diversità delle figure e degli angoli, cioè dalla diversità delle portioni de’ cerchi sotto le quali si truovano.</s> </p> <p type="main"> <s>Si è detto ‘l cerchio dividersi dalla corda, la quale è quella linea retta che ‘l taglia non passando pel pel centro in due portioni disuguali per la diciannovesima def. del primo degli Elementi. </s> <s>Si appella corda dagli Architetti e dagli artefici, perché si servano delle cose geometriche adattandole alla materia. </s> <s>Questa linea tagliando ‘l cerchio è cagione degli angoli retti e degli acuti, mentre sopra i tagliamenti fatti nella sua circonferenza si formano i quadrati e triangoli equilateri, come si vede qui nell’essempio. </s> <s>Ciascun lato di queste figure si appella corda perciochè taglia ‘l cerchio in due portioni minori e maggiori AB e CDAB. overo ACBD e DB. nel primo essempio DEF. ED. nel secondo. </s> <s>Si divide anchora ‘l cerchio per la diversa division del diametro. </s> <s>Cioè dividendo prima ‘l diametro in vintiquattro particelle uguali, e per formare ‘l triangolo equilatero, acciò vediamo gli angoli acuti, <pb pagenum="folios 10v-11r"></pb>bisogna porre un piè del compasso in una estremità del diametro, allargando l’altro tanto che prenda la vintuna particella, e con tal misura facendo l’arco, si tagli da ogni banda la circonferenza; onde il detto taglio determinarà la grandezza d’un lato del triangolo, la quale replicata due volte su pel cerchio, chiuderà la figura e determinarà gli angoli. </s> <s>Ma volendo noi formare il quadrato per trarne gli angoli retti, nella medesima guisa si prendano vintidue portioni ponendo prima un piè del compasso di sotto e formando l’arco di sopra, che tagli ‘l cerchio da ogni parte: e poi ponendolo di sopra per formar l’arco di sotto, che tagli ‘l cerchio; si truovano due misure che stabiliscano la uguale longhezza di due lati d’un quadrato, onde segue la grandezza degli altri. </s> <s>Con 14 portioni si truova un arco che tagliando ‘l cerchio stabilisce un lato del pentagono onde risultano cinque angoli ottusi. </s> <s>Con 9 l’ottangolo, con 12 l’esagono, con 10 e _ la figura di sette angoli. </s> <s>E così si potrebbe con diverso numero di parti del diametro truovare il tagliamento del cerchio per formar ciascuna maniera di figure moltilatere, per costituir le diversità degli angoli: o in specie, come retti, ottusi e acuti: o in grandezza, come più o meno acuti o ottusi; che i retti non variano nella grandezza già mai, essendo sempre uguali; son ben diversi di specie dagli altri. </s> <s>Ma proponiamo l’essempio sensato, acciò più facilmente s’intenda la pratica di questo tagliamento del cerchio. <lb></lb>//<lb></lb>AB. è ‘l diametro diviso in 24 parti uguali. </s> <s>CD è un arco che costituisce un lato del triangolo. </s> <s>EF. GH. sono due archi che formano un quadro. </s> <s>IK. è un arco che determina due lati del pentagono. </s> <s>LM. un altro che determina due lati dell’esagono. </s> <s>NO un altro che stabilisce due lati dell’ettangolo. </s> <s>PQ un altro che forma due lati dell’ottangolo. </s> <s>Di modo che congiognendosi in qualsivoglia tagliamento del cerchio i punti de’ tagli si formano le figure di molti lati, e si stabiliscano anchora gli angoli. </s> <s>Con la division del diametro si divide ‘l cerchio dallo ‘ntero diametro perciòchè sempre si taglia per mezzo ad angoli misti nella superficie piana, e ad angoli retti sferici nella superficie sferica, sì come è ‘l tagliamento <pb pagenum="folios 11v-12r"></pb>che si fa dall’equinottiale, che divide tutta la Sfera in due parti uguali, come dice Giovanni Sacrobosco nel cap. 2° della Sfera. </s> <s>Ma in apparenza sono misti; perciochè quantunque l’equinottiale sia un cerchio, ansi uno de’ cerchij maggiori della Sfera, contuttociò per la lontananza, essendo parallelo al diametro e per la dirittura con esso insieme con l’occhio nostro, per ragion di Prospettiva, si mostra il diametro stesso della sfera. </s> <s>Perciochè, talvolta, secondo Prospettiva lo spatio di qual si voglia cosa si vede in una linea, cioè per la digradatione e scemamento all’apparenza ridotto ad una linea. </s> <s>Come suol fare il Sole collocato nel cerchio equinottiale, il cui centro della circonferenza dell’equatore, e ‘l cui raggio dallo stesso piano giognendo al centro del Mondo non si ripiega per far conio, ma si ferma secondo ‘l suo movimento in esso, cioè descrivendo il cerchio stesso equatore, poiché fra la medesima circonferenza e ‘l centro del Sole è per diritto del centro del Mondo. </s> <s>Ma ciò si può vedere nella terza prop. della Gnomonica del Padre Cristoforo Clavio: overo ciò accade perché ‘l cerchio equatore rappresenta una superficie, che taglia ‘l corpo sferico passando pel centro; che i corpi non si dividono se non dalla superficie immediatamente e per cagion di essa dalla linea anchora: e le linee da punti, e le superficie dalle linee; onde ‘l taglio della detta superficie sarà fatto dal diametro. </s> <s>Tal che in apparenza gli angoli che dal suo tagliamento son prodotti, si mostrano parte rettilinei e pate curvilinei; che nella circonferenza della Sfera col mezzo del taglio fatto dalla superficie divisa dal suo diametro, si costituiscano dal diametro stesso, che è una linea retta, e della circonferen <lb></lb>//<lb></lb>za del detto cerchio posto a dirittura del diametro, che in apparenza è una linea retta. </s> <s>Oltre che i paralleli considerati da’ Cosmografi, sono cerchi fra loro ugualmente lontani, che dividono la circonferenza della sfera ad angoli retti, e con tutto ciò si descrivano in forma di linee rette per imitar l’apparenza loro. </s> <s>Onde, non fuor di proposito, sono appellati questi cerchi paralleli retti, come si vede appo ‘l Clavio, nella Gnomonica, ed appresso Teodosio nella prop. quindicesima del primo lib. benchè per altra ragione da esso sieno appellati retti. </s> <s>Si divide oltre acciò dalla metà del diametro, quando già formato cerchio, si applica sei volte sopra la circonferenza, e si taglia il cerchio in sei parti uguali, che la metà del diametro è la sesta parte della circonferenza: e con questo tagliamento si forma l’esagono, figura di sei angoli . Onde per questa divisione si producano gli angoli ottusi nella portion minore del cerchio. </s> <s>E queste son tutte le differenze de’ tagliamenti del cerchio, per le quali si formano varie specie di figure rettilinee, che non sono senza le diverse maniere d’angoli; e però mentre il vario tagliamento del cerchio c’insegna a formar diverse ragioni di figure, ci mostra anchora il nascimento degli angoli diversi; onde con ragione possiamo affermare, che per la varia division del cerchio si truovano tutte le specie degli angoli. </s> <s>Dalle cose dette si prende occasion di dubbitare che se ‘l triangolo è principio d’ogni composition di figura rettilinea, sì come dice il Commandino, nella Prop. trentaduesima del primo di Euclide, perciochè ciascuna si risolve ne’ triangoli, come ‘l composto naturale secondo ‘ Filosofi si risolve nelle parti componenti, che sono gli elementi della cosa composta; <pb pagenum="folios 12v-13r"></pb>per qual cagione s’è dimostrato la division del cerchio esserne principio, mentre costituisce diverse specie d’angoli? Si risponde che n’è principio ‘l triangolo e ne è principio ‘l tagliamento del cerchio, o per dir meglio il cerchio variamente diviso; ma in diversa maniera. </s> <s>Perciochè ‘l cerchio tagliato è principio della figura, in quanto stabiliti diversi tagliamenti nella sua circonferenza ugualmente lontani fra loro; e di esso molte portioni uguali scambievolmente per la congiuntion de’ detti tagli fatta con più linee rette, da esso risulta qualunque figura e per conseguenza ogni maniera d’angolo: e ‘l triangolo è principio dell’altre figure, cioè non della formatione, ma della compositione e congiognimento delle parti della figura già formata; e che ciò sia vero vi si scorgano virtualmente i triangoli, come si vede nel quadrato e nell’altre figure. </s> <s>Né per questo dobbiamo dire il triangolo esser figura simplice perfetta e indipendente; da cui piglino origine tutte l’altre; perciochè ‘l triangolo non dà ‘l nascimento delle figure; ma l’accozzamento delle parti componenti nelle quali esse si risolvano.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>D’altri tagliamenti del cerchio onde risultano diverse ragioni d’angoli</s> </p> <p type="head"> <s>Cap. 5</s> </p> <p type="main"> <s>I tagliamenti del cerchio nel precedente cap. dimostrati si fanno dalla linea retta e dalla scambievole intersegation de’ cerchij; onde scaturiscono diverse maniere d’angoli. </s> <s>Ed oltre a questi altri ne sono, che si eseguiscano dalle superficie coniche, cagionati dal percotimento de’ raggi del <lb></lb>//<lb></lb>Sole sopra la terra e da essi parimente nel taglio del cerchio si costituiscano varie specie d’angoli. </s> <s>Di questi ragionaremo hora in questo luogo per maggior notitia della propositione del precedente cap. </s> <s>Le superficie coniche formate da’ raggi del Sole, essendo lui collocato fuor dell’equatore, che hanno per punto verticale commune il centro del Mondo, tagliano diversamente ‘l cerchio Meridiano e in uno stesso tempo costituiscono gli angoli perciochè o noi consideriamo i cerchij che sono le basi delle dette superficie esser in fra loro paralleli e retti e tagliare ancho ad angoli retti sferali: overo consideriamo gli angoli formati dal contatto de’ lati del conio insieme con la base, e così ritruovaremo gli angoli acuti: o veramente que che son formati dal concorso e dal contatto de’ raggi del Sole nel centro del Mondo, come in loro orizzonte; ed in questo modo trovaremo gli angoli talvolta acuti e talvolta ottusi, secondo che ‘l centro del Sole sarà più vicino o più lontano all’equinottiale e al centro del Mondo, come si può osservare nella Gnomonica del Clavio in uno Analemma descritto per dimostrar la terza Prop. </s> <s>Così anchora dal segamento delle superficie coniche, e dal piano dell’Horiuolo parallelo al maggior cerchio, si può trarre ‘l nascimento degli angoli, sì come si può vedere appo ‘l medesimo Autore nella quinta Prop. </s> <s>Perciochè se noi riguardiamo bene, vedremo, che nel tagliamento di due piani paralleli, che sono due cerchi, fatto da un gran cerchio, si costituiscano gli angoli <pb pagenum="folios 13v-14r"></pb>retti; quantunque si mostrino acuti overo ottusi; ma si dee ciò intendere, supposto che l’occhio nostro sia nel centro del Sole, che co’ suo’ raggi forma le superficie coniche. </s> <s>Il qual cerchio girando intorno all’asse della Sfera, forma un triangolo, tagliando la linea retta del piano dell’altro cerchio maggiore, che è uguale ad esso, la qual linea è asse di lui; e così ancho la settion conica è ‘l piano dell’Horiuolo, e ‘l piano ch’è base del conio inferiore con una settion commune; onde si vede sorger la settion detta Parabola, che si forma ad angoli retti; perciochè si fa col mezzo delle linee rette, e dell’asse che è perpendicolare. </s> <s>Di modo che rispetto al triangolo noi truovaremo la costitution degli angoli acuti. </s> <s>Similmente nella settione Iperbole e nella Ellipse truovaremo ‘l nascimento degli angoli retti, acuti e ottusi, come si può vedere appresso ‘l Clavio nela Prop. sesta e settima del medesimo libro. </s> <s>In un’altra guisa si ritruova ‘l nascimento degli angoli, cioè col partimento del cerchio in trecentosessanta gradi; onde nella circonferenza con la misura d’un arco si può truovare la misura della longhezza d’un lato di qualsivoglia figura; perciochè ‘l quadrato ha un arco di 90 gradi; il triangolo di 120; il pentagono di 72 e così del lato di qualunque altra figura si dee dire; di maniera che truovato un lato, sono anche truovati gli altri; perciochè un lato solo più volte applicato nella circonferenza è misura di tutti; poiché si mol <lb></lb>//<lb></lb>tiplica tanto che si chiuda la figura e dal congiognimento di tutti i lati risultano gli angoli o retti, o acuti, o ottusi. </s> <s>Talchè finalmente si conclude, per le cose dette, che da tutti questi diversi tagliamenti del cerchio hanno origine tutte le maniere degli angoli, il che è quanto nel precedente cap. si prese a ricercare. </s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"><s>Se ogn’angolo sia divisibile</s></p> <p type="head"><s>Cap. 6</s></p> <p type="main"><s>Se si riguarda al 4° Problema del primo d’Euclide, non havremo dubbio, che ogn’angolo si divida; poiché quivi così afferma. </s><s>Divider per mezzo un angolo rettilineo dato. </s><s>E per dimostrare il problema forma una descrittione con un angolo acuto incontro al quale costituisce un altro angolo acuto d’un triangolo equilatero collocato sopra la linea retta del taglio. </s><s>Onde se l’angolo acuto che è minor d’ogn’altro si divide, è cosa ragionevole che si possa dividere e con più facilità l’angolo ottuso e ‘l retto; perciochè Euclide dicendo divider per mezzo ogn’angolo rettilineo, non intende solamente dell’acuto, ma anchora dell’ottuso e del retto poiché l’angolo rettilineo è genere e l’acuto e l’ottuso e ‘l retto son le specie. </s><s>Oltre che quel che Euclide mostra nell’angolo acuto col mezzo del triangolo equilatero o con l’equicrure. </s><s>L’amendue hanno gli angoli acuti, si può mostrar col mezzo del triangolo ottusangolo e del rettangolo, come se due squadre s’intersegassero insieme, ma con lo scaleno non sai. </s><s>Ma se riguardiamo a quel che ne dice Teone o pure Euclide stesso nella sua Prospettiva nella notation e sopra ‘l 3° Teorema vedremo l’angolo esser indivisibile. </s><s>Poiché dice l’angolo del contatto esser indivisibile, citando la decima del 3°. Onde conclude la grandezza <pb pagenum="folios 14v-15r"></pb>che si vede sotto così fatto angolo non potersi vedere sotto angolo minore. </s><s>Questo, a parer mio si è detto o da Euclide o da Teone, non per mostrar che l’angolo del contatto geometricamente non si possa dividere, ma per dimostrar che se la grandezza fosse collocata sotto un angolo minore di quello del contatto, non si potrebbe vedere. </s><s>Overo diciamo questa esser un’altra questione appartenente agli angoli curvilinei, uno de’ quali è l’angolo del contatto. </s><s>Ma noi qui favelliamo degli angoli rettilinei. </s><s>E rimirando a quel che sopra ciò dica ‘l Commandino, intenderemo che per la istitutione elementare non si può divider per mezzo ogn’angolo, che questo da lui si pone in dubbio e si certifica poi nella specie degli angoli acuti, dicendo potersi divider in tre parti l’angolo retto, ma non già l’acuto. </s><s>Di modo che l’istitutione elementare che forza che ogn’angolo non si divida e per conseguenza l’angolo acuto esser indivisibile, non è altro che le specie degli angoli sieno elementi e parti componenti altri angoli di specie diversa da loro; tal che quegli angoli che d’altra specie d’angoli son composti nelle medesime specie dividendosi si risolvano delle quali composti sono; ma gli angoli acuti non son composti d’altra maniera d’angoli diversa da loro adunque non si potran risolvere in altra specie d’angoli adunque neancho si possan dividere; perché oltre agli angoli acuti non ci dà altra specie. </s><s>E questa si è la ragione onde ‘l Commandino si mosse a dibbitare se ogn’angolo si dividesse e ad affermare che gli angoli acuti non si possan dividere. </s><s>Ma tal ragione si potreb<lb></lb>//<lb></lb>be per avventura rispondere in questa guisa. </s><s>Altra cosa è la divisione, altra la risolution delle cose, quando si dice che l’angolo ottuso si divide in un retto in un acuto overo in più angoli acuti; si dee intender che tal partimento sia piutosto risolution nelle parti componenti, che divisione. </s><s>Quando si dice l’angolo acuto dividersi, all’hora s’intende il partimento esser veramente divisione e non risolutione; perciochè è più universale la divisione, che la risolutione; perché la risolutione è solamente quella che distruggendo discioglie le parti componenti, che prima erano insieme collegate e continue; onde poi risulta la cognition delle parti e degli elementi, che concorreno a formare il tutto: e questi sono specie diverse dalla specie della cosa, che di essi è composta. </s><s>La divisione poi non solamente comprende la risolutione, ma tutte le maniere de’ separamenti delle cose congionte, o appartenghino alla sustanza del composto o alla quantità, o alla qualità, o alle diverse nature, o a generi ed alle specie. </s><s>Però ‘l partimento degli angoli acuti è solamente divisione non risolutione in altre specie d’angoli; poiché, essendo divisi, non si risolvano in altri angoli che sieno elementi loro. </s><s>Perciochè gli angoli, che sono elementi degli angoli, o sono acuti, o son retti; ma gli elementi degli angoli acuti non si danno; onde avviene che essi non si possino risolvere negli elementi componenti; e per questo il tagliamento degli angoli acuti non sarà risolutione, ma simplice divisione. </s><s>Oltre acciò sì come un tutto composto è realmente diverso dalle parti componenti, come tutto l’ottusangolo è diverso dal retto e dall’acuto, o da più acuti, che ‘l compongono; e così l’angolo <pb pagenum="folios 15v-16r"></pb>retto dagli acuti che ‘l costituiscono; così le parti e gli elementi, dico i detti angoli componenti sono da esso diversi realmente e ciò si manifesta per la risolutione; per la qual cosa essendo gli angoli ne’ quali l’angolo acuto si divide acuti, segue che ‘l suo partimento non sia risolutione, poiché sono della medesima specie; adunque sarà simplice divisione, la quale non appartiene agli angoli in specie, come la risolutione per la quale si costituiscano diverse maniere d’angoli; ma appartiene agli angoli in quanto sono di quantità e di grado differente secondo ‘l difetto e ll’escesso, questo in acutezza e quello il misura. </s><s>Perciò che gli angoli acuti si dividono in angoli acuti, che son diversi da tutto l’angolo solamente nell’esser più acuti e nell’haver misura più stretta; che l’angolo intero acuto è ‘l doppio di due, come si può ritrarre dalla dimostratione XI del IIII di Euclide, o triplo di tre, o quadruplo di quattro angoli acutissimi. </s><s>Che quantunque l’angolo sia acuto, con tutto ciò, sempre si può far più acuto, sì come l’ottuso sempre si può far più ottuso; perciochè queste ragion d’angoli sono imperfette e indeterminate. </s><s>Ma l’angolo retto non può già mai farsi più o men retto, che essendo retto e ‘n quanto retto è invariabile e non ha bisogno di perfettione. </s><s>Oltre acciò la divisione o più tosto la risolutione degli angoli si mostra simile a quella del genere nelle specie e delle specie negli individui, dove (secondo Platone) bisogna fermarsi, come se gli angoli acuti sieno simiglianti agli individui; che sì come gli individui non si risolvano in altra specie; ma si dividano in parti componenti integrali, così gli angoli acuti non si risolvano in altra specie d’angoli; ma in parti integrali, che son della medesima natura. </s><s>Perciochè <lb></lb>//<lb></lb>gli angoli retti e gli ottusi, che per natura loro sono interi, hanno simiglianza di generi, e gli angoli che sono elementi e parti componenti hanno simiglianza di specie; ma gli angoli acuti, presi separatamente, non essendo interi non possono haver simiglianza di generi, e perciò non si possan risolvere in altre specie. </s><s>Hanno sì bene apparenza di specie e perciò si dividono in parti, che rassomigliano gl’individui. </s><s>Di più il ripartimento degli angoli sta in quella guisa, che si divide un arbolo; perciochè esso si divide nel tronco e ne’ rami e ‘rami non si dividano in altri rami, ma in particelle di rami, che si appellano ramuscelli: e ‘l tronco non si divide in altri tronchi, ma in parti di esso, le quali non son il tronco. </s><s>E gli angoli acuti altresì non si dividano in altra specie d’angoli; che non sono altri angoli oltre ad essi; ma si dividano in parti, overo in angoli più acuti ed acutissimi. </s><s>E questa si è diversa divisione da quella degli altri angoli; poiché la division degli altri angoli costituisce angoli diversi in specie dall’angolo diviso, come gli acuti da’ retti, e gli acuti e retti dagli ottusi; ma dalla division degli acuti non si costituiscano angoli diversi di specie, ma differenti in quantità e in grado. </s><s>Ma in altro modo si può pruovare ogn’angolo esser divisibile, e questo si eseguisce in tre maniere: la prima si prende dalla proprietà di chiuder lo spatio; la seconda dall’esser parte di quantità continua e di spatio terminato da varie figure angolari; la terza dall’esser diferente dal punto. </s><s>Tutta la difficoltà è collocata nell’angolo acuto; che degli altri non si dubbita se sieno divisibili, e perciò tutto ‘l discorso, come da principio si può vedere infine a questo luogo, si raggira intorno all’angolo acuto, e così quindi infine al fine, seguirà la medesima intentione. </s><s>E per cominciare dirò. Se l’angolo acuto fusse indivisi <pb pagenum="folios 16v-17r"></pb>bile non sarebbe niente differente dal punto il che non essendo vero, non sarà vero l’angolo acuto esser indivisibile. </s><s>E vero che sì come ‘l punto è principio della linea, così l’angolo della figura; perciochè non solamente le dà la forza, ma ancho la denominatione; ma non è vero che sì come ‘l punto di maniera è principio della linea, che non divien già mai parte di essa, così l’angolo sia principio della figura; perciochè è parte di essa, e che ciò sia vero, levisi alla figura rettilinea l’angolo, tosto vedremo essa non solamente scemarsi, ma distruggersi. </s><s>Si ‘ngannano coloro che fanno l’angolo acuto indivisibile, non distinguendolo forse dal ponto e credendo che ‘l ponto del contatto di due linee sia l’angolo. </s><s>Ma non il ponto è l’angolo, ma ‘l contatto e ‘l concorso di due linee in un medesimo ponto, come si è già mostrato nel cap. 3. e non può farsi questo concorso se da una banda le linee non sono separate l’una dall’altra, altramente non sarebbero due linee, ma una sola, o una cadente sopra l’altra, terminando ne’ medesimi punti in quella guisa che Euclide dimostra nella IIII del primo. </s><s>Adunque ‘l punto è termine del contatto, e l’angolo non è altro che ‘l contatto delle linee, che si fa nel modo già detto. </s><s>Dunque, per esser termine, non può far che l’angolo acuto non si divida. </s><s>Da queste ragioni si ritrahe ‘l punto e l’angolo esser cose differenti, come ‘l termine è differente dalla cosa terminata: e da questa differenza si conclude l’angolo esser divisibile. </s><s>Ma passiamo più avanti. </s><s>La medisima ragione hanno le parti che ha ‘l tutto, sì come è sentenza commune de’ Filosofi, adunque se l’angolo acuto è parte della figura o dello spatio figurato, che è divisibile, ancho esso sarà divisibile. </s><s>Non si può negare che la figura sia divisibile; perciochè è quantità continua, la quale per Aristotile, nella <lb></lb>//<lb></lb>Filosofia Naturale, è sempre divisibile: e conferma con la commune autorità de’ Matematici e de’ Filosofi; perciochè alla quantità continua convengon le tre misure. </s><s>Che l’angolo sia parte della figura, oltre che la sperienza 'l mostra, l’afferma Averroe nel 5 della Metafisica, dicendo: l’angolo è parte della figura col mezzo della quantità e della qualità e si conferma da Henrigo Glareano nel capitolo primo della Geografia, il quale dice l’angolo esser una particella della figura, che dal contatto della linea sorge nella larghezza. </s><s>Essendo adunque l’angolo acuto parte della figura, per essempio del triangolo, e la figura terminando lo spatio, segue che l’angolo acuto sia anchora parte dello spatio, e posta la figura divisibile, e lo spatio divisibile, l’angolo acuto anchora necessariamente sarà divisibile, il che si pruova; perciochè Euclide in molte dimostrationi del primo, dove si fa comparatione de’ lati del triangolo, e degli angoli, si dice sempre. </s><s>Angoli sottoposti a lati uguali. </s><s>Onde si ritrahe che l’angolo non sarebbe stato sottoposto a’ lati, se in fra essi non si trovasse lo spatio: né i lati si possono chiamar lati, se toccandosi insieme in un punto (non per diritto di ciascuno) non determinano la figura o quadrata o triangolare e perciochè diventano lati costituendo la figura con gli angoli di essa. </s><s>Però, dovunque è angolo, è anchora spatio, ed ogni spatio è divisibile, adunque ciascun angolo è divisibile, adunque anchora l’angolo acuto sarà divisibile. </s><s>Perciochè onde avviene che l’angolo si divida, se non perché si divide lo spatio? Finalmente onde procede la verità dell’ultimo Assioma del primo d’Euclide. </s><s>Che due linee rette non chiudino <pb pagenum="folios 17v-18r"></pb>spatio se non dal non formare angolo alcuno? E che ciò sia vero si osservi Euclide che si dichiara espressamente, dicendo, due linee rette; che se havesse detto simplicemente, due linee non chiudono spatio, non sarebbe vero; che due linee curve chiudono spatio, formando la figura binangola; poiché si toccano e formano due angoli. </s><s>Adunque ‘l chiuder lo spatio nasce dalla costitution degli angoli adunque dagli angoli nasce lo spatio rettilineo e curvilineo anchora. </s><s>Però se avverrà che si divida lo spatio si divideranno anchora gli angoli qualunque sieno. </s><s>AGGIUNTA (1)</s></p> <p type="main"><s>Vediamo ora se da altri luoghi di Euclide si può ritrarre, che ogn’angolo sia divisibile. </s><s>Nella trentaquattresima del primo afferma, che gli spatij parallelogrammi si segano per mezzo dal diametro, il quale taglia ancho gli angoli pel mezzo: e nella descrittione si formano acuti. </s><s>Dove dal segamento degli angoli si conclude ‘l segamento dello spatio parallelogrammo; così convertendo questo detto si può affermare, che dal segamento dello spatio parallelogrammo si può venire al segamento degli angoli opposti. </s><s>La qual cosa serva in confermation delle cose dette. </s><s>Nella dimostratione undicesima del 4° si conclude due angoli di un triangolo equicrure simile a un altro triangolo dato, esser doppij di un angolo del medesimo triangolo posto nella cima, cioè ciascuno per se stesso esser doppio del detto angolo: e non si mostra altramente, che col segamento degli angoli, che sono acuti. </s><s>Così anchora nella dimostratione della dodicesima si vede un angolo acuto diviso esser doppio d’un altro acuto, tutti collocati nel centro d’un cerchio, intorno al quale si è descritto un pentagono. </s><s>Nella prop. terza del 6° suppone, che un’angolo </s></p> <lb></lb>//<lb></lb><p type="main"><s>AGGIUNTA (1) [Oltre acciò habbiamo ‘l Padre Clavio dalla nostra, il quale nello scholio sopra la IX del I dice: quindi è manifesto che l’angolo rettilineo si può dividere in quattro angoli uguali, in 8, in 32, in 64 e così negli altri sempre procedendo per l’agumento del doppio. </s><s>Ma Euclide none insegnò in verun luogo in che modo si potesse dividere un angolo rettilineo in tante parti uguali, in quante altri voglia; perciò che fine al suo tempo non era stato dimostrato. </s><s>E pertanto, per dimostrarlo ci serviremo della dimostratione di Pappo Alessandrino, il quale prende per mezzo una certa linea curva, nel fine del sesto libro. </s><s>Ma se fra tanto con l’operatione del compasso vorremo dividere qualunque angolo rettilineo si propunga l’angolo BAC. da dividersi in cinque angoli uguali. </s><s>Dal ponto A. centro si descriva l’arco BC. il quale tagli le rette linee AB. AC. in qualunque intervallo. </s><s>Si divida l’arco BC. con le seste in tante parti uguali, in quante si ha da dividere l’angolo BAC e per essempio si divida in cinque parti uguali ne’ ponti D. E. F. G. Hora se dal centro A. a’ detti punti si tirino le linee rette, sarà diviso l’angolo BAC. in cinque parti uguali, essendo uguali gl’intervalli presi col compasso, cioè BD. DE. EF. FG. GC. saranno tutte queste linee uguali. </s><s>Adunque i due lati BA. AD. del triangolo BAD. l’uno all’altro saranno uguali, essendo tutti tirati dal centro per la quindicesima def. del primo d’Euclide. </s><s>E la base BD. e la DE. essendo uguali, come s’è detto; gli angoli DAB. ed EAD. saranno uguali e per la medesima ragione l’angolo EAD. è uguale all’angolo EAF. e così gli altri. </s><s>Adunque tutti gli angoli appresso all’A. sono fra loro uguali per la ventisettesima del 3°; perciochè le circonferenze BD e DE etc. si son poste uguali. //]</s></p> <pb pagenum="folios 18v-19r"></pb><lb></lb>//<lb></lb><p type="main"><s>d’un triangolo si seghi pel mezzo. </s><s>Dunque lo suppon per noto, o come già dimostrato; in qualunque modo diciamo, sempre potremo quindi ritrarre ogn’angolo esser divisibile, non esser cosa dubbiosa. </s><s>E ‘l Commandino stesso dice l’angolo dividersi perché è grandezza non lineale; onde molto meno sarà simile al punto. </s><s>Oltre acciò dice anchora, l’angolo esser divisibile, atto a ricever l’ugualità e la disugualità secondo la quantità che in esso si truova; el parlar suo è indifferente, e perciò non esclude gli angoli acuti. </s><s>Nell’ultimo luogo bisogna vedere, se l’oppinion del Commandino si può difender in qualche modo. </s><s>E però si osservi, che altro è dire assolutamente l’angolo acuto esser indivisibile; altro è dire esser indivisibile, cioè che appena si possa dividere per la sua molta strettezza, in mezzo alla quale, benchè con grande esquisitezza si tirino le linee; con tutto ciò, con grandissima fadiga si possan produrre in maniera le linee in mezzo, che non si confondano, e non si congiongano con l’angolo: e ciò avviene nelle linee disegnate e negli angoli sensati e materiali. </s><s>Ciò s’intende sempre degli acuti, che i retti e gli ottusi, benchè materiali e sensati, si possan divider per mezzo come si ritrahe da Euclide nella Prospettiva degli specchi, nella dimostrazione del 20° Teorema. </s></p> <p type="main"><s>Ma negli angoli non sensati né materiali, ma intelligibili e puramente matematici, sempre sarà vero, che ogni angolo sia divisibile: e truovandosi angolo quanto più acuto e quanto più stretto si possa immaginare, sempre si truovarà ancho una linea che ‘l tagliarà per mezzo. </s><s>Ed è questo negotio come quello della Sfera che tocchi ‘l piano in un punto; là dove la Sfera materiale nel piano materiale nol fa; ma amendue puri matematici sempre mostran vera questa prop. </s><s>Ma ritornando al nostro proposito, diciamo intendendosi nel primo modo non esser vera l’oppinion del Commandino. </s><s>Ma intendendosi nel 2° esser vera; ma non è conforme al puro Matematico, né allo stile <pb pagenum="folios 19v-20r"></pb>d’Euclide; perciochè tutto quel che considera ne’ libri degli Elementi il considera, come cosa pura matematica, non vi adattando conditione alcuna, altramente non si porrebbe differenza fra le scienze pure matematiche, e fra quelle che son miste, o scienze di mezzo.</s></p> <p type="main"><s></s></p> </chap> <chap> <p type="head"> <s>Se si dà l’angolo indeterminato sì come il determinato</s> </p> <p type="head"> <s>Cap. 7</s> </p> <p type="main"> <s>L’angolo determinato non è altro che ‘l retto; perciochè aggiognendosi o scemandosi non è più retto; che se si aggiogne diventa ottuso, e se si diminuisce, diviene acuto. </s> <s>Però ragionevolmente si dice determinato; che non richiede gionta né scemamento: e sì come quelle cose che variano nella specie per la mutatione, si dicano determinate, così l’angolo retto si appella determinato; perché si muta in altra specie, essendo aggionto o scemato. </s> <s>Oltre acciò si dice determinato; perciochè nel definir Euclide gli altri angoli, dice l’angolo acuto esser minore e l’ottuso maggior del retto; ma dell’angolo retto non si dice esser maggiore o minore; ma uguale; perciochè avanti all’angolo retto non si dà altra maniera d’angolo alla quale si possa assimigliare. </s> <s>Di modo che gli altri angoli si determinano nel retto, ma ‘l retto non si determina in altri angoli. </s> <s>Gli angoli indeterminati sono gli acuti e gli ottusi; perciochè, secondo Proclo, l’angolo retto si forma dalla ragione che nasce dal fine e l’ottuso e l’acuto da quella che procede all’infinito. </s> <s>Questi si appellano indeterminati perché sempre si possano diminuire od accrescere; talchè per questo sono indeterminati; perché non è determinata la grandezza loro, sì come per questo si dice l’angolo retto determinato, perché è determinata la sua grandezza. </s> <s>Oltre acciò si dicano indeterminati; perciochè, secondo Proclo, hanno gran forza di accrescere e diminuire gli spati; onde per essi sono indeterminati. </s> <s>Ma si avvertisca che quando si dice l’angolo ottuso indeterminato; perciochè sempre si può aggiognere <lb></lb>//<lb></lb>e l’acuto altresì; perché sempre si può scemare, non si dee intendere che l’uno infinitamente si aggionga e l’altro infinitamente si scemi; perciochè o lo scemamento o ‘l crescimento non può proceder in infinito; poiché se si cresce in infinito l’ottuso necessariamente bisogna venire a un caso, che non vi sia più angolo; perciochè quanto più si cresce l’angolo, tanto più si avvicina a perdersi. </s> <s>E però per forza di Prospettiva, e per ragion di lontananza dalla vista l’angolo ottuso si perde, e si converte in una linea: e quanto più si scema l’angolo, tanto più si accosta al non esser più angolo; perciochè finalmente si perde sì come al suo luogo ne ragionaremo a pieno. </s> <s>Onde quando diciamo l’ottuso sempre potersi crescere, e l’acuto sempre scemare, bisogna intender secondo qualche tempo determinato; di modo che sempre non voglia dir altro che molte volte si possa o l’uno aggiognere o l’altro scemare. </s> <s>E di questo per hora non diremo altro.</s> </p> <p type="main"> <s></s> </p> <p type="main"> <s>Se l’angolo si riduca al tutto alla pianezza e ugualità della linea retta, </s> </p> </chap> <chap> <p type="head"> <s>overo alla curvità della circolare</s> </p> <p type="head"> <s>Cap.8</s> </p> <p type="main"> <s>Per ritruovar la verità di questo problema bisogna considerare, che l’angolo può esser riguardato in due maniere, in quanto si truova in un piano, overo in una linea piana: e in quanto si truova nella superficie d’una Sfera, o nella linea circolare, sì come si vede nelle figure moltilatere. </s> <s>Onde considerato l’angolo come in superficie sferica, o ‘n linea circolare, noi ‘l vedremo che si ridurrà alla linea circolare o all’incurvamento della Sfera: e considerato l’angolo come in un piano, o ‘n’una linea piana il vedremo ridursi allo stesso piano, overo alla stessa linea piana, la qual cosa si può confermar con Marsilio Ficino nel cap. 41° del Timeo di Platone, là dove dice l’angolo ottusissimo esser tosto seguito da <pb pagenum="folios 20v-21r"></pb>due angoli retti; perciochè se oltre a questo allongando procederai, subbito perverrai a un piano nel quale cadendo una linea retta perpendicolarmente formarà due angoli retti, come si può osservare in questo essempio segnato A. Il medesimo per avventura havrebbe possuto dire adattando l’angolo ottusissimo alla Sfera, od alla circonferenza; perciochè così fatto angolo può esser seguito da due angoli retti sferali; che allargandosi continuamente l’angolo ottusissimo posto o fuore o dentro la circonferenza, o sopra la superficie della Sfera, finalmente bisogna che si riduca alla stessa circonferenza, overo alla superficie sferica, come si può osservar nella formation delle figure di molti lati, là dove moltiplicati i tagliamenti e costituiti tuttavia archi minori, per la trentunesima del 3° d’Euclide si formano sempre angoli maggiori ed ognhora più ottusi, come si accenna con questa figura, segnata B. Ma per mostrar questo con qualche ragione, bisogna dire che ‘l riducimento dell’angolo alla linea si può fare in due maniere, cioè o aggiognendo all’angolo o diminuendo. </s> <s>Diminuendo si fa tirando sempre nuove linee che si tocchino nel medesimo punto dentro alle prime, che chiudono l’angolo proposto, come dentro le AB. BC. che formano l’angolo retto ABC. Si tirino le linee DG. che scemandolo il fanno diventar acuto; di poi le EF., le quali scemano l’angolo acuto e ‘l fanno più acuto di prima; quindi tirate le linee GH, l’angolo scemato diverrà ancho più acuto: tirate poi le IK si scemarà l’angolo acuto e si farà molto più acuto, e così continuando di tirar nuove linee dentro al detto angolo, si tagliarà talmente e talmente si diminuirà che posta in mezzo ad esso la linea LB. l’angolo continuamen scemandosi si risolverà in essa. </s> <s>Perciochè le linee dello scemamento tanto si moltiplicano che si accostano alla linea LB e quanto più si accostano tanto più si perde dello spatio; di modo che finalmente tutto lo spatio contenuto dall’angolo si converte in una linea, quasi nella <lb></lb>//<lb></lb>stessa guisa che fa un piano opposto dirittamente all’occhio, come si dimostra dalla Prospettiva. </s> <s>Overo si dimostra in questa maniera. </s> <s>Essendo già tirato le due linee AB. BC. che terminino l’angolo ABC. si tirino le linee DG dentro l’angolo proposto, le quali scemino l’angolo retto e ‘l risolvino in acuto: di poi tirate dentro a queste due altre linee, ciò sono le EF, si scemi l’angolo acuto DBG. onde divenga via più acuto che ‘l primo che sarà l’angolo EBF. e tirate dentro a quest’angolo le linee GH. venga scemato l’angolo EBF. e fattone un angolo anchora più acuto che è GBH. Tirinsi in fra esso due altre linee IK, necessariamente verrà scemato l’angolo GBH e fattone l’angolo IBK. In mezzo al quale si faccia una linea retta LB. che divida l’angolo IBK per mezzo; pel 4° problema del primo d’Euclide e per le ragioni dette nel cap. 6°. Di modo che si truovi l’angolo LBI assai minore. </s> <s>Sia l’angolo LBI uguale all’angolo GBC. dividasi dalla linea NB per mezzo; adunque ‘l rimanente angolo NBC sarà assai minore d’ogn’altro: taglisi per mezzo l’angolo NBC. dalla linea OB. seguirà che l’angolo OBC sia molte volte più minore degli altri angoli, essendo minore tante volte dell’angolo precedente NBC. il quale è minore d’ogni altro angolo contenuto nella division dell’angolo ABC. E se sia possibile con la pratica dividersi l’angolo OBC. da una linea retta PB. dico che l’angolo PBC sarà molte volte più minore del precedente e de’ rimanenti angoli; onde procedendo continuamente in questa guisa, se non con la pratica, almeno con la speculativa si divida tante volte l’angolo che si pervenga finalmente alla stessa linea, e così si vedrà l’angolo necessariamente in essa essersi risoluto; che per la division continua, l’angolo continuamente si ristregne. </s> <s>Questo medesimo possiamo dimostrare con la ventunesima del primo d’Euclide là dove si mostra che costituite due linee rette dentro un triangolo dato, le quali sieno tirate dai termini de’ lati d’esso, essendo minori de’ lati del <pb pagenum="folios 21v-22r"></pb>triangolo formano un angolo maggiore; di modo che, quante più linee si faranno dentro al triangolo proposto, gli angoli da esse contenuti tanto più maggiori diverranno; onde saranno ogni volta più ottusi; di maniera che se dentro al detto triangolo continuamente si produrranno angoli più ottusi, bisognarà alfine condursi allo svanimento dell’angolo e nel riducimento alla linea. </s> <s>Perciochè ‘l triangolo è a figura e però è grandezza determinata; onde non si può infinitamente scemare: e così gli angoli dentro a essa non possono esser infiniti, né infinitamente allargati e fatti ottusi; ma tanto si scema l’angolo proposto e tanti angoli si formano dentro ‘l dato triangolo e tanto si allargano e si ottusano; che finalmente si perviene al riducimento di tutto l’angolo alla linea. </s> <s>Ciò non si dimostra essendo provato in gran parte da Euclide nella detta prop. </s> <s>Solamente per facilitar la intelligenza formaremo qui appresso la figura. </s> <s>Ma l’angolo si riduce alla linea retta aggiognendosi in questa maniera, cioè costituite due linee che formino l’angolo retto ABC. si tirano di fuore due linee rette DE. che terminino nel segno B. e così le IG. HI. e le KL. tanto che si pervenga a due linee che sieno per diritto l’una all’altra, ciò sono la MB. BN. Sia l’angolo MBA. uguale al NBC. perché dalle cose uguali levandosi cose uguali, le rimanenti sono uguali pel 3° Assioma del primo d’Euclide. </s> <s>Traggasi dall’angolo MBA l’angolo IBC. e dall’NBC. l’angolo GBA. Adunque ‘l rimanente HBK è uguale al rimanente IBL. Dividasi l’angolo GBA. e l’HBK. dalla linea OP. adunque OBK è uguale all’angolo LBP. Ma l’angolo OBK è minore di tutti gli angoli costituiti per le gionte, adunque ancho l’angolo LBP. sarà minore altresì di tutti gli altri; ma quello è minore che più si ristregne e si accosta alla linea per la ragion precedente; adunque i detti <lb></lb>//<lb></lb>angoli son minori e ristregnendosi continuamente si convertano in una linea. </s> <s>Adunque l’angolo OBK. od altro minore, che per la division si succeda sarà fatto la linea MB. e l’angolo LBP. od altro minore, che li succeda col mezzo della divisione si sarà convertito nella linea BN. la quale essendo per diritto della MB. non conterrà più angolo; adunque l’angolo ABC allargato continuamente, overo aggionto con le linee DE. FG. HI. KL. e così seguendo finchè la larghezza dell’angolo proposto si accosti alla linea MN. ed aggiognendo sempre all’angolo precedente si converte finalmente nella intera linea MN. Overo si dimostri così. Aggiontato l’angolo ABC. dalle dette linee e diviso l’angolo KBN. dalla linea OB. e l’angolo IBL. della linea BP. segue che l’angolo MBK. (essendo già tirata la linea MBN. sia uguale all’angolo NBO. Ma l’angolo NBO. è ‘l più piccolo, e ‘l più vicino d’ong’altro a convertirsi nella linea, adunque l’angolo MBK. vi sarà ancho vicino essendo amendue parti dell’angolo MBH. Così anchora l’angolo NBL. è uguale all’angolo LBP. ma l’angolo LBP. è ‘l più stretto e ‘l più vicino d’ogn’altro a farsi linea, adunque, adunque l’angolo NBL. vi sarà ancho vicino, esendo amendue parti dell’angolo NBI. Ma i detti angoli son parti dell’angolo intero KBL. che è la gionta dell’angolo ABC. adunque tutto l’angolo intero si è già avvicinato a ridursi alla linea retta MN. alla quale all’hora giognerà quando si farà nuova gionta, tanto che si facciano due linee che sieno per diritto in fra loro come la MB alla BN. e questo è quel che si cercava di mostrare. </s> <s>Oltre acciò nella figura che si fa appo Euclide per dimostrar la trentacinquesima del primo, la quale è così formata, ponendo due parallelogrammi sopra la medesima base, e fra le medesime parallele; apparisce un corpo solido con sei angoli e cinque superficie, cioè tre parallelogra <pb pagenum="folios 22v-23r"></pb>mme e due triangolari, una delle quali superficie parallelogramme si vede apertamente risoluta in una linea e due parallele opposte con due altre parallele pure opposte dalle quali è contenuto e terminato lo spatio parallelogrammo ADEF. che è una delle dette superficie: e così anchora gli angoli del detto spatio si vedano risoluti in una linea. </s> <s>Ma si può dimostrare in questo modo. </s> <s>Sia il lato AB. uguale al lato DC. perciochè ABCD è parallelogrammo, come si dimostra da Euclide, el lato AE. è uguale al lato AB. Adunque è ancho uguale al lato DC. Ma DC. è lato di parallelogrammo, adunque ancho AE sarà lato di parallelogrammo. </s> <s>Onde si è già truovato un lato del parallelogrammo. </s> <s>Oltre acciò sia il lato FC. uguale al lato BE. per la medesima ragione el lato FD è uguale al lato FC. adunque FD è uguale a BE. Ma il lato BE. è lato di parallelogrammo, adunque FD anchora sarà lato di parallelogrammo. </s> <s>E però si sarà truovato un altro lato del parallelogrammo. </s> <s>Onde nella linea AF già sono i lati AE e DF del parallelogrammo. </s> <s>I quali son fra loro uguali; perciochè sono uguali a’ medesimi lati de’ parallelogrammi posti fra le medesime parallele. </s> <s>Oltre acciò sia il lato AD uguale al BC. el lato EI. anchora al BC. non possono non essere uguali fra loro pel primo assioma d’Euclide. </s> <s>Ma il lato BC. è lato di parallelogrammo, adunque AD, ed EF sono due lati di parallelogrammo fra loro uguali, come s’è già dimostrato; di maniera che si son già truovati due altri lati del parallelogrammo, ciò sono AD. EF. Congiongansi i lati AE. DF. FE. DA. e formino quattro angoli ADF. DFE. DAE. EFD. si sarà formato il parallelogrammo AEFD uguale a’ parallelogrammi rimanenti pel medesimo assioma. </s> <s>Ma tutti i lati di questo <lb></lb>//<lb></lb>parallelogrammo sono nella linea AF e così gli angoli e per conseguenza ancho lo spatio; adunque il parallelogrammo AEDF è tutto nella linea AF. E però sarà vero che la superficie parallelogramma e le quattro parallele opposte e gli angoli parimente opposti terminanti lo spatio, sieno tutti risoluti in una linea. </s> <s>Il che intendevamo dimostrare. </s> <s>Nella Prospettiva anchora non è dubbio alcuno, che l’angolo si risolve in una linea; perciochè lo spatio di qualunque figura, secondo la position dell’occhio nostro talvolta apparisce in una linea, cioè quando l’occhio e parallelo all’obbietto piano o quando amendue sono per diritta linea opposti; perciochè nella Prospettiva si danno tre positioni dell’occhio, cioè o sopra l’obbietto o sotto, o rincontra ad esso e per diritto. </s> <s>Quando è sopra l’obbietto, si vede tutto ‘l piano; quando è sotto non si vede, ma quando è rincontra per linea diritta, si vede in una linea. </s> <s>Ma ritorniamo alle ragion geometriche e diciamo che Euclide nel dimostrar la tredicesima del primo mostra la differenza, che è fra le linee che sono per diritto, e quelle che non sono, cioè che quelle che non sono, cadendovi sopra una linea formano due angoli non uguali a due retti, e per conseguenza non uguali fra loro: ed esse formano un angolo ottuso; ma quelle che son per diritto insieme, cadendovi sopra una perpendicolare costituiscano due angoli uguali a due retti; e uguali fra loro. </s> <s>Facciasi il detto angolo più largo, continuamente le due linee verranno per diritto e però le due linee diverranno una linea, e l’angolo si risolvarà in essa. </s> <s>Da queste cose adunque si rende chiaro che l’angolo posto nel piano o nella linea retta si risolva nella linea retta e nel piano. </s> <s>Hora bisogna vedere se l’angolo posto <pb pagenum="folios 23v-24r"></pb>nella superficie sferica e nella circonferenza, si risolva in essa. </s> <s>Con la continua division del cerchio e con la diversa forma di figure moltilatere, che per l’altra divisione si formano, si può venire in cognitione di questo effetto. </s> <s>Perciochè in quante più parti si divide ‘l cerchio e di quanti più lati e di quanti più angoli si forma la figura, tanto più si accosta alla ugualità circolare, sì come di sopra già si è accennato; perciochè multiplicati i lati, sempre divengono più corti e quanto più corti, tanto più gli angoli sono ottusi; di maniera che col estesimento del numero de’ lati e degli angoli, scema la longhezza de’ lati e cresce la larghezza degli angoli; onde si fanno tuttavia più ottusi; e finalmente si perdano e si risolvano nella circonferenza del cerchio. </s> <s>E questo si potrebbe pruovare adducendo l’essempio di tutte le figure che dentro o fuor d’un medesimo cerchio si possan formare; cominciando dalla figura di tre lati infine a quella di cento; onde vedremo in ciascuna figura l’angolo pigliare agumento, ed ognhora più allargandosi accostarsi alla tondezza del cerchio o del globo. </s> <s>Da tutte queste cose si possan ritrarre due propositioni convesse in fra loro, ciò sono:</s> </p> <p type="main"> <s>Quanto più si ristregne l’angolo retto, l’ottuso e l’acuto, tanto più si accosta alla linea e finalmente in essa si converte.</s> </p> <p type="main"> <s>Quanto più si allarga l’angolo retto, l’ottuso e l’acuto, tanto più si accosta alla linea, e finalmente divien la stessa linea.</s> </p> <p type="main"> <s>Queste medesime propositioni non solamente hanno luogo negli angoli posti nel piano; ma anchora in quelli che si pongano nella circonferenza o nella sfera. </s> <s>Ma ciò basti haver detto intorno a questo proposito. </s> </p> </chap> <p type="main"> <s>//</s> </p> <chap> <p type="head"> <s>Per qual cagione alcune volte il cerchio sia detto tutto angolo</s> </p> <p type="head"> <s>Cap. 9</s> </p> <p type="main"> <s>Se questo detto che’l cerchio sia tutto angolo, fusse assolutamente vero, bisognarebbe ancho affermare che fra la curvità della linea circolare e la rettitudine della retta fusse qualche proportione; onde avverrebbe che non fusse tanto difficile la quadratura del cerchio, se non impossibile. </s> <s>Perciochè l’angolo è costituito dal contatto delle linee rette, adunque la circonferenza, che costituisce il cerchio doverebbe ad esse esser proportionata; che una misura si dice esser tutta un’altra misura, che non solo le è proportionata, ma ancho uguale; perciochè per l’ottavo Assioma del primo d’Euclide quelle cose sono uguali, che si adattano bene insieme. </s> <s>E non si dice una misura esser tutta un’altra se non le si adatta; adunque adattandosi l’angolo al cerchio e così per opposito il cerchio all’angolo, el cerchio sarà tutto angolo e l’angolo tutto cerchio; che non è altro che dire, che sieno uguali: la qual cosa è impossibile. </s> <s>Oltre acciò per la terza def. del 5° d’Euclide essendo la proportione una certa convenienza di due grandezze del medesimo genere; in quanto appartiene alla quantità è cosa certissima, che ‘l cerchio non ha proportione alcuna con l’angolo; perciochè e l’angolo e ‘l cerchio sono due generi diversi; onde avviene che non abbiano fra loro convenienza alcuna, e perciò non hanno proportione. </s> <s>E per questo si conclude che non possino adattarsi insieme in maniera che’l cerchio vaglia quanto l’angolo e l’angolo quanto ‘l cerchio. </s> <s>Quando si dice il cerchio esser tutto angolo, o s’intende esser in quanto alla proprietà, o in quanto alla misura. </s> <s>Non in quanto alla misura come già s’è pruovato e si pruova di nuovo; perciochè una misura si dice tutta un’altra, quando i termini dell’una cadano sopra <pb pagenum="folios 24v-25r"></pb>‘termini dell’altra; che tanto è longa o larga o alta una cosa quanto è lo stendimento della sua misura, cioè di quella linea che si applica alla cosa che si misura secondo la positiva delle tre misure della quantità continua. </s> <s>Ma ‘l cerchio per questa ragione non si può dir tutto angolo; perciochè se è collocato ‘l cerchio dentro all’angolo, l’angolo avanza; se fuore il cerchio avanza l’angolo. </s> <s>Non in quanto alla proprietà perciochè altre sono le proprietà del cerchio altre dell’angolo; che l’angolo comunque si consideri può sempre toccare ‘l piano in un ponto e ‘l cerchio non può toccarlo se non considerato in una maniera pura geometrica, come s’è detto altre volte: il cerchio è più atto a muoversi e forma ‘l movimento continuo ed uniforme; ma l’angolo è impedimento al muoversi e se cosa alcuna angolare si muove, l’angolo gli interrompe il movimento o ‘l ritarda, o li toglie l’esser continuo uguale e uniforme, e fa che ‘l mobile si muova saltando. </s> <s>Come dunque sarà vera la propositione che ‘l cerchio sia tutt’angolo? Di più se ‘l cerchio fusse tutt’angolo, sarebbe escesso e mancamento di se medesimo; ma non è alcuno di questi; perciochè ‘l cerchio è una figura perfetta, e finita: e non si dà una figura che avanzi se medesima e da se stessa sia manchevole o che sia maggiore o minore in uno stesso tempo e secondo uno stesso rispetto. </s> <s>Adunque ‘l cerchio non dee chiamarsi tutto angolo, che l’angolo a riguardo del cerchio sia escesso di esso o mancamento, si vede per esperienza. </s> <s>Però facendosi ‘l cerchio tutto angolo verrà tutto escesso e tutto mancamento il quale si dice dell’angolo rispetto al cerchio; adunque verrà fatto tutto escesso e tutto mancamento di se medesimo, il che è impossibile. </s> <s>Per la qual cosa senza ragione il cerchio sarà appellato tutt’angolo. </s> <s>Ma vediamo hora <lb></lb>//<lb></lb>se si può truovar modo da salvar questa propositione.</s> </p> <p type="main"> <s>Il cerchio potersi dir tutt’angolo si può intender in più maniere, cioè o che sia tutt’angolo, cioè che habbia ‘l medesimo estendimento e misura che ha l’angolo, e questo è impossibile; poiché ‘l cerchio e l’angolo non son proportionali né uguali: o che sia tutt’angolo, cioè in quanto a qualche proprietà e simiglianza, e questo è probabile. </s> <s>Perciochè se l’angolo raccoglie ristregne e termina lo spatio secondo qualunque specie di figura a differenza delle linee rette e parallele che non ‘l chiudono, ma ‘l possono allongare in infinito; il cerchio raccoglie, ristregne e determina lo spatio seconda la sua specie, mentre è contenuto da una sola linea e col mezzo di essa a differenza delle rette linee che nol posson terminare. </s> <s>Se l’angolo essendo mosso in giro ha proprietà di formare ‘l cerchio; il cerchio pel vario e moltiplicato taglio fatto su per la circonferenza, produce varie specie d’angoli. </s> <s>Se l’angolo prende principio da un punto e in un punto finisce; perciochè ha origine dal punto del contatto di due linee, il quale è termine commune di esse, o pure congiognimento, e finisce nel medesimo punto; perciò che è termine del contatto e per conseguenza dell’angolo; il cerchio comincia da un punto, che è quello onde sorge la sua circonferenza, il quale è l’istesso, che ‘l termine dell’intervallo: e in un punto finisce; perciochè, finito il rivolgimento, quivi termina la circonferenza. </s> <s>Overo diciamo che comincia dal centro e nel centro finisce; perciochè ‘l nascimento del cerchio procede dal movimento dell’intervallo, il quale fissato nel centro, comincia il giramento da esso (che se non havesse principio dal centro non potrebbe formare ‘l cerchio) muovendo tutta la sua longhezza e con essa anchora il ponto <pb pagenum="folios 25v-26r"></pb>che la termina, e ritornando di nuovo nel ponto onde si partì, si ferma appresso tal centro, e nel medesimo termine. </s> <s>Se l’angolo si fa per contatto el cerchio anchora si fa per contatto; perciochè l’angolo si forma per contatto di due linee rette (come più volte s’è detto) el cerchio per contatto di linee curve, cioè delle portioni della sua circonferenza. </s> <s>Overo diciamo che ‘l cerchio sia tutt’angolo, non in atto; ma in potenza, cioè che’l cerchio in qualche modo si possa ridurre all’angolo; e ciò dico perciochè assolutamente non si può affermare; che non è riducimento perfetto, ma vicino al perfetto; che è impossibile a truovarsi; ma ha qualche apparenza, sì come dimostraremo al suo luogo: o che sia tutto contatto, quasi che sia formato per un continuo contatto delle parti succedenti. </s> <s>E questo si vede chiaro perché in qualunque parte, col mezzo del tagliamento si può ritruovare il punto del contatto. </s> <s>Ma avanti che si dimostri il riducimento del cerchio all’angolo, bisogna preporre alcune notitie che renderanno più facile la dimostratione di questo problema.</s> </p> <p type="main"> <s>Ridurre all’angolo il cerchio; onde si possa chiamar tutt’angolo.</s> </p> <p type="main"> <s>Suppongansi dunque prima queste positioni.</s> </p> <p type="main"> <s>1. In ogni circonferenza è qualche parte di linea, cha ha similitudine con la retta, e quanto più la circonferenza è grande, tanto più saranno grandi le dette parti.</s> </p> <p type="main"> <s>2. Congionte insieme quelle parti della circonferenza che hanno simiglianza con la linea retta, necessariamente formano una linea quasi retta.</s> </p> <p type="main"> <s>3. Tutti i cerchi fra loro son simili, e così le circonferenze.</s> </p> <p type="main"> <s>4. I cerchi minori hanno corrispondenza a’ maggiori e così le parti.</s> </p> <p type="main"> <s>5. Che tutte le parti di diverse circonferenze hanno il lor centro particolare.</s> </p> <p type="main"> <s>6. Che tutte sono simiglianti in fra loro, e congionte insieme tanto che formino l’angolo, tanto spatio abbracciano quan <lb></lb>//<lb></lb>to le linee rette. </s> </p> <p type="main"> <s>La prima positione si potrebbe dichiarar con la minuta division del cerchio e con la sperienza del globo della terra e dell’acqua, che apparisce piana, e con tutto ciò (come dice il Sacrobosco nella Sfera) vedendonvi nel mare da lontano venir una nave a poco a poco, la vediamo quasi sorgere, e ciò accade per la tondezza di tutto ‘l corpo dell’acqua e della terra; ma apparisce piana; perciochè nelle parti vi è qualche dirittura e pianezza e quella si può in parte piana; poiché vi si posano e caminano in piano gli animali e perché fra due ponti Zenit e Nadir, detti verticali, cade la linea perpendicolare che forma angoli retti nel taglio della palla del mondo inferiore: oltre acciò vi si fermano in piano le piante, e gli edificij. </s> <s>Ciò si può confermar con Tolomeo nel cap. 10° del 2° dell’Almagesto, là dove facendo conferenza delle proportioni degli archi maggiori e minori alle corde loro, mostra un arco minore che è una piccola portione d’un gran cerchio esser gradi 60 e la sua corda esser parti 0 gradi 60. Si è detto nella circonferenza esser qualche parte di linea simigliante alla retta; perciochè quantunque sia uguale ed habbia qualche dirittura, con tutto ciò, non si può dir linea retta; perché non si produce da un punto all’altro, come insegna Euclide nel primo Postulato; ma si produce dal movimento d’un punto intorno ad un altro punto immobile col mezzo dell’intervallo, sì come si produce tutta la circonferenza.</s> </p> <p type="main"> <s>La seconda dichiara, che sì come di più linee rette minori si può formare una retta maggiore (che ‘l risolvere è contrario del comporre; onde vediamo, che d’una linea maggiore se ne taglia altra minore, com’è manifesto pel 3° Prob. del primo d’Euclide), così di più particelle di circonferenza si può formar una linea intera quasi retta per la medesima ragione addotta. <pb pagenum="folios 26v-27r"></pb></s> </p> <p type="main"> <s>La terza e la quarta si possano dichiarar con la definitione II del 3° d’Euclide e con la dodicesima aggionta dal Commandino.</s> </p> <p type="main"> <s>La quinta è manifesta; perciochè ciascuna volta che più portioni di circonfernza sono accozzate insieme non possono haver un centro commune non essendo collocate in guisa che sieno concentriche; ma essendo eccentriche. </s> <s>Però necessariamente avviene che ciascuna habbia ‘l suo centro particolare.</s> </p> <p type="main"> <s>La sesta si può dichiarare per la simiglianza delle parti e degli angoli o linee contenute; ansi per la uguaglianza degli angoli che si truovano nella circonferenza, come si vede nella XI def. del 3° d’Euclide e nella XII aggionta dal Commandino. </s> </p> <p type="main"> <s>Proponiamo due essempij d’angoli e ciascuno de’ quali sia adattato al cerchio, cioè l’angolo ABC. applicato al cerchio DFE. e l’angolo GFH. adattato al cerchio GHI. e formiamo la dimostratione in questo modo. </s> </p> <p type="main"> <s>Sieno nella prima figura due linee rette AB. e BC. che facciano l’angolo ABC. e dal punto B. del contatto dell’angolo, si tiri una perpendicolare, cioè la BF. costituiscasi in essa il punto M.(pel 3° Post. di Euclide) e fatto l’intervallo MF. si formi ‘l cerchio DFE che tocchi i termini delle linee rette AB. e BC., così sopra il centro M. posto il centro T alto 3 portioni della linea MF. si faccia il cerchio GHI. e sopra il punto T pongasi ‘l centro R. di uguale altezza, e si descriva il cerchio KML. ed ascendendo quasi per due portioni si ponga il centro V e si disegni ‘l cerchio NO e nel taglio fatto dal cerchio KLM. nella perpendicolare, dico nel segno S. si determini un altro centro e si descriva il cerchio piccolo PQX. di maniera che si saranno formati cinque cerchi i quali col convesso della circonferenza loro si congiogneranno con le linee <lb></lb>//<lb></lb>AB. e BC. formando con esse l’angolo ABC. I quali cerchi sono fra loro proportionali perciochè tutti sono di proportione sesquialtera; poiché per la diminutione loro si palesa la commune proportione in fra essi. </s> <s>Sono ancho simili come è manifesto per la terza positione e come si può confermare con la undicesima def. del 3° d’Euclide, perciochè le portioni di loro prendono angoli uguali, o sopra esse si fanno angoli uguali; che l’angolo ABM. è uguale all’angolo ABC. essendo uguali le basi AM. MC. *(Per la contrapositione della ventiquattresima e venticinquesima del primo d’Euclide) [nota in margine] <pb pagenum="folios 27v-28r"></pb></s> </p> <p type="main"> <s>I detti cerchi diminuiti proportionalmente formano quasi un angolo simile all’angolo ABC. Onde essendo tutti simili e prortionali, la medesima possanza di formare un angolo converrà communemente a tutti insieme uniti ed incatenati e perciò potrà dirsi assolutamente il cerchio farsi tutto angolo; poiché le parti di ciascun cerchio concorreno a formarlo: che per la quarta positione i cerchi maggiori e minori e le parti loro hanno in fra lor corrispondenza, con la quale si accordano a far l’angolo. </s> <s>Ma pruoviamo l’angolo costituito da questi cerchi; esser uguale al rettilineo. </s> <s>Congionti i ponti AM. MC. onde seguono due triangoli ABM. MBC. Perché per Euclide, nella diciannovesima prop. del primo al maggior angolo è sottoposto maggior lato; per opposito all’angolo uguale sarà sottoposto lato uguale. </s> <s>Ma ‘l lato AM. del triangolo ABM è uguale al lato MC. del triangolo CBM. Adunque l’angolo ABM. sarà uguale all’angolo CBM. Ma il lato IM. è ancho lato del triangolo formato con la forza delle portioni de’ cerchij GKNPB e ‘l CM. è lato del triangolo ILOQB adunque l’angolo GBM sarà uguale all’angolo IBM. Ma l’angolo GBM è uguale al ABM e ll’angolo IBM è uguale all’angolo MBC. Adunque tutto l’angolo IBA è uguale a tutto l’angolo ABC. Ma l’angolo IBA è formato di portioni di cerchij, nelle quali si son risoluti, e l’ABC. è fatto di linee rette; adunque l’angolo di portioni di cerchij sarà uguale all’angolo rettilineo; onde seguirà che ‘l cerchio si sia ridotto tutto l’angolo. </s> <s>Si potrebbe ancho dimostrare facendo comparation degli spatij contenuti da amendue gli angoli; ma passia <lb></lb>//<lb></lb>mo ad un’altra dimostratione, nella quale apparisca l’angolo formato di parti di cerchij; onde si formi un angolo almeno simile se non uguale interamente all’angolo rettilineo. </s> <s>Essendo già proposto il cerchio GHST e in esso tirati il diametro QR. e la perpendicolare FI che pasi pel centro, sopra la quale sieno tirati tre cerchij minori proportionali, cioè KL. MN. OP. essendo già diviso in quattro quarte uguali dal diametro detto e dalla perpendicolare, si tirino due linee che dividino pel mezzo le dette quarte, ciò sono GT. HS. onde tutto ‘l cerchio maggiore venga diviso in otto parti. </s> <s>Si divida la metà del diametro QR. in quattro parti uguali e presa la quarta si ponga nell’altra metà del diametro appresso alla circonfrenza nel segno QX. come nella prima metà di esso è collocata nel segno VR. Di poi preso per centro il punto T. e per intervallo TG. si faccia col compasso una portion di cerchio cominciando a muover il piede dal termine G. tanto che ‘l cerchio maggiore si congionga col primo minore terminando nel segno W. Così ancho posto ‘l centro S. e l’intervallo SH. si formi una portion di cerchio che terminando nel segno S congionga il primo cerchio minore col maggiore. </s> <s>Quindi preso per centro ‘l taglio V. si faccia con le seste l’arco WF. e preso ‘l taglio X. si descriva l’arco SF. i quali archi si tagliaranno nella perpendicolare nel punto F e formaranno l’angolo. </s> <s>Finalmente stabilito centro il termine della terza portione in ogni metà del diametro si formi un arco che agguagli e congionga i due archi GW. WF. ne’ punti DY. e così un altro che spiani e colleghi i due archi HS.SF. ne’ punti LZ. Di modo che di tutte le portioni de’ cerchi e di tutti gli archi si son formate due linee quasi rette, cioè GF. FH. le quali toccandosi nel punto F. formano l’angolo GFH. uguale all’angolo ABC. la qual cosa si dimostrarà qui appresso. <pb pagenum="folios 28v-29r"></pb>Perché le linee GF. FH. sono uguali alle linee AB. BC. le quali sono linee rette, segue che ancho le GF. FH. sieno rette. </s> <s>Perciochè per la prima suppositione son formate di parti di circonferenze di cerchij, le quali son rette. </s> <s>Oltre acciò per 8° Assioma el primo d’Euclide, quelle cose sono iguali, che si adattano bene insieme. </s> <s>Ma le linee GF. FH. assai ben si adattano con le linee AB. BC. adunque le GF. e FH. sono uguali alle AB. e BC. e le grandezze, che sono uguali fra loro, sono ancho simili, essendo della medesima specie; adunque GF. FH. saranno simili ad AB. e BC. ma AB. e BC. son linee rette, adunque GF. ed FH. saranno simiglianti a linee rette, ma sono ancho linee rette, come s’è dimostrato. </s> <s>E con tutto ciò son composte di particelle di circonferenze. </s> <s>Hora congionti i punti HA. AF. AG. GF. FH. nella seconda: e nella prima congionti CM. MB. MA. AB. BC. si saranno due triangoli, cioè nella prima ABM. BCM. e nella seconda FGA. FHA. Perché due lati del primo triangolo della prima dico dico BA. ed AM. sono uguali a due lati del 2° BC. e CM. segue il triangolo ABM esser uguale al triangolo MBC. Così anchora nella seconda figura per due lati del primo traingolo, cioè GF. e GA. sono uguali a’ due lati del 2° FH. HA. segue parimente i due triangoli GFA. FAH esser uguali. </s> <s>Perciochè hanno anchora gli angoli uguali e la base commune. </s> <s>Ma mutando ordine il lato GF è uguale al lato AB. come già si è dimostrato el lato GA. al lato AM, perciochè *(Per la quindicesima def. del primo d’Euclide) [nota in margine] escono del centro di due cerchi uguali; così anchora per la medesima ragione il lato FH sarà uguale al lato BC. ed HA. a CM. e la base FA: commune alla base BM. Adunque i triangoli FGA. FHA. saranno uguali a’ triangoli BAM. BCM. Adunque l’angolo <lb></lb>//<lb></lb>GFA. è uguale all’angolo ABM. e l’angolo AFH è uguale all’angolo MBC. Ma gli angoli GFA. e AFH son parti dell’angolo GFH. costituito dalle linee formate di particelle di circonferenze: e gli angoli ABM. e MBC. sono parti dell’angolo ABC. fatto col toccamento di due linee rette. </s> <s>Adunque tutto l’angolo GFH. è uguale a tutto l’angolo ABC. ma è ancho simile. </s> <s>Adunque l’angolo GFH fatto di linee circolari sarà simile e uguale all’angolo ABC. che è retto. </s> <s>Il che bisognava dimostrare.</s> </p> <p type="main"> <s>E le dette linee circolari, che per la prima supp. son simiglianti alle rette, non son separate dal cerchio (perciochè o sono attualmente costituite da tagliamenti di quattro cerchij disuguali o sono formate stabiliti altri centri, come già si è dimostrato nella pratica) ed essendo congionte insieme formano le linee GF. FH. le quali toccandosi nel ponto F. formano l’angolo GFH. Adunque insieme col cerchio lo formano. </s> <s>Onde segue che ‘l cerchio si riduca all’angolo, mentre le parti della sua circonferenza si son fatte linee, che pel contatto formano l’angolo. </s> <s>Però segue anchora ‘l cerchio potersi chiamar tutto angolo, perché risolvendosi in parti tutto si converte in angolo. </s> <s>Si potrebbe ancho dimostrare facendosi comparatione fra i cerchij, cioè fra maggiori e maggiori, e fra minori e minori e fra segamenti e fra le portion de’ cerchij, e quindi trahendo l’ugualità delle linee che sopra i convessi loro son tirate, e di poi l’uguaglianza degli angoli. </s> <s>Ma che le linee rassembrino le rette benchè sieno parti di circonferenze, quindi è manifesto; perché si formano stabilito ‘l centro, *(come si è supposto nella .5. positione) [nota in margine] e perché essendo parti di circonferenze di cerchi maggiori hanno qualche rettitudine, come è chiaro per la prima positione. </s> <s>Che le parti delle circonferenze sieno fra loro simili, come si è supposto nella positione terza e sesta <pb pagenum="folios 29v-30r"></pb>ed uguali e proportionate quindi si può ritrarre che son linee curve; si perché sopra esse si possan formare angoli uguali per la def. aggionta dal Commandino nel 3° di Euclide. </s> <s>Si disse che questo riducimento del cerchio all’angolo non è perfetto, riserbandosi la ragione a questo luogo. </s> <s>Si ritragga dunque la ragione dalla pratica e dalla teorica di questo problema. </s> <s>Che ‘l congiognimento de’ cerchij non è tanto esatto che tolga l’angolo rimanente in fra’ segamenti che si fanno da essi scambievolmente: nella linea, che gli congiogne è talmente diritta e piana, che non habbia qualche incurvamento per natura sua, benchè sia incognoscibile. </s> <s>O vero diciamo intendersi il cerchio tutto angolo; perciò che l’angolo è più vicino a diventar cerchio, che a farsi linea retta; come si vede nelle figure di molti lati; che quanto più, che quanto più vi si moltiplicano i lati e gli angoli, tanto più l’angolo si spiana, e finalmente diventa tutto cerchio. </s> <s>E queste son tutte le maniere con le quali per me s’è possuto sostentare che ‘l cerchio sia tutto angolo.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>A che serva l’angolo nell’Universo</s> </p> <p type="head"> <s>Cap. 10</s> </p> <p type="main"> <s>Nell’Universo non si trova cosa che sia indarno come si ritrahe dal Filosofo. </s> <s>Però se l’angolo è cosa reale overo intelligibile, bisogna vedere qual fine habbia. </s> <s>Primieramente per ispedirmi più tosto che sia possibile dirò che sì come ‘l cerchio è stato truovato specialmente pel movimento locale circolare (onde avviene che ‘l Cielo sia inclinevole al giramento e la sua inclinatione sia detta da’ Filosofi principio passivo interno e quasi natura) così la figura angolare per la quiete e col mezzo di essa l’angolo anchora; perciochè gli angoli impediscono ‘l movimento continuo; che <lb></lb>//<lb></lb>(sì come dice Pietro da Medina nel 3° cap. del primo lib. dell’Arte del navigare) la propria operatione del Cielo è muoversi continuamente e circolarmente e perciò conviene che habbia la figura accomodata a tal movimento e questa si è la figura ritonda perciochè è privata d’angoli, che sono impedimento al muoversi. </s> <s>Oltre acciò non si possano costituir le quattro quarte del Mondo senza gli angoli perciochè o prendansi da’ quattro venti, in quanto son cagionati da quattro Pianeti secondo gli Astrologi, cioè da Giove, dal Sole, da Marte e dalla Luna. </s> <s>Onde son detti dalla Luna occidentali da Marte meridionali, dal Sole orientali e da Giove settentrionali: overo si prendano da’ quattro triplicità del Zodiaco, come dall’Ariete, dal Leone, dal Sagittario, tutti di qualità calda e secca, e Segni orientali onde vengono i venti orientali: o dal Toro, dalla Vergine, e dal Capricorno, tutti di qualità fredda e secca e meridionali; onde nascono i venti australi: o da Gemelli, dalla Libbra e dall’Aquario, tutti di qualità calda e humida e occidentali; onde procedono i venti occidentali di maniera che queste son quattro triplicità di Segni, appellate ignea, terrea, aerea, aquea, le quali costituiscono quattro venti, che danno nome a quattro parti e a quattro angoli del Mondo; perciochè come dice l’istesso autore nel 5° cap. la tondezza della terra ha quattro parti, angoli e regioni principali non solamente avvertite da’ Filosofi e dagli Astrologi, ma ancho dalla Sagra Scrittura, come appo San Luca nel 14° cap. e appresso David nel Salmo 106. E queste si nominano e conoscano pe’ quattro venti cardinali, ciò sono Levante, Ponente, Tramontana, Mezzodì. Da’ quali tirate <pb pagenum="folios 30v-31r"></pb>le linee rette che passino pel centro del Mondo, necessariamente tagliano la sua circonferenza ad angoli retti e costituiti in fra quattro altri venti collaterali e congionti con le linee loro formaranno gli angoli della medesima maniera. </s> <s>Così nella carta da navigare i 32 venti che stabiliscono altretanti rombi dividono la tondezza del Mondo in tante parti e ne’ tagliamenti costituiscono gli angoli, i quali si possano ancho ritruovar nella sua circonferenza per cagion delle posture destra, sinistra, e alta e bassa, che dal Filosofo sono adattate al Cielo in riguardo della postura dell’huomo, come si può vedere nel lib. del Cielo e del Mondo. </s> <s>Là dove la parte destra del Cielo è detta oriente e la sinistra occidente, e l’alta e la bassa i due poli. </s> <s>O vero potremo truovar gli angoli per la differenza del movimento, osservando ciò che dice San Tommaso nel 3° della Filosofia Naturale nella particella 54. Nel cielo sono le differenze del movimento; avanti l’emisfero di sopra, dietro l’emisfero di sotto; a destra l’oriente; a sinistra l’occidente; di sopra mezzo giorno; di sotto settentrione. </s> <s>E ‘l reflettimento de’ raggi del Sole sopra la terra, formando diversità d’angoli, produce più o men gradi di calore, secondo che con l’angolo si unisce e si disgrega la forza del lor calore; onde nasce ‘l vario temperamento delle stagioni dell’anno. </s> <s>Perciochè ‘l ripiegamento de’ raggi è cagione del calore; perciochè molti raggi si riducano in un punto vero, dove per la caldezza unita e rinvigorita il luogo si riscalda e s’abbracia, sì come disse Alberto Magno nel tratt. 3° del 2° del Cielo. </s> <s>Nel qual luogo bisogna osservare che i raggi della luce si dirizzano a un vero punto, <lb></lb>//<lb></lb>cioè non sensibile e materiale, ma geometrico; e questo da Alberto si aggiogne per mostrare in quanta acutezza talvolta per virtù degli angoli formati dal percotimento e piegamento de’ raggi del Sole; si raccolga e si ristrenga il calore; che si uniscano in una cosa indivisibile; onde non è maraviglia, che la virtù si multiplichi insieme col calore; onde il luogo venga o caldo overo acceso; che perciò disse Averroè nella parte 72 del 3° del Cielo, che l’acutezza degli angoli è cagion di bruciamento. </s> <s>Oltre acciò di tre movimenti locali, che si vedano continuamente nell’universo: il retto e ‘l reflesso non potran formarsi già mai senza ‘l producimento degli angoli. </s> <s>Perciochè ‘l movimento delle cose gravi o delle leggiere si fa sempre per linea retta; che mentre le cose gravi si muovano formano una linea che ferisce ‘l piano della terra, ad angoli retti, come si può pruovare con l’autorità di Tolomeo nel 7° cap. del primo dell’Almagesto e se potessero passar la terra la detta linea fermandosi nel centro tagliarebbe ad angoli retti il diametro del Mondo. </s> <s>Onde con ragione disse Ammonio sopra Porfirio, che gli angoli retti non si possan tirare altrove, che al centro. </s> <s>E mentre le cose lievi si muovano verso ‘l concavo della luna, fanno una linea che divide la circonferenza del convesso della sfera del fuoco e ferisce la concavità di quella della luna ad angoli retti, ed amendue le linee sono perpendicolari solide; perciochè sopra la superficie nella quale cadono formano angoli conseguenti uguali e si formano dentro ‘l corpo sferico. </s> <s>Il movimento reflesso, o ripiegato che si dica, necessariamente si fa col producimento degli angoli in quella guisa che farebbe una palla da giocare cadente <pb pagenum="folios 31v-32r"></pb>sopra ‘l piano, la quale tosto percossolo risaltesse e tornasse a muoversi al contrario che ‘l movimento reflesso, secondo Aristotile nella parte 64 dell’8° della Nat. </s> <s>Filosofia è composto di contrarij movimenti; i quali si discernano per la quiete infra posta e la quiete pel percotimento della cosa mobile nel piano, e ‘l percotimento per l’angolo si conosce; onde il Filosofo stesso nella settion 16 parte 4 de’ Problemi disse: quelle cose che cadono nella terra formano angoli nel piano da ogni banda del punto nel quale habbiano toccato ‘l piano perciochè tutte per lor natura si muovano per linea retta; ma quelle che desiderano muoversi a un luogo uguale, quando dalla linea perpendicolare e del diametro feriranno nel piano, perciò faranno col balzamento loro angoli uguali perché devidono in parti uguali ‘l diametro. </s> <s>Ma quelle cose che per lato cadono, perché feriscono la terra non per linea perpendicolare ma da un punto sopra accade che ribattute dal luogo si muovano nella contraria parte. </s> <s>Ma per qual cagione Aristotile mostra che le cose che fanno ‘l movimento reflesso si muovano dalla perpendicolare e dal diametro? Non solamente per la ragione detta, ma per accennare che qualunque cosa, che si muova verso ‘l piano, mentre a poco a poco si stacca dalla linea perpendicolare, si parte ancho dal diametro; perciochè ovunque sia, sempre costituisce una linea che se fusse continuata, e passasse pel centro col mezzo della cosa mobile grave e quindi si allongasse fine alla parte opposta necessariamente formarebbe ‘l diametro. </s> <s>E che ‘l reflettimento non si faccia <lb></lb>//<lb></lb>senza angoli si conferma col detto del Commentatore nella parte 49 e nel .C. 61 del 2° del Cielo. </s> <s>Che ‘l reflettimento non si fa se non secondo gli angoli terminati. </s> <s>Ma poiché si è ragionato de’ movimenti non par fuor di proposito ragionar ancho di quelle cose che gli facilitano. </s> <s>Gran differenza è da una cosa mobile di figura piana ad una di figura angolare, e così da una che habbia angoli retti overo ottusi ad una che gli habbia acuti. </s> <s>Perciochè le cose che sono di figura piana, quantunque sieno gravi difficilmente fendono ‘l mezzo e perciò con difficoltà si muovano, il che si vede per la tardanza del movimento loro: e poste sopra l’acqua stanno a gallo invece di calar nel fondo. </s> <s>Le cose angolari, quantunque minori e talvolta ancho men gravi, più facilmente tagliando l’aria, si muovano verso la terra: e nell’acqua tosto cadano nel fondo. </s> <s>Quelle che hanno angoli retti e ottusi tagliano sì bene l’aria e per essa si muovano scendendo al lor centro; ma non senza qualche resistenza del mezzo. </s> <s>Finalmente le cose che hanno angoli acuti, facilmente trapassano ‘l mezzo, e per esso si muovano velocissimamente giognendo al centro. </s> <s>Ma se per avventura prenderemo due cose gravi o sieno angolari, o sieno ritonde o qualunque altra figura; ma sieno disuguali di grandezza e di peso e da qualche luogo eminente le lassaremo cadere in un medesimo tempo in terra, onde avviene che amendue vi arrivino in un medesimo tempo?Dovrebbe pur giognervi più presto la minore poiché occupando meno spatio e havendo men diametro è più atta a fendere ‘l mezzo; tanto più che truova minor <pb pagenum="folios 32v-33r"></pb>resistenza per la sua poca larghezza: overo vi havrebbe prima ad arrivar la maggiore, come più grave; poiché quanta più gravezza vi è tanta più velocità di movimento vi doverebbe essere. </s> <s>A questo si dee dire che amendue in uno stesso tempo arrivano al piano della terra perciochè se nella minore non è tanta gravezza che possa dar tanta forza al movimento vi è la brevità della figura in supplimento; talchè le toglie la resistenza dell’aria e le dà la facilità al trapassarla: e se nella maggiore è tanta larghezza che dà occasione alla troppa resistenza dell’aria vi è la gravezza che che supplisce; di modo che la poca grandezza e la men resistenza dà alla minore quella parte di virtù che le mancava per muoversi: e la gravità della maggiore supplisce la parte della forza, che dalla figura e dalla resistenza veniva scemata. </s> <s>Di maniera che amendue supplimenti agguagliano le forze d’amendue, onde vengano a muoversi e ad arrivare in un medesimo tempo in terra. </s> <s>Oltre acciò l’acqua che ha da muover qualunque maniera di macchina non ha forza nel muovere se cadendo non fa angoli simiglianti a’ retti; perciochè se scorre per linea obliqua vien trattenuta a poco a poco dalla superficie del piano che sia ritorto onde perde in gran parte l’impeto del corso; ma se cade a piombo segue col’impeto suo naturale fine al fondo della fossa, anzi dall’angolo della caduta piglia forza maggiore; poiché quivi diviene abbandonata da ogni maniera di sostentamento e lassata tutta nel dominio della sua gravità. E per trapassare a essempij più nobili <lb></lb>//<lb></lb>l’huomo anchora, o stando in piedi o sedendo forma l’angolo; perciochè essendo creato di figura diritta per sua natura costituisce la linea perpendicolare, la quale cadendo nel piano del suo posamento forma da ogni banda l’angolo retto, non solamente stando in piedi, ma sedendo anchora, sì come si può vedere appresso Arist. nella question 30. delle Meccaniche, là dove si mostra, che sedendo si costituisce in due luoghi l’angolo retto e stando in piedi in un sol luogo e la ragione si è perciochè l’angolo retto, che è uguale ovunque sia è cagione della quiete e dello stare in piedi, e quest’angolo è prodotto dalla linea perpendicolare non sopra ‘l pavimento uguale, ma sopra la circonferenza della terra, e per conseguenza da una linea che ha riguardo al centro di essa, ond’avviene che ‘l detto angolo sia cagion di quiete; perciochè è formato da una linea che si termina nel centro, che è la cagion principale della quiete. </s> <s>Nella stessa maniera e per la medesima cagione le piante stanno fisse in terra, formando angolo retto, o nel terreno piano per natura o nel piano del suo posamento, cioè quando sono in spiagge o su per un colle o vero in un terreno al tutto erto e scosceso. </s> <s>Perciochè quantunque la terra sia obliqua, con tutto ciò le piante stanno sopra essa diritte; di maniera che tutte in quanto all’apparenza feriscono ‘l terreno ad angoli impari, ottusi e acuti; ma in quanto appartiene al natural posamento loro feriscono la terra ad angoli retti, di modo che ciascun arbolo per se stesso col suo diritto nascimento riduce in piano e livella il terreno vicino nella stessa guisa che si suol fare misurando le superficie delle colline, tenendo in aria la misura da una banda <pb pagenum="folios 33v-34r"></pb>e ponendo in piano continuamente e tirando le linee a piombo da uno estremo della misura sopra la terra, cme se havessemo a formar molti gradi uguali. </s> <s>Di maniera che da tutto questo discorso possiamo trarre l’uso dell’angolo esser collocato in molte parti dell’Universo, cioè o rispetto al movimento delle parti loro o rispetto alla quiete e al posamento e a’ compartimenti della Sfera del Mondo fatti secondo la position de’ venti: o riguardando l’effetto de’ raggi solari: o le differenze del luogo e del movimento: o pure la differenza delle figure delle cose che si muovono verso ‘l centro e finalmente l’impeto e la forza delle cose cadenti.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Quali sieno gli usi dell’angolo nella Geometria, nell’Astronomia, nella Prospettiva</s> </p> <p type="head"> <s>Nella Geometria</s> </p> <p type="head"> <s>Cap. 11</s> </p> <p type="main"> <s>In tutta la Geometria gli angoli son di tanta virtù che senza essi non solamente non potrebbe fabbricar le figure e ‘corpi regolari o irregolari; ma ancho quando pur ciò potesse fare, non potrebbe formarne dimostratione alcuna, per pruovar la nesità delle propositioni alle figure e a’ corpi appartenenti. </s> <s>Perciochè chi bene osservarà Euclide, vedrà non esser quasi dimostratione alcuna, che non prenda forza d’illatione dalla conferenza degli angoli. </s> <s>Ma questa fin qui è la minor parte del giovamento, che la Geometria riceve dall’uso degli angoli. </s> <s>Però per darne più copioso ragguaglio, in quanto comporta questo luogo; aggiognerò che gli angoli son <lb></lb>//<lb></lb>cagione di molti effetti; perciochè o sono l’origine o ‘l termine delle figure e de’ solidi: o da essi depende l’accrescimento e la diminutione degli spatij e delle figure e de’ corpi solidi: o l’accrescimento o lo scemamento de’ lati e delle basi: o per essi si conoscano le linee parallele, o le perpendicolari, o le linee per diritto. </s> <s>E per esplicar a parte a parte tutti questi effetti degli angoli senza i quali la Geometria sarebbe vana, con ordine retrogrado mi farò da quest’ultimo, il quale si manifesta da Euclide nella quattordicesima prop. del primo, cioè che per gli angoli si venga in cognitione di due linee, che sieno poste per diritto fra loro. </s> <s>Perciochè conosciuto che due linee rette tirate da diverse parti a una linea retta, e ad un ponto dato in essa, cioè ad uno de’ suoi termini, formano due angoli conseguenti uguali a due retti, anzi due angoli retti stessi; tosto si conosce che le dette linee son per diritto fra loro. </s> <s>E che questo sia vero, quindi apparisce, che quando gli angoli che da esse si costituiscano non sono uguali a due retti, e amendue retti le dette linee non possono esser per diritto fra loro, sì come chiarissimamente apparisce nella dimostration d’Euclide nel luogo citato. </s> <s>Di modo che sì come delle linee poste per diritto fra loro, le quali convengono nella stremità d’una linea retta che stia a piombo nascono gli angoli conseguenti non solamente uguali a due retti, ma ancho retti, così all’incontro, da questi angoli conseguenti retti e uguali a due retti si costituiscano le linee per diritto fra loro. </s> <s>E ciò avviene perciochè la linea alla quale e dal ponto della quale da diverse bande son tirate altre linee è perpendicolare, ed esse, essendo per diritto formano una stessa linea retta piana, che <pb pagenum="folios 34v-35r"></pb>determina i detti angoli. </s> <s>La notitia degli angoli è ancho cagione che si conoschino le linee perpendicolari; e questo si può vedere manifestamente nella decima def. del primo d’Euclide e nella prop. dodicesima. </s> <s>Perciochè nella definition degli angoli retti si mostra che dalla costitution degli angoli segue la linea, che sta sopra a un’ altra linea retta esser perpendicolare; che facendo angoli conseguenti e dalle bande uguali, gli forma ancho retti; poiché l’esser uguale è proprio degli angoli retti, ed essendo retti necessariamente segue la linea che sovrasta alla giacente esser perpendicolare. </s> <s>E che questo sia vero rimiriamo con diligenza la definitione.</s> </p> <p type="main"> <s>“Quando la linea retta stando sopra un’altra fa gli angoli da lati fra loro uguali, sono amendue retti e la linea che sta sopra si chiama perpendicolare a quella a cui sovrasta”.</s> </p> <p type="main"> <s>Questa definitione, al parer mio, procede in questa maniera; perciochè è composta di due parti, una si è ‘l caso che si propone, e l’altra si è tutto quello che da esso, in guisa di fratto si cagiona. </s> <s>Primieramente adunque propone. </s> <s>Quando la linea retta, stando sopra un’altra, fa gli angoli da lati fra loro uguali. </s> <s>E questo si è ‘l caso che si compone d’una parte indeterminata e universale, che è una linea retta star sopra un’altra: ed una parte che è ‘l ristregnimento e la condition che determina la prima, cioè fare gli angoli da’ lati fra loro uguali. </s> <s>Di poi immediatamente aggiogne. </s> <s>Sono amendue retti e la linea, che sta sopra si chiama perpendicolare. </s> <s>E questa si è la seconda parte, che nasce dalla prima, come ‘l frutto dalla pianta, la quale ancho è composta di due <lb></lb>//<lb></lb>portioni, la prima che detti angoli sieno amendue retti, la seconda che nasce dalla prima, che la linea che le sovrasta sia perpendicolare. </s> <s>Onde si vede chiaramente che dalla formation degli angoli per la sovrastante linea nascono gli angoli retti; e per gli angoli retti s’acquista la certezza, che la linea che sta sopra a un’altra sia perpendicolare: e che dagli angoli uguali si concludano gli angoli retti e da essi si concluda la linea perpendicolare; ma non si potrebbe concludere se non si procedesse dalle cose note alle non conosciute, si come è chiaro appresso ‘Filosofi; adunque bisogna dire che dalla cognition degli angoli uguali, e retti necessariamente si viene alla cognition della linea perpendicolare; come si manifesta dalla detta definitione. </s> <s>Ma nella prop. dodicesima si vede manifestamente, nel formar la sua descrittione dalla forza degli angoli stabilirsi necessariamente la perpendicolare. </s> <s>E questo più chiaro si vede nella sua dimostratione; perciochè poste due linee rette uguali fra loro, le quali sono ancho per diritto ed una commune in fra esse, e due linee uguali a due linee, cioè una di quelle insieme con la commune a un’altra presa insieme con la commune: così ancho la base uguale alla base (perciochè per la descrittion del problema si formano due triangoli) seguono due angoli costituiti dalle due rette linee, e dalla commune conseguenti e uguali, e la detta linea commune esser quella, che sovrasta alla retta linea data, ed esser perpendicolare, come si conferma per la decima def. già addotta. </s> <s>Quindi adunque si conclude che truovata <pb pagenum="folios 35v-36r"></pb>l’ugualità degli angoli conseguenti, tosto veniamo a conoscere la perpendicolare, la quale dagli antichi fu detta Gnomone; perciochè lo gnomone è una linea retta, che cade ad angoli retti sopra l’orizzonte; onde vediamo che dal cader ad angoli retti si forma questa linea così detta; che se non cadesse ad angoli retti; cioè se cadendo non formasse angoli retti, non si potrebbe appellar Gnomone, né perpendicolare. </s> <s>Dagli architetti si chiama Catetto, overo Piombo o linea a piombo, la quale sopra ‘l piano forma gli angoli a squadra, che geometricamente si chiamano retti; che se detti angoli non fossero a squadra la linea non sarebbe a piombo. </s> <s>Di maniera che l’esser a squadra sia cagion, che ella sia a piombo e perpendicolare. </s> <s>Ma passiamo più avanti. </s> <s>Dalla cognition degli angoli si cagiona la cognition delle parallele sì come chiaramente vedrà chiunque osserva Euclide nel 5° postulato e dalla prop. ventisettesima, infine alle trentaduesima. </s> <s>In tutte queste propositioni e nelle dimostrationi loro, dalla ugualità degli angoli alterni, degli esterni, e degli interni e opposti e dall’esser uguali a due retti si ritrahe la cognition delle parallele. </s> <s>Il che si vede osservando gli angoli che si costituiscano da una linea, che sia tirata sopra le parallele in tal guisa, che le taglino ad angoli retti. </s> <s>La qual cosa potrà da ciascuno esser intesa in Euclide nelle dimostrationi delle accennate prop. </s> <s>Ma nel postulato 5° dagli angoli fatti per la linea cadente sopra due rette linee, i quali son minori di due retti, s’impara a conoscer qua’ linee non sieno <lb></lb>//<lb></lb>parallele, di maniera che, per opposito, se la linea cadente sopra esse formasse gli angoli uguali a due retti, noi dalla notitia di quest’angoli potremo cavar la cognitione delle parallele. </s> <s>Gli angoli oltre acciò son cagione dello scemamento, e dell’accrescimento de’ lati e delle basi delle figure triangolari e parallelogramme così ancho della ugualità e della disugualità, come si può vedere appo Euclide nelle prop. .6., .19., .21. e .24. e .25. del primo. </s> <s>Perciochè nella .6. da due angoli d’un triangolo si traggano i lati sottoposti a essi esser uguali, il che è certissimo per la dimostratione; sì come pel contrario posti due angoli disuguali d’un medesimo triangolo, anchora i lati diverranno disuguali. </s> <s>Nella .19. si mostra, che sotto maggior angolo di ciascun triangolo è collocato maggior lato. </s> <s>E ciò non avvien per altro se non perché mentre cresce l’angolo cresce ‘l lato e però nella .18. prop. disse, il maggior lato di ciascun triangolo esser sotto a maggior angolo. </s> <s>Però possiamo dire che gli angoli col mezzo della grandezza loro son cagione della grandezza de’ lati. </s> <s>Si potrebbe ancho ciò ritrarre dalla prop. ventesima. </s> <s>Ma questo si dee intendere del lato che è opposto all’angolo nelle figure di tre lati. </s> <s>Vediamo hora se si può ritrarre ‘l medesimo effetto degli angoli ne’ parallelogrammi. </s> <s>Dice Euclide nella prop. trentacinquesima. </s> <s>I parallelogrammi posti nella medesima base e nelle medesime parallele esser fra loro uguali: là dove per dimostrarla si forma questa descrittione ABCDEF e sopra essa si dimostrano i due parallelogrammi ABCD e BEFC. esser uguali essendo nella medesima <pb pagenum="folios 36v-37r"></pb>base BC e nelle parallele BC. FA. Ma questo è vero in quanto alla teorica, e secondo la dimostratione fondata in questo supposto, che sieno nella medesima base, e fra le medesime parallele. </s> <s>Ma secondo la pratica apparisce ‘l contrario, il che non doverebbe accadere; perciochè la teorica e la pratica nelle scienze e nell’arti doverebbero convenire insieme; poiché amendue sono quasi due gambe, con le quali la scienza e l’arte procede; che se per avventura mancasse e l’una e l’altra, andarebbe zoppa. </s> <s>Ansi benchè vi sieno amendue, con tutto ciò essendo con qualche sproportione, conviene che amendue in qualche parte vadano zoppicando. </s> <s>Apparisce dico il contrario perciochè il lato EB. del secondo parallelogrammo è maggiore del lato AB. del primo e ‘l lato FC. del secondo è maggiore del alto DC. del primo. </s> <s>Perciochè posto ‘l centro B. per la .3. supp. del primo d’Euclide e lo intervallo BE. si descriva ‘l cerchio GE. e posto ‘l centro C. e lo intervallo CF. si descriva ‘l cerchio FH. vedremo espressamente i detti cerchi avanzare i lati del primo parallelogrammo e per conseguenza essendo i due intervalli uguali, che son due lati del secondo parallelogrammo, saran cagione che ‘l secondo parallelogrammo sia maggiore del primo, che se fosse uguale il cerchio EG. toccarebbe il lato EC. nel segno E. Di modo che per la def. del cerchio i lati de’ parallelogrammi verrebbero a esser tutti uguali. </s> <s>Ma i due lati BE. FC del 2° parallelogrammo son maggiori e contengono angoli maggiori; onde dagli angoli maggiori BEF. FCB. dependono i lati maggiori BE. FC. che se fussero <lb></lb>//<lb></lb>uguali agli angoli del primo parallelogrammo, i lati anchora sarebbero uguali. </s> <s>Oltre acciò teoricamente anchora apparisce ciò esser vero, supposti tre triangoli uguali, ne’ quali si sieno risoluti i parallelogrammi, cioè ADB. DBC. ECF. che per la .19. del primo, sotto a ciascuno angolo maggiore de’ tre triangoli è posto un lato maggiore; che ciascun triangolo ha due angoli minori e un maggiore; ma il parallelogrammo BEFC. è composto di due triangoli DBC. ECF., adunque tutto l’angolo BEF. sarà composto dell’angolo BEC. minore e dell’angolo CEF.maggiore, e però seguirà che BEF. sia molto maggiore dell’angolo BAD., così anchora per la medesima ragione, tutto l’angolo BCF. sarà molto maggiore dell’angolo ABC. e così si potrebbe dire facendo comparatione di questi angoli maggiori agli altri angoli rimanenti; onde seguirebbe che per cagion degli angoli maggiori i lati del secondo parallelogrammo sieno maggiori de’ lati del primo. </s> <s>Oltre acciò se ‘l lato BE. fusse uguale al lato AB. bisognarebbe affermare che ‘l diametro fusse commensurabile col lato del parallelogrammo, la qual cosa è impossibile. </s> <s>Ma ritorniamo alle figure di tre lati. </s> <s>Euclide nella .21. prop. del primo dimostra che dall’angolo maggiore contenuto da linee rette tirate dentro un dato triangolo da’ termini di esso, cioè dagli angoli presso alla base hanno dependenza a due linee minori de’ due lati del triangolo dato; di maniera che si può dire che quanto più sarà maggiore l’angolo, tanto più saranno minori i lati, e così per opposito; e quanto più saranno mag <pb pagenum="folios 37v-38r"></pb>giori, tanto più sarà minore l’angolo e così allo ‘ncontro. </s> <s>Perciochè sopra la base del triangolo dato non si può collocar un altro triangolo che non sia minore, dico dentro ‘l triangolo dato; ma uguale non mai; perciochè sarebbe ‘l medesimo triangolo, come si potrebbe dimostrare per la .7. del medesimo libro. </s> <s>Di modo che fin qui avemo dimostrato, che dall’esser maggiori gli angoli considerati, o come superiori o come accanto a’ lati del triangolo, si fanno i lati minori o maggiori. </s> <s>Ma vediamo ora se dalla grandezza degli angoli de’ triangoli si possan trar le grandezze delle basi. </s> <s>Questo è facil cosa a veder; perciochè ‘l dimostra Euclide nella .4. e nella .24. e .25. del primo. </s> <s>Nella quarta si vede chiaramente; perciochè da’ due lati supposti uguali a’ due lati, seguendo due angoli uguali a due angoli contenuti da linee rette uguali nascono le basi uguali alle basi; ma ciò si certifica per la sua dimostratione. </s> <s>Nella .24. supposti nella medesima guisa due lati de’ triangoli uguali, ma l’angolo esser maggior dell’angolo contenuto dalle linee rette uguali, si dimostra le basi de’ triangoli dover esser disuguali, cioè una maggior dell’altra, la qual cosa si rende certa per la dimostratione come se neccessariamente dagli angoli maggiori e minori contenuti da linee rette uguali si produchino le basi maggiori e minori. </s> <s>E che questo sia vero si vede espressamente per la venticinquesima che è sua conversa. </s> <s>Da tutte queste ragioni dunque si mostrano le grandezze maggiori o minori e gli agumenti e gli scemamenti de’ lati e delle basi de’ triangoli nascer dalle grandezze maggiori o mi <lb></lb>//<lb></lb>nori e dagli accrescimenti e scemamenti degli angoli de’ triangoli; così anchora i lati de’ parallelogrammi farsi maggiori o minori dalla grandezza degli angoli. </s> <s>Hora preposte tutte queste cognitioni, sarà facil cosa cercare se sia vero che dagli angoli dependa l’accrescimento e la diminution delle figure e degli spatij. </s> <s>Se è vero che ‘lati, e le basi de’ triangoli all’hora sieno maggiori, quando son maggiori gli angoli, all’hora minori, quando son minori gli angoli; all’hora uguali quando sono uguali; sarà ancho vero che gli spatij e le figure che gli terminano sieno maggiori o minori o uguali secondo che gli angoli son maggiori, o minori o uguali. </s> <s>Il che espressamente si pruova nella .4a. del primo. </s> <s>Oltre acciò, ancho Proclo nel 4° lib. sopra Euclide, nel com. 9° dice che l’ugualità e la disugualità degli angoli ha gran forza di accrescere e scemare gli spatij. </s> <s>E questo avviene perciochè ne’ triangoli quanto più si dà un angolo maggiore, tanto più si dà maggior lo spatio: né si può conceder spatio maggiore che non si dia ancho almeno un lato maggiore; ma in verità si danno ancho due lati maggiori, uno de’ quali sarà sottoposto all’angolo maggiore, e l’altro conseguente al primo lato maggiore; anzi anchor esso è sottoposto all’angolo maggiore, come è manifesto per la diciottesima e diciannovesima del primo. </s> <s>Questo medesimo si potrebbe adattare a’ corpi solidi, perciochè in essi anchora si truova l’angolo, che secondo Euclide nella .21. definitione dell’11° è differente dall’angolo piano; perciochè è l’inclinatione di più di due linee che si toccano non essendo nella medesima superficie: ed è compreso da più di due <pb pagenum="folios 38v-39r"></pb>angoli piani, come ‘l dimostra nella prop. .21. dell’11°. Di maniera che se si agumentano gli angoli piani, le superficie anchora ricevono agumento: e se si agumentano amendue, necessariamente si accresce l’angolo solido. </s> <s>Ma veniamo hora a considerare se sia vero che gli angoli sieno il termine e ‘l principio delle figure piane e solide. </s> <s>Avanti che si esplichi se ciò sia vero, fa di mestiero avvertire che da questa questione s’escluda la figura circolare, l’ovata, la sferica ed altre figure terminate da una linea e da una superficie sola. </s> <s>Se è vero che ‘l ponto sia principio e termine della linea, e la linea sia principio e termine della superficie, che non è senza figura: e la superficie è termine e principio del corpo, come sarà vero che gli angoli sieno principio e termine delle figure piane e solide? Bisogna avvertire che nella Geometria o ‘l punto, o la linea, o la superfice, o l’angolo possa dirsi principio delle figure in due maniere, cioè che sia principio del producimento loro considerato assolutamente e ‘n generale, cioè che dal movimento loro intelligibile risultino le figure e ‘ solidi: overo che sia principio speciale in maniera che per esso si determini la specie della figura considerandolo in quanto che senza esso la figura non può haver sussistenza. </s> <s>Nel primo modo si vede ‘l principio quasi materiale e nel 2° quasi formale. </s> <s>Nel primo modo l’angolo non è principio delle figure; ma ‘l punto, la linea e la superficie: nel 2° l’angolo solamente è principio di esse. </s> <s>Perciochè quel medesimo principio che dà la denominatione alle figure, gli dà ancho l’essere; che la de <lb></lb>//<lb></lb>nominatione nasce dalla determination della specie e tale determinatione si produce dagli angoli i quali non sono altro che la differenza costitutiva e divisiva delle figure. </s> <s>Di maniera che l’angolo sia lo stesso atto della figura, il quale non altramente che forma ponga separamento le figure. </s> <s>Oltre acciò non si possan formar le figure piane, senza chiuder spatio, né le solide senza terminar grossezza; perciochè sì come due linee rette non chiudono spatio come si vede nella quarta del primo d’Euclide, così due superficie non chiudono grossezza, ne si può già mai chiuder spatio alcuno, né alcuna grossezza di corpo solido senza formar gli angoli. </s> <s>Onde le figure piane e le solide non si dicano figure avanti che sien formati gli angoli; ma subbito formati, che tosto ne risultano le specie determinate delle figure. </s> <s>E però, secondo ‘l numero degli angoli prodotti si specificano e si denominano le figure. </s> <s>Anzi ancho le linee curve non posson chiudere spatio senza gli angoli, come avviene nella figura triangola. </s> <s>Ne si possan formar le figure curvilinee se prima non si costituiscano gli angoli con l’intersegamento de’ cerchij, come si vede nella prima del 1°. Nella medesima guisa si dee dire che l’angolo sia termine delle figure, perciochè se produce la figura, la termina anchora; e mentre per l’angolo si chiude lo spatio e la larghezza e la grossezza, si termina anchora. </s> <s>Overo diciamo l’angolo esser principio delle figure come origine di esse, e termine in quanto che per gli angoli vengono specificate. </s> <s>Gli angoli oltre acciò stabiliscono l’origine e ‘l termine de’ dia <pb pagenum="folios 39v-40r"></pb>metri de’ cerchi, così ancho delle diagonali, che dividono ‘l parallelogrammo: fanno discerner le linee finite dall’infinite; perciochè solamente si formano nelle finite le quali hanno i punti che le determinano; che non si formano se non pe’l contatto in un punto collocato già in esse: e l’angolo non retto ridotto al retto facilita la misura delle figure irregolari; come si ritrahe dalla tredicesima del primo. </s> <s>Gli angoli sono in fra loro misura perciochè l’angolo retto è misura de’ non retti, ma non per opposito, se forse non riguardiamo fra i retti e gli acuti. </s> <s>Percioche ‘l retto è misura dell’acuto; che fattane comparatione si vede quanto l’acuto sia minore del retto e così per opposito quanto ‘l retto sia maggior dell’acuto: overo si dice l’acuto poter esser misura del retto; perciochè nel retto tante volte si replica la sua grandezza, finchè si misuri tutto; che ‘l retto è moltiplice dell’acuto e l’acuto come grandezza minore lo minore lo misura, come è manifesto per la seconda def. del 5°. Ed applicando i detti angoli a’ numeri, facendo l’angolo retto esser quanto ‘l numero maggiore e l’acuto quanto ‘l minore, potremo dire l’acuto angolo esser misura del retto; perché in quanto numero minore è parte del maggiore perciochè lo misura, com’è chiaro per la terza def. del 5° e di questo non si dee dubbitare, perché l’angolo retto è composto almeno di quattro angoli acuti: e si potrebbe per avventura ciò confermare con la definition del misuramento addotta da Niccolò Tartaglia nel primo cap. del suo primo lib. di Geometria, cioè:</s> </p> <p type="main"> <s>Misurare è un voler truovar quante volte in una quantità si ritruovi alcuna quantità famosa, overo alcuna parte: overo sapere quante parti sieno di detta famosa quantità. </s> </p> <p type="main"> <s>Gli angoli, oltre acciò collocati dentro a’ cerchi quantunque <lb></lb>//<lb></lb>disuguali, con la quantità loro mostrano le portioni simili, come insegna da Euclide nella .11. def. del .3°.. L’angolo retto posto nel cerchio ci dà inditio del mezzo cerchio, come si vede nella .10. def. del 3°. Questi son tutti effetti ed utilità che gli angoli producono nella Geometria, ed altre anchora de’ quali non si favellerà per hora in questo luogo per non allongar troppo il discorso; ma se ne lassarà la cura a qualunque osservarà i rimanenti libri di Euclide bastandoci solo haver accennato in parte l’uso degli angoli nella Geometria.</s> </p> <p type="main"> <s> </s> </p> </chap> <chap> <p type="head"> <s>Nell’Astronomia</s> </p> <p type="head"> <s>Cap. 12</s> </p> <p type="main"> <s>Se ‘l cielo dee sempre muovarsi e con facil giramento intorno alla Terra, e se ‘l suo girare non dee esser interrotto da intervallo alcuno di piccolissima quiete, né proceder saltando, necessariamente li conviene la figura in tutto ed ogni maniera d’angoli privata. </s> <s>E se di questa maniera si fa obbietto degli Astronomi per qual cagione nelle contemplationi loro si ricercano le considerationi e gli usi delle figure angolari e degli angoli stessi? Tolomeo nel 3° cap. del primo lib. dell’Almagesto afferma che alla struttura degli stromenti del movimento celeste non conviene altra figura che la Sferica; perciochè ‘l movimento delle cose celesti, non essendo impedito da alcuna cosa avviene che via più d’ogni altra cosa facilissimamente giri. </s> <s>E di tutte le figure dico nelle superficie la circolare: e ne’ solidi la sferica facilmente si muove. </s> <s>Ma che occorre dubbitare della natural figura del Cielo; se gli Astronomi oltre ad essa con la varia consideratione loro e per cagion di pruovare alcuni effetti con dimostrationi matematiche, non solo hanno diviso ‘l Cielo in molti cerchi; fondati però nelle diversità de’ movi <pb pagenum="folios 40v-41r"></pb>menti, come si può veder appo ‘l Clavio nella Sfera, e vi sia collocato anchora altri cerchij nuovi, col mezzo de’ quali mostrano la diversità dell’apparenza de’ pianeti; ma ancho vi hanno formato diverse figure angolari, e diversi intersegamenti d’archi e per conseguenza anchora molte e diverse maniere d’angoli per dimostrare alcune cose, che con ragioni naturali pruovar non si potevano. </s> <s>Di modo che supposta la figura natutale de’ Cieli gli Astronomi forzati a dimostrar i varij effetti di essi, e non havendo altre dimostrationi, che le matematiche, e non convenendogli usar quelle de’ Filosofi naturali, oltre all’altre cose costituirono nelle Celesti Sfere diversi cerchij posti in diversissimi modi, ed in guisa che tagliandosi insieme, formano angoli tal hora retti, tal hora ottusi e tal hora acuti; onde risultano ancho diverse specie di figure e varie portioni d’archi. </s> <s>Né questa fu solamente pura immaginatione loro, ma imaginatione ritratta dalla osservation de’ corpi celesti i quali fra loro muovendosi diversamente e secondo diverse positure, formano varij giri che tagliandosi variamente insieme, costituiscono diverse specie d’angoli e di figure. </s> <s>Onde non è maraviglia se Proclo nel 2° lib. d’Euclide, nel cap. 11° dica che agli Dei delle progressioni e del movimento, donatori delle varietà delle potenze sì come dano gli angoli ottusi e gli acuti. </s> <s>E che sono altro questi Dei che l’Intelligenze, che assistono a’ corpi Celesti per dargli ‘l movimento sì come è commun parere de’ Peripatetici? Anzi sono l’anime de’ Cieli, come è oppinione di essi e de’ Platonici anchora, sì come afferma Marsilio Ficino nella Teologia di <lb></lb>//<lb></lb>Platone e ‘l Cardinal Bessarione nelle calunnie dell’istesso cap. 10° del 2° lib. </s> <s>Queste secondo l’oppinion di Guglielmo Postello nel lib. dell’Origine della Toscana etc. sono i Genij stessi così appellati; perciochè sovrastano alla generatione in tutti que’ luoghi a quali assistono: e conservono tutte le generationi degli animali e delle piante nelle nature loro determinate a una specie, ad un luogo e ad una maniera di gente. </s> <s>Onde chiunque è nato in Grecia, grecheggia, o ‘n Italia, italianeggia, o ‘n Francia, franceseggia per natura e ciò si potrebbe confermare col detto di Marsilio Ficino sopra Plotino nella En. seconda. </s> <s>Egli dice in quel luogo, che i Celesti Dei provedono ciascuna cosa agl’ inferiori, overo, con l’autorità del Ficino sopra Protagora e nel 12° cap. del Timeo di Platone, diciamo che gli Dei suddetti non sieno altro che le stelle, le quali col movimento loro formano varij cerchij che intersegandosi insieme producono angoli diversi, come si è detto. </s> <s>E perché si chiamano donatori delle potenze delle varietà, se non perché infondono nelle cose inferiori varie dispositioni e diverse ragioni di forme sostantiali e son cagione in noi di quasi innumerabili inclinationi? Tutti effetti maravigliosi cagionati da’ Cieli in queste basse cose. </s> <s>Ma poniamo da parte queste filosofiche speculationi e torniamo al proposito nostro. </s> <s>Gli angoli adunque delle figure celesti che appo Jacomo Carpentario, nel cap. 11° sopra Alcinoo, dagli Oracoli son chiamati congiognimenti e nodi; perciochè sono immagini delle unioni ristregnementi e delle divine congiuntioni per le quali quelle cose che per natura loro son separate scambievolmente si accos <pb pagenum="folios 41v-42r"></pb>tano per molti rispetti son cagione di varij giovamenti nell’Astronomia. </s> <s>Per gli oracoli si intende la ragion del pronosticare; e le divine congiuntioni; sono de’ corpi celesti e così le figure. </s> <s>Perciochè i cerchij della Sfera maggiori e minori, secondo la positura loro, o retta od obliqua, s’intersegano scambievolmente, e ‘l tagliamento loro si determina e si manifesta dalla costitution degli angoli retti o non retti; perciochè quando le uguali circonferenze o le disuguali di maniera si tagliano nella superficie della Sfera, che intorno al ponto del tagliamento commune si cagionino angoli uguali il che non accade se non ne’ cerchij grandi: o solamente da una banda due angoli collaterali, e gli altri due rimanenti dall’altra son fatti uguali; la qual cosa non suole accader se non nelle circonferenze uguali, allhora due cerchij si tagliano ad angoli retti e così per opposito. </s> <s>E questo si vede espressamente del tagliamento dell’Orizzonte fatto dall’Equinottiale. </s> <s>Perciochè quelli che hanno la Sfera retta vedono l’Equinottiale esser tagliato dall’Orizzonte ad angoli retti: e que’ che l’hanno obliqua ‘l vedono tagliato dall’Orizzonte ad angoli disuguali e obliqui; perciochè in questo caso discernono solamente gli angoli opposti uguali, o veruno esser uguale; onde i cerchij e le circonferenze loro obliquamente si segano insieme e l’una declina dall’altra. </s> <s>Oltre acciò nel nascimento degli angoli nella Sfera si fa la division dell’Eclittica e dell’Equatore in quattro quarte, fatta dall’Orizzonte e dal Cerchio Meridiano, come si può ritrarre dal 10° cap. del 2° dell’Almagesto di Tolomeo. </s> <s>E ‘punti termi <lb></lb>//<lb></lb>nanti le dette quarte sogliono esser chiamati cardini ed angoli; poiché il tagliamento di queste quarte non si può eseguir senza formar gli angoli nella sfera. </s> <s>Neancho senza formar’ angoli il Zodiaco, uno de’ cerchij maggiori sega l’Equinottiale in due parti uguali, ma obliquamente. </s> <s>Così anchora i coluri mentre passano pe’ Poli del Mondo non tagliano i cerchij maggiori e minori senza formare angoli retti. </s> <s>Si aggiogne che ‘l Zodiaco stesso non si divide in dodici Segni senza la formation degli angoli retti; onde da alcuni sono stati detti quadrati o quadrangolari: ansi ciascun segno non per altro si appella piramide quadrilatera, se non perché essendo esso la base, ha quattro lati sorgenti da quattro angoli che terminano nella terra e formano angolo acuto. </s> <s>E tutti questi tagliamenti, benchè si mostrino nella Sfera materiale, con tutto ciò s’immaginano ancho nella celeste; poiché la materiale si forma per la intelligenza della celeste. </s> <s>Oltre acciò ‘l nascimento e l’uso degli angoli è giovevole espressamente alla figura della dispositione del Cielo. </s> <s>Perciochè (sì come si ritrahe da Gio. da Monte Regio nella disputa contra ‘l Cremonese) il cielo si divide in tre modi. </s> <s>Il primo si è quello che si fa da sei cerchij grandi, che si tagliano insieme sopra i punti opposti, o sieno poli del Zodiaco o poli del Mondo, o ‘due punti dell’Orizzonte e del Meridiano. </s> <s>Il diametro de’ quali si parte dal Settentrione, e arriva al Mezzogiorno, ed essi passano sopra dodici tagli, immaginati nell’Equinottiale, a’ quali corrispondono XII parti del Zodiaco, dette immagini, case habi <pb pagenum="folios 42v-44r"></pb>tacoli, torri, borghi, hospitij, ricettacoli, luoghi, immagini e castelli: e dividono qualunque cerchio per due punti opposti, e tutto ‘l Cielo e tutta la macchina del Mondo in dodici parti; di maniera che ciò che è nel Mondo si ritruovi in alcuna di queste parti dette case. </s> <s>Tal che ‘l primo modo si vedrà descrivendo due circonferenze concentriche, e dividendole in XII parti uguali: overo descrivendo la Sfera e costituito l’Equinottiale, i poli e l’asse: e formandovi sei cerchij che si taglino ne’ poli e dividino in XII parti il cerchio Equinottiale. </s> <s>Il 2° modo risolve tutta la disposition del Cielo in un quadrato, dividendo ciascun de’ suoi lati in tre parti uguali; onde congiognendo tutti i punti della divisione tutta l’area del quadrato vien divisa in nove quadrati uguali e divisi i quadrati che son negli angoli della figura col mezzo del diametro tosto apparisce la division di tutto ‘l cielo in XII parti intorno al quadrato di mezzo. </s> <s>Quattro delle quali opposte per diritto formano una croce, e si appellano cardini overo angoli, cioè la I. la IIII. la VII. e la X. e le otto parti rimanenti sono otto triangoli, che raffigurano la II. la III. la V. la VI. l’VIII. la IX. l’XI. e la XII. parte del Cielo. </s> <s>Il terzo modo si è quando si formano due quadrati, uno dentro all’altro, un maggiore e l’altro minore, ed un altro quadrato, i cui lati toccando gli angoli del quadrato minore, formino gli angoli nel mezzo de’ lati del maggiore, dividendosi ciascun lato in due parti uguali: e tirati i diametri del maggiore, apparisce tutta la <lb></lb>//<lb></lb>figura divisa in XII triangoli posti intorno al quadrato minore, i quali corrispondono alle XII parti del Cielo. </s> <s>Di modo che qualunque sia di queste divisioni, che rappresenti ‘n che maniera sia disposta la figura del Cielo non si può formar senza cagionar angoli diversi; poiché quindi apparisce risolversi il Cielo in XII. parti raffigurate da altretante figure angolari. </s> <s>E per qual cagione Marsilio Ficino nel cap. 12 sopra ‘l Timeo di Paltone ragionando de’ corpi celesti, disse che in maniera concorreno che insieme nascono o veramente quasi per linea perpendicolare si congiongano, come si può vedere ancho appo Girolamo Fracastoro nel cap. 21° dell’ Homocentrica, se non perde dalla cognition degli angoli haveva già ritratta la notitia di tal cagionamento? Che non poteva la congiuntione in modo alcuno farsi per linea perpendicolare senza gli angoli perciochè (sì come è manifesto per la Geometria) non è linea alcuna che si possa chiamar perpendicolare senza formar angoli da ogni banda, cadendo sopra altra linea. </s> <s>E se riguardiamo alla Sfera materiale, la quale corrisponde alla naturale, noi vedremo chiarissimamente, in fra ‘tagliamenti de’ cerchij maggiori e de’ minori, adoperarsi l’uso degli angoli. </s> <s>Perciochè, osservando noi i cerchi maggiori vediamo il Meridiano e l’Orizzonte tagliarsi ad angoli retti, stando la Sfera retta, ne’ poli del Mondo; e nel ponto Verticale appellato Zenit, cadente sopra ‘l nostro capo: e in uno stesso tempo amendue tagliare l’Equinottiale pur ancho ad angoli retti. </s> <s>Ma considerata la Sfera <pb pagenum="folios 43v-44r"></pb>obliqua, l’Orizzonte col Meridiano si tagliano pure ancho ad angoli retti; ma fuor del polo del Mondo; perché a que’ che hanno la Sfera torta il polo s’inalza sopra l’Orizzonte, e ‘n questo caso l’Equinottiale è segato dall’Orizzonte ad angoli disuguali e obliqui, cioè ottusi, ed acuti. </s> <s>E se rimiriamo ‘l cerchio Equinottiale, vedremo che taglia formando angoli retti sferici il primo mobile; onde è dela sua cintura come riferisce M° Mauro Fiorentino Servita nelle Notationi della Sfera. </s> <s>Né può esserne cintura senza formare angoli; perciochè, mentre ‘l cegne il taglia, segando ancho ‘l Zodiaco ad angoli disuguali. </s> <s>Onde possiamo concludere che dalla qualità degli angoli si conosca la positura della Sfera, cioè se sia o retta o obliqua e così ancho la maniera de’ tagliamenti fatti da’ cerchij maggiori. </s> <s>Oltre acciò se osserviamo i cerchij Coluri, vedremo che si tagliano fra loro ad angoli retti ne’poli Mondo: e mentre tagliano ad angoli disuguali il Zodiaco, passando pe’ poli Solstiali, cioè del Cancro e del Capricorno, la State e l’Inverno, determinare i due Solstitij e due Equinotij. </s> <s>E questi punti non si possan costituir senza la formation degli angoli, che necessariamente procede da’ communi tagliamenti de’ cerchij. </s> <s>I medesimi Coluri tagliono ad angoli retti i due cerchij minori chiamati Tropici, del Cancro e del Capricorno; onde il Sole si rivolge e si accosta di State al nostro Zenit, e di Verno a quello degli Antipodi. </s> <s>Nel medesimo modo tagliono l’Orizzonte e ‘l Meridiano cioè nella Sfera retta, ma nell’obliqua ad an <lb></lb>//<lb></lb>goli disuguali. </s> <s>E quindi ancora si può imparare a discerner la Sfera diritta dalla torta. </s> <s>E questo basti per dimostrar l’utilità degli angoli nel commun segamento de’ cerchij maggiori della Sfera. </s> <s>Hora se ci voltiamo a’ minori, i quali sono il cerchio Artico e Antartico, il Tropico del Cancro e del Capricorno, vedremo espressamente che tutti tagliano ‘l Meridiano l’Orizzonte retto e ‘Coluri ad angoli retti: e così l’asse del Mondo; ma l’Orizzonte obliquo e l’asse del Zodiaco ad angoli obliqui e disuguali. </s> <s>Fra ‘ due Tropici muovendosi ‘l Sole, forma 182 spire, overo cerchij involuti i quali raddoppiati nel suo ritorno e computato due volte l’Equinottiale sul quale gira due volte si formano giorni 365 in tutto l’anno, e quasi 6 hore e 21. M. meno, che sono la centesima parte d’un giorno. </s> <s>E le dette spire tagliando l’Eclittica, il Meridiano, l’Orizzonte retto e obliquo e ‘Coluri e gli assi del Mondo e del Zodiaco, formano varie specie d’angoli. </s> <s>Di modo che le dette spire tante sieno quanti i tagliamenti e ‘tagliamenti quanti gli angoli e quanti gli angoli o tagliamenti o le spire, tanti sieno i giorni. </s> <s>Queste dall’Orizzonte nella Sfera diritta son tagliate in parti uguali, mentre egli passa sopra ‘ Poli del Mondo; onde risultano i giorni uguali. </s> <s>Ma essendo divise da esso nella Sfera torta, stando esso obliquamente, si cagiona la disuguaglianza e la diversità de’ giorni e delle notti in diverse parti del Mondo. </s> <s>Di maniera che queste divisioni non si possendo eseguir senza formare angoli e giorni cagionandosi uguali o disuguali col mezzo di esse seguirà che da <pb pagenum="folios 44v-45r"></pb>gli angoli anchora si cagionino l’ugualità e le disuguaglianze de’ giorni. </s> <s>E se è lecito dalle celesti Sfere trapassar scendendo al globo della terra, riputata consorte del Cielo da Leone Helode, nel 2° Dialogo d’Amore, potremo dire che ‘quattro cerchij che la dividono in cinque zone, ciò sono i due cerchietti, Artico e Antartico: e’ due Tropici del Cancro e del Capricorno; formano nella circonferenza della terra angoli retti sferali, nella stessa guisa che ne’ cerchij Celesti per la Commune proportione che è fra ‘ cerchijj che sono intorno al medesimo centro. </s> <s>Il che avviene come nel divider in quarte o in quadrati ‘l Cielo e la Terra; là dove essendo il quadrante della Terra al quadrante del Cielo simile e proportionale, e l’angolo che si fa nel quadrante di amendue havendo proportione, ansi essendo uguale, segue che ‘l cerchio della Terra contenuto da quello del Cielo sia proportionato e simile al cerchio del Cielo,suo continente, sì come si dimostra dal Glareano nel 12° cap. della Geografia, di modo che e la Terra el Cielo parimente si divide in quattro parti. </s> <s>Così i detti quattro cerchij che dividono il cielo in cinque parti dividono ancho la Terra in altretante, contenute sotto le dette portioni del Cielo, e a esse proportionali e simili. </s> <s>E come nella circonferenza celeste i detti cerchij formano angoli retti sferali, così nella circonferenza terrena. </s> <s>Ma poiché di sopra si è fatta mentione del Zenit, non si dee tralassare, che anch’esso, essendo un punto collocato nella Sfera diritta, ne poli del Mondo, da esso cade una linea perpendicolare sopra ‘l <lb></lb>//<lb></lb>nostro capo, overo in quella parte della superficie della Terra dove terminano i piedi, onde essendo il nostro capo di figura sferica, ma alquanto più longa che larga, sì come afferma Galeno nel cap. 17° del 9° lib. </s> <s>Dell’uso delle parti e nel 1° cap. dove favella dell’ossa, e per la sperienza si conferma: e la detta portione essendo parte della circonferenza della terra, non può non produr angoli misti nello stesso contatto, sì come ancho partendosi del concavo della Sfera. </s> <s>Oltre acciò le dimostrationi sferiche, con le quali si pruovano le grandezze degli archi massimi che si descrivono sopra i poli dell’equinottiale, i quali son collocati fra esso e l’eclittica, mentre in esse si fa la conferenza delle proportioni composte delle linee, non si fanno senza formare angoli nelle descrittioni loro; poiché si ordinano producendo linee, delle quali si compongano triangoli con lati proportionali, sì come si vede appresso Tolomeo nel cap. 12° del primo lib. dell’Almagesto, là dove si mostra la proportion d’una linea maggiore ad una minore, che è parte di essa, esser composta delle proportioni delle linee minori alle parti loro, la qual cosa concludendosi nelle linee si conlcuderà anchora negli angoli e negli spatij triangolari contenuti da esse. </s> <s>Il che è quasi lo stesso che quel che afferma de’ triangoli Euclide nella quarta del primo, che non vi è altra differenza che dall’esser uguale all’esser proportionale. </s> <s>Come se dicessimo: se due triangoli hanno due lati proportionali a due lati, e un angolo proportionale a un angolo contenuto da linee rette proportionali; haveranno ancho la base proportionale alla <pb pagenum="folios 45v-46r"></pb>base e ‘l triangolo al triangolo e gli altri angoli agli altri angoli, e l’uno all’altro, a quali soggiacciono lati proportionali. </s> <s>Di modo che quindi vediamo, che ‘l medesimo effetto, che si cagiona da’ lati proportionali de’ triangoli, si cagiona anchora dagli angoli, cioè ‘l far le basi, i triangoli e gli altri angoli proportionali; onde segue la proportionalità degli archi a’ quali son sottoposti per l’.XI. def. del 3° d’Euclide e per la XII del Commandino e ciò vi si vede espressamente; poiché si fa comparatione fra le corde degli archi proposti e si mostra tale esser la proportione in fra gli archi, che è in fra le corde, le quali son linee rette e fanno angoli nelle circonferenze e ne’ tagliamenti loro. </s> <s>Ma ‘l tutto si dichiara da Tolomeo con sette dimostrationi nel med. lib. </s> <s>Ma nel cap. 13, per gli mostrar la declinatione di qual si voglia punto dell’eclittica, cerca la proportione d’un arco minore e truovata la doppia proportione della corda d’un arco maggiore alla doppia proportione della corda d’una parte di lui, cioè d’un arco minore, esser composta di proportioni delle corde degli archi minori e del doppio della misura loro; procede mostrando la proportion degli archi in fra loro, facendo sempre comparation degli archi agli archi e delle corde alle corde; onde con scemamento di gradi si perviene finalmente all’arco della declinatione dell’eclittica, e al termine di essa. </s> <s>E ciò si vede in una descrittion del cerchio Meridiano, diviso da due mezzi cerchij ad angoli retti sferici che si tagliano nel centro di esso ad angoli disuguali, dove costituiscono il ponto dell’Equinottio dell’Inverno. </s> <s>Uno de’ <lb></lb>//<lb></lb>quali cerchij è obliquo ed è la stessa eclittica, e l’altro diritto, ed è l’Equatore: ed amendue terminano in due punti della circonferenza del Meridiano, là dove sono i due Tropici del Cancro e del Capricorno, e a man destra nella medesima circonferenza è ‘l polo del Zodiaco da cui nasce un arco che taglia l’eclittica e l’equinottiale ad angoli disuguali e ne’ tagliamenti costituisce la portion d’un arco minore, dove si vede collocarsi il punto della declinatione della eclittica. </s> <s>Di maniera che in questa descrittione dimostrando Tolomeo la declination di qualunque punto dell’eclittica; ci fa conoscere che i detti cerchij fra loro si tagliano proportionalmente; onde seguono diversi archi proportionali, e così diverse corde altresì proportionali: e quindi ancho quattro triangoli proportionali; onde avviene che faccia la circonferenza delle proportioni loro. </s> <s>Ne’ triangoli, ne’ lati di essi, i quali sono specie d’archi, possono esser proportionali, se gli angoli anchora non son proportionali. </s> <s>Il medesimo si può dire che si faccia nella seconda dimostratione, dove si mostra, che ‘l tagliamento retto di qualunque arco dell’eclittica si comincia dalla settion di essa e dell’equatore. </s> <s>Oltre acciò (come si ritrahe dal 10° cap. del 2° dell’Almagesto) per virtù degli angoli si viene in cognition de’ cerchij massimi della Sfera Celeste, che gli contengono, mentre da essi sopra ‘poli si prende lo spatio commune: e della quarta parte del cerchio costituita dall’arco infraposto alle parti che fanno angolo. </s> <s>Di più dalla misura e proportion dell’angolo sottoposto all’arco detto, e <pb pagenum="folios 46v-47r"></pb>paragonata a quattro angoli retti possiamo conoscere la proportion dell’arco interposto a tutto ‘l cerchio. </s> <s>Onde se l’angolo formato nel tagliamento del cerchio sarà di 90 parti, di tante sarà l’arco, ed essendo nel taglio quattro angoli retti de’ quali ciascuno sia 90 parti, seguirà che tutti insieme sieno 360: ed essendo quattro archi infraposti fra ‘termini degli angoli detti corrispondenti agli angoli retti, ciascuno per sé sarà 90 parti, e tutti insieme 360. E per seguir più avanti recitano la sentenza di Tolomeo. </s> <s>Adunque degli angoli che si formano secondo ‘l cerchio obliquo, quegli a questa speculatione (astronomica) grandemente son giovevoli che son compresi dal tagliamento di esso, del Meridiano e dell’Orizzonte in qual si voglia sito. </s> <s>Così anchora que’ che son contenuti dalla settion di esso e dal cerchio massimo descritto sopra ‘poli dell’Orizzonte. </s> <s>E con quest’angoli insieme si dimostrano gli archi, che s’interpongono fra ‘l tagliamento el polo dell’orizzonte, cioè del ponto verticale, overo Zenit; perciochè ciascuna di queste cose dimostrata conferisce molto alla stessa speculatione, e a tutto quel che si cerca nelle diversità degli aspetti della Luna. </s> <s>E questo basti haver riferito per dimostrar l’utilità degli angoli. </s> <s>Chi ne desidera le dimostrationi ricorra a Tolomeo nel medesimo luogo citato. </s> <s>Ma poiché si sono accennati gli aspetti, questo solo appartenente a essi aggiognerò, che la diversità di essi, la quale apparisce nelle quadrature, non si può far senza <lb></lb>//<lb></lb>formare angoli. </s> <s>Perciochè (sì come si vede appresso Girolamo Fracastoro nel cap. 21° dell’Omocentrica) le quadrature, l’oppositioni, e le congiuntioni si formano in due linee, una delle quali si appella asse e l’altra antasse: e amendue tagliando i deferenti, e l’eccentrico, mentre si segano insieme ad angoli retti costituiscono i punti dell’oppositione e della congiuntione, ne’ quali il Pianeta ritruovandosi apparisce due volte vicino e due lontano. </s> <s>Di maniera che nella congiuntione e nell’oppositione, la Luna essendo sempre nell’asse; e nelle quadrature, essendo nell’antasse, dimostra la varietà degli aspetti per mezzo degli angoli costituiti da’ tagliamenti di due linee rette. </s> <s>Questi aspetti della Luna nelle quadrature non si cagionano ne’ cerchij eccentrici deferenti (detti Draghi, i cui tagliamenti ad angoli retti sferali son chiamati il capo e la coda del Drago) senza gli angoli perciochè se riguardiamo all’asse, che determina la maggior longhezza e distanza della Luna dal centro del Mondo: e dell’antasse, che dimostra la vicinanza, vedremo che ‘n fra loro segandosi formano per ogni verso angoli retti: e così anchora se riguardiamo a’ termini nel taglio dell’ovato vi scorgeremo quattro angoli retti, in quanto all’esser loro; ma in quanto all’apparenza due ottusi e due acuti, dove si termina tutto l’ovato, cioè immaginandoci che ‘ due punti della congiuntione e dell’oppositione e degli accostamenti si congionghino e in questa guisa si vedrà formata la figura quadrata, che in <pb pagenum="folios 47v-48r"></pb>apparenza si mostra un rombo dentro l’ovato prodotto dal ristregnimento de’ deferenti. </s> <s>Ma si avvertisca che di questo cerchio, che dal Fracastoro è appellato ovato, avviene come degli altri cerchij; che per la varia positura loro, opposti alla veduta nostra, si mostrano di figura difettuosa, come è l’ellipse o l’ovato, come si potrebbe pruovare per ragion di Prospettiva; che questo cerchio non per altro è detto cerchio ovato, se non che in apparenza si mostra ovato, come è ‘l taglio obliquo della piramide o del cilindro; ma nell’esser suo è cerchio perfetto, come gli altri della Sfera, che opposti all’occhio, dimostrano così fatta apparenza. </s> <s>Gli angoli finalmente servono negli aspetti triangolari, quadrati, e sestili, ne’ quali si ritruovano le proportioni armoniche, cioè tripla, sesquialtera, sesquitertia, come nelle voci si formano le consonanze: Diapente, Diatesseron, come riferisce Cesare Cesariano nel cap. primo del 1° di Vitruvio, ritrahendolo dal 2° del Quadripartito di Tolomeo. </s> <s>E questo basti in quanto a questa parte dell’uso degli angoli nell’Astronomia; che per hora non intendiamo dirne altro, rimettendoci in tutto a’ diligenti osservatori delle cose Celesti.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Nella Prospettiva</s> </p> <p type="head"> <s>Cap. 13</s> </p> <p type="main"> <s>Non è dubbio alcuno che se consideraremo bene tutte le parti della Prospettiva, o appartenghino all’ombre, dico alle scene, o a reflettimenti de’ raggi del Sole, o agli specchi, overo alle linee e raggi visuali, noi vi conosceremo tutta la forza delle pratiche e tutta l’efficacia delle dimos<lb></lb>//<lb></lb>trationi nascen dalla diversa aplicatione degli angoli, la qual cosa prenderemo a dimostrare. </s> <s>Ma perché ci si offerisce la parte Ottica, la quale appartiene al modo con cui si eseguisce la visione, però prima ragionaremo di questa, ma avanti si dee supporre che Euclide, essendo Filosofo Platonico, nella sua Prospettiva dimostra esser dell’oppinion del suo maestro Platone, in quanto appartiene al modo col quale si fa la visione ed è che si faccia col mezzo di raggi visuali, ch’escono dall’occhio e vanno a truovarsi l’obbietto. </s> <s>Il che da lui si fece chiaro quando nella prima suppositione della Prospettiva disse i raggi visuali uscir dall’occhio e andar a truovar l’obbietto. </s> <s>Della quale oppinione sono stati i suoi seguaci e commentatori, contro a’ quali si è mostrato Giovanni Arcivescovo di Cantauria nella prima parte della sua Prospettiva comune nella 44 e 45 conclusione, là dove disse: i matematici indarno affermare che la visione si faccia pe’ raggi che escono dall’occhio: ed esser cosa impossibile che i detti raggi uscendo dall’occhio alla cosa veduta sieno bastevoli a formar la visione *(e ‘l Cavalier Lorenzo Sirigatti nel 3° cap. del primo libro della Pratica di Prospettiva) [nota in margine]. E benchè l’oppinion d’Euclide non sia conforme alla dottrina Peripatetica, contuttociò trattandosi di Prospettiva insieme con esso non si disconverrà seguirla; quantunque secondo l’oppinion Aristotile non si mostri sconvenevole ragionarne; poiché, con altro adattamento di raggi visuali affermaremo farsi la visione, discendo che invece di uscir dal centro dell’occhio e andar afferir nell’obbietto, scaturischino dall’estremi<pb pagenum="folios 48v-49r"></pb>tà dell’obbietto e si terminano tornandosi nel centro dell’occhio, acciochè la specie della cosa veduta, passando pe’ raggi visuali, arrivi all’occhio, onde si faccia la perfetta visione. </s> <s>Ma per dar principio ad eseguir ciò che si è proposto si dee dire che l’uso degli Angoli nella Prospettiva Ottica si scuopre in ogni risguardamento d’obbietto ed in ciascuna dimostratione degli effetti che secondo diverse apparenze e secondo varia posizione si mostrano diversi. </s> <s>Non è dubbio alcuno che seguendo noi in questo luogo il parer de’ Platonici nel producimento della visione ci faccia bisogno affermare con Euclide che non possiamo veder obbietto alcuno se i raggi visuali non escono dall’occhio per andar dirittamente a trovar l’obbietto e se non hanno nella parte più lontana intervallo, sicome si vede appresso la prima supposizione della Prospettiva. </s> <s>E mentre i detti raggi, che non sono altro che linee rette escono dall’occhio, quanto più si allontanano da esso e si avvicinano alla cosa veduta, tanto più si discostano; onde necessariamente formano angolo; ansi non possan haver tra loro intervallo senza formare angolo, che uscendo dal centro dell’humor cristallino ed essendo l’obbietto molte volte maggiore dell’occhio e dovendo terminarsi i raggi nella estremità di esso, acciò si faccia la visione, non possan costituir intervallo secondo la misura dell’obbietto: e questo non si può fare se prima no si forma l’angolo nel centro dell’occhio. </s> <s>E però Euclide, nella seconda supposizione disse la figura formata da’ raggi visuali esser un conio la cui ponta è nell’occhio e la base nella <lb></lb>//<lb></lb>stremità della cosa veduta (al mezzo dentro a questo conio si forma la visione, quasi che la virtù apprensiva dell’occhio si muova su pe’ raggi visuali per accostarsi all’obbietto. </s> <s>Ma se questo conio è una piramide ritonda come vuole Euclide nella def. 16 dell’XI e come piace a Vitellione nel 4° libro) e se essendosi tonda ha per base ‘l cerchio; onde è che si fa la visione anchora quando l’obbietto non è di figura circolare o sferica? Non conviene affermare che sotto ‘l nome del Conio si contenga ogni specie di piramide, come alcuni erroneamente hanno pensato; poiché dall’esser del conio sono al tutto lontane le figure angolari. </s> <s>Né si dee dire che ‘l conio, benchè ritondo possa abbracciar le figure angolari; perciochè la figura ritonda non può tutta ugualmente accostarsi all'’ngolare, non vi essendo commune proportione alcuna; perciochè la tondezza del conio o escede la figura angolare o ‘l conio da essa è superato.Neancho è ragionevole ‘l pensar che ‘l conio si adatti in maniera ad ogni figura angolare, che di tondo si faccia angulare; perciochè, tosto che diviene angolare, non è più conio; perciochè, acciochè si faccia angolare bisogna che a ciascuno angolo della figura vada il raggio visuale, il che non si vede nel conio. </s> <s>Questa difficoltà facilmente si toglie avvertendo che ogni volta che miriamo alcuna cosa e sia di qualunque figura non vediamo solamente l’obbietto, ma ancho assai maggiore spatio intorno a esso, il quale è come un cerchio che è base del conio la cui ponta è nel centro dell’occhio, come insegna Eliodoro Larisseo nella Prospettiva. </s> <s>Il qual conio è tutto ripieno di <pb pagenum="folios 49v-50r"></pb>luce, onde forse per questa ragione Averroe nel lib. de sen. et sen. disse esser propio del vedere haver la presenza del cono lucido, cioè dell’obbietto illuminato con questo conio luminoso. </s> <s>La qual cosa non intendendo bene alcuni Filosofi antichi, secondo che racconta Aristotile nel lib. </s> <s>De sen. et sen., cap. 2, dissero ‘l vedere esser fuoco. </s> <s>Il che appresso di esso nel med. lib. nel cap. 2 dimostra Empedocle, facendo comparatione del fuoco del vedere, cioè della luce degli occhij al lume racchiuso nella lanterna, come si vede ne’ suo’ versi.</s> </p> <p type="main"> <s>Seu casu quis progredi meditans lanternam preparavit</s> </p> <p type="main"> <s>Hybernam per noctem ignis ardentis lumen </s> </p> <p type="main"> <s>accendens splendidum cornu omnimodi obstaculum flatus,</s> </p> <p type="main"> <s>Quod ventorum quidem flantium dissipat spiritum.</s> </p> <p type="main"> <s>Extrinsecus autem extensum exiliens quantum procurrevit lumen,</s> </p> <p type="main"> <s>Splendevit per pavimentum indomitis radijs,</s> </p> <p type="main"> <s>Sic quod in tuniculis costringitur antiquis ignis.</s> </p> <p type="main"> <s>Subtilibus velis rotundam complectitur pupillam</s> </p> <p type="main"> <s>Que circumfluentis profundum continet aque.</s> </p> <p type="main"> <s>Ove si vede espressamente che ‘l fuoco o lo splendore compreso nella pupilla dell’occhio è quasi un lume racchiuso nella lanterna essendo contenuto dalle torricelle dell’occhio: e come la luce chiusa nella lanterna penetra l’osso, il talco, o ‘l vetro e illumina il luogo oscurato dalle tenebre della notte l’Inverno; così la <lb></lb>//<lb></lb>detta luce natia dell’occhio trapassa oltre agli altri humori, ancho l’humor aque; perciochè ancho ‘l Filosofo nel med. luogo disse la pupilla e l’occhio esser acque per cagion degli humori che concorreno a formarlo; ma riguardando alla luce contenuta in esso che da Empedocle e dagli antichi è appellata fuoco, disse pur nello stesso luogo la pupilla essere come il lume della lanterna, la quale rotta si fanno le tenebre. </s> <s>Onde possiamo ritrarre che quella cosa luminosa che si contiene nel conio della vista sia ‘l lume della pupilla. </s> <s>Quindi adunque si ritrahe lo scioglimento del dubbio che se s’apre il conio de’ raggi visuali ha per base il cerchio, benchè spesse volte la figura veduta sia angolare perciochè lo spazio che circonda l’obbietto è sempre di questa figura. </s> <s>Ma per tornare al proposito nostro diciamo pure che l’uso dell’angolo nella prosp. si conosce per l’uso del conio, che è effetto dell’angolo, perciochè dall’angolo prende origine. </s> <s>E che ciò sia vero si rimiri, che tutto quel che si vede dagli occhi nostri si vede per virtù del conio fatto da’ raggi visuali; onde, perché talvolta ha l’angolo minore, talvolta maggiore e tal’hora ancho uguale, perciò è cagione che l’obbietto ci apparisca tal’hora uguale, tal’hora maggiore e talvolta minore. </s> <s>Onde Euclide nella .5a. supposizione disse: Quelle cose che sotto maggior angolo si veggono ci appariscono maggiori. </s> <s>E nella .6a.: Quelle che sotto minor angolo si veggano appariscono minori. </s> <s>E nella .7.: Quelle che sotto uguale angolo si veggano appariscono uguali. </s> <s>Di modo che quindi si cono<pb pagenum="folios 50v-51r"></pb>sce che tutto quel che si vede apparire sotto qualche angolo; che la figura formata da’ raggi visuali e dall’obbietto per cagionar la visione, non è altro che ‘l conio sicome s’è detto. </s> <s>Il quale ci mostra l’obbietto maggiore o minore, secondo la grandezza dell’angolo da esso costituito, perciochè la grandezza degli angoli e la proportione che è fra un angolo e l’altro ritratta dalle portioni delle circonferenze del conio è cagione che l’obbietto si mostri hora minore ed hore maggiore. </s> <s>E sicome del crescimento dell’angolo si cagiona il crescimento dell’apparenza della grandezza dell’obbietto, così all’incontro, dallo scemamento, si cagiona lo scemamento della grandezza; perciochè la medesima forza che hanno gli angoli nella Geometria, di aggiongere o scemar gli spatij (sì come altre volte s’è detto) ritengono anchora nella Prospettiva; ma si adatta a diverso fine perciochè quivi si applica solamente per ritruovar l’uguaglianza o la disuguaglianza delle grandezze pure geometriche e qui si adatta solamente alla diversa apparenza degli obbietti visibili. </s> <s>Oltre acciò, quando l’angolo è maggiore o minore, comprende più o meno dello spatio dello splendor del conio posto intorno all’obbietto; di maniera che quando l’angolo sarà uguale le cose saranno vedute con ugual lume; onde necessariamente ci appariranno uguali. </s> <s>Talchè quindi anchora si ritrahe che la grandezza dell’angolo porge la grandezza dell’apparenze dell’obbietti. </s> <s>Il medesimo effetto si può vedere, collocato l’<lb></lb>//<lb></lb>obbietto in varie positure, come si chiarisce da Euclide nell’ottava supposizione, là dove si vede che quelle cose che poste in alto fanno conio con maggior angolo si mostrano più basse, cosa che si rende certa per l’esperienza del Sole, che nell’inverno apparisce più basso. </s> <s>Di più, non solamente l’angolo del conio con la varia sua grandezza è cagione d’apparenze diverse, ma ancho quanto più è diviso o moltiplicato in più angoli secondo che un conio maggiore si divide in più conij minori, tanto più è cagione di più perfetta visione. </s> <s>E però nella dodicesima supp. disse: quelle cose che si vedono sotto più angoli si vedono più distintamente. </s> <s>E la ragione si è che ciò che si vede non si può veder se non sotto qualche angolo, come è già manifesto, allhora si vedrà più perfettamente, quando sarà contenuto sotto più angoli, perciò che essendo più angoli saranno più conij ed essendo più conij saranno ancho molti più raggi visuali, onde se i primi raggi visuali arrivano alla estremità dell’obbietto, i secondi, i terzi, i quarti, i quinti e così altri anchora feriranno l’obbietto dentro l’estremità, di maniera che non vi sarà parte d’obbietto alla quale non gionghino i detti raggi. </s> <s>Onde se quelle cose si veggono alle quali giongonsi i raggi visuali, come disse nella terza supp. quelle più esattamente si vedranno alle quali giungeranno più raggi visuali; come se tuttta la vista apprensiva del vedere proceda muovendosi sopra ‘ raggi visuali; perciochè tutta la virtù del vedere secondo Eliodoro è collocata in <pb pagenum="folios 51v-52r"></pb>quelle cose che li son davanti, le quali sono i raggi visuali, il cono e l’ obbietto in quanto è base del conio conforme a questa sentenza è ‘l detto del Filosofo nel 2° delle parti degli animali, cioè che ‘l vedere è stato collocato davanti perciochè si discerne dirittamente el movimento del vedere si fa davanti e nel medesimo modo si vede che si fa ‘l movimento. </s> <s>Ma per provar l’uso degli angoli nelle dimostrazioni de’ diversi effetti dell’apparenze proporremo, se non tutti, almeno alquanti accidenti, che avvengono alle varie apparenze. </s> <s>Con ciò sono l’obbietto non potersi veder tutto in una volta, il che si dimostra da Euclide nel 1° teorema della Prosp. e la ragione di questo accidente si è lo intervallo infraposto a’ raggi visuali; onde avviene essi non ferischino l’obbietto continuamente. </s> <s>Né ciò può accadere senza l’aiuto dell’angolo, perciochè se i raggi visuali non formassero angolo nel mezzo dell’occhio non farebbero ‘l conio e non facendolo non riceverebbero la distanza in fra loro, per la quale l’obbietto non si vedesse tutto in un girar d’occhio. </s> <s>Se adunque l’angolo è cagion dell’intervallo de’ raggi visuali e l’intervallo è cagione che l’obbietto non sia veduto tutto in una volta; ma prima una parte e poi l’altra come accade nel rimirar leggendo qualche scrittura, l’angolo anchora ne sarà cagione e tanto più, quanto più sarà maggiore perciochè sarà cagion di maggior discostamento de’ raggi visuali. </s> <s>E però, acciò più esattamente <lb></lb>//<lb></lb>si veda alcuna cosa, si divide tutto lo intero conio in più conij minori come s’insegna nella dodicesima supp. </s> <s>Onde poi tante volte si vede un obbietto in quanti conij si divide il conio maggiore ed in quante parti lo stesso obbietto è diviso; perciochè tante basi son quanti conij; Di modo che la prima vista forma un conio che riceve la base in una parte dell’obbietto: la seconda ne forma un altro che si fa base d’un’altra parte dell’obbietto e così seguendo finchè si scorra co’ raggi visuali per tutta la sua grandezza: o vero diciamo che ‘l medesimo conio venga replicato e mosso sopra a ciascuna parte dell’obbietto successivamente. </s> <s>Onde segue che tutto l’obbietto in un solo aprir d’occhio non si possa veder; ma faccia bisognio o moltiplicar i conij e così ancho gli angoli o muover sopra l’obbietto il medesimo conio più volte anzi insieme col conio replicar il raggio visuale che è asse di esso e ferir continuamente nel mezzo della base del conio, cioè nel mezzo della parte dell’obbietto che è veduta a guisa di coloro che tirano di mira che non feriscono ‘l bersaglio se non trovano ‘l suo centro. </s> <s>Le grandezze di qualsivoglia obbietto benchè uguali non apparire uguali. </s> <s>Accidente che si mostra da Euclide nel 5° teorema. </s> <s>Supposto che disugualmente sieno lontane dall’occhio e tutta la ragione di esso consiste nella grandezza degli angoli sotto ‘quali son vedute e colla conferenza degli angoli si dimostra. </s> <s>Il medesimo anchora si dimostra nel 7° teorema. </s> <s>Gli angoli delle grandezze uguali, poste disugualmente lontane all’occhio non tengono la medesima proportione che è nelle distanze. </s> <s>Il che si vede nel teorema 8°. Perciochè è minore la proportione del maggior <pb pagenum="folios 52v-53r"></pb>angolo sotto cui è veduta la grandezza più vicina; al minore sotto ‘l quale è veduta la grandezza più lontana, che non è l’intervallo maggiore della grandezza lontana. </s> <s>Non potersi veder alcuna ragion d’angoli posti lontani alla vista. </s> <s>Quest’accidente si manifesta nel teorema 9°, là dove si legge: Le grandezze rettangole che di lontano son vedute appariscono ritonde. </s> <s>La cagione di quest’effetto si cerca da Alessandro Afrodiseo nel 37° Pub. </s> <s>Ma se ne rende la ragione più filosofica che matematica ed è che l’occhio non può da lontano veder gli angoli, essendo sotto li iguali levati, ciò che rimane apparisce ritondo. </s> <s>Ma secondo la prospettiva avviene altramente; perciochè questo effetto avviene sì per cagion della distanza che è fra l’obbietto e l’occhio, sì anchora per cagion della figura angolare. </s> <s>Dalla figura angolare nasce ‘l non si poter vedere gli angoli di lontano, perciochè sempre la larghezza della figura è minore appresso gli angoli che altrove; onde non è maraviglia se di lontano gli angoli non apparischino e così ancho le parti vicine agli angoli; e quindi è che di due linee se ne forma una sola. </s> <s>Dalla distanza procede lo svanimento degli angoli perciochè ogni cosa visibile ha una determinata distanza la quale passata non si può più vedere e ciò facilmente avviene agli angoli che sono di pochissimo estendimento. </s> <s>Perciochè fra le conditioni che si richiedono a formar la visione una si è la debita lontananza che è fra l’obbietto e l’occhio e però ‘l Filosofo nel cap. 8° De sen. et sen. disse: Il vedere <lb></lb>//<lb></lb>cagionarsi in una distanza tale che in essa sia collocata qualche cosa che sia prima e qualche cosa che sia ultima oltre alla quale non si possa discernere che non è altro che dire che alla perfetta visione si richieda la determinata distanza dell’occhio all’obbietto. </s> <s>E per questa ragione Alessandro Afrodiseo nel 4° della Metafisica, nel comm. .55. c.115, diceva che allhora il vedere dimostra la verità quando si rimira da un intervallo mediocre non quando da lontano. </s> <s>Ma torniamo a considerar gli accidenti dell’apparenze. </s> <s>L’occhio che dirittamente alla linea retta, od a cosa che per linea retta si erga sopra ‘l piano non poter veder la sua longhezza. </s> <s>La qual cosa si afferma da Egnatio Danti con l’autorità di Vitellione nell’annotatione del 22° teorema della Prosp. </s> <s>Ma la ragione di questo si è perciochè l’occhio vede la linea in un punto, onde concorrendo i raggi visuali in esso e confondendosi insieme, non posson formare il conio e per conseguenza neancho l’angolo; perciochè ‘l punto non è atto ad esser base del conio, essendo indivisibile; e se pure è divisibile, essendo punto di Prosp., cioè visuale, in ogni modo non può essere bastevole ad esser base del conio, onde segua un angolo che sia buono a far discerner l’obbietto essendo d’un’acutissima e strettissima grandezza. </s> <s>E se per avventura l’occhio starà a piombo sopra una colonna, non potrà veder la sua longhezza, poiché la vedrà tutta nella superficie più vicina, la quale sarà al base conio. </s> <s>Il medesimo effetto si potrebbe sperimentare collocando l’occhio e la linea e la colonna nel me<pb pagenum="folios 53v-54r"></pb>desimo piano, in maniera che stieno per diritto fra loro. </s> <s>Oltre acciò talvolta essendo opposta una palla all’occhio nostro, vedremo di lei la parte minore parerci maggiore e per opposito la parte maggiore dimostrarsi minore e ‘l tutto accade per la forza dell’angolo del conio che si fa maggiore o minore discostandosi od accostandosi all’obbietto, come si dimostra nel teor. 24°. L’istesso accade alla colonna, come si dimostra nel 30°. Così ancho avviene al conio alzato o abbassato l’occhio, come apparisce nella dimostratione del 34°. E tutti questi casi son cagionati dalla grandezza dell’angolo, sì chome ancho l’ugualità degli angoli formati da’ raggi visuali nel mirar i diametri d’un cerchio il cui centro stia a piombo sotto l’occhio è cagione che si vedano i detti diametri uguali, sì come è manifesto per la dimostratione del 36° teor. e tutta la ragione di questo effetto consiste nella linea perpendicolare dall’occhio sopra ‘l piano del cerchio la qual faccia angoli retti per ogni verso; onde segue che l’occhio col suo centro ferisca ‘l centro del cerchio e perciò discerna tutto ugualmente; poiché ciascuna parte di esso li è ugualmente lontana. </s> <s>La qual cosa non succederebbe se l’occhio non fusse collocato sopra la linea perpendicolare perciochè gli angoli del conio sarebbero disuguali sì come anchora quegli del piano del cerchio e gli angoli disuguali (come si mostra nel Teor. 37°) son cagione delle disuguali apparenze dei deti diametri. </s> <s>Quindi adunque è manifesto che le diversità dell’ apparenze si dimostrano con l’aiuto <lb></lb>//<lb></lb>degli angoli sicome ancho della virtù degli angoli si cagionano. </s> <s>Perciochè ovunque si fa dimostration di Prosp. sempre si presuppongono l’occhio e ‘ raggi visuali terminanti nella grandezza veduta. </s> <s>Onde necessariamente risulta l’angolo e ‘l conio di grandezza minore o maggiore o uguale secondo la varia positura dell’obbietto e dell’occhio.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Se ‘l colore è propio obbietto del vedere ond’è che Euclide mostra la grandezza esser obbietto?</s> </p> <p type="head"> <s>Cap. 14</s> </p> <p type="main"> <s>Par cosa non convenevole, affermar nella Prospettiva, che l’obbietto propio del vedere sia la grandezza e non il colore; perciochè ella suole adattar alle cose da lei considerate, l’esser visibili, onde avviene che si distingua dall’altre parti della Matematica. </s> <s>La qual cosa essendo avvertita dall’arcivescovo di Cantauria nella Prosp. </s> <s>Commune, nella 2a conclusione, disse che ‘l colore illuminato opera impressivamente nel vedere, stimando il colore esser obbietto propio di esso. </s> <s>E sse gli obbietti di qualunque specie sieno si fanno visibili col mezzo del colore, per qual cagione ancho il colore non è obbietto propio del vedere? Ansi più propio sarà il colore che primiero le si afferisce; ma la grandezza, se però è propio obbietto del vedere, dopo ‘l colore si apprende. </s> <s>Oltre acciò, se Arist. nella part..63. del 2° dell’Anima afferma l’obbietto propio d’un senso esser quello che non si apprende da altro senso, la grandezza non potrà dirsi obbietto propio del vedere; perciochè com’egli stesso nella part. .64. dice è obbietto sensibile commune. </s> <s>Oltre acciò, se la grandezza <pb pagenum="folios 54v-55r"></pb>non si apprende dagli occhi nostri se non con l’aiuto del colore di cui è vestita, segue che non possa esser obbietto sensibile per se stesso e perciò non sarà propiamente obbietto sensibile del vedere, come si può confermare con le parole della particella .65.. Ma a così fatto quesito potremo per avventura rispondere in più modi. </s> <s>Perciochè ‘l secondo Filosofo nel lib. </s> <s>De sen. e sen., nel 3° capitolo. </s> <s>Il colore o è nella stremità del corpo overo è la stessa estremità sua. </s> <s>Onde i Pittagorici appellavano la superficie colore. </s> <s>Di modo che, in questa maniera il colore essendo la stessa estremità, cioè la superficie del corpo, sarà ‘l medesimo che la grandezza; onde Euclide, ponendo la grandezza per obbietto del vedere, non escluderà ‘l colore; che ‘l colore (come ‘l Filosofo stesso afferma nel medesimo luogo) è l’estremità della cosa luminosa in un corpo terminato. </s> <s>Oltre acciò si dee considerare che ‘l colore e la grandezza appariscono una stessa cosa, perciochè si vedano insieme; che ‘l colore non si dicendone separatamente, né la grandezza è invisibile, come sarebbe se non havesse ‘l colore; onde se amendue si mostrano una stessa cosa, per questa ragione non sidee credere che dicendo Euclide, la grandezza è l’obbietto del vedere, non includa ancho il colore; che qualunque faccia mesione solamente della grandezza, virtualmente intende ancho il colore, ponendo la grandezza per obbietto principale in quanto appartiene alla prospettiva e ‘l colore come obbietto congionto, cioè quello stimando suo principale obbietto che più li è proportionato. </s> <s>Finalmente Euclide considera solamente la grandezza come <lb></lb>//<lb></lb>visibile e non riguarda ‘l colore come obbietto; ma come mezzo, pel quale ella si apprende; perciochè non si cura della qualità, che non è propio obbietto della Prospettiva, ma della quantità e si serve della qualità, dico del colore, come di strumento; perciochè Aristotele, nel .p°. cap. de sen. et sen., dice tutte le cose corporee esser colorate; di maniera che gli obbietti communi col mezzo del colore si conoscano, in fra ‘ quali principalmente sono le figure e le grandezze e in questo modo mi par che abastanza si sia risposto alla questione proposta.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>L’uso degli angoli nella prospettiva appartenente agli specchij e a’ reflettimenti de’ raggi del sole</s> </p> <p type="head"> <s>Cap. 15</s> </p> <p type="main"> <s>Sì come la stessa Prospettiva servendosi delle cose geometriche e specialmente della linea e del ponto, dal modo di considerare prende ‘l nome col quale determina e specifica (per così dire) le dette cose a se stessa applicate, così mentre si serve degli angoli, dalla maniera di considerare e dall’adattamento di essi a’ suoi propri termini si trahe ‘l nome; di modo che, sì come nomina le linee e punti visuali, così appella gli angoli non solamente visuali, ma dell’Incidenza e del Reflettimento. </s> <s>Di modo che, come vedo nella Prosp. le cose della geometria con la gionta di qualche nome, o conditione, si comprende l’uso di esse: così dall’Incidenza e dal Reflettimento adattato agli angoli si prende la cognition dell’uso loro nella Prosp. </s> <s>Perciochè ciascun arte o scienza che si serve de’ termini, o della materia d’altra scienza od arte superiore appropiandogli a se stessa, gli veste di nuovi nomi e di nuove conditioni, come si può<pb pagenum="folios 55v-56r"></pb>vedere esaminando tutte le scienze ed arti. </s> <s>Gli angoli adunque che si adoperano in questa parte di Prosp.son que’ che si appellano dell’Incidenza e del Reflettimento, i quali si formano da’ raggi del Sole altresì negli specchi, nell’acqua ed in altra maniera di corpi ne’ quali s’incontrano. </s> <s>Di questi ragionaremo in questo luogo, ma avanti che cominciamo, per continuar in parte il presente ragionamento, con le cose dette, bisogna avvertire che la parte della Prosp. ottica ragiona de’ raggi visuali in quanto appartengono al vedere gli obbietti dirittamente nel mezzo illuminato; che pertanto Euclide nella sua Prosp. fece questo supposto, che ‘raggi che escon dall’occhio si muovono per retta linea, como ancho afferma Eliodoro Larisseo e l’Arcivescovo di Cantauria nella concl. 48 della prima parte della Prop. </s> <s>Comm., ove dice, facilmente vedersi gli obbietti dirittamente. </s> <s>Ma nel mezzo illuminato; perciochè questo stesso autore nella concl. 49 afferma che nissuna cosa si può vedere senza lume. </s> <s>Il che oltre alla sperienza, si conferma dal filosofo nel 2° dell’Anima nella part. 67 là dove afferma che ‘l colore non si può veder senza ‘l lume el mezzo nel quale si vede non essendo illuminato non può render colore alcuno. </s> <s>Oltre che secondo Francesco Piccolomini nel Duodo nel cap. 5 del 6. Il lume si trova negli obbietti per cagion del mezzo, pel quale passa; che non può giogner da uno estremo, cioè dal corpo luminoso all’altro estremo, cioè all’obbietto senza passar pel mezzo. </s> <s>La parte specularia è tutta volta alla consideratione e all’uso <lb></lb>//<lb></lb>de’ raggi del lume e de’ visuali. </s> <s>Perciochè se l’ottica considera i raggi visuali in quanto si adoperano alla vista dirittamente fatta, poiché procede secondo la dirittura de’ raggi che uscendo dall’occhio e terminando nell’obbietto formano ‘l conio; la specularia all’incontro ragiona de’ raggi visuali in quanto appartengono al vedere per modo reflesso negli specchi illuminati insieme col mezzo. </s> <s>Per modo reflesso dico, perciochè la vista in essi vien regolata secondo ‘l reflettimento de’ raggi che si partano dall’occhio per andar a trovar l’obbietto e la sua immagine impressa negli specchi, ne’ quali, essendo piani o ritondi e convessi e concavi (secondo Eucl. nel primo teor.) i raggi visuali si ripiegano ad angoli uguali, come s’avvertisce da Ignatio Danti sopra la terza supp. d’Eucl. negli specchi e per la gran conformità posta in fra la luce e gli occhi (onde è che da pochi sono stati chiamati lumi, splendori, stelle e soli) ciò che de’ raggi visuali s’è detto dirsi dee de’ raggi della luce del Sole, anzi principalmente di questi. </s> <s>Perciochè ancho essi non solamente negli specchi qualunque sieno; ma anchora ne’ corpi lucidi, come nelle gioie e nelle pietre egregiamente polite anzi nella terra anchora, benchè opaca e varia di colori e di superficie; si reflettano con varie specie d’angoli ed in ogni ripiegamento formano dall’una e l’altra banda angoli uguali, poiché sempre si ripiegano in un punto. </s> <s>Ma bisogna avvertire che benchè i raggi visuali e ‘ raggi del Sole riflettendosi negli specchi formino angoli uguali, con tutto ciò in fra essi è qualche differenza, che altro è l’angolo della Reflession de’ raggi visuali <pb pagenum="folios 56v-57r"></pb>altro de’ raggi del Sole negli specchi, e la differenza si prende dal fine , il quale in essi è diverso; perciochè l’angolo de’ raggi visuali appartiene al veder le cose negli specchi; ma l’angolo de’ raggi del Sole appartiene al raddoppiamento degli splendori de’ lumi e della vista che è atta ad aveder le materie oppostegli. </s> <s>Ma per cominciar a scoprir l’uso degli angoli nella Specularia, seguendo di farne comparatione all’ottica, diremo che se nell’ottica si ritrahe l’uso degli angoli in quanto son fatti da’ coni de’ raggi visuali procedenti dirittamente pel mezzo dell’obbietto, acciò si formi la visione: nella specularia si conosce l’uso di essi come formati da’ raggi visuali, che non vanno dirittamente all’obbietto, overo alla immagine sua; ma per modo ripiegato; perciochè vanno dirittamente fine alla superficie dello specchio, là dove formano gli angoli dell’Incidenza; e dallo specchio si riperquotano e fanno gli angoli reflessi, acciochè i raggi del conio della veduta prendano l’immagine dell’obbietto che è fuor dello specchio, acciò la vista habbia l’uso del suo conio nell’obbietto reale e nell’immagine, talmente che la base che è nell’immagine corrisponda alla base che è nell’obbietto; perciochè ripiegandosi i raggi visuali nello specchio e nella simiglianza stessa dell’obbietto finalmente si terminino nell’obbietto, onde prende origine tutto quel, che apparisce nello specchio. </s> <s>La qual cosa vedrà apertissimamente chiunque osservarà Eucl. nella Spec. e specialmente nelle dimostrationi de’ teoremi .4.7.11.17. e .20. Di modo che nell’ottica si scorgano solamente gli angoli dell’Incidenza, che son quelli che si <lb></lb>//<lb></lb>formano da’ raggi visuali mentre perquotono nell’obbietto, costituendolo base del conio loro. </s> <s>E se favellando alla peripatetica, dicessemo i raggi visuali nascer dall’obbietto , costituiremo un angolo solo dell’Incidenza formato nel centro dell’occhio. </s> <s>Però si dice che la vista procede per linea retta. </s> <s>Ma nella Prosp. degli specchi oltre agli angoli dell’Incidenza si veggono quegli del reflettimento. </s> <s>Perciochè ogni volta che i raggi visuali si riperquotano negli specchi fanno due angoli, l’uno dell’Incidenza, che è ‘l primo che si fa nel ferir lo specchio e l’altro della reflessione, che è ‘l secondo fatto tosto doppo ‘l primo sorgendo e muovendosi finchè gionghino al termine loro, che è l’obbietto reale. </s> <s>L’angolo dell’Incidenza è quello che si comprende dalla superficie dello specchio e della linea che esce dall’occhio. </s> <s>L’angolo della Reflessione è quello che si abbraccia dalla superficie dello specchio e dalla linea che si ripiega dal punto dell’incidenza che è quello in cui ferisce il raggio visuale e si muove verso altro punto in cui si ferma. </s> <s>Di maniera che amendue questi angoli si formano sopra la superficie dello specchio e si suppongano uguali acciò possano servire alle dimostrationi , come si può vedere, osservando Eucl. </s> <s>Oltre a questi angoli vi è ancho ‘l terzo esterno e maggiore d’amendue, il quale è contenuto dal raggio diritto e dal raggio piegato. </s> <s>Senza i quali angoli non si può veder cosa alcuna negli specchi; perciochè, come dice Eucl. nel Teor. 18° e 27°, le immagini delle cose visibili negli specchi, si vedano in quel luogo ove concorre il raggio che esce dall'’cchio con la linea, che dalla cosa visibile si è tirata fine al centro dello specchio <pb pagenum="folios 57v-58r"></pb>concavo; che sì come afferma Ignatio Danti sopra la quarta supp. e apparenza prima, le cose che si vedono negli specchi non si vedano per quella linea con cui s’improntano, ma nel concorso di detta linea e del raggio che esce dall’occhio e ciò avviene quando il raggio non si ripiega in se stesso come si mostra da Eucl. nel 2° Teor. perciochè in questo caso la linea dell’incidenza e della reflessione è l’istessa, onde non vi si scorge se non l’angolo dell’incidenza. </s> <s>Questo concorso di raggi visuali con le linee degli obbietti non si fa senza i detti angoli ed in esso si vedano le immagini degli obbietti visibili e però negli angoli dell’Incidenza si forma la visione di ritratti delle cose visibili negli specchi; perciochè (come nel precedente capitolo si è dimostrato), ciò che si vede apparisce sempre sotto qualche maniera d’angolo; che se nell’ottica ciò si dimostra nel conio diritto, nella Prosp. degli specchi si dimostra nel conio reflesso. </s> <s>Ma si avvertisca che nell’ottica solamente si fa conto dell’angolo costituito da’ raggi visuali nel centro dell’humor cristallino e nella specularia oltre a questo si stimano ancho gli angoli formati nella superficie dello specchio nella stremità dell’immagine dell’obbietto, là dove si forma la visione reflessa. </s> <s>E per intender meglio questo ripiegamento de’ ragi visuali negli specchi fa di mestiero considerare che può dirsi che ‘raggi visuali che escono dagli occhi e feriscono lo specchio ne’ termini del diametro dell’immagine impressavi si ripieghino partendosi da’ punti dell’Incidenza per muoversi tanto che gionghino all’obbietto rappresentato dell’imagine dello specchio, acciochè si formi un conio reflesso colla forza del <lb></lb>//<lb></lb>quale si faccia la visione della detta immagine conforme alla vision dell’obbietto reale; perciochè la base del conio che è nell’immagine impressa nelo specchio sarà simile e proportionale alla base che si ritruovarà nello stesso obbietto. </s> <s>Onde si vede che i raggi visuali che fanno gli angoli dell’incidenza e della reflessione per cagion del conio appariscono gli stessi raggi che escon dagli occhi per formare ‘l conio diritto che la natura loro sarebbe di andar dirittamente, ma l’incontro dello secchio gli fa piegare; che tale sia la natura loro, si vede quando s’incontrano in un corpo lucido e trasparente come è l’acqua che è quasi uno specchio piano in cui naturalmente i raggi che verso la cosa rappresentata si reflettano per trarne l'’mmagine e trasportarla nell'’cqua benchè a rivescio, si vanno dirittamente continuando tanto che trasportino la figura dell’obbietto nella stessa perpendicolare, come si vede nelle due dimostrationi del 7° Teor. </s> <s>Ma si dee avvertire che’ detti raggi che si continuano nell’acqua non formano linee perpendicolari, ma più tosto inclinate; che se facessero linee perpendicolari si ripiegarebbero in loro stesse e non formarebbero altro angolo che dell'incidenza e però nol farebbero Reflesso; perciochè avviene de’ raggi del vedere non altramente che avvenga di quegli del Sole i quali non si ripiegano sopra ‘l corpo ove perquotono se non quando non son perpendicolari, come si vede appresso l’Autor della Prosp. </s> <s>Comm. nella quindicesima concl. della prima parte. </s> <s>Oltre acciò quando Eucl. nel Teor. 18° e 27° dimostra negli specchi all’hora vedersi le immagini delle cose visibili quando concorreno i raggi che escono dagli occhi insieme con le linee, che dalla cosa visibile si tirano; par che accenni che l’obbietto stesso imprime la sua immagine nello spec<pb pagenum="folios 58v-59r"></pb>chio perciochè a qual fine tirar le linee dall’obbietto allo specchio, se l’obbietto non v’imprime la sua figura col mezzo di esse. </s> <s>Adunque l’obbietto con la forza delle linee che da esso nascono cagiona la sua figura nello specchio; Onde pare che in queste Eucl. segua l’oppinion de’ Peripatetici e in parte quella de’ Platonici, questa col mostrar che le cose che si discernano nello specchio si vedano per la forza de’ raggi visuali uscenti dagli occhi e quella col dimostrar che le linee nascono dall’obbietto. </s> <s>Ma se l’obbietto ha virtù d’imprimer la sua effigie negli specchi, come non harà la medesima possanza ancho negli occhi, che sono simigliantissimi agli specchi, sicome si vede per esperienza e come si ritrahe da Francesco Petrarca nella Canz. </s> <s>Perché la vita è breve, là dove si legge:</s> </p> <p type="main"> <s>Luci beate e liete,</s> </p> <p type="main"> <s>Se non che ‘l veder voi stesse v’è tolto,</s> </p> <p type="main"> <s>Ma quante volte a me vi rivolgete,</s> </p> <p type="main"> <s>Conoscete in altrui quel che voi sete.</s> </p> <p type="main"> <s>Adunque sì come l’obbietto opera nello specchio, così potrà operare negli occhi. </s> <s>Nello specchio principalmente e negli occhi col mezzo dello specchio. </s> <s>Ma si avvertisca che Eucl. non dice che le linee nascano dall’obbietto, ma che si tirano dall’obbietto, acciò si faccia ‘l detto concorso e perciò non potrà dirsi, che la veduta si faccia per ricevimento di specie insieme co’ Peripatetici. </s> <s>Perciochè anchorchè la vista che si fa intorno all’immagini rappresentate dagli specchi si formi col mezzo del ricevimento delle impronte di esse, contuttociò non è ‘l ricevimento considerato da Aristotile perciochè non si fa negli occhi, ma nello specchio, là dove fanno angoli i raggi visuali e si congiongano con le linee degli obbietti. </s> <s>Perciochè in tal congiognimento si riceve la figura dell’<lb></lb>//<lb></lb>obbietto da raggi visuali. </s> <s>Né per l’uscir che si faria da raggi visuali secondo i Platonici si cagiona la visione negli specchi perciochè non è l’uscir di essi cagion della visione, ma ‘l detto concorso de’ raggi visuali con le linee degli obbietti. </s> <s>Oltre acciò i raggi visuali per loro stessi giognerebbero all’obbietto se non s’incontrassero nelle linee che si partano da esso per trasportar l’immagine sua nello specchio. </s> <s>E però l’oppinion d’Eucl. sarà schiettamente Platonica, cioè al tutto lontana dal ricevimento dell’immagine visibile. </s> <s>Ma per non tralassar quel che appartiene agli specchi dirò che altri son piani, altri tondi de’ quali altri sono concavi ed altri convessi. </s> <s>Gli specchi tondi, o sieno convessi o concavi, sono tutti portioni di cerchi considerati nella superficie piana, in cui si figurano; ma considerati nell’esser loro non sono altro che tante portioni di corpi sferici, o concavi o convessi, non altramente che se noi tagliassemo una palla vota di vetro o di rame. </s> <s>Che quantunque Eucl. gli dimostri con portioni di cerchij, come si vede nel Teor. 5.6.8.11.etc. contuttociò si dee intender che le figure da lui formate nel piano della superficie non sien fatte per mostrar simplicemente le portioni de’ cerchij; ma per raffigurar sotto tali portioni i frammenti de’ corpi sferici. </s> <s>Sono adunque gli specchi portioni di corpi sferici e l’occhio per lo più vien collocato dentro la circonferenza loro, o nel centro o infra essa e ‘l centro, onde muovendo i raggi visuali verso lo specchio, finalmente con essi vi forma l’angolo dell’Incidenza e del Reflesso, mentre vicostituisce ‘l conio ripiegato. </s> <s>Gli specchi piani son quegli la cui superficie è in modo <pb pagenum="folios 59v-60r"></pb>giacente e diritta che dall’obbietto la linea vi cade ad angoli retti, sì come vede nelle dimostrationi del Teor. 7.9.16 e 19 ed essendo allongati fine alla perpendicolare dell’obbietto i raggi visuali che già si ripegavano (perciochè secondo il Teor. 16 ciascuna cosa visibile negli specchi piani si vede nella perpendicolare tirata dalla cosa visibile fine allo specchio) la superficie dello specchio vien tagliata ad angoli disuguali e diversi di specie. </s> <s>Onde l’immagine dell’obbietto si trasporta nello specchio al contrario; perciochè essendo già nello specchio impressa l’immagine per mezzo delle linee che da esso procedono i raggi visuali che prima si ripiegavano prendendo la detta immagine seguono di muoversi finchè quasi penetrando l’obbietto trasportino la detta immagine nella perpendicolare, collocandola a rivescio in quella guisa che ci appariscono le cose nell’acqua, cioè secondo la diversa positurae contraria delle parti; perciochè i raggi visuali mutan sito, come si può vedere nel Teor. 7. 19. 11. 12. etc., benchè sì l’obbietto e sì l’immagine appariscano sotto ‘l medesimo angolo e dentro al medesimo conio; che’l conio piegato è equalmente al conio diritto, come si può vedere nella dimos. del 7° Teor. </s> <s>Ma poiché di sopra si accennò la conformità nella quale convergono gli occhi con la luce del Sole e perciò gli splendori della detta luce in diversi incontri di corpi se reverberando formar gli angoli della Reflessione e dell’Incidenza e quivi cagionar l’ aurescimento del colore e della virtù infiammate. </s> <s>Pertanto fa di mestiero considerare diligentemente tutte le maniere degli angoli che da’ raggi del Sole si producano e la ragion del modo del producimento loro e l’effetto che <lb></lb>//<lb></lb>ne risulta ed insieme anchora l’uso. </s> <s>Tre sono le maniere degli angoli che si possano produrre da’ raggi del Sole nella ripercussione o nel concorso o nello ‘ncontro loro. </s> <s>Ciò sono gli angoli dell’Incidenza del Reverberamento e dello ‘ntersegamento. </s> <s>Gli angoli dello Intersegamento son quegli che si formano mentre i raggi del Sole trapassano per uno spiraglio d’un muro o d’una finestra chuisa o d’una vetrata rotta, perciochè dal corpo luminoso del Sole si partano i raggi i quali passando pel detto spiraglio s’incontrano insieme e si tagliano formando due coni opposti e per conseguenza due angoli opposti uguali. </s> <s>Perciochè i raggi che dal Sole si partano, in qualunque punto terminati secondo l’Autor della Prosp. </s> <s>Comm. nella Concl. 5 della prima parte, non possono fermarsi in esso, ma tagliandosi insieme vanno più oltre; onde quando passano per lo spiraglio angolare s’intersegano in esso, allongandosi dirittamente e pervenendo da una parte a tanta lontananza quanta è la lontananza del Sole dall’altra parte, si allargano secondo la larghezza del Sole perciochè gli angoli opposti essendo uguali e ‘ lati altresì uguali, la base anchora sarà uguale, come si potrebbe confermar per la quarta del primo d’Eucl.: Che ‘ raggi necessariamente debbano passar più oltre e tagliarsi insieme; quindi è manifesto perciochè non truovano incontro di corpo denso, che necessariamente si richiede alla ripercussione de’ raggi, come dimostra ‘l medesimo Autore nella concl. .15. della medesima parte: che lo ‘ncontro che truovano non è altro che un punto del mezzo, cioè dell’aria contenuta nello spiraglio. </s> <s>Onde è come punto e come corpo penetrabile da’ raggi del Sole (perciochè l’aria non sola <pb pagenum="folios 60v-61r"></pb>mente è corpo trasparente, ma raro), non è tale che vi si possan fermar o riperquoter i raggi del Sole.Gli angoli del reverberamento de’ raggi del Sole son simiglianti agli angoli del Reflesso de’ raggi visuali negli specchi; perciochè son compresi dalla superficie del corpo denso e dalle linee che nascono dal corpo luminoso e incontrandosi nel corpo denso o lucido o chiaro e polito o variamente colorato, com’è la superficie della terra percotendolo nel punto dell’Incidenza, quindi si partano e vanno a terminare altrove, cioè in contraria parte. </s> <s>E questo avviene perciochè non si può far riflettimento alcuno, così angoli. </s> <s>E questo accade perciochè non si può far riflettimento alcuno con angoli acuti interni, sì come ancho si vede ne raggi visuali; che la linea ripiegata non termini in contraria parte obliquamente; perciochè i raggi si ripiegarebbero in se stessi e cadrebbero nel punto dell’Incidenza perpendicolarmente e non formarebbero angoli acuti; non retti come si può ritrarre dal 2° Teor. degli specchi. </s> <s>E l’autor della Prosp. </s> <s>Comm. nella quindicesima conclusione della prima parte mostra che ‘l raggio della luce icontrandosi nel mezzo denso si ripiega, non essendogli perpendicolare. </s> <s>Gli angoli dell’Incidenza de’ raggi del Sole non son diversi da quegli de’ raggi visuali, in quanto al nascimento loro; periochè si producano nel percotimento de’ raggi in qualche corpo denso sì come quegli si cagionano nel ferir lo specchio. </s> <s>Onde son que’ che si comprendano dalla superficie del corpo denso e dalla linea che nasce dallo stesso corpo luminoso. </s> <s>Nel concorso di questi raggi raddoppiata <lb></lb>//<lb></lb>la virtù infiammante si accende ‘l fuoco, come si può vedere nella dimostration del Teor. 3 degli specchi d’Euclide. </s> <s>E ciò basti intorno a questa parte della Prospettiva.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Dell’uso degli angoli nella Prospettiva scenografica</s> </p> <p type="head"> <s>Cap. 16</s> </p> <p type="main"> <s>E’ tanto universale l’uso degli angoli che non è parte di Matematica o pura o non pura che non si serva di essi. </s> <s>Il che si è dimostrato ne’ precedenti cap. come ancho in questo si mostrarà.</s> </p> <p type="main"> <s> Perciochè nella prospettiva appartenente alle scene l’uso degli angoli non solamente è necessario nell’operationi sue; ma ancho nelle ragioni sopra le quali tutta la pratica si sostiene. </s> <s>E perché ‘l detto uso degli angoli si applica a diverse cose, perciò si distingue in varie maniere, dalle quali procedano effetti diversi, come nell’operare facilissimamente si conosce.</s> </p> <p type="main"> <s>L’uso adunque degli angoli nella scenografia si impiega: nell’accomodamento delle cose vere con le finte e nel dar rilievo e dirittura nelle cose poste negli angoli concavi e convessi e nel far saltar sopra ‘l piano le cose come rilevate: negli intersegamenti varij: nel trasportamento delle figure geometriche: e nelle linee del taglio.</s> </p> <p type="main"> <s>La linea del taglio che ( come si ritrahe dal primo libro del Cavalier Lorenzo Sirigatti nel cap. 3 e 5) altramente è detta delle misure, non si può generar senza gli angoli; perciochè è una linea retta perpendicolare, che cadendo sopra la linea del piano, la taglia ad angoli retti, sì come è natura delle linee perpendicolari: e in un medesimo tempo, tagliando i raggi visuali, e le linee che si par<pb pagenum="folios 61v-62r"></pb>tano dal piè dell’huomo riguardante, che terminano negli angoli dell’obbietto, dico negli angoli della pianta; i visuali ne’ superiori, e le linee pedali negli inferiori: e ne’ tagliamenti formano angoli alterni, e opposti uguali, come si può vedere appo Euclide nelle propositioni 28 e 29 del primo e ciò si vede chiaro; percioché le linee che escono dall’occhio e dal piede, le quali toccano gli angoli della figura dell’obbietto, sono fra loro parallele, anzi formano due parallelogrammi proportionali, i quali hanno un lato commune terminante nel punto dell’occhio e del piede gli angoli de’ quali non solamente son simili ma ancho uguali perciochè per la trentacinquesima del primo.</s> </p> <p type="main"> <s>I parallelogrammi costituiti nella medesima base e nelle medesime parallele sono fra loro uguali. </s> <s>La linea del taglio è ancho quella che si chiama orizzontale, che (sì come si vede appresso Baldassar Peruzzi nel Serlio e appo ’l Vignuova e appresso Giovan Cusino) partendosi dal punto principale, in cui si termina la lontananza della veduta, e terminando nella linea del piano, il taglia proportionalmente e ad angoli uguali, in apparenza ottusi, od acuti ma in essenza retti.</s> </p> <p type="main"> <s>Può dirsi anchora esser la linea perpendicolare, che si tira nella figura geometrica, la quale terminando nella linea del piano, che è base del triangolo, dentro ‘l quale si dispone, e comparte il piano iscorciato, la taglia ad angoli retti, e da’ tagliamenti nascendo le linee si congiongano in un ponto e intersegano le diagonali con proportion di tagliamenti, formando insieme angoli uguali, interni ed esterni e cos <lb></lb>//<lb></lb>tituendo col taglio loro il producimento delle parallele. </s> <s>Di modo che la linea del taglio, intendasi come altri vuole, non potrà esser linea del taglio, se non costituisce varie maniere d’angoli, che una linea retta non può già mai tagliare un’altra che non formi qualche maniera d’angolo; che l’angolo (sì come può esser noto per la geometria) non si può formar senza contatto o inclinatione, o declinatione, né l’inclinatione, o la declinatione, il contatto può farsi già mai senza qualche tagliamento di linea e così per opposito.</s> </p> <p type="main"> <s>Adunque la linea del taglio si produce nel medesimo tempo, che si formano gli angoli.</s> </p> <p type="main"> <s>Oltre acciò per la linea del taglio si può prendere la linea del piano, o della terra; perciochè in essa si terminano tutte le perpendicolari che nascono dagli angoli delle figure geometriche, o delle piante de’ corpi, che si hanno a levare in prospettiva.</s> </p> <p type="main"> <s>Onde la detta linea venendo tagliata diversamente, porge le misure di ciò che si dee fare nel piano iscorciato. </s> <s>Si adoperano gli angoli nel trasportamento delle figure geometriche; perciò che elle non si potran trasportare se con le linee e co’ raggi visuali non si truovano gli angoli delle figure, o le linee perpendicolari, che nascono da essi, il che depende dalla ragione ottica (sì come si può vedere osservando le regole, con le quali si trasportano: e la suppositione prima e seconda della Prosp. di Eucl.).</s> </p> <p type="main"> <s>Negli intersegamenti delle linee col mezzo de’ quali si dispongano e compartiscano i piani digradati, e le figure sopra essi, e vi si ergano i corpi regolari, o irregolari, apparisce espressamente l’uso degli angoli; perciochè non si <pb pagenum="folios 62v-63r"></pb>possan già mai formar gli intersegamenti senza costituir varie specie d’angoli secondo la diversa applicatione delle regole di Prosp..</s> </p> <p type="main"> <s>Né le cose vere con le finte si possan bene accordare insieme senza servirsi degli angoli, perciochè se riguardiamo i due piani d’una scena, l’uno finto che si ritruova nella maestà, e l’altro vero, che è ‘l piano del palco, dove si posano le cose sode e dove si rappresenta l’attion della commedia vi vedremo l’intersegamento di due triangoli, mentre i lati dell’uno son tagliati da’ lati dell’altro e ‘l tagliamento farsi ad angoli ottusi; perciochè ‘l fine di questo intersegamento è che i due piani si convertano in uno stesso piano, né ciò possono fare se non si tagliano ad angoli ottusi i lati di amendue ‘triangoli, come si vede in questa figura qui appresso.</s> </p> <p type="main"> <s>Che acciochè di due triangoli se ne faccia uno bisogna che ‘quattro lati di essi divenghino due, e non posson già mai i quttro lati de’ due triangoli nell’apparenza diventar due lati d’un sol triangolo, se tagliandosi insieme non costituiscono angoli ottusi; perciochè a voler di due lati far uno in apparenza,bisogna far che, tagliandosi, formino una maniera d’angoli che opposti alla vista nostra di lontano si perdano, e questi non sono altro che gli ottusi, i quali vie più degli altri, per lor natura si accostano alla linea retta, come altrove si è dimostrato.</s> </p> <p type="main"> <s>Di modo che mentre si tagliano i lati de’ due triangoli formando angoli ottusi, i quali svaniscono dalla vista nostra, di maniera si <lb></lb>//<lb></lb>uniscano, che di essi si forma una stessa linea.</s> </p> <p type="main"> <s>Onde venendo fatti due lati, i quattro de’ due triangoli, necessariamente diventeranno un solo, e così di due piani terminati da essi, se ne sarà formato un piano solo, intendendo sempre in riguardo dell’apparenza; e nell’esser loro sì come saranno quattro lati di due triangoli così saranno due triangoli e per conseguenza due diversi piani come si potrebbe dimostrare nella presente figura.</s> </p> <p type="main"> <s>Ma lasciamo queste considerationi alle teoriche della scenografia. </s> <s>Nella stessa guisa si scuopre l’uso degli angoli nelle cose finte negli angoli della stanza; perciochè si costituiscano delle intersegation delle figure stesse, che vi si dipengano.</s> </p> <p type="main"> <s>Perciochè per le medesime ragioni vi si fa l’intersegatione ad angoli ottusi, come si vede nel quadrolongo ABCD, al quale corrisponde nell’apparenza AEBCFD.</s> </p> <p type="main"> <s>Perciochè perdendosi l’angolo AEB e l’angolo CFD la linea AEB si fa equivalente alla linea AB e la CFD alla CD.</s> </p> <p type="main"> <s>Finalmente nella disposition de’ piani, e delle elevation delle faccie e de’ coprimenti si vede espressamente l’uso degli angoli; perciochè non si possono iscorciare senza la figura triangolare, che si costituisce dalle linee orizzontali, le quali congiognendosi in un ponto formano angolo per lo più ottuso, come si vede chiarissimamente nella pratica della Scenografia. </s> <s>E la ragione di ciò è fondata nella suppositione 8 e 9 e nel teorema 4 e 5 della Prospettiva d’Euclide, che mentre i raggi visuali son più alti e gli intervalli di essi, benchè uguali da lontano apparendo minori, e gli obbietti vicini all’occhio essendo maggiori son cagione che necessariamente <pb pagenum="folios 63v-64r"></pb>i detti raggi visuali e le linee dell' orizzonte costituiscono qualche maniera d’angoli, come può esser noto a chiunque osserva la Prospettiva.</s> </p> <p type="main"> <s>E questo sia a bastanza per raccontar l’uso degli angoli nella prospettiva Scenografica; perciochè tutto l’uso di essi per quanto s’è fin qui da me osservato consiste in questi quattro capi o in altri che ad essi si riducano; che se altri con più diligenza di me osservando le ragioni e le pratiche di questa parte di Prospettiva trovarà altri usi d’angoli in essa ve l’aggionghi che a me è bastato solamente trovarne tanti quanti fussero bastevoli a pruovar la mia ‘ntentione.</s> </p> <p type="main"> <s></s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Nelle Meccaniche</s> </p> <p type="head"> <s>Cap. 17</s> </p> <p type="main"> <s>Tutta la somma delle Meccaniche si riduce alla lieva, alla bilanca ed alla statera, sì come apertamente potrà vedere qualunque con diligenza osservarà Le Questioni Mec. d’Arist. e ‘n queste è manifesto l’uso degli angoli; perciochè la lieva ( stromento col mezzo del quale qualunque movente, benchè di poca virtù, muove, con gran facilità ogni peso, al cui movimento naturalmente si richiedano più moventi) non muove peso alcuno se posata nel piano della terra non forma angoli disuguali e se fermata nel centro che è ‘l sotto lieva in esso non costituisce angoli uguali o simili: finalmente se ella con la sua estremità formando una portion di circonferenza in essa non costituisce angoli retti sferali, come si può vedere in A. B. Perciochè nel movimento della lieva si presuppongano cinque cose, come si può ritrar da Arist. nelle Mecc. <lb></lb>//<lb></lb>ciò sono el termine che sta immobile; perciochè tutto quello che si muove si muove sopra qualche cosa immobile e questo si è ‘l centro detto sotto lieva o quel che è posto in sua vece; la linea della terra perciochè le cose gravi che si muovano dalla forza della lieva si muovano per lo spatio che è nella superficie della terra; la lieva stessa che è lo stromento che si adatta al movimento e supplisce la forza manchevole nel movente; la linea che scaturisce da centro che rappresenta la stessa linea; le circonferenze formate dalle due stremità della lieva. </s> <s>Vi si aggiogne il termine mobile che sono le stremità della lieva: el movente congionto con la lieva, che prende la virtù di molti moventi insieme uniti. </s> <s>E per ripigliar da capo tutte le sopra dette cose bisogna dire che posta la linea della terra e ‘l termine immobile sul quale si regge e si muove la lieva e la linea della stessa lieva necessariamente bisogna che passando pel detto punto e terminando nella linea della terra formi da una banda un angolo ottuso e dall’altra un acuto; perciochè la natura della lieva non è d’esser secondo la linea perpendicolare ma secondo la inclinata alla giacente del piano che tosto che la lieva è fatta perpendicolare sul piano della terra ha finito insieme col suo corso tutta la sua forza; sì come movendosi la linea A.D. fermata ne centro C. infine al B. di maniera che si converta nella B.D. perpendicolare, segue di muovere il peso C.F. ma se più oltre si muovesse, non havrebbe<pb pagenum="folios 64v-65r"></pb>più alcuna forza di muovere il peso non andando incontra ‘l peso e non spegnendolo; ma mutando la lieva e fermandola nel centro C. e sopra il piano della terra sul ponto F. e muovendosi F. E. verso G.. Adunque se tutta la forza della linea consiste nello spegnere il peso bisogna necessariamente che ogni volta che la linea è fatta perpendicolare non habbia più forza. </s> <s>Adunque l’angolo della costitution della lieva non è ‘l retto, ma l’ottuso e l’acuto. </s> <s>Così anchora per la medesima ragione tosto che la lieva A. C. è pervenuta alla linea F. E. giacente che è la linea della terra perde tutta la forza che haveva nell’alzare ‘l peso D. mentre si muove sopra la sottolieva B.perciochè all’hora insieme con la linea del peso forma l’angolo retto. </s> <s>Overamente si aggionga che all’hora accostandosi alla F. E. non forma più angolo e perciò perde la forza di muovere. </s> <s>Adunque bisogna che la lieva si costituisca non ad angoli retti, ma acuti ed ottusi. </s> <s>Oltre acciò non si può negar che nel collocar la lieva non formino gli angoli; che ella non è altro che una linea materiale posata nel piano o inclinante verso esso o da esso declinante. </s> <s>E se la lieva è uno stromento manuale che mentre si muove di movimento circolare forma più linee, che si partano dal centro immobile e terminano nella circonferenza formata da lui, perché non si dee credere che nel centro costituisca più angoli ed ottusi ed acuti e nella circonferenza più angoli retti? Ciò si persuade a bastanza dalla ragione e dalla sperienza, come apertamente vedrà chi osser <lb></lb>//<lb></lb>varà ’l movimento della lieva espresso in questo essempio: e come si può veder sensatamente nelle taglie e nelle ruote: Senza che quel che si muove proportionatamente sopra la linea retta si muove secondo ‘l diametro e quanti sono i movimenti tanti i diametri i quali tagliandosi nel centro formano angoli. </s> <s>Finalmente la bilancia se riguardiamo alla postura dell’equilibrio, essendo fermata nel suo centro con la lenguetta che sopra esso si eleva dentro all’aco la qual da Greci si appella Carcone e da’ Latini Correttore et Virgula, forma angoli retti insieme con esso pervenendo nella stessa linea perpendicolare in quella guisa che fa l’Archipendolo de’ muratori, mentre ‘l filo s’affronta nelle due tacche come si vede qui appresso nell’essempio e dalla cognition di quest’ angoli retti subbito si viene alla notitia dell’equilibrio. </s> <s>Ma osservata la bilancia fuor di questa positura, in qualunque modo sia, essendo fermata nel medesimo centro, si vedrà che la detta lenguetta insieme co l’aco non si accorda non havendo commune la perpendicolare e perciò non forma più l’angolo retto, ma l’acuto e l’ottuso. </s> <s>Di modo che tosto veduto un angolo diverso dal retto e non esser commune la perpendicolare si conosce la bilancia esser fuor dell’equilibrio onde si scorge la differenza del peso delle cose poste in amendue le bilance. </s> <s>Ma nella statera è necessario osservare gli angoli retti nell’aggiustamento della lenguetta con l’aco e gli angoli acuti per conoscer l’alzamento del contrapeso e la quantità della cosa che si pesa con la statera, come qui si vede. e questo <pb pagenum="folios 65v-66r"></pb>è quanto m’occoreva dire intorno all’uso degli angoli nelle Meccaniche lassando molt’altre cose da dirti che pienamente si trattano da Arist. e da altri autori moderni a’ quali mi rimetto, che in questo luogo m’è paruto più convenevole il considerar le parti principali della scienza meccanica che tutte le cose ad essa appartenenti, non essendo questo luogo convenevole.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Nell’Architettura e particolarmente nella ornata e nella militare</s> </p> <p type="head"> <s>Cap. 18</s> </p> <p type="main"> <s>Non è dubbio alcuno che l’Architettura generalmente considerata e specialmente l’Architettura ornata e la militare ha principalmente dependenza dalla Geometria; perciochè ciascuna di queste prende i punti, le linee e le superficie diversamente figurate e corpi altresì variamente terminati e li adatta alla materia trattabile; che nella pura Geometria si considerano separati da ogni materia. </s> <s>Però, sì come nella Geometria pura per cagion delle dette cose si scorge l’uso degli angoli, così col mezzo delle medesime si potrà comprender nell’Architettura ornata e militare. </s> <s>E per dimostrare il detto uso in ciascuna di queste proporrò primieramente tutte le parti dell’Architettura ornata nelle quali apparisce espressamente l’uso degli angoli. </s> <s>Perciochè essendo più propria della pace che della guerra par che ragionevolmente le convenga il primo luogo: e alla militare convenendo molto più alla guerra che alla pace con ragione si adatta il secondo luogo; che la pace in quanto che è ‘l conservamento delle cose ed in quanto è fine della guerra merita d’esser tenuta più degna, e perciò l’Architettura <lb></lb>//<lb></lb>ornata appartenendo alla pace doverà ottenere il primo luogo in questo nostro discorso. </s> </p> <p type="main"> <s>Mostriamo adunque prima le parti nelle quali consiste l’uso degli angoli appresso l’ornata e consideriamole a parte a parte. </s> <s>L’uso degli angoli in quest’Architettura si comprende nelle figure angolari di varie maniere, o sieno regolari o irregolari, le quali si adoperano nel formar le piante delle fabbriche e nel compartirle.</s> </p> <p type="main"> <s>Nel compartimento delle facciate degli edifitij.</s> </p> <p type="main"> <s>Nel ridurre in piano le cose.</s> </p> <p type="main"> <s>Nel tirar a piombo e dirittamente le muraglie.</s> </p> <p type="main"> <s>Nell’adoperar la squadra e l’archipendolo.</s> </p> <p type="main"> <s>Nel formar i disegni delle fabbriche.</s> </p> <p type="main"> <s>Ne’compartimenti della superficie de’ piani e delle città.</s> </p> <p type="main"> <s>Ne’ compartimenti delle città si vede apertissimamente l’uso degli angoli; perciochè eletta già l’area in cui si debba fabbricar la città (supposto che si habbia da edificar tutta in un tempo e non in più tempi successivi ed in più diversi aggiognimenti, come spesso accade) e collocato in mezzo ad essa uno stromento di marmo da Vitruvio appellato Amusio, nel mezzo del quale sia collocato lo Gnomone e secondo l’osservation dell’ombra tante hore avanti quante doppo mezzogiorno, ritrovata la linea meridiana e quindi anchora ritrovati tutti i venti secondo la pratica insegnataci da Vitruvio si divide tutta la sua circonferenza overo (come modernamente si costuma) con l’uso della Bossola detta Agucchia, secondo ‘l numero de’ venti, cioè in XXIIII parti, e ‘n tutta l’area si disegnano due quadrati perfetti che s’interseghino insieme i quali cos <pb pagenum="folios 66v-67r"></pb>tituiscano gli VIII venti: così anchora vi si formano due ottangoli i quali pur’ancho si taglino fra loro e da amendue si divide la circonferenza in sedici parti di maniera che ogni quarta vien divisa in quattro parti uguali. </s> <s>Ma di nuovo si divide in tre parti tali che tutta essendo divisa ugualmente costituisce XXIIII venti. </s> <s>Di maniera che tagliandosi insieme i lati delle suddette figure si costituiranno angoli diversi e disuguali come ottusi e acuti, più acuti o meno acuti, interni ed esterni ed opposti, come si potrebbe render chiaro per la figura. </s> <s>E secondo la disposition di queste linee prodotte da’ venti e secondo ‘l tagliamento de’ lati delle dette figure si forma tutto ‘l compartimento della città, cioè le piazze, le strade, i vicoli, gli angiporti, l’isole, i borghi e ‘siti de’ tempij e d’altri edificij pubblici, e delle porte e delle torri. </s> <s>Onde essendo queste parti della città formate dentro alle linee de’ venti e dentro gl’intersegamenti loro, da’ quali si costituiscano gli angoli, haveranno necessariamente il sito loro determinato dentro qualche maniera d’angoli; per la qual cosa ci si manifesta l’uso degli angoli nel fabbricar le città, come molto meglio ciascuno si potrà certificare che osservi Cesare Cesariano Milanese nel Comm. di Vitruvio; là dove e con parole e con disegno dimostra eccellentissimamente il compartimento della città.</s> </p> <p type="main"> <s>Nel compartir le superficie de’ piani, cioè di pavimenti degli edificij e delle piazze e di giardini è necessario servirsi degli angoli; perciochè sieno le dette superficie di qualunque figura per for <lb></lb>//<lb></lb>mar in esse qualsivoglia maniera di compartimento bisogna risolverla in più figure o della medesima specie o di specie diversa, regolari o irregolari; né si possan formarvi le dette figure senza costituir varie ragioni d’angoli. </s> <s>Ed in questo si manifesta l’uso loro.</s> </p> <p type="main"> <s>I disegni delle fabbriche, o sieno di piante, o di elevationi non si possan far senza servirsi degli angoli perciochè si formano tirando prima una linea retta giacente e di poi una perpendicolare nel mezzo dello spatio, cadente sopra essa e quindi altre linee parallele piane e di poi altre perpendicolari parimente parallele alla prima perpendicolare le quali cadendo sopra le linee piane per la .12. definitione e per la .13. e .14. prop. del primo di Euclide, non posson non formare angoli retti o uguali a più retti in fra di loro conseguenti. </s> <s>E questo si è l’uso degli angoli che in questa parte necessariamente si richiede; perciochè in tale operatione si adoperano le linee piane e le linee a piombo, con le quali naturalmente si accompagnano gli angoli retti.</s> </p> <p type="main"> <s>Nell’adoperar la squadra si vede manifestamente l’uso degli angoli perciochè la squadra non è altro che l’angolo retto congionto con la materia; onde è che dagli Architetti l’angolo retto è detto angolo a squadra. </s> <s>Perciò che la squadra si compon di due linee rette una cadente sopra l’altra ad angoli retti, e così anchora l’angolo retto come è manifesto per 12. def. già citata. </s> <s>E nell’uso dell’archipendolo si adopra l’angolo; perciochè all’hora sta diritto e giusto e (come si dice) a livella, quando ‘l filo e ‘l piombino, detto linea della fiducia, cade perpendicolarmente e forma nella traversa da ogni banda angoli retti, dividen <pb pagenum="folios 67v-68r"></pb>dola in due parti uguali, come ancho dividendo l’angolo in parti uguali, come è manifesto per la nona, per la decima, per l’undicesima e per la dodicesima propositione del primo di Euclide, onde nasce la fabbrica di tale stromento.</s> </p> <p type="main"> <s>Nel tirar a piombo e addirizzar le muraglie si vede l’uso degli angoli; perciochè la linea del piombo sopra quella del piano o sopra qualunque altra giacente o sopra riga o regolo forma angoli retti onde quando gli angoli non appariscano retti all’hora si conosce la cosa non esser in piano, così anchora quando ‘l filo che guida l’elevation d’una muraglia non forma angoliretti con quello del piano, dà segno che la fabbrica pende; e riducendola col mezzo del filo si addirizza e si accommoda a piombo.</s> </p> <p type="main"> <s>Nel ridurre in piano ciascuna cosa bisogna adoperare gli angoli; perciochè si usa l’archipendolo col mezzo del quale all’hora si conosce la cosa esser in piano, quando la linea della fiducia tagliando ugualmente la linea trasverale dello stromento forma da ogni parte angoli retti.</s> </p> <p type="main"> <s>Ancho nel compartir le facciate delle fabbriche concorre l’uso degli angoli; perciochè primieramente non è facciata di qualunque sia fabbrica che non sia di qualche figura angolare, levatane la circolare e l’ornata, la quale anchora non può non servirsi degli angoli acciochè habbia posamento nel terreno. </s> <s>E stando sopra ‘l piano ciascuna fabbrica perpendicolarmente bisogna che formi sopra esso angoli retti, come si potrebbe pruovare per la diciottesima dell’undecimo d’Euclide. </s> <s>Ed havendo a riempier d’ornamenti tutto lo spatio delle facciate è necessario risolverle in molte figure, le quali non <lb></lb>//<lb></lb>saranno senza qualche maniera d’angoli; perciochè tutti i vani, come porte, finestre, quadri, nicchie, intervali ed altre cose simili non si possan formar se non con gli angoli.</s> </p> <p type="main"> <s>Finalmente nelle figure angolari, le quali concorreno nel componimento di tutte le fabbriche si vede espressamente l’uso degli angoli; perciochè di qualunque figura sieno è necessario per compartir tutto il loro spatio formarvi molte figure angolari secondo che richiede la varia qualità loro. </s> <s>Di qui adunque in tutte queste cose appartenenti all’Architettura Ornata si manifesta l’uso degli angoli, sì come habbiamo accennato. </s> </p> <p type="main"> <s>Resta hora a considerare in ciò che sia collocato l’uso degli angoli nella Militare.</s> </p> <p type="main"> <s>L’uso degli angoli nella Fortificatione si dimostra in più maniere; perciochè principalmente sopra gli angoli di qualunque figura regolare o irregolare si fonda tutta la fortification delle fortezze e delle città, perciochè l’angolo per sua strettezza essendo debile in maniera che non può molto resister alla forza e all’impeto de’ colpi delle batterie, ha bisogno d’esser fortificato: e perché lo spatio che si racchiude in ciascuno angolo è troppo breve, onde non è atto a dar commodità a’ difensori ed all’artigliarie che vi si richiedano per difesa del luogo (che la fortezza degli angoli consiste nel poter haver le piazze larghe) né alle ritirate, che vi fusse bisogno di fare, però fa di mestiero allargarlo. </s> <s>Onde non è maraviglia se Bonaiuto Lorini dica che ogni maniera di fortificatione si formi sopra gli angoli alla qual sentenza si sottoscriverebbe ciascuno <pb pagenum="folios 68v-69r"></pb>Ingegnero; perciochè questo dalla sperienza si conferma. </s> <s>Oltre acciò quest’uso si manifesta nel formar i disegni della fortificatione; poiché si formano col mezzo delle linee delle difese e co’ tiri dell’Artigliarie che si descrivano mentre si formano i profili delle fortezze. </s> <s>Le qua’ linee e qua’ tiri si tagliano in fra loro e formano varie maniere d’angoli. </s> <s>Finalmente apparisce quest’uso nel levar le piante delle fortezze o delle città o con istromenti o senza ritrahendo al costume de’pittori e de’ disegnatori; perciochè non si possan levar tali piante senza trovar prima gli angoli loro. </s> <s>Ma se da alcuno si ricercasse per avventura quali angoli sieno più frequentati nel fortificare e quali ragionevolmente convenghino alla buona Architettura militare, si potrebbe risponder esser gli angoli ottusi, intendendo però per il più che di rado si serve degli acuti e de’retti. </s> <s>E in questo luogo intendiamo degli angoli de’ ricinti delle città e delle fortezze, sopra ‘ quali si fonda la fortificatione; perciochè gli angoli usati ne’ baluardi sogliono essere acuti, benchè spesso sieno ottusi; che si variano non solamente per cagion della figura di più o di men lati, ma per la varietà del luogo de’ punti delle difese, preso o nel mezzo della cortina, o a’ due terzi, o nel collo del baluardo. </s> <s>Potermmo aggiognere che l’uso degli angoli si manifesta con le linee che formano i fianchi del baluardo; perciochè tagliano la linea della cortina ad angoli retti, il che si vede chiaramente; perciochè queste linee o si formano <lb></lb>//<lb></lb>dalla squadra o da quella operatione del compasso per la quale si produce la linea perpendicolare. </s> <s>E questo basti haver detto dell’uso degli angoli nell’Architettura militare; che in essa non hanno altri usi che questi e se pur si hanno degli altri, almen non son così principali.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Nell’arte militare</s> </p> <p type="head"> <s>Cap. 19</s> </p> <p type="main"> <s>Fra tutte le cose che al’arte della guerra appartengono una ve n’è nella quale apparisce l’uso degli angoli e questa si è ‘l modo di accommodar in campo i battaglioni degli eserciti. </s> <s>Ma in essa non si vede tale uso per sé, ma per accidente, cioè per mezzo delle figure costituite da’ soldati; perciochè con essi si formano varie maniere di figure angolari, ciò sono quadrate, quadre longhe, triangolari, pentagone; ma inregolari, che da altri son dette triangolari; ma per essergli levati due angoli appresso alla base con due tagliamenti, vi si costituiscano quattro angoli onde insieme con l’angolo superiore del triangolo risultano cinque angolie la figura di cinque lati: a lunetta e a forbice, tutte figure simplici e angolari. </s> <s>Altre sono figure composte o con più figure quadre congionte insieme con una parte di lunetta, sì come è la figura simigliante allo scorpione: overo un quadro congionto con due lunette, posto in fra ‘due convessi di esse, come si costuma colle picche: o un quadro perfetto con due braccia da una fronte, che son due metà di due lunette solito farci con le picche: o con quattro quadri perfetti accomodati a modo di croce, nel cui mezzo si pongano le bandiere, le lance spezzate e <pb pagenum="folios 69v-70r"></pb>gli huomini grandi guardati in ogni angolo dalle bombarde: similmente con tre lunette, le cui fronti si formano da’ cannoni loro, di modo che in mezzo rimanga uno spatio triangolare di linee curve, dove sieno in mezzo le insegne e le persone principali guardate da ogn’angolo con l’artigliaria: o con una figura triangolare accompagnata da ogni banda da un’ala di soldati fatta con una portione di lunetta, sì che amedue l’ale servono per braccia; perciochè la base di tal figura è una lunetta, che avanzando da ogni banda ugualmente, fa fronte verso l’angolo superiore, che è la fronte principale del battaglione, come si suol fare con le picche: Finalmente con due romboidi e con un rombo, che le congionga in modo che formino la lettera V e col rimanente fatto in forma di lunetta stesa infine alle romboidi, la quale rassembri un mezzo circo, nel mezzo del quale si racchiuda un quadro, figura che per la simiglianza si domanda a forbice: e con altre figure composte le quali contengono in sé diverse specie d’angoli. </s> <s>Di modo che questi battaglioni sieno di figure così simplici, come di composte, non possano già mai formarsi senza servirsi degli angoli come a chiunque considera bene qualsivoglia figura di battaglione, ansi le file stesse de’ soldati in fra loro (che queste formano que’ medesimi angoli che le figure formate da loro) apparirà chiarissimo l’uso degli angoli. <lb></lb>//<lb></lb>Ma quel che appartiene al modo di ordinar le file secondo i numeri determinati, acciò produchino le figure, che s’intende di formare, si lassarà considerare a que’ che esattamente insegnano l’arte militare e specialmente il modo di ordinar i battaglioni e d’accampar gli eserciti alla campagna; onde rimettendomi agli autori che ne trattano, e specialmente a Girolamo Maggi e all’autore del libro intitolato Il Vallo e a tutti quelli che n’hanno scritto, porrò fine a questo presente capitolo.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Nella Navigatoria</s> </p> <p type="head"> <s>Cap. 20</s> </p> <p type="main"> <s>Non è dubbio alcuno che chi riguarda bene l’arte del navigare vedrà chiarissimamente l’uso degli angoli per se stesso non le esser necessario; ma risultare dalle cose necessarie a tale arte. </s> <s>Perciochè dalla costitution de’ venti rispetto alla sfera del mondo e dalle linee formate dal movimento de’ Navilij, alle quali corrisponde la carta da navigare e la bossola, stromenti necessarissimi de’ naviganti, scaturisce l’uso degli angoli; poiché considerinsi venti in quanto formano le quarte e ‘n quanto costituiscono i rombi, sempre apparirà chiarissimo l’uso degli angoli; perciochè sì le quarte e sì i rombi non si formano senza angoli, considerinsi rispetto al centro del mondo dove concorreno o rispetto alla circonferenza che è tagliata ad angoli retti da ciascun vento; perciochè ogni vento forma una linea non retta; ma curva e più tosto circolare secondo <pb pagenum="folios 70v-71r"></pb>la quale si muove e questa linea taglia ad angoli retti la circonferenza della sfera della terra e dell’acqua: E benchè i venti nella carta da navigare costituischino le linee rette, con tutto ciò non si considerano né s’adoprano come rette; ma come circolari; onde è che la medesima quantità di miglia e di leghe che nella tondezza della terra e dell’acqua insieme prese si dimostra, apparisce ancho in piano per la carta da navigare, così in terra come in acqua (sì come si ritrahe dall’Arte del navigare di Pietro da Medina, nel cap. 7 del 3° libro) segnandosi con leghe e gradi la lontananza, che habbia qualsivoglia cosa nella tondezza del mondo, senza scemarle parte alcuna e ciò non è inconveniente, perché a un corpo ritondo si può conceder la medesima proportione nel piano, come si dimostra da Tolomeo nel Planisferio e da Giordano nel trattato del medesimo. </s> <s>Ma come questa proportione si dimostri e per qual cagione le misure delle distanze delle cose, così in piano, come in tondo, sieno le medesime non si dee ragionar in questo luogo, bastando solamente dimostrare in che consista l’uso degli angoli nella Navigatoria, e questo esser il tagliamento della circonferenza della sfera, la costitution delle varie portioni di essa, essendo già divisa da’ venti in quarte e ‘n rombi. </s> <s>E si avvertisca che si considerano gli angoli nella Navigatoria non in quanto all’utilità, che a tale arte apportar possino, perciochè in essa non si fa conto alcuno degli angoli; ma in quanto che ci si trovano <lb></lb>//<lb></lb>quelle cose che alla Navigatoria sono di grandissimo giovamento; che non si riguarda quali angoli formino i venti nella carta da navigare, o nella circonferenza della terra e dell’acqua, ma si considera la positura loro, acciò si possa vedere per qual vento sia meglio navigare sicuramente e con prestezza. </s> <s>E questo basti intorno al proposito della materia di questo capitolo.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Nell’agricoltura</s> </p> <p type="head"> <s>Cap. 21</s> </p> <p type="main"> <s>Non è arte alcuna che servendosi in qualche maniera de’ lineamenti o delle superficie in vario modo terminate non si serva ancho degli angoli. </s> <s>Questo si vede apertissimamente non solo per le cose dette ma ancho per la osservatione dell’agricoltura; perciochè, nel porre gli arboli con ordine, come nel far gli oliveti, le selve, le lame, e nel piantar le vigne e gli anguillacci e finalmente ne’ compartimenti degli ordini, tale uso apparisce chiarissimamente. </s> <s>E che altro dimostrano gli ordini degli arboli piantati (come si dice) a Quincunce, de’ quali fece mention Cicerone nel dialogo Della vecchiezza, se non l’uso degli angoli? Quest’ordine così appellato da Cicerone stesso si espone in tal guisa:</s> </p> <p type="main"> <s>Cum autem admiraretur Lysander et processitates arborum et disectos in quincuncem ordines. </s> </p> <p type="main"> <s>Come ancho dicendo:</s> </p> <p type="main"> <s>Quid de pratorum viriditate, aut arborum ordinibus, aut vinearium olivetorium ne specie dicam?</s> </p> <p type="main"> <s>E Quintiliano nel cap. 3° dell’8° libro:</s> </p> <p type="main"> <s>Quid illo quincunce speciosius qui in quacumque partem spectaveris vetus est? <pb pagenum="folios 71v-72r"></pb></s> </p> <p type="main"> <s>Ma Ambrogio Calepino così ‘l dichiara:</s> </p> <p type="main"> <s>Quincumque dicitur, cum ita disposite sunt arbores, ut bine cum tertia proximi ordinis equali intervallo sibi opposita quo’ quo’ te vertis quincuncialem referunt.</s> </p> <p type="main"> <s>Il quale ordine si dimostra con tale essempio sensato qui appresso. </s> <s>Ma per intender meglio questa voce Quincunce si dee avvertire che appresso gli antichi il numero cinque, come in se stesso e come numerante si mostra con questa quinta lettera vocale V, come dice Valerio Probo nel lib. </s> <s>De Notis Romanorum e questo si era ‘l segno dimostrativo di tal numero quinario e considerato come numero numerato, cioè in quanto che è applicato al peso d’alcuna cosa, che pesi cinque once, si dice Quineunx e Quincunce, come si vede appresso Pietro Diacono “De minutijs”, onde e dalla quantità del peso e dalla sua misura, che è ‘l numero cinque riceve ‘l nome Quincunce e si nota col medesimo segno, sì come ‘l segno dimostrativo del numero Denario era la decima lettera X consonante e sì come ‘l cinque V è la metà del dieci X, così ‘l segno Quincunce V è la metà del segno denario X. Oltre acciò si avverta che la figura del Quincunce si forma col mezzo dell’intersegatione di due linee; perciochè, poste alcune piante con uguale intervallo, a due a due, gli si oppone sempre la terza del secondo ordine, come si vede <lb></lb>//<lb></lb>qui appresso, che alle piante AB si oppone la pianta C e alle EC la D e alle CF la G e alle DG la H e mentre da A al G e da B a D si tirano le linee, che si tagliano insieme nel punto C, si forma ‘l Quincunce da ogni banda, come ACB. DCG. DCG. DHG. EAC. EDC. CBF. CGF e però disse Ambrogio Calepino: Quòquò te vertas quincuncialem referunt. </s> <s>E adunque ‘l Quincunce degli antichi un ordine usato nel piantar gli arboli e le viti, nel quale corrispondendo dirittamente le file di essi ed intersegandosi da ogni parte le linee loro, mostrano da ogni banda la figura di questo segno V Quincunce, in quella stessa guisa che si vede in Roma quell’edifitio vulgarmente appellato le “sette sale”, che anticamente era una conserva d’acque, detta “castello dell’acque”. E perché quest’ordine non può formar la detta figura senza costituir angoli, quindi è che possiamo dire che nel formarsi ‘l piantamento a Quincunce, cioè a mandorla, si adoperano gli angoli. </s> <s>Perciochè non è altro l’ordine a quincunce, che ‘l tirar due linee, le quali si taglino in fra loro in quella stessa guisa che si dimostra da Euclide nella quindicesima propositione del primo libro: e nel tirar dette linee si costituiscano quattro angoli che che da lui si appellano “angoli ad verticem”. I quali, secondo che egli dimostra sono fra loro uguali. </s> <s>E che ciò sia vero rimiriamo con diligenza il Quincunce e restaremo accertati esser ‘l tagliamento di due linee rette AB. CD. nel ponto E e delle GH. GB. nel ponto F., e di più vi potremo osservare il Quincunce non esser altro che l’angolo AEC. CFG. DEB. BFH. AED. CFB. CEB. GFH, onde si può concludere che ‘l piantare a Quincunce non sia altro che’l formar gli angoli ne’ tagliamenti delle linee rette in fra loro opposti direttamente e per diametro uguali, cioè AEC. all’angolo DEB. AED <pb pagenum="folios 72v-73r"></pb>a CEB. e così degli altri. </s> <s>Che ‘l Quincunce AEC è l’angolo stesso AEC. così ‘l DEB. Si potrebbe ancho, che nel Quincunce sia l’uso degli angoli esterni, interni e opposti formati dal tagliamento delle parallele sopra le quali cada una linea retta, in quella maniera che si dimostra da Euclide nella ventinovesima del primo, la qual cosa si vede apertamente nell’osservar la figura dell’ordine a Quincunce, la quale si compone con linee parallele, le quali si tagliano fra loro e nel taglio costituiscono i detti angoli come si vede qui appresso e come vediamo nelle ferrate delle finestre o nelle grate delle prigioni o de’ paralatorij delle monache. </s> <s>Si vede oltre acciò l’uso degli angoli ne’ compartimenti de’ campi, fatti per distribuir a ciascuna parte di terreno la sementa ordinata per succession di tempo e per iscambiamento di seme; perciochè mentre si dividano i campi in più parti si risolvano in molte e diverse figure regolari, secondo che comporta tutta la disposition della figura del campo intero. </s> <s>Si può ancho vedere nelle varie positure delle fosse e de’ solchi e degli acquedotti che si fanno per cavar l’acque de’ campi; perciochè ne’ piani il più delle volte si fano ad angoli retti nella estremità di essi: e nelle coste e nelle colline si fanno terminanti nel fine e nel principio del campo ad angoli disuguali, cioè ad angoli ottusi ed acuti per dar la pendina e lo scolamento più facile e più sicuro all’acqua e con meno violenza e con maggior fermezza di terreno. </s> <s>E si dispongano gli acquedotti e le fosse in guisa che formino tal’hora angoli retti, tal’hora ottusi ed acuti, tal’hora più o meno ottusi ed acuti ne’ termini de’ campi secondo la dispositione e secondo ‘l sito e secondo la figura loro. </s> <s>Oltre acciò si dee osservare che acciochè le sponde de’ fossi sieno più stabili e non isgrottino e dilombino e riquoprino ‘l fondo o riempiano le fosse fa di bisogno che le sponde loro non sieno tirate a piombo <lb></lb>//<lb></lb>e per linea perpendicolare e ad angoli retti sopra ‘l piano del fondo; ma per linea inclinante verso le parti del campo e declinante dal vano della fossa, la quale faccia nel fondo l’angolo ottuso, come s’usa ne’ fossi delle fortezze. </s> <s>Io dico l’angolo interno, non l’esterno, il quale è acuto come l’angolo ABC. e BCD. è ottuso e interno, e l’angolo ABE. è acuto ed esterno, il quale si costituisce nella base della sponda, ed essendo acuto le linee che lo comprendono, cioè AB. BE. inclinandosi e ristregnendosi l’una verso l’altra, son cagione che ‘l detto angolo ha proprietà di semare e ritenere ‘l terreno. </s> <s>Senza che pell’inclinamento della linea AB. verso la BE. si fa sì che la base del terreno della sponda è maggiore e perciò più atto a ritenere il peso della terra; perciò ch’è più grande ‘l pesamento di tutta la cosa grave che vi si ferma sopra come si vedrebbe tirando una perpendicolare dal ponto A sopra la linea EC. E come si potrebbe mostrare per l’assioma 15 posto da Cristofan Clavio nel primo libro d’Euclide, ritrahendolo da Proclo. </s> <s>Perciochè, poste uguali le tre grandezze GF. ID. AB. e aggiunte due grandezze disuguali alle GF. e alle ID., cioè EG. alla GF. e CI. alla ID. di modo che EG. sia maggiore e CI. minore e dalla EG. si tagli la GH. uguale alla CI. di modo che EH. sia l’avanzo in cui la grandezza GE. avanza la CI. aggionta alla ID. e perché pel 2° assioma alla GF. e alla ID. supposte uguali, si sono aggionte CI. ed HG. uguali, tutta la CD. sarà uguale a tutta la HF. Adunque, poste le gionte disuguali tutta la EF. avanza tutta la CD. col medesimo avanzo che la gionta EG. avanza la CI. Ma si è detto la AB. esser uguale alla ID. GF. adunque tutta la EF. avanza maggiormente la AB., cioè con l’escesso di tutta la EF. verso tutta la CD. e della gionta EG. verso CI. preso due volte. </s> <s>Adunque la base EF. essendo maggiore della CD. sarà molto maggiore della AB. Ma CD. ed AB. sia ‘l peso, adunque sarà minore del posamento EF. adunque sarà molto minore il peso AB. Oltre acciò qua quale è l’angolo nel fondo della fossa <pb pagenum="folios 73v-74r"></pb>tale dee esser nella sponda del campo come B. ed A. E questa maniera di tagliare ‘l terreno non a piombo ne’ cavamenti delle fosse si dice tagliare a scarpa e a sperone, o vero a barbacane ed è assai in uso nella fortificatione poiché non si potrebbero tenere intere le sponde de’ fossi delle fortezze se no’ si tagliassero a scarpa, né ‘bastioni, né ‘terrapieni potrebbero già mai mantenersi in piedi se non fussero fatti a scarpa. </s> <s>Ma per tornare al compartimento de’ campi non si dee tralassar che in essi si comprendano i compartimenti degli horti e de’ giardini i quali si fanno per commodità di seminar diverse specie di semi e di piantar varie ragion di piante e d’arboli e ordinare la distribution delle parti di qualunque superficie di terreno con misura, con ordine, con accozzamento di più varie figure angolari, per haver in uno stesso tempo congionta con la commodità, la vaghezza e l’ornamento che risulta dalla varietà delle figure e dalla distintione e diversità dell’herbe e de’ fiori e dal numero dell’accoppiamento, dalla corrispondenza e dalla proportione e dal collegamento del tutto insieme con le parti e delle parti in fra loro. </s> <s>Serve oltre acciò l’uso degli angoli nell’accommodar le steccate, dette altramente scotatoie, ne’ letti de’ fiumi per riparar i campi dal corso dell’acque tenendole lontane; perciochè debbono collocarsi in maniera appresso le rive de’ fiumi, che con la linea retta loro facciano due angoli, cioè l’esterno ottuso e l’interno acuto; avvertendo sempre di opporre al corso dell’acque l’angolo ottuso, il quale essendo contenuto da una linea inclinante verso la parte opposta al corso del fiume e declinante dalla parte onde viene ‘l corso, ed essendo l’angolo ottuso per sua natura non atto a ritenere per la sua larghezza ‘l cagion dello sfuggimento dell’acque; che se li si opponesse l’angolo acuto non <lb></lb>//<lb></lb>si cagionarebbe lo sfuggimento, ansi perché per sua natura doverebbe dar ricetto all’acque ristregnendosi dentro le sue linee con poco intervallo non essendo impedito dalla debilezza della materia, non fa resistenza alcuna all’impeto dell’acque e pertanto non le ritiene ma tosto cede alla forza loro lassandosi disfare in tutto. </s> <s>Che la linea inclinata la quale forma l’angolo acuto rende il riparo indabile al resistere al corso ed all’impeto de’ fiumi. </s> <s>Nella maniera e per la ragion medesima si fanno i pilastri de’ ponti dalla parte che è contro al corso del fiume, che gli si fanno alcuni contra forti, o denti i quali co’ pilastri fanno angoli ottusi per cagionar lo sfuggimento dell’acqua da ogni banda, ma vi si forma ancho un angolo acuto in faccia il quale stando incontra al corso dell’acque divide la forza loro, dividendo ancho l’onde. </s> <s>Dalla qual cosa si rende sicura e stabile qualunque fabbrica di ponti fatta nell’acque de’ fiumi. </s> <s>Il che si renderà più chiaro con l’esempio sensato //</s> </p> <p type="main"> <s></s> </p> </chap> <pb pagenum="folios 74v-75r"></pb> <chap> <p type="head"> <s>L’uso degli angoli nel disegno</s> </p> <p type="head"> <s>Cap. 22</s> </p> <p type="main"> <s>Trovandosi l’uso degli angoli principalmente nella Geometria, sì come abastanza da Euclide in diversi luoghi del suo volume degli “Elementi” e come in parte è stato accennato da noi nel cap. 21 di questo lib. ed essendo la scienza pratica del disegno subalternata alla Geometria; perciò che si serve di ponti, di linee, di superficie e di corpi, come sensibili e materiali e tutte cose che si dichiarano e si provano nella Geometria, come in scienza superiore; in essa anchora apparirà manifestamente l’uso degli angoli. </s> <s>Che ciò sia vero rimirinsi le posture, le movenze e varij aspetti delle figure di ciascuno animale, l’elevaioni delle piante sopra ‘l piano e delle cose inanimate, e vedremo chiarissimamente l’uso degli angoli. </s> <s>Perciò che la figura dell’huomo posta in piedi sopra ‘l piano forma la perpendicolare, la quale, cadendo su piano forma angoli retti, e secondo che si muove varia l’angolo, ed in se stessa piegandosi moltiplica gli angoli. </s> <s>Onde segue che per disegnar tal figura nel piano sia di bisogno formar solamente prima una perpendicolare e sopra essa poi disegnar la figura od almeno primieramente immaginarvela, la qual linea necessariamente farà gli angoli retti, sì come è manifesto per la decima def. del primo di Euclide. </s> <s>Ma avanti che si venga all’intera dichiaratione di quest’uso è necessario mostrar tutte le specie degli angoli che appartengono alla scienza, ed alla pratica del Disegno. </s> <s>Tre sono le ragioni degli angoli, delle quali si serve il disegno, ciò sono rettilinei, curvilinei, misti. </s> </p> <p type="main"> <s>I rettilinei non si adoperano se non con le linee inter <lb></lb>//<lb></lb>ne le quali son rette. </s> <s>E questi sono o retti o ottusi od acuti, i quali si veggano secondo la varia positione e secondo la diversa movenza della figura.</s> </p> <p type="main"> <s>I curvilinei si adoperano in fra le parti e fra ‘congiognimenti delle membra e de’ muscoli delle figure. </s> <s>E questi o sono nelle parti concave al più o si trovano nelle convesse. </s> </p> <p type="main"> <s>Finalmente gli angoli misti saranno que’ ch’im parte son formati da linee rette e in parte da linee curve, e si trovano altresì ne’ luoghi concavi e ne’ convessi delle membra.</s> </p> <p type="main"> <s>Dopo la consideration degli angoli è necessario considerar tre stati secondo quali si possan trovar le figure degli animali e specialmente dell’huomo: o come ferma e quieta, overo come movente se stessa: come ferma e quieta, sarà o posta in ginocchioni o ‘n piedi o sedente: come movente se stessa, cioè o correndo o saltando o camminando o muovendo braccia, mani, o piedi.</s> </p> <p type="main"> <s>Si è detto avanti doversi stabilire una linea retta interna e occulta che cada perpendicolarmente sopra la linea del piano e ‘n torno ad essa disegnarsi qualunque figura in piedi. </s> <s>Col mezzo di questa linea la figura posta in piedi farà due angoli retti sopra ‘l piano; né senza quest’uso di tali angoli si può regolarmente disegnar una figura, non possendosi disegnar senza la linea perpendicolare che si parta dal mezzo della fontanella della gola e passando pel mezzo del collo del piede ferisca ‘l piano ad angoli retti. </s> <s>Come si vede nella presente figura, dove la linea AB. è ‘l piano e la CD. <pb pagenum="folios 75v-76r"></pb>è la perpendicolare che forma due angoli retti da ogni banda, cioè CDB. e ADC. disegnando la figura in faccia come ancho in profilo. </s> <s>Quando Aristotile nella trentesima quistione delle Meccaniche rendendo la ragione perché mentre l’huomo postosi a sedere e levandosi in piedi faccia con la gamba e con la coscia e col petto l’angolo acuto dice che tutto quel che è uguale è cagion di quiete in qualunque luogo e l’angolo retto è angolo d’egualità, pertanto esser cagione dello star fermo in piedi. </s> <s>E quando dice angolo retto non intende l’angolo retto formato sopra ‘l pavimento, benchè anchora così lo potesse intendere, ma quello che si forma dalla linea perpendicolare sopra la terra; perciochè considerà l’huomo come corpo grave il quale con la sua linea perpendicolare si muova verso la superficie convessa e sferica della terra ad angoli simili, i quali sono angoli retti sferali, non che si muova ad angoli retti verso ‘l pavimento; perciochè considera la principale intensione delle cose gravi, che ricevono la qiuete loro nel centro, o vicino al centro; che poi l’huomo interpostosili lo impedimento del piano non possa andar a posarsi sopra la circonferenza della terra; per accidente si ferma sopra esso ad angoli retti, e seguendo questo come seconda intensione nel qual piano è necessario che stia fermo ad angoli retti; perciochè qualunque l’huomo non sia nella circonferenza o nel centro della terra, con tutto ciò la sua perpendicolare è collo <lb></lb>//<lb></lb>cata ad angoli retti , come si può vedere in questa figura , la dove KBL è la stessa N. il centro CD ‘l pavimento nella terra EF. il pavimento sopra terra: e così GE: AB. la perpendicolare dell’huomo posto in piedi; e così AM. ed HI. le quali perpendicolari formano angoli retti nelle linee de’ piani loro, e questi son tutti uguali; perciochè ovunque sieno son sempre uguali, come è manifesto per la decima definitione del primo d’Euclide e pel 12° assioma. </s> <s>Ed è natura degli angoli retti esser sempre uguali. </s> <s>Ma nella circonferenza della terra formano angoli simili a’ retti sì come dice Aristotile; ma in vero si appellano retti sferali, e si dice che le linee vi cadono sopra ad angoli retti sferici. </s> <s>Quando adunque ‘l Filosofo dà principio allo scioglimento della questione, accenna due principij in fra loro contrarij, ciò sono:</s> </p> <p type="main"> <s>L’ugualità degli angoli retti in qualsivoglia cosa è cagion di quiete.</s> </p> <p type="main"> <s>La mutation degli angoli retti negli acuti e negli ottusi, ed in più o meno acuti e ottusi è cagion di movimento.</s> </p> <p type="main"> <s>Adunque sì come gli angoli retti son cagione di riposo, il quale si fa o stando in piedi (sì come si è accennato) o stando a sedere o giacendo, così l’uso di essi può esser cagione di disegnare o dipegnere o scolpire le figure ritte, giacenti e sedenti. </s> <s>Ma avanti che si dimostri tale uso, la regola di far le figure sedenti, bisogna osservar ciò che dice Arist. nella medesima questione. </s> <s>Egli facendo comparatione dello stare in piedi con lo stare a sedere dice che standosi in piedi la testa e ’l piede son collocati <pb pagenum="folios 76v-77r"></pb>nella medesima linea perpendicolare. </s> <s>Come la linea AC posta nel piano AB in cui C. rappresenti la testa e l’A il piede. </s> <s>Ma sedendo il capo non è nella medesima linea che ‘l piede, ma il piede e ‘l capo sono nelle linee parallele, come essendo DE. ed FG. perpendicolari nel piano AB. in fra loro parallele, la testa dell’huomo sarà nel punto D. della DE. e ‘l piede nel punto G. della GF., di modo che sì come gli angoli CAE. e AH fanno star fermo in piedi l’huomo, son cagione che si disegni in tal positura, così gli angoli DEG. DLF. il fanno posarsi a sedere e danno la regola per disegnarlo regolatamente e con facilità in tal positura. </s> <s>Proposta la linea del piano AB. perciochè si dee sempre proporre ‘l piano avanti che si facciano le figure, o sieno sole, o molte insieme unite, come accade ne’ componimenti delle historie, si dee tirare una perpendicolare d’altezza uguale all’altezza che si mostra dall’huomo sedente e sia la linea CD. cha faccia angoli retti sopra ‘l piano, di poi presa la misura della coscia, cioè dal ginocchio fine all’estremità della natica, che si fa replicando la misura della testa tante volte quante formano la sua lunghezza, sì come è costume de’ pittori e degli scultori: e si tagli la linea del piano secondo la misura detta; per esempio nel ponto F. Quindi nel medesimo modo presa la misura della gamba e si determini nel ponto E. e da esso si tiri la perpendicolare EF. parallela alla CD. e per la terza del primo d’Euclide, della CD, si tagli una linea uguale alla EF., cioè nel ponto B. di modo che GD. sia uguale alla EF. e fianalme si congionga EG. di maniera che <lb></lb>//<lb></lb>vi sieno due angoli retti, detti angoli della quiete, ciò sono CGE., EFB. Perciochè l’huomo sedendo fa due piegature, una concava nell’anca, il quale si rappresenta dall’angolo retto CGE. e l’altra convessa nel ginocchio, la quale si rappresenta per l’angolo GEF. Hora per far la figura sopra le linee CG. GE. EF. è necessario tirare una linea dal ponto G. al ponto E, di modo che si formi un triangolo rettangolo CGE. Si divide poi il lato GE in due parti uguali nel ponto .I. si tiri una linea dal ponto C. che termini nella GE. in detto ponto. </s> <s>Finalmente, sopra questi lineamenti si disegni la figura ponendo ‘l ponto C. nel mezzo della testa e ‘l ponto .I. nel collegamento della coscia e del corpo e ‘l ponto E. nel ginocchio e ‘l ponto F. nel piede, cosa apparisce nella figura. </s> <s>Se adunque l’angolo retto è cagion di quiete e ci dà commodità di poter disegnar le figure nella lor quiete, per opposito segue, che l’angolo non retto sia cagion di movimento e dia la regola di formar le figure in atto di mostrar qualche movenza; adunque, mutandosi positura, si muta angoli e scambiandosi angoli si muta positura. </s> <s>Perciò che stando l’huomo ritto forma gli angoli retti nel pian della terra, così ancho stando a sedere forma angoli retti nel piano e nella sedia; ma quando si leva da sedere muta angoli; perciochè ‘l rizzarsi è una maniera di movimento.Ma per intender meglio questo movimento bisogna osservare, che se standosi ritto ‘l capo e ‘l piede sono nella medesima linea, e sedendosi sono <pb pagenum="folios 77v-78r"></pb>nelle linee parallele ed in amendue le maniere si formano angoli retti nel piano della terra: e levandosi da seder bisogna che si muti l’angolo retto in acuto; perciochè colui che siede per levarsi da sedere dee accommodarsi in maniera, che’piedi venghino nella medesima linea perpendicolare, né può fagli tornar nella medesima linea, se prima non accorda i piedi in modo che venghino a dirittura del capo, ritirandoli indietro e se non spegne ‘l capo e ‘l petto avanti; e così di due angoli retti si fanno due acuti nel piegamento concavo e nel convesso; e fatti tali angoli tosto l’huomo si leva in piedi. </s> <s>Di modo che quelle linee moltiplicate nel piegamento si fanno una linea sola nel levarsi da sedere e come nel posarsi a sedere una sola linea divien più linee, sì come insegna Arist. nel primo cap. del lib. </s> <s>Del movimento degli animali dicendo in questo tenore:</s> </p> <p type="main"> <s>E’ impossibile, che si muova alcuno animale, non havendo in sé alcuna parte immobile.</s> </p> <p type="main"> <s>Perciochè nelle Meccaniche insegna che ciò che si muove sempre si dee muovere sopra una cosa immobile. </s> <s>Onde avviene che gli animali habbiano i piegamenti che gli servono in luogo di centro e tutta la parte dov’è ‘l piegamento è ‘n potenza e in alto è una e due: è retta e piegata: e quella che si muta è in potenza avanti ‘l piegamento. </s> <s>E quando alcuno animale si muove e si piega, una parte si muove ed un’altra rimane immobile, come si può vedere con l’esperienza ed in questo essempio.La parte B. che è nel centro è immobile, la DB. si muove verso BE. e la AB. verso BC. e di nuovo la EB. torna in DB. tanto che vien per dritto BC. Onde <lb></lb>//<lb></lb>di più linee se ne fa una sola in alto dove prima era in potenza e di più piegate si forma una diritta: così avverrebbe se fusse la figura piegata verso G. perciochè dirizzandosi BG. si muova verso BD. e se ne farebbe una stessa linea. </s> <s>Di qui possiamo ritrarre che piegandosi l’huomo od altro animale con la parte che si muove e con l’altra che rimane immobile si formano due linee che fanno nel piegamento là dove è ‘l contanto commune, l’angolo retto, l’ottuso o l’acuto, secondo la maniera del piegamento. </s> <s>Né si può far piegatura alcuna senza l’angolo. </s> <s>E questo è fatto conforme alla lieva , dove fa di mestiero porre una parte immobile per sotto lieva. </s> <s>Quindi adunque si ritrahe che formandosi gli angoli che fanno ‘l piegamento proprio della movenza del levarsi in piedi facilmente si faranno le figure in atto di levarsi da sedere, in quella guisa che insegna Arist. nelle “Meccaniche”, la qual cosa dimostraremo con la figura qui appresso.</s> </p> <p type="main"> <s>Volendo adunque tal figura levarsi da sedere, bisogna che in uno stesso tempo si muova la linea AB. e venga in FB. e la CD. e venga in CE. in modo che vi resti una linea immobile e sia CB. intorno la quale sia ‘l movimento perciochè muovasi la linea AB. sopra ‘l centro immobile B. si muove secondo la portione del cerchio AF. muovasi la CD. sopra ‘l centro immobile E. si <pb pagenum="folio 78v - attached piece 79r"></pb>muove secondo la portion del cerchio DE. di modo che come la AB. vien nella FB. così la CD. nella CE. e come B. è centro immobile, così C. è centro immobile; adunque tutta la linea che si è tirata da un centro all’altro, cioè CB. è immobile, adunque come intorno a’ centri così intorno a tal linea si fanno questi movimenti. </s> <s>Ma perché i detti centri son posti in ugual distanza, perciò le linee FB. e CD. son parallele; oltre che son parallele anchora perché fanno gli angoli alterni FBC. e BCE. uguali, sì come è manifesto per la ventinovesima del primo d’Euclide. </s> <s>E perché qui son due triangoli ABF. CGE i quali hanno le basi uguali e perciò ancho i lati uguali, cioè ciascuno a ciascuno, come sarebbe chiaro convertendo la quarta del .po. sì come si converse dal Clavio sopra lo scholio dell’ottava del .po.. Adunque AB. sarà uguale a GE. ed FB. a CE. ma CE. è uguale ad EG. adunque CE. sarà uguale ad AB. ed AB. e GE. sono una medesima linea AE. che GB. è una parte di mezzo che le congiogne dirittamente. </s> <s>Adunque il termine della CE. verrà al diritto sotto ‘l ponto A nella medesima linea. </s> <s>Ma habbiamo detto nel ponto A. esser la testa e nel ponto E. ‘l piede; adunque ‘l piede verrà sotto ‘l capo nella medesima linea muovendosi, mossa la CD. in CE. e BE. ritornando in BA. Adunque all’hora si levarà in piedi, quando’l ponto D verrà nel ponto E. e ‘l ponto F. tornarà nel ponto A. cominciando ‘l movimento della figura dal movimento innanzi di AB. verso FB. al quale succede ‘l ritiramento indietro di CD. verso BE. e questo è movimento naturale; perciochè naturalmente sempre due membra <lb></lb>// [Attached piece (folio 79) not transcribed.] <lb></lb><pb pagenum="attached drawing 79v - folio 80r"></pb>si muovano in parti contrarie, mentre si muove ‘l tutto, come si vede nel caminare, che spegnendosi avanti la gamba destra, si tira adietro la spalla e ‘l braccio sinistro: e spegnendosi avanti la gamba sinistra si muove indietro la spalla e ‘l braccio destro: e spegnendosi avanti la spalla diritta, si manda indietro la gamba manca: e sporgendo innansi la spalla manca si tira indietro la gamba dritta, come si può veder ancho nel caminar degli animali di quattro piedi e si può ritrar da Arist. nel lib. </s> <s>Del caminar degli animali, che negli huomini le spalle e le braccia e le gambe fanno quello che le gambe davanti, con quelle di dietro degli animali di quattro piedi. </s> <s>La forza di questo movimento non è altro che la forza della lieva; perciochè la linea FB. è ‘l movente e la GE. e CB. la lieva, e la DE. il sottolieva; che mentre FB. si muove innanzi e CE. indietro premendo ‘l piano DE. CB. si solleva e inalza tutto ‘l peso. </s> <s>Tutti questi piegamenti di linee e tutti questi angoli che son cagion di questa movenza ci danno la regola con la quale agevolmente possiamo formare una figura humana in atto di levarsi da sedere, sì come si vede chiaro osservando la figura e tirando i suoi contorni intorno alle linee che fanno gli angoli acuti FBC. BCE. Ma perché havemo anchor accennato che giacendosi si formano gli angoli retti; perciochè in tal positura si sta in riposo e ‘n quiete, perciò si dee mostrar come la figura humana giacente faccia angoli retti sopra la superficie della terra. </s> <s>E’ cosa chiara che tal figura dimostra un corpo che ha tutte le sue misure in fra le quali la misura della grossezza forma la linea perpendicolare in più parti, la quale cade nel <pb pagenum="folios 80v-81r"></pb>piano ad angoli retti, come per essempio nel piano AB. giacendo la figura D. con la testa con l’anche, con le ginocchia e co’ piedi fa tante linee perpendicolari, le quali stanno ad angoli retti sopra ‘l piano AB. e se si havesse a muovere bisognarebbe che la linea FH venisse nella perpendicolare GH. che così sarebbe sedente nel piano; e volendo rizzarsi bisognarebbe che la GH. si muovesse verso IL. e la IL. verso la MN. e la MN. e la QR. verso OP. mutando sempre gli angoli retti in acuti, sì come già s’è detto e secondo i lineamenti del giacere e del muoversi a sedere e del rizzarsi; e secondo la forza degli angoli contenuti sotto tante linee fra loro variamente piegate si potrà con facilità disegnar le figure in tutte le muovenze e ‘n tutte le positure, sempre procedendo dalla figura della quiete a quella del movimento, da quella del movimento a quella della quieta e quindi un’altra volta a quella della quiete, cioè dal giacere al rizzarsi a sedere; dal rizzarsi al sedere, dal sedere al levarsi in piedi, dal levarsi in piedi allo star ritto. </s> <s>Ma avanti che poniamo fine a questo discorso, porremo alcune figure di movenze insieme co’ lineamenti e con le regole e colli angoli ne’ quali consistono le diversità delle movenze, dalle quali agevolmente si potrà imparare a mettere insieme le figure.</s> </p> <p type="main"> <s>La muovenza di questa figura si fa col mezzo di due triangoli almeno uno acut’angolo, cioè ACE <lb></lb>//<lb></lb>e l’altro ottus’angolo, cioè ECD, aggiontivi due altri triangoli, cioè uno scaleno DFE. ed uno equicure GDF. per l’accomodamento delle braccia. </s> <s>Dove nel muover il passo AE. si fa la base del triangolo acutangolo e nello spegnere ‘n fuore la testa col petto si forma ‘l triangolo ottusangolo la cui base è la perpendicolare ED. perciochè acciochè si faccia che la figura non sia cadente, bisogna che ‘l capo sia appiombo sopra ‘l piede.</s> </p> <p type="main"> <s>Questa figura forma una movenza che si può disegnare col mezzo di tre triangoli, due rettangoli AFD. FCG. ed uno ottuso angolo DEF. Il passo AD. è la base del triangolo ADF. la base dell’ottusangolo è FD. nella quale si regge l’angolo del piegamento E. la qual base è per dritto del lato FC. del triangolo FGC. nel cui angolo retto è ‘l sito della testa e nell’angolo acuto appresso alla base, cioè cioè nel CGF. el termine dello stendimento del braccio.<pb pagenum="folios 81v-82r and attached piece"></pb></s> </p> <p type="main"> <s>Questa movenza non solamente si serve di triangoli rettangoli, acutangoli, ottusangoli, ancho d’un parallelogrammo. </s> <s>Perciò che vi sono sei rettangoli, ciò sono FCD. DEF. FCH. DHG. DHI. CHI. vi è uno acutangolo CID. uno ottusangolo CFH. e due acutangoli, uno equilatero CDI. e uno equicruro NPE ed un parallelogrammo CDEF. Perciochè la distanza dal luogodel ginocchio D. alla mano E. si fa dalla longhezza del parallelogrammo, così la distanza dell’anca al collo o al mento od alla spalla o l’altezza che dee porsi fra’l ginocchio al termine dell’anca e così fra la palma della mano all’ultimo della spalla si costituisce dall’altezza del parallelogrammo: il luogo della testa si pone fra due angoli del triangolo CFH. Il quale si forma tirata la linea L sopra gli angoli DF. ed allongata fine ad N quanto è la meta della CD. Il luogo dell’anche e delle natiche sarà nell’angolo C. lo ingrossamento del corpo sarà sopra la base del triangolo HCF. e le spalle e reni G sopra ‘l lato CN del triangolo CNF. e divisa la linea FD in due parti uguali nel ponto G. si tiri da esso, sopra ‘l ponto H una linea tanto longa che la G. vada al ponto I. Di poi tirata dal ponto D. una linea retta che passi pel ponto L. infine al ponto K. il quale sia parallelo col ponto C. di modo che da I. a K. sia una linea uguale alla DC. e tirata una linea dal .L. al C. che formi un triangolo rettangolo nella linea CI. che sia la <lb></lb>//<lb></lb>metà d’uno triangolo equilatero CDI. Nella qual linea sarà la coscia manca e nel punto I il ginocchio e nella linea KI la gamba e ‘l piede e nella rimanente DL uguale alla IK. la gamba e ‘l piè destro: e nella CD la coscia dritta. </s> <s>Di modo che CDL. fanno ‘l piegamento destro della gamba con la coscia: e CIK il piegamento sinistro: ed FCD ‘l piegamento del corpo con le cosce e con le gambe e lati del triangolo EUP. con la base EU allongata verso M. danno i luoghi alle braccia.</s> </p> <p type="main"> <s>Questa figura benchè sopra ‘l piano faccia gli angoli retti e di quiete, con tutto ciò non è rappresentativa di quiete ma d’atto di riverenza e d’adoratione; perciochè gli stati di quiete non sono più di tre giacendo, sedendo e stando ritto. </s> <s>Si disegna sopra la superficie del piano CD, tirata la perpendicolare AB e la parallela GD. e la FG. ed allontanandosi dal ponto B. quanto è naturalmente longa una gamba si formi un triangolo rettangolo nella CD. facendo la base EB. el maggior lato opposto al maggiore angolo B. e ‘l lato mezzano la perpendicolare FB. che è parte della AB. e trovata la longhezza dal ginocchio all’estremo della natica dal ponto F. si tiri una linea di tal longhezza cioè FG. e dalla CD. si tagli una parte uguale cioè BD. Finalmente dal ponto G si tiri una perpendicolare al ponto D. formando un parallelogrammo di modo che nella perpendicolare AB. si ponga la testa, l’anca o ‘l ginocchio: nelle linee che fanno l’angolo retto FGD opposto all’FBD si accomodi la coscia e la gamba manca: e ne’ due lati del triangolo FB. BE. si disegni la coscia, e la gamba dritta.</s> </p> <p type="main"> <s>Alla precedente figura appartiene un’altra, con la quale si mostra quella movenza che si forma nel porsi <pb pagenum="folios 82v-83r"></pb>in ginocchioni pertanto la possono qui appresso insieme co’ lineamenti suoi, acciò si veda la regola per disegnarla e l’uso degli angoli.</s> </p> <p type="main"> <s>La figura di questa movenza dimostra l’attitudine di porsi in ginocchioni, e si fa posta la linea del piano AB. e sopra essa tirata la perpendicolare CD. da cui allontanandosi quanto è l’altezza della sesta si faccia ‘l segno F. sopra cui si tiri un’altra perpendicolare EF. e divisa la perpendicolare EF. in sette parti uguali, presene quattro si formi ‘l parallelogrammo DFHI. il quale si divida per mezzo con la linea infinita LN. quindi presa la longhezza DF. e replicata nella linea AB., cioè nel ponto G. dal quale si tiri una linea che passi sopra ‘l ponto L. e termini nel ponto H. e preso ‘l compasso con esso si pigli la longhezza GD. e si trasporti in GO. tirando l’arcuatione: e presa la longhezza LG., fatto centro L. si faccia un’altra arcuatione che tagli la prima nel segno O. dal quale si tirino le linee OL. OG. onde si formi un triangolo acutangolo la cui base divisa in tre parti uguali dal taglio della terza, cioè dal P. si tiri la linea PL. finalmente sopra la linea IH. si costituisca un triangolo rettangolo tirata la linea IE. e presa una delle .7. portioni con essa si tagli la linea MN. nel ponto N. dal quale a ponti FH. si tirino due linee rette, le quali chiudino un triango <lb></lb>//<lb></lb>lo ottusangolo HNF. la cui base nella perpendicolare EF. ed è FH. La retta si disegnarà nell’angolo E. facendo che l’estremo della perpendicolare sia nell’orecchio, la spalla, il corpo e parte della natica e dell’anca si farà dentro ‘l triangolo IEH. la coscia e la gamba dritta nella linea nelle linee HL. LP. e nelle linee HN. NF. la coscia e la gamba manca. </s> <s>Il calcagno del piè destro sarà nel ponto P. e la ponta del piede nel ponto A. e ‘l ginocchio nel ponto L.: il principio della coscia sinistra sarà nel ponto H. il ginocchio nel ponto N., la pianta del piede nel ponto F. Il braccio cominciarà dal terzo della linea EI. presso al ponto E. Nella qual figura si vede che tutti i piegamenti delle parti si fanno negli angoli, i quali parte sono ottusi, e parte acuti. </s> <s>Ma perché con l’essempio di queste poche movenze ciascun diligente disegnatore , o pittore ,o scultore potrà facilissimamente disegnar qualunque figura di qualsivoglia animale secondo qualunque maniera di movenza, di gesto e di positura; perciò non pigliarò cura di porvi davanti altri essempi. </s> <s>Ma solamente aggiognerò questa consideratione degna d’esser da ciascuno osservata che a tutte queste regole si dee anteppor la cognition della misura dell’altezza della figura humana e delle sue membra, acciò che con essa possiamo determinare l’altezza delle perpendicolari le grossezze e longhezze de’ corpi e delle braccia, delle cosce e delle gambe e de’ piedi. </s> <s>Pertanto acciò che’l nostro discorso non sia manchevole, ed acciò altri non si habbia a dolere di noi, quando non insegnassemo tal misura, ne faremo un racconto nell’ultimo di questo capitolo.</s> </p> <p type="main"> <s>La statura della figura humana, come si ritrahe da Vitruvio, nel cap. primo del 3° libro, può ricever tre maniere di misure secondo tre ragioni di misurare. </s> <s>Perciò che <pb pagenum="folios 83v-84r"></pb>la prima si prende dalla testa, cominciandosi dalla estremità del mento e terminando nelle radici dei capelli; la seconda si piglia dal mento alla sommità della testa; la terza dalla sommità del petto alle barbe de’ capelli.</s> </p> <p type="main"> <s>Se si prende nel terzo modo, si dividarà tutta l’altezza della figura humana in sei parti uguali e ciascuna di esse sarà un piede antico formato con quattro palmi, ciascuno de’ quali è quattro dita, il quale è alquanto maggiore del nostro mezzo braccio.</s> </p> <p type="main"> <s>Se nel secondo compartiremo l’intera altezza della statura dell’huomo in otto parti, che saranno otto teste, e ciascuna testa corrisponderà a nove once del nostro braccio e tre quarti d’oncia, od al più ad un terzo.</s> </p> <p type="main"> <s>Se nel primo partiremo tutta l’altezza sua in dieci parti uguali, cioè in dieci teste, e ciascuna si dimostra dallo spiegamento della mano, cominciando dall’attaccatura, e terminando nella estremità del dito di mezzo: e corrisponde a sette once del nostro braccio e un quarto d’oncia, ed al più ad un terzo.</s> </p> <p type="main"> <s>Ma si avverta oltre acciò, che secondo altri si può divider tutta l’altezza dell’huomo in quattro parti uguali, cioè in quattro cubiti, di modo che ella arrivi all’altezza di dodici piedi, la quale, al modo nostro, si riduce a tre braccia o poco più.</s> </p> <p type="main"> <s>Pomponio Gaurico, nel libro Della scoltura dell’huomo e Girolamo Cardano, De subtilitate rerum, dividono l’altezza dell’huomo in nove teste, cioè presa l’altezza della testa dalla estremità del mento fine ad un capello della cima del capo: e ciò osservano negli huomini perfetti; che l’altezza de’ fanciulli è solamente quattro facce. </s> <s>Vitruvio nello stesso luogo citato dice; il petto esser la quarta parte di tutto ‘l corpo humano; ma non esplica in che maniera si faccia tal misura, cioè donde prenda principio e dove termini. </s> <s>Solamente egli dice, quasi <lb></lb>//<lb></lb>per dichiaration di questo che ‘l cubito sia la quarta parte, ma non applica tal misura alla larghezza né alla longhezza del petto. </s> <s>Ma io tengo per certo il petto cominciar dal fine delle costole e terminar nella fontanella della gola, là dove concorrono le clavicole; perciochè tanto si estende il petto, quanto spatio si contiene sopra ‘l diaframma e fia l’intervallo dal termine delle costole alla fontanella della gola è la quarta parte dell’altezza dell’huomo. </s> <s>Benchè Guglielmo Filandro non creda che ‘l petto sia la quarta parte; perciochè dice non esser la quarta parte, ma un poco meno della quinta. </s> <s>Ma per salvare l’oppinion di Vitruvio soggiogne che quando egli dice il cubito esser la quarta parte la prende non secondo ‘l costume quasi di tutti gli autori, dal congiognimento del braccio, cioè dal gombito fine al corpo o bracciale, cioè dal congiognimento del braccio con la mano, che altramente si dice collo della mano; ma fine all’estremo del dito di mezzo. </s> <s>Vi sono altre maniere di misure, le quali si esplicano molto bene dal Filandro nell’ Annotatione sopra Vitruvio, nel cap. primo del 3° libro. </s> <s>Solamente aggiognerò per beneficio delle regole proposte doversi haver cognitione delle misure delle braccia, delle cosce e delle gambe per poter meglio ed a misura proportionatamente dispor i lineamenti co’ quali si compongano e si mettano insieme le parti delle figure.</s> </p> <p type="main"> <s>Il braccio cominciando dalla sua congiuntura infine al collo della mano è due teste, presa la misura nel primo modo e compresavi la mano secondo ‘l suo maggiore stendimento sarà tre teste. </s> <s>La coscia parimente è due teste, cioè dall’attaccamento, nell’ anguinaia infine al congiognimento del ginocchio che è nel mezzo di esso. </s> <s>La gamba cominciando da questo collegamento, e seguendo infine al mezzo delle <pb pagenum="folios 84v-85r"></pb>cappolle o talloni;altresì è due teste, ma compresa tutta la longhezza del piede è tre teste e considerata l’altezza di tutto ‘l piede, cioè dalla pianta al mezzo delle cappolle è due teste e mezzo. </s> <s>Ma perché l’huomo, come perfettissimo di tutti gli animali, è misura di tutti, sì come si ritrahe da Vitruvio e come si confermarebbe da’ Filosofi, che vogliono che in ogni università cose, se ne ritrovi una, la quale essendo perfettissima sia misura di tutte l’altre, pertanto non è maraviglia che dalle regole della sua figura, che consistono nell’uso degli angoli, si possino ritrarre le regole da formare le figure degli altri animali, servendosi altresì degli angoli, come nel disegnar le figure humane secondo varie positure, e secondo diverse movenze. </s> <s>Le quali, da qualunque giuditiosissimo disegnatore osservase potranno esser cagione del ritrovamento de’ lineamenti e degli angoli, sopra quali, con agevolezza e con giustissima regola ciascuno potrà disegnar qualsivoglia figura d’animale. </s> <s>Onde non prenderemo cura di mostarne essempio alcuno, rimettendoci in tutto alla diligenza di quantunque ingegnoso ed eccellente disegnatore.</s> </p> <p type="main"> <s></s> </p> </chap> <chap> <p type="head"> <s>Dell’uso degli angoli nell’arti fabrili</s> </p> <p type="head"> <s>Cap. 23</s> </p> <p type="main"> <s>Perciochè le cose che nell’uso humano sono più frequenti, sono molto meno osservate e conosciute; onde bene spesso si vede che l’huomo non osserva né avvertisce le cose che seco accompagnate sono e sue proprie e che non solamente nel corpo ma nell’animo anchora si truovano per la qual cosa ne segue una ignoranza delle cose a lui più fami<lb></lb>//<lb></lb>liari; pertanto la ragione, la natura, e la cagion della forza e proprietà di molti stromenti communissimamente adoprati nell’arti fabrili è al tutto ignota, non per altra cagione se non perché nella frequenza dell’uso si nasconde il difetto dell’osservanza e della consideratione della proprietà e delle forze loro e quindi sorge l’ignoranza dell’origine delle loro operationi e della dipendenza e dell’utilità che prendono dalle Matematiche e specialmente dalla Geometria. </s> <s>Per levare adunque la detta ignoranza, formeremo alcune considerationi sopra alquanti stromenti dell’arti fabrili e ne trarremo la cognition dell’utilità, che ricevono dalla Geometria, cioè dall’uso degli angoli. </s> <s>Perciochè gli stromenti fabrili, o tagliano o forano o segano o incidono o radono o limano o pianano o puliscono o rompono o premono o spengano o cacciano o scagliano o scalzano ne possono mandare ad effetto dette operationi se non col mezzo degli angoli acuti, o retti, o misti. </s> <s>Oltre acciò gli angoli servono ancho per collegamento e così serve l’angolo retto. </s> <s>I quali stromenti apparte si consideraranno, riguardando minutamente qualunque sia di loro. </s> <s>E quindi vedremo di quanta utilità sia la Geometria verso l’arti fabrili: che chi negasse tale utilità levarebbe uno degli usi principali delle matematiche. </s> <s>Ma avanti che esaminiamo tutti gli stromenti dell’arti fabrili bisogna notare che mentre ragioniamo degli angoli si debba intendere degli angoli solidi, non de’ piani perciochè tali stromenti son corpi solidi ma irregolari. </s> <s>Gli stromenti che servono per tagliare o si adattano a tagliar la pietra o ‘l legno; que’ che servono a tagliare il legno son questi: l’accetta, l’ascia, lo scarpello, la pialla e ‘l coltello. </s> <s>Il coltello o con due manichi che si chiama coltello a petto o con un manico, che oltre al tagliare il legno serve ancho a diversi usi e a varie arte. </s> <s>La pialla non è differente se non secondo <pb pagenum="folios 85v-86r"></pb>la grandezza e secondo la figura e secondo la varietà dell’uso, che altra serve per far piane ed uguali le tavole, altra serve per far le cornici e questa ragion di pialla che comunemente si appella pialluzzo ha tante differenze quante sono le maniere delle membra di qualsivoglia cornice secondo qualunque de’ cinque ordini dell’architettura ed oltre all’havere l’angolo solito nel taglio hanno ancho talhora a’ triangoli concavi necessarij alla formatica degli angoli convessi delle membra che da esse si debbano formare. </s> <s>Gli stromenti che servono a tagliare la pietra sono gli scarpelli che sono o acuti o piani o dentati o spartiti. </s> <s>Lo scarpello da legno overo è piano od acuto ed aognato overo incurvato, come sono le sgobbie: e questi o si adoperano a lavorar di quadro o d’intaglio overo a tornire. </s> <s>L’ascia altramente detta mana non ha altre differenze che quelle della grandezza e dell’uso; che altre asce servono per abbozzare le tavole e ‘ regoli ed altre servono per incavare, le quali più longhe, ma di taglio più stretto dell’altre, le quali si adoperano ad incavar travi e a fare ombuti. </s> <s>L’accetta, chiamata scure, altre si nota altre differenze che della grandezza; ciascuno di questi stromenti riceve l’essere e l’operation sua della natura degli angoli secondo ‘quali è formato. </s> <s>Non senza ragione Aristotile ne’ libri dell’ Anima dice per essempio che la scure non è veramente scure se non ha ‘l taglio, ma è scure equivocamente, e ‘l taglio consiste nell’angolo il quale è acuto, che quanto più è acuto, tanto più ha forza di tagliare perciochè più agevolmente entra nel legno, penetrando con forza di lieva e a guisa di conio, ad entrar da un ponto indivisibile, acquistando ogn’hora più di legno ed ogn’hora più ne leva. </s> <s>Onde vediamo espressamente che ‘l taglio è collocato nell’angolo acuto, e quando l’angolo è acuminato dalla pietra d’arrotare, allhora più taglia e con più agevolezza ricide ogni legno, benchè sia duro, ed ogni <lb></lb>//<lb></lb>pietra benchè dura. </s> <s>Il medesimo accade all’ascia e ad ogni maniera di scarpello, di pialla, o di coltello. </s> <s>Gli stromenti che servono per forare o la pietra, o ‘l metallo, o l’osso, o ‘l legno sono il trapano o ‘l succhiello, detto in altra maniera trivello. </s> <s>Il succhiello non ha altre differenze che quelle le quali si prendano dalla grandezza; che altri sono succhielli grossi, altri minuti, altri mezzani, secondo la grandezza de’ chiodi, delle bullette, de’ cavicchi, o de perni, che si adoperano per collegamento del legname, o per altro fine. </s> <s>L’operation di questo stromento consiste nel passar il legno rodendo e bucando acciochè meglio faccia l’effetto suo ha bisogno di due maniere d’angoli; uno è acuto, ma alquanto curvo, come è quello della ponta onde comincia formarsi la vite, secondo la quale si gira, acciochè si ficchi dentro ‘l legno, ed è parte convesso e parte concavo e l’altro altresì è acuto, su per lato e su per la vita, ed è parte convesso e parte concavo, acciochè mentre si volta tagli e roda il legno, e riceva la tagliatura nella concavità della vite. </s> <s>Si trova ancho un’altra maniera di succhielli, la quale si adopra nell’agricoltura per far li buchi nel terreno sodo, per mettervi le colonne da pergole. </s> <s>Il trapano è di varie maniere; perciochè altro è ‘l trapano da legno, altro da pietra, altro da metallo. </s> <s>Il trapano da metallo si fa a ponta di diamnte ad angolo retto o ad angolo acuto e quadrato o triangolare; il trapano da pietra si fa con angolo acuto nel taglio, ma ‘l taglio è curvo, di modo che quando comincia a rodere la pietra, muovendosi di moto circolare reflesso e ripiegato comincia in un ponto; quello da legno o da osso è formato con due angoli simili ma acutissimi, uno de’ quali serve per centro, in fra ‘ quali è ‘l taglio a modo di lunetta. </s> <s>Gli stromenti che adoperano a segar o la pietra, o ‘l legno, o l’osso. </s> <s>Quelli che servono a segare il legno, o l’osso sono tutte le ragioni di seghe, cioè grandi per segar i modelli per far le tavole le quali sono grosse e lunghe, e queste sono di due maniere; percio<pb pagenum="folios 86v-87r"></pb>chè altre hanno i denti più spessi ed angoli più acuti ed altre gli hanno radi, e d’ angoli meno acuti; quelle che s’usano a segar le pietre sono senza denti, dritte e fatte con due angoli retti nel filo che sega e nella grossezza quasi a modo di riga. </s> <s>Queste sono di due maniere, secondo due ragion di materie da segarsi; perciochè altre sono le seghe da segar le pietre tenere, altre quelle da segar le dure. </s> <s>Le seghe che si adoperano a segar le pietre dure, e specialmente le gioie, sono di rame, le quali per loro stesse non hanno forza di segar tali pietre, ma insieme con la polvere di smeriglio, sì come ‘l solo smeriglio non può segarle , ma amendue insieme, perciò che ‘l rame essendo dolce cede allo smeriglio e muovendosi il conduce da qua e là su pper la pietra, tanto che l’uno e l’altro entra e rode e sega le pietre benchè durissime. </s> <s>Quelle che si adoperano a segar le pietre tenere sono di ferro, le quali parimente per loro stesse non segano, ma insieme con la rena di fiume, per la medesima ragione che segano quelle di rame. </s> <s>Gli stromenti che si adattano a incidere sono i bulini, che si distinguono in due maniere, perciochè altri servono a intagliar le lamine di rame per le stampe de’ disegni, altri si usano dagli orefici per intagliar l’oro o l’argento, in qualunque maniera di lavoro. </s> <s>E questi sono angolari, o ad angolo più o meno acuto o ad angolo retto. </s> <s>Per radere o raschiare servono i rasoi o coltelli, i quali son fatti acutissimi. </s> <s>Per limare, o raspare, o pulire il metallo, il legno, la pietra e l’osso, servono le lime, che sono angolari, cioè o con angoli retti o con angoli acuti rettilinei, o con angoli acuti curvilinei, o con angoli misti, ma sono dritte o ritorte secondo che si richiede nella materia che si ha da polire e da spianare, come con le lime e le raspe che si adoperano per ri<lb></lb>//<lb></lb>metter le figure che si fanno di gitto o quelle che si scolpiscano in marmo o in altra pietra tenera. </s> <s>Oltreacciò le lime e le raspe, rodendo, spianano e puliscono la materia col mezzo della multiplication degli angoli minutissimi che sono ne’ denti loro fatti nelle superficie. </s> <s>Gli stromenti che si adoperano per rompere la pietra o ‘l legno sono i conj o le zeppe, le quali son fatte o di ferro o di legno, i quali acciochè rompino e spacchino le materie con forza di lieva, bisogna che habbiano l’angolo acuto. </s> <s>Quelli che premono e spengono e cacciano i chiodi, conficcandosi son fatti ad angoli retti a squadra ed alcune volte ancho non a squadra, ma a sottosquadra, cioè ad angoli acuti appresso alla superficie piana. </s> <s>Tali sono i martelli, i magli, le mazze, che servono ancho per spianare e tirar ed agguagliare il ferro o ‘l rame; i modelli de’ torchij e delle viti degli strettoi e parimente i ciselli che adoperano gli orefici per iscolpire con bassirilievi ed opra che gli antichi Romani si appellava celata. </s> <s>Gli strumenti co’ quali si scaglia, cioè co’ quali si tagliano le pietre levandone a scaglie a scaglie sono le subbie, cioè una ragion di scarpelli fatti non con taglio, ma con ponta quadrata e con angolo acuto nella ponta. </s> <s>Finalmente gli stromenti co’ quali si scalza e si solleva le pietre, la terra, le muraglie, scalzando e smurando i mattoni e le pietre, sono gli zapponi, i picconi, i pali di ferro, i martelli e le zappe. </s> <s>Ma le zappe e gli zapponi si adoprano ancho nell’agricoltura per fare scassati e per far fosse e forme; e nell’arte del fabbricare servono per cavar fondamenti, pozzi, citerne e cantine; <pb pagenum="folios 87v-88r"></pb>ed oltre acciò ancho la vanga e la pala serve a’ medesimi usi. </s> <s>I picconi, i pali di ferro e ‘ martelli si adoperano per tagliare le muraglie, per ispianarle, per rompere e cavar le pietre e tagliarle, facendo fondamenti, incavando la terra per far cantine, pozzi, cisterne, mine, acquedotti, o bottini, o cavando le miniere, facendo ogni maniera di cavamento. </s> <s>E tutti questi stromenti si adattano a tutte queste operationi con l’aiuto dell’angolo acuto che si trova o nella ponta o nel taglio loro. </s> <s>Perciochè gli zapponi, le zappe e le vanghe hanno l’angolo acuto ma non quadrato nella ponta, ma taglio dritto e piano, come ancho i martelli da muratori, nel tagilo della penna; i picconi e ‘ pali di ferro hanno l’angolo acuto e quadrato nella ponta, a guisa di subbia, ma i picconi hanno ancho da un’altra banda un angolo acuto non quadrato, ma triangolare non in ponta ma in linea piana, col quale si suol far la lieva alle pietre scalzate per muoverle dal luogo od a sollevare altro peso. </s> <s>Oltreacciò, nel dar forza al movimento delle macchine, come degli argani, delle viti o d’altro, servono gli angoli retti, che vi si formano i luoghi, dove si debbano por le stanghe, o manovelle, o lieve per muover gli argani e questi si fanno ad angoli retti. </s> <s>Ma nelle viti che hanno la testa quadrata servono ancho gli angoli retti perciochè ponendo la testa loro in una chiave quadrata perfettamente in modo che la testa della vite molto bene s’incastri nel vuoto della chiave; sì che gli angoli convessi dell’una occupino interamente gli angoli concavi dell’altra; onde muovendosi la chiave per la resistenza e contra <lb></lb>//<lb></lb>lieva degli angoli si fa muover la vite e li argani. </s> <s>Per la medesima ragione e per la medesima forza si fanno girar le viti che nel mezzo della testa loro hanno un taglio fatto a linea retta , la quale colli estremi di essa, e col suo diametro fa angoli retti, i quali danno vigore al movimento. </s> <s>E così altri stromenti ed altre macchine fabrili si potrebbono mostrare, i quali si servano del benefitio degli angoli, ma per non far maggior volume, si tralassano, dando luogo a qualunque ingegnoso ne volesse far gionta a questo mio trattato, e così facendo fine sigillaremo con quest’ultime righe l’opra nostra a gloria e ad honore della infinita Sapienza e Providenza, onde procede tutto l’ingegno e tutte le nostre questioni.</s> </p> </chap> </body> </text> </archimedes>