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<?xml version="1.0"?><!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" > <archimedes> <info><author>Monte, Guidobaldo del</author><title>Le Mechaniche</title><date>1581</date><place>Venezia</place><translator>Pigafetta, Filippo</translator><lang>it</lang><cvs_file>monte_mecha_02_it_1581</cvs_file><cvs_version>2643.17</cvs_version><locator>037.xml</locator></info> <text id="id.0.0.0.0.3"> <pb id="p.0001" xlink:href="037/01/001.jpg"/> <front id="id.1.0.0.0.0"> <section> <s id="id.2.1.2.1.0">LE <lb/>MECHANICHE <lb/>DELL'ILLVSTRISS SIG. <lb/>GVIDO VBALDO <lb/>DE' MARCHESI DEL <lb/>MONTE: <lb/></s> <s id="id.2.1.3.1.0">TRADOTTE IN VOLGARE <lb/>DAL SIG. FILIPPO PIGAFETTA: <lb/>Nellequali &longs;i contiene la vera Dottrina di tutti gli I&longs;trumenti <lb/>principali da mouer pe&longs;i grandis&longs;imi con <lb/>picciola forza. <lb/></s> <s id="id.2.1.4.1.0"><emph type="italics"/>A beneficio di chi &longs;i diletta di que&longs;ta nobili&longs;&longs;ima scienza; & ma&longs;&longs;imamente <lb/>di Capitani di guerra, Ingegnieri, Architetti, & d'ogni <lb/>Artefice, che intenda per via di Machine <lb/>far opre marauiglio&longs;e, e qua&longs;i <lb/>&longs;opra naturali. <emph.end type="italics"/><lb/></s> <s id="id.2.1.5.1.0">Et &longs;i dichiarano i vocaboli, & luoghi più difficili. <lb/></s> <figure id="id.037.01.001.1.jpg" xlink:href="037/01/001/1.jpg"> </figure> <s id="id.2.1.7.1.0"><emph type="italics"/>In Venetia, Appre&longs;&longs;o France&longs;co di France&longs;chi Sane&longs;e. </s> <s id="id.2.1.7.2.0">MD LXXXI. <emph.end type="italics"/></s> </section> <pb xlink:href="037/01/002.jpg"/> <pb xlink:href="037/01/003.jpg"/> <section> <s id="id.2.1.10.1.0">ALL'ILLVSTRISSIMO <lb/>SIGNOR GIVLIO <lb/>SAVORGNANO, <lb/>CONTE DI BELGRADO. &c. <lb/>Signore o&longs;&longs;eruandi&longs;&longs;imo. </s> <s id="id.2.1.13.1.0">C<emph type="italics"/>onciosia co&longs;a, che la &longs;cienza delle Mecha<lb/>niche gioui &longs;ommamente à molte, & importan<lb/>ti attioni della no&longs;tra vita, à gran ragione fu ella <lb/>da i Filo&longs;ofi, & da i Rè antichi &longs;timata degna di <lb/>laudi &longs;ingularißime; & i Matematici vi han<lb/>no impiegato lo &longs;tudio, & l'opera più che meza<lb/>namente, & i Principi fauoriti gl'ingegnieri ec<lb/>cellenti, & arricchiti. </s> <s id="id.2.1.13.2.0">Ben è per certo di alti&longs;<lb/>&longs;ima &longs;peculatione, & di &longs;ottile manifattura; imperoche tocca quella par<lb/>te della Filo&longs;ofia, che tratta de gli elementi in vniuer&longs;ale, & del moto, & <lb/>della quiete de' corpi, &longs;econdo i luoghi &longs;uoi, a&longs;&longs;egnando la cagione in certo <lb/>modo de' loro mouimenti naturali; & anco sforzandoli, per via di machi<lb/>ne à partir&longs;i da proprij &longs;iti, gli tra&longs;porta all'insù, & per ogni lato in mo<lb/>uimenti contrari alla natura loro. <emph.end type="italics"/></s> <s id="id.2.1.14.1.0"><emph type="italics"/>Mena ella ad effetto ambedue que&longs;te intentioni con le propo&longs;itioni che <lb/>na&longs;cono, & &longs;ono congiunte con la materia &longs;te&longs;&longs;a, & co' difici, & i&longs;trumen<lb/>ti, che forma artificialmente. </s> <s id="id.2.1.14.2.0">La onde egli è dibi&longs;ogno con&longs;iderare que&longs;ta<emph.end type="italics"/><pb xlink:href="037/01/004.jpg"/><emph type="italics"/>dottrina in due manìere; l'vna ìn quanto và &longs;peculando, & con ragione <lb/>di&longs;correndo &longs;opra le co&longs;e, che s'hanno à fare, &longs;eruendo&longs;i dell' Arithmetica, <lb/>della Geometria, dell' A&longs;trologia, & della Filo&longs;ofia naturale: & l'altra che <lb/>po&longs;cia le manda ad e&longs;ecutione, & haue nece&longs;sita dell'e&longs;&longs;ercitio, & lauoro <lb/>delle mani, v&longs;ando l'Architettura, la Pittura, il di&longs;egno, l'arte de' fabri, <lb/>de'legnaiuoli, de'muratori, & d'altri me&longs;tieri tali, per modo che ella vie<lb/>ne ad e&longs;&longs;ere me&longs;colata, & in parte compo&longs;ta della naturale Filo&longs;ofia, delle <lb/>Matematiche, & delle arti manuali. </s> <s id="id.2.1.14.3.0">Per laqual co&longs;a chiunque &longs;i troua <lb/>dotato d'ingegno acuto, & da fanciullo hà incominciato ad apprendere le <lb/>già dette &longs;cienze, & &longs;a di&longs;egnare, & lauorare di &longs;ua mano, potrà nel vero <lb/>ottimo Mechanico, & <expan abbr="inu&etilde;tore">inuentore</expan>, & facitore di opere marauiglio&longs;e riu&longs;cire. <emph.end type="italics"/></s> <s id="id.2.1.15.1.0"><emph type="italics"/>Infinite parti, & vtilißime à gli huomini comprende que&longs;ta noti<lb/>tia, & in guerra, & in pace, ne i commodi della città, della villa, & della <lb/>mercatantia, & in altri; peroche la Medicina toglie da lei i difici per ri<lb/>porre le o&longs;&longs;a &longs;mo&longs;&longs;e, & rotte ne i &longs;iti &longs;uoi. </s> <s id="id.2.1.15.2.0">Onde pone Oriba&longs;io nel libro delle <lb/>Machine, diuer&longs;i i&longs;trumenti pre&longs;i dalla Mechanica, & <expan abbr="cõuertiti">conuertiti</expan> nell'v&longs;o del<lb/>la Medicina, come il Tri&longs;pa&longs;ton di Archimede: l'arte del nauigare ricono<lb/>&longs;ce anco diuer&longs;i aiuti, come il timone, co'l quale, collocato di dietro, ouero <lb/>alle bande del nauilio ageuolmente lo moue, & dirizza, quantunque per <lb/>ri&longs;petto à tutto il corpo del va&longs;ello piccioli&longs;simo &longs;ia. </s> <s id="id.2.1.15.3.0">I remi, che à gui&longs;a di <lb/>leua lo &longs;pingono innanzi, & l'arbore, & la vela &longs;ono pur di &longs;ua inuentio<lb/>ne. </s> <s id="id.2.1.15.4.0">I molini, i quali &longs;i girano co'l vento, con l'acqua, & con la forza vi<lb/>ua: & i pi&longs;trini, le carra, gli aratri, & altri ordigni di villa; il pe&longs;are con <lb/>la bilancia, & con la &longs;tadera; il cauare l'acqua da pozzi con le grù, ouero <lb/>cicogne, dette da latini to&longs;senoni, che &longs;ono come grandis&longs;ime bilancie, & <lb/>con le rote, & altre co&longs;e tali &longs;i riducono alla Mechanica. </s> <s id="id.2.1.15.5.0">La ragione pa<lb/>rimente del condurre le acque, & da profondis&longs;ime valli in alto farle &longs;ur<lb/>gere uà &longs;otto lei. </s> <s id="id.2.1.15.6.0">Chiamarono gli antichi coloro Mechanici ancora, i quali <lb/>co'l fiato, ò vento, ouero acqua, ò corde, ò nerui faceuano vedere, & vdi<lb/>re effetti miracolo&longs;i; come &longs;uoni diuer&longs;i, & canti d'augelli, & fin ad e&longs;pri<lb/>mere la voce humana in parole: & quelli che con horologi, i quali &longs;i mo<lb/>uono da &longs;e &longs;tes&longs;i con rote, ò da acqua, ò da &longs;ole il tempo mi&longs;urarono, & di<lb/>&longs;tin&longs;ero in hore. </s> <s id="id.2.1.15.7.0">Appartengono alla Mechanica gli facitori delle Sfere <lb/>compartite ne'&longs;uoi cieli, co'l mouimento de'Pianeti, & di tutti i corpi <lb/>cele&longs;tiali à &longs;embianza dell'vniuer&longs;o mondo, & ciò mediante il mouimen<lb/>to eguale, & in giro, che loro daua l'acqua, di cui la fama &longs;uona e&longs;&longs;ere <lb/>&longs;tato Archimede Siracu&longs;ano il primo mae&longs;tro. </s> <s id="id.2.1.15.8.0">il mouere etiandio con poca <emph.end type="italics"/><pb xlink:href="037/01/005.jpg"/><emph type="italics"/>forza pe&longs;i grandis&longs;imi con i&longs;trumenti, & ingegni diuer&longs;i è principale of<lb/>ficio della Mechanica, come Bilancie, Stadere, Leue, Taglie, Cunei, Moli<lb/>nelli, Rote co' denti & &longs;enza, Viti d'ogni &longs;orte, Argani, Mangani, Triuel<lb/>le, & altri molti, i quali da que&longs;ti &longs;i compongono: & &longs;econdo Ari&longs;totele <lb/>tutti &longs;i riducono alla Leua, & al cerchio, & alla machina ritonda, laquale <lb/>quanto è maggiore, tanto più velocemente &longs;i moue. </s> <s id="id.2.1.15.9.0">L'arte del fortificare <lb/>le piazze, & i &longs;iti, & del difendergli, laquale acconciamente &longs;i puote chia<lb/>mare Architettura militare, è profrees&longs;ione Mechanica: peroche per via di <lb/>Cortine, & di Baloardi, & d'altri ripari, qua&longs;i con machine, & i&longs;trumen<lb/>ti s'ingegna l'huomo con pochi &longs;oldati di ributtarne in dietro molti, & <lb/>mantener&longs;i con vantaggio. </s> <s id="id.2.1.15.10.0">Il fabricare, & adoprare oltre à ciò gli i&longs;tru<lb/>menti da guerra è proprio dono di que&longs;ta &longs;cienza, come Bali&longs;te, ò Bale&longs;tre, <lb/>Catapulte, Scorpioni, Fionde, & &longs;imili, che da lontano gittano foco, & &longs;aßi, <lb/>& ma&longs;&longs;e di ferro pe&longs;anti dugento cinquanta, & più libre, & Moli da <lb/>molino &longs;econdo Silio Italico, & Vitruuio, per di&longs;tanza di for&longs;e<emph.end type="italics"/> 300. <emph type="italics"/>pas&longs;i <lb/>à mi&longs;ura con ruino&longs;o colpo; & &longs;aette, & verettoni, & falariche grandi <lb/>à gui&longs;a di traui: & quelli che percoteuano con l'vrto da pre&longs;&longs;o, come Arie<lb/>ti, Onagri, Te&longs;tugini, & &longs;imili; & in altri v&longs;i, come <expan abbr="Sãbuche">Sambuche</expan>, Corui, Mani <lb/>di ferro, & gli altri maritimi, & Angoni, Monangoni, Tollenoni, &longs;cale &longs;no<lb/>date, ponti, torri mobili, & &longs;imili difici antichi, i quali &longs;ono &longs;tati poi ri <lb/>fiutati, &longs;uccedendo in &longs;uo luogo le Artiglierie, da e&longs;&longs;ere anch'e&longs;&longs;e ordinate <lb/>nell' ampiezza della con&longs;ideratione Mechanica, facendo elle còn sì poca ma<lb/>teria acce&longs;a, tanto horribile perco&longs;&longs;a. <emph.end type="italics"/></s> <s id="id.2.1.16.1.0"><emph type="italics"/>Que&longs;ta &longs;cienza, che fuor di quanto &longs;i è detto, abbraccia innumerabili <lb/>altri v&longs;i, & diletteuoli, & nece&longs;&longs;ari à mortali, in diuer&longs;i tempi hebbe in <lb/>&longs;orte vari &longs;tati, per ri&longs;petto à gli artefici, che la e&longs;ercitarono: peroche, <lb/>di là cominciando, ne gli antichis&longs;imi &longs;ecoli, che pa&longs;&longs;arono auanti la guer<lb/>ra di Troia vi&longs;&longs;e Dedalo Athenie&longs;e gran mae&longs;tro di Mechanica, ilquale <lb/>trouò il primiero la &longs;ega, l'a&longs;cia, il piombino da torre le diritture, la tri<lb/>uella, l'albero, l'antenna, la vela, & altri or digni: di&longs;egnò in Creta poi <lb/>quell'intricato labirinto, & alla fine gli conuenne fabricare per &longs;e, & per <lb/>Icaro &longs;uo figlio due paia d'ali, & volar&longs;ene via per l'aere à gui&longs;a d'au<lb/>gelli, come cantano i Poeti. <emph.end type="italics"/></s> <s id="id.2.1.17.1.0"><emph type="italics"/>Nella fabrica del tempio di Salomone, che fu la maggiore per grandez<lb/>za, per mae&longs;tria d' Architettura, & ornamento, di quante ne &longs;iano &longs;tate <lb/>fatte giamai; & delle piramidi, & di tanti altri difici di quei &longs;eco'i, che <lb/>hanno riempito il mondo di &longs;tupore, egli &longs;i può credere, che interueni&longs;&longs;ero <emph.end type="italics"/><pb xlink:href="037/01/006.jpg"/><emph type="italics"/>eccellenti Mechanici, per leuare in alto le pietre &longs;mi&longs;urate, & per altre <lb/>opere, lequali à condurgli à fine &longs;i ricercauano. </s> <s id="id.2.1.17.2.0">Nacquero dapoi Eudo&longs;<lb/>&longs;o, & Archita Tarentino, ambidue valenti ingegnieri; & di Archita &longs;i <lb/>legge, che lauorò di legno vna colomba con tanta mae&longs;tria temperata, & <lb/>gonfiata, che da &longs;e volaua per l'aria à gui&longs;a di viua colomba. </s> <s id="id.2.1.17.3.0">Seguì co<lb/>&longs;toro il Filo&longs;ofo Ari&longs;totele, ilquale certe poche, ma bellis&longs;ime que&longs;tioni Me<lb/>chaniche, la&longs;ciò &longs;critte. </s> <s id="id.2.1.17.4.0">A lui venne appre&longs;&longs;o Demetrio Rè, nominato il <lb/>pigliatore, ò di&longs;truggitore delle città, peroche fabricaua machine, & difi<lb/>ci, co' quali per di&longs;opra vi montaua, & &longs;e ne faceua padrone, lequali per <lb/>auentura furono &longs;imiglianti alla machina detta Cauallo, con cui li Greci <lb/>pre&longs;ero la famo&longs;a Troia; di che ragionando Pau&longs;ania nell' Attica, dice che <lb/>giudica e&longs;pre&longs;&longs;a mattezza il credere, che fo&longs;&longs;e vn cauallo, & non machina <lb/>bellico&longs;a per acco&longs;tare alle muraglie, & prenderle. </s> <s id="id.2.1.17.5.0">Que&longs;to Rè cominciò <lb/>ad aumentare la Mechanica in qualche honore. </s> <s id="id.2.1.17.6.0">Ma Archimede, che fù <lb/>il megliore artefice di quanti fecero giamai que&longs;ta profes&longs;ione innanzi, & <lb/>dopo lui, & qua&longs;i vn lume, che poi ha illu&longs;trato tutto il mondo, accrebbe <lb/>in colmo la riputatione della Mechanica, & di pouera arte, & vile, che pri<lb/>ma era, come vuole Plutarco nella vita di Marcello, nel numero delle arti <lb/>nobili, & pregiate alla militia pertinenti la ripo&longs;e. </s> <s id="id.2.1.17.7.0">Imperoche combat<lb/>tendo Marcello Siracu&longs;a patria &longs;ua per mare, & per terra con grande <lb/>ho&longs;te di Romani, egli co'&longs;uoi diuer&longs;i ingegni, & machine differenti, ribut<lb/>tò &longs;empre gli sforzi, con graue lor danno, & vergogna; come Liuio, Plutar<lb/>co, & altri nominando i difici che v&longs;aua, diffu&longs;amente raccontano. </s> <s id="id.2.1.17.8.0">Per<lb/>cioche quando Marcello s'auicinaua aile muraglie per conqui&longs;tar le con la <lb/>Sambuca, il buon Archimede co'l Tollenone, & con le mani di ferro la al<lb/>zaua di pe&longs;o in aere, & poi &longs;nodando quegli vncini &longs;uoi, la faceua cadere <lb/>da alto, in mare &longs;ommergendola; il mede&longs;imo effetto adoprando contra gli <lb/>altri nauili, sì fattamente, che gli conuenne allontanare l'armata ben to<lb/>&longs;to dalle mura. </s> <s id="id.2.1.17.9.0">Ne ce&longs;sò tuttauia d'infe&longs;tare il nemico: ma &longs;i come nota <lb/>Galeno nel terzo libro de' temperamenti, & Giouanni Zonara, & Tze&longs;es <lb/>confermano, allegando Diodoro, & Dione, compo&longs;e certi &longs;pecchi grandi <lb/>& concaui, &longs;econdo la proportione della di&longs;tanza di quei va&longs;elli dalla mu<lb/>raglia, & opponendogli à raggi del Sole in diritta linea qua&longs;i per miraco<lb/>lo, gli bru&longs;ciaua. </s> <s id="id.2.1.17.10.0">Dalla parte della terra &longs;imilmente offendeua gli aduer<lb/>&longs;ari con arme diuer&longs;e da gittare. </s> <s id="id.2.1.17.11.0">Per laqual co&longs;a nè in mare, nè in terra <lb/>da gl'ingegni di quell'eccellente Mechanico &longs;i poteua egli &longs;chermire, nuoui <lb/>ripari, & horribili offe&longs;e apparecchiando &longs;empre. </s> <s id="id.2.1.17.12.0">Pappo Ale&longs;&longs;andrino <emph.end type="italics"/><pb xlink:href="037/01/007.jpg"/><emph type="italics"/>allega il quarante&longs;imo trouato di Archimede, per dichiarare, che almeno <lb/>i &longs;uoi difici al numero di quaranta a&longs;cendeuano. </s> <s id="id.2.1.17.13.0">La onde Marcello, veg<lb/>gendo, che niuno profitto apportauano all'impre&longs;a gli a&longs;&longs;alti &longs;uoi, & che <lb/>erano vn mettere le genti ad euidente pericolo, per cagione di quel &longs;olo <lb/>valoro&longs;o vecchio, gli nacque vna tal opinione, & à tutto l'e&longs;ercito, che da <lb/>po&longs;&longs;anza diuina fo&longs;&longs;e gouernato in quella dife&longs;a, & mutò la ragione del <lb/>guerreggiare, dando&longs;i all'aßedio, & al vietare &longs;trettißimamente le vitto<lb/>uaglie a quella città. <emph.end type="italics"/></s> <s id="id.2.1.18.1.0"><emph type="italics"/>Que&longs;te furono le cagioni, che la Mechanica &longs;alì in tanta gloria, & che <lb/>i Romani le a&longs;&longs;egnarono dapoi grado honoreuolißimo ne gli e&longs;erciti lo<lb/>ro, come &longs;i legge nel primo libro della guerra ciuile, che Ce&longs;are &longs;e prigione <lb/>il Capitano de' fabri di Pompeio, nomato Magio Cremona, & Vitruuio fu <lb/>Capitano delle Bali&longs;te di Ce&longs;are Augu&longs;to, che &longs;arebbe nella militia moder<lb/>na, come Capitano generale dell' artiglieria. </s> <s id="id.2.1.18.2.0">La qual gloria &longs;ucceßiua<lb/>mente le fu mantenuta poi da molti dottißimi &longs;crittori, & mae&longs;tri di <lb/>Mechanica, come da Cte&longs;ibio Ale&longs;&longs;andrino, da Herone Ale&longs;&longs;andrino, da <lb/>un'altro Herone, da Ateneo, da Bione, da Pappo Ale&longs;&longs;andrino, che allega <lb/>Carpo di Antiochia, da Eliodoro, da Oriba&longs;io, & da altri Greci, i quali fio<lb/>rirono in diuer&longs;i tempi, in&longs;egnando la ragione, la mi&longs;ura, & l'v&longs;o de gli <lb/>i&longs;trumenti bellico&longs;i non &longs;olo, ma di tutti gli altri, che le pertengono. </s> <s id="id.2.1.18.3.0">Fra <lb/>Latini antichi Varrone &longs;cri&longs;&longs;e dell' Architettura, & per con&longs;eguente douet<lb/>te anco far mentione della Mechanica: & Vitruuio, & Vegetio, & qual<lb/>che altro hanno fauellato d'intorno alla fabrica delle machine militari, <lb/>& da mouer pe&longs;i, & aiutato à con&longs;eruare fra gli huomini viua la digni<lb/>tà della Mechanica. <emph.end type="italics"/></s> <s id="id.2.1.19.1.0"><emph type="italics"/>Ma ruinando l'Imperio di Romani, & &longs;uccedendo i barbari in Italia, <lb/>in Grecia, in Egitto, & in ogni contrada, oue &longs;i e&longs;ercita&longs;&longs;ero le buone lette<lb/>re, caddero mi&longs;erabilmente, & &longs;i perderono qua&longs;i del tutto le &longs;cienze, & in <lb/>&longs;pecialità re&longs;tò la Mechanica lunghißimo tempo negletta, non cono&longs;cen<lb/>do&longs;i in guerra altri difici, che Bricole, Trabucchi, Mangani, Martinelli, & <lb/>certi i&longs;trumenti tali, finche &longs;ouragiun&longs;e l'artiglieria, laquale à poco à po<lb/>co gli &longs;e di&longs;u&longs;are à fatto: & di quella parte altre&longs;i della Mechanica, laqua<lb/>le s'adopra al mouer pe&longs;i, ben picciolo intendimento rima&longs;e. </s> <s id="id.2.1.19.2.0">Vera co&longs;a è, <lb/>che &longs;embra da vn tempo in quà le arti, & le dottrine più nobili, come le <lb/>belle lettere appellate humane, la Filo&longs;ofia, la Medicina, l'A&longs;trologia, l'A<lb/>rithmetica con la Mu&longs;ica, la Geometria, l'Architettura, la Scoltura, la Pit<lb/>tura con molte altre: & &longs;pecialmente la Mechanica e&longs;&longs;ere dalle o&longs;cure te<emph.end type="italics"/><pb xlink:href="037/01/008.jpg"/><emph type="italics"/>nebre, oue giaccuano &longs;epolte, alla chiara luce ri&longs;u&longs;citate: Percioche ri<lb/>&longs;tringendomi alle Mechaniche Giordano, che &longs;cri&longs;&longs;e de' pe&longs;i, la incominciò <lb/>à &longs;olleuare alquanto, & poi Leon Batti&longs;ta Alberti nella &longs;ua Architettu<lb/>ra: il Tartaglia aper&longs;e anco la via à molte &longs;peculationi Mechaniche: Vitto <lb/>rio Fau&longs;to nell' Arzanà di Venetia mo&longs;trò d'e&longs;&longs;ere buon Mechanico: Mon<lb/>&longs;ig. Reuerendiß. barbaro eletto d' Aquileia ne' Commentari del decimo di <lb/>Vitruuio nominò gli i&longs;trumenti da mouer pe&longs;i: Georgio Agricola nel &longs;e&longs;to <lb/>de' Metalli raccol&longs;e a&longs;&longs;ai machine da leuar pe&longs;i, & qualche d'vn'altro: & <lb/><expan abbr="noauam&etilde;te">nuouamente</expan> l'Autore di que&longs;t'opera, ilquale ben d'altra maniera in ciò pro<lb/>cedette, che gli autori nominati, peroche con ordine ammirabile, & con <lb/>vere, & certe ragioni ha in&longs;egnato &longs;olo fra Latini ottimamente que&longs;ta <lb/>&longs;cienza tutta da mouer pe&longs;i. <emph.end type="italics"/></s> <s id="id.2.1.20.1.0"><emph type="italics"/>Ma &longs;i come i moderni da me ricordati, & principalmente l'Autore del <lb/>pre&longs;ente libro hanno ornata & e&longs;altata la Mechanica con le parole, & co i <lb/>volumi; co&longs;i V. S. Illu&longs;triß. l'hà celebrata, & magnificata co' di&longs;cor&longs;i, & <lb/>con le operationi i&longs;te&longs;&longs;e, & co' fatti re&longs;a famigliare, & dome&longs;tica, diuer&longs;e <lb/>machine fabricando con profondißima dottrina, & facendone e&longs;perten<lb/>ze nel mouere qualunque gran pe&longs;o, di cui &longs;i po&longs;&longs;a l'huomo in ogni bi&longs;ogno <lb/>&longs;eruire. </s> <s id="id.2.1.20.3.0">Talche ben &longs;i puote con verità affermare, che per vna parte e&longs;&longs;a, <lb/>& l'Autore di que&longs;ti trattati per l'altra, habbiate alla Mechanica il pri&longs;ti<lb/>no honore re&longs;tituito, che da i tempi antichi in quà le cra &longs;marrito. <emph.end type="italics"/></s> <s id="id.2.1.21.1.0"><emph type="italics"/>Sono &longs;or&longs;e quaranta anni già &longs;cor&longs;i, che per i&longs;cherzare con Nicolò Tar<lb/>taglia, per&longs;ona à &longs;uoi tempi molto &longs;timata in que&longs;ta profeßone, & che &longs;i <lb/>dilettaua di andare &longs;oluendo que&longs;tioni &longs;ottili di Mechanica, & di Mathe<lb/>matica, & ne' &longs;uoi dialoghi introduceua à fauellare per&longs;onaggi grandi: <lb/>& alcuna fiata gli faceua dire qualche co&longs;a, di cui eßi prendeuano onta, <lb/>V. S. Illu&longs;triß. gliene propo&longs;e for&longs;e quaranta Mechaniche qua&longs;i tutte, & <lb/>difficili: alcune delle quali egli prouò di &longs;oluere, delle altre &longs;i &longs;cusò con di<lb/>re, che à cia&longs;cheduna di loro &longs;arebbe &longs;tato me&longs;tieri vn volume intero, co<lb/>me &longs;i legge ne' &longs;uoi libri &longs;tampati della noua &longs;cientia. <emph.end type="italics"/></s> <s id="id.2.1.22.1.0"><emph type="italics"/>Hor non è punto di marauiglia, che ella habbia penetrato con l'inten<lb/>dimento <expan abbr="tãto">tanto</expan> dentro, & &longs;aputo co&longs;i bene operare nelle Mechaniche, & &longs;ia <lb/>fatta padrona in tutto dell'arte del fortificare i &longs;iti, & d'ogni altra parte <lb/>della militia: peroche fu dall'ottimo &longs;uo padre alleuata in compagnia di <lb/>huomini &longs;cientiati, & d'alio affare, tra quali fu vn tempo Con&longs;tantino <lb/>La&longs;cari nobilißimo huomo Greco, & pieno di dottrina, da cui &longs;ucceßi<lb/>uamente imparò, oltra le altre lettere, Arithmetica, Geometria, A&longs;tro<emph.end type="italics"/><pb xlink:href="037/01/009.jpg"/><emph type="italics"/>logia, Geografia; à ài&longs;egnare, & lauorare manualmente in me&longs;tieri diuer<lb/>&longs;i; à caualcare, à maneggiare le arme, à tirare d'archibugio, & d'artiglie<lb/>ria, & à <expan abbr="cõporre">comporre</expan> fochi artificiati, & l'arte per eccellenza detta del bom<lb/>bardiero; à viuere &longs;obriamente, & le fatiche tolerare al caldo, al freddo, <lb/>& ad ogni di&longs;agio, co&longs;e tutte, che di&longs;pongono l'animo, & indurano il corpo <lb/>alla militia. </s> <s id="id.2.1.22.2.0">Giunta poi all' età di &longs;edici anni, fu inutata con dodici caual<lb/>li qua&longs;i tutti Turchi, & con prouedimento conueneuole di denari à vede<lb/>re tutta quella guerra, che pa&longs;sò in Italia dalla pre&longs;ura del Rè France&longs;co <lb/>Primo di Francia, fin alla pace generale, che &longs;eguì l'anno<emph.end type="italics"/> 1529. </s> <s><emph type="italics"/>Nella<lb/>quale interuennero qua&longs;i tutti i mouimenti militari, che &longs;i po&longs;&longs;ano ima<lb/>ginare, sì per gli e&longs;erciti grandi, che erano à fronte l'vn contra l'altro, sì <lb/>per la qualità, & quantità delle impre&longs;e fatte, & per mille altri acciden<lb/>ti importantißimi, & &longs;tratagemi auenuti, & sì principalmente; percio<lb/>che nell'vn campo, & l'altro in varie &longs;tagioni militarono i primi guer<lb/>rieri del mondo, & in gran numero, i quali con prudenza, a&longs;tutia, & <lb/>brauura contendeuano à gara, & per honore di &longs;oura&longs;tare, & e&longs;&longs;ere vinci <lb/>tori. </s> <s id="id.2.1.22.3.0">Et veramente chi ben con&longs;idera, fin da i tempi antichi, rarißime vol<lb/>te è &longs;tato con numero maggiore di Capitani famo&longs;i, ò con più copia d'im<lb/>pre&longs;e grandi guerreggiato, che in quegli anni: Peroche furono fatti prigio<lb/>ni due de' maggiori Prencipi del mondo, &longs;i a&longs;&longs;ediò Milano, & per forza fu<lb/>rono pre&longs;e tre città, Roma, Cremona, & Pauia; &longs;i videro più fatti d'ar<lb/>me, & gli e&longs;erciti &longs;i andarono per&longs;eguitando da Milano à Roma; &longs;i che Pia<lb/>cenza, Parma, Bologna, & Fiorenza guardaròn&longs;i dalle armi nemiche. <emph.end type="italics"/></s> <s id="id.2.1.23.1.0"><emph type="italics"/>Nello &longs;plendore dunque della &longs;cola del Duca France&longs;co Maria d'Vrbino, <lb/>ilquale era Capitano generale della Lega, & di quegli altri valentißimi <lb/>Capitani, andaua V. S. Illu&longs;triß. come di &longs;ua libertà, & benißimo à ca<lb/>uallo, con chi le piaceua, & &longs;i trouaua à quelle fattioni, che volea, &longs;eguen<lb/>do le più volte il Sig. Giouanni de' Medici, & Paulo Luzza&longs;co, che erano <lb/>&longs;empre de&longs;ti, & arditi, & come l'occhio dell'e&longs;ercito. </s> <s id="id.2.1.23.4.0">Quì non è mia in<lb/>tentione di narrare gli auenimenti di quella guerra, ma &longs;i bene di auerti<lb/>re, che chi la vide, & appre&longs;e da buon &longs;enno i &longs;uoi moti; & &longs;eppe manda<lb/>re à memoria quei &longs;atti marauiglio&longs;i, ben puote meritamente vantar&longs;i <lb/>di hauer mirato ca&longs;i memorabili, i quali nè anche in migliata d'anni &longs;o<lb/>gliono accadere; come ella, che e&longs;&longs;endo giouine di viuace &longs;pirito, & am<lb/>mae&longs;trata nelle arti nece&longs;&longs;arie al &longs;oldato, & volentero&longs;ißima d'imparare, <lb/>hebbe opportuna occa&longs;ione di far&longs;i prattica dell'ordinare, dell'e&longs;ercitare, <lb/>del far marciare in battaglia, dell'alloggiare in campagna gli e&longs;erciti &longs;i<emph.end type="italics"/><pb xlink:href="037/01/010.jpg"/><emph type="italics"/>curamente: & del pre&longs;entare al nemico il fatto d'arme con vantaggio: <lb/>Del fortificare, & difendere i &longs;iti, & offenderli con le mine, con le trin<lb/>cee, con le artiglierie, con gli a&longs;&longs;alti, & con tutti gli altri sforzi; & d'o<lb/>gni parte della militare &longs;cienza. <emph.end type="italics"/></s> <s id="id.2.1.24.1.0"><emph type="italics"/>Ritornati in pace i Prencipi Chri&longs;tiani, &longs;i dedicò al &longs;eruigio de' Sereniß. <lb/>&longs;uoi Signori, oue ne i più importanti carichi, & maggiori, & in due guer<lb/>re haue e&longs;&longs;a aggiunto cinquanta anni di noua, & ottima &longs;eruitù all'an<lb/>tica di qua&longs;i dugento anni, continua, & fedeli&longs;s. fattagli da i &longs;uoi pre<lb/>dece&longs;&longs;ori Sauorgnani, fabricando nello &longs;patio di que&longs;to tempo in diuer&longs;e pro<lb/>uincie de' &longs;uoi &longs;tati pre&longs;&longs;o che cinquanta Baloardi, con eccellentißima ra<lb/>gione inte&longs;i, & con vero magi&longs;terio lauorati, & notabilißimo ri&longs;parmio <lb/>del publico danaro. <emph.end type="italics"/></s> <s id="id.2.1.25.1.0"><emph type="italics"/>Ma per tornare alle Mechaniche dico, che quando gli anni pa&longs;&longs;ati io <lb/>venni à vi&longs;itarla ad O&longs;opo &longs;ua fortezza, &longs;entì &longs;ommo piacere in &longs;corgere <lb/>quel monte, che circonda più d'un miglio, &longs;ituato alla foce del fiume Ta<lb/>gliamento, oue dalle &longs;trettezze di quei gioghi s'allarga nelle pianure del <lb/>Friuli, d'ogn'intorno alto pre&longs;&longs;o che &longs;e&longs;&longs;anta paßi à mi&longs;ura, tutto di ma<lb/>cigno duro, & di&longs;co&longs;ce&longs;e, & erto sì, che rende la &longs;alita impoßibile fornito <lb/>attorno di baloardi cauati nel &longs;a&longs;&longs;o, & di molti tagli, & canoniere per <lb/>ferire gli aduer&longs;ari, & di artiglierie, & d'arme d'ogni &longs;orte à &longs;ufficienza, <lb/>da cui &longs;i hà vi&longs;ta di qua&longs;i tutto il Friuli, & è &longs;cudo, & riparo, come al<lb/>tra volta fù, contra l'empito delle genti nemiche, lequali in Italia tenta&longs;<lb/>&longs;ero di &longs;cendere da quella parte; po&longs;to di co&longs;ta alla &longs;trada principale, che <lb/>conduce in Lamagna, per laqual vanno, & vengono Signori, & Principi, <lb/>& Amba&longs;ciadori, & infinite mercatantie; onde ella, che tiene &longs;empre le <lb/>guardie, & vedette sù quel monte, quando pa&longs;&longs;ano Signori principali, <lb/>hà per co&longs;tume di &longs;alutargli con le &longs;ue artiglierie, & conuitarg'i anco <lb/>nel &longs;uo alloggiamento d'O&longs;opo, oue tutto l'anno &longs;oggiorna, quantunque <lb/>habbia & Belgrado, & Aris, & Ca&longs;telnouo, & Sauorgnano, & villaggi <lb/>a&longs;&longs;ai: percioche l'aere vi è purißimo, & &longs;pende il &longs;uo tempo in ocio con ne<lb/>gocio, di continuo vi&longs;itata da Gentil'huomini, & Signori diuer&longs;i; talche la <lb/>&longs;ua ca&longs;a viene ad e&longs;&longs;ere vn ridotto di per&longs;one virtuo&longs;e, & vn'albergo di &longs;ol <lb/>dati, & di dottori. </s> <s id="id.2.1.25.2.0">lui &longs;i caualca, tenendo ella vna &longs;talla piena di buoni&longs;<lb/>&longs;imi caualli, &longs;i armeggia, &longs;i và alla caccia, & in ogni attione &longs;i e&longs;ercita vi<lb/>ta cauallere&longs;ca. </s> <s id="id.2.1.25.3.0">Oltre à quanto hò diui&longs;ato, pre&longs;i anco diletto in vedere la <lb/>&longs;ua habitatione e&longs;&longs;ere à gui&longs;a d'vna bottega d'arme politamente à &longs;uoi <lb/>luoghi &longs;erbate: & vn magazino di machine bellico&longs;e, & da mouer pe&longs;i,<emph.end type="italics"/><pb xlink:href="037/01/011.jpg"/><emph type="italics"/>hauendone ella fabricate di &longs;ua indu&longs;tria for&longs;e dodici di maniere differen<lb/>ti, parte da &longs;tra&longs;cinare, & parte da alzare con pochißima forza &longs;mi&longs;u<lb/>rati pe&longs;i: come quella, che hà vna &longs;ola rota co' denti, & ali'erta tira cin<lb/>que de' &longs;uoi canoni con la po&longs;&longs;anza di Grada&longs;&longs;o &longs;uo Nano: & quell'altra, la <lb/>quale con vna oncia di forza &longs;ola, po&longs;ta nel manico, che la volge, dà il me<lb/>to à quattordici mila libre di pe&longs;o: che &longs;e al detto manico &longs;i attribui&longs;ce <lb/>la forza, che comunalmente haue l'huomo con la mano, cioè libre cinquan<lb/>ta, egli è manife&longs;to la predetta machina hauere po&longs;&longs;anza di mouere, co&longs;a <lb/>incredibile, molto più di otto millioni di libre. </s> <s id="id.2.1.25.4.0">Que&longs;te machine portabi<lb/>li da vn mulo, & alcune anche da vn'huomo &longs;ono à diuer&longs;i affari nece&longs;&longs;a<lb/>rijßime, & maßimamente à maneggiare, & condurrei pezzi großi del<lb/>l'artiglieria. </s> <s id="id.2.1.25.5.0">& per certo &longs;e l'anno<emph.end type="italics"/> 1529. <emph type="italics"/>il Conte di San Polo Capitano <lb/>France&longs;e nel ritirar&longs;i dall'a&longs;&longs;edio di Milano inuer&longs;o Piemonte con l'e&longs;er<lb/>cito, & con l'artiglieria, haue&longs;&longs;e portato &longs;eco vno de' minimi i&longs;trumenti <lb/>d'O&longs;opo, non &longs;arebbe &longs;cor&longs;o in quello &longs;tremo infortunio, percioche in mar<lb/>ciando fu da vn graue canone rotto il ponte, che trauer&longs;aua il fo&longs;&longs;o della <lb/>&longs;trada, & il pezzo cadè nel fango. </s> <s id="id.2.1.25.6.0">Onde formoßi il campo per non la<lb/>&longs;ciarlo à dietro, & non hauendo ingegno da cauarlo fuori, &longs;i con&longs;umò tan<lb/>to tempo, che &longs;opragiun&longs;e Antonio da Leua con le &longs;ue genti, & ritrouan<lb/>do l'e&longs;&longs;ercito nemico &longs;eparato, & in quel di&longs;ordine, lo mi&longs;e in rotta, & fè <lb/>preda delle bagaglie, delle artiglierie, & del Capitano mede&longs;mo. </s> <s id="id.2.1.25.7.0">Non hà <lb/>troppo tempo, che il Duca France&longs;co di Gui&longs;a, allhor che di Francia guidò <lb/>l'e&longs;ercito in Abruzzo, douendo partire, volle &longs;piegare prima la fanteria, <lb/>& cauælleria &longs;ua in ordinanza à fronte del nemico, qua&longs;i à battaglia sfi<lb/>dandolo; ma poi nel ritorno &longs;caualco&longs;si vn pezzo d'artiglieria, & s'arre<lb/>&longs;tò tutta la ma&longs;&longs;a delle genti, & quei Prencipi France&longs;i &longs;montati da ca<lb/>uallo, penarono buona pezza auanti, che lo ripone&longs;&longs;ero &longs;u le rote, con ri&longs;chio <lb/>di patir danno da gli aduer&longs;ari, che haue&longs;&longs;ero con quella occa&longs;ione &longs;pinto <lb/>innanzi. </s> <s id="id.2.1.25.8.0">Di que&longs;ti e&longs;empi non mancano per l'hi&longs;torie. <emph.end type="italics"/></s> <s id="id.2.1.26.1.0"><emph type="italics"/>Hora che è pace V. S. Illu&longs;triß. è andata inue&longs;tigando per &longs;uo diporto <lb/>molte, & varie &longs;orti di ordigni da mouer pe&longs;i, affine di valer&longs;ene nelle <lb/>fabriche, & nell'argine di pietre, che fa per ritenere l'impeto del Taglia<lb/>mento, che non gua&longs;ti i colti di O&longs;opo, & per douer&longs;ene anco &longs;eruire, quan<lb/>do che &longs;ia in guerra. </s> <s id="id.2.1.26.3.0">Si come fece Archimede, ilquale, &longs;econdo Plutarco, &longs;tan<lb/>do in pace à petitione di Hierone Rè, compo&longs;e quelle <expan abbr="tãte">tante</expan> Machine per giuo<lb/>co, & i&longs;cherzo di Geometria, lequali poi &longs;oprauenendo la guerra, le &longs;eppe <expan abbr="cõuertire">con<lb/>uertire</expan> opportunamente contra Romani. </s> <s id="id.2.1.26.4.0">Et &longs;e egli, come te&longs;tificano diuer&longs;i <emph.end type="italics"/><pb xlink:href="037/01/012.jpg"/><emph type="italics"/>autori, &longs;edendo con certa machina detta, &longs;econdo Oriba&longs;io, Tri&longs;pa&longs;ton, per <lb/>che &longs;i maneggiaua con tre corde, tirò dal mare in terra quella gran naue <lb/>del Rè &longs;uo; & con la forza della mano &longs;ini&longs;tra mo&longs;&longs;e mediante l'i&longs;tru<lb/>mento vn pe&longs;o di cinque mila &longs;taia ò moggia, sì fattamente che diputan<lb/>do à cia&longs;cuno &longs;taio quarantacinque libre di pe&longs;o, a&longs;cenderebbono alla &longs;om<lb/>ma di dugento venticinque mila libre; & pre&longs;umeua&longs;i di hauer potuto <lb/>mouere la terra, trouando doue fermar&longs;i con la leua, ò con quella &longs;ua ma<lb/>china de&longs;critta da Pappo nell'ottauo libro delle raccolte matematiche, la <lb/>quale hauea cinque rote co' &longs;uoi as&longs;i, & vna vite perpetua co'l manico: Io <lb/>mi rendo certo, che ella s'ingegnerebbe di formare i&longs;trumenti per adoprare <lb/>altretanto. <emph.end type="italics"/></s> <s id="id.2.1.27.1.0"><emph type="italics"/>Hauendo io dunque veduti, & i&longs;perimentati que&longs;ti vari difici ad O&longs;o<lb/>po; & es&longs;endomi &longs;tato da lei mo&longs;trato la prima volta il pre&longs;ente libro, & <lb/>commendato &longs;ommamente, mi propo&longs;i nell' animo, che vtile &longs;arebbe il ri<lb/>durlo in volgare, accioche coloro i quali &longs;ono atti per altro ad intenderlo, <lb/>ma non hanno cono&longs;cenza del Latino, pote&longs;sero, farne &longs;uo profitto. </s> <s id="id.2.1.27.2.0">Co&longs;i <lb/>compiuta l'opera, & fattala &longs;tampare, la mando à V. S. Illu&longs;tri&longs;s. che po&longs;<lb/>&longs;ede e&longs;qui&longs;itamente que&longs;ta materia, & &longs;econda i &longs;tudi delle buone lette<lb/>re, i quali, &longs;e dopo Iddio, non vengono fauoriti da i gran Signori, nulla va<lb/>gliono. </s> <s id="id.2.1.27.4.0">Che &longs;e in qualche parte haurò à gli amatori delle Mechaniche re<lb/>cata ageuolezza, & vtilità con le mie fatiche, douranno eglino &longs;aper' à <lb/>lei buon grado, che di que&longs;ta fattura è &longs;tata cagione. <lb/></s> <s id="id.2.1.27.5.0">Di Venetia à<emph.end type="italics"/> 28. <emph type="italics"/>di Giugno<emph.end type="italics"/> 1581. </s> <s id="id.2.1.28.1.0"><emph type="italics"/>Di V. S. Illu&longs;tri&longs;s. <lb/>Affettionatiß. &longs;eruidore <lb/>Filippo Pigafetta. <emph.end type="italics"/></s> </section> <pb xlink:href="037/01/013.jpg"/> <section> <s id="id.2.1.31.1.0">AI LETTORI</s> <s id="id.2.1.32.1.0">Il pre&longs;ente libro contiene &longs;ei trattati, il primo de <lb/>quali è della Bilancia con la Stadera, l'altro della <lb/>Leua, il terzo della Taglia, il quarto dell' A&longs;&longs;e nel<lb/>la rota, il quinto del Cuneo, & l'vltimo della Vite, <lb/>che tutti &longs;ono i&longs;trumenti Mechanici. </s> <s id="id.2.1.32.2.0">Intitula&longs;i le <lb/>Mechaniche. </s> <s id="id.2.1.32.3.0">Ma percioche que&longs;ta parola Mechaniche non ver<lb/>rà for&longs;e inte&longs;a da cia&longs;cheduno per lo &longs;uo vero &longs;ignificato, anzi <lb/>troueran&longs;i di quelli, che &longs;timeranno lei e&longs;&longs;ere voce d'ingiuria, <lb/>&longs;olendo&longs;i in molte parti d'Italia dire ad altrui Mechanico per <lb/>i&longs;cherno, & villania; & alcuni per e&longs;&longs;ere chiamati Ingegnieri &longs;i <lb/>prendono &longs;degno: non &longs;arà per auentura fuori di propo&longs;ito il <lb/>ricordare, che Mechanico è vocabolo honorati&longs;&longs;imo, dimo&longs;tran<lb/>te, &longs;econdo Plutarco, me&longs;tiero alla Militia pertinente, & conue<lb/>neuole ad huomo di alto affare, & che &longs;appia con le &longs;ue mani, <lb/>& co'l &longs;enno mandare ad e&longs;ecutione opre marauiglio&longs;e à &longs;ingu<lb/>lare vtilità, & diletto del viuere humano. </s> <s id="id.2.1.33.1.0">Fù, per nominarne alcuno tra molti Filo&longs;ofi, & Prencipi de' <lb/>preteriti &longs;ecoli, Archita Tarentino, & Eudo&longs;&longs;o <expan abbr="cõpagni">compagni</expan> di Pla<lb/>tone, & valenti&longs;&longs;imi Ingegnieri, & Mechanici, che &longs;ono vna me<lb/>de&longs;ma co&longs;a, di cui fa Plutarco mentione nella vita di Marcello: <lb/>& Demetrio Rè, inuentore &longs;ottili&longs;&longs;imo di Machine bellico&longs;e, <lb/>& ne lauoraua di &longs;ua mano ancora: & fra Greci di Sicilia Me<lb/>chanico, & Ingegniere famo&longs;is&longs;imo Archimede Siracu&longs;ano, il <lb/>quale era di <expan abbr="grã">gran</expan> legnaggio, & parente di Hierone Rè di Sicilia. </s> <s id="id.2.1.34.1.0">Et quantunque Plutarco nell'i&longs;te&longs;&longs;a vita affermi, che egli di <lb/>&longs;pregia&longs;&longs;e le Mechaniche, come bas&longs;i & vili, & materiali, nè di <lb/>loro degna&longs;&longs;e &longs;criuere giamai, & che non per opera principale, <lb/>ma per vn cotale &longs;ollazzo, & giuoco di Geometria impiegaua <lb/>la fatica nelle Mechaniche, pregato da quel Rè; sì leggiamo <lb/>noi tuttauia in altri autori, lui hauere dettato vn libro della mi <lb/>&longs;ura, & proportione d'ogni maniera di va&longs;ello, diui&longs;ando la for<lb/>ma della gran naue fabricata da Hierone, à cui nulla manca<lb/>ua: & Pappo Ale&longs;&longs;andrino allega il libro della Bilancia di Ar<lb/>chimede, che è pur Mechanico tutto: & l'i&longs;te&longs;&longs;o nell'ottauo del<lb/>le raccolte Matematiche pone vn'i&longs;trumento da mouer pe&longs;i, <pb xlink:href="037/01/014.jpg"/>mo&longs;trando e&longs;&longs;ere il quarante&longs;imo trouato d' Archimede, per cui <lb/>di&longs;&longs;e; Dami oue io mi fermi, ch'io mouerò la terra; & Carpo <lb/>Mechanico &longs;cri&longs;&longs;e, che Archimede compo&longs;e vn libro del modo <lb/>del fare le Sfere, che è fattura Mechanica. </s> <s id="id.2.1.39.2.0">Ma più il mede&longs;imo <lb/>Archimede, non vna &longs;ola volta cita &longs;e &longs;te&longs;&longs;o, nel libro della Qua<lb/>dratura della Parabola, con parole tali. </s> <s id="id.2.1.39.3.0">Imperoche egli è dimo<lb/>&longs;trato nelle Mechaniche; accennando alcune propo&longs;itioni del <lb/>&longs;uo libro delle co&longs;e, che egualmente pe&longs;ano, ilquale è tutto Me<lb/>chanico. </s> <s id="id.2.1.39.4.0">Oltre à ciò vna parte del libro della Quadratura della <lb/>Parabola, & il &longs;econdo delle co&longs;e, che &longs;tanno &longs;opra l'acqua, oue<lb/>ro à galla &longs;ono Mechanici. </s> <s id="id.2.1.39.5.0">Da que&longs;ti luoghi vede&longs;i e&longs;pre&longs;&longs;o, che <lb/>non &longs;olamente Archimede fece opre Mechaniche, ma ne &longs;cri&longs;&longs;e <lb/>anco molti trattati; & confe&longs;&longs;a Plutarco per niuna altra dottri<lb/>na e&longs;&longs;ere tanto in riputatione &longs;alito Archimede, quanto per le <lb/>impre&longs;e Mechaniche; anzi veramente co'l mezo loro hauer&longs;i egli <lb/>all'hora procacciato fama non di &longs;cienza humana, ma di &longs;apien<lb/>za diuina. </s> <s id="id.2.1.39.6.0">Per la qual co&longs;a egli è ben da con&longs;iderare, come Plu<lb/>tarco &longs;i &longs;ia la&longs;ciato tra&longs;correr' à dire, che Archimede le Mechani<lb/>che di&longs;preggia&longs;&longs;e, nè di loro degna&longs;&longs;e &longs;criuere: & per certo egli <lb/>forte d'opinione &longs;arebbe&longs;i <expan abbr="ingãnato">ingannato</expan>, &longs;e haue&longs;&longs;e poco &longs;timata quel <lb/>la facultà, che lo fè guadagnare gloria di gran lunga maggio<lb/>re, che qualunque altra &longs;cienza &longs;i po&longs;&longs;ede&longs;&longs;e. </s> <s id="id.2.1.39.7.0">Vitruuio de i <lb/>Latini fù buon Mechanico, & &longs;eruì per Capitano delle Bali&longs;te, <lb/>& delle altre machine da guerra Ottauiano Ce&longs;are, & gli intitu<lb/>lò le &longs;ue fatiche dell' Architettura, & ne diuenne ricco. </s> <s id="id.2.1.40.1.0">L'e&longs;&longs;ere Mechanico dunque, & Ingegniero con l'e&longs;empio di <lb/>tanti valent'huomini, è officio da per&longs;ona degna, & &longs;ignorile: <lb/>& Mechanica è voce Greca &longs;ignificante co&longs;a fatta con artificio <lb/>da mouere, come per miracolo, & fuori dell'humana po&longs;&longs;anza <lb/>grandis&longs;imi pe&longs;i con picciola forza, & in generale comprende <lb/>cia&longs;cun Dificio, Ordigno, I&longs;trumento, Argano, Mangano, oue<lb/>ro ingegno mae&longs;treuolmente ritrouato, & lauorato per cotali ef<lb/>fetti, & &longs;imili altri infiniti in qual &longs;i voglia &longs;cienza, arte, & e&longs;er<lb/>citio. </s> <s id="id.2.1.40.2.0">Laquale hò de&longs;critta co&longs;i materialmente per darne vn cer<lb/>to &longs;aggio accommodato al gu&longs;to del più de gli huomini; trala<lb/>&longs;ciando le accurate di&longs;&longs;initioni à miglior tempo. </s> <s id="id.2.1.41.1.0">Aggiunga&longs;i, che &longs;otto que&longs;to vniuer&longs;ali&longs;&longs;imo titolo &longs;i è con<pb xlink:href="037/01/015.jpg"/>tentato l'Autore di manife&longs;tare per hora, & il primo de Latini <lb/>con dimo&longs;trationi ageuoli, & piane, in&longs;egnare &longs;olamente la ra<lb/>gion dello intendere, & maneggiare gli &longs;ei predetti I&longs;trumenti <lb/>Mechanici; à quali &longs;i riducono tutti gli altri, come à &longs;uoi prin<lb/>cipii, & fondamenti; & da'quali &longs;i po&longs;&longs;ono comporne diuer&longs;e ma<lb/>niere, accozzandone in&longs;ieme due, tre, & più, come l'A&longs;&longs;e nella <lb/>rota con la Taglia, la Vite co'l detto A&longs;&longs;e, & con la Leua, & &longs;uc<lb/>ces&longs;iuamente de gli altri ad arbitrio di chiunque in varie opre &longs;e <lb/>ne sà con giudicio valere, come nota l'Autore nel fine di que&longs;to <lb/>volume. </s> <s id="id.2.1.35.1.0">Hor come che l'Autore con bella via, & chiara, & con ordine <lb/>ammirabile di que&longs;ti difici habbia ragionato, & la co&longs;a per &longs;e <lb/>molto o&longs;cura non &longs;ia ad intender&longs;i: nondimeno ben ricerca ella <lb/>tutto l'intelletto dell'huomo, & che con &longs;i&longs;&longs;a &longs;peculatione &longs;i leg<lb/>gano attentis&longs;imamente più d'vna volta le dimo&longs;trationi. </s> <s id="id.2.1.36.1.0">Doue &longs;i vede in alcuni luoghi di que&longs;ti trattati cotale &longs;orte di <lb/>lettere picciole, differente dalle altre, come la pre&longs;ente; auer<lb/>ta&longs;i che non vi &longs;ono co&longs;e dettate dall' Autore di que&longs;to libro di <lb/>Mechaniche, ma notate da colui che l'hà volgarizato, à fine di <lb/>chiarire qualche pa&longs;&longs;o difficile, & ageuolare l'intendimento à' <lb/>Lettori non co&longs;i prattichi nelle Scole de' Filo&longs;ofi. </s> <s id="id.2.1.37.1.0">Ponga&longs;i anco mente, che à carte 121. nel trattato della Vite, <lb/>è po&longs;to fra i detti dell' Autore il Problema di Pappo, ilquale do<lb/>uea e&longs;&longs;ere &longs;tampato con lettere differenti dalle altre, ma per in<lb/>auertenza è &longs;tato me&longs;&longs;o co' caratteri &longs;te&longs;&longs;i delle propo&longs;itioni del<lb/>l' Autore, che è difetto. </s> <s id="id.2.1.37.2.0">Non è &longs;tato pos&longs;ibile &longs;chiuare alcuni <lb/>falli nello &longs;tampare. </s> <s id="id.2.1.37.3.0">Onde correggan&longs;i in que&longs;ta maniera. </s> <s id="id.2.1.37.4.0">Nel <lb/>la Lettera à carte 1. faccia 2. ver&longs;i 25. to&longs;&longs;enoni, leggi tollenoni. </s> <s><lb/>car. 43. ver. 22. dell'angolo, all'angolo. </s> <s>carte 48. f. 2. nella po<lb/>&longs;tilla, per la 2. di que&longs;to; della 2. di que&longs;to. </s> <s>carte 87. f. 2. <lb/>ver. 14. dalla, alla. </s> <s>carte 93. ver. 32. cni, cui. </s> <s>carte 115. ver. 20. <lb/>Hlici, Helici. </s> <s id="id.2.1.37.17.0">Gli altri errori di lettere meno importanti, & che <lb/>non mouono il &longs;en&longs;o alla, di&longs;cretione del giudicio&longs;o Lettore &longs;i ri <lb/>mettono. </s> </section> <pb xlink:href="037/01/016.jpg"/> <section> <s id="id.2.1.68.1.0">TRATTATI IN QVEST'OPERA <lb/>CONTENVTI.<lb/></s> <table> <row> <cell>I. </cell> <cell>Della Bilancia, con la Stadera à carte </cell> <cell>1 </cell> </row> <row> <cell>II. </cell> <cell>Della Leua. </cell> <cell>35 </cell> </row> <row> <cell>III. </cell> <cell>Della Taglia. </cell> <cell>56 </cell> </row> <row> <cell>IIII. </cell> <cell>Dell' A&longs;&longs;e nella Rota. </cell> <cell>102 </cell> </row> <row> <cell>V. </cell> <cell>Del Cuneo. </cell> <cell>107 </cell> </row> <row> <cell>VI. </cell> <cell>Della Vite. </cell> <cell>115 </cell> </row> </table> </section> <pb pagenum="1" xlink:href="037/01/017.jpg"/> <section> <s id="id.2.1.43.1.0">LIBRO DI <lb/>MECHANICHE, <lb/>DELL'ILLVSTRISSIMO <lb/>SIGNORE, <lb/>II. S. GVIDO VBALDO DE' MARCHESI <lb/>DEL MONTE. </s> </section> </front> <body id="id.2.0.0.0.0"> <chap> <s id="id.2.1.45.1.0">Diffinitioni. </s> <s id="id.2.1.46.1.0">Il centro della grauezza di cia&longs;cun corpo e vn <lb/>certo punto po&longs;to dentro, dal quale &longs;e con la <lb/>imaginatione s'intende e&longs;&longs;erui appe&longs;o il gra<lb/>ue, mentre è portato &longs;ta fermo, & mantiene <lb/>quel &longs;ito, che egli hauea da principio, ne in <lb/>quel portamento &longs;i và riuolgendo. </s> <s id="id.2.1.47.1.0"><emph type="italics"/>Que&longs;ta diffinitione del centro della grauezza in&longs;egnò <lb/>Pappo Ale&longs;&longs;andrino nell'ottauo libro delle raccolte ma<lb/>thematiche. </s> <s id="id.2.1.47.2.0">Ma Federico Comandino nel libro del cen<lb/>tro della grauezza de' corpi &longs;olidi dichiarò l'i&longs;te&longs;&longs;o centro in questa maniera de&longs;cri<lb/>uendolo. <emph.end type="italics"/></s> <s id="id.2.1.48.1.0">Il centro della grauezza di cia&longs;cuna figura &longs;olida è quel punto po&longs;to <lb/>dentro, d'intorno alquale le parti di momenti eguali da ogni parte <lb/>&longs;i fermano. </s> <s id="id.2.1.48.2.0">Peroche &longs;e per tale centro &longs;arà condotto vn piano, che <lb/>&longs;eghi in qual &longs;i voglia modo la figura, &longs;empre la diuiderà in parti, <lb/>che pe&longs;eranno egualmente. </s> <pb xlink:href="037/01/018.jpg"/> <s id="id.2.1.50.1.0">NOTITIE COMVNI. </s> <s id="id.2.1.51.1.0">I. </s> <s id="id.2.1.52.1.0">Se da co&longs;e egualmente pe&longs;anti &longs;i leneranno co&longs;e, che pur egualmente <lb/>pe&longs;ino, le re&longs;tanti pe&longs;eranno egualmente. </s> <s id="id.2.1.53.1.0">II. </s> <s id="id.2.1.54.1.0">Se à co&longs;e egualmente pe&longs;anti &longs;i aggiungeranno co&longs;e, che pur <expan abbr="egualm&etilde;te">egualmen<lb/>te</expan> pe&longs;ino, tutte in&longs;ieme pe&longs;eranno egualmente. </s> <s id="id.2.1.55.1.0">III. </s> <s id="id.2.1.56.1.0">Le co&longs;e, che all'i&longs;te&longs;&longs;o &longs;ono eguali in pe&longs;o, &longs;ono tra loro anco gra<lb/>ui egualmente. </s> <s id="id.2.1.57.1.0">PRESVPPOSTE. </s> <s id="id.2.1.58.1.0">I. </s> <s id="id.2.1.59.1.0">Di vno corpo è vn &longs;olo centro della grauezza. </s> <s id="id.2.1.60.1.0">II. </s> <s id="id.2.1.61.1.0">Il centro della grauezza di vn corpo è &longs;empre nel mede&longs;imo &longs;ito per <lb/>ri&longs;petto al &longs;uo corpo. </s> <s id="id.2.1.62.1.0">III. </s> <s id="id.2.1.63.1.0">I Pe&longs;i &longs;ono portati in giu &longs;econdo il centro della grauezza. </s> <s id="id.2.1.64.1.0">DIFFINITIONI. </s> <s id="id.2.1.64.2.0">La diffinitione è vn breue parlare, che manife&longs;ta, & inte<lb/>ramente dichiara la co&longs;a propo&longs;ta, &longs;i fattamente che non &longs;i po&longs;&longs;a trouare condi<lb/>tione, ouero accidente alcuno principale in e&longs;&longs;a co&longs;a, &longs;e la diffinitione è buona, <lb/>che non &longs;ia in virtù compre&longs;a, & detta da lui; come per e&longs;empio l'Autore qui di <lb/>&longs;opra da ad intendere che &longs;ia il centro della grauezza con due diffinitioni. </s> <s id="id.2.1.65.1.0">Le Notitie comuni poi &longs;ono certe &longs;entenze manife&longs;te al &longs;en&longs;o comune de gli huomi<lb/>ni, lequali pur che vi &longs;i ponga mente, &longs;ubito vdite, &longs;i intendono, & &longs;e le pre&longs;ta il <lb/>con&longs;entimento. </s> <s id="id.2.1.66.1.0">Ma la Pre&longs;uppo&longs;ta è diuer&longs;a, peroche mette per vero la co&longs;a co&longs;i e&longs;&longs;ere, come &longs;i pro<lb/>pone &longs;enza altro di&longs;cor&longs;o per farla cono&longs;cere. </s> </chap> <pb pagenum="2" xlink:href="037/01/019.jpg"/> <chap> <s id="id.2.1.70.1.0">DELLA BILANCIA</s> <s id="id.2.1.71.1.0">Avanti che &longs;i faccia mentione della Bilancia, accioche la <lb/>co&longs;a re&longs;ti più chiara, &longs;ia la Bilancia AB in linea diritta, & <lb/>CD la Truttina della Bilancia, laquale &longs;econdo la con&longs;uetu<lb/>dine comune &longs;tà &longs;empre à piombo dell'orizonte. </s> <s id="id.2.1.71.2.0">& il punto C im<lb/>mobile, d'intorno alquale &longs;i volge la Bilancia, &longs;i chiami il centro del<lb/>la bilancia, &longs;ia pur collo<lb/>cato di &longs;opra della bilan<lb/>cia, ò di &longs;otto, benche <lb/>non propriamente, che <lb/>non fa nulla Ma il CA, <lb/>& il CB &longs;iano le di&longs;tan<lb/>ze, & braccia della Bilan<lb/>cia, co&longs;i nomate. </s> <s id="id.2.1.71.3.0">& &longs;e <lb/>dal centro della bilancia <lb/>collocato di &longs;opra, ò di <lb/>&longs;otto della Bilancia, &longs;arà <lb/>tirata vna linea à piom<lb/>bo di AB, que&longs;ta &longs;i chia<lb/>merà perpendicolo, che <lb/>&longs;o&longs;terrà la Bilancia AB, <lb/>& &longs;empre &longs;tarà à piom<lb/>bo di e&longs;&longs;a Bilancia, mo<lb/>ua&longs;i ella in qual &longs;i voglia <lb/>modo. </s> <figure id="id.037.01.019.1.jpg" xlink:href="037/01/019/1.jpg"> </figure> <s id="id.2.1.73.1.0">Concio&longs;ia che in que&longs;to trattato della Bilancia, & ne gli altri ancora l'Autore v&longs;i <lb/>alcune parole, lequali non &longs;i &longs;ono potute tra&longs;portare commodamente in volga<lb/>re, per non e&longs;&longs;ere e&longs;&longs;e anco &longs;tate accettate in que&longs;ta lingua, ne inte&longs;e da ognuno, <lb/>io le ho la&longs;ciate co&longs;i latine. </s> <s id="id.2.1.73.2.0">Ma accioche non facciano difficultà à coloro, i quali <lb/>non intendono il latino, le andrò per tutto à fuoi luoghi dichiarando. </s> <s id="id.2.1.74.1.0">Nel re&longs;to poi delle parole mi &longs;ono attenuto più al chiaro, & all'v&longs;ato, che &longs;ia pos&longs;i<lb/>bile, & ho po&longs;to angolo retto, & linea retta in cambio di angolo diritto, & linea <lb/>diritta, & linea della direttione in lo co di linea della dirittura, & co&longs;i diretto per <lb/>diritto, & alcuna volta magnitudine in vece di grandezza, & angolo mi&longs;to per <lb/>me&longs;colato, & angolo curuilineo per di linee torte, & linea curua per torta, & &longs;oli<lb/>do per &longs;odo, & for&longs;e qualche altro vocabolo poco v&longs;ato in que&longs;ta no&longs;tra fauella, <lb/>&longs;timando che cote&longs;te parole &longs;iano per dimo&longs;trare maggiormente la co&longs;a, & la in<lb/>tentione dell' Autore: & etiandio de&longs;iderando, che &longs;i rendano famigliari, & dome<lb/>&longs;tiche in que&longs;ta &longs;cienza, talche ognuno le po&longs;&longs;a ageuolmente intendere. </s> <s id="id.2.1.75.1.0">Trutina è quella co&longs;a, che &longs;o&longs;tiene tutta la Bilancia, laquale Trutina pigli a il Perno, <lb/>ouero l'A&longs;&longs;etto, & noma&longs;i in que&longs;ti pae&longs;i Gioa, altroue Giouola, ouero l'orecchie <lb/>della Bilancia, & in altre contrade Scocca, talche non &longs;i troua &longs;in hora vocabolo, <pb xlink:href="037/01/020.jpg"/>che in Italia communemente vi &longs;i confaccia, ne alcuno di <expan abbr="qne&longs;ti">que&longs;ti</expan> &longs;arebbe inte&longs;o <lb/>per tutto. </s> <s id="id.2.1.75.2.0">Onde io ho &longs;critto co&longs;i la Trutina, &longs;perando, che &longs;i habbia à fare termi <lb/>ne, & parola generale à tutte le nationi d'Italia. </s> <s id="id.2.1.76.1.0">Perpendicolo vuol dire quella linea, che &longs;porge in fuori dal centro della Bilancia al <lb/>mezo di detta Bilancia, ilqual Perpendicolo è &longs;olamente nelle Bilancie, lequali han<lb/>no il centro di fuori della Bilancia, o &longs;ia di &longs;otto, ò &longs;ia di &longs;opra. </s> <s id="id.2.1.76.2.0">Ma quando il cen<lb/>tro della Bilancia è nel mezo di e&longs;&longs;a, all'hora non vi è que&longs;to Perpendicolo per e&longs;<lb/>&longs;ere il centro della Bilancia, & il mezo di e&longs;&longs;a vn'i&longs;te&longs;&longs;o punto. </s> <s id="id.2.1.76.3.0">Et que&longs;to Perpen<lb/>dicolo è co&longs;a imaginata dall' Autore &longs;olamente, & non da altri, per ageuolare al<lb/>cune dimo&longs;trationi della Bilancia, che di nouo ha inue&longs;tigate: & non è la linguet<lb/>ta, ne meno la linea della direttione, ò dirittura che &longs;i habbia à dire. </s> <s id="id.2.1.77.1.0">LEMMA. </s> <s id="id.2.1.78.1.0">Sia la linea AB à piombo dell'orizonte, & col diametro AB &longs;i de&longs;cri<lb/>ua il cerchio AEBD, il cui centro &longs;ia C. </s> <s id="id.2.1.78.2.0">Dico il punto B e&longs;&longs;ere <lb/>l'infimo luogo della circonferenza del cerchio AEBD, & il pun<lb/>to A il piu alto, & quali &longs;i voglian punti, come DE, i quali &longs;iano <lb/>però egualmente di&longs;tanti da A e&longs;&longs;ere egualmente po&longs;ti di &longs;otto, & <lb/>quelli che &longs;tanno piu da pre&longs;&longs;o ad e&longs;&longs;o A, e&longs;&longs;ere più alti di quelli, che <lb/>&longs;ono più da lunge. </s> <s id="id.2.1.79.1.0"><arrow.to.target n="note1"></arrow.to.target><emph type="italics"/>Allunghi&longs;i la linea AB fin al centro del mondo, <lb/>che &longs;ia F. </s> <s id="id.2.1.79.2.0">Dapoi &longs;ia pre&longs;o nella circonferenza <lb/>del cerchio qual &longs;i voglia punto, come G, & &longs;i <lb/>congiungano le linee FG FD FE. </s> <s id="id.2.1.79.3.0">Hor per<lb/>cioche BF è la minima linea di tutte quelle, <lb/>che dal punto F &longs;ono tirate alla circonferenza <lb/>AEBD, &longs;arà la BF minore della FG. </s> <s id="id.2.1.79.4.0">Per <lb/>laqual co&longs;a il punto B &longs;arà piu da pre&longs;&longs;o al pun<lb/>to F, che il G. </s> <s id="id.2.1.79.5.0">Et per cotesta ragione &longs;i dimo<lb/>strerà, che il punto B &longs;ta più da pre&longs;&longs;o al centro <lb/>del mondo di qual &longs;i voglia altro punto della cir<lb/>conferenza del cerchio AEBD. </s> <s id="id.2.1.79.6.0">Sarà dunque <lb/>il punto B l'infimo luogo della circonferenza del <lb/>cerchio AEBD. </s> <s id="id.2.1.79.7.0">Dapoi perche AF tirata <lb/>per lo centro è maggiore di GF, &longs;arà il punto A <lb/>più alto non &longs;olamente di G, ma etiandio di qual <lb/>&longs;i voglia altro punto della circonferenza del cer<lb/>chio AEBD. </s> <s id="id.2.1.79.8.0">Oltre à ciò perche DF, & FE <lb/>&longs;ono eguali, i punti DE &longs;aranno egualmente di <lb/>stanti dal centro del mondo. </s> <s id="id.2.1.79.9.0">Et e&longs;&longs;endo DF <lb/>maggiore di FG, &longs;arà il punto D, che è più da<lb/>pre&longs;&longs;o al punto A, più alto del punto G, lequali <lb/>co&longs;e tutte erano da mo&longs;trar&longs;i. <emph.end type="italics"/></s> <s id="id.2.1.80.1.0"><margin.target id="note1"></margin.target><emph type="italics"/>Per la ottaua del terzo. <emph.end type="italics"/></s> <figure id="id.037.01.020.1.jpg" xlink:href="037/01/020/1.jpg"> </figure> <pb pagenum="3" xlink:href="037/01/021.jpg"/> <s id="id.2.1.83.1.0">Que&longs;to vocabolo Lemma greco v&longs;ato da tutti i volgarizatori di Euclide, & da gli <lb/>altri Scrittori di Mathematica ancora, hò accettato anch'io. </s> <s id="id.2.1.83.2.0">Ma ben con tutto ciò <lb/>&longs;timo che egli habbia me&longs;tieri di vn poco di lume per e&longs;&longs;er inte&longs;o; & viene à dire, <lb/>&longs;i come nota Cicerone nel &longs;econdo della Diuinatione, co&longs;a che prima &longs;i prende <lb/>per render facile l'intendimento delle co&longs;e, lequali &longs;i hanno dapoi à mo&longs;trare, & <lb/><expan abbr="nõ">non</expan> è Pre&longs;uppo&longs;ta, perche ella <expan abbr="nõ">non</expan> &longs;i proua <expan abbr="cõ">con</expan> ragione, ma &longs;uppon&longs;i; ma il Lemma <lb/>&longs;i dimo&longs;tra, come in que&longs;to luogo, che prende il punto B e&longs;&longs;ere po&longs;to nell'infimo <lb/>&longs;ito della circonferenza del cerchio, & lo proua per douer&longs;ene valere nelle &longs;eguen<lb/>ti dimo&longs;trationi. </s> <s id="id.2.1.84.1.0">Doue in que&longs;to Lemma &longs;i dice, che la linea AB è à piombo dell'orizonte, intenda&longs;i <lb/>per orizonte il piano della campagna, & del terreno &longs;ottopo&longs;to, volendo dire ori<lb/>zonte parola greca vn cerchio, che termina la no&longs;tra veduta, & abbraccia & diui <lb/>de la metà della terra tutta. </s> <s id="id.2.1.84.2.0">Quando dunque &longs;i troua in que&longs;ti libri vna linea, oue<lb/>ro altra quantità e&longs;&longs;ere à piombo, ouero egualmente di&longs;tante, ò inchinata all'ori<lb/>zonte, intenda&longs;i per l'orizonte il piano della campagna, ò del terreno. </s> <s id="id.2.1.85.1.0">PROPOSITIONE I. </s> <s id="id.2.1.86.1.0">Se il pe&longs;o &longs;arà &longs;o&longs;tenuto nel centro della &longs;ua grauezza da linea diritta <lb/>non &longs;i fermerà giamai, &longs;e quella i&longs;te&longs;&longs;a linea non &longs;arà à piombo del<lb/>l'orizonte. </s> <figure id="id.037.01.021.1.jpg" xlink:href="037/01/021/1.jpg"> </figure> <s id="id.2.1.88.1.0"><emph type="italics"/>Sia il pe&longs;o A, & il centro della &longs;ua <lb/>grauezza B, ilqual pe&longs;o venga &longs;o <lb/>&longs;tenuto dalla linea CB. </s> <s id="id.2.1.88.2.0">Dico che <lb/>il pe&longs;o non è per fermar&longs;i giamai, <lb/>&longs;e CB non &longs;arà à piombo dell'o<lb/>rizonte. </s> <s id="id.2.1.88.3.0">Sia il punto C immobi<lb/>le, e&longs;&longs;endo co&longs;i nece&longs;&longs;ario, accio il <lb/>pe&longs;o &longs;ia &longs;o&longs;tenuto: & e&longs;&longs;endo il pun<lb/>to C immobile, &longs;e il pe&longs;o A de<lb/>ue&longs;i mouere, il punto B de&longs;criuerà <lb/>la circonferenza di vn cerchio, il <lb/>cui mezo diametro &longs;arà CB. </s> <s id="id.2.1.88.4.0">Per <lb/>laqual co&longs;a &longs;u'l centro A & con <lb/>lo &longs;patio BC &longs;i de&longs;criua il cerchio <lb/>BFDE. </s> <s id="id.2.1.88.5.0">& &longs;ia di prima BC à <lb/>piombo dell'orizonte, & &longs;ia tirata <lb/>&longs;in à D, & il punto C &longs;tia di &longs;ot<lb/>to al punto B. </s> <s id="id.2.1.88.6.0">Hor percioche il pe&longs;o A &longs;i moue in giù &longs;econdo il centro della gra<emph.end type="italics"/><arrow.to.target n="note2"></arrow.to.target><lb/><emph type="italics"/>uezza, il punto B &longs;i mouerà in giù, oue naturalmente inchina ver&longs;o il centro del mon<lb/>do per la linea diritta BD: tutto il pe&longs;o A dunque con B &longs;uo centro della gra<lb/>uezza, grauerà &longs;opra la linea diritta BC, & concio&longs;ia che il pe&longs;o venga &longs;o&longs;tenuto <lb/>dalla linea CB, la linea CB &longs;o&longs;terrà tutto il pe&longs;o A, &longs;opra laquale non puote mo<emph.end type="italics"/><pb xlink:href="037/01/022.jpg"/><emph type="italics"/>uer&longs;i in giù, e&longs;&longs;endogliene da e&longs;&longs;a <lb/>vietato. </s> <s id="id.2.1.88.7.0">Per la diffinitione dun<lb/>que del centro della grauezza, il <lb/>punto B & il pe&longs;o A &longs;taranno <lb/>in que&longs;to &longs;ito. </s> <s id="id.2.1.88.8.0">& quantunque il <lb/>B &longs;ia piu alto di qual &longs;i voglia al<lb/>tro punto del cerchio, tuttauia non <lb/>&longs;i mouerà in giù da que&longs;to &longs;ito per <lb/>la circonferenza del cerchio, pero<lb/>che non &longs;i inchinerà più ver&longs;o lo F, <lb/>che ver&longs;o lo E, per e&longs;&longs;ere nell'vna <lb/>parte & nell'altra eguale la di&longs;ce<lb/>&longs;a: ne il pe&longs;a A piu &longs;tà pendente <lb/>in vna parte che nell'altra, ilche <lb/>non auiene in qual &longs;i voglia altro <lb/>punto della circonferenza del cer<lb/>chio, eccettuato il D. </s> <s id="id.2.1.88.9.0">Sia il centro <lb/><figure id="id.037.01.022.1.jpg" xlink:href="037/01/022/1.jpg"></figure><lb/>della grauezza dell'i&longs;te&longs;&longs;o pe&longs;o, come in F, concio&longs;ia che la di&longs;ce&longs;a &longs;ia dal punto <lb/>F ver&longs;o il D, & la a&longs;ce&longs;a ver&longs;o il B, però il punto F mouera&longs;&longs;i in giù: & per<lb/>cioche non &longs;i puote mouere al centro del mondo per linea diritta, per e&longs;&longs;ere impe<lb/>dito dal punto C immobile per cau&longs;a della linea CF, ma ben &longs;i mouerà &longs;empre <lb/>in giù come richiede la &longs;ua natura: & e&longs;&longs;endo il D il luogo infimo, &longs;i mouerà per <lb/>la circonferenza FD finche peruenga in D, nelqual &longs;ito fermera&longs;&longs;i il pe&longs;o, & <lb/>re&longs;terà immobile, sì perche non &longs;i puote più mouere in giù per e&longs;&longs;ere attaccato al <lb/>punto C, sì anche percioche egli è &longs;o&longs;tenuto nel &longs;uo centro della grauezza. </s> <s id="id.2.1.88.10.0">Et <lb/>quando F &longs;arà in D, &longs;arà &longs;imilmente la FC in DC, & in&longs;ieme à piombo <lb/>dell'orizonte. </s> <s id="id.2.1.88.11.0">il pe&longs;o dunque non &longs;i fermerà giamai finche la linea CF non &longs;tia <lb/>à piombo dell'orizonte, che bi&longs;ognaua prouare. <emph.end type="italics"/></s> <s id="id.2.1.90.1.0"><margin.target id="note2"></margin.target><emph type="italics"/>Per la terza pre&longs;upposta di questo. <emph.end type="italics"/></s> <s id="id.2.1.91.1.0">Di quì &longs;i puote cauare, che il pe&longs;o &longs;ia pur &longs;o&longs;tenuto in vn dato punto <lb/>in qual &longs;i voglia modo, non &longs;tarà fermo giamai, &longs;e non quando la <lb/>linea tirata dal centro della grauezza del pe&longs;o à quel punto, &longs;tia à <lb/>piombo dell'orizonte. </s> <pb pagenum="4" xlink:href="037/01/023.jpg"/> <s id="id.2.1.93.1.0"><emph type="italics"/>Come, po&longs;te le co&longs;e i&longs;te&longs;&longs;e, &longs;ia &longs;o&longs;tenuto <lb/>il pe&longs;o dalle linee CG CH. </s> <s id="id.2.1.93.2.0">Dico <lb/>che &longs;e la tirata linea BC &longs;arà à <lb/>piombo dell'orizonte, il pe&longs;o &longs;tarà <lb/>fermo: ma &longs;e la tirata linea CF <lb/>non &longs;arà à piombo dell'orizonte, il <lb/>punto F &longs;imouerà in giù fin al D, <lb/>nel qual &longs;ito &longs;tarà fermo il pe&longs;o, <lb/>& la tirata linea CD &longs;arà à piom<lb/>bo dell'orizonte. </s> <s id="id.2.1.93.3.0">Le quali co&longs;e <lb/>tutte con la ragione mede&longs;ima &longs;i pro<lb/>uerebbono. <emph.end type="italics"/></s> <figure id="id.037.01.023.1.jpg" xlink:href="037/01/023/1.jpg"> </figure> <s id="id.2.1.95.1.0">PROPOSITIONE II. </s> <s id="id.2.1.96.1.0">La bilancia egualmente di&longs;tante dall'orizonte, il cui centro &longs;tia &longs;opra <lb/>la detta bilancia, & che habbia i pe&longs;i eguali nelle &longs;tremità, & egual<lb/>mente di&longs;tanti dal perpendicolo, &longs;e da cotale &longs;ito &longs;arà mo&longs;&longs;a, & <lb/>nell'i&longs;te&longs;&longs;o di nuouo la&longs;ciata, ritornerà, & iui re&longs;terà. </s> <figure id="id.037.01.023.2.jpg" xlink:href="037/01/023/2.jpg"> </figure> <s id="id.2.1.98.1.0"><emph type="italics"/>Sia la bilancia AB in <lb/>linea diritta egualmen<lb/>te di&longs;tante dall'orizon<lb/>te, il cui centro C &longs;ia <lb/>&longs;opra la bilancia, & <lb/>&longs;ia CD il perpendi<lb/>colo, il quale &longs;arà à <lb/>piombo dell'orizonte: <lb/>& la di&longs;tanza DA <lb/>&longs;ia eguale alla di&longs;tan<lb/>za DB: & &longs;iano i <lb/>pe&longs;i in AB eguali, <lb/>i centri della grauez<lb/>za de' quali &longs;iano ne i <lb/>punti AB. </s> <s id="id.2.1.98.2.0">Moua&longs;i <lb/>da que&longs;to &longs;ito la bi<lb/>lancia AB come in EF, dapoi &longs;ia la&longs;ciata. </s> <s id="id.2.1.98.3.0">Dico che la bilancia EF ritor<lb/>neràin AB di&longs;tante egualmente dall'orizonte, & iui rimanerà. </s> <s id="id.2.1.98.4.0">Hora percioche <emph.end type="italics"/><pb xlink:href="037/01/024.jpg"/><emph type="italics"/>il punto C &longs;tà immobì <lb/>le mentre la bilancia &longs;i <lb/>moue, il punto D veni <lb/>rà à de&longs;criuere vna cir<lb/>conferenza di cerchio, il <lb/>cui mezo diametro &longs;a<lb/>rà CD. </s> <s id="id.2.1.98.5.0">Per laqual <lb/>co&longs;a co'l centro D, & <lb/>lo &longs;patio CD de&longs;cri<lb/>ua&longs;i il cerchio DGH. <lb/></s> <s id="id.2.1.98.6.0">Et perche CD &longs;empre <lb/>&longs;tà à piombo della bi<lb/>lancia, mentre la bilan<lb/>cia &longs;arà in EF, la li<lb/>nea CD &longs;arà in CG <lb/>&longs;i fattamente, che CG <lb/><figure id="id.037.01.024.1.jpg" xlink:href="037/01/024/1.jpg"></figure><lb/>venga ad e&longs;&longs;ere à piombo di EF: & concio&longs;ia che AB &longs;ia diui&longs;a in due parti <emph.end type="italics"/><lb/><arrow.to.target n="note3"></arrow.to.target><emph type="italics"/>eguali nel punto D, & i pe&longs;i in AB &longs;iano eguali, &longs;arà etiandio il centro della <lb/>grauezza della magnitudine compo&longs;ta di que&longs;ti due corpi AB nel mezo, cioè in <lb/>D: & quando la bilancia in&longs;ieme co i pe&longs;i &longs;arà in EF, &longs;arà parimente G il cen<lb/>tro della grauezza della magnitudine compo&longs;ta di e&longs;&longs;i AB: & percioche CG <lb/>non è à piombo dell'orizonte, la grandezza compo&longs;ta de i pe&longs;i EF non rimarrà <emph.end type="italics"/><lb/><arrow.to.target n="note4"></arrow.to.target><emph type="italics"/>in questo &longs;ito, ma &longs;i mouerà in giù &longs;econdo il centro della grauezza &longs;ua, che è in <lb/>G, per la circonferenza GD, finche &longs;i faccia à piombo dell'orizonte, cioè finche <lb/>CG ritorni in CD. </s> <s id="id.2.1.98.7.0">Et quando CG &longs;arà in CD, la linea EF (perche &longs;em<lb/>pre &longs;tà ad angoli retti con CG) &longs;arà in AB, nelqual &longs;ito &longs;tarà ferma. </s> <s id="id.2.1.98.8.0">La bi<emph.end type="italics"/><lb/><arrow.to.target n="note5"></arrow.to.target><emph type="italics"/>lancia dunque EF ritornerà in AB, laquale è di&longs;tante egualmente dall'orizon<lb/>te, & iui rimarrà, che bi&longs;ognaua dimo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.100.1.0"><margin.target id="note3"></margin.target><emph type="italics"/>Per la quarta del primo di Archimede delle co&longs;e che pe&longs;ano egualmente. <emph.end type="italics"/></s> <s id="id.2.1.101.1.0"><margin.target id="note4"></margin.target><emph type="italics"/>Per la prima di questo. <emph.end type="italics"/></s> <s id="id.2.1.102.1.0"><margin.target id="note5"></margin.target><emph type="italics"/>Per la prima di questo. <emph.end type="italics"/></s> <s id="id.2.1.103.1.0">PROPOSITIONE III. </s> <s id="id.2.1.104.1.0">La bilancia egualmente di&longs;tante dall'orizonte, che habbia nelle &longs;tre<lb/>mità pe&longs;i eguali, & egualmente lontani dal perpendicolo, e&longs;&longs;endo <lb/>collocato il centro di &longs;otto, rimarrà in que&longs;to &longs;ito. </s> <s id="id.2.1.104.2.0">Ma &longs;e indi &longs;arà <lb/>mo&longs;&longs;a, & la&longs;ciata à ba&longs;&longs;o, &longs;i mouerà &longs;econdo la parte piu ba&longs;&longs;a. </s> <pb pagenum="5" xlink:href="037/01/025.jpg"/> <figure id="id.037.01.025.1.jpg" xlink:href="037/01/025/1.jpg"> </figure> <s id="id.2.1.107.1.0"><emph type="italics"/>Sia la bilancia AB in <lb/>linea diritta, egual<lb/>mente di&longs;tante dall'ori<lb/>zonte, il cui centro C <lb/>&longs;ia di &longs;otto alla bilan<lb/>cia, & &longs;ia CD il per<lb/>pendicolo, ilquale &longs;arà <lb/>à piombo dell'orizon<lb/>te, & la di&longs;tanza AD <lb/>&longs;ia eguale alla distan<lb/>za DB, & &longs;iano in <lb/>AB pe&longs;i eguali, i cen<lb/>tri della grauezza de' <lb/>quali &longs;iano ne' punti <lb/>AB. </s> <s id="id.2.1.107.2.0">Dico primiera<lb/>mente che la bilancia <lb/>AB &longs;tarà ferma in <lb/>que&longs;to &longs;ito. </s> <s id="id.2.1.107.3.0">Hor percioche AB &longs;i diuide in parti eguali nel punto D, & i <lb/>pe&longs;i po&longs;ti in AB &longs;ono eguali, &longs;egue, che il punto D &longs;ia il centro della grauez<lb/>za della magnitudine compo&longs;ta di ambedue i corpi me&longs;&longs;i in AB; & il CD che <emph.end type="italics"/><arrow.to.target n="note6"></arrow.to.target><lb/><emph type="italics"/>&longs;ostiene la bilancia &longs;tà à piombo dell'orizonte: Adunque la bilancia AB in <lb/>que&longs;to &longs;ito rimarrà ferma. </s> <s id="id.2.1.107.4.0">Ma da que&longs;to &longs;ito moua&longs;i la bilancia AB come in <lb/>EF, & la&longs;ci&longs;i dapoi. </s> <s id="id.2.1.107.5.0">Dico che la bilancia EF &longs;i mouerà dalla parte dello F. <lb/></s> <s id="id.2.1.107.6.0">Et percioche il CD &longs;tà &longs;empre à piombo della bilancia, mentre la bilancia &longs;arà <lb/>in EF verrà ad e&longs;&longs;ere anche il CD in CG à piombo di EF, & il punto <emph.end type="italics"/><arrow.to.target n="note7"></arrow.to.target><lb/><emph type="italics"/>G della magnitudine composta di EF &longs;arà il centro della grauezza, ilquale men<lb/>tre &longs;i moue de&longs;criuerà la circonferenza del cerchio DGH, il cui mezo diametro <lb/>è CD, & il centro C. </s> <s id="id.2.1.107.7.0">Ma perche CG non &longs;tà à piombo dell'orizonte, la <lb/>grandezza compo&longs;ta de i pe&longs;i EF non rimarrà in questo &longs;ito, ma &longs;econdo il cen<lb/>tro della &longs;ua grauezza &longs;i mouerà in giù per la circonferenza GH. </s> <s id="id.2.1.107.8.0">La bilancia <lb/>dunque EF &longs;i mouerà in giù dalla parte dello F, che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.108.1.0"><margin.target id="note6"></margin.target><emph type="italics"/>Per la quarta del primo d' Archimede delle co&longs;e che pe&longs;ano <expan abbr="egualm&etilde;te">egualmente</expan>. <emph.end type="italics"/></s> <s id="id.2.1.109.1.0"><margin.target id="note7"></margin.target><emph type="italics"/>Per la prima di questo. <emph.end type="italics"/></s> <s id="id.2.1.110.1.0">PROPOSITIONE IIII. </s> <s id="id.2.1.111.1.0">La bilancia egualmente di&longs;tante dall'orizonte, & che habbia nelle &longs;tre<lb/>mità pe&longs;i eguali, & egualmente di&longs;tanti dal centro collocato in e&longs;&longs;a <lb/>bilancia. </s> <s id="id.2.1.111.2.0">Se ella indi &longs;arà mo&longs;&longs;a, ò non, douunque ella &longs;arà la&longs;cia<lb/>ta, rimarrà. </s> <pb xlink:href="037/01/026.jpg"/> <s id="id.2.1.113.1.0"><emph type="italics"/>Sia la bilancia nella linea <lb/>diritta AB egualmen<lb/>te di&longs;tante dall'orizon<lb/>te, il cui centro C &longs;ia <lb/>nella i&longs;te&longs;&longs;a linea AB, <lb/>& la di&longs;tanza CA &longs;ia <lb/>eguale alla distanza <lb/>CB, & &longs;iano i pe&longs;i <lb/>AB eguali, i cui cen<lb/>tri della grauezza &longs;tia <lb/>no ne i punti AB. </s> <s id="id.2.1.113.2.0">Mo <lb/>ua&longs;i la bilancia come in <lb/>DE, & iui &longs;ia la&longs;cia<lb/>ta. </s> <s id="id.2.1.113.3.0">Dico primamen<lb/><figure id="id.037.01.026.1.jpg" xlink:href="037/01/026/1.jpg"></figure><lb/>te che la bilancia DE non &longs;i mouerà, & rimarrà in quel &longs;ito. </s> <s id="id.2.1.113.4.0">Hor percioche i <lb/>pe&longs;i AB &longs;ono eguali, &longs;arà il centro della grauezza della magnitudine compo&longs;ta <lb/>delli due pe&longs;i A & B in C. </s> <s id="id.2.1.113.5.0">Per laqual co&longs;a l'i&longs;te&longs;&longs;o punto C &longs;arà il centro <lb/>della bilancia, & il centro della grauezza di tutto il pe&longs;o. </s> <s id="id.2.1.113.6.0">Et percioche il centro <lb/>della bilancia che è C, mentre la bilancia AB in&longs;ieme co'pe&longs;i &longs;i moue in DE, <lb/>rimane immobile, non &longs;i mouerà ne anche il centro della grauezza, che è l'i&longs;te&longs;&longs;o C. <lb/></s> <s id="id.2.1.113.7.0">Adunque ne anche la bilancia DE &longs;i mouerà per la diffinitione del centro della <lb/>grauezza, e&longs;&longs;endo in e&longs;&longs;o appiccata. </s> <s id="id.2.1.113.8.0">L'i&longs;te&longs;&longs;o accade parimente &longs;tando la bilancia <lb/>AB egualmente di&longs;tante dall'orizonte, ouero e&longs;&longs;endo in qual &longs;i voglia altro &longs;ito. <lb/></s> <s id="id.2.1.113.9.0">Rimarrà dunque la bilancia oue &longs;arà la&longs;ciata, che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.115.1.0"><emph type="italics"/>Benche habbiamo con&longs;iderato nelle co&longs;e predette le grauezze &longs;olamente delle magni<lb/>tudini, le quali &longs;ono po&longs;te nelle &longs;tremità della bilancia, &longs;enza la grauezza della bi<lb/>lancia; niente di manco per e&longs;&longs;ere anche le braccia della bilancia eguali, auenirà lo <lb/>i&longs;te&longs;&longs;o alla bilancia, con&longs;iderata la &longs;ua grauezza in&longs;ieme co' pe&longs;i, ouero &longs;enza pe&longs;i, <lb/>percioche il centro iste&longs;&longs;o della grauezza &longs;enza pe&longs;i &longs;arà anco centro della grauez<lb/>za della bilancia &longs;ola. </s> <s id="id.2.1.115.2.0">Similmente &longs;e li pe&longs;i &longs;aranno appiccati nelle &longs;tremità del<lb/>la bilancia, come &longs;uole far&longs;i, auerrà l'iste&longs;&longs;o, purche le linee tirate da i punti oue &longs;o<lb/>no attaccati i pe&longs;i ver&longs;o i centri delle grauezze, (moua&longs;i la bilancia in qual &longs;i vo<lb/>glia modo) vadano à concorrere nel centro del mondo, peroche doue &longs;ono attaccati <lb/>i pe&longs;i in questa maniera, iui grauano, come &longs;e in quegli &longs;te&longs;&longs;i punti baue&longs;&longs;ero i cen<lb/>tri delle grauezze. </s> <s id="id.2.1.115.3.0">Oltre à ciò poßiamo con&longs;iderare le co&longs;e che &longs;eguono in tut<lb/>to al modo i&longs;te&longs;&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.116.1.0"><arrow.to.target n="note8"></arrow.to.target><emph type="italics"/>Ma percioche à que&longs;ta vltima conchiu&longs;ione molte co&longs;e dette da alcuni, che &longs;entono al<lb/>tramente, paiono contra&longs;tare; però in cote&longs;ta parte egli &longs;arà bi&longs;ogno dimorare <lb/>alquanto, & &longs;econdo le mie forze non &longs;olo farò opra di difendere la propria <lb/>mia &longs;entenza, ma Archimede ancora, ilquale &longs;embra e&longs;&longs;ere &longs;tato in que&longs;to i&longs;te&longs;<lb/>&longs;o parere. <emph.end type="italics"/></s> <s id="id.2.1.118.1.0"><margin.target id="note8"></margin.target><emph type="italics"/>Giord. </s> <s id="id.2.1.118.2.0">de' pe&longs;i. </s> <s id="id.2.1.118.3.0">Il Cardano della &longs;ottigliezza. </s> <s id="id.2.1.118.4.0">Il Tartaglia de' que&longs;iti, & <expan abbr="inu&etilde;tioni">inuentioni</expan><emph.end type="italics"/></s> <pb pagenum="6" xlink:href="037/01/027.jpg"/> <s id="id.2.1.119.1.0"><emph type="italics"/>Po&longs;te le co&longs;e i&longs;te&longs;&longs;e, &longs;ia <lb/>tirata la linea FCG <lb/>à piombo di AB, & <lb/>dell'orizonte: & col <lb/>centro C, & lo &longs;pa<lb/>tio CA &longs;ia de&longs;crit<lb/>to il cerchio ADFB <lb/>EG: &longs;aranno i punti <lb/>ADBE nella circon<lb/>ferenza del cerchio, <lb/>per e&longs;&longs;ere le braccia <lb/>della bilancia eguali. </s> <s id="id.2.1.119.2.0">& percioche conuen<lb/>gono que&longs;ti autori in <lb/>vna &longs;entenza, affer<lb/>mando, che la bilan<lb/>cia DE non &longs;i moue <lb/>in FG, ne rimane in<emph.end type="italics"/><lb/><figure id="id.037.01.027.1.jpg" xlink:href="037/01/027/1.jpg"></figure><lb/><emph type="italics"/>DE, ma ritorna nella linea AB egualmente di&longs;tante dall'orizonte, mo&longs;trerò que<lb/>&longs;ta loro opinione non potere à modo alcuno &longs;tare. </s> <s id="id.2.1.119.3.0">Percioche &longs;e egli è vero quel <lb/>che dicono, ouero auenirà questo effetto per e&longs;&longs;ere il pe&longs;o D più graue del pe&longs;o E, <lb/>ouero &longs;e li pe&longs;i &longs;ono eguali, le di&longs;tanze nelle quali &longs;ono po&longs;ti, non &longs;aranno eguali, <lb/>cioè la CD non &longs;arà eguale alla CE, ma più grande. </s> <s id="id.2.1.119.4.0">Ma che i pe&longs;i col<lb/>locati in DE &longs;iano eguali, & la di&longs;tanza CD &longs;ia eguale alla di&longs;tanza CE, è <lb/>chiaro dalla pre&longs;uppo&longs;ta. </s> <s id="id.2.1.119.5.0">Hor perche dicono che il pe&longs;o po&longs;to in D in quel &longs;i<lb/>to è più graue del pe&longs;o po&longs;to in E nell altro &longs;ito da ba&longs;&longs;o: mentre i pe&longs;i &longs;ono in <lb/>DE, non &longs;arà il punto C piu centro della grauezza, imperoche non stanno fer<lb/>mi &longs;e &longs;ono attaccati al C, ma &longs;arà nella linea CD per la terza del primo di Ar<lb/>chimede delle co&longs;e che pe&longs;ano egualmente. </s> <s id="id.2.1.119.6.0">Non &longs;arà già nella CE per e&longs;&longs;ere il <lb/>pe&longs;o D più graue del pe&longs;o E: &longs;ia dunque in H, nelquale &longs;e &longs;aranno attacca<lb/>ti, rimarranno. </s> <s id="id.2.1.119.7.0">Et percioche il centro della grauezza de' pe&longs;i congiunti in AB <lb/>&longs;tà nel punto C: ma de' pe&longs;i po&longs;ti in DE il punto è H: mentre dunque i pe&longs;i <lb/>AB &longs;i muouono in DE, il centro della grauezza C moueraßi ver&longs;o D, & <lb/>s'appre&longs;&longs;erà più da vicino al D, ilche è impoßibile, per mantenere i pe&longs;i vna me<lb/>de&longs;ima di&longs;tanza fra loro: peroche il centro della grauezza di cia&longs;cun corpo &longs;tà &longs;em<lb/>pre nel mede&longs;imo &longs;ito per ri&longs;petto al &longs;uo corpo. </s> <s id="id.2.1.119.8.0">Et quantunque il punto C &longs;ia il <emph.end type="italics"/><arrow.to.target n="note9"></arrow.to.target><lb/><emph type="italics"/>centro della grauezza di due corpi A. & B, tuttauia per e&longs;&longs;ere mediante la bi<lb/>lancia co&longs;i giunti in&longs;ieme, che &longs;empre &longs;i trouano nell'iste&longs;&longs;o modo; però il punto C <emph.end type="italics"/><arrow.to.target n="note10"></arrow.to.target><lb/><emph type="italics"/>&longs;arà co&longs;i centro della grauezza loro, come &longs;e fo&longs;&longs;e vna &longs;ola magnitudine; percio<lb/>che la bilancia in&longs;ieme co' pe&longs;i fa vn &longs;olo corpo continuo, il cui centro della grauez<lb/>za &longs;empre &longs;tarà nel mezo. </s> <s id="id.2.1.119.9.0">Non è dunque il pe&longs;o po&longs;to in D più graue del pe<lb/>&longs;o po&longs;to in E. </s> <s id="id.2.1.119.10.0">Che &longs;e dice&longs;&longs;ero il centro della grauezza non nella linea CD, ma<emph.end type="italics"/><pb xlink:href="037/01/028.jpg"/><emph type="italics"/>nella CE douer e&longs;&longs;ere, auerrà l'i&longs;te&longs;&longs;o fallo. <emph.end type="italics"/></s> <s id="id.2.1.121.1.0"><margin.target id="note9"></margin.target><emph type="italics"/>Per la &longs;econda &longs;upposta di questo. <emph.end type="italics"/></s> <s id="id.2.1.122.1.0"><margin.target id="note10"></margin.target><emph type="italics"/>Per la quarta del primo di Archime de delle co&longs;e che pe&longs;ano egualmente. <emph.end type="italics"/></s> <s id="id.2.1.123.1.0"><emph type="italics"/>Di più &longs;e il pe&longs;o D &longs;i mouerà in giù, mouerà il pe&longs;o E in sù. </s> <s id="id.2.1.123.2.0">Adunque vn pe&longs;o <lb/>più graue di E nel mede&longs;imo &longs;ito pe&longs;erà tanto quanto il pe&longs;o D, & auerrà che <lb/>co&longs;e graui di&longs;uguali, po&longs;te in eguale distanza pe&longs;eranno egualmente. </s> <s id="id.2.1.123.3.0">Aggiun<lb/>ga&longs;i dunque al pe&longs;o E qualche co&longs;a graue, &longs;i fattamente, che contrape&longs;i al D &longs;e <lb/>nel C &longs;aranno attac<lb/>cati. </s> <s id="id.2.1.123.4.0">Ma e&longs;&longs;endo &longs;ta<lb/>to di &longs;opra mo&longs;trato <lb/>il punto C e&longs;&longs;ere il c&etilde;<emph.end type="italics"/><lb/><arrow.to.target n="note11"></arrow.to.target><emph type="italics"/>tro della grauezza di <lb/>pe&longs;i eguali po&longs;ti in <lb/>DE; &longs;e dunque il pe<lb/>&longs;o. </s> <s id="id.2.1.123.5.0">E &longs;arà più graue <lb/>del pe&longs;o D, &longs;arà anche <lb/>il centro della grauez<lb/>za nella linea C E. </s> <s id="id.2.1.123.6.0">& &longs;ia que&longs;to centro <lb/>il<emph.end type="italics"/> K. </s> <s><emph type="italics"/>Ma per la diffi<lb/>nitione del centro del<lb/>la grauezza, &longs;e li pe&longs;i <lb/>&longs;aranno appiccati al <emph.end type="italics"/><lb/>K, <emph type="italics"/>staranno fermi. <lb/></s> <s id="id.2.1.123.7.0">Dunque &longs;e &longs;aranno <emph.end type="italics"/><lb/><figure id="id.037.01.028.1.jpg" xlink:href="037/01/028/1.jpg"></figure><lb/><emph type="italics"/>appiccati al C, non &longs;taranno fermi, che è contra la pre&longs;uppo&longs;ta: ma il pe&longs;o E &longs;i <emph.end type="italics"/><lb/><arrow.to.target n="note12"></arrow.to.target><emph type="italics"/>mouerà in giù. </s> <s id="id.2.1.123.8.0">Che &longs;e appiccati al C pe&longs;a&longs;&longs;ero ancora egualmente, na&longs;cerebbe <lb/>che di vna magnitudine, due &longs;arebbono i centri della grauezza, che è impo&longs;&longs;ibile. <lb/></s> <s id="id.2.1.123.9.0">Adunque il pe&longs;o po&longs;to in E più graue di quello che è in D, non pe&longs;erà tanto <lb/>quanto il D attaccando&longs;i al punto C. </s> <s id="id.2.1.123.10.0">I pe&longs;i dunque eguali po&longs;ti in DE, attac<lb/>cati nel centro della loro grauezza pe&longs;eranno egualmente, & &longs;taranno immobili, <lb/>che &longs;u proposto di mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.125.1.0"><margin.target id="note11"></margin.target><emph type="italics"/>Per la terza del primo di Archimede delle co&longs;e che pe&longs;ano egualmente. <emph.end type="italics"/></s> <s id="id.2.1.126.1.0"><margin.target id="note12"></margin.target><emph type="italics"/>Per la prima &longs;upposta di questo. <emph.end type="italics"/></s> <s id="id.2.1.127.1.0"><arrow.to.target n="note13"></arrow.to.target><emph type="italics"/>A que&longs;ta vltima &longs;conueneuolezza ri&longs;pondono, dicendo e&longs;&longs;ere impo&longs;&longs;ibile aggiungere al <lb/>lo E &longs;i picciolo pe&longs;o, che in ogni modo &longs;e ben &longs;i appiccano al C, il pe&longs;o E non <lb/>&longs;i moua &longs;empre in giù ver&longs;o il G. </s> <s id="id.2.1.127.2.0">La qual co&longs;a habbiamo noi pre&longs;uppo&longs;to poter&longs;i <lb/>fare, & credeuamo poter&longs;i fare: Peroche quel che è di più del pe&longs;o D &longs;opra <lb/>il pe&longs;o E, hauendo ragione, & parte di quantità, &longs;i imaginauamo non &longs;olamente <lb/>e&longs;&longs;ere minimo, ma ancora poter&longs;i diuidere in infinito, il che eßi per certo non &longs;ola<lb/>mente minimo, ma ne anche e&longs;&longs;ere minimo, non potendo&longs;i ritrouare, &longs;i sforzano di <lb/>mo&longs;trare in que&longs;ta maniera. <emph.end type="italics"/></s> <s id="id.2.1.129.1.0"><margin.target id="note13"></margin.target><emph type="italics"/>Il Tartaglia nella &longs;esta propo&longs;itione del quarto libro. <emph.end type="italics"/></s> <pb pagenum="7" xlink:href="037/01/029.jpg"/> <s id="id.2.1.130.1.0"><emph type="italics"/>Pongan&longs;i le co&longs;e iste&longs;&longs;e <lb/>& da i punti DE <lb/>&longs;iano tirate le linee <lb/>DHE<emph.end type="italics"/>K <emph type="italics"/>à piombo <lb/>dell'orizonte, & &longs;ia <lb/>vn'altro cerchio L <lb/>DM, il cui centro <lb/>&longs;ia N, ilquale toc<emph.end type="italics"/><arrow.to.target n="note14"></arrow.to.target><lb/><emph type="italics"/>chi FDG nel pun<lb/>to D, & &longs;ia eguale <emph.end type="italics"/><arrow.to.target n="note15"></arrow.to.target><lb/><emph type="italics"/>ad FDG. </s> <s id="id.2.1.130.2.0">Sarà <lb/>NC linea retta: & <lb/>perche l'angolo<emph.end type="italics"/> K <lb/><emph type="italics"/>EC è eguale all'an<lb/>golo HDN, & <lb/>l'angolo CEG è pa<lb/>rimente eguale al<lb/>l'angolo NDM,<emph.end type="italics"/><lb/><figure id="id.037.01.029.1.jpg" xlink:href="037/01/029/1.jpg"></figure><lb/><emph type="italics"/>peroche egli è contenuto da mezi diametri, & da circonferenze eguali: &longs;arà il re<lb/>stante angolo & mi&longs;to<emph.end type="italics"/> K<emph type="italics"/>EG eguale al re&longs;tante angolo & mi&longs;to HDM. </s> <s id="id.2.1.130.3.0">Et per<lb/>cioche pre&longs;uppongono, che quanto è minore l'angolo contenuto dalla linea tirata à <lb/>piombo dell'orizonte, & dalla circonferenza, tanto in quel &longs;ito e&longs;&longs;ere anco più gra<lb/>ue il pe&longs;o. </s> <s id="id.2.1.130.4.0">Talche &longs;i come l'angolo contenuto da HD, & dalla circonferenza <lb/>DG, è minore dell'angolo<emph.end type="italics"/> K<emph type="italics"/>EG, cioè dell'angolo HDM, co&longs;i &longs;econdo que&longs;ta <lb/>proportione il pe&longs;o po&longs;to in D &longs;ia più graue di quello che &longs;tà in E. </s> <s id="id.2.1.130.5.0">Mala pro<lb/>portione dell'angolo MHD all'angolo HDG è minore di qual &longs;i voglia altra <lb/>proportione, che &longs;i troui tra la maggiore, & minore quantità: Adunque la pro<lb/>portione de i pe&longs;i DE &longs;arà la minima di tutte le proportioni, anzi non &longs;arà qua&longs;i <lb/>ne anche proportione, e&longs;&longs;endo la minima di tutte le proportioni. </s> <s id="id.2.1.130.6.0">Che la propor<lb/>tione di MDH ver&longs;o HDG &longs;ia di tutte la minima, mo&longs;trano con que&longs;ta ne<lb/>ce&longs;&longs;aria ragione, peroche MHD &longs;upera HDG con angolo di linea curua, che <lb/>è MGD, ilquale angolo è il minimo di tutti gli angoli fatti di linee rette: ne po<lb/>tendo&longs;i dare angolo minore di MGD &longs;arà la proportione di MDH ver&longs;o HDG <lb/>la minima di tutte le proportioni. </s> <s id="id.2.1.130.7.0">Laqual ragione pare e&longs;&longs;ere grandemente friuo<lb/>la, peroche quantunque l'angolo MDG &longs;ia di tutti gli angoli fatti di linee rette <lb/>il minore, non perciò &longs;egue totalmente egli e&longs;&longs;ere di tutti gli angoli il minimo, im<emph.end type="italics"/><arrow.to.target n="note16"></arrow.to.target><lb/><emph type="italics"/>peroche &longs;ia dal punto D tirata la linea DO à piombo di NC, ambedue que<lb/>ste toccheranno le circonferenze LDMFDG nel punto D. </s> <s id="id.2.1.130.8.0">Ma percioche le <lb/>circonferenze &longs;ono eguali, &longs;arà l'angolo MDO misto eguale all'angolo ODG mi<lb/>&longs;to. </s> <s id="id.2.1.130.9.0">L'vno de gli angoli dunque, cioè ODG &longs;arà minore di MDG, cioè minore <emph.end type="italics"/><arrow.to.target n="note17"></arrow.to.target><lb/><emph type="italics"/>del minimo. </s> <s id="id.2.1.130.10.0">Dapoi l'angolo ODH &longs;arà minore dell'angolo MDH. </s> <s id="id.2.1.130.11.0">Per laqual co&longs;a <lb/>ODH haurà proportione minore all'angolo HDG, che MDH all'i&longs;te&longs;&longs;o <emph.end type="italics"/><pb xlink:href="037/01/030.jpg"/><emph type="italics"/>HDG. </s> <s id="id.2.1.130.12.0">Dara&longs;&longs;i dunque la proportione anco minore della minima, laquale mostre<lb/>remo dauantaggio in infinito minore in questo modo. </s> <s id="id.2.1.130.13.0">De&longs;criua&longs;i il cerchio DR, <lb/>il cui centro &longs;ia E, & il mezo diametro ED, la circonferentia DR tocche<emph.end type="italics"/><lb/><arrow.to.target n="note18"></arrow.to.target><emph type="italics"/>rà la circonferenza <lb/>DG nel punto D, <lb/>& la linea DO nel <emph.end type="italics"/><lb/><arrow.to.target n="note19"></arrow.to.target><emph type="italics"/>punto D. </s> <s id="id.2.1.130.14.0">Per laqual <lb/>co&longs;a minore &longs;arà l'an<lb/>golo RDG dell'an<lb/>golo ODG, & &longs;i<lb/>milmente l'angolo R <lb/>DH dell'angolo O <lb/>DH. </s> <s id="id.2.1.130.15.0">Adunque ha<lb/>uerà minore propor<lb/>tione RDH ad HD <lb/>G di quel che haurà <lb/>ODH ad HDG. <lb/></s> <s id="id.2.1.130.16.0">Pigli&longs;i dapoi tra E <lb/>& C, come &longs;i vuo<lb/>le, il punto P, dal<lb/>quale nella di&longs;tanza <emph.end type="italics"/><lb/><figure id="id.037.01.030.1.jpg" xlink:href="037/01/030/1.jpg"></figure><lb/><emph type="italics"/>di PD &longs;i de&longs;criua vn'altra circonferenza DQ, laquale toccherà la circonferen<lb/>tia DR, & la circonferentia DG nel punto D, & l'angolo QDH &longs;arà mi <lb/>nore dell'angolo RDH. </s> <s id="id.2.1.130.17.0">Adunque QDH haurà proportione minore ad HDG <lb/>che RDH ad HDG, & nell'i&longs;te&longs;&longs;o modo in tutto, &longs;e tra il C & il P &longs;i tor<lb/>rà vn'altro punto, & tra que&longs;to, & il C vn'altro, & co&longs;i &longs;ucceßiuamente &longs;i de<lb/>&longs;criueranno infinite circonferentie tra DO, & la circonferenza DG: dalle quali <lb/>troueremo &longs;empre la proportione minore in infinito: & co&longs;i &longs;egue, che la propor<lb/>tione del pe&longs;o po&longs;to in D al pe&longs;o po&longs;to in E non &longs;ia tanto picciola, che non &longs;i <lb/>po&longs;&longs;a ritrouarla &longs;empre minore in infinito. </s> <s id="id.2.1.130.18.0">Et perche l'angolo MDG &longs;i puote <lb/>diuidere in infinito, &longs;i potrà anche diuidere quel più di grauezza che ha il D &longs;o<lb/>pra lo E in infinito. <emph.end type="italics"/></s> <s id="id.2.1.134.1.0"><margin.target id="note14"></margin.target><emph type="italics"/>Per la &longs;econda del terzo. <emph.end type="italics"/></s> <s id="id.2.1.135.1.0"><margin.target id="note15"></margin.target><emph type="italics"/>Per la vige&longs;imanona del primo. <emph.end type="italics"/></s> <s id="id.2.1.136.1.0"><margin.target id="note16"></margin.target><emph type="italics"/>Per la decima ottaua del terzo. <emph.end type="italics"/></s> <s id="id.2.1.137.1.0"><margin.target id="note17"></margin.target><emph type="italics"/>Per la ottaua del quinto. <emph.end type="italics"/></s> <s id="id.2.1.138.1.0"><margin.target id="note18"></margin.target><emph type="italics"/>Per la vndecima del terzo. <emph.end type="italics"/></s> <s id="id.2.1.139.1.0"><margin.target id="note19"></margin.target><emph type="italics"/>Per la decima ottaua del terzo. <emph.end type="italics"/></s> <pb pagenum="8" xlink:href="037/01/031.jpg"/> <s id="id.2.1.140.1.0"><emph type="italics"/>Ne bi&longs;ogna trala&longs;ciare, che <lb/>eglino hanno pre&longs;uppo&longs;to <lb/>nella demo&longs;tratione l'ango <lb/>lo<emph.end type="italics"/> K<emph type="italics"/>EG e&longs;&longs;er maggiore del<lb/>l'angolo HDC, come co<lb/>&longs;a nota: il che ben è vero &longs;e <lb/>DHE<emph.end type="italics"/>K <emph type="italics"/>&longs;ono fra loro e<lb/>gualmente di&longs;tanti. </s> <s id="id.2.1.140.2.0">Ma <lb/>percioche, come eßi pari<lb/>mente pre&longs;uppongono, le <lb/>linee DHE<emph.end type="italics"/>K <emph type="italics"/>&longs;i vanno à <lb/>trouare nel centro del mon<lb/>do, le linee DHE<emph.end type="italics"/>K <emph type="italics"/>non <lb/>&longs;aranno egualmente di&longs;tan<lb/>ti giamai, et <expan abbr="l'ãgolo">l'angolo</expan><emph.end type="italics"/> K<emph type="italics"/>EG <lb/>non &longs;olo non &longs;arà maggio<lb/>re dall'angolo HDG, ma <lb/>minore. </s> <s id="id.2.1.140.3.0">Come per gra<lb/>tia di e&longs;&longs;empio, &longs;ia tirata la <lb/>linea FG &longs;in al centro del <lb/>mondo, che &longs;ia S, & con <lb/>giungan&longs;i DS ES. </s> <s id="id.2.1.140.4.0">Egli <lb/>è da mostrare l'angolo SE <lb/>G e&longs;&longs;ere minore dell'ango <lb/>lo SDG. </s> <s id="id.2.1.140.5.0">Tiri&longs;i dal punto <lb/>E la linea ET, che toc<lb/>chi il cerchio DGEF, & <lb/>dall'i&longs;te&longs;&longs;o punto &longs;ia tirata <lb/>la EV egualmente di&longs;tan<emph.end type="italics"/><lb/><figure id="id.037.01.031.1.jpg" xlink:href="037/01/031/1.jpg"></figure><lb/><emph type="italics"/>te da DS: Percioche dunque EVDS &longs;ono tra loro egualmente di&longs;tanti, &longs;imil<lb/>mente ET DO &longs;ono egualmente di&longs;tanti: &longs;arà l'angolo VET eguale all'ango<lb/>lo SDO: & l'angolo TEG eguale all'angolo ODM, per e&longs;&longs;ere contenuto da <lb/>linee toccanti la circonferenza, & da circonferenze eguali. </s> <s id="id.2.1.140.6.0">Tutto l'angolo dun<lb/>que VEG &longs;arà eguale all'angolo SDM. </s> <s id="id.2.1.140.7.0">Leui&longs;i via dall'angolo SDM l'ango <lb/>lo di linee curue MDG: & dall'angolo VEG leui&longs;i via l'angolo VES, & <lb/>l'angolo VES fatto di linee rette è maggiore dell'angolo MDG fatto di linee <lb/>curue; &longs;arà il re&longs;tante angolo SEG minore dell'angolo SDG. </s> <s id="id.2.1.140.8.0">Per laqual co&longs;a <lb/>dalle pre&longs;uppo&longs;te loro non &longs;olo il pe&longs;o posto in D &longs;arà più graue del pe&longs;o po&longs;to <lb/>in E, ma per lo contrario il pe&longs;o E &longs;arà più graue dell'i&longs;te&longs;&longs;o D. <emph.end type="italics"/></s> <pb xlink:href="037/01/032.jpg"/> <s id="id.2.1.143.1.0"><emph type="italics"/>Producono tutta via <lb/>ragioni con le quali <lb/>&longs;i sforzano di mo<lb/>&longs;trare, che la bilan<lb/>cia DE ritorna per <lb/>neceßità in AB e<lb/>gualmente distante <lb/>dall'orizonte. </s> <s id="id.2.1.143.2.0">Pri<lb/>ma dimo&longs;trano l'i<lb/>&longs;te&longs;&longs;o pe&longs;o e&longs;&longs;ere più <lb/>graue in A, che <lb/>in altro &longs;ito, che <lb/>chiamano &longs;ito della <lb/>egualità, e&longs;&longs;endo la <lb/>linea AB egual<lb/>mente di&longs;tante dal<lb/>l'orizonte. </s> <s id="id.2.1.143.3.0">Da<lb/>poi quanto è più da<emph.end type="italics"/><lb/><figure id="id.037.01.032.1.jpg" xlink:href="037/01/032/1.jpg"></figure><lb/><emph type="italics"/>pre&longs;&longs;o allo A, tanto e&longs;&longs;ere piu graue di qual &longs;i voglia altro più da lontano, cioè <lb/>il pe&longs;o po&longs;to in A e&longs;&longs;ere più graue, che in D; & in D, che in L: & &longs;imil<lb/>mente in A più graue, che in N; & in N più graue, che in M. </s> <s id="id.2.1.143.4.0">Con&longs;ide<emph.end type="italics"/><lb/><arrow.to.target n="note20"></arrow.to.target><emph type="italics"/>rando &longs;olamente vn pe&longs;o in vno delle braccia in sù, ouero in giù mo&longs;&longs;o. </s> <s id="id.2.1.143.5.0">Percio<lb/>che dicono, po&longs;ta la trutina della bilancia in CF, il pe&longs;o me&longs;&longs;o in A è più lunge <lb/>dalla trutina che in D; & in D più lunge, che in L: peroche tirate le linee DO <emph.end type="italics"/><lb/><arrow.to.target n="note21"></arrow.to.target><emph type="italics"/>LP à piombo di CF, la linea AC re&longs;ta maggiore di DO, & DO di e&longs;&longs;a LP, <lb/>& auiene l'i&longs;te&longs;&longs;o ne i punti NM. </s> <s id="id.2.1.143.6.0">Dapoi dicono da qual luogo il pe&longs;o &longs;i mo<lb/>ue più velocemente, iui è più graue: ma egli &longs;i moue più velocemente dallo <emph.end type="italics"/><lb/><arrow.to.target n="note22"></arrow.to.target><emph type="italics"/>A, che da altro &longs;ito; adunque egli è più graue nello A. </s> <s id="id.2.1.143.7.0">Con &longs;imile mo<lb/>do, quanto più egli è da pre&longs;&longs;o allo A, tanto più velocemente &longs;i moue: <lb/>adunque nel D &longs;arà più graue, che in L. </s> <s id="id.2.1.143.8.0">L'altra cagione poi che cauano dal mo<emph.end type="italics"/><lb/><arrow.to.target n="note23"></arrow.to.target><emph type="italics"/>uimento più diritto, & più torto è, che quanto il pe&longs;o di&longs;cende più diritto in archi <lb/>eguali, pare e&longs;&longs;er anco più graue; concio&longs;ia che il pe&longs;o e&longs;&longs;endo libero, & &longs;ciolto, &longs;i <lb/>moua di &longs;ua propria natura per lo diritto; ma in A egli di&longs;cende più dirittamen<emph.end type="italics"/><lb/><arrow.to.target n="note24"></arrow.to.target><emph type="italics"/>te; dunque in A &longs;arà più graue, & dimo&longs;trano ciò pigliando l'arco AN egua<lb/>le all'arco LD. </s> <s>& da i punti NL &longs;iano tirate le linee NRLQ egualmente di<lb/>&longs;tanti dalla linea FG, laquale chiamano anche della direttione; & quelle altre &longs;e<lb/>gheranno le linee ABDO in QR, & dal punto N &longs;ia tirata la NT à piombo <lb/>di FG: Dimo&longs;trano veramente LQ e&longs;&longs;ere eguale à PO, & NR ad e&longs;&longs;a CT, <lb/>& la linea NR e&longs;&longs;er maggiore di Lq. </s> <s>Hor percioche la di&longs;ce&longs;a del pe&longs;o dallo A <lb/>fin ad N per la circonferentia di AN trapa&longs;&longs;a maggior parte della linea FG, <lb/>(che eßi chiamano pigliare di diritto) che la di&longs;ce&longs;a di L in D per la circonferenza <lb/>LD; concio&longs;ia che la di&longs;ce&longs;a AN trapaßi la linea CT, ma la di&longs;ce&longs;a LD la linea <emph.end type="italics"/><pb pagenum="9" xlink:href="037/01/033.jpg"/><emph type="italics"/>PO, & CT è maggiore di PO, la di&longs;ce&longs;a di AN &longs;arà più diritta, che la di<lb/>&longs;ce&longs;a di LD: &longs;arà dunque più graue il pe&longs;o po&longs;to in A, che in L, ouero in qual <lb/>&longs;i voglia altro &longs;ito, & nell'i&longs;te&longs;&longs;o modo dimo&longs;trano, che quanto il pe&longs;o è più vicino <lb/>allo A, è più graue; cioè &longs;iano le circonferenze LD DA tra loro eguali, & <lb/>dal punto D &longs;ia tirata la linea DR à piombo di AB; &longs;arà la DR eguale al<emph.end type="italics"/><arrow.to.target n="note25"></arrow.to.target><lb/><emph type="italics"/>la CO. </s> <s id="id.2.1.143.9.0">& dimo<lb/>&longs;trano po&longs;cia, che <lb/>la linea DR è mag<lb/>giore della LQ, & <lb/>dicono che la &longs;ce&longs;a <lb/>di DA prende più <lb/>di &longs;ce&longs;a diritta, che <lb/>non fa LD, pe<lb/>roche è maggiore <lb/>la linea CO, che <lb/>la OT: Per la<lb/>qual co&longs;a il pe&longs;o &longs;a<lb/>rà più graue in D, <lb/>che in L, ilche pa<lb/>rimente auiene ne <lb/>punti NM. </s> <s id="id.2.1.143.10.0">& <lb/>co&longs;i il pre&longs;uppo&longs;to, <lb/>per loquale dimo<emph.end type="italics"/><lb/><figure id="id.037.01.033.1.jpg" xlink:href="037/01/033/1.jpg"></figure><lb/><emph type="italics"/>&longs;trano la bilancia DE ritornare in AB a&longs;&longs;ermano come noto, & manife&longs;to; cioè <emph.end type="italics"/><arrow.to.target n="note26"></arrow.to.target><lb/><emph type="italics"/>che &longs;econdo il &longs;ito il pe&longs;o è tanto più graue, quanto nel mede&longs;imo &longs;ito manco tor<lb/>ta è la &longs;ce&longs;a: & la cagione di cotal ritorno dicono e&longs;&longs;ere que&longs;ta; peroche la &longs;ce&longs;a del <lb/>pe&longs;o po&longs;to in D è più diritta della &longs;ce&longs;a del pe&longs;o po&longs;to in E, per pigliare il pe&longs;o <emph.end type="italics"/><arrow.to.target n="note27"></arrow.to.target><lb/><emph type="italics"/>di E manco della direttione in de&longs;cendendo che non fa il pe&longs;o di D pur nel di&longs;cen<lb/>dere: Come &longs;e l'arco EV &longs;ia eguale à DA, & &longs;iano tirate VHET à piom<lb/>bo di FG; &longs;arà maggiore DR di TH. </s> <s id="id.2.1.143.11.0">Per laqual co&longs;a per la pre&longs;uppo&longs;ta il pe<emph.end type="italics"/><arrow.to.target n="note28"></arrow.to.target><lb/><emph type="italics"/>&longs;o me&longs;&longs;o in D per ri&longs;petto al &longs;ito &longs;arà più graue del pe&longs;o me&longs;&longs;o in E. </s> <s id="id.2.1.143.12.0">Adunque <lb/>il pe&longs;o me&longs;&longs;o in D e&longs;&longs;endo più graue &longs;i mouerà in giù, & il pe&longs;o po&longs;to in E in <lb/>&longs;u fin che la bilancia DE ritorni in AB. <emph.end type="italics"/></s> <s id="id.2.1.146.1.0"><margin.target id="note20"></margin.target><emph type="italics"/>Il Cardano nel primo della &longs;ottigliezza. <emph.end type="italics"/></s> <s id="id.2.1.147.1.0"><margin.target id="note21"></margin.target><emph type="italics"/>Giordano nella quarta propo&longs;itione. <emph.end type="italics"/></s> <s id="id.2.1.148.1.0"><margin.target id="note22"></margin.target><emph type="italics"/>Il Tartaglia nella quinta propo&longs;itione. <emph.end type="italics"/></s> <s id="id.2.1.149.1.0"><margin.target id="note23"></margin.target><emph type="italics"/>Il Cardano. </s> <s id="id.2.1.149.2.0">Giordano alla propo&longs;itione quarta. <emph.end type="italics"/></s> <s id="id.2.1.150.1.0"><margin.target id="note24"></margin.target><emph type="italics"/>Il Tartaglia alla propo&longs;itione. <emph.end type="italics"/> 5. </s> <s id="id.2.1.151.1.0"><margin.target id="note25"></margin.target><emph type="italics"/>Per la trige&longs;imaquarta del primo. <emph.end type="italics"/></s> <s id="id.2.1.152.1.0"><margin.target id="note26"></margin.target><emph type="italics"/>Giordane nella quarta pre&longs;upposta. <emph.end type="italics"/></s> <s id="id.2.1.153.1.0"><margin.target id="note27"></margin.target><emph type="italics"/>Giordano nella &longs;econda propo&longs;itione. <emph.end type="italics"/></s> <s id="id.2.1.154.1.0"><margin.target id="note28"></margin.target><emph type="italics"/>Il Tartaglia nella quinta propo&longs;itione. <emph.end type="italics"/></s> <s id="id.2.1.155.1.0"><emph type="italics"/>L'altra ragione ancora di que&longs;to ritorno è, che <expan abbr="quãdo">quando</expan> la trutina della bilancia è &longs;opra <emph.end type="italics"/><arrow.to.target n="note29"></arrow.to.target><lb/><emph type="italics"/>di lei in CF; la linea CG è la meta: & percio che l'angolo GCD è maggiore <lb/>dell'angolo GCE, & l'angolo maggiore dalla meta rende più graue il pe&longs;o: adun<lb/>que &longs;tando la trutina della bilancia di &longs;opra &longs;arà più graue il pe&longs;o in D, che in E, <lb/>& perciò il D ritornerà nello A, & lo E nel B. <emph.end type="italics"/></s> <s id="id.2.1.156.1.0"><margin.target id="note29"></margin.target><emph type="italics"/>Il Cardano. <emph.end type="italics"/></s> <s id="id.2.1.157.1.0">Meta è pur voce Latina co&longs;tumata da gli antichi ne i giuo chi, & conte&longs;e fatte ne i cer<lb/>chi murati, & ne i Theatri, percio che il principio, oue &longs;i dauano le mo&longs;&longs;e a' corri<lb/>tori, &longs;i chiamaua Carcere, & il fine Meta; di modo, che meta viene à dire termine <lb/>& fine: & piu in altro &longs;ignificato il luogo piu ba&longs;&longs;o, & infimo. </s> <s id="id.2.1.157.2.0">Hor qui &longs;i puote <pb xlink:href="037/01/034.jpg"/>intendere ad ambidue i modi, cioè che la linea CG &longs;ia la meta, cioè il termine <lb/>& fine, nelquale ha da peruenire il pe&longs;o collocato nella bilancia; ouero il luogo <lb/>infimo della circonferenza, alquale capita il pe&longs;o per natura. </s> <s id="id.2.1.157.3.0">Doue &longs;criue l'Auto<lb/>re l'angolo maggiore dalla Meta, vuol dire l'angolo, che fa il braccio della bilan<lb/>cia con la Meta CG. </s> <s id="id.2.1.158.1.0"><emph type="italics"/>Et co&longs;i <expan abbr="cõ">con</expan> que&longs;te ragioni &longs;i sforzano dimo&longs;trare la bilancia DE ritornare in AB; le <lb/>quali al parer mio &longs;i po&longs;&longs;ono ageuolmente &longs;oluere. <emph.end type="italics"/></s> <s id="id.2.1.159.1.0"><emph type="italics"/>Primieramente dunque in quanto s'appartiene alle ragioni, che dicono il pe&longs;o me&longs;&longs;o <lb/>in A e&longs;&longs;ere piu graue, che in altro &longs;ito, lequali cauano dalla di&longs;tanza piu da lonta<lb/>no, & piu da pre&longs;&longs;o della linea FG, & dal mouimento piu veloce, & piu diritto <lb/>dal punto A. </s> <s id="id.2.1.159.2.0">In prima non dimo&longs;trano veramente perche il pe&longs;o &longs;i moua piu velo<lb/>cemente dallo A, che da altro &longs;ito. </s> <s id="id.2.1.159.3.0">ne perche &longs;ia maggiore CA di DO, & DO <lb/>di LP, per que&longs;to, come per vera cagione, &longs;egue il pe&longs;o po&longs;to in A e&longs;&longs;ere piu gra<lb/>ue di quello, che è in D, & quello di D, di quel che &longs;tà in L, percioche non &longs;i queta <lb/>l'intelletto, &longs;e di ciò altra cagione non &longs;i dimo&longs;tra, parendo &longs;egno piu to&longs;to, che vera <lb/>cagione. </s> <s id="id.2.1.159.4.0">Quello ste&longs;&longs;o accade parimente all'altra ragione, laquale adducono dal <lb/>mouimento piu diritto, & piu torto. </s> <s id="id.2.1.159.5.0">Oltre à ciò tutte quelle co&longs;e, che per&longs;uadono <lb/>per via del <expan abbr="mouim&etilde;to">mouimen<lb/>to</expan> piu veloce, & <lb/>piu tardo il pe&longs;o in <lb/>A e&longs;&longs;ere piu graue, <lb/>che in D, non per<lb/>ciò dimo&longs;trano, che <lb/>il pe&longs;o in A, in <expan abbr="quãto">quan<lb/>to</expan> è in A, &longs;ia piu <lb/>graue del pe&longs;o D, in <lb/>quanto è in D, ma <lb/>in quanto &longs;i parte <lb/>da i punti D A. </s> <s id="id.2.1.159.6.0">Onde, <expan abbr="auãti">auanti</expan> che piu <lb/>oltre &longs;i proceda, pri<lb/>ma dimo&longs;trerò, che <lb/>il pe&longs;o quanto egli <lb/>è piu da pre&longs;&longs;o ad <lb/>FG manco graua, <lb/>&longs;i in quanto egli &longs;tà <lb/>nel &longs;ito, oue &longs;i ritro<emph.end type="italics"/><lb/><figure id="id.037.01.034.1.jpg" xlink:href="037/01/034/1.jpg"></figure><lb/><emph type="italics"/>ua, come anche in quanto &longs;i parte da quello: & in&longs;ieme, che egli è fal&longs;o il pe&longs;o e&longs;&longs;ere <lb/>piu graue in A, che in altro &longs;ito. <emph.end type="italics"/></s> <s id="id.2.1.161.1.0"><emph type="italics"/>Tiri&longs;i la FG fin al centro del mondo, che &longs;ia in S, & dal punto S tiri&longs;i anco vna linea, <lb/>che tocchi il cerchio AFBG. </s> <s>non potrà già questa linea tirata dal punto S toc<lb/>care il cerchio nel punto A; imperoche tirata la linea AS, il triangolo ACS ver<emph.end type="italics"/><pb pagenum="10" xlink:href="037/01/035.jpg"/><emph type="italics"/>rebbe ad hauere due angoli retti, cioè SAC, & ACS, che è impoßibile: ne me<lb/>no toccherà &longs;opra il punto A nella circonferenza AF; peroche &longs;egherebbe il cer<emph.end type="italics"/><arrow.to.target n="note30"></arrow.to.target><lb/><emph type="italics"/>chio. </s> <s id="id.2.1.161.2.0">Toccherà dunque &longs;otto, & &longs;ia SO: &longs;iano dapoi congiunte le lince SD SL, <lb/>lequali &longs;eghino la circonferenza AOG ne' punti<emph.end type="italics"/> K<emph type="italics"/>H, & &longs;iano ancho congiunte le <lb/>linee C<emph.end type="italics"/>K <emph type="italics"/>CH. </s> <s id="id.2.1.161.3.0">Et percioche il pe&longs;o, quanto egli è piu da pre&longs;&longs;o di F, tanto piu an<lb/>co &longs;tà &longs;opra il centro; come il pe&longs;o in D preme, & &longs;tà piu &longs;opra il punto del volgi<lb/>mento C, come à centro, cioè in D piu graua &longs;opra la linea CD, che &longs;e egli fo&longs;&longs;e in A <lb/>&longs;opra la linea CA: & dauantaggio piu in L &longs;opra la linea CL. </s> <s id="id.2.1.161.4.0">imperoche e&longs;&longs;endo <lb/>li tre angoli di cia&longs;cun triangolo eguali à due angoli retti, & l'angolo DC<emph.end type="italics"/>K <emph type="italics"/>del <lb/>triangolo DC<emph.end type="italics"/>K, <emph type="italics"/>che è di due lati eguali &longs;ia <lb/>minore dell'angolo LCH del <expan abbr="triãgolo">triangolo</expan> LCH, <lb/>che è pur di due lati eguali: &longs;aranno gli altri <lb/>alla ba&longs;e, cioè CD<emph.end type="italics"/>K <emph type="italics"/>C<emph.end type="italics"/>K<emph type="italics"/>D in&longs;ieme pre&longs;i <lb/>maggiori de gli altri CLH CHL; & le <lb/>metà di que&longs;ti, cioè l'angolo CDS &longs;arà mag<lb/>giore dell'angolo CLS. </s> <s id="id.2.1.161.5.0">E&longs;&longs;endo adunque <lb/>CLS minore, la linea CL piu &longs;i acco&longs;terà <lb/>al mouimento naturale del pe&longs;o me&longs;&longs;o in L <lb/>del tutto &longs;ciolto; cioè à dire alla linea LS, <lb/>che CD al mouimento DS: percioche il pe<lb/>&longs;o po&longs;to in L libero, & &longs;ciolto &longs;i mouerebbe <lb/>ver&longs;o il centro del mondo per LS, & il pe<lb/>&longs;o po&longs;to in D per DS. </s> <s id="id.2.1.161.6.0">Ma perche il pe&longs;o <lb/>me&longs;&longs;o in L graua tutto &longs;opra LS, & quello <lb/>che è in D &longs;opra DS, il pe&longs;o in L grauerà <lb/>piu &longs;opra la linea CL, che quello, che &longs;tà in <lb/>D &longs;opra la linea DC. </s> <s id="id.2.1.161.7.0">Adunque la linea <lb/>CL &longs;o&longs;terrà piu il pe&longs;o, che la linea CD, & <lb/>nel modo iste&longs;&longs;o quanto piu il pe&longs;o &longs;arà da<lb/>pre&longs;&longs;o ad F, &longs;i dimostrerà piu e&longs;&longs;er &longs;o&longs;tenuto <lb/>dalla linea CL per cotesta cagione, peroche <lb/>&longs;empre l'angolo CLS &longs;arebbe minore, la<lb/>qual co&longs;a etiandio èmanife&longs;ta; perche &longs;e le li<lb/>nee CL, & LS s'incontra&longs;&longs;ero in vna li<lb/>nea, ilche auiene in FCS, all'hora la linea <lb/>CF &longs;o&longs;terrebbe tutto il pe&longs;o, che è in F, & <lb/>lo renderebbe immobile, nè haurebbe niuna <lb/>grauezza in tutto nella circonferenza del cer<lb/>chio. </s> <s id="id.2.1.161.8.0">L'i&longs;te&longs;&longs;o pe&longs;o dunque per la diuer&longs;ità<emph.end type="italics"/><lb/><figure id="id.037.01.035.1.jpg" xlink:href="037/01/035/1.jpg"></figure><lb/><emph type="italics"/>de' &longs;iti &longs;arà piu graue, & piu lieue. </s> <s id="id.2.1.161.9.0">& que&longs;to non già percio che per ragione del &longs;ito <lb/>alcuna volta egli acqui&longs;ti veramente grauezza maggiore, & alcuna volta la perda, <lb/>e&longs;&longs;endo &longs;empre della i&longs;te&longs;&longs;a grauezza, troui&longs;i douunque &longs;i voglia: ma percioche egli <emph.end type="italics"/><pb xlink:href="037/01/036.jpg"/><emph type="italics"/>graua piu, & meno nella circonferenza, come in D piu graua &longs;opra la circonferenza <lb/>DA, che in L &longs;opra la circonferenza LD: cioè &longs;e il pe&longs;o &longs;arà &longs;o&longs;tenuto dalle circon<lb/>ferenze, & dalle linee diritte; la circonferenza AD &longs;o&longs;terrà piu il pe&longs;o po&longs;to in D, <lb/>che la circonferenza DL, &longs;tando il pe&longs;o in L; peroche meno aiuta CD, che CL. <lb/></s> <s id="id.2.1.161.10.0">Oltre à ciò quando il pe&longs;o è in L, &longs;e egli fo&longs;&longs;e del tutto libero & &longs;ciolto, &longs;i mouerebbe <lb/>in giu per LS, &longs;e non gliene fu&longs;&longs;e vietato dalla linea CL, laquale sforza il pe&longs;o po&longs;to <lb/>in L à mouer&longs;i oltre la linea LS per la circonferenza LD, & lo caccia in certo mo<lb/>do, & in cacciandolo viene in parte à &longs;o&longs;tenerlo; percioche &longs;e non lo &longs;o&longs;tene&longs;&longs;e, & <lb/>gli face&longs;&longs;e re&longs;i&longs;tenza, &longs;i mouerebbe in giu per la linea LS, ma non già per la cir<lb/>conferenza LD. </s> <s id="id.2.1.161.11.0">Similmente la CD fa re&longs;i&longs;tenza al pe&longs;o po&longs;to in D, sforzan<lb/>dolo à mouer&longs;i per la circonferenza DA. </s> <s id="id.2.1.161.12.0">Nell'iste&longs;&longs;o modo &longs;tando il pe&longs;o in A, <lb/>la linea CA con&longs;tringerà il pe&longs;o à mouer&longs;i <lb/>oltre la linea AS per la circonferenza AO; <lb/>peroche l'angolo CAS è acuto, e&longs;&longs;endo lo <lb/>angolo ACS retto. </s> <s id="id.2.1.161.13.0">Adunque le linee <lb/>CA CD in qualche parte, ma non già e<lb/>gualmente fanno re&longs;istenza al pe&longs;o. </s> <s id="id.2.1.161.14.0">& qua<lb/>lunque volta l'angolo, che è nella circonfe<lb/>renza del cerchio fatto dalle linee che e&longs;cono <lb/>dal centro del monde S, & dal centro C &longs;a<lb/>rà acuto, dimo&longs;treremo auenire l'i&longs;te&longs;&longs;o. </s> <s id="id.2.1.161.15.0">Hor <lb/>percioche l'angolo mi&longs;to CLD è eguale à <lb/>l'angolo CDA, per e&longs;&longs;ere conteuuto da <lb/>mezi diametri, & dall'i&longs;te&longs;&longs;a circonferenza; <lb/>& l'angolo CLS è minore dell'angolo <lb/>CDS; &longs;arà il re&longs;tante SLD maggiore <lb/>del re&longs;tante SDA. </s> <s id="id.2.1.161.16.0">Per laqual co&longs;a la cir<lb/>conferenza DA, cioè la di&longs;ce&longs;a del pe&longs;o <lb/>in D &longs;ara piu da pre&longs;&longs;o al mouimento natu<lb/>rale del pe&longs;o &longs;ciolto me&longs;&longs;o in D, cioè della li<lb/>nea DS, che la circonferenza LD della <lb/>linea LS. </s> <s id="id.2.1.161.17.0">Meno dunque farà re&longs;i&longs;tenza la <lb/>linea CD al pe&longs;o po&longs;to in D, che la linea <lb/>CL al pe&longs;o po&longs;to in L. </s> <s id="id.2.1.161.18.0">Però la linea CD <lb/>&longs;o&longs;terrà meno, che CL, & il pe&longs;o &longs;arà <lb/>piu libero in D, che in L: mouendo&longs;i piu <lb/>naturalmente il pe&longs;o per DA, che per LD. <lb/></s> <s id="id.2.1.161.19.0">Per laqual co&longs;a piu graue &longs;arà in D, che in <lb/>L. </s> <s id="id.2.1.161.20.0">Similmente dimo&longs;treremo, che CA man<lb/>co &longs;o&longs;tiene, che CD & che il pe&longs;o piu in A, <lb/>che in D è libero, & piu graue. </s> <s id="id.2.1.161.21.0">Dopo dalla <emph.end type="italics"/><lb/><figure id="id.037.01.037.1.jpg" xlink:href="037/01/037/1.jpg"></figure><lb/><emph type="italics"/>parte di &longs;otto per l'i&longs;te&longs;&longs;e cagioni, quanto il pe&longs;o &longs;arà piu da pre&longs;&longs;o al G, &longs;arà piu ri<emph.end type="italics"/><pb pagenum="11" xlink:href="037/01/037.jpg"/><emph type="italics"/>tenuto, come in H dalla linea CH, che in<emph.end type="italics"/> K <emph type="italics"/>dalla linea C<emph.end type="italics"/>K: <emph type="italics"/>percioche e&longs;&longs;en<emph.end type="italics"/><arrow.to.target n="note31"></arrow.to.target><lb/><emph type="italics"/>do l'angolo CHS maggiore dell'angolo C<emph.end type="italics"/>K<emph type="italics"/>S, le linee CH HS, &longs;i acco&longs;te<lb/>ranno piu alla direttione, che C<emph.end type="italics"/>K K<emph type="italics"/>S. </s> <s id="id.2.1.161.22.0">& per que&longs;to &longs;arà piu ritenuto il pe&longs;o da <lb/>CH, che da C<emph.end type="italics"/>K; <emph type="italics"/>percioche &longs;e CH HS &longs;i incontra&longs;&longs;ero in vna linea, come auie<lb/>ne &longs;tando il pe&longs;o in G, allhora la linea CG &longs;o&longs;terrebbe tutto il pe&longs;o in G, per <lb/>modo che &longs;tarebbe immobile. </s> <s id="id.2.1.161.23.0">Quanto minore dunque &longs;arà l'angolo contenuto dal <lb/>la linea CH, & dalla di&longs;ce&longs;a del pe&longs;o &longs;ciolto, cioè dalla linea HS, tanto meno <lb/>anco quella linea ritenirà il pe&longs;o, & doue &longs;arà manco ritenuto, iui &longs;arà piu libero, & <lb/>piu graue. </s> <s id="id.2.1.161.24.0">Oltre à ciò &longs;e il pe&longs;o fo&longs;&longs;e libero in K, & &longs;ciolto, &longs;i mouerebbe per la li<lb/>nea KS, ma egli è impedito dalla linea CK, laquale sforza il pe&longs;o a mouer&longs;i di <lb/>qua dalla linea KS per la circonferenza KH; percio che lo ritira in certo modo, <lb/>& in ritirandolo viene a &longs;o&longs;tenerlo, peroche &longs;e non lo &longs;o&longs;tene&longs;&longs;e, &longs;i mouerebbe il pe<lb/>&longs;o in giu per la linea diritta KS, ma non per la circonferenza KH. </s> <s id="id.2.1.161.25.0">Similmente <lb/>la CH ritiene il pe&longs;o, sforzandolo a mouer&longs;i per la circonferenza HG. </s> <s id="id.2.1.161.26.0">Et percio<lb/>che l'angolo CHS è maggiore dell'angolo CKS, leuati via gli angoli eguali <lb/>CHG, C<emph.end type="italics"/>K<emph type="italics"/>H, &longs;arà il re&longs;tante SHG maggiore del re&longs;tante S<emph.end type="italics"/>K<emph type="italics"/>H. </s> <s id="id.2.1.161.27.0">Adunque <lb/>la circonferenza KH, cioè la di&longs;ce&longs;a del pe&longs;o po&longs;to in K &longs;arà piu da pre&longs;&longs;o al mo<lb/>uimento naturale del pe&longs;o po&longs;to in K &longs;ciolto, cioè alla linea KS, che la circonfe<lb/>renza HG alla linea HS. </s> <s id="id.2.1.161.28.0">Per laqual co&longs;a meno ritiene la linea CK, che CH, <lb/>mouendo &longs;i il pe&longs;o piu naturalmente per KH, che per HG, Con ragione &longs;imile <lb/>anco &longs;i mo&longs;trerà, che quanto minore &longs;arà l'angolo SKH, la linea CK &longs;o&longs;terrà <lb/>meno. </s> <s id="id.2.1.161.29.0">Stando dunque il pe&longs;o in O, percioche l'angolo SOC non &longs;olamente è <lb/>minore dell'angolo CKS, ma anco il minimo di tutti gli angoli, che e&longs;con da i pun<lb/>ti CS, & hanno la cima nella circonferenza OKG; &longs;arà l'angolo SOK il mi <lb/>nimo &longs;i dell'angolo SKH, come de tutti gli altri co&longs;i fatti. </s> <s id="id.2.1.161.30.0">Adunque la di&longs;ce&longs;a <lb/>del pe&longs;o po&longs;to in O &longs;arà piu da pre&longs;&longs;o al mouimento naturale di e&longs;&longs;o pe&longs;o &longs;ciolto in <lb/>O, che in altro &longs;ito della circonferenza OKG: & la linea CO meno &longs;o&longs;tenirà <lb/>il pe&longs;o, che &longs;e egli fo&longs;&longs;e in qual &longs;i voglia altro &longs;ito della i&longs;te&longs;&longs;a circonferenza OG. <lb/></s> <s id="id.2.1.161.31.0">Similmente perche l'angolo del toccamento SOK è minore &longs;i dell'angolo SDA, <lb/>&longs;i dello SAO, & &longs;i di qual &longs;i voglia altro &longs;imile; &longs;arà la &longs;ce&longs;a del pe&longs;o me&longs;&longs;o in O <lb/>piu da pre&longs;&longs;o al mouimento naturale di e&longs;&longs;o pe&longs;o &longs;ciolto in O, che in altro &longs;ito del<lb/>la <expan abbr="circõfer&etilde;za">circonferenza</expan> ODF. </s> <s id="id.2.1.161.32.0">Oltre a ciò perche la linea CO <expan abbr="nõ">non</expan> puote &longs;pingere il pe&longs;o po&longs;to <lb/>in O mentre egli &longs;i moue in giu, per modo che egli &longs;i moua oltre la linea OS, per <lb/>cioche la linea OS non taglia il cerchio, ma lo tocca; & l'angolo SOC è retto <lb/>& non acuto, il pe&longs;o po&longs;to in O non grauerà niente &longs;opra la linea CO, ne &longs;tarà <lb/>&longs;opra il centro, come accaderebbe in qual &longs;i voglia altro punto &longs;opra l'O. </s> <s id="id.2.1.161.33.0">Sarà dun<lb/>que il pe&longs;o po&longs;to in O per que&longs;te cagioni libero, & &longs;ciolto piu in que&longs;to &longs;ito, che in <lb/>qual &longs;i voglia altro della circonferenza FOG; & perciò in que&longs;to &longs;arà piu graue, <lb/>cioè a dire piu grauerà, che in altro &longs;ito. </s> <s id="id.2.1.161.34.0">Et quanto &longs;arà piu da pre&longs;&longs;o ad O, &longs;arà <lb/>piu graue di quello, che &longs;e fo&longs;&longs;e piu da lunge: & la linea CO &longs;arà egualmente di<lb/>&longs;tante dall'orizonte: non pero all'orizonte del punto C (come &longs;timano e&longs;&longs;i) ma <lb/>del pe&longs;o po&longs;to in O, douendo&longs;i prendere l'orizonte dal centro della grauezza del pe<lb/>&longs;o. </s> <s id="id.2.1.161.35.0">Lequali co&longs;e tutte bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.165.1.0"><margin.target id="note30"></margin.target><emph type="italics"/>Per la decima ottaua del terzo<emph.end type="italics"/></s> <s id="id.2.1.166.1.0"><margin.target id="note31"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 21. <emph type="italics"/>del prim. <emph.end type="italics"/></s> <pb xlink:href="037/01/038.jpg"/> <s id="id.2.1.167.1.0"><emph type="italics"/>Ma &longs;e il braccio della bilancia fo&longs;&longs;e maggiore <lb/>di CO, come per la quantità di CD; <lb/>&longs;arà parimente il pe&longs;o me&longs;&longs;o in O piu gra<lb/>ue. </s> <s id="id.2.1.167.2.0">De&longs;criua&longs;i il cerchio OH, il cui <lb/>centro &longs;ia D, & il mezo diametro D</s> <s id="id.2.1.167.3.0">il cerchio OH toccherà il cerchio FOG <lb/>nel punto O, & toccherà anche la linea <emph.end type="italics"/><lb/><arrow.to.target n="note32"></arrow.to.target><emph type="italics"/>OS nel punto mede&longs;imo, laquale è la &longs;ce<lb/>&longs;a naturale, & diritta del pe&longs;o po&longs;to in O.<emph.end type="italics"/><lb/><arrow.to.target n="note33"></arrow.to.target></s> <s><emph type="italics"/>Et percioche l'angolo SOH è minore del<lb/>l'angolo SOG, &longs;arà la &longs;ce&longs;a del pe&longs;o po&longs;to <lb/>in O per la circonferenza OH piu dapre&longs;<lb/>&longs;o al mouimento naturale OS, che per la <lb/>circonferenza OG. </s> <s id="id.2.1.167.4.0">Piu libero dunque <lb/>& &longs;ciolto, & per con&longs;equente piu graue &longs;a<lb/>ràin O, &longs;tante il centro della bilancia in <lb/>D, che in C. </s> <s id="id.2.1.167.5.0">Similmente &longs;i mo&longs;trerà, <lb/>che quanto piu grande &longs;arà il braccio DO, <lb/>il pe&longs;o po&longs;to in O &longs;arà d'auantaggio piu <lb/>graue. <emph.end type="italics"/></s> <s id="id.2.1.168.1.0"><margin.target id="note32"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <s id="id.2.1.169.1.0"><margin.target id="note33"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <figure id="id.037.01.038.1.jpg" xlink:href="037/01/038/1.jpg"> </figure> <s id="id.2.1.171.1.0"><emph type="italics"/>Ma &longs;e l'iste&longs;&longs;o cerchio AFBG co'l &longs;uo centro R &longs;arà piu da pre&longs;&longs;o ad S centro <lb/>del mondo, & dal punto S &longs;ia tirata vna linea, che tocchi il cerchio ST, il pun<lb/>to T, (doue il pe&longs;o è piu graue) &longs;arà piu lontano dal punto A, che il punto O: <lb/>percioche &longs;iano tirate da i punti OT le linee OMTN à piombo di CS, & <lb/>congiungan&longs;i RT, & &longs;ia il centro R nella linea CS, & la linea ARB &longs;ia <lb/>egualmente di&longs;tante ad ACB. </s> <s id="id.2.1.171.2.0">Percioche dunque i triangoli COS RTS &longs;ono <emph.end type="italics"/><lb/><arrow.to.target n="note34"></arrow.to.target><emph type="italics"/>di angoli retti, &longs;arà SC à CO, come CO à CM. </s> <s id="id.2.1.171.3.0">Similmente SR ad <emph.end type="italics"/><lb/><arrow.to.target n="note35"></arrow.to.target><emph type="italics"/>RT, come RT ad RN. </s> <s id="id.2.1.171.4.0">E&longs;&longs;endo dunque RT eguale à CO, & SC mag<lb/>giore di RS: haurà proportione maggiore SC à CO, che SR ad RT. </s> <s id="id.2.1.171.5.0">on<emph.end type="italics"/><lb/><arrow.to.target n="note36"></arrow.to.target><emph type="italics"/>de baurà parimente proportione maggiore CO à CM, che RT ad RN. </s> <s id="id.2.1.171.6.0">&longs;a <lb/>rà dunque minore CM, che RN. </s> <s id="id.2.1.171.7.0">Tagli&longs;i dunque RN in P &longs;i fattamen<emph.end type="italics"/><pb pagenum="12" xlink:href="037/01/039.jpg"/><emph type="italics"/>te, che RP &longs;ia eguale à CM; & dal <lb/><expan abbr="p&utilde;to">punto</expan> P &longs;ia tirata la linea PQ egual<lb/>mente di&longs;tante dalle linee MONT, <lb/>laquale tagli la <expan abbr="circõfer&etilde;za">circonferenza</expan> AT in Q, <lb/>& in fine <expan abbr="cõgiõgan&longs;i">congiongan&longs;i</expan> la RQ. </s> <s>Hor per <lb/>cioche le due CO CM &longs;ono eguali à <lb/>le due RQ RP, & l'angolo CMO <emph.end type="italics"/><arrow.to.target n="note37"></arrow.to.target><lb/><emph type="italics"/>è eguale all'angolo RPQ. </s> <s>&longs;arà an<lb/>che l'angolo MCO eguale all'angolo <lb/>PRQ. </s> <s>Ma l'angolo MCA retto <lb/>è eguale all'angolo PRA retto; a<lb/>dunque il re&longs;tante OCA al restante <emph.end type="italics"/><arrow.to.target n="note38"></arrow.to.target><lb/><emph type="italics"/>QRA &longs;arà eguale, & la circonferen<lb/>za OA parimente eguale alla circon<lb/>ferenza QA. </s> <s id="id.2.1.171.8.0">Però il punto T per <lb/>e&longs;&longs;ere piu di&longs;tante dal punto A, che <lb/>Q, &longs;arà anco piu di&longs;tante dal punto <lb/>A, che il punto O. </s> <s id="id.2.1.171.9.0">Dimo&longs;trera&longs;&longs;i pa<lb/>rimente, che quanto piu il cerchio &longs;arà <lb/>vicino al centro del mondo, che egli &longs;a <lb/>rà anco piu lontano. </s> <s id="id.2.1.171.10.0">Et co&longs;i come pri<lb/>ma dimo&longs;trera&longs;&longs;i il pe&longs;o nella circonfe<lb/>renza TAF &longs;tar &longs;opra il centro R, <lb/>ma nella circonferenza TG e&longs;&longs;ere ri<lb/>tenuto dalla linea, & ritrouar&longs;i piu gra<lb/>ue nel punto T. <emph.end type="italics"/></s> <s id="id.2.1.172.1.0"><margin.target id="note34"></margin.target><emph type="italics"/>Per la ottaua del &longs;esto. <emph.end type="italics"/></s> <s id="id.2.1.173.1.0"><margin.target id="note35"></margin.target><emph type="italics"/>Per la ottaua del <expan abbr="qu&itilde;to">quinto</expan>. <emph.end type="italics"/></s> <s id="id.2.1.174.1.0"><margin.target id="note36"></margin.target><emph type="italics"/>Per la decima del quinto. <emph.end type="italics"/></s> <s id="id.2.1.175.1.0"><margin.target id="note37"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>del &longs;esto. <emph.end type="italics"/></s> <s id="id.2.1.176.1.0"><margin.target id="note38"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 26. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <figure id="id.037.01.039.1.jpg" xlink:href="037/01/039/1.jpg"> </figure> <pb xlink:href="037/01/040.jpg"/> <s id="id.2.1.179.1.0"><emph type="italics"/>Che &longs;e il punto G fo&longs;&longs;e nel centro del mondo; allhora quanto piu il pe&longs;o &longs;arà da pre&longs;&longs;o al <lb/>G, &longs;arà piu graue: & douunque &longs;ia po&longs;to il pe&longs;o, fuor che nel G &longs;empre &longs;tarà &longs;opra <lb/>il centro C, come in<emph.end type="italics"/> K<emph type="italics"/>: Imperoche tirata la linea G<emph.end type="italics"/>K; <emph type="italics"/>que&longs;ta (&longs;e condo laqua<lb/>le &longs;i fa il mouimento naturale del pe&longs;o) in&longs;ieme co'l braccio della bilancia<emph.end type="italics"/> K<emph type="italics"/>C <lb/>farà vn'angolo acuto, peroche <lb/>gli angoli posti alla ba&longs;e in<emph.end type="italics"/> K <lb/><emph type="italics"/>& G del triangolo di due la<lb/>ti eguali C<emph.end type="italics"/>K<emph type="italics"/>G &longs;ono &longs;empre <lb/>acuti. </s> <s id="id.2.1.179.2.0">Hor &longs;iano paragonate <lb/>in&longs;ieme que&longs;te due co&longs;e, cioè il <lb/>pe&longs;o posto in<emph.end type="italics"/> K, <emph type="italics"/>& quello, <lb/>che è po&longs;to in D, &longs;arà il pe&longs;o <lb/>in K piu graue, che quello <lb/>in D; imperoche tirata la li<lb/>nea DG, e&longs;&longs;endo che li tre an<lb/>goli di cia&longs;cuno triangolo &longs;iano <lb/>eguali à due angoli retti, & <lb/>l'angolo DCG del triangolo <lb/>CDG di due lati eguali &longs;ia <lb/>maggiore dell'angolo KCG <lb/>del triangolo CKG di due <lb/>lati eguali; &longs;aranno gli altri an<lb/>goli alla ba&longs;e DGC GDC <lb/>pre&longs;i in&longs;ieme minori de gli al<lb/>tri KGC GKC pre&longs;i in&longs;ie<emph.end type="italics"/><lb/><figure id="id.037.01.040.1.jpg" xlink:href="037/01/040/1.jpg"></figure><lb/><emph type="italics"/>me; & la metà di questi, cioè l'angolo CDG &longs;arà minore dell'angolo CKG: <lb/>Per laqual co&longs;a mouendo&longs;i il pe&longs;o po&longs;to in K &longs;ciolto naturalmente per KG, & <lb/>il pe&longs;o po&longs;to in D per DG come per &longs;patij, per i quali &longs;ono portati nel centro del <lb/>mondo; la linea CD, cioè il braccio della bilancia &longs;i acco&longs;terà piu al mouimento <lb/>naturale del pe&longs;o po&longs;to in D <expan abbr="totalm&etilde;te">totalmente</expan> &longs;ciolto, alla linea cioè DG, che CK al <lb/>mouimento &longs;atto &longs;econdo KG. </s> <s id="id.2.1.179.3.0">So&longs;tenterà dunque piu la linea CD, che C K. </s> <s id="id.2.1.179.4.0">& perciò il pe&longs;o po&longs;to in K per le co&longs;e di &longs;opra dette &longs;arà piu graue, che in D. </s> <s id="id.2.1.179.5.0">Ol<lb/>tre à ciò, perche &longs;e il pe&longs;o po&longs;to in K fo&longs;&longs;e del tutto libero, & &longs;ciolto, &longs;i mouerebbe <lb/>in giu per KG, &longs;e egli non fo&longs;&longs;e impedito dalla linea CK, laquale sforza il pe&longs;o <lb/>à mouer&longs;i oltra la linea KG per la circonferenza KH; la linea KG &longs;o&longs;tente<lb/>rà il pe&longs;o in parte, & gli farà re&longs;istenza, sforzandolo à mouer&longs;i per la circonferenza <lb/>KH. </s> <s id="id.2.1.179.6.0">Et percioche l'angolo CDG è minore dell'angolo CKG, & l'angolo <lb/>CDK è eguale all'angolo CKH, &longs;arà l'angolo re&longs;tante GDK maggiore del re <lb/>&longs;tante GKH. </s> <s id="id.2.1.179.7.0">Dunque la circonferenza KH &longs;arà piu da pre&longs;&longs;o al mouimento <lb/>naturale del pe&longs;o &longs;ciolto po&longs;to in K, cioè alla linea KG, che la circonferenza <lb/>DK alla linea DG. </s> <s id="id.2.1.179.8.0">Per laqual co&longs;a la linea CD &longs;a piu re&longs;i&longs;tenza al pe&longs;o po&longs;to <lb/>in D, che la linea CK al pe&longs;o posto in K. </s> <s id="id.2.1.179.9.0">Adunque il pe&longs;o po&longs;to in K &longs;arà<emph.end type="italics"/><pb pagenum="13" xlink:href="037/01/041.jpg"/><emph type="italics"/>piu graue, che in D. </s> <s id="id.2.1.179.10.0">Similmente mostrera&longs;&longs;i, che quanto il pe&longs;o &longs;arà piu da pre&longs;&longs;o <lb/>ad F, come in L manco grauerà; ma quanto piu da pre&longs;&longs;o &longs;i trouerà al G, co<lb/>me in H, e&longs;&longs;ere piu graue. <emph.end type="italics"/></s> <s id="id.2.1.181.1.0"><emph type="italics"/>Che &longs;e il centro del mondo fo&longs;&longs;e in S fra i punti CG; Primieramente &longs;i mo&longs;trerà nel <lb/>modo i&longs;te&longs;&longs;o, che il pe&longs;o in qualunque luogo po&longs;to starà &longs;opra il centro C, come in <lb/>H: peroche tirate le li<lb/>nee HG HS, l'angolo <lb/>che è alla ba&longs;e GHC del <lb/><expan abbr="triãgolo">triangolo</expan> di due lati eguali <lb/>CHG è &longs;empre acuto: <lb/>Per laqual co&longs;a anco SHC <lb/>minor di lui &longs;arà parimen<lb/>te &longs;empre acuto. </s> <s id="id.2.1.181.2.0">ma &longs;ia ti <lb/>rata dal punto S la linea <lb/>SK à piombo di CS. <lb/></s> <s id="id.2.1.181.3.0">Dico che il pe&longs;o è piu gra<lb/>ue in<emph.end type="italics"/> K, <emph type="italics"/>che in alcun'al<lb/>tro &longs;ito della circonferen<lb/>za FKG; & quanto <lb/>piu da pre&longs;&longs;o &longs;arà allo F, <lb/>ouero al G meno graue<lb/>rà. </s> <s id="id.2.1.181.4.0">Prendan&longs;i ver&longs;o lo <lb/>F i punti DL, & con <lb/><expan abbr="giungã&longs;i">giungan&longs;i</expan> le linee LC LS <lb/>DC DS, & &longs;iano al<emph.end type="italics"/><lb/><figure id="id.037.01.041.1.jpg" xlink:href="037/01/041/1.jpg"></figure><lb/><emph type="italics"/>lungate le linee LS DS KS HS fin'alla <expan abbr="circõferenza">circonferenza</expan> del cerchio in EM NO; <lb/>& &longs;iano <expan abbr="cõgiunte">congiunte</expan> CE, CM, CN, CO. </s> <s id="id.2.1.181.5.0">Hor percioche LE DM &longs;i taglia<lb/>no in&longs;ieme in S, &longs;arà il rettangolo LSE eguale al rettangolo DSM. </s> <s id="id.2.1.181.6.0">Onde &longs;i co<emph.end type="italics"/><arrow.to.target n="note39"></arrow.to.target><lb/><emph type="italics"/>me è la LS ver&longs;o la DS, co&longs;i &longs;arà la SM ver&longs;ola SE; ma è maggior la LS <lb/>della DS; & la SM di e&longs;&longs;a SE. </s> <s id="id.2.1.181.7.0">Dunque LS SE pre&longs;e in&longs;ieme &longs;aranno mag<emph.end type="italics"/><arrow.to.target n="note40"></arrow.to.target><lb/><emph type="italics"/>giori delle DS SM. </s> <s id="id.2.1.181.8.0">& per la ragion i&longs;te&longs;&longs;a &longs;i mo&longs;trerà la KN e&longs;&longs;er minore di DM. <lb/></s> <s id="id.2.1.181.9.0">Di piu percioche il rettangolo OSH è eguale al rett'angolo KSN; per la mede&longs;i<emph.end type="italics"/><arrow.to.target n="note41"></arrow.to.target><lb/><emph type="italics"/>ma ragione la HO &longs;arà maggiore della KN. </s> <s>& nell'i&longs;te&longs;&longs;o modo in tutto la <emph.end type="italics"/><arrow.to.target n="note42"></arrow.to.target><lb/><emph type="italics"/> KN &longs;i dimostrerà minore di tutte le altre linee, che pa&longs;&longs;ino per lo punto S. </s> <s id="id.2.1.181.10.0">Et <lb/>percioche de i triangoli di due lati eguali CLE DCM i lati LC CE &longs;ono e<lb/>guali a i lati DC CM; & la ba&longs;e LE è maggiore di DM: &longs;arà l'angolo <lb/>LCE maggiore dell'angolo DCM. </s> <s id="id.2.1.181.11.0">Per laqual co&longs;a gli angoli CLE CEL po<emph.end type="italics"/><arrow.to.target n="note43"></arrow.to.target><lb/><emph type="italics"/>sti alla ba&longs;e tolti in&longs;ieme &longs;aranno minori de gli angoli CDM CMD; & le me<lb/>tà di que&longs;ti, cioè l'angolo CLS &longs;arà minore dell'angolo CDS. </s> <s id="id.2.1.181.12.0">Dunque il pe&longs;o po<lb/>&longs;to in L &longs;opra la linea LC grauerà piu, che po&longs;to in D &longs;opra la DC; & piu <lb/>&longs;tarà &longs;opra il centro in L, che in D. </s> <s id="id.2.1.181.13.0">Similmente &longs;i mo&longs;trerà, che il pe&longs;o in D<emph.end type="italics"/><pb xlink:href="037/01/042.jpg"/><emph type="italics"/>&longs;tarà piu &longs;opra il centro C, che in K. </s> <s id="id.2.1.181.14.0">Adunque il pe&longs;o po&longs;to in K &longs;arà piu <lb/>graue, che in D, & in D, che in L. </s> <s>& con la mede&longs;ima ragione in tutto, pero<lb/>che KN è minore di HO, &longs;arà l'angolo CKS maggiore dell'angolo CHS. <lb/></s> <s id="id.2.1.181.15.0">Per laqual co&longs;a il pe&longs;o posto in H &longs;tarà piu &longs;opra il centro C, che in<emph.end type="italics"/> K; <emph type="italics"/>& in que<lb/>&longs;ta maniera &longs;i mostrerà, che douunque &longs;ia il pe&longs;o nella circonferenza FDG, manco <lb/>starà &longs;opra il centro quando &longs;arà po&longs;to in K, che in altro &longs;ito: & quanto piu da<lb/>pre&longs;&longs;o egli &longs;arà ad F, ouero à G piu &longs;tarà &longs;opra. </s> <s id="id.2.1.181.16.0">Dopo percioche l'angolo CKS <lb/>è maggiore del CDS, & CDK è eguale à CKH: &longs;arà il re&longs;tante SKH mi<lb/>nore del re&longs;tante SDK. </s> <s id="id.2.1.181.17.0">Per laqual co&longs;a la circonferenza KH &longs;arà piu da pre&longs;&longs;o <lb/>al mouimento naturale <lb/>diritto del pe&longs;o po&longs;to in <lb/>K &longs;ciolto, cioè alla li<lb/>nea KS, che la circon<lb/>ferenza DK al moui<lb/>mento DS. </s> <s id="id.2.1.181.18.0">& perciò <lb/>la linea CD &longs;a piu re&longs;i<lb/>&longs;tenza al pe&longs;o po&longs;to in D <lb/>che la CK al pe&longs;o me&longs;<lb/>&longs;o in<emph.end type="italics"/> K. </s> <s><emph type="italics"/>& per que&longs;ta <lb/>ragione &longs;i mo&longs;trera l'an<lb/>golo SHG e&longs;&longs;er mag<lb/>giore dello SKH; & <lb/>per con&longs;equente la linea <lb/>CH &longs;are piu re&longs;i&longs;tenza <lb/>al pe&longs;o po&longs;to in H, che <lb/>CK al pe&longs;o me&longs;&longs;o in K. <lb/></s> <s id="id.2.1.181.19.0">Similmente dimo&longs;trera&longs;&longs;i <lb/>che la linea CL piu &longs;o<lb/>&longs;tenterà il pe&longs;o, che CD:<emph.end type="italics"/><lb/><figure id="id.037.01.042.1.jpg" xlink:href="037/01/042/1.jpg"></figure><lb/><emph type="italics"/>& per le, cagioni i&longs;te&longs;&longs;e &longs;i prouerà, che il pe&longs;o me&longs;&longs;o in K grauerà meno &longs;opra la li<lb/>nea CK, che in qual &longs;i voglia altro &longs;ito della circonferenza FDG: & quanto <lb/>piu da pre&longs;&longs;o &longs;arà ad F, ouero à G, manco grauerà. </s> <s id="id.2.1.181.20.0">dunque piu graue &longs;ara in K, <lb/>che in altro &longs;ito: & &longs;arà meno graue quanto piu da pre&longs;&longs;o &longs;tara ad F, ouero a G. <emph.end type="italics"/></s> <s id="id.2.1.184.1.0"><margin.target id="note39"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 35. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <s id="id.2.1.185.1.0"><margin.target id="note40"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del &longs;esto. <emph.end type="italics"/></s> <s id="id.2.1.186.1.0"><margin.target id="note41"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <s id="id.2.1.187.1.0"><margin.target id="note42"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 25. <emph type="italics"/>del quinte. <emph.end type="italics"/></s> <s id="id.2.1.188.1.0"><margin.target id="note43"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 25. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.189.1.0"><emph type="italics"/>Se in fine il centro C fo&longs;&longs;e nel centro del mondo, egli è manife&longs;to, che il pe&longs;o po&longs;to doue <emph.end type="italics"/><lb/><arrow.to.target n="note44"></arrow.to.target><emph type="italics"/>&longs;i voglia &longs;tarà fermo. </s> <s id="id.2.1.189.2.0">Come posto il pe&longs;o in D la linea CD &longs;o&longs;terrà tutto il pe&longs;o, <lb/>per e&longs;&longs;er a piombo dell'orizonte di e&longs;&longs;o pe&longs;o po&longs;to in D. </s> <s id="id.2.1.189.3.0">Dunque &longs;tarà fermo <lb/>il pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.190.1.0"><margin.target id="note44"></margin.target><emph type="italics"/>Per la prima di questo. <emph.end type="italics"/></s> <s id="id.2.1.191.1.0"><emph type="italics"/>Hor percioche nelle co&longs;e, che fin qui &longs;ono &longs;tate dimostrate non habbiamo fatto mentio<lb/>ne alcuna della grauezza del braccio della bilancia, però &longs;e vorremo anco con&longs;idera<lb/>re la grauezza del detto braccio, &longs;i potrà ritrouare il centro della grauezza della ma<emph.end type="italics"/><pb pagenum="14" xlink:href="037/01/043.jpg"/><emph type="italics"/>gnitudine fatta dal pe&longs;o, & dal braccio, & &longs;i <expan abbr="de&longs;criuerãno">de&longs;criueranno</expan> le circonferenze de' cerchi <lb/>&longs;econdo la di&longs;tanza dal centro della bilancia ad e&longs;&longs;o centro della grauezza, come &longs;e <lb/>in e&longs;&longs;o (come è veramente) fo&longs;&longs;e posto il pe&longs;o, Et le co&longs;e che &longs;enza la con&longs;ideratio <lb/>ne della grauezza del braccio della bilancia habbiamo trouato, tutte nell'i&longs;te&longs;&longs;o mo<lb/>do con&longs;iderando ancora tal grauità le ritrouaremo. <emph.end type="italics"/></s> <s id="id.2.1.192.1.0"><emph type="italics"/>Dalle co&longs;e dette dunque, <expan abbr="con&longs;iderãdo">con&longs;iderando</expan> la bilancia, <lb/>come ella è lontana dal centro del mondo <lb/>nel modo che e&longs;&longs;i hanno fatto, come etiandio <lb/>è in atto, appare la fal&longs;ità di coloro, che dico<lb/>no il pe&longs;o po&longs;to in A e&longs;&longs;ere piu graue, che <lb/>in altro &longs;ito; & in&longs;ieme e&longs;&longs;er fal&longs;o, che quan<lb/>to piu il pe&longs;o è lontano dalla linea FG, tan<lb/>to e&longs;&longs;ere piu graue: imperoche il punto O <lb/>è piu da pre&longs;&longs;o alla FG, che il punto A; <lb/>percioche la linea tirata a piombo dal pun<emph.end type="italics"/><arrow.to.target n="note45"></arrow.to.target><lb/><emph type="italics"/>to O ad FG è minore della CA. </s> <s id="id.2.1.192.2.0">Da poi <lb/>egli è parimente fal&longs;o, che il pe&longs;o dal punto <lb/>A &longs;i moua piu velocemente, che da altro <lb/>&longs;ito. </s> <s id="id.2.1.192.3.0">peroche dal punto O &longs;i mouerà piu ve<lb/>locemente, che dal punto A, concio&longs;ia che <lb/>in O &longs;ia piu libero ve&longs;ciolto, che in altro &longs;ito; <lb/>& la &longs;ce&longs;a dal punto O &longs;ia piu da pre&longs;&longs;o al <lb/>mouimento naturale diritto, che qual &longs;i vo<lb/>glia altra di&longs;ce&longs;a. <emph.end type="italics"/></s> <s id="id.2.1.193.1.0"><margin.target id="note45"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <figure id="id.037.01.043.1.jpg" xlink:href="037/01/043/1.jpg"> </figure> <pb xlink:href="037/01/044.jpg"/> <s id="id.2.1.196.1.0"><emph type="italics"/>Oltre a ciò quando mo&longs;trano per via della piu diritta, & della piu torta di&longs;ce&longs;a, che il pe<lb/>&longs;o è piu graue in A, che in D, & in D, che in L. </s> <s id="id.2.1.196.2.0">Primieramente per certo e&longs;tima <lb/>no il fal&longs;o, che &longs;e alcun pe&longs;o &longs;arà collocato in qual &longs;i voglia &longs;ito della circonferenza, <lb/>come in D, la &longs;ua vera di&longs;ce&longs;a douer&longs;i fare per la linea diritta DR egualmente di<lb/>&longs;tante da e&longs;&longs;a FG, come &longs;econdo il mouimento naturale, &longs;i come prima è &longs;tato det<lb/>to. </s> <s id="id.2.1.196.3.0">Percioche in qual &longs;i voglia &longs;ito &longs;i collochi alcun pe&longs;o, &longs;e riguardiamo il mouimen<lb/>to &longs;uo naturale al proprio luogo, alquale &longs;i moue dirittamente per &longs;ua natura, pre&longs;up<lb/>po&longs;ta tutta la figura dell'vniuer&longs;o mondo, &longs;arà tale, che &longs;empre lo &longs;patio, per lo qua<lb/>le &longs;i moue naturalmente, parerà hauere ragione di linea tirata dalla circonferenza al <lb/>centro. </s> <s id="id.2.1.196.4.0">Adunque le na<lb/>turali di&longs;ce&longs;e diritte di <lb/>qual &longs;i voglia pe&longs;o &longs;ciol<lb/>to non &longs;i po&longs;&longs;ono fare <lb/>per linee tra loro egual<lb/>mente di&longs;tanti, per an<lb/>dar&longs;i à trouar tutte nel <lb/>centro del mondo. </s> <s id="id.2.1.196.5.0">pre<lb/>&longs;uppongono da poi, che <lb/>il pe&longs;o mo&longs;&longs;o da D in <lb/>A per linea diritta ver<lb/>&longs;o il centro del mondo <lb/>&longs;ia della <expan abbr="quãtità">quantità</expan> i&longs;te&longs;&longs;a, <lb/>come &longs;e egli fo&longs;&longs;e da O <lb/>in C &longs;i fattamente, <lb/>che il <expan abbr="p&utilde;to">punto</expan> A &longs;ia egual<lb/>mente di&longs;tante dal cen<lb/>tro del mondo, come C; <lb/>ilche è parimente fal&longs;o: <emph.end type="italics"/><lb/><figure id="id.037.01.044.1.jpg" xlink:href="037/01/044/1.jpg"></figure><lb/><arrow.to.target n="note46"></arrow.to.target><emph type="italics"/>Imperoche il punto A è piu da lontano dal centro del mondo, che C: percioche <lb/>maggior è la linea tirata dal centro del mondo fin ad A, che quella del centro del <lb/>mondo fin a C, concio&longs;ia che vna linea dal centro del mondo fin ad A &longs;i di&longs;tenda <lb/>&longs;otto vn'angolo retto contenuto dalle linee AC, & dal punto C al centro del <lb/>mondo. </s> <s id="id.2.1.196.6.0">Dalle quali co&longs;e non &longs;olo rie&longs;ce vana quella pre&longs;uppo&longs;ta, laquale dimostra, <lb/>che la bilancia DE ritorna in AB, ma anco cadono tutte le loro dimo&longs;trationi; <lb/>&longs;e for&longs;e non dice&longs;&longs;ero, che que&longs;te co&longs;e tutte per la grandi&longs;&longs;ima di&longs;tanza, che è fra il cen<lb/>tro del mondo, & noi &longs;ono co&longs;i in&longs;en&longs;ibili, che per cagione di que&longs;ta in&longs;en&longs;ibilità, <lb/>&longs;i po&longs;&longs;ano pre&longs;upponere, come vere; concio&longs;ia, che tutti quelli, iquali hanno trattato <lb/>que&longs;te co&longs;e, le habbiano pre&longs;uppo&longs;te, come note; ma&longs;&longs;imamente, percioche quello <lb/>e&longs;&longs;ere in&longs;en&longs;ibile non fà, che la di&longs;ce&longs;a del pe&longs;o da L in D (per v&longs;are le loro paro<lb/>le) non pigli meno del diretto, che la di&longs;ce&longs;a DA. </s> <s id="id.2.1.196.7.0">Similmente l'arco DA piglie<lb/>rà piu del diretto, che la circonferenza EV. </s> <s id="id.2.1.196.8.0">onde &longs;arà vera la pre&longs;uppo&longs;ta, & le <lb/>altre dimo&longs;trationi rimarranno nella &longs;ua &longs;ua forza. </s> <s id="id.2.1.196.9.0">Concediamo etiandio, che il pe<emph.end type="italics"/><pb pagenum="15" xlink:href="037/01/045.jpg"/><emph type="italics"/>&longs;o po&longs;to in A &longs;ia piu graue, che in altro &longs;ito; & che la di&longs;ce&longs;a diritta del pe&longs;o &longs;i deb<lb/>ba fare per linea diritta egualmente di&longs;tante da FG, & quali &longs;i voglian punti pre&longs;i <lb/>nelle linee egualmente di&longs;tanti dall'orizonte e&longs;&longs;ere egualmente lontani dal centro <lb/>del mondo: non &longs;eguiterà gia per que&longs;to, che la loro dimostratione &longs;ia vera, con la<lb/>quale vengono a dire, che il pe&longs;o posto in A è piu grane, che in altro &longs;ito, come in <lb/>L. </s> <s id="id.2.1.196.10.0">Percioche &longs;e egli fo&longs;&longs;e vero, che quanto piu il pe&longs;o in que&longs;ta maniera di&longs;cende <lb/>piu al diritto, iui fo&longs;&longs;e piu graue; &longs;eguirebbe etiandio, che quanto l'iste&longs;&longs;o pe&longs;o de<lb/>&longs;cende&longs;&longs;e egualmente in archi eguali al diritto, che ne i luoghi mede&longs;imi haue&longs;&longs;e gra<lb/>uezza eguale, ilche in que&longs;to modo e&longs;&longs;er fal&longs;o &longs;i dimo&longs;tra. <emph.end type="italics"/></s> <s id="id.2.1.198.1.0"><margin.target id="note46"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.199.1.0"><emph type="italics"/>Siano le circonferenze AL AM tra loro eguali, & congiunga&longs;i LM, laquale ta<lb/>gli AB in X; &longs;arà LM egualmente di&longs;tante da FG, & à piombo di AB, <emph.end type="italics"/><arrow.to.target n="note47"></arrow.to.target><lb/><emph type="italics"/>& XM &longs;arà eguale ad XL. </s> <s id="id.2.1.199.2.0">Se dunque il pe&longs;o da L &longs;arà mo&longs;&longs;o in A per la cir<lb/>conferenza LA, il mouimento &longs;uo diritto &longs;arà &longs;econdo la linea LX. </s> <s id="id.2.1.199.3.0">Ma &longs;e egli &longs;i <lb/>mouerà da A in M per la circonferenza AM, il &longs;uo mouimento &longs;arà &longs;econdo <lb/>la linea diritta XM. </s> <s id="id.2.1.199.4.0">Per laqual co&longs;a la &longs;ce&longs;a da L in A &longs;arà eguale alla &longs;ce&longs;a da <lb/>A in M, &longs;i per cau&longs;a delle circonferenze eguali, & &longs;i per le linee rette eguali, & à <lb/>piombo di e&longs;&longs;a AB. </s> <s id="id.2.1.199.5.0">Adunque il pe&longs;o mede&longs;imo po&longs;to in L grauerà egualmente, <lb/>come in A, ilche è fal&longs;o, concio&longs;ia, che egli è di gran lunga piu graue in A, che in L. <emph.end type="italics"/></s> <s id="id.2.1.200.1.0"><margin.target id="note47"></margin.target><emph type="italics"/>Per la terza del terzo. <emph.end type="italics"/></s> <s id="id.2.1.201.1.0"><emph type="italics"/>Et benche AMLA prendano, &longs;econdo e&longs;&longs;i, egualmente del diretto, diranno for&longs;e, <lb/>nondimeno perche il principio della &longs;ce&longs;a da L, cioè LD piglia meno del diretto, che <lb/>il principio della &longs;ce&longs;a da A, cioè AN, il pe&longs;o in A &longs;arà piu graue, che in L. <lb/></s> <s id="id.2.1.201.2.0">Imperoche e&longs;&longs;endo (come è &longs;tato di &longs;opra po&longs;to) la circonferenza AN eguale ad <lb/>LD, laquale (&longs;econdo eßi) piglia di diretto CT; ma LD piglia di diretto PO, <lb/>però il pe&longs;o &longs;arà piu graue in A, che in L. </s> <s>ilche &longs;e fo&longs;&longs;e vero, &longs;eguirebbe, che l'i&longs;te&longs;<lb/>&longs;o pe&longs;o nel mede&longs;imo &longs;ito, in diuer&longs;o modo &longs;olamente con&longs;iderato, ver&longs;o il mede&longs;imo <lb/>&longs;ito fo&longs;&longs;e & piu graue, & piu lieue; ilche è impo&longs;&longs;ibile. </s> <s id="id.2.1.201.3.0">cioè &longs;e con&longs;ideriamo la &longs;ce&longs;a <lb/>del pe&longs;o po&longs;to in L in quanto egli de&longs;cende da L in A &longs;arà piu graue, che &longs;e con&longs;ide<lb/>reremo la &longs;ce&longs;a del pe&longs;o i&longs;te&longs;&longs;o da L in D &longs;olamente. </s> <s id="id.2.1.201.4.0">ne po&longs;&longs;ono negare per i mede<lb/>&longs;imi detti &longs;uoi, che la di&longs;ce&longs;a del pe&longs;o da L in A non pigli del diretto LX, ouero PC. <lb/></s> <s id="id.2.1.201.5.0">Et che &longs;imilmente la &longs;ce&longs;a AM non prenda di diretto XM: pigliando eßi ancora <lb/>à que&longs;to modo, & co&longs;i nece&longs;&longs;ario &longs;ia di pigliare. </s> <s id="id.2.1.201.6.0">percioche &longs;e vogliono dimo&longs;trare, <lb/>che la bilancia DE ritorni in AB paragonando la &longs;ce&longs;a del pe&longs;o po&longs;to in D con <lb/>la &longs;ce&longs;a del pe&longs;o posto in E, egli è nece&longs;&longs;ario, che mo&longs;trino, che la diritta &longs;ce&longs;a OC <lb/>ri&longs;pondente alla circonferenza DA &longs;ia maggiore della &longs;ce&longs;a diritta TH ri&longs;ponden<lb/>te alla circonferenza EV. </s> <s>peroche &longs;e piglia&longs;&longs;ero &longs;olamente vna parte di tutta la &longs;ce <lb/>&longs;a da D in A, come D<emph.end type="italics"/>K, <emph type="italics"/>& dimo&longs;tra&longs;&longs;ero, che piu di diretto piglia la &longs;ce&longs;a D<emph.end type="italics"/>K, <lb/><emph type="italics"/>che la eguale portione della &longs;ce&longs;a dal punto E, &longs;eguirebbe il pe&longs;o po&longs;to in D, &longs;econ<lb/>do eßi, e&longs;&longs;ere piu graue del pe&longs;o po&longs;to in E, & mouer&longs;i in giu fin al K &longs;olamente. <lb/></s> <s id="id.2.1.201.7.0">per modo che la bilancia &longs;ia mo&longs;&longs;a in KI. </s> <s id="id.2.1.201.8.0">Similmente &longs;e vogliono mo&longs;trare, che la <lb/>bilancia KI ritorni in AB pigliando vna portione della &longs;ce&longs;a da K in A, cioè KS, <lb/>& mo&longs;tra&longs;&longs;ero, che KS pigli piu di diretto, che la &longs;ce&longs;a eguale, che è dirimpetto dal <lb/>punto I: &longs;eguirebbe con &longs;imile modo il pe&longs;o po&longs;to in K e&longs;&longs;ere piu graue, che in I, & <emph.end type="italics"/><pb xlink:href="037/01/046.jpg"/><emph type="italics"/>mouer&longs;i &longs;olamente fin ad S. </s> <s id="id.2.1.201.9.0">Et &longs;e di nouo mo&longs;tra&longs;&longs;ero vna portione della &longs;ce&longs;a da S <lb/>in A, & co&longs;i &longs;ucceßiuamente e&longs;&longs;ere piu diritta della &longs;ce&longs;a eguale del pe&longs;o oppo&longs;to; <lb/>&longs;empre &longs;eguirà, che la bilancia SI andarà piu da pre&longs;&longs;o ad AB, ma non <expan abbr="dimostrerãno">dimostre<lb/>ranno</expan> giamai che per <lb/>uenga in AB. </s> <s id="id.2.1.201.10.0">Se <lb/>dunque vogliono di <lb/>mo&longs;trare, che la <expan abbr="bilãcia">bilan<lb/>cia</expan> DE ritorni in <lb/>AB, egli è nece&longs;&longs;a<lb/>rio, che pre&longs;upponga<lb/>no, che la &longs;ce&longs;a del <lb/>pe&longs;o da D in A <expan abbr="pr&etilde;da">pren<lb/>da</expan> di diretto la quan<lb/>tità della linea tira<lb/>ta dal punto D ad <lb/>AB ad angoli ret<lb/>ti; & co&longs;i, &longs;e para<lb/>goneremo le &longs;ce&longs;e e<lb/>guali di DA AN <lb/>fra loro, lequali <expan abbr="pr&etilde;dono">pren<lb/>dono</expan> di diretto OC <lb/>CT, accaderà, che <emph.end type="italics"/><lb/><figure id="id.037.01.046.1.jpg" xlink:href="037/01/046/1.jpg"></figure><lb/><emph type="italics"/>il pe&longs;o i&longs;te&longs;&longs;o &longs;arà in D graue egualmente, come in A. </s> <s id="id.2.1.201.11.0">Ma &longs;e le portioni &longs;olamente <lb/>piglieremo da DA, &longs;arà piu graue in A, che in D. </s> <s id="id.2.1.201.12.0">Adunque dalla diuer&longs;ità &longs;o<lb/>lamente del modo del con&longs;iderare, auerrà, che il pe&longs;o mede&longs;imo &longs;arà & piu graue, <lb/>& piu leggiero; & non per la natura della co&longs;a. </s> <s id="id.2.1.201.13.0">Di piu la pre&longs;uppo&longs;ta loro non <lb/>afferma, che il pe&longs;o &longs;econdo il &longs;ito &longs;ia piu graue, quanto nel &longs;ito mede&longs;imo il principio <lb/>della &longs;ua di&longs;ce&longs;a è meno obliquo. </s> <s id="id.2.1.201.14.0">La pre&longs;upposta dunque di &longs;opra addotta, cioè che <lb/>&longs;econdo il &longs;ito il pe&longs;o è piu graue quanto nell'i&longs;te&longs;&longs;o &longs;ito meno obliqua è la di&longs;ce&longs;a, non <lb/>&longs;olamente non &longs;i puote concedere à modo alcuno, per le co&longs;e, che habbiamo dette; <lb/>ma anco percioche non è co&longs;a difficile il dimo&longs;trare tutto l'oppo&longs;to, cioè il pe&longs;o mede&longs;i<lb/>mo in eguali circonferenze quanto meno obliqua è la di&longs;ce&longs;a, iui meno grauare. <emph.end type="italics"/></s> <s id="id.2.1.203.1.0"><emph type="italics"/>Siano come prima le circonferenze AL AM tra loro eguali; & &longs;ia il punto L vici <lb/>no ad F, & congiunga&longs;i LM, la quale &longs;arà à piombo di AB & LX &longs;arà anco <lb/>eguale ad XM. </s> <s id="id.2.1.203.2.0">Dapoi pre&longs;&longs;o ad M tra M & G &longs;ia pre&longs;o come &longs;i vuole, il pun<emph.end type="italics"/><lb/><arrow.to.target n="note48"></arrow.to.target><emph type="italics"/>to P, & &longs;ia fatta la circonferenza PO eguale alla circonferenza AM, &longs;arà il <lb/>punto O pre&longs;&longs;o ad A. </s> <s id="id.2.1.203.3.0">& &longs;iano congiunte le linee CL, CO, CM, CP, OP<emph.end type="italics"/><lb/><arrow.to.target n="note49"></arrow.to.target></s> <s><emph type="italics"/>& dal punto P tiri&longs;i la PN a piombo di OC. </s> <s id="id.2.1.203.4.0">& percioche la circonferenza. <lb/></s> <s id="id.2.1.203.5.0">AM è eguale alla circonferentia OP; &longs;arà l'angolo ACM eguale all'angolo <emph.end type="italics"/><lb/><arrow.to.target n="note50"></arrow.to.target><emph type="italics"/>OCP, & l'angolo CXM retto eguale al retto CNP, &longs;arà anco il re&longs;tante angolo <lb/>XMC del triangolo MXC eguale al re&longs;tante NPC del triangolo PCN. <emph.end type="italics"/></s><pb pagenum="16" xlink:href="037/01/047.jpg"/> <s><emph type="italics"/>Ma il lato ancora CM è eguale al lato CP, dunque il triangolo MCX è egua<lb/>le al triangolo PCN, & il lato MX eguale al lato NP. </s> <s id="id.2.1.203.6.0">Onde la linea PN <lb/>&longs;arà eguale ad LX. </s> <s id="id.2.1.203.7.0">Tiri&longs;i oltre a ciò dal punto O la linea OT egualmente di<lb/>&longs;tante da AC, laquale tagli NP in V. </s> <s>& &longs;ia anco tirata dal punto P vna <lb/>linea a piombo di OT, <lb/>la quale per certo non <lb/>puote cadere tra OV, <lb/>perche e&longs;&longs;endo l'angolo <lb/>ONV retto, &longs;arà acu<emph.end type="italics"/><arrow.to.target n="note51"></arrow.to.target><lb/><emph type="italics"/>to lo OVN. </s> <s id="id.2.1.203.8.0">Per la <lb/>qualco&longs;a OVP &longs;arà <lb/>ottu&longs;o. </s> <s id="id.2.1.203.9.0">Non caderà <lb/>dunque la linea tirata <lb/>dal punto P tra OV <lb/>à piombo di OT: pe<lb/>roche due angoli d'uno <lb/><expan abbr="triãgolo">triangolo</expan> &longs;arebbono l'u<lb/>no retto, & l'altro ot<lb/>tu&longs;o, che è impoßibile. <lb/></s> <s id="id.2.1.203.10.0">Caderà dunque nella li<lb/>nea OT nella parte di <lb/>VT, et &longs;ia PT. </s> <s id="id.2.1.203.11.0">&longs;arà &longs;e <lb/>condo e&longs;&longs;i, PT la di<emph.end type="italics"/><lb/><figure id="id.037.01.047.1.jpg" xlink:href="037/01/047/1.jpg"></figure><lb/><emph type="italics"/>ritta &longs;ce&longs;a della circonferenza OP. </s> <s id="id.2.1.203.12.0">Percioche dunque l'angolo ONV è retto, <emph.end type="italics"/><arrow.to.target n="note52"></arrow.to.target><lb/><emph type="italics"/>&longs;arà la linea OV maggiore della ON. </s> <s id="id.2.1.203.13.0">Onde la OT &longs;arà parimente maggiore <lb/>della ON. </s> <s id="id.2.1.203.14.0">& co&longs;i di&longs;tendendo&longs;i la linea OP &longs;otto gli angoli retti ONP, <lb/>OTP, &longs;arà il quadrato di OP eguale alli quadrati ON NP in&longs;ieme pre&longs;i, &longs;i<emph.end type="italics"/><arrow.to.target n="note53"></arrow.to.target><lb/><emph type="italics"/>milmente eguale a i quadrati di OT TP in&longs;ieme. </s> <s id="id.2.1.203.15.0">per laqual co&longs;a li quadrati in&longs;ie<lb/>me di ON NP &longs;aranno eguali a i quadrati in&longs;ieme di OT TP. </s> <s id="id.2.1.203.16.0">Ma il quadrato <lb/>di OT è maggiore del quadrato di ON, per e&longs;&longs;ere maggiore la linea OT della <lb/>ON. </s> <s id="id.2.1.203.17.0">Adunque il quadrato di NP &longs;ara maggiore del quadrato TP & perciò la <lb/>linea TP &longs;arà minore della linea PN, & della linea LX. </s> <s id="id.2.1.203.18.0">Meno obliqua <lb/>dunque &longs;arà la &longs;ce&longs;a dell'arco LA, che dell'arco OP. </s> <s id="id.2.1.203.19.0">Dunque il pe&longs;o po<lb/>sto in L, per i loro detti, &longs;arà piu graue, che in O, il che, per le co&longs;e, che di <lb/>&longs;opra habbiamo detto, è manife&longs;tamente fal&longs;o. </s> <s id="id.2.1.203.20.0">concio&longs;ia, che il pe&longs;o po&longs;to in O <lb/>&longs;ia piu graue, che in L. </s> <s id="id.2.1.203.21.0">Non &longs;i puote dunque raccogliere dal piu diritto, & <lb/>piu torto mouimento in quel modo pigliato, e&longs;&longs;ere il pe&longs;o tanto piu graue &longs;econ<lb/>do il &longs;ito, quanto nel mede&longs;imo &longs;ito è meno torta la &longs;ce&longs;a. </s> <s id="id.2.1.203.22.0">& quinci na&longs;ce tutto <lb/>qua&longs;i il &longs;uo errore & inganno in cote&longs;ta co&longs;a. </s> <s id="id.2.1.203.23.0">Imperoche quantunque per acciden<lb/>te alle volte dalle co&longs;e fal&longs;e ne &longs;egua il vero, tutta via per &longs;e &longs;te&longs;&longs;e principalmente <lb/>dalle fal&longs;e ne &longs;egue il fal&longs;o, &longs;i come dalle vere &longs;empre il vero ne &longs;egue. </s> <s id="id.2.1.203.24.0">Non è pero <lb/>da marauigliar&longs;i, &longs;e mentre e&longs;&longs;i prendono co&longs;e fal&longs;e, & &longs;tanno &longs;opra quelle, come ve<emph.end type="italics"/><pb xlink:href="037/01/048.jpg"/><emph type="italics"/>ri&longs;&longs;ime, raccolgono, & conchiudono co&longs;e in tutto fal&longs;i&longs;&longs;ime. </s> <s id="id.2.1.203.25.0">&longs;ono oltre a ciò inganna<lb/>ti, mentre pigliano a contemplare la bilancia &longs;emplicemente per via di matematica, <lb/>e&longs;&longs;endo la con&longs;ideratione &longs;ua mechanica affatto, ne di lei &longs;i po&longs;&longs;a ragionare a modo al<lb/>cuno &longs;enza il vero mouimento, & &longs;enza i pe&longs;i, che &longs;ono in tutto co&longs;e naturali, &longs;en<lb/>za le quali non &longs;i po&longs;&longs;ono ritrouare per niuna maniera le vere cagioni di quelle co&longs;e, <lb/>che accadono alla bilancia. <emph.end type="italics"/></s> <s id="id.2.1.205.1.0"><margin.target id="note48"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 27. <emph type="italics"/>del terzo<emph.end type="italics"/></s> <s id="id.2.1.206.1.0"><margin.target id="note49"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 32. <emph type="italics"/>del primo<emph.end type="italics"/></s> <s id="id.2.1.207.1.0"><margin.target id="note50"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 26. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.208.1.0"><margin.target id="note51"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 13. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.209.1.0"><margin.target id="note52"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 19. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.210.1.0"><margin.target id="note53"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 47. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.211.1.0"><emph type="italics"/>Oltre a ciò &longs;e anche con<lb/>cederemo la pre&longs;up<lb/>po&longs;ta, &longs;i partono tut<lb/>tauia molto <expan abbr="l&utilde;ge">lunge</expan> dal <lb/>la <expan abbr="cõ&longs;ideratione">con&longs;ideratione</expan> della <lb/>bilancia, mentre di<lb/>&longs;corrono; che in quel <lb/>la maniera debba la <lb/>bilancia DE ritor<lb/>nare in AB: percio <lb/>che &longs;empre pigliano <lb/>vn di due pe&longs;i &longs;epara<lb/>tamente come D, <lb/>ouero E, come &longs;e hor <lb/>l'uno, hor l'altro fo&longs;<lb/>&longs;e po&longs;to nella bilan<lb/>cia, non congiunti in <lb/>&longs;ieme ambidue in <lb/>modo veruno, e&longs;&longs;en<emph.end type="italics"/><lb/><figure id="id.037.01.048.1.jpg" xlink:href="037/01/048/1.jpg"></figure><lb/><emph type="italics"/>doche nondimeno bi&longs;ogni fare tutto all'oppo&longs;ito di ciò, ne &longs;i puote con&longs;iderare dirit<lb/>tamente l'uno &longs;enza l'altro, e&longs;&longs;endoche &longs;i ragiona di loro nella bilancia collocati. <lb/></s> <s id="id.2.1.211.2.0">Concio&longs;ia che quando dicono la di&longs;ce&longs;a del pe&longs;o po&longs;to in D e&longs;&longs;ere meno torta, che <lb/>la di&longs;ce&longs;a del pe&longs;o po&longs;to in E, co&longs;i &longs;arà il pe&longs;o in D, per la pre&longs;uppo&longs;ta, piu graue <lb/>del pe&longs;o po&longs;to in E; onde per e&longs;&longs;ere piu graue, eglie nece&longs;&longs;ario, che &longs;i moua in giu, <lb/>& che la bilancia DE ritorni in AB: Cote&longs;to di&longs;cor&longs;o non è di momento alcu<lb/>no. </s> <s id="id.2.1.211.3.0">Primieramente &longs;empre argomentano come &longs;e i pe&longs;i in DE debbano &longs;cende<lb/>re, con&longs;iderando la &longs;ce&longs;a di vno &longs;olamente &longs;enza la compagnia, & congiungimen<lb/>to dell'altro. </s> <s id="id.2.1.211.4.0">Vltimamente nondimeno e&longs;&longs;i per la comparatione delle di&longs;ce&longs;e de'pe<lb/>&longs;i conchiudono il pe&longs;o posto in D mouer&longs;i in giu, & il po&longs;to in E in &longs;u, prenden<lb/>do l'uno, & l'altro pe&longs;o congiunti in&longs;ieme fra loro nella bilancia. </s> <s id="id.2.1.211.5.0">Ma da &longs;uoi me<lb/>de&longs;imi principij, i quali v&longs;ano, & dalle &longs;ue dimo&longs;trationi &longs;i puote cauare ageuoli&longs;&longs;i<emph.end type="italics"/><lb/><arrow.to.target n="note54"></arrow.to.target><emph type="italics"/>mamente l'oppo&longs;ito di quel che &longs;i faticano di difendere. </s> <s id="id.2.1.211.6.0">Imperoche &longs;e &longs;i paragona <lb/>la di&longs;ce&longs;a del pe&longs;o po&longs;to in D con la &longs;alita del pe&longs;o po&longs;to in E, come tirate le'linee <lb/>E<emph.end type="italics"/>K <emph type="italics"/>DH a piombo di AB, e&longs;&longs;endo l'angolo DCH eguale all'angolo ECK, <emph.end type="italics"/><lb/><arrow.to.target n="note55"></arrow.to.target><emph type="italics"/>& l'angolo DHC retto eguale al retto E<emph.end type="italics"/>K<emph type="italics"/>C, & il lato DC eguale al lato <lb/>CE; &longs;arà il triangolo CDH eguale al triangolo CEK, & il lato DH egua<emph.end type="italics"/><pb pagenum="17" xlink:href="037/01/049.jpg"/><emph type="italics"/>le al lato EK: & e&longs;&longs;endo l'angolo DCA eguale all'angolo ECB, &longs;arà anco <lb/>la circonferenza DA eguale alla circonferenza BE. </s> <s id="id.2.1.211.7.0">Mentre dunque il pe&longs;o po<lb/>sto in D &longs;cende per la circonferenza DA, il pe&longs;o po&longs;to in E &longs;ale per la circon<lb/>ferenza EB eguale a DA, & la &longs;ce&longs;a del pe&longs;o po&longs;to in D prenderà, (&longs;econdo <lb/>il co&longs;tume loro) di diretto DH: & la &longs;alita del pe&longs;o E prenderà di diretto EK <lb/>eguale a DH: &longs;arà dunque la &longs;ce&longs;a del pe&longs;o posto in D eguale alla &longs;alita del pe&longs;o <lb/>po&longs;to in E: & quale &longs;arà la inclinatione d'uno al mouimento in giù, tale &longs;arà etian<lb/>dio la re&longs;i&longs;tenza dell'altro al mouimento in sù, cioè la re&longs;istentia della violenza del <lb/>pe&longs;o po&longs;to in E nella a&longs;ce&longs;a, contra&longs;tando &longs;i oppone alla naturale po&longs;&longs;anza del pe<lb/>&longs;o po&longs;to in D per e&longs;&longs;ere a lei eguale; percioche quanto il pe&longs;o po&longs;to in D per la na<lb/>tural po&longs;&longs;anza de&longs;cende piu velocemente in giù, in tanto il pe&longs;o po&longs;to in E più tar<lb/>do &longs;ale violentemente. </s> <s id="id.2.1.211.8.0">Per laqual co&longs;a niuno di loro due pe&longs;era piu dell'altro, non <lb/>procedendo attione da eguale. </s> <s id="id.2.1.211.9.0">il pe&longs;o po&longs;to in D dunque non mouerà il pe&longs;o po&longs;to <lb/>in E in &longs;u&longs;o, peroche &longs;e lo moue&longs;&longs;e, &longs;arebbe nece&longs;&longs;ario, che il pe&longs;o po&longs;to in D ha<lb/>ue&longs;&longs;e virtu maggiore in di&longs;cendendo, che il pe&longs;o po&longs;to in E in &longs;alendo, ma que&longs;te co<lb/>&longs;e &longs;ono eguali: adunque &longs;taranno &longs;ermi i pe&longs;i, & la grauezza del pe&longs;o po&longs;to in D &longs;a<lb/>rà eguale alla grauezza del pe&longs;o po&longs;to in E. </s> <s id="id.2.1.211.10.0">Oltre a ciò perche pre&longs;uppongono, che <lb/>quanto il pe&longs;o è piu di&longs;tante dalla linea FG della dirittura, tanto e&longs;&longs;ere piu graue. <lb/></s> <s id="id.2.1.211.11.0">però tirate parimente da i punti DE le linee DO, EI a piombo di FG, con <lb/>modo &longs;imile &longs;i dimostrerà il triangolo CDO e&longs;&longs;ere eguale al triangolo CEI: & <lb/>la linea DO e&longs;&longs;ere eguale ad EI. </s> <s id="id.2.1.211.12.0">Tanto dunque è di&longs;tante il pe&longs;o po&longs;to in D <lb/>dalla linea FG, quanto il pe&longs;o po&longs;to in E. </s> <s id="id.2.1.211.13.0">Dalle ragioni loro dunque, & dalle &longs;ue <lb/>pre&longs;uppo&longs;te li pe&longs;i me&longs;&longs;i in DE &longs;ono graui egualmente. </s> <s id="id.2.1.211.14.0">Di piu, che vieta che non &longs;i di <lb/>mo&longs;tri la bi lancia DE mouer&longs;i per nece&longs;&longs;ità in FG con &longs;imile ragione? </s> <s id="id.2.1.211.15.0">Primie<lb/>ramente &longs;i puote raccogliere dalle loro mede&longs;ime dimo&longs;trationi, la &longs;alita del pe&longs;o po<lb/>&longs;to in E ver&longs;o il B e&longs;&longs;ere piu diritta della &longs;alita del pe&longs;o po&longs;to in D ver&longs;o lo F, <lb/>cioè manco prendere di diretto la &longs;alita del pe&longs;o po&longs;to in D in archi eguali, che la <lb/>&longs;alita del pe&longs;o po&longs;to in E. </s> <s id="id.2.1.211.16.0">Pre&longs;upponga&longs;i dunque, che il pe&longs;o &longs;ia piu leggiero &longs;econ<lb/>do il &longs;ito tanto quanto nel &longs;ito mede&longs;imo meno diritta è la &longs;ua &longs;alita: Laqual pre<lb/>&longs;upposta pare tanto manife&longs;ta, quanto l'altra loro. </s> <s id="id.2.1.211.17.0">percioche dunque la &longs;alita del <lb/>pe&longs;o po&longs;to in E è piu diritta della &longs;alita del pe&longs;o po&longs;to in D, per la pre&longs;uppo&longs;ta il <lb/>pe&longs;o po&longs;to in D &longs;arà piu leggiero del pe&longs;o po&longs;to in E. </s> <s id="id.2.1.211.18.0">Adunque il pe&longs;o po&longs;to in D <lb/>&longs;i mouerà in sù dal pe&longs;o po&longs;to in E, &longs;i fattamente che la bilancia peruenga in FG, <lb/>& co&longs;i potra&longs;si dimo&longs;trare la bilancia DE mouer&longs;i in FG, laqual dimo&longs;tratio<lb/>ne è del tutto veramente friuola, & pati&longs;ce le difficultà mede&longs;ime. </s> <s id="id.2.1.211.19.0">Percioche quan<lb/>tunque &longs;i conceda, come vero, che il pe&longs;o po&longs;to in E &longs;alendo &longs;ia piu graue del pe&longs;o <lb/>in D &longs;imilmente &longs;alendo, non perciò da que&longs;to &longs;egue, che il pe&longs;o po&longs;to in E de<lb/>&longs;cendendo &longs;ia piu graue del pe&longs;o posto in D &longs;alendo. </s> <s id="id.2.1.211.20.0">Niuna dunque di que&longs;te due <lb/>dimo&longs;trationi, che dicono la bilancia DE ritornare in AB, ouero mouer&longs;i in <lb/>FG, è vera. <emph.end type="italics"/></s> <s id="id.2.1.213.1.0"><margin.target id="note54"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.214.1.0"><margin.target id="note55"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 25. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.215.1.0"><emph type="italics"/>Oltre a ciò &longs;e e&longs;amineremo la loro pre&longs;uppo&longs;ta, & la forza delle loro parole, vedremo <lb/>per certo che altro &longs;entimento hanno. </s> <s id="id.2.1.215.2.0">Imperoche e&longs;&longs;endo che &longs;empre lo &longs;patio per lo<emph.end type="italics"/><pb xlink:href="037/01/050.jpg"/><emph type="italics"/>quale il pe&longs;o <expan abbr="naturalm&etilde;te">naturalmente</expan> &longs;i moue, &longs;i deue prendere dal centro della grauezza di e&longs;<lb/>&longs;o pe&longs;o ver&longs;o il centro del mondo à &longs;embianza di vna linea diritta tirata dal centro <lb/>della grauezza al centro del mondo, tanto &longs;i dirà que&longs;ta co&longs;i fatta di&longs;ce&longs;a del pe&longs;o <lb/>piu, & meno obliqua, quanto, &longs;econdo lo &longs;patio di&longs;&longs;egnato, a &longs;embianza della pre<lb/>detta linea piu ò meno &longs;i mouerà, (andando pero &longs;empre a trouare il luogo &longs;uo natu<lb/>rale, & vie piu &longs;empre auicinandoui&longs;i.) talche tanto piu obliqua &longs;i dica la &longs;ce&longs;a <expan abbr="quãto">quan<lb/>to</expan> &longs;i parte da cotale &longs;patio: & piu diritta quanto a lui &longs;i acco&longs;ta. </s> <s id="id.2.1.215.3.0">& in que&longs;to <lb/>&longs;entimento quella pre&longs;upposta non deue partorire difficulta ad alcuno, percioche co<lb/>&longs;i è la verita &longs;ua chiara, & conforme alla ragione, che non pare hauer me&longs;tieri di e&longs;<lb/>&longs;er fatta in alcun modo manife&longs;ta. <emph.end type="italics"/></s> <s id="id.2.1.216.1.0"><emph type="italics"/>Se dunque il pe&longs;o &longs;ciolto, collocato nel &longs;i<lb/>to di D &longs;i deue mouere al luogo pro<lb/>prio, &longs;enza dubbio, po&longs;to S centro del <lb/>mondo, &longs;i mouerà per la linea DS, <expan abbr="&longs;imilm&etilde;te">&longs;i<lb/>milmente</expan> il pe&longs;o po&longs;to in E &longs;ciolto &longs;i mo<lb/>uerà per la linea ES. </s> <s id="id.2.1.216.2.0">Per laqual co<lb/>&longs;a &longs;e, (come è vero) la &longs;ce&longs;a del pe&longs;o &longs;i <lb/>dirà piu, ò meno obliqua, &longs;econdo lo al <lb/>lontanar&longs;i, ouero appre&longs;&longs;ar&longs;i a gli &longs;patij <lb/>di&longs;segnati per le linee DS ES, per ri <lb/>&longs;petto a'loro naturali mouimenti ver&longs;o <lb/>i proprij luoghi, egli è chiaro, che meno <lb/>obliqua è la &longs;ce&longs;a di E per EG, che <lb/>di D per DA, per e&longs;&longs;ere stato di <lb/>&longs;opra mo&longs;trato che l'angolo SEG è <lb/>minore dell'angolo SDA. </s> <s id="id.2.1.216.3.0">Per laqual <lb/>co&longs;a piu grauerà il pe&longs;o in E, che in D, <lb/>il che totalmente è il contrario di quel<lb/>lo, che e&longs;si &longs;i &longs;ono sforzati di prouare. <lb/></s> <s id="id.2.1.216.4.0">Leueran&longs;i per auuentura contra di noi <lb/>dicendo. </s> <s id="id.2.1.216.5.0">Se dundue il pe&longs;o po&longs;to in E è <lb/>piu graue del pe&longs;o po&longs;to in D, la bi<lb/>lancia DE non &longs;tarà giamai in que<lb/>&longs;to &longs;ito, laqual co&longs;a noi habbiamo pro<lb/>po&longs;to di mantenere, ma &longs;i mouerà in F <lb/>G. </s> <s id="id.2.1.216.6.0">Allequali co&longs;e ri&longs;pondiamo. </s> <s id="id.2.1.216.7.0">che im<lb/>porta a&longs;&longs;ai, &longs;e noi con&longs;ideriamo i pe&longs;i o<lb/>uero in quanto &longs;ono &longs;eparati l'uno dal<lb/>l'altro, ouero in quanto &longs;ono tra loro <lb/>congiunti: perche altra è la ragione del <emph.end type="italics"/><lb/><figure id="id.037.01.050.1.jpg" xlink:href="037/01/050/1.jpg"></figure><lb/><emph type="italics"/>pe&longs;o po&longs;to in E &longs;enza il congiungimento del pe&longs;o po&longs;to in D, & altra di lui con <lb/>l'altro pe&longs;o congiunto, &longs;i fattamente che l'uno &longs;enza l'altro non &longs;i po&longs;&longs;a mouere. </s> <s id="id.2.1.216.8.0">Im<emph.end type="italics"/><pb pagenum="18" xlink:href="037/01/051.jpg"/><emph type="italics"/>peroche la diritta, & naturale di&longs;ce&longs;a dal pe&longs;o po&longs;to in E, inquanto egli è &longs;enza al<lb/>tro congiungimento di pe&longs;o, &longs;i fa per la linea ES. </s> <s>ma inquanto egli è congiunto <lb/>col pe&longs;o D, la &longs;ua naturale di&longs;ce&longs;a non &longs;arà piu per la linea ES, ma per vna li<lb/>nea egualmente di&longs;tante da CS. </s> <s id="id.2.1.216.9.0">percioche la magnitudine compo&longs;ta de i pe&longs;i ED. </s> <s><lb/>& della bilancia DE il cui centro della grauezza è C, &longs;e in ne&longs;&longs;un luogo non &longs;a<lb/>rà &longs;o&longs;tenuta, &longs;i mouerà naturalmente in giu nel modo che &longs;i troua, &longs;econdo la gra<lb/>uezza del centro per la linea diritta tirata dal centro della grauezza C al centro <lb/>del mondo S, finche il centro C peruenga nel centro S. </s> <s id="id.2.1.216.10.0">La bilancia dunque DE <lb/>in&longs;ieme co'pe&longs;i, in quella maniera, che &longs;i troua &longs;i mouerà in giu per modo tale, che il <lb/>punto C &longs;i moua per la linea CS, fin che C peruenga in S, & la bilancia <lb/>DE in HK; & habbia la bilancia in HK la po&longs;itione i&longs;te&longs;&longs;a, che prima hauea; <lb/>cio è, che la HK &longs;ia egualmente distante da DE. </s> <s id="id.2.1.216.11.0">Congiungan&longs;i dunque DH <lb/>EK. </s> <s id="id.2.1.216.12.0">egli è manife&longs;to, che mentre la bilancia DE &longs;i moue in HK, mouer&longs;i an<lb/>che i punti DE per le linee DH EK, come quelle che &longs;ono & fra &longs;e, & ad <emph.end type="italics"/><arrow.to.target n="note56"></arrow.to.target><lb/><emph type="italics"/>e&longs;&longs;a CS eguali, & egualmente di&longs;tanti. </s> <s id="id.2.1.216.13.0">Per la qual co&longs;a i pe&longs;i posti in DE, in <lb/>quanto &longs;ono fra loro congiunti, &longs;e riguarderemo il mouimento loro naturale &longs;i moue<lb/>ranno non &longs;econdo le linee DS, ES, ma &longs;econdo LDH MEK egualmente <lb/>di&longs;tanti da e&longs;&longs;a CS. </s> <s id="id.2.1.216.14.0">Ma la naturale inclinatione del pe&longs;o po&longs;to in E libero, & <lb/>&longs;ciolto &longs;arà per ES, & del pe&longs;o po&longs;to in D <expan abbr="&longs;imilm&etilde;te">&longs;imilmente</expan> &longs;ciolto &longs;arà per DS. </s> <s>& per<lb/>cio non è &longs;conueneuole, che il pe&longs;o mede&longs;imo hora in E, hora in D, &longs;ia piu graue <lb/>in E, che in D. </s> <s id="id.2.1.216.15.0">Ma &longs;e i pe&longs;i po&longs;ti in ED &longs;ono l'un l'altro fra &longs;e congiunti, & gli <lb/>con&longs;idereremo in quanto &longs;ono congiunti, &longs;arà la naturale inclinatione del pe<lb/>&longs;o po&longs;to in E per la linea MEK, percioche la grauezza dell'altro pe&longs;o po&longs;to <lb/>in D fa &longs;i, che il pe&longs;o po&longs;to in E non graui &longs;opra la linea ES, ma nella EK. <lb/></s> <s id="id.2.1.216.16.0">Ilche fa parimente la grauezza del pe&longs;o po&longs;to in E, cioè, che il pe&longs;o po&longs;to in D <lb/>non graui per la linea retta DS, ma &longs;econdo DH, per impedir&longs;i ambedue l'uno <lb/>l'altro che non vadino à propri luoghi. </s> <s id="id.2.1.216.17.0">Concio&longs;ia dunque che la naturale &longs;ce&longs;a dirit<lb/>ta de i pe&longs;i po&longs;ti in DE &longs;ia &longs;econdo LDH, MEK, &longs;arà &longs;imilmente la naturale <lb/>&longs;alita diritta loro &longs;econdo le i&longs;te&longs;&longs;e linee HDL KEM. </s> <s>& la naturale &longs;alita del <lb/>pe&longs;o po&longs;to in E &longs;i dirà più, & meno torta, quanto che &longs;econdo lo &longs;patio &longs;i mouerà <lb/>più, & meno pre&longs;&longs;o la linea MK. </s> <s id="id.2.1.216.18.0">& a que&longs;to modo in tutto &longs;i ha da pigliare & la &longs;a <lb/>lita & la di&longs;ce&longs;a del pe&longs;o po&longs;to in D &longs;econdo la linea LH, &longs;e dunque il pe&longs;o po&longs;to <lb/>in E &longs;i moue&longs;&longs;e in giù per la linea EG, mouerebbe il pe&longs;o po&longs;to in D in sù per <lb/>DF. </s> <s>& percioche l'angolo CEK è eguale all'angolo CDL, & l'angolo CEG <emph.end type="italics"/><arrow.to.target n="note57"></arrow.to.target><lb/><emph type="italics"/>è eguale all'angolo CDF; &longs;arà il <expan abbr="re&longs;tãte">re&longs;tante</expan> angolo GEK al re&longs;tante LDF egua<lb/>le. </s> <s id="id.2.1.216.19.0">& e&longs;&longs;endo quella pre&longs;uppo&longs;ta, che dice il pe&longs;o e&longs;&longs;er più graue &longs;econdo il &longs;ito, <lb/>quanto in quel mede&longs;imo &longs;ito la di&longs;ce&longs;a è meno obliqua per chiara, & manife&longs;ta ri<lb/>ceuuta, &longs;arà anche da e&longs;&longs;ere accettata &longs;enza dubbio que&longs;t' altra, cioè, che il pe&longs;o &longs;arà <lb/>più graue &longs;econdo il &longs;ito, quanto nel &longs;ito mede&longs;imo meno obliqua &longs;arà la &longs;alita; per <lb/>non e&longs;&longs;ere manco manife&longs;ta, ne meno conforme alla ragione. </s> <s id="id.2.1.216.20.0">&longs;arà dunque eguale <lb/>la &longs;ce&longs;a del pe&longs;o po&longs;to in E alla &longs;alita del pe&longs;o po&longs;to in D, percioche la &longs;ce&longs;a del pe<lb/>&longs;o po&longs;to in E tiene tanto di obliquo, quanto la &longs;alita del pe&longs;o po&longs;to in D. </s> <s>& quale<emph.end type="italics"/><pb xlink:href="037/01/052.jpg"/>&longs;arà la inclinatione dell'vno al moui<lb/>mento in giù, tale parimente &longs;arà la re<lb/>&longs;istenza dell'altro al mouimento in sù. <lb/></s> <s id="id.2.1.377.8.0">Adunque il pe&longs;o po&longs;to in E non mo<lb/>uerà in sù il pe&longs;o po&longs;to in D: ne il pe&longs;o <lb/>po&longs;to in D: &longs;i mouerà in giù &longs;i fatta<lb/>mente, che moua in sù il pe&longs;o po&longs;to in <lb/>E. </s> <s id="id.2.1.377.9.0">imperoche e&longs;&longs;endo l'angolo CEB <lb/>eguale a CDA, & l'angolo CEM <lb/>&longs;ia eguale all'angolo CDH; &longs;arà il <lb/>re&longs;tante MEB eguale al re&longs;tante <emph.end type="italics"/><lb/><arrow.to.target n="note116"></arrow.to.target><emph type="italics"/>HDA. </s> <s id="id.2.1.377.10.0">La &longs;ce&longs;a dunque del pe&longs;o po<lb/>&longs;to in D &longs;arà eguale alla &longs;alita del pe<lb/>&longs;o po&longs;to in E. </s> <s id="id.2.1.377.11.0">Adunque il pe&longs;o po&longs;to <lb/>in D non mouerà in sù il pe&longs;o po&longs;to <lb/>in E. </s> <s id="id.2.1.377.12.0">Dalle quali co&longs;e &longs;egue che i pe&longs;i <lb/>po&longs;ti in DE, in quanto tra loro &longs;o<lb/>no congiunti, &longs;ono egualmente graui. <emph.end type="italics"/></s> <s id="id.2.1.219.1.0"><margin.target id="note56"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 33. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.220.1.0"><margin.target id="note57"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.379.1.0"><margin.target id="note116"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del prime. <emph.end type="italics"/></s> <figure id="id.037.01.052.1.jpg" xlink:href="037/01/052/1.jpg"> </figure> <s id="id.2.1.381.1.0"><emph type="italics"/>L'altra ragione po&longs;cia, con laquale vorrebbono mo&longs;trare, che &longs;imilmente la bilancia <lb/>DE ritorna in AB, con dire, che e&longs;&longs;endo la trutina della bilancia CF, la méta <lb/>viene ad e&longs;&longs;er CG. </s> <s>& percioche l'angolo DCG è maggiore dell'angolo ECG, <lb/>il pe&longs;o po&longs;to in D &longs;arà più graue del po&longs;to in E; dunque la bilancia DE ritorne<lb/>ra in AB; non conchiude nulla al parer mio; & que&longs;ta fintione della trutina, & <lb/>della méta è più to&longs;to da trala&longs;ciare, & pa&longs;&longs;arla con &longs;ilentio, che farne pur vna paro <lb/>la per confonderla, e&longs;&longs;endo del tutto co&longs;a volontaria, percioche la nece&longs;&longs;aria ragione <lb/>per laquale il pe&longs;o po&longs;to in D dall' angolo maggiore &longs;ia più graue, & perche il mag<lb/>giore angolo &longs;ia cagione di grauezza maggiore non appare in niun loco. </s> <s id="id.2.1.381.2.0">che &longs;e gli <lb/>angoli &longs;aranno tra loro paragonati, e&longs;&longs;endo l'angolo GCD eguale all'angolo <lb/>FCE; &longs;e l'angolo GCD è cau&longs;a della grauezza, perche l'angolo FCE &longs;imil<pb pagenum="19" xlink:href="037/01/053.jpg"/><emph type="italics"/>mente non è della grauez<lb/>za cagione? </s> <s id="id.2.1.375.5.0">Di questo ef<lb/>fetto mostrano di produ<lb/>cere in mezo que&longs;ta cagio<lb/>ne, perche CG è la mé<lb/>ta, & CF la trutina; <lb/>&longs;e (dicono e&longs;&longs;i) CG fo&longs;<lb/>&longs;e la trutina, & CF la <lb/>méta, all'hora l'angolo <lb/>FCE &longs;arebbe cagione <lb/>della grauezza, ma non <lb/>già il DCG ad e&longs;&longs;o e<lb/>guale laquale ragione è al <lb/>tutto fatta con la imagi<lb/>natione, & di voglia pro<lb/>pria. </s> <s id="id.2.1.375.6.0">Peroche, che puote <lb/>importare che la trutina <lb/>&longs;ia ouero in CF, ouero <lb/>in CG, e&longs;&longs;endo la bilan<lb/>cia DE &longs;empre &longs;o&longs;ten<lb/>tata nell'i&longs;te&longs;&longs;o punto C? </s> <s id="id.2.1.375.7.0">Ma affine che l'inganno loro re&longs;ti più chiaro. <emph.end type="italics"/></s> <figure id="id.037.01.053.1.jpg" xlink:href="037/01/053/1.jpg"> </figure> <s id="id.2.1.377.1.0"><emph type="italics"/>Sia la mede&longs;ima bilancia AB, il cui mezo C. </s> <s id="id.2.1.377.2.0">dapoi tutta la FG &longs;ia la trutina, <lb/>laquale &longs;tia immobile, & &longs;o&longs;tenga la bilancia AB nel punto C. </s> <s id="id.2.1.377.3.0">& moua&longs;i la <lb/>bilancia in DE. </s> <s>& per<lb/>cioche la trutina è &longs;opra, & <lb/>&longs;otto la bilancia, quale ango <lb/>lo &longs;arà cagione della grauez<lb/>za, e&longs;&longs;endo &longs;o&longs;tenuta la bi<lb/>lancia DE &longs;empre nel pun<lb/>to mede&longs;imo? </s> <s id="id.2.1.377.4.0">Diranno for<lb/>&longs;e &longs;e la trutina &longs;arà &longs;o&longs;tenu<lb/>ta dalla po&longs;&longs;anza po&longs;ta in <lb/>F, allhora CG &longs;arà tan<lb/>to quanto la méta, & l'an<lb/>golo DCG &longs;arà della gra<lb/>uezza cagione. </s> <s id="id.2.1.377.5.0">Ma &longs;e <emph.end type="italics"/><lb/><figure id="id.037.01.053.2.jpg" xlink:href="037/01/053/2.jpg"></figure><lb/><emph type="italics"/>egli &longs;arà &longs;ostenuto in G, allhora FCE &longs;arà cagione della grauezza, & la CF <lb/>&longs;arà tanto quanto la méta. </s> <s id="id.2.1.377.6.0">della qual co&longs;a niuna cagione pare poter&longs;i addurre, <lb/>&longs;e <expan abbr="nõ">non</expan> imaginata; peroche la méta (che dicono) non pare hauere à modo veruno nien<lb/>te di virtù che tiri dalla parte dell'angolo maggiore alcuna volta, & alcuna dalla <lb/>parte del minore. </s> <s id="id.2.1.377.7.0">Ma &longs;ia &longs;o&longs;tenuta la trutina da due po&longs;&longs;anze in F cioè, & in G, <pb xlink:href="037/01/054.jpg"/>ilche &longs;i puote fare per nece&longs;&longs;ità, come &longs;e la po&longs;&longs;anza posta in F fo&longs;&longs;e tanto debile, <lb/>che per &longs;e &longs;te&longs;&longs;a pote&longs;&longs;e &longs;o&longs;tentare &longs;olamente la metà del pe&longs;o & &longs;ia la po&longs;&longs;anza <lb/>posta in G eguale alla po&longs;&longs;anza po&longs;ta in F, & ambedue in&longs;ieme co' pe&longs;i &longs;o&longs;tenga<lb/>no la bilancia. </s> <s id="id.2.1.381.3.0">all'hora quale angolo &longs;arà cagione della grauezza? </s> <s id="id.2.1.381.4.0">non gia <lb/>FCE, peroche la trutina è <lb/>in CF, & è &longs;o&longs;tentata in <lb/>F: ne meno il DCG, e&longs;&longs;en<lb/>do la trutina in CG, & pa<lb/>rimente &longs;o&longs;tentata in G. <lb/></s> <s id="id.2.1.381.5.0">Non &longs;aranno dunque gli an<lb/>goli della grauezza cagione. <lb/></s> <s id="id.2.1.381.6.0">Co&longs;i ne anche la bilancia <lb/>DE da que&longs;to &longs;ito per que<lb/>&longs;ta cagione &longs;i mouerà. </s> <s id="id.2.1.381.7.0">Ma <emph.end type="italics"/><lb/><arrow.to.target n="note117"></arrow.to.target><emph type="italics"/>que&longs;ta loro &longs;entenza pare <lb/>e&longs;&longs;ere confermata da e&longs;&longs;i in <lb/>due modi. </s> <s id="id.2.1.381.8.0">Primieramente <emph.end type="italics"/><lb/><figure id="id.037.01.054.1.jpg" xlink:href="037/01/054/1.jpg"></figure><lb/><emph type="italics"/>dicono Ari&longs;totele nelle que&longs;tioni mecaniche hauere propo&longs;to que&longs;te due que&longs;tioni &longs;o<lb/>lamente, & le &longs;ue dimo&longs;trationi e&longs;&longs;ere fondate &longs;i nel maggiore, & nel minore <lb/>angolo, & &longs;i nella giacitura della trutina della bilancia. </s> <s id="id.2.1.381.9.0">Affermano dapoi que&longs;to <lb/>iste&longs;&longs;o in&longs;egnare la e&longs;perientia ancora, cioè, che la bilancia DE, &longs;tando la &longs;ua <lb/>trutina in CF, ritorna in AB egualmente di&longs;tante dall'orizonte. </s> <s id="id.2.1.381.10.0">& quando <lb/>la trutina &longs;tà in CG, mouer&longs;i in FG. </s> <s id="id.2.1.381.11.0">Mane Ari&longs;totele, ne la e&longs;perienza fauo<lb/>ri&longs;cono que&longs;ta loro opinione, anzi più to&longs;to le &longs;ono contrarij. </s> <s id="id.2.1.381.12.0">Peroche in quan<lb/>to appartiene alla e&longs;perienza &longs;i ingannano, e&longs;&longs;endo manife&longs;to ciò per e&longs;perienza <lb/>accadere, all'hor che il centro ancora della bilancia &longs;arà collocato ò &longs;opra, ò &longs;ot<lb/>to della bilancia, ma non già auenire que&longs;to stando la trutina ò &longs;opra &longs;olamente, <lb/>è &longs;otto. <emph.end type="italics"/></s> <s id="id.2.1.384.1.0"><margin.target id="note117"></margin.target><emph type="italics"/>il Cardano. <emph.end type="italics"/></s> <pb pagenum="20" xlink:href="037/01/055.jpg"/> <s id="id.2.1.385.1.0"><emph type="italics"/>Imperoche &longs;e la bilancia A <lb/>B haue&longs;&longs;e il centro C <lb/>&longs;opra la bilancia, & fo&longs;<lb/>&longs;e la trutina CD &longs;otto <lb/>la bilancia, & &longs;i moue&longs;<lb/>&longs;e la bilancia in EF, al<lb/>lhora EF di nouo ri<lb/>tornerà in AB. </s> <s>egual<lb/>mente di&longs;tante dall'o<lb/>rizonte. </s> <s id="id.2.1.385.2.0">&longs;imilmente &longs;e la <lb/>bilancia haue&longs;&longs;e il cen<lb/>tro C &longs;otto la bilancia, <lb/>& fo&longs;&longs;e la trutina CD <lb/>&longs;opra la bilancia, et &longs;i mo<lb/>ue&longs;&longs;e la bilancia in EF, <emph.end type="italics"/><arrow.to.target n="note118"></arrow.to.target><lb/><emph type="italics"/>egli è manife&longs;to, che la bi<lb/>lancia &longs;i mouerà in giu <lb/>dalla parte di F, &longs;tan<lb/>do la trutina &longs;opra la bi<lb/>lancia. </s> <s id="id.2.1.385.3.0">& in qual &longs;i vo<lb/>glia altro &longs;ito che &longs;ia la <lb/>trutina, auerrà &longs;empre il <lb/>mede&longs;imo. </s> <s id="id.2.1.385.4.0">Adunque <expan abbr="nõ">non</expan><lb/> è la trutina, ma il centro <lb/>della bilancia cagione di <lb/>cotali diuer&longs;i effetti. <emph.end type="italics"/></s> <s id="id.2.1.386.1.0"><margin.target id="note118"></margin.target><emph type="italics"/>Per la terza di questo. <emph.end type="italics"/></s> <figure id="id.037.01.055.1.jpg" xlink:href="037/01/055/1.jpg"> </figure> <figure id="id.037.01.055.2.jpg" xlink:href="037/01/055/2.jpg"> </figure> <s id="id.2.1.388.1.0"><emph type="italics"/>Egli è pero d'auertire in que&longs;ta parte che con di&longs;&longs;icultà &longs;i puote lauora re vna bilancia <lb/>materiale, che in vno punto &longs;olamente &longs;ia &longs;o&longs;tenuta, &longs;i come con la mente la imagi<lb/>niamo, & habbia le braccia dal centro co&longs;i eguali non &longs;olamente in lunghezza, ma <lb/>in larghezza, & in profundità, ò gro&longs;&longs;ezza, che tutte le parti di quà, & di là pe&longs;i<lb/>no a punto egualmente. </s> <s id="id.2.1.388.2.0">percio che la materia di&longs;&longs;icili&longs;&longs;imamente pati&longs;ce cotale giu<lb/>&longs;ta mi&longs;ura. </s> <s id="id.2.1.388.3.0">Per laqual co&longs;a &longs;e con&longs;idereremo il centro e&longs;&longs;ere in e&longs;&longs;a bilancia, non bi<lb/>&longs;ogna ricorrere al &longs;en&longs;o, concio&longs;ia, che le co&longs;e artificiate non &longs;i po&longs;&longs;ano ridurre a quel <lb/>&longs;ommo grado di perfettione. </s> <s id="id.2.1.388.4.0">Ma nelle altre co&longs;e la e&longs;perienza veramente potrà in&longs;e<lb/>gnare le co&longs;e che appaiono percioche <expan abbr="quãtunque">quantunque</expan> il centro della <expan abbr="bilãcia">bilancia</expan> &longs;empre &longs;ia vn <lb/>punto, nondimeno quando egli &longs;arà &longs;opra la bilancia, poco importa, &longs;e ben la bilancia <lb/>non &longs;ara &longs;o&longs;tenuta in quel punto co&longs;i puntalmente però che per e&longs;&longs;ere &longs;empre &longs;opra la <lb/>bilancia auerrà &longs;empre il mede&longs;imo. </s> <s id="id.2.1.388.5.0">Con &longs;imile modo, quando egli anco è &longs;otto la bi<lb/>lancia, ilche tuttauia non accade stando il centro in e&longs;&longs;a bilancia, per che &longs;e egli non <lb/>&longs;arà &longs;o&longs;tenuto &longs;empre in quel mezo accuratamente, &longs;ara differenza, e&longs;&longs;endo co&longs;a faci <lb/>li&longs;&longs;ima, che quel centro, muti il proprio &longs;ito, mentre &longs;i moue la bilancia. <emph.end type="italics"/></s> <pb xlink:href="037/01/056.jpg"/> <s id="id.2.1.390.1.0"><emph type="italics"/>Ma che Ari&longs;totele habbia <lb/>propo&longs;to due que&longs;tioni &longs;o<lb/>lamente, cioè perche la <lb/>trutina &longs;tando &longs;opra, &longs;e <lb/>la bilancia <expan abbr="nõ">non</expan> &longs;arà egual<lb/>mente di&longs;tante dall'ori<lb/>zonte in equilibrio, cioè <lb/>egualmente di&longs;tante dal <lb/>orizonte ritorna, ma &longs;e la <lb/>trutina &longs;ara po&longs;ta &longs;otto <lb/>non ritorna, ma di piu &longs;i <lb/>moue <expan abbr="&longs;ecõdo">&longs;econdo</expan> la parte ba&longs;<lb/>&longs;a: egli è verò per certo. <lb/></s> <s id="id.2.1.390.2.0">Ma non già per que&longs;to le <lb/>dimo&longs;trationi &longs;ue &longs;ono <lb/>fondate nell'angolo mag<lb/>giore, ò minore, & nella <lb/>giacitura della trutina, <lb/>come e&longs;&longs;i dicono: per cio<lb/>che in questo non com<lb/>prendono la <expan abbr="m&etilde;te">mente</expan> del filo<lb/>&longs;ofo, che a&longs;&longs;egna la ragio<lb/>ne de gli effetti diuer&longs;i <lb/>de'mouimenti della bilan<lb/>cia. </s> <s id="id.2.1.390.3.0">peroche tanto è lon<lb/>tano, che il filo&longs;ofo attri<lb/>bui&longs;ca que&longs;ti diuer&longs;i effet<emph.end type="italics"/><lb/><figure id="id.037.01.056.1.jpg" xlink:href="037/01/056/1.jpg"></figure><figure id="id.037.01.056.2.jpg" xlink:href="037/01/056/2.jpg"></figure><lb/><emph type="italics"/>ti à gli angoli, che piu to&longs;to dica e&longs;&longs;ere cagione l'ecce&longs;&longs;o, & quel &longs;opra più della gran<lb/>dezza che è dal perpendicolo dell'uno delle braccia della bilancia hor dall'una parte, <lb/>hora dall'altra. <emph.end type="italics"/></s> <s id="id.2.1.392.1.0"><emph type="italics"/>Come stando la trutina &longs;opra in CF, il perpendicolo &longs;arà FCG, il quale &longs;em<lb/>pre inchina, &longs;econdo lui, ver&longs;o il centro del mondo, il quale anco diuide la bilancia mo&longs;<lb/>&longs;a in DE in parti di&longs;uguali: & la parte maggiore è ver&longs;o il D, & quel che è piu, <lb/>inchina in giu. </s> <s id="id.2.1.392.2.0">Adunque dalla parte di D la bilancia &longs;i mouerà in giu fin che ri<lb/>torni in AB. </s> <s id="id.2.1.392.3.0">Ma &longs;e la trutina &longs;arà in CG di &longs;otto, &longs;arà GCF il perpendico<lb/>lo, ilquale diuiderà parimente la bilancia DE in parte di&longs;uguali, & la parte mag<lb/>giore &longs;arà ver&longs;o E; Per laqual co&longs;a la bilancia &longs;i mouerà in giu dalla parte di </s> <s id="id.2.1.392.4.0">& accioche que&longs;to &longs;ia dirittamente compre&longs;o, &longs;appia&longs;i, che quando la trutina è &longs;o<lb/>pra la bilancia, &longs;i ha da intendere, che anche il centro della bilancia &longs;ia &longs;opra la bi<lb/>lancia, & &longs;e di &longs;otto, anche il centro deue &longs;tare di &longs;otto, come piu a ba&longs;&longs;o manife&longs;te<lb/>ra&longs;&longs;i. </s> <s>Altramente la dimo&longs;tratione di Ari&longs;totele non conchiuderebbe nulla, pero <lb/>che stando il centro in e&longs;&longs;a bilancia, come in C moua&longs;i la bilancia in qual &longs;i voglia <emph.end type="italics"/><pb pagenum="21" xlink:href="037/01/057.jpg"/><emph type="italics"/>modo, il perpendicolo FG non diuiderà giamai la bilancia &longs;e non nel punto C, et <lb/>in parti eguali. </s> <s id="id.2.1.392.5.0">Onde la &longs;entenza di Ari&longs;totele non &longs;olamente non gli fauori&longs;ce, ma <lb/>gli fa anche grandi&longs;sima <lb/>mente contra. </s> <s id="id.2.1.392.6.0">il che <lb/>non &longs;olamente è chiaro <lb/>dalla &longs;econda & terza <lb/>propo&longs;itione di que&longs;to li<lb/>bro, ma anco percioche <lb/>&longs;tando il centro &longs;opra <lb/>la bilancia, il pe&longs;o alzato <lb/>acqui&longs;ta grauezza mag<lb/>giore per cau&longs;a del &longs;ito. <lb/></s> <s id="id.2.1.392.7.0">Dalla qual co&longs;a accade il <lb/>ritorno della bilancia ad <lb/>eguale di&longs;tanza dall'ori<lb/>zonte. </s> <s id="id.2.1.392.8.0">Ma per lo con<lb/>trario auiene quando il <lb/>centro è &longs;otto la bilan<lb/>cia. </s> <s id="id.2.1.392.9.0">Le quali co&longs;e tutte <lb/>&longs;i dimo&longs;treranno in que<lb/>&longs;ta maniera, pre&longs;uppo<lb/>nendo le co&longs;e, che di &longs;o<emph.end type="italics"/><lb/><figure id="id.037.01.057.1.jpg" xlink:href="037/01/057/1.jpg"></figure><lb/><emph type="italics"/>pra furono dechiarate, cioè il pe&longs;o &longs;ar&longs;i più graue da quel loco dal quale &longs;cende piu <lb/>dirittamente, & da quello che egli &longs;ale piu dirittamente far&longs;i parimente piu <lb/>graue. <emph.end type="italics"/></s> <pb xlink:href="037/01/058.jpg"/> <s id="id.2.1.395.1.0"><emph type="italics"/>Sia la bilancia AB egualmente di&longs;tante dall'orizonte, il cui centro C &longs;ia &longs;opra la <lb/>bilancia, & &longs;ia il perpendicolo CD: & &longs;iano i centri della grauezza di pe&longs;i eguali <lb/>po&longs;ti in AB: & la bilancia &longs;ia mo&longs;&longs;a in EF. </s> <s id="id.2.1.395.2.0">Dico, che il pe&longs;o posto in E ha <lb/>grauezza maggiore, che il <lb/>pe&longs;o posto in F. </s> <s>& per<lb/>ciò la bilancia EF e&longs;&longs;e<lb/>re per ritornare in A B. </s> <s id="id.2.1.395.3.0">&longs;ia allungata prima la linea <lb/>CD fin'al centro del mon<lb/>do, che &longs;ia S. </s> <s id="id.2.1.395.4.0">Dapoi &longs;ia<lb/>no congiunte le linee AC, <lb/>CB, EC, CF, HS; <lb/>& dai punti EF &longs;iano ti<lb/>rate le linee EKGFL egual<lb/><expan abbr="m&etilde;te">mente</expan> di&longs;tanti da HS. </s> <s id="id.2.1.395.5.0">Per<lb/>cioche dunque la di&longs;ce&longs;a na<lb/>turale diritta di tutta la <lb/>grandezza, cioè della bilan<lb/>cia EF co&longs;i di&longs;po&longs;ta in&longs;ie<lb/>me co'pe&longs;i è &longs;econdo la gra<lb/>uezza del centro H per la <lb/>diritta linea HS; &longs;arà <expan abbr="parim&etilde;te">pa<lb/>rimente</expan> la di&longs;ce&longs;a de'pe&longs;i me&longs;<lb/>&longs;i in EF co&longs;i di&longs;po&longs;ti &longs;econ<lb/>do le linee diritte E<emph.end type="italics"/>K <lb/><emph type="italics"/>FL egualmente distanti <lb/>da HS, &longs;i come di &longs;opra <lb/>habbiamo dimo&longs;trato. </s> <s id="id.2.1.395.6.0">La <lb/>di&longs;ce&longs;a dunque, & la &longs;ali<lb/>ta de i pe&longs;i po&longs;ti in EF &longs;i <lb/>dirà più, & meno obliqua <lb/>&longs;econdo la vicinanza, ò lon<lb/>tananza diputata &longs;econdo <emph.end type="italics"/><lb/><arrow.to.target n="note119"></arrow.to.target><emph type="italics"/>le linee EK FL. </s> <s id="id.2.1.395.7.0">& per<lb/>cioche li due lati AD DC <lb/>&longs;ono eguali a i due lati BD<emph.end type="italics"/><lb/><figure id="id.037.01.058.1.jpg" xlink:href="037/01/058/1.jpg"></figure><lb/><emph type="italics"/>DC; & gli angoli al D &longs;ono retti, &longs;arà il lato AC eguale al lato CB. </s> <s>& e&longs;<lb/>&longs;endo il punto C immobile; mentre, che i punti AB &longs;i moueranno, de &longs;criueran<lb/>no la circonferenza di vno cerchio, il cui mezo diametro &longs;arà AC. </s> <s id="id.2.1.395.8.0">Per laqual co<emph.end type="italics"/><lb/><arrow.to.target n="note120"></arrow.to.target><emph type="italics"/>&longs;a co'l centro C &longs;ia de&longs;critto il cerchio AE BF, i punti AB E<emph.end type="italics"/>F <emph type="italics"/>&longs;aranno nel <lb/>la circonferenza del cerchio. </s> <s id="id.2.1.395.9.0">ma e&longs;&longs;endo EF eguale ad AB, &longs;arà la circonfe<lb/>renza EAF eguale alla circonferenza AFB. </s> <s id="id.2.1.395.10.0">Onde tolta via la comune AF<emph.end type="italics"/><pb pagenum="22" xlink:href="037/01/059.jpg"/><emph type="italics"/>&longs;arà la circonferenza EA eguale alla circonferenza FB. </s> <s id="id.2.1.395.11.0">Hor percioche l'ango<lb/>lo mi&longs;to CEA è eguale al mi&longs;to CFB, & HFB è maggiore di CFB, & <emph.end type="italics"/><arrow.to.target n="note121"></arrow.to.target><lb/><emph type="italics"/>l'angolo HEA è minore di CEA; &longs;arà l'angolo HFB maggiore dell'angolo <lb/>HEA. </s> <s id="id.2.1.395.12.0">Da quali &longs;e &longs;aranno leuati via gli angoli HFG HEK eguali, &longs;arà l'an<lb/>golo GFB maggiore dell'angolo KEA. </s> <s id="id.2.1.395.13.0">Adunque la di&longs;ce&longs;a del pe&longs;o po&longs;to in <lb/>E &longs;arà meno obliqua della &longs;alita del pe&longs;o po&longs;to in F. </s> <s id="id.2.1.395.14.0">& <expan abbr="quãtunque">quantunque</expan> il pe&longs;o po&longs;to in E <lb/>de&longs;cendendo, & il pe&longs;o po&longs;to in F &longs;alendo &longs;i mouino per eguali circonferenze, nondi<lb/>meno percioche il pe&longs;o po&longs;to in E da que&longs;to luogo di&longs;cende piu dirittamente di quel <lb/>che il pe&longs;o F <expan abbr="a&longs;c&etilde;de">a&longs;cende</expan>:pero la naturale po&longs;&longs;anza del pe&longs;o po&longs;to in E &longs;upererà la <expan abbr="re&longs;i&longs;t&etilde;za">re&longs;i&longs;ten<lb/>za</expan> della violentia del pe&longs;o F. </s> <s id="id.2.1.395.15.0">Onde grauezza maggiore hauerà il pe&longs;o posto in E, <lb/>che il pe&longs;o po&longs;to in F. </s> <s id="id.2.1.395.16.0">Adunque il pe&longs;o po&longs;to in E &longs;i mouerà in giù & il pe&longs;o po&longs;to <lb/>in F in sù, fin che la bilancia EF ritorni in AB, che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.397.1.0"><margin.target id="note119"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.398.1.0"><margin.target id="note120"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 28. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <s id="id.2.1.399.1.0"><margin.target id="note121"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.400.1.0"><emph type="italics"/>La ragione di que&longs;to effetto po&longs;ta da Ari&longs;totele qui &longs;i puote vedere manife&longs;ta. </s> <s id="id.2.1.400.2.0">Percio<emph.end type="italics"/><arrow.to.target n="note122"></arrow.to.target><lb/><emph type="italics"/>che &longs;ia il punto N doue le linee CS EF &longs;i tagliano in&longs;ieme. </s> <s id="id.2.1.400.3.0">& percioche HE <lb/>è eguale ad HF; &longs;arà NE maggiore di NF. </s> <s id="id.2.1.400.4.0">adunque la linea CS, che no<lb/>ma perpendicolo, diuiderà la bilancia EF in parti di&longs;uguali. </s> <s id="id.2.1.400.5.0">concio&longs;ia dunque, che <lb/>la parte della bilancia NE &longs;ia maggiore della NF, & quel che è di più bi&longs;o<lb/>gni, che &longs;ia portato in giù, la bilancia EF dalla parte di E &longs;i mouerà in giu finche <lb/>ritorni in AB. <emph.end type="italics"/></s> <s id="id.2.1.401.1.0"><margin.target id="note122"></margin.target><emph type="italics"/>Ragione de Aristotele. <emph.end type="italics"/></s> <s id="id.2.1.402.1.0"><emph type="italics"/>Oltre à cio da quelle co&longs;e, che <lb/>fin hora &longs;ono &longs;tate dette, <lb/>&longs;i puote affermare, la bilan<lb/>cia EF da quel &longs;ito mo<lb/>uer&longs;i piu velocemente in <lb/>AB; d'onde la linea EF <lb/>allungata a dirittura per<lb/>uenga nel centro del mon<lb/>do. </s> <s id="id.2.1.402.2.0">come &longs;ia EFS vna <lb/>linea diritta. </s> <s id="id.2.1.402.3.0">& percioche <lb/>CD CK &longs;ono tra loro <lb/>eguali. </s> <s id="id.2.1.402.4.0">&longs;e dunque col cen<lb/>tro C, & con lo &longs;patio <lb/>CD &longs;i de&longs;criuerà il cerchio <lb/>DHM, &longs;aranno i punti <lb/>DH nella circonferenza <lb/>del cerchio. </s> <s id="id.2.1.402.5.0">Ma perche la <lb/>CH è à piombo di EF, <lb/>toccherà la EHS il cer<lb/>chio DHM nel punto <lb/>H. </s> <s id="id.2.1.402.6.0">il pe&longs;o dunque po&longs;to in <lb/>H, (&longs;i come di &longs;opra hab<lb/>biamo prouato) &longs;arà piu <emph.end type="italics"/><lb/><figure id="id.037.01.060.1.jpg" xlink:href="037/01/060/1.jpg"></figure><pb xlink:href="037/01/060.jpg"/><emph type="italics"/>graue che in verun altro &longs;ito del cerchio DHM. </s> <s id="id.2.1.402.7.0">Adunque la grandezza fatta de' <lb/>pe&longs;i EF, & della bilancia EF, il cui centro della grauezza sta in H, in cote&longs;to <lb/>&longs;ito grauerà più, che in qual &longs;i voglia altro &longs;ito del cerchio &longs;i troui il punto H. </s> <s id="id.2.1.402.8.0">Da <lb/>que&longs;to &longs;ito adunque &longs;i mouera piu velocemente che da qualunque altro. </s> <s id="id.2.1.402.9.0">& &longs;e lo H <lb/>&longs;arà piu da pre&longs;&longs;o al D <lb/>manco grauerà, & me<lb/>no &longs;i mouerà da quel &longs;ito; <lb/>peroche &longs;empreè piu torta <lb/>la &longs;ce&longs;a, & meno diritta. <lb/></s> <s id="id.2.1.402.10.0">La bilancia dunque EF <lb/>&longs;i mouerà più velocemen<lb/>te da que&longs;to &longs;ito, che da <lb/>altro &longs;ito, & &longs;e piu dapre&longs;<lb/>&longs;o acco&longs;teraßi ad AB, <lb/>d'indi &longs;i mouerà meno poi <lb/>quanto piu da lunge &longs;arà <lb/>di&longs;tante il punto H dal <lb/>punto C &longs;i mouerà più ve<lb/>locemente, il che non &longs;olo <lb/>da Ari&longs;totele nel principio <lb/>delle que&longs;tioni mecaniche, <lb/>& dai detti di &longs;opra è ma<lb/>nife&longs;to, ma ancora da quel <lb/>le co&longs;e, che di &longs;otto nella <lb/>&longs;e&longs;ta propo&longs;itione &longs;iamo <lb/>per dire, apparerà chiaro. <lb/></s> <s id="id.2.1.402.11.0">La bilancia dunque EF <lb/>quanto più &longs;arà lontana <lb/>dal &longs;uo centro, &longs;i mouerà anche piu velocemente. <emph.end type="italics"/></s> <figure id="id.037.01.060.2.jpg" xlink:href="037/01/060/2.jpg"> </figure> <pb pagenum="23" xlink:href="037/01/061.jpg"/> <s id="id.2.1.406.1.0"><emph type="italics"/>Sia poi la bilancia AB, il cui centro C stia &longs;otto la bilancia, & &longs;iano in AB <lb/>pe&longs;i eguali, & &longs;ia mo&longs;&longs;a la bilancia in EF. </s> <s id="id.2.1.406.2.0">Dico che il pe&longs;o ha grauezza maggio<lb/>re in F, che in E. </s> <s>& <lb/>perciò la bilancia EF <lb/>e&longs;&longs;ere per mouer&longs;i in giù <lb/>dalla parte di F. </s> <s id="id.2.1.406.3.0">&longs;ia allun<lb/>gata la linea DC dall'una <lb/>parte, & dall'altra fin <lb/>nel centro del mondo S, <lb/>& fin ad O, & &longs;ia tira<lb/>ta la linea HS, alla qua<lb/>le dai punti EF &longs;iano ti<lb/>rate le linee GEK FL <lb/>egualmente di&longs;tanti, & <lb/>&longs;iano congiunte le CE <lb/>CF: & dal centro C <expan abbr="cõ">con</expan> <lb/>lo &longs;patio CE de&longs;criua&longs;i <lb/>il cerchio AEO B</s> <s id="id.2.1.406.4.0">&longs;i dimo&longs;trerà &longs;imilmente <lb/>i punti AB EF e&longs;&longs;e<lb/>re nella circonferenza del <lb/>cerchio, & che la di&longs;ce&longs;a <lb/>della bilancia EF in&longs;ie<lb/>me co'pe&longs;i &longs;i fà diritta &longs;e <lb/>condo la linea HS: & <lb/>de i pe&longs;i po&longs;ti in EF &longs;e<lb/>condo le linee GK FL <lb/>egualmente di&longs;tanti da <lb/>HS. </s> <s id="id.2.1.406.5.0">Et percioche l'ango<lb/>lo CFP è eguale all'an<lb/>golo CEO &longs;arà l'ango<lb/>lo HFP maggiore del<lb/>l'angolo HEO. </s> <s id="id.2.1.406.6.0">ma l'an<emph.end type="italics"/><arrow.to.target n="note123"></arrow.to.target><lb/><emph type="italics"/>golo HFL è eguale al<lb/>l'angolo HEG. </s> <s id="id.2.1.406.7.0">Da qua<lb/>li &longs;e &longs;aranno leuati via <lb/>gli angoli HFP HEO,<emph.end type="italics"/><lb/><figure id="id.037.01.061.1.jpg" xlink:href="037/01/061/1.jpg"></figure><lb/><emph type="italics"/>&longs;arà l'angolo LFP minore dell' angolo GEO. </s> <s id="id.2.1.406.8.0">Per laqual co&longs;a la &longs;ce&longs;a del pe&longs;o <lb/>po&longs;to in F &longs;arà piu diritta della a&longs;ce&longs;a del pe&longs;o po&longs;to in E. </s> <s id="id.2.1.406.9.0">Adunque la po&longs;&longs;anza <lb/>naturale del pe&longs;o po&longs;to in F &longs;upererà la re&longs;i&longs;tenza della violentia del pe&longs;o po&longs;to in <lb/>E. </s> <s>& percio hauerà maggior grauezza il pe&longs;o di F, che il pe&longs;o di E. </s> <s id="id.2.1.406.10.0">Adunque <lb/>il pe&longs;o di F &longs;i mouerà in giù, & il pe&longs;o di E &longs;i mouerà in sù. <emph.end type="italics"/></s> <s id="id.2.1.409.1.0"><margin.target id="note123"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <pb xlink:href="037/01/062.jpg"/> <s id="id.2.1.410.1.0"><arrow.to.target n="note124"></arrow.to.target><emph type="italics"/>La ragione di Ari&longs;totele parimente qui è chiara. </s> <s id="id.2.1.410.2.0">Percioche &longs;ia il punto N doue le <lb/>linee CO EF &longs;i tagliano in&longs;ieme. </s> <s id="id.2.1.410.3.0">&longs;arà la NF maggiore della NE. </s> <s>& perche <lb/>il perpendicolo CO, &longs;e<lb/>condo lui, diuide in parti <lb/>di&longs;uguali la bilancia, & <lb/>la parte maggiore è ver&longs;o <lb/>F, cioè NF; la bilan<lb/>cia EF &longs;i mouerà in giù <lb/>dalla parte di F, concio <lb/>&longs;ia che quel che è di piu <lb/>venga portato à ba&longs;&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.411.1.0"><margin.target id="note124"></margin.target><emph type="italics"/>Ragione di Aristotele. <emph.end type="italics"/></s> <s id="id.2.1.412.1.0"><emph type="italics"/>Similmente dalle co&longs;e dette <lb/>caueremo, che <expan abbr="quãto">quanto</expan> piu <lb/>la bilancia EF tenente <lb/>il centro &longs;otto la bilancia, <lb/>&longs;arà <expan abbr="lõtana">lontana</expan> dal &longs;ito AB <lb/>&longs;i mouerà piu velocemen<lb/>te, percioche il centro del<lb/>la grauezza H, quanto <lb/>piu è di&longs;tante dal punto <lb/>D, tanto piu velocemen<lb/>te il pe&longs;o compo&longs;to de' pe<lb/>&longs;i EF, & della bilancia <lb/>EF &longs;i mouerà, finche <lb/>l'angolo CHS diuenga <lb/>retto. </s> <s id="id.2.1.412.2.0">& dauantaggio &longs;i <lb/>mouerà anche piu veloce<lb/>mente quanto la bilancia <lb/>&longs;arà piu lontana dal cen<lb/>tro C. <emph.end type="italics"/></s> <s id="id.2.1.413.1.0"><emph type="italics"/>Oltre à ciò ne piace dalle &longs;ue <lb/>ragioni, & fal&longs;e pre&longs;uppo<lb/>&longs;te manife&longs;tare, & pro<lb/>durre gli effetti, & i moti <lb/>già dichiarati della bilan<lb/>cia, affine che appaia <expan abbr="quãta">quan<lb/>ta</expan> &longs;ia la efficacia della ve<lb/>rità, come quella, che dalle co&longs;e fal&longs;e ancora &longs;i sforza di ri&longs;plendere. <emph.end type="italics"/></s> <figure id="id.037.01.062.1.jpg" xlink:href="037/01/062/1.jpg"> </figure> <s id="id.2.1.415.1.0"><emph type="italics"/>Pongan&longs;i le co&longs;e iste&longs;&longs;e, cioè &longs;ia il cerchio AE BF, & la bilancia AB, il cui cen<lb/>tro C &longs;ia &longs;opra la bilancia, moua&longs;i in EF. </s> <s id="id.2.1.415.2.0">Dico che il pe&longs;o po&longs;to in E hà iui <lb/>grauezza maggiore, che il pe&longs;o po&longs;to in F; & che la <expan abbr="bilãcia">bilancia</expan> EF ritornerà in AB <emph.end type="italics"/><pb pagenum="42" xlink:href="037/01/063.jpg"/><emph type="italics"/>&longs;iano tirate dai punti EF le linee EL FM à piombo di AB, le quali &longs;aran<emph.end type="italics"/><arrow.to.target n="note125"></arrow.to.target><lb/><emph type="italics"/>no tra loro egualmente di&longs;tanti, & &longs;ia il punto N doue la AB, & la EF &longs;i <lb/>tagliano fra loro. </s> <s id="id.2.1.415.3.0">Percioche dunque l'angolo FNM è eguale all'angolo ENL, <emph.end type="italics"/><arrow.to.target n="note126"></arrow.to.target><lb/><emph type="italics"/>& l'angolo FMN ret<lb/>to è eguale ad ELN <lb/>retto, & il re&longs;tante <lb/>NFM al re&longs;tante <emph.end type="italics"/><arrow.to.target n="note127"></arrow.to.target><lb/><emph type="italics"/>NEL è etiandio egua<lb/>le; &longs;arà il triangolo NLE <lb/>&longs;imile al triangolo NMF. <emph.end type="italics"/><arrow.to.target n="note128"></arrow.to.target><lb/></s> <s><emph type="italics"/>Si come dunque è la NE <lb/>ver&longs;o la EL, co&longs;i NF <emph.end type="italics"/><arrow.to.target n="note129"></arrow.to.target><lb/><emph type="italics"/>ad FM; & permutan<lb/>do, &longs;i come EN ad NF, <lb/>co&longs;i EL ad FM. </s> <s id="id.2.1.415.4.0">Ma <lb/>e&longs;&longs;endo HE eguale ad <lb/>HF, &longs;arà EN mag<lb/>gior di NF. </s> <s id="id.2.1.415.5.0">Per laqual <lb/>co&longs;a anco EL &longs;arà mag<emph.end type="italics"/><lb/><figure id="id.037.01.063.1.jpg" xlink:href="037/01/063/1.jpg"></figure><lb/><emph type="italics"/>giore di FM. </s> <s id="id.2.1.415.6.0">& percioche mentre il pe&longs;o po&longs;to in E de&longs;cende per la circonferen<lb/>za EA, il pe&longs;o po&longs;to in F &longs;ale per la circonferenza FB eguale alla circonferen<lb/>za EA, & la di&longs;ce&longs;a del pe&longs;o po&longs;to in E piglia (come e&longs;&longs;i dicono) di diretto EL: <lb/>& la &longs;alita del pe&longs;o po&longs;to in F piglia di diretto FM, meno di diretto verrà a pi<lb/>gliare la &longs;alita del pe&longs;o po&longs;to in F, che la di&longs;ce&longs;a del pe&longs;o po&longs;to in E. </s> <s id="id.2.1.415.7.0">Dunque il pe<lb/>&longs;o po&longs;to in E haurà grauezza maggiore, che il pe&longs;o po&longs;to in F. <emph.end type="italics"/></s> <s id="id.2.1.417.1.0"><margin.target id="note125"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 28. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.418.1.0"><margin.target id="note126"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.419.1.0"><margin.target id="note127"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.420.1.0"><margin.target id="note128"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>del &longs;esto. <emph.end type="italics"/></s> <s id="id.2.1.421.1.0"><margin.target id="note129"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.422.1.0"><emph type="italics"/>Sia allungata la linea CD dall una parte, & dall'altra in OP, laquale tagli la linea <lb/>EF nel punto S. </s> <s id="id.2.1.422.2.0">& percioche (come dicono) quanto piu è lontano il pe&longs;o dalla <lb/>linea della direttione OP, tanto &longs;i fa piu graue; però con que&longs;to mezo ancora pro<lb/>uera&longs;&longs;i il pe&longs;o po&longs;to in E hauer grauezza maggiore del pe&longs;o po&longs;to in F. </s> <s id="id.2.1.422.3.0">Siano dai <lb/>punti EF tirate le linee EQ FR a piombo di OP. </s> <s id="id.2.1.422.4.0">Con &longs;imile ragione mo&longs;tre<lb/>ra&longs;&longs;i, che il triangolo QES è &longs;imile al triangolo RFS; & che la linea EQ è <lb/>maggiore di RF. </s> <s>& co&longs;i il pe&longs;o po&longs;to in E &longs;arà piu lontano dalla linea OP, che <lb/>il pe&longs;o po&longs;to in F; & per ciò il pe&longs;o po&longs;to in E hauerà grauezza maggiore del pe<lb/>&longs;o po&longs;to in<emph.end type="italics"/> F. </s> <s><emph type="italics"/>Dallequali co&longs;e appare euidente il ritorno della bilancia E<emph.end type="italics"/>F <emph type="italics"/>in AB. <emph.end type="italics"/></s> <pb xlink:href="037/01/064.jpg"/> <s id="id.2.1.424.1.0"><emph type="italics"/>Ma &longs;e il centro della bilancia &longs;arà &longs;otto la bilancia, allhora &longs;i mo&longs;trerà con gli i&longs;te&longs;&longs;i me <lb/>zi, che il pe&longs;o abba&longs;&longs;ato hauerà grauezza maggiore dall'alzato. </s> <s id="id.2.1.424.2.0">&longs;iano tirate da pun<lb/>ti EF le linee EL FM <lb/>a piombo di AB. </s> <s id="id.2.1.424.3.0">&longs;imil<lb/>mente &longs;i prouerà EL e&longs;<lb/>&longs;ere maggiore di FM; et <lb/>perciò la &longs;ce&longs;a del pe&longs;o po<lb/>sto in F prenderà meno <lb/>di dirittura, che la &longs;alita <lb/>del pe&longs;o po&longs;to in E. </s> <s id="id.2.1.424.4.0">On<lb/>de la re&longs;i&longs;tenza della vio<lb/>lentia del pe&longs;o po&longs;to in E <lb/>&longs;upererà la naturale incli<lb/>natione del pe&longs;o po&longs;to in <lb/>F. </s> <s id="id.2.1.424.5.0">Adunque il pe&longs;o po&longs;to <lb/>in E &longs;arà piu graue del <lb/>pe&longs;o posto in F. <emph.end type="italics"/></s> <s id="id.2.1.425.1.0"><emph type="italics"/>Sia <expan abbr="all&utilde;gata">allungata</expan> etiandio la CD <lb/>dall'una parte & l'altra <emph.end type="italics"/><lb/><figure id="id.037.01.064.1.jpg" xlink:href="037/01/064/1.jpg"></figure><lb/><emph type="italics"/>in OP, & &longs;iano tirate dai punti EF le linee EQ FR à piombo dilei. </s> <s id="id.2.1.425.2.0">&longs;i pro <lb/>verà con l'i&longs;te&longs;&longs;o modo in tutto, che la linea EQ è maggiore di FR. </s> <s>& percio il <lb/>pe&longs;o po&longs;to in E &longs;arà piu lontano dalla linea della dirittura OP, che il pe&longs;o po&longs;to <lb/>in F. </s> <s id="id.2.1.425.3.0">Adunque il pe&longs;o po&longs;to in E haurà grauezza maggiore del pe&longs;o po&longs;to in F. <lb/></s> <s id="id.2.1.425.4.0">Dalle quali co&longs;e &longs;egue, che la bilancia EF &longs;i moue in giù dalla parte di E. <emph.end type="italics"/></s> <s id="id.2.1.427.1.0"><emph type="italics"/>Si che Aristotele propo&longs;e que&longs;te due que&longs;tioni &longs;olamente, & la&longs;ciò la terza, cioè quando <lb/>il centro della bilancia &longs;tà nella bilancia i&longs;te&longs;&longs;a. </s> <s id="id.2.1.427.2.0">Que&longs;ta però trala&longs;ciò egli, co<lb/>me nota, &longs;i come egli &longs;ole trala&longs;ciare le co&longs;e molto note. </s> <s id="id.2.1.427.3.0">Imperoche à chi puote <lb/>far dubbio, che &longs;e il pe&longs;o &longs;arà &longs;o&longs;tentato nel centro della grauezza &longs;ua, che non i&longs;tia <lb/>fermo? </s> <s id="id.2.1.427.4.0">Ma potrebbe for&longs;e alcuno riprendere quelle co&longs;e che per &longs;ua &longs;ententia hab<lb/>biamo propo&longs;to, affermando noi non hauere prodotto in mezo tutta la intera &longs;enten<lb/>za &longs;ua. </s> <s id="id.2.1.427.5.0">Imperoche proponendo egli nella &longs;econda parte della que&longs;tione &longs;econda. </s> <s><lb/>“Perche la bilancia e&longs;&longs;endo posta la trutina di &longs;otto, quando, portato il pe&longs;o in giu, al<lb/>cuno lo rimoue, non a&longs;cende, ma rimane?” non afferma perciò la bilancia mouer&longs;i in <lb/>giù, ma rimanere, il che pare &longs;imilmente hauere nella vltima conclu&longs;ione raccolto. <lb/></s> <s id="id.2.1.427.6.0">Ma que&longs;to non &longs;olamente non ci fa contra, ma &longs;e egli è ben' inte&longs;o grandi&longs;&longs;imamen<lb/>te aiuta. <emph.end type="italics"/></s> <s id="id.2.1.428.1.0"><emph type="italics"/>Percioche &longs;ia la bilancia AB egualmente di&longs;tante dall'orizonte, il cui centro E &longs;ia <lb/>&longs;otto la bilancia. </s> <s id="id.2.1.428.2.0">& perche Ari&longs;totele con&longs;idera la bilancia come ella è in fatto, però <lb/>egli è nece&longs;&longs;ario collocare la trutina, ouero qualche altra co&longs;a &longs;otto il centro E, co<lb/>me EF, che in ogni modo &longs;arà trutina, per modo, che &longs;o&longs;tenga il centro E. </s> <s>& &longs;ia <lb/>ECD il perpendicolo. </s> <s id="id.2.1.428.3.0">& accioche la bilancia AB &longs;i moua da que&longs;to &longs;ito, dice <emph.end type="italics"/><pb pagenum="25" xlink:href="037/01/065.jpg"/><emph type="italics"/>Ari&longs;totele, ponga&longs;i il pe&longs;o in B, ilquale e&longs;&longs;endo graue mouerà la bilancia dalla par<lb/>te B in giù, come in G, talche per l'impedimento non potrà egli piu mouer&longs;i in <lb/>giu, ma non dice gia Ari&longs;totele, che &longs;i moua la bilancia in giu dalla parte di B fin <lb/>tanto che parerà, da <lb/>poi &longs;i la&longs;ci, come noi <lb/>di cemmo: ma ordina <lb/>che &longs;ia posto il pe&longs;o <lb/>in B, il quale di &longs;ua <lb/>natura &longs;i mouera <lb/>&longs;empre in giù finche <lb/>la bilancia &longs;i appog<lb/>gi alla trutina, ouerò <lb/>a qualche altra co&longs;a. </s> <s id="id.2.1.428.4.0">& quando il B &longs;a<lb/>rà nel G, la bilan<lb/>cia &longs;arà in GH, nel <lb/>qual &longs;ite leuato via <lb/>il pe&longs;o, rimarrà: per <lb/>e&longs;&longs;ere la maggior par<lb/>te della bilancia dal <lb/>perpendicolo uer&longs;o il <emph.end type="italics"/><lb/><figure id="id.037.01.065.1.jpg" xlink:href="037/01/065/1.jpg"></figure><lb/><emph type="italics"/>G, che è DG, che DH. </s> <s>ne piu mouera&longs;&longs;i in giu, imperoche la bilancia &longs;tarà &longs;opra <lb/>la trutina, ouero qualche altra co&longs;a, che &longs;o&longs;tenga il centro della bilancia. </s> <s id="id.2.1.428.5.0">peroche &longs;e a <lb/>cote&longs;ta non &longs;i appoggia&longs;&longs;e, verrebbe la bilancia à mouer&longs;i, &longs;econdo la &longs;ua opinione, <lb/>in giù dalla parte di G, concio&longs;ia, che quello che è di piu, cioè DG debba e&longs;&longs;ere <lb/>per nece&longs;&longs;ità in giu portato. <emph.end type="italics"/></s> <s id="id.2.1.430.1.0"><emph type="italics"/>Ma potrebbe dauantagio dire alcuno, &longs;e in B &longs;arà collocato vn pe&longs;o picciolo, &longs;i mo<lb/>uerà ben la bilancia in giu, ma non gia fin al G; nel qual &longs;ito, &longs;econdo Aristo<lb/>tele, leuato via il pe&longs;o, deue remanere. </s> <s id="id.2.1.430.2.0">ilche è manife&longs;to per la e&longs;perientia, inchi<lb/>nando&longs;i la <expan abbr="bilãcia">bilancia</expan> più, & meno, quando in vna e&longs;tremita della bilancia &longs;olamente <lb/>vi è po&longs;to il pe&longs;o, che &longs;ia ò maggiore, ò minore. </s> <s id="id.2.1.430.3.0">ilche è veri&longs;&longs;imo allhora che il centro <lb/>è collocato &longs;opra la bilancia, ma non già &longs;otto, ne in e&longs;&longs;a bilancia, come per gratia <lb/>di e&longs;empio. <emph.end type="italics"/></s> <pb xlink:href="037/01/066.jpg"/> <s id="id.2.1.432.1.0"><emph type="italics"/>Sia la bilancia AB egualmente di&longs;tante dall'orizonte, il cui centro C &longs;ia &longs;opra la bi<lb/>lancia, & il perpendicolo <lb/>CD a piombo dell' ori<lb/>zonte, il quale da la par<lb/>te D &longs;ia allungato in H. <lb/></s> <s id="id.2.1.432.2.0">Hor percioche con&longs;idera<lb/>ta la grauezza della bi<lb/>lancia, &longs;arà il punto D <lb/>il centro della grauezza <lb/>della bilancia. </s> <s id="id.2.1.432.3.0">&longs;e dunque <lb/>vn piccolo pe&longs;o &longs;arà po<lb/>&longs;to nel B, il cui centro <lb/>della grauezza &longs;ia nel <expan abbr="p&utilde;to">pun<lb/>to</expan> B; gia piu non &longs;arà <lb/>il centro della grauezza <lb/>D della magnitudine <emph.end type="italics"/><lb/><arrow.to.target n="note130"></arrow.to.target><emph type="italics"/>compo&longs;ta della bilancia <emph.end type="italics"/><lb/><figure id="id.037.01.066.1.jpg" xlink:href="037/01/066/1.jpg"></figure><lb/><emph type="italics"/>AB, & del pe&longs;o po&longs;to in B, ma &longs;arà nella linea DB, come in K: per modo <lb/>che DE ad EB &longs;ia come il pe&longs;o po&longs;to in B alla grauezza della bilancia AB. <lb/></s> <s id="id.2.1.432.4.0">congiunga&longs;i la CE. </s> <s id="id.2.1.432.5.0">& percioche il punto C è immobile, mentre la bilancia &longs;i <lb/>moue, il punto E de&longs;criuerà la circonferenza del cerchio EFG, il cui mezo dia<lb/>metro è CE, & il centro C. </s> <s id="id.2.1.432.6.0">Ma perche CD &longs;tà a piombo dell' orizonte, la li<lb/>nea CE non &longs;arà gia ella à piombo dell' orizonte. </s> <s id="id.2.1.432.7.0">Per laqual co&longs;a la grandez<lb/>za composta di AB, & del pe&longs;o po&longs;to in B non rimarrà in questo &longs;ito; ma &longs;i <lb/>mouerà in giu &longs;econdo il centro E della &longs;ua grauezza per la circonferenza EFG, <emph.end type="italics"/><lb/><arrow.to.target n="note131"></arrow.to.target><emph type="italics"/>finche CE diuenti a piombo dell' orizonte, cioè finche la CE peruenga in CDF. <lb/></s> <s id="id.2.1.432.8.0">& allhora la bilancia AB &longs;arà mo&longs;&longs;a in KL, nel qual &longs;ito la bilancia rimarrà <lb/>in&longs;ieme co'l pe&longs;o, ne d'auantaggio &longs;i mouerà in giù. </s> <s id="id.2.1.432.9.0">che &longs;e in B &longs;arà po&longs;to vn pe&longs;o <lb/>piu graue, il centro'della grauezza di tutta la magnitudine &longs;arà piu dappre&longs;&longs;o al B, <lb/>come in M. </s> <s>& allhora la bilancia &longs;i mouerà in giu, finche la congiunta linea CM <lb/>peruenga nella linea CDH. </s> <s id="id.2.1.432.10.0">Dal por&longs;i dunque pe&longs;o maggiore ò minore in B, la <lb/>bilancia &longs;i inchinerà piu ò meno. </s> <s id="id.2.1.432.11.0">Da che &longs;egue che il pe&longs;o B de&longs;criuerà &longs;empre vna <lb/>circonferenza minore della quarta parte d'un cerchio, per e&longs;&longs;ere l'angolo FCE &longs;em<lb/>pre acuto:ne il punto B peruenirà gia mai fin alla linea CH, percioche &longs;empre il <lb/>centro della grauezza del pe&longs;o, & dalla bilancia in&longs;ieme &longs;arà fra BD. </s> <s id="id.2.1.432.12.0">tuttauia <expan abbr="quãto">quan<lb/>to</expan> &longs;arà il pe&longs;o po&longs;to in B piu graue, de&longs;criuerà anche circonferenza maggiore, ve<lb/>nendo&longs;i per que&longs;to il punto B ad acco&longs;tare piu alla linea CH. <emph.end type="italics"/></s> <s id="id.2.1.434.1.0"><margin.target id="note130"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>del primo. di. Arch. delle co&longs;e <expan abbr="egualm&etilde;te">egualmente</expan> <expan abbr="pesãti">pesanti</expan>. <emph.end type="italics"/></s> <s id="id.2.1.435.1.0"><margin.target id="note131"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.436.1.0"><emph type="italics"/>Mi habbia la bilancia AB il centro C nella i&longs;te&longs;&longs;a bilancia, & nel &longs;uo mezo, <lb/>&longs;arà il C centro ancora della grauezza della bilancia, dal quale &longs;ia tirata la li<lb/>nea FCG a piombo di e&longs;&longs;a AB, & dell' orizonte. </s> <s id="id.2.1.436.2.0">Ponga&longs;i dapoi in B qual <lb/>pe&longs;o &longs;i voglia; &longs;arà il centro di tutta la grauezza, come in E; &longs;i fattamente che <lb/>la CE ver&longs;o EB &longs;ia come il pe&longs;o po&longs;to in B alla grauezza della bilancia. </s> <s id="id.2.1.436.3.0">& per<emph.end type="italics"/><pb pagenum="26" xlink:href="037/01/067.jpg"/><emph type="italics"/>cioche la CE non è a piombo dell' orizonte, la bilancia AB, & il pe&longs;o po&longs;to in <lb/>B non rimaranno in que<lb/>&longs;to &longs;ito gia mai; ma &longs;i mo<lb/>ueranno in giu dalla par<lb/>te di B, fin che CE &longs;i <lb/>faccia à piombo dell' ori<lb/>zonte; cioè fin che la bilan<lb/>cia AB peruenga in FG. <lb/></s> <s id="id.2.1.436.4.0">Onde è chiaro, che cia&longs;cun <lb/>pe&longs;o po&longs;to in B, &longs;empre <lb/>de&longs;criue la quarta parte <lb/>d'un cerchio. <emph.end type="italics"/></s> <figure id="id.037.01.067.1.jpg" xlink:href="037/01/067/1.jpg"> </figure> <s id="id.2.1.438.1.0"><emph type="italics"/>Ma &longs;ia il centro C &longs;otto la bilancia AB, & &longs;ia DCE il perpendicolo. </s> <s id="id.2.1.438.2.0">&longs;imilmente <lb/>per e&longs;&longs;er il pe&longs;o posto in B, &longs;arà il centro della grauezza della magnitudine compe <lb/>&longs;ta di AB bilancia, & del pe&longs;o po&longs;to in B nella linea DB, come in F; &longs;i <expan abbr="fattam&etilde;te">fattamen<lb/>te </expan>che come DF &longs;i ha ver&longs;o FB co&longs;i &longs;ia il pe&longs;o po&longs;to in B al pe&longs;o della bilan<lb/>cia. </s> <s id="id.2.1.438.3.0">congiunga&longs;i CF. </s> <s>& <lb/>percioche CD è a piombo <lb/>dell' orizonte, non &longs;arà gia <lb/>la linea CF a piombo del<lb/>l'orizonte. </s> <s id="id.2.1.438.4.0">Per laqual co&longs;a <lb/>la magnitudine compo&longs;ta <lb/>della bilancia AB, & del <lb/>pe&longs;o po&longs;to in B in que&longs;to <lb/>&longs;ito non &longs;tarà mai ferma; <lb/>ma in giu mouera&longs;&longs;i &longs;e alcu<lb/>na co&longs;a non la impedi&longs;ce, <lb/>finche CF peruenga in <lb/>DCE, nel qual &longs;ito la bi<lb/>lancia rimarrà in&longs;ieme co'l <emph.end type="italics"/><lb/><figure id="id.037.01.067.2.jpg" xlink:href="037/01/067/2.jpg"></figure><lb/><emph type="italics"/>pe&longs;o. </s> <s id="id.2.1.438.5.0">& il punto B &longs;arà come in G, & il punto A in H, & la bilancia GH <lb/>non hauerà piu il centro di &longs;otto, ma &longs;opra e&longs;&longs;a. </s> <s id="id.2.1.438.6.0">La qual co&longs;a hauerà &longs;empre, quan<lb/>tunque &longs;i ponga vn minimo pe&longs;o in B. </s> <s id="id.2.1.438.7.0">Auanti che dunque il B peruenga al G, <lb/>egli è nece&longs;&longs;ario, che la bilancia incontri la trutina po&longs;ta di &longs;otto, ouero alcuna altra <lb/>co&longs;a, che &longs;o&longs;tenti il centro C, & iui s'appoggi. </s> <s id="id.2.1.438.8.0">Da que&longs;to &longs;egue, che il pe&longs;o B &longs;em<lb/>pre &longs;i moue oltre la linea DK, & de&longs;criue &longs;empre vna circonferenza maggiore del<lb/>la quarta parte del cerchio, per e&longs;&longs;ere l'angolo FCE &longs;empre ottu&longs;o, & l'angolo <lb/>DCF &longs;empre acuto. </s> <s id="id.2.1.438.9.0">& quanto il pe&longs;o posto in B &longs;arà piu leggiero, de&longs;criuerà tut<lb/>tauia anche circonferenza maggiore. </s> <s id="id.2.1.438.10.0">Imperoche quanto il pe&longs;o po&longs;to in G &longs;arà piu <lb/>leggiero, tanto piu il pe&longs;o detto posto in G &longs;i alzerà; & la bilancia GA s'acco&longs;te <emph.end type="italics"/><pb xlink:href="037/01/068.jpg"/><emph type="italics"/>rà piu pre&longs;&longs;o al &longs;ito egualmente di&longs;tante dall'orizonte. </s> <s id="id.2.1.444.2.0">Le quali co&longs;e tutte re&longs;tano ma<lb/>nife&longs;te da quelle che di &longs;opra &longs;ono &longs;tate dette. <emph.end type="italics"/></s> <s id="id.2.1.447.1.0"><emph type="italics"/>Prouate que&longs;te co&longs;e, egli è chia<lb/>ro, che il centro della bilan<lb/>cia è cagione de gli effetti di<lb/>uer&longs;i della bilancia. </s> <s id="id.2.1.447.2.0">& &longs;i ve<lb/>de ancora che tutte le pro<lb/>po&longs;itioni di Archimede del<lb/>le co&longs;e, che egualmente pe&longs;a<lb/>no, a ciò pertinenti, in ogni <lb/>&longs;ito &longs;ono vere. </s> <s id="id.2.1.447.3.0">cioè, &longs;ia pur <lb/>la bilancia di&longs;tante <expan abbr="egualm&etilde;te">egualmen<lb/>te</expan> dall'orizonte, ouero non, <lb/>pur che il centro della bilan<lb/>cia &longs;ia collocato in e&longs;&longs;a <expan abbr="bilãcia">bilan<lb/>cia</expan>, &longs;i come egli la con&longs;ide<emph.end type="italics"/><lb/><figure id="id.037.01.068.1.jpg" xlink:href="037/01/068/1.jpg"></figure><lb/><emph type="italics"/>rà. </s> <s id="id.2.1.447.4.0">& quantunque la bilancia habbia di&longs;uguali le braccia, auerrà tuttauia l'i&longs;te&longs;&longs;o, & <lb/>&longs;i dimo&longs;trerà co'l modo i&longs;te&longs;&longs;o in tutto, che il centro della bilancia collocato in diuer<lb/>&longs;e maniere produrrà vari effetti. <emph.end type="italics"/></s> <s id="id.2.1.449.1.0"><emph type="italics"/>Percioche &longs;ia la bilancia <lb/>AB egualmente di&longs;tan<lb/>te dall'orizonte; & &longs;iano <lb/>in AB pe&longs;i di&longs;uguali, il <lb/>centro della grauezza <lb/>dei quali &longs;ia in C, & <lb/>&longs;ia attacata la bilancia <lb/>nell'i&longs;te&longs;&longs;o punto di C, <lb/>& moua&longs;i la bilancia in <emph.end type="italics"/><lb/><arrow.to.target n="note134"></arrow.to.target><emph type="italics"/>DE; egliè manife&longs;to, <lb/>che la bilancia rimarrà <lb/>non &longs;olamente in DE, <lb/>ma in qual &longs;i voglia altre <lb/>&longs;ito. <emph.end type="italics"/></s> <s id="id.2.1.450.1.0"><margin.target id="note134"></margin.target><emph type="italics"/>Per la diffinitione del centro della grauezza. <emph.end type="italics"/></s> <figure id="id.037.01.068.2.jpg" xlink:href="037/01/068/2.jpg"> </figure> <pb pagenum="27" xlink:href="037/01/069.jpg"/> <s id="id.2.1.441.1.0"><emph type="italics"/>Ma &longs;ia il centro della bilancia AB &longs;opra il C in F; & &longs;ia FC à piombo di AB, <lb/>& dell' orizonte: & &longs;e <lb/>la bilancia &longs;arà mo&longs;&longs;a in <emph.end type="italics"/><arrow.to.target n="note132"></arrow.to.target><lb/><emph type="italics"/>DE, la linea CF &longs;arà <lb/>mo&longs;&longs;a in FG, la quale <lb/>per non e&longs;&longs;ere à piombo <lb/>dell' orizonte, la bilancia <lb/>DE &longs;imouerà in giu dalla <lb/>parte di D, finche FG <lb/>ritorni in FC: & allho<lb/>ra la bilancia DE &longs;arà <lb/>in AB, nel qual &longs;ito an <lb/>che rimarrà. <emph.end type="italics"/></s> <s id="id.2.1.442.1.0"><margin.target id="note132"></margin.target><emph type="italics"/>Per la prima di questo. <emph.end type="italics"/></s> <figure id="id.037.01.069.1.jpg" xlink:href="037/01/069/1.jpg"> </figure> <s id="id.2.1.444.1.0"><emph type="italics"/>Che &longs;e il centro F della bi<lb/>lancia &longs;arà &longs;otto la <expan abbr="bilãcia">bilan<lb/>cia</expan>, & &longs;ia la bilancia mo&longs;<emph.end type="italics"/><arrow.to.target n="note133"></arrow.to.target><lb/><emph type="italics"/>&longs;a in DE primieramen<lb/>te egli è manife&longs;to che la <lb/>bilancia rimarrà in AB: <lb/>& in DE mouera&longs;&longs;i in <lb/>giu dalla parte di E, per <lb/>non e&longs;&longs;ere la linea FG <lb/>à piombo dell' orizonte. <emph.end type="italics"/><lb/><figure id="id.037.01.069.2.jpg" xlink:href="037/01/069/2.jpg"></figure></s> <s id="id.2.1.446.1.0"><margin.target id="note133"></margin.target><emph type="italics"/>Per la prima di questo. <emph.end type="italics"/></s> <pb xlink:href="037/01/070.jpg"/> <s id="id.2.1.452.1.0"><emph type="italics"/>Da que&longs;te co&longs;e co&longs;i terminate, &longs;e la bilancia fo&longs;&longs;e inarcata, ouero, che le braccia della bi <lb/>lancia forma&longs;&longs;ero vn'angolo, & &longs;i di&longs;poneße il centro diuer&longs;amente, (ben che que<lb/>&longs;ta propriamente non &longs;arebbe bilancia,) potremo nondimeno anche dimo&longs;trare di lei <lb/>varij effetti. </s> <s id="id.2.1.452.2.0">Come &longs;ia la bilancia ACB, il cui centro, d'intorno al quale &longs;i volge, <lb/>&longs;i a C, & tiratala linea AB, &longs;ia <lb/>l'arco ouerò l'angolo ACB &longs;opra <lb/>la linea AB; & pongan&longs;i in AB <lb/>i centri della grauezza de'pe&longs;i, i quali <lb/>rimangano in que&longs;to &longs;ito. </s> <s id="id.2.1.452.3.0">Moua&longs;i poi <lb/>la <expan abbr="bilãcia">bilancia</expan> da que&longs;to &longs;ito, come in ECF. <lb/></s> <s id="id.2.1.452.4.0">Dico che la bilancia ECF ritornerà <lb/>in ACB. </s> <s id="id.2.1.452.5.0">Ritroui&longs;i il centro della <lb/>grauezza di tutta la magnitudine D, <lb/>& &longs;ia congiunta la CD. </s> <s id="id.2.1.452.6.0">Hor percio<emph.end type="italics"/><lb/><arrow.to.target n="note135"></arrow.to.target><emph type="italics"/>che i pe&longs;i AB stanno fermi, la li<lb/>nea CD &longs;arà à piombo dell'orizon<emph.end type="italics"/><lb/><figure id="id.037.01.070.1.jpg" xlink:href="037/01/070/1.jpg"></figure><lb/><emph type="italics"/>te. </s> <s id="id.2.1.452.7.0">Quando dunque la bilancia &longs;arà in ECF, la linea CD &longs;arà come in CG; <lb/>la quale per non e&longs;&longs;ere à piombo dell' orizonte, la bilancia ECF ritornerà in <lb/>ACB. </s> <s>ilche parimente auenirà, &longs;e il centro C &longs;arà me&longs;&longs;o &longs;opra la bilancia, co<lb/>me in H. <emph.end type="italics"/></s> <s id="id.2.1.454.1.0"><margin.target id="note135"></margin.target><emph type="italics"/>Per la prima di questo. <emph.end type="italics"/></s> <s id="id.2.1.455.1.0"><emph type="italics"/>Che &longs;e l'arco, ouero l'angolo ACB <lb/>&longs;arà &longs;otto la linea AB, nel <lb/>modo i&longs;te&longs;&longs;o mo&longs;treremo, la bi<lb/>lancia ECF, il cui centro &longs;ia <lb/>ouero in C, ouero in H, do<lb/>uer&longs;i mouere in giu dalla parte <lb/>di F. <emph.end type="italics"/><lb/><figure id="id.037.01.070.2.jpg" xlink:href="037/01/070/2.jpg"></figure></s> <pb pagenum="28" xlink:href="037/01/071.jpg"/> <figure id="id.037.01.071.1.jpg" xlink:href="037/01/071/1.jpg"> </figure> <figure id="id.037.01.071.2.jpg" xlink:href="037/01/071/2.jpg"> </figure> <figure id="id.037.01.071.3.jpg" xlink:href="037/01/071/3.jpg"> </figure> <s id="id.2.1.459.1.0"><emph type="italics"/>Et &longs;e l'angolo ACB fo&longs;&longs;e &longs;oprala linea AB, & il centro della bilancia H; & <lb/>& la linea CH &longs;o&longs;tene&longs;&longs;e la bilancia; & &longs;i moue&longs;&longs;e la bilancia in EKF; la bilan<lb/>cia EKF ritornerà in ACB. <emph.end type="italics"/></s> <s id="id.2.1.460.1.0"><emph type="italics"/>Ma &longs;e il centro della bilancia &longs;arà D, moua&longs;i in qualunque modo la bilancia, doue &longs;i <lb/>la&longs;cierà, lui rimarrà. <emph.end type="italics"/></s> <s id="id.2.1.461.1.0"><emph type="italics"/>Se poi il punto H &longs;arà &longs;otto la linea AB; allhora la bilancia EKF &longs;i mouerà in <lb/>giu dalla parte di F. <emph.end type="italics"/></s> <s id="id.2.1.462.1.0"><emph type="italics"/>Et con &longs;imile ragione in tutto, &longs;e l'ango<lb/>lo ACB &longs;arà &longs;otto la linea AB; <lb/>& &longs;ia il centro della bilancia H, & <lb/>&longs;ia la bilancia &longs;o&longs;tentata dalla linea <lb/>CH; &longs;e la bilancia moueraßi da que&longs;to <lb/>&longs;ito, &longs;i mouerà in giu dalla parte del pe<lb/>&longs;o più ba&longs;&longs;o. </s> <s id="id.2.1.462.2.0">& &longs;e il centro della bilan<lb/>cia &longs;ia D; rimarrà doue &longs;i la&longs;cierà. </s> <s id="id.2.1.462.3.0">che <lb/>&longs;e &longs;arà in K; & da cotale &longs;ito &longs;i mo<lb/>uerà, ritornerà ad ogni modo nello i&longs;te&longs;<lb/>&longs;o. </s> <s id="id.2.1.462.4.0">Le quali co&longs;e tutte da quel che in<emph.end type="italics"/><lb/><figure id="id.037.01.071.4.jpg" xlink:href="037/01/071/4.jpg"></figure><lb/><emph type="italics"/>principio dicemmo &longs;ono manifeste. </s> <s id="id.2.1.462.5.0">&longs;imilmente &longs;e il centro della bilancia &longs;arà po&longs;to <lb/>in vno della bracia della bilancia, ò dentro, ò fuori, ò in qual &longs;i voglia modo trouere<lb/>mo le co&longs;e i&longs;te&longs;&longs;e. <emph.end type="italics"/></s> <pb xlink:href="037/01/072.jpg"/> <s id="id.2.1.464.1.0">In que&longs;to luogo egli conuiene auertire, il che poteua&longs;i anco fare di &longs;opra à carte cin<lb/>que pre&longs;&longs;o la fine della &longs;econda faccia oue è &longs;critto. </s> <s id="id.2.1.464.2.0">oltre à ciò po&longs;siamo con&longs;ide<lb/>rare le co&longs;e che &longs;eguono in tutto al modo i&longs;te&longs;&longs;o. </s> <s id="id.2.1.464.3.0">Che que&longs;to autore è &longs;tato il <lb/>primo à con&longs;iderare e&longs;qui&longs;itamente la bilancia, & intenderla dalla natura, & dal <lb/>vero e&longs;&longs;er &longs;uo; pero che egli il primiero di tutti ha manife&longs;tato chiaramente il mo<lb/>do del trattarla, & in&longs;egnarla, con proporre tre centri da e&longs;&longs;ere con&longs;iderati in que<lb/>&longs;ta &longs;peculatione; l'uno è il centro del mondo, l'altro il centro della bilancia, & il <lb/>terzo il centro della grauezza della bilancia, che in e&longs;&longs;a era vn na&longs;co&longs;to &longs;ecreto di <lb/>natura. </s> <s id="id.2.1.464.4.0">Senza que&longs;ti tre centri, chiara co&longs;a è, che non &longs;i puote venire in cono&longs;ci<lb/>mento perfetto, ne dimo&longs;trare gli effetti varij della bilancia, i quali na&longs;cono dalla <lb/>diuer&longs;ità del collo care il centro della bilancia in tre modi, cioè quando il <lb/>centro della bilancia &longs;ta &longs;opra il centro della grauezza di e&longs;&longs;a, ouero quando è <lb/>di &longs;otto, o pure allhorche il centro della bilancia è nell'i&longs;te&longs;&longs;o centro della gra<lb/>uezza di lei; &longs;i come l'autore in&longs;egna nella tre precedenti dimo&longs;trationi, cioè <lb/>nella <expan abbr="&longs;ecõda">&longs;econda</expan>, nella terza, & nella quarta propo&longs;itione: peroche nella &longs;econda mo<lb/>&longs;tra quando la bilancia torna &longs;empre egualmente di&longs;tante dall'orizonte; nella ter<lb/>za quando non &longs;olo non ritorna, ma &longs;i moue al contrario; nella quarta, che <lb/>e&longs;&longs;endo la bilancia &longs;o&longs;tenuta nel &longs;uo centro dalla grauezza &longs;ta ferma douunque el<lb/>la &longs;i troua, il quale effetto in particolare non è piu &longs;tato tocco, ne veduto, ne man<lb/>co da niuno manife&longs;tato, fuor che dall'autore: anzi fin hora tenuto fal&longs;o, & impo&longs;<lb/>&longs;ibile da tutti gli predece&longs;&longs;ori no&longs;tri; i quali con molte ragioni &longs;i &longs;ono sforzati di <lb/>prouare non &longs;olamente il contrario, ma hanno etiandio affermato per certo, che <lb/>la &longs;períenza mo&longs;tra la bilancia non dimorare gia mai ferma &longs;e non quando ella è <lb/>egualmente di&longs;tante dall'orizonte. </s> <s id="id.2.1.464.5.0">Laqual co&longs;a in tutto è contraria alla ragione <lb/>prima, per e&longs;&longs;ere la dimo&longs;tratione della &longs;udetta quarta propo&longs;itione tanto chiara, <lb/>facile, & vera, che non sò, come &longs;e le po&longs;&longs;a in modo alcuno contradire: & poi al<lb/>l'e&longs;perienza concio&longs;ia che l'autore habbia fatto &longs;ottili&longs;simamente lauorare bilan<lb/>cie giu&longs;te a po&longs;ta per chiarire que&longs;ta verità, vna delle quali hò io veduto in mano <lb/>dell'Illu&longs;tre Signor Gio. Vicenzo Pinello, mandatagli dall'i&longs;te&longs;&longs;o autore, la quale <lb/>per e&longs;&longs;ere &longs;o&longs;tenuta nel centro della &longs;u a grauezza, mo&longs;&longs;a douunque &longs;i vuole, & poi <lb/>la&longs;ciata, &longs;tà ferma in ogni &longs;ito doue ella vien la&longs;ciata. </s> <s id="id.2.1.464.7.0">Ben è egli vero, che non bi <lb/>&longs;ogna, nel fare cote&longs;ta e&longs;perienza, correr co&longs;i a furia, per e&longs;&longs;ere co&longs;a oltra modo <lb/>difficile, come dice l'áutore di &longs;opra, il fare vna bilancia, la quale &longs;ia nel mezo del<lb/>le &longs;ue braccia &longs;o&longs;tenuta à punto, & nel centro proprio della &longs;ua grauezza. </s> <s id="id.2.1.464.8.0">Per la <lb/>qual co&longs;a egli è da por <expan abbr="m&etilde;te">mente</expan>, che qual'hora alcuno &longs;i mette&longs;&longs;e à far cotale e&longs;perien<lb/>za, & non gli riu&longs;ci&longs;&longs;e, non perciò &longs;i deue &longs;gomentare, anzi dica pur fermamente <lb/>di non hauer bene operato, & vn'altra volta ritorni à farne la &longs;perienza, fin che la <lb/>bilancia &longs;ia giu&longs;ta, & eguale, & venga &longs;o&longs;tenuta à punto nel centro della grauez<lb/>za &longs;ua. </s> <s id="id.2.1.464.9.0">Et benche da altri &longs;iano &longs;tate tocche le altre due predette &longs;peculationi, cioè <lb/>quando la bilancia ritorna &longs;empre egualmente di&longs;tante dall'orizonte, & quando <lb/>&longs;i moue al contrario di que&longs;to &longs;ito, tuttauia non &longs;i è piu inte&longs;a que&longs;ta verità gia <lb/>mai apertamente, &longs;e non dall'autore no&longs;tro; peroche gli altri non hanno co'l &longs;en<lb/>no penetrato in ciò tanto auanti, che habbiano &longs;aputo con di&longs;tintione con&longs;idera<lb/>re il centro della bilancia in tre modi, come hò narrato. </s> <s id="id.2.1.464.10.0">Che &longs;e hanno pur diui&longs;a<lb/>to qualche co&longs;a d'intorno à que&longs;to, l'hanno fatto confu&longs;i&longs;simamente, & con ma<lb/>le dimo&longs;trationi, dalle quali non &longs;i puote cauare ferma <expan abbr="cõchiu&longs;ione">conchiu&longs;ione</expan>, & chiara. </s> <s id="id.2.1.464.11.0">Que<lb/>sti predece&longs;&longs;ori no&longs;tri han&longs;i da intendere i moderni &longs;crittori di cotal materia alle<lb/>gati in diuer&longs;i luoghi dall'autore, fra quali Giordano, che &longs;cri&longs;&longs;e de'pe&longs;i fù riputa<pb pagenum="29" xlink:href="037/01/073.jpg"/>to a&longs;&longs;ai, & &longs;in qui è &longs;tato &longs;eguito molto nella &longs;ua dottrina. </s> <s id="id.2.1.464.12.0">Hor l'autore no&longs;tro hà <lb/>procurato con ogni &longs;tudi o di caminare per la via de' buoni Greci antichi, <lb/>mae&longs;tri delle &longs;cienze, & in particolare di Archimede Siracu&longs;ano prencipe delle ma <lb/>thematiche famo &longs;i&longs;simo, & di Pappo Ale&longs;&longs;andrino, come egli dice, leggendogli <lb/>nella &longs;ua propria fauella, non tradotti; peroche il piu delle volte &longs;ono co&longs;i mal<lb/>trattati, che à gran pena &longs;i puote trarre da loro frutto veruno. </s> <s id="id.2.1.464.13.0">& affine che que&longs;ta <lb/>noua opinion &longs;ua, dimo&longs;trata à pieno nella predetta quarta propo&longs;itione, re&longs;ti to<lb/>talmente chiara, non &longs;i è gia <expan abbr="cont&etilde;tato">contentato</expan> egli d'hauerla dimo&longs;trata con viue ragioni, <lb/>& certe &longs;olamente, ma come buon filo&longs;ofo, procedente con via di reale dottrina, <lb/>& di fondata &longs;cienza, (imitando Ari&longs;totele, ilqual ne' principii de &longs;uoi libri, inue<lb/>&longs;tigando dottrina migliore, hà datto contra la opinione de gli antichi, &longs;oluendo <lb/>le ragioni addotte da loro:) hà ben voluto, e&longs;&longs;endo la verità vna &longs;ola, proporre le <lb/>opinioni de'&longs;uoi predece&longs;&longs;ori, & e&longs;aminare le loro ragioni, lequali &longs;embrano pro<lb/>uar il contrario, & &longs;oluerle, la loro fallenza <expan abbr="dimo&longs;trãdo">dimo&longs;trando</expan> co'l pre&longs;ente di&longs;cor&longs;o, che <lb/>incomincia, come è detto à carte cinque nella faccia &longs;econda, & qui fini&longs;ce il qua<lb/>le di&longs;cor&longs;o &longs;eruirà in que&longs;ta materia, &longs;econdo che &longs;i &longs;uole dire per la opinione de <lb/>gli antichi. </s> <s id="id.2.1.464.14.0">Et percio che egli contiene co&longs;e di alti&longs;sima &longs;peculatione, ma&longs;simamen<lb/>te d'intorno al con&longs;iderare doue &longs;ia piu graue vn pe&longs;o &longs;olo po&longs;to in vno braccio <lb/>della bilancia, bi&longs;ogna in ogni modo, per bene intendere, leggerlo, & i&longs;tudiarlo <lb/>con accurati&longs;sima diligenza. </s> <s id="id.2.1.464.15.0">Ma per certo l'autore è &longs;tato non &longs;olo il primo à tro<lb/>uare que&longs;ta verità, ma il primo etiandio a dimo&longs;trare in qual maniera &longs;ia me&longs;tieri <lb/>con&longs;iderare, & &longs;peculare interamente la pre&longs;ente materia tutta. </s> <s id="id.2.1.464.16.0">Con laquale &longs;pecu<lb/>latione proua di nouo, & confermai varij effetti, & accidenti della bilancia già di <lb/>mo&longs;trati nelle pro&longs;sime tre propo&longs;itioni; mo&longs;trando ancora, come &longs;in qui cote&longs;te <lb/>co&longs;e &longs;iano da gli altri &longs;tate malamente con&longs;iderate, & con principij fal&longs;i. </s> <s id="id.2.1.464.17.0">Anzi di <lb/>piu per confermatione della verità &longs;oggiunge, che que&longs;ti tali non hanno &longs;aputo fa <lb/>re le loro demo&longs;trationi; poi che co'l proprio modo di &longs;peculare v&longs;ato da loro, <lb/>& con le loro mede&longs;ime ragioni proua la &longs;ua intentione, & &longs;entenza e&longs;&longs;ere veri&longs;si<lb/>ma, appoggiando &longs;i alla dottrina di Ari&longs;totele &longs;empre, & facendo toccar con ma<lb/>no, che egli con e&longs;&longs;o lui è d'accordo nelle que&longs;tioni mechaniche. </s> <s id="id.2.1.464.18.0">In trattando <lb/>que&longs;ta materia moue l'autore alcuni dubbi molto belli, & curio&longs;i, & poi chiara<lb/>mente gli &longs;olue. </s> <s id="id.2.1.464.19.0">In vltimo, accioche non manca&longs;&longs;e nulla al compiuto cono&longs;cimen<lb/>to di que&longs;to &longs;oggetto, egli hà trattato delle bilancie, che hanno le braccia di&longs;ugua<lb/>li, & di quelle che hanno le dette braccia piegate, & torte. </s> <s id="id.2.1.464.20.0">In &longs;omma &longs;i può ben <lb/>affermare, che in cote&longs;to di&longs;cor&longs;o &longs;iano compre&longs;e tutte quelle co&longs;e, che po&longs;&longs;ono e&longs;<lb/>&longs;ere diui&longs;ate d'intorno à materia tale. </s> <s id="id.2.1.464.21.0">Le quali &longs;ono di belli&longs;sima & &longs;ottili&longs;sima &longs;pe<lb/>culatione, & à chiunque &longs;i diletta, & attende à que&longs;ti no bili &longs;tudi nece&longs;&longs;arij&longs;sime, <lb/>& da e&longs;&longs;ere, come hò ricordato piu d'una volta, con molta attentione vedute, & <lb/>con&longs;iderate. </s> <s id="id.2.1.465.1.0">Doue &longs;i legge que&longs;to vocabolo latino Equilibrio, intenda&longs;i per eguale contrape&longs;o, <lb/>cioè che pe&longs;a tanto da vna banda, quanto dallaltra in pari lance, ò libra, ò bilancia <lb/>che &longs;i dica. </s> <s id="id.2.1.466.1.0"><emph type="italics"/>Librar congiu&longs;te lance. <emph.end type="italics"/></s> <s id="id.2.1.467.1.0">Di&longs;&longs;e il Petrarcha. </s> <pb xlink:href="037/01/074.jpg"/> <s id="id.2.1.221.1.0">PROPOSITIONE V. </s> <s id="id.2.1.222.1.0">Due pe&longs;i attaccati nella bilancia, &longs;e la bilancia &longs;arà tra loro in modo <lb/>diui&longs;a, chele parti ri&longs;pondano &longs;cambieuolmente à pe&longs;i; pe&longs;eranno <lb/>tanto ne'punti doue &longs;ono attaccati, quanto &longs;el'uno & l'altro fo&longs;&longs;e <lb/>pendente dal punto della diui&longs;ione. </s> <figure id="id.037.01.074.1.jpg" xlink:href="037/01/074/1.jpg"> </figure> <s id="id.2.1.224.1.0"><emph type="italics"/>Siala bilancia AB, il cui centro &longs;ia C, & &longs;iano due pe&longs;i EF pendenti da' punti <lb/>BG: & diuida&longs;i BG in H, &longs;i fattamente, che BH ad HG habbia la pro<lb/>portione iste&longs;&longs;a, che hà il pe&longs;o E al pe&longs;o F. </s> <s id="id.2.1.224.2.0">Dico i pe&longs;i EF pe&longs;are tanto in BG, <lb/>quanto &longs;e amendue pende&longs;&longs;ero dal punto H. </s> <s id="id.2.1.224.3.0">faccia&longs;i AC eguale à CH. </s> <s>& &longs;i <lb/>come AC à CG, co&longs;i faccia&longs;i il pe&longs;o E al pe&longs;o L. </s> <s>&longs;imilmente come AC à <lb/>CB, co&longs;i faccia&longs;i il pe&longs;o F al pe&longs;o M. </s> <s>& &longs;iano attaccati i pe&longs;i LM al punto <lb/>A. </s> <s id="id.2.1.224.4.0">Hor percioche AC è eguale à CH, &longs;arà BC ver&longs;o CH come il pe&longs;o <lb/>M al pe&longs;o F. </s> <s>& percioche piu grande è BC di CH; &longs;arà anche il pe&longs;o M <emph.end type="italics"/><lb/><arrow.to.target n="note58"></arrow.to.target><emph type="italics"/>maggiore di F. </s> <s id="id.2.1.224.5.0">Diuida&longs;i dunque il pe&longs;o M in due parti QR, & &longs;iala parte di <emph.end type="italics"/><lb/><arrow.to.target n="note59"></arrow.to.target><emph type="italics"/>Q eguale ad F; &longs;arà BC à CH, come RQ à Q: & diuidendo, come BH <lb/>ad HC, co&longs;i R à Q. </s> <s>Dapoi conuertendo, come CH ad HB, co&longs;i Q ad <emph.end type="italics"/><lb/><arrow.to.target n="note60"></arrow.to.target><emph type="italics"/>R. </s> <s id="id.2.1.224.6.0">Oltre à ciò perche CH è eguale à CA, &longs;arà HC ver&longs;o CG come il pe&longs;o <lb/>E al pe&longs;o L: maè piu grande HC di CG, però &longs;arà anche il pe&longs;o E maggio<emph.end type="italics"/><lb/><arrow.to.target n="note61"></arrow.to.target><emph type="italics"/>re del pe&longs;o L. </s> <s id="id.2.1.224.7.0">Onde diuida&longs;i il pe&longs;o E in due parti NO, &longs;i fattamente, che la <lb/>parte di O &longs;ia eguale ad L, &longs;arà HC à CG come tutto lo NO ad O; & <lb/>diuidendo, come HG à GC, co&longs;i N ad O. </s> <s>& conuertendo, come CG à <emph.end type="italics"/><lb/><arrow.to.target n="note62"></arrow.to.target><emph type="italics"/>GH, co&longs;i O ad N. </s> <s>& di nuouo componendo, come CH ad HG, co&longs;i ON <emph.end type="italics"/><lb/><arrow.to.target n="note63"></arrow.to.target><emph type="italics"/>ad N. </s> <s id="id.2.1.224.8.0">& come GH ad HB, co&longs;i è F ad ON. </s> <s id="id.2.1.224.9.0">Per la qual co&longs;a per la pro<lb/>portione vguale come CH ad HB, co&longs;i F ad N. </s> <s id="id.2.1.224.10.0">Ma come CH ad HB <emph.end type="italics"/><lb/><arrow.to.target n="note64"></arrow.to.target><emph type="italics"/>co&longs;i è Q ad R: &longs;arà dunque Q ad R come F ad N. </s> <s>& permutando co<lb/>me Q ad F; co&longs;i R ad N. </s> <s id="id.2.1.224.11.0">ma la parte di Q è egual ad e&longs;&longs;o F. </s> <s>per la qual <emph.end type="italics"/><lb/><arrow.to.target n="note65"></arrow.to.target><emph type="italics"/>co&longs;a la parte di R ancora &longs;arà eguale ad N. </s> <s id="id.2.1.224.12.0">e&longs;&longs;endo dunque il pe&longs;o L eguale <emph.end type="italics"/><pb pagenum="30" xlink:href="037/01/075.jpg"/><emph type="italics"/>ad O, & il pe&longs;o F eguale parimente al Q, & la parte di R eguale ad N; &longs;a <lb/>ranno i pe&longs;i LM eguali a i pe&longs;i E\1</s> <s id="id.2.1.224.13.0">& percioche &longs;i come AC ver&longs;o CG, co<emph.end type="italics"/><arrow.to.target n="note66"></arrow.to.target><lb/><emph type="italics"/>&longs;i è il pe&longs;o E al pe&longs;o L, i pe&longs;i EL pe&longs;eranno egualmente. </s> <s id="id.2.1.224.14.0">&longs;imilmente percioche <lb/>&longs;i come AC è ver&longs;o CB, co&longs;i il pe&longs;o F è al pe&longs;o M, i pe&longs;i FM pe&longs;eranno <lb/>anco egualmente. </s> <s id="id.2.1.224.15.0">i pe&longs;i dunque LM pe&longs;eranno egualmente co'pe&longs;i EF attacca<lb/>ti in BG. </s> <s id="id.2.1.224.16.0">& e&longs;&longs;endo la di&longs;tanza CA eguale alla di&longs;tanza CH, &longs;e dunque am<lb/>bidue i pe&longs;i EF &longs;aranno attaccati in H, i pe&longs;i LM pe&longs;eranno egualmente co' <emph.end type="italics"/><arrow.to.target n="note67"></arrow.to.target><lb/><emph type="italics"/>pe&longs;i EF attaccati in H. </s> <s id="id.2.1.224.17.0">Ma LM pe&longs;a ancora egualmente con EF in GB. <lb/></s> <s id="id.2.1.224.18.0">Adunque &longs;aranno egualmente graui i pe&longs;i EF in GB attaccati come in H. </s> <s id="id.2.1.224.19.0">pe<emph.end type="italics"/><arrow.to.target n="note68"></arrow.to.target><lb/><emph type="italics"/>&longs;eranno dunque tanto in BG quanto attaccati in H. <emph.end type="italics"/></s> <s id="id.2.1.225.1.0"><margin.target id="note58"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.226.1.0"><margin.target id="note59"></margin.target><emph type="italics"/>Per la <expan abbr="con&longs;egu&etilde;za">con&longs;eguenza</expan> della<emph.end type="italics"/> 4. <emph type="italics"/>del<emph.end type="italics"/> 5. </s> <s id="id.2.1.227.1.0"><margin.target id="note60"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.228.1.0"><margin.target id="note61"></margin.target><emph type="italics"/>Per la <expan abbr="con&longs;egu&etilde;za">con&longs;eguenza</expan> della<emph.end type="italics"/> 4. <emph type="italics"/>del<emph.end type="italics"/> 5. </s> <s id="id.2.1.229.1.0"><margin.target id="note62"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.230.1.0"><margin.target id="note63"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.231.1.0"><margin.target id="note64"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.232.1.0"><margin.target id="note65"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.233.1.0"><margin.target id="note66"></margin.target><emph type="italics"/>Perla<emph.end type="italics"/> 6. <emph type="italics"/>del primo di Archimede delle co&longs;e che pe&longs;ano egualmente. <emph.end type="italics"/></s> <s id="id.2.1.234.1.0"><margin.target id="note67"></margin.target><emph type="italics"/>Per lo<emph.end type="italics"/> 2. <emph type="italics"/><expan abbr="cõ.">con.</expan> della not di questo. <emph.end type="italics"/></s> <s id="id.2.1.235.1.0"><margin.target id="note68"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/><expan abbr="cõ.">con.</expan> della not ai questo. <emph.end type="italics"/></s> <figure id="id.037.01.075.1.jpg" xlink:href="037/01/075/1.jpg"> </figure> <s id="id.2.1.237.1.0"><emph type="italics"/>Ma &longs;iano i pe&longs;i EF attaccati in CB; & &longs;ia C il centro della bilancia, & diuida&longs;i <lb/>CB in H, per modo che CH ver&longs;o HB &longs;ia come il pe&longs;o F al pe&longs;o E. </s> <s id="id.2.1.237.2.0">Dico <lb/>che i pe&longs;i EF pe&longs;eranno tanto in CB quanto nel punto H. </s> <s id="id.2.1.237.3.0">faccia&longs;i CA egua<lb/>le à CH, & come CA ver&longs;o CB; co&longs;i faccia&longs;i il pe&longs;o F ver&longs;o vn'altro, che <lb/>&longs;ia D, ilquale &longs;i appicchi in A. </s> <s id="id.2.1.237.4.0">Hor percioche CH è eguale à CA, &longs;arà CH <lb/>ver&longs;o CB, come F à D; & ben è maggiore CB di CH, però il pe&longs;o D &longs;a <lb/>rà maggiore del pe&longs;o F. </s> <s id="id.2.1.237.5.0">Diuida&longs;i dunque il D in due parti GK, & &longs;ia il G <emph.end type="italics"/><arrow.to.target n="note69"></arrow.to.target><lb/><emph type="italics"/>eguale allo F; &longs;arà BC à CH come GK ver&longs;o il G; et diuidendo, come BH <lb/>ad HC, co&longs;i K ver&longs;o G; & conuertendo come CH ad HB, co&longs;i G ver<emph.end type="italics"/><arrow.to.target n="note70"></arrow.to.target><lb/><emph type="italics"/>&longs;o K. </s> <s id="id.2.1.237.6.0">& come CH ad HB, co&longs;i è F ver&longs;o E. </s> <s id="id.2.1.237.7.0">Dunque come G ver<lb/>&longs;o K co&longs;i è F ad E. </s> <s>& permutando come G ad F, co&longs;i K ad E. </s> <s>& per<emph.end type="italics"/><arrow.to.target n="note71"></arrow.to.target><lb/><emph type="italics"/>che GF &longs;ono eguali, &longs;aranno anche KE tra loro eguali. </s> <s id="id.2.1.237.8.0">Concio&longs;ia dunque che <lb/>la parte G &longs;ia eguale ad F, & il K ad e&longs;&longs;o E; &longs;arà tutto il GK eguale a i pe<emph.end type="italics"/><arrow.to.target n="note72"></arrow.to.target><lb/><emph type="italics"/>&longs;i EF. </s> <s>& percioche AC è eguale à CH; &longs;e dunque i pe&longs;i EF &longs;aranno penden<lb/>ti dal punto H, il pe&longs;o D pe&longs;erà egualmente co'pe&longs;i EF attaccati in H. </s> <s id="id.2.1.237.9.0">Ma <lb/>pe&longs;a anche egualmente con eßi in CB, cioè F in B, & E in C; per e&longs;&longs;ere <lb/>come AC ver&longs;o CB, co&longs;i F ver&longs;o D: percioche il pe&longs;o E pendente da C <lb/>centro della bilancia non è cau&longs;a, che la bilancia &longs;i moua in alcuna delle due parti. <lb/></s> <s id="id.2.1.237.10.0">tanto &longs;aranno dunque graui i pe&longs;i EF in CB, quanto in H appicati. <emph.end type="italics"/></s> <s id="id.2.1.239.1.0"><margin.target id="note69"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.240.1.0"><margin.target id="note70"></margin.target><emph type="italics"/>Per la <expan abbr="con&longs;egu&etilde;za">con&longs;eguenza</expan> della<emph.end type="italics"/> 4. <emph type="italics"/>del<emph.end type="italics"/> 5. </s> <s id="id.2.1.241.1.0"><margin.target id="note71"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.242.1.0"><margin.target id="note72"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <pb xlink:href="037/01/076.jpg"/> <figure id="id.037.01.076.1.jpg" xlink:href="037/01/076/1.jpg"> </figure> <s id="id.2.1.244.1.0"><emph type="italics"/>Sia <expan abbr="finalm&etilde;te">finalmente</expan> la <expan abbr="bilãcia">bilancia</expan> AB, & da i <expan abbr="p&utilde;ti">punti</expan> AB &longs;iano <expan abbr="p&etilde;denti">pendenti</expan> i pe&longs;i EF, & &longs;ia il centro <lb/>della bilancia C fra i pe&longs;i, & diuida&longs;i la AB in D, talche AD ver&longs;o DB <lb/>&longs;ia come il pe&longs;o F al pe&longs;o E. </s> <s id="id.2.1.244.2.0">Dico che i pe&longs;i EF pe&longs;ano tanto in AB, quan<lb/>to &longs;e ambidue fo&longs;&longs;ero pendenti dal punto D. </s> <s id="id.2.1.244.3.0">faccia&longs;i CG eguale à CD; & co<lb/>me DC à CA, co&longs;i faccia&longs;i il pe&longs;o E ad vn'altro pe&longs;o H, ilquale &longs;ia attac<lb/>cato in D. </s> <s>& come GC ver&longs;o CB, co&longs;i faccia&longs;i il pe&longs;o F ad vn'altro che <lb/>&longs;ia K, & attachi&longs;i K in G. </s> <s id="id.2.1.244.4.0">Hor percioche, come il BC è ver&longs;o il CG, cioè <lb/>ver&longs;o il CD, co&longs;i il pe&longs;o K ad F; &longs;arà il K maggiore del pe&longs;o F. </s> <s id="id.2.1.244.5.0">Per laqual <lb/>co&longs;a diuida&longs;i il pe&longs;o K in L & in MN, & faccia&longs;i la parte L eguale ad F, <emph.end type="italics"/><lb/><arrow.to.target n="note73"></arrow.to.target><emph type="italics"/>&longs;arà come BC à CD, co&longs;i tutto LMN ad L; & diuidendo, come BD <emph.end type="italics"/><lb/><arrow.to.target n="note74"></arrow.to.target><emph type="italics"/>ver&longs;o DC, co&longs;i la parte MN alla parte L. </s> <s>come dunque BD à DC, co&longs;i <lb/>la parte MN ad F. </s> <s>& come AD à DB, co&longs;i F ad E. </s> <s id="id.2.1.244.6.0">Per laqual co&longs;a <emph.end type="italics"/><lb/><arrow.to.target n="note75"></arrow.to.target><emph type="italics"/>per la egual proportione, come AD ver&longs;o DC, co&longs;i MN ad E. </s> <s>& e&longs;&longs;endo AD <emph.end type="italics"/><lb/><arrow.to.target n="note76"></arrow.to.target><emph type="italics"/>maggiore di CD; &longs;arà anco la parte MN maggiore del pe&longs;o E. </s> <s id="id.2.1.244.7.0">Diuida&longs;i dun<lb/>que MN in due parti MN, & &longs;ia M eguale ad E. </s> <s id="id.2.1.244.8.0">&longs;arà come AD à <emph.end type="italics"/><lb/><arrow.to.target n="note77"></arrow.to.target><emph type="italics"/>DC, co&longs;i NM ad M; & diuidendo, come AC ver&longs;o CD, co&longs;i N ad M: <emph.end type="italics"/><lb/><arrow.to.target n="note78"></arrow.to.target><emph type="italics"/>& conuertendo, come DC ver&longs;o CA, co&longs;i M ad N. </s> <s id="id.2.1.244.9.0">& come DC à <emph.end type="italics"/><lb/><arrow.to.target n="note79"></arrow.to.target><emph type="italics"/>CA, co&longs;i è E ad H; &longs;arà dunque M ad N come E ad H; & permutan<lb/>do come M ad E, co&longs;i N ad H. </s> <s id="id.2.1.244.10.0">Ma per e&longs;&longs;ere ME tra loro eguali, &longs;aran<lb/>no anche NH tra &longs;e eguali. </s> <s id="id.2.1.244.11.0">& percioche co&longs;i è AC ver&longs;o CD, come H <lb/>ad E: i pe&longs;i HE pe&longs;eranno egualmente. </s> <s id="id.2.1.244.12.0">&longs;imilmente percioche, come è GC à CB, <lb/>co&longs;i F ver&longs;o K, i pe&longs;i etiandio KF pe&longs;eranno egualmente. </s> <s id="id.2.1.244.13.0">Adunque i pe&longs;i <emph.end type="italics"/><lb/><arrow.to.target n="note80"></arrow.to.target><emph type="italics"/>EK HF nella bilancia AB, il cui centro &longs;ia C pe&longs;eranno egualmente. </s> <s id="id.2.1.244.14.0">& con<lb/>cio&longs;ia che GC &longs;ia eguale à CD, & il pe&longs;o H &longs;ia pur eguale ad N, i pe&longs;i NH <emph.end type="italics"/><pb pagenum="31" xlink:href="037/01/077.jpg"/><emph type="italics"/>pe&longs;eranno egualmente. </s> <s id="id.2.1.244.15.0">& percioche tutti pe&longs;ano egualmente, tolti via i pe&longs;i HN, <lb/>iquali pe&longs;ano egualmente, i re&longs;tanti pe&longs;eranno egualmente; cioè i pe&longs;i EF, & il pe<emph.end type="italics"/><arrow.to.target n="note81"></arrow.to.target><lb/><emph type="italics"/>&longs;o LM pendenti dal centro C della bilancia. </s> <s id="id.2.1.244.16.0">Ma percioche la parte L è egua<lb/>le ad F, & la parte M è eguale alla parte E; &longs;arà tutto LM eguale a i pe&longs;i <lb/>FE in&longs;ieme pre&longs;i. </s> <s id="id.2.1.244.17.0">& e&longs;&longs;endo CG eguale à CD, &longs;e i pe&longs;i EF &longs;aranno &longs;atti <lb/>pendenti dal punto D, i pe&longs;i EF appiccati in D pe&longs;eranno <expan abbr="egualm&etilde;te">egualmente</expan> con LM. <lb/></s> <s id="id.2.1.244.18.0">Per laqual co&longs;a LM pe&longs;erà <expan abbr="egualm&etilde;te">egualmente</expan> <expan abbr="tãto">tanto</expan> ad eßi EF appiccati in AB, <expan abbr="quãto">quan<lb/>to</expan> &longs;e fo&longs;&longs;ero appiccati nel punto D; peroche la bilancia rimane &longs;empre nell'i&longs;te&longs;&longs;o <emph.end type="italics"/><arrow.to.target n="note82"></arrow.to.target><lb/><emph type="italics"/>modo. </s> <s id="id.2.1.244.19.0">Adunque i pe&longs;i EF pe&longs;eranno tanto in AB quanto nel punto D; che <lb/>bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.245.1.0"><margin.target id="note73"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.246.1.0"><margin.target id="note74"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 23. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.247.1.0"><margin.target id="note75"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.248.1.0"><margin.target id="note76"></margin.target><emph type="italics"/>Corollario della quarta del quinto. <emph.end type="italics"/></s> <s id="id.2.1.249.1.0"><margin.target id="note77"></margin.target>11. <emph type="italics"/>del<emph.end type="italics"/> 5. </s> <s id="id.2.1.250.1.0"><margin.target id="note78"></margin.target>16. <emph type="italics"/>del<emph.end type="italics"/> 5. </s> <s id="id.2.1.251.1.0"><margin.target id="note79"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>del<emph.end type="italics"/> 1. <emph type="italics"/>di Archimede delle co&longs;a che <expan abbr="egualm&etilde;te">egualmente</expan> pe&longs;ano. <emph.end type="italics"/></s> <s id="id.2.1.252.1.0"><margin.target id="note80"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>notitia commune di que&longs;to. <emph.end type="italics"/></s> <s id="id.2.1.253.1.0"><margin.target id="note81"></margin.target><emph type="italics"/>Per la commune notitia di questo. <emph.end type="italics"/></s> <s id="id.2.1.254.1.0"><margin.target id="note82"></margin.target><emph type="italics"/>Per la commune notitia di questo. <emph.end type="italics"/></s> <s id="id.2.1.255.1.0">Ma que&longs;te co&longs;e tutte dimo&longs;treremo in altra maniera, & piu Mechani<lb/>camente. </s> <figure id="id.037.01.077.1.jpg" xlink:href="037/01/077/1.jpg"> </figure> <s id="id.2.1.257.1.0"><emph type="italics"/>Sia la bilancia AB, & il &longs;uo centro C, & &longs;iano, come nel primo ca&longs;o, due pe&longs;i EF <lb/>pendenti da i punti BG: & &longs;ia GH ad HB, come il pe&longs;o F al pe&longs;o E. </s> <s id="id.2.1.257.2.0">Di<lb/>co che i pe&longs;i EF pe&longs;eranno tanto in GB, quanto &longs;e ambidue &longs;te&longs;&longs;ero pendenti <lb/>dal punto H della diui&longs;ione. </s> <s id="id.2.1.257.3.0">Siano di&longs;po&longs;te le mede&longs;ime co&longs;e, cioè faccia&longs;i AC <lb/>eguale à CH, & dal punto A &longs;iano appe&longs;i due pe&longs;i LM, per modo che il pe<lb/>&longs;o E ver&longs;o il pe&longs;o L &longs;ia come CA ver&longs;o CG; & come CB ver&longs;o CA, co <lb/>&longs;i &longs;ia il pe&longs;o M ver&longs;o il pe&longs;o F. </s> <s id="id.2.1.257.4.0">I pe&longs;i LM pe&longs;eranno egualmente (come è detto <lb/>di &longs;opra) con li pe&longs;i EF appiccati in GB. </s> <s id="id.2.1.257.5.0">Siano dapoi due punti NO li centri <lb/>della grauezza de' pe&longs;i EF; & &longs;iano congiunte le linee GN BO; & &longs;ia con<lb/>giunta NO, laquale &longs;arà come bilancia; laquale etiandio faccia sì, che le linee <lb/>GN BO &longs;iano tra loro egualmente di&longs;tanti; & dal punto H &longs;ia tirata la HP <lb/>à piombo dell'orizonte, laquale tagli NO nel P, & &longs;ia egualmente distante dal <lb/>le linee GN BO. </s> <s id="id.2.1.257.6.0">In fine congiunga&longs;i GO, laquale tagli HP in R. </s> <s id="id.2.1.257.7.0">Percio<emph.end type="italics"/><arrow.to.target n="note83"></arrow.to.target><lb/><emph type="italics"/>che dunque HR è egualmente di&longs;tante dal lato BO del triangolo GBO; &longs;arà <lb/>la GH ver&longs;ola HB, come GR ad RO. </s> <s id="id.2.1.257.8.0">Similmente percioche RP è egual<emph.end type="italics"/><pb xlink:href="037/01/078.jpg"/><figure id="id.037.01.078.1.jpg" xlink:href="037/01/078/1.jpg"></figure><lb/><emph type="italics"/>mente di&longs;tante dal lato GN del triangolo OGN; &longs;arà GR ver&longs;o RO, come <lb/>NP ver&longs;o PO. </s> <s id="id.2.1.257.9.0">Per laqual co&longs;a come GH ad HB, così è NP ver&longs;o PO. <emph.end type="italics"/><lb/><arrow.to.target n="note84"></arrow.to.target></s> <s><emph type="italics"/>Ma come GH ver&longs;o HB, così è il pe&longs;o F ver&longs;o il pe&longs;o E; adunque come NP <lb/>ver&longs;o PO, così è il pe&longs;o F ver&longs;o il pe&longs;o E. </s> <s id="id.2.1.257.10.0">Dunque il punto P &longs;arà il centro <emph.end type="italics"/><lb/><arrow.to.target n="note85"></arrow.to.target><emph type="italics"/>della grauezza della magnitudine compo&longs;ta di ambidue i pe&longs;i EF. </s> <s id="id.2.1.257.11.0">Intendan&longs;i <lb/>dunque i pe&longs;i EF e&longs;&longs;ere in maniera dalla bilancia NO annodati, come &longs;e fo&longs;&longs;e vna <lb/>grandezza &longs;ola d'ambidue i pe&longs;i EF composta, & attacata ne i punti BG, &longs;e dun<lb/>que &longs;aranno &longs;ciolti i legamenti BG de' pe&longs;i; rimarranno i pe&longs;i EF <expan abbr="p&etilde;denti">pendenti</expan> da HP; <lb/>&longs;i come prima &longs;tauane in GB. </s> <s id="id.2.1.257.12.0">Ma i pe&longs;i EF appiccati in GB pe&longs;ano egualmente <lb/>co'i pe&longs;i LM, & i pe&longs;i EF pendenti dal punto H hanno l'i&longs;te&longs;&longs;a di&longs;po&longs;itione ver<emph.end type="italics"/><lb/><arrow.to.target n="note86"></arrow.to.target><emph type="italics"/>&longs;o la bilancia AB, come &longs;e fo&longs;&longs;ero appiccati in BG: Gli isteßi pe&longs;i dunque EF <lb/>pendenti da H pe&longs;aranno egualmente con gli i&longs;te&longs;&longs;i pe&longs;i LM. </s> <s id="id.2.1.257.13.0">Sono dunque egual<lb/>mente graui i pe&longs;i EF attaccati in GB, come attaccati in H. <emph.end type="italics"/></s> <s id="id.2.1.259.1.0"><margin.target id="note83"></margin.target><emph type="italics"/>Per la &longs;econda del &longs;esta. <emph.end type="italics"/></s> <s id="id.2.1.260.1.0"><margin.target id="note84"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>del quinto<emph.end type="italics"/></s> <s id="id.2.1.261.1.0"><margin.target id="note85"></margin.target><emph type="italics"/>Per la &longs;esta del primo di Archimede delle co&longs;e, che pe&longs;ano egualmente. <emph.end type="italics"/></s> <s id="id.2.1.262.1.0"><margin.target id="note86"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.078.2.jpg" xlink:href="037/01/078/2.jpg"> </figure> <figure id="id.037.01.078.3.jpg" xlink:href="037/01/078/3.jpg"> </figure> <pb pagenum="32" xlink:href="037/01/079.jpg"/> <s id="id.2.1.265.1.0"><emph type="italics"/>Similmente dimo&longs;treraßi, che i pe&longs;i EF pe&longs;eranno tanto appiccati in qual &longs;i voglia al<lb/>tro punto, quanto &longs;e l'vno, & l'altro fo&longs;&longs;e pendente dal punto H della diui&longs;ione. <lb/></s> <s id="id.2.1.265.2.0">Percioche &longs;e, come di &longs;opra habbiamo in&longs;egnato, &longs;i troueranno i pe&longs;i nella bilancia, à <lb/>i quali i pe&longs;i EF pe&longs;ino egualmente; gli isteßi pe&longs;i EF pendenti da H pe&longs;eranno <lb/>egualmente co' mede&longs;imi pe&longs;i trouati; per e&longs;&longs;ere il punto P &longs;empre il centro della <lb/>grauezza loro; & la HP a piombo dell'orizonte. <emph.end type="italics"/></s> <s id="id.2.1.266.1.0">PROPOSITIONE VI. </s> <s id="id.2.1.267.1.0">I pe&longs;i eguali nella bilancia appiccati hanno in grauezza quella pro<lb/>portione, che hanno le di&longs;tanze, dalle quali &longs;tanno pendenti. </s> <figure id="id.037.01.079.1.jpg" xlink:href="037/01/079/1.jpg"> </figure> <s id="id.2.1.269.1.0"><emph type="italics"/>Sia la bilancia BAC &longs;o&longs;pe&longs;a nel punto A; & &longs;ia &longs;egata la AC, come pare in D. </s> <s>& <lb/>da i punti DC &longs;iano attaccati EF pe&longs;i eguali. </s> <s id="id.2.1.269.2.0">Dico, che il pe&longs;o F ver&longs;o il pe&longs;o E ba <lb/>quella proportione in grauezza, che hala di&longs;tanza CA alla di&longs;tanza AD. </s> <s id="id.2.1.269.3.0">Per<lb/>cioche faccia&longs;i come CA ver&longs;o AD, co&longs;i il pe&longs;o F ver&longs;o vn'altro pe&longs;o, che &longs;ia G. <lb/></s> <s id="id.2.1.269.4.0">Dico prima i pe&longs;i GF pendenti dal punto C tanto pe&longs;are, quanto i pe&longs;i EF penden<lb/>ti da punti DC. </s> <s id="id.2.1.269.5.0">Tagli&longs;i DC in due parti eguali in H, & da H &longs;iano fatti pendere <lb/>ambidue i pe&longs;i EF. </s> <s id="id.2.1.269.6.0">Pe&longs;eranno EF pre&longs;i in&longs;ieme in quel &longs;ito tanto quanto pe&longs;ano <emph.end type="italics"/><arrow.to.target n="note87"></arrow.to.target><lb/><emph type="italics"/>in DC. </s> <s id="id.2.1.269.7.0">Ponga&longs;i BA eguale ad AH, & &longs;itagli BA in K, di modo, che KA <lb/>&longs;ia eguale ad AD: dapoi dal punto B &longs;ia &longs;atto pendente il pe&longs;o L, ilquale &longs;ia il dop<lb/>pio del pe&longs;o F, cioè eguale a i due pe&longs;i EF, ilqual pe&longs;erà egualmente co'pe&longs;i EF ap<lb/>piccati in H, cioè appiccati in DC. </s> <s id="id.2.1.269.8.0">Percioche dunque, come CA ver&longs;o AD, così è <lb/>il pe&longs;o F ver&longs;o il pe&longs;o G, &longs;arà componendo come CA AD ver&longs;o AD, cioè come <lb/>CK ver&longs;o AD, così i pe&longs;i FG ver&longs;o il pe&longs;o G. </s> <s id="id.2.1.269.9.0">Ma per e&longs;&longs;er come CA ver&longs;o AD, <emph.end type="italics"/><arrow.to.target n="note88"></arrow.to.target><lb/><emph type="italics"/>così il pe&longs;o F al pe&longs;o G, &longs;arà anche conuertendo, come DA ver&longs;o AC, così il pe&longs;o <lb/>G ver&longs;o il pe&longs;o F; & i doppi dei con&longs;eguenti, come DA alla doppia di e&longs;&longs;a AC, <lb/>così il pe&longs;o G al doppio del pe&longs;o F, cioè al pe&longs;o L. </s> <s id="id.2.1.269.10.0">Per laqual co&longs;a come CK ver&longs;o <emph.end type="italics"/><arrow.to.target n="note89"></arrow.to.target><lb/><emph type="italics"/>DA, così i pe&longs;i FG al pe&longs;o G; & come AD alla doppia di AC, così il pe&longs;o G al <lb/>pe&longs;o L, adunque dalla egual proportione come CK alla doppia di AC, così i pe&longs;i FG <lb/>al pe&longs;o L. </s> <s id="id.2.1.269.11.0">Ma come CK alla doppia di AC, così la metà di CK, cioè AH, cioè <emph.end type="italics"/><arrow.to.target n="note90"></arrow.to.target><lb/><emph type="italics"/>BA ver&longs;o AC. </s> <s id="id.2.1.269.12.0">Adunque come BA ver&longs;o AC, così FG pe&longs;i al pe&longs;o L. </s> <s id="id.2.1.269.13.0">Per laqual <emph.end type="italics"/><pb xlink:href="037/01/080.jpg"/><emph type="italics"/>co&longs;a per la &longs;e&longs;ta dell'i&longs;te&longs;&longs;o primo di Archimede, i due pe&longs;i FG pendenti dal punto C <emph.end type="italics"/><lb/><arrow.to.target n="note91"></arrow.to.target><emph type="italics"/>pe&longs;eranno tanto, quanto il pe&longs;o L pendente dal B; cioè quanto i pe&longs;i EF pen<lb/>denti da i punti DC. </s> <s id="id.2.1.269.14.0">Così percioche i pe&longs;i FG tanto pe&longs;ano quanto i pe&longs;i EF, <lb/>leuato via il pe&longs;o comune F, tanto pe&longs;erà il pe&longs;o G appicato in C, quanto il pe<emph.end type="italics"/><lb/><figure id="id.037.01.080.1.jpg" xlink:href="037/01/080/1.jpg"></figure><lb/><emph type="italics"/>&longs;o E in D. </s> <s id="id.2.1.269.15.0">Et perciò il pe&longs;o F al pe&longs;o E hà quella proportione in grauezza, <lb/>che hà al pe&longs;o G. </s> <s id="id.2.1.269.16.0">Ma il pe&longs;o F ver&longs;o il G era come CA ver&longs;o AD. </s> <s>adun<lb/>que il pe&longs;o F ancora ver&longs;o il pe&longs;o E hauerà quella proportione in grauezza, che <lb/>ha CA ver&longs;o AD che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.271.1.0"><margin.target id="note87"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.272.1.0"><margin.target id="note88"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.273.1.0"><margin.target id="note89"></margin.target><emph type="italics"/>Per la con&longs;eguenza della quarta del quinto. <emph.end type="italics"/></s> <s id="id.2.1.274.1.0"><margin.target id="note90"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 22. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.275.1.0"><margin.target id="note91"></margin.target><emph type="italics"/>Per la &longs;ettima del<emph.end type="italics"/> 5. </s> <s id="id.2.1.276.1.0"><emph type="italics"/>Ma &longs;e nella bilancia BAC &longs;i faranno pendenti da i punti BC, i pe&longs;i EF eguali; <lb/>Dico &longs;imilmente, che il pe&longs;o E ver&longs;o il pe&longs;o F hà quella proportione in grauezza, <lb/>che ha la di&longs;tanza <lb/>CA alla di&longs;tanza <lb/>AB. </s> <s id="id.2.1.276.2.0">faccia&longs;i AD <lb/>eguale ad AB, & <lb/>dal punto D &longs;ia <lb/>fatto <expan abbr="p&etilde;dente">pendente</expan> il pe<lb/>&longs;o G eguale al pe<lb/>&longs;o F, ilquale <expan abbr="etiã">etian</expan><emph.end type="italics"/><lb/><figure id="id.037.01.080.2.jpg" xlink:href="037/01/080/2.jpg"></figure><lb/><emph type="italics"/>dio &longs;arà eguale ad E. </s> <s id="id.2.1.276.3.0">Et percioche AD è eguale ad AB; i pe&longs;i FG pe&longs;eran<lb/>no egualmente, & hauranno la mede&longs;ima grauezza. </s> <s id="id.2.1.276.4.0">Et concio&longs;ia, che la grauezza <lb/>del pe&longs;o E ver&longs;o la grauezza del pe&longs;o G &longs;ia come CA ad AD; &longs;arà la gra<lb/>uezza del pe&longs;o E ver&longs;o la grauezza del pe&longs;o F, come CA ad AD, cioè CA <lb/>ad AB, che parimente era da mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.278.1.0">Altramente. </s> <s id="id.2.1.279.1.0"><emph type="italics"/>Sia la bilancia BAC, col &longs;uo centro A: & ne i punti BC &longs;iano appiccati pe&longs;i <lb/>eguali GF, & &longs;ia prima il centro A, come &longs;i vuole, fra B, & C. </s> <s id="id.2.1.279.2.0">Dico, che <lb/>il pe&longs;o F ver&longs;o il pe&longs;o G hà quella proportione in grauezza, che ha la di&longs;tanza <lb/>CA alla di&longs;tanza AB. </s> <s id="id.2.1.279.3.0">Faccia&longs;i come BA ver&longs;o AC, co&longs;i il pe&longs;o F ad vn<emph.end type="italics"/><pb pagenum="33" xlink:href="037/01/081.jpg"/><emph type="italics"/>altro H, ilquale &longs;ia appiccato in B: i pe&longs;i HF pe&longs;eranno egualmente de A. <emph.end type="italics"/><arrow.to.target n="note92"></arrow.to.target><lb/></s> <s><emph type="italics"/>Ma e&longs;&longs;endo i pe&longs;i FG eguali, haurà il pe&longs;o H ver&longs;o il pe&longs;o G la proportione me<lb/>de&longs;ima, che ha ad F. </s> <s id="id.2.1.279.4.0">Come dunque CA ver&longs;o AB, co&longs;i è H ver&longs;o G: & <lb/>come H ver&longs;o G, co&longs;i è la grauezza di H alla grauezza di G, per e&longs;&longs;ere attac<lb/>cati nell i&longs;te&longs;&longs;o punto B. </s> <s id="id.2.1.279.5.0">Per laqual co&longs;a come CA ad AB, co&longs;i la grauezza <lb/>del pe&longs;o H alla grauezza del pe&longs;o G. </s> <s id="id.2.1.279.6.0">Et concio&longs;ia che la grauezza del pe&longs;o F <lb/>attacato in G &longs;ia <emph.end type="italics"/><arrow.to.target n="note93"></arrow.to.target><lb/><emph type="italics"/>eguale alla grauez<lb/>za del pe&longs;o H attac<lb/>cato in B, &longs;arà la <lb/>grauezza del pe&longs;o F <lb/>ver&longs;o la grauezza <lb/>del pe&longs;o G, come <lb/>CA ver&longs;o AB, <lb/>cioè come la di&longs;tan<lb/>za alla di&longs;tanza, che <lb/>bi&longs;ognaua mostrare. <emph.end type="italics"/></s> <s id="id.2.1.280.1.0"><margin.target id="note92"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>del prime di Archimede delle co&longs;e che pe&longs;ano egualmente. <emph.end type="italics"/></s> <s id="id.2.1.281.1.0"><margin.target id="note93"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <figure id="id.037.01.081.1.jpg" xlink:href="037/01/081/1.jpg"> </figure> <s id="id.2.1.283.1.0"><emph type="italics"/>Ma &longs;e la bilancia BAC fo&longs;&longs;e tagliata, come &longs;i vuole in D, & appicchin&longs;i in DC <lb/>i pe&longs;i EF eguali. </s> <s id="id.2.1.283.2.0">Dico &longs;imilmente co&longs;i e&longs;&longs;ere la grauezza del pe&longs;o F alla gra<lb/>uezza del pe&longs;o E, come la di&longs;tanza CA alla di&longs;tanza AD. </s> <s id="id.2.1.283.3.0">Faccia&longs;i AB <lb/>eguale ad AD <lb/>& &longs;ia appicca<lb/>to in B il pe&longs;o <lb/>G eguale al pe<lb/>&longs;o E, & al pe<lb/>&longs;o F. </s> <s id="id.2.1.283.4.0">Hor <lb/>percioche AB <lb/>è eguale ad A <lb/>D; i pe&longs;i GE <emph.end type="italics"/><lb/><figure id="id.037.01.081.2.jpg" xlink:href="037/01/081/2.jpg"></figure><lb/><emph type="italics"/>pe&longs;eranno egualmente. </s> <s id="id.2.1.283.5.0">Ma per e&longs;&longs;ere la grauezza del pe&longs;o F ver&longs;o la grauezza <lb/>del pe&longs;o G, come CA ad AB, & la grauezza del pe&longs;o E &longs;ia eguale alla <lb/>grauezza del pe&longs;o G; &longs;arà la grauezza del pe&longs;o F ver&longs;o la grauezza del pe&longs;o E, <lb/>come CA ad AB, cioè CA ad AD, che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.285.1.0">COROLLARIO. </s> <s id="id.2.1.286.1.0">Da que&longs;to è manife&longs;to, che quanto il pe&longs;o è piu di&longs;tante dal centro <lb/>della bilancia, tanto egli è anco piu graue, & per con&longs;eguente mo<lb/>uer&longs;i piu velocemente. </s> <s id="id.2.1.287.1.0">Quinci oltre à ciò &longs;i mo&longs;trerà facilmente anche la ragione della Sta<lb/>dera. </s> <pb xlink:href="037/01/082.jpg"/> <s id="id.2.1.289.1.0">Corollario vocabolo Latino co&longs;tumato da tutti gli altri Scrittori Italiani in cotal ma<lb/>teria, nè di&longs;piacque à Dante nel 28. cap. del Purgatorio. </s> <s id="id.2.1.289.3.0">“Dirotti vn corollario an<lb/>co per gratia.” vuol dire, &longs;econdo Varrone nel primo libro della lingua Latina, <lb/>quella giunta, & quel &longs;opra piu, che &longs;i dà oltre al pagamento, quando &longs;i compera <lb/>qualche co&longs;a. </s> <s id="id.2.1.289.4.0">Al tempo antico allhor che i recitatori di Tragedie, Comedie, & <lb/>altri Poemi nelle &longs;cene &longs;i portauano bene, & piaceuano à gli vditori, era loro do<lb/>nato oltra al prezzo a&longs;&longs;egnato, vn corollario per cia&longs;cuno, cioè vna piccola coro <lb/>na per douer&longs;ene ornare le tempie per giunta, & &longs;opra piu delle &longs;ue mercedi. </s> <s id="id.2.1.289.5.0">Co&longs;i <lb/>nelle &longs;cienze matematiche v&longs;a&longs;i di aggiungere certe co&longs;e, oltra le propo&longs;itioni, <lb/>qua&longs;i giunte & con&longs;equenze, le quali na&longs;cono dalle co&longs;e primieramente dimo&longs;tra<lb/>te, & &longs;ono loro corri&longs;pondenti, & non &longs;ono però nè propo&longs;itioni, nè problemi, <lb/>nè lemmi, ma alla &longs;embianza predetta chiaman&longs;i corollarij, molti de i quali han<lb/>no congiunta la &longs;ua dimo&longs;tratione. </s> <s id="id.2.1.290.1.0"><emph type="italics"/>Hor &longs;ia AB il fusto della Stadera, la cui trutina &longs;ia in C; & &longs;ia il marco della &longs;ta<emph.end type="italics"/><lb/><arrow.to.target n="note94"></arrow.to.target><emph type="italics"/>dera E. </s> <s id="id.2.1.290.2.0">Appicchi&longs;i in A il pe&longs;o D, che pe&longs;i egualmente col marco E appic<lb/>cato in F. </s> <s id="id.2.1.290.3.0">Appicchi&longs;i parimente vn'altro pe&longs;o G in A, ilqual anco pe&longs;i egual<lb/>mente col marco E appiccato in B. </s> <s id="id.2.1.290.4.0">Dico, la grauezza del pe&longs;o D ver&longs;o la gra<lb/>uezza del <lb/>G e&longs;&longs;ere co <lb/>&longs;i, come CF <lb/>ver&longs;o CB. <lb/></s> <s id="id.2.1.290.5.0">Hor per<lb/>cioche la <lb/>grauezza <lb/>del pe&longs;o D <lb/>è eguale al <lb/>la grauez<lb/>za del pe<lb/>&longs;o E at<lb/>taccato in <lb/>F, & la <emph.end type="italics"/><lb/><figure id="id.037.01.082.1.jpg" xlink:href="037/01/082/1.jpg"></figure><lb/><emph type="italics"/>grauezza del pe&longs;o G è eguale alla grauezza del pe&longs;o E po&longs;to in B; &longs;arà la grauez<lb/>za del pe&longs;o D alla grauezza del pe&longs;o E po&longs;to in F, come la grauezza del pe&longs;o G alla <lb/>grauezza del pe&longs;o E po&longs;to in B; & permutando come la grauezza del pe&longs;o D alla <lb/>grauezza del pe&longs;o G, co&longs;i la grauezza di E po&longs;to in F alla grauezza di E po&longs;to in B; <lb/>ma la grauezza del pe&longs;o E in F alla grauezza di E in B po&longs;to è come CF <lb/>ver&longs;o CB; come dunque la grauezza del pe&longs;o D alla grauezza del pe&longs;o G, co&longs;i <lb/>è CF ver&longs;o CB. </s> <s id="id.2.1.290.6.0">Se dunque la parte del fu&longs;to CB diuidera&longs;&longs;i in parti eguali, po<lb/>&longs;to &longs;olo il pe&longs;o E & piu da pre&longs;&longs;o, & piu da lontano dal punto C; le grauezze de <lb/>pe&longs;i, lequali &longs;tanno pendenti dal punto A &longs;aranno tra loro manife&longs;te & note. </s> <s id="id.2.1.290.7.0">Co<lb/>me&longs;e la di&longs;tanza CB &longs;arà tripla della di&longs;tanza CF, &longs;arà parimente la grauezza <lb/>di e&longs;&longs;o G tripla della grauezza di D, che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.293.1.0"><margin.target id="note94"></margin.target><emph type="italics"/>Ragione del la stadera. <emph.end type="italics"/></s> <pb pagenum="34" xlink:href="037/01/083.jpg"/> <s id="id.2.1.294.1.0">In altro modo po&longs;&longs;iamo anco v&longs;are la &longs;tadera, affine che le grauezze <lb/>de i pe&longs;i &longs;i facciano note. </s> <s id="id.2.1.295.1.0"><emph type="italics"/>Sia il fu&longs;to della stadera AB, la cui trutina &longs;ia in C, & &longs;ia il marco della stadera <lb/>E, ilquale &longs;ia appiccato in A; & &longs;iano i pe&longs;i DG di&longs;uguali, le proportioni delle <lb/>grauezze de quali cerchia<lb/>mo: &longs;ia appiccato il pe&longs;o D <lb/>in B talche pe&longs;i egual<lb/>mente con E. </s> <s id="id.2.1.295.2.0">Similmente <lb/>appicchi&longs;i il pe&longs;o G in F, <lb/>ilquale pe&longs;i egualmente con <lb/>l'i&longs;te&longs;&longs;o pe&longs;o E. </s> <s id="id.2.1.295.3.0">Dico D <lb/>ver&longs;o G co&longs;i e&longs;&longs;ere; come <emph.end type="italics"/><arrow.to.target n="note95"></arrow.to.target><lb/><emph type="italics"/>CF ver&longs;o CB. </s> <s id="id.2.1.295.4.0">Hor perche <lb/>i pe&longs;i DE pe&longs;ano <expan abbr="egualm&etilde;">egualmen</expan><emph.end type="italics"/><lb/><figure id="id.037.01.083.1.jpg" xlink:href="037/01/083/1.jpg"></figure><lb/><emph type="italics"/>te, &longs;arà D ad E, come CA à CB. </s> <s id="id.2.1.295.5.0">& concio&longs;ia, che anche i pe&longs;i GE pe&longs;i<lb/>no egualmente, &longs;arà il pe&longs;o E ver&longs;o il pe&longs;o G, come FC à CA; Per laqual <lb/>co&longs;a per la proportion eguale il pe&longs;o D al pe&longs;o G, co&longs;i &longs;arà, come CF à CB. <lb/></s> <s>che parimente bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/><arrow.to.target n="note96"></arrow.to.target></s> <s id="id.2.1.297.1.0"><margin.target id="note95"></margin.target><emph type="italics"/>Per la &longs;esta del primo di Archimede delle co&longs;e, che pe&longs;ano egualmente. <emph.end type="italics"/></s> <s id="id.2.1.298.1.0"><margin.target id="note96"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 23. <emph type="italics"/>del quinto<emph.end type="italics"/></s> <s id="id.2.1.299.1.0">PROPOSITIONE VII. </s> <s id="id.2.1.300.1.0">PROBLEMA. </s> <s id="id.2.1.301.1.0">Dati quanti &longs;i vogliano pe&longs;i nella bilancia, appiccati in qual luogo &longs;i <lb/>&longs;ia, ritrouare il centro della bilancia, dal quale &longs;e &longs;arà fatta penden<lb/>te la bilancia, i dati pe&longs;i &longs;taranno fermi. </s> <s id="id.2.1.302.1.0">PROBLEMA. </s> <s id="id.2.1.302.2.0">Sotto il nome di Propo&longs;itione &longs;i contiene il Problema ancora vo<lb/>cabolo greco; ma il Problema ha dauantaggio della Propo&longs;itione in particolare, <lb/>che ordina, & in&longs;egna ad operare qualche effetto; doue la Propo&longs;itione &longs;uole &longs;ta <lb/>re nella nuda &longs;peculatione &longs;olamente. </s> <s id="id.2.1.302.3.0">Et que&longs;ta è la differenza tra la Propo&longs;itio<lb/>ne, & il Problema. </s> <pb xlink:href="037/01/084.jpg"/> <figure id="id.037.01.084.1.jpg" xlink:href="037/01/084/1.jpg"> </figure> <s id="id.2.1.305.1.0"><emph type="italics"/>Sia la bilancia AB, & &longs;iano dati quanti &longs;i vogliano pe&longs;i CDEFG prendan&longs;i nel <lb/>la bilancia, a piacere i punti AHKLB, da quali &longs;ian fatti pendenti i dati pe&longs;i. <lb/></s> <s id="id.2.1.305.2.0">Bi&longs;ogna ritrouar il centro della bilancia, dal quale &longs;e &longs;i farà l'appiccamento, rimanga<lb/>no i dati pe&longs;i. </s> <s id="id.2.1.305.3.0">Diuida&longs;i AH in M, &longs;i che HM ad MA &longs;ia come la grauezza <lb/>del pe&longs;o C alla grauezza del pe&longs;o D. </s> <s id="id.2.1.305.4.0">Dapoi diuida&longs;i anco BL in N, &longs;i che <lb/>LN ad NB &longs;ia come la grauezza pe&longs;o G alla grauezza del pe&longs;o F. </s> <s id="id.2.1.305.5.0">Et di<lb/>uida&longs;i MN in O, &longs;i che MO ver&longs;o ON &longs;ia come la grauezza de pe&longs;i FG <lb/>alla grauezza de'pe&longs;i CD. </s> <s id="id.2.1.305.6.0">Et in fine diuida&longs;i KO in P, &longs;i che KP ver&longs;o PO <lb/>&longs;ia come la grauezza de'pe&longs;i CD FG alla grauezza del pe&longs;o E. </s> <s id="id.2.1.305.7.0">Hor percio<emph.end type="italics"/><lb/><arrow.to.target n="note97"></arrow.to.target><emph type="italics"/>che i pe&longs;i CDFG tanto pe&longs;ano in O, quanto CD in M, & FG in N; <lb/>pe&longs;eranno egualmente i pe&longs;i CD in M, & FG in N, & il pe&longs;o E in K, <lb/>&longs;e &longs;aranno &longs;o&longs;pe&longs;i nel punto P. </s> <s id="id.2.1.305.8.0">Et concio&longs;ia, che i pe&longs;i CD tanto pe&longs;ino in M, <lb/>quanto in AH, & FG in N quanto in LB; i pe&longs;i CDFG pendenti da' <lb/>punti AHLB, & il pe&longs;o E da K, &longs;e da P &longs;aranno &longs;o&longs;pe&longs;i, pe&longs;eranno egual<lb/>mente, & rimarranno. </s> <s id="id.2.1.305.9.0">egli è dunque trouato il P centro della bilancia, dalquale <lb/>rimangono i pe&longs;i dati. </s> <s id="id.2.1.305.10.0">Che bi&longs;ogna operare. <emph.end type="italics"/></s> <s id="id.2.1.306.1.0"><margin.target id="note97"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.307.1.0">COROLLARIO. </s> <s id="id.2.1.308.1.0">Da que&longs;to è chiaro, che &longs;ei centri della grauezza de' pe&longs;i CDEFG <lb/>fo&longs;&longs;ero ne' punti AHKLB, &longs;arebbe il punto P il centro della <lb/>grauezza della magnitudine compo&longs;ta di tutti i pe&longs;i CDEFG. </s> <s id="id.2.1.309.1.0"><emph type="italics"/>Que&longs;to è manife&longs;to dalla diffinitione del centro della grauezza, concio&longs;ia che i pe&longs;i ri<lb/>mangano, &longs;e &longs;ono &longs;o&longs;tenuti dal punto P. <emph.end type="italics"/></s> <s id="id.2.1.310.1.0"><emph type="italics"/>Il fine della Bilancia. <emph.end type="italics"/></s> </chap> <pb pagenum="35" xlink:href="037/01/085.jpg"/> <chap> <s id="id.2.1.312.1.0">DELLA LEVA. </s> <s id="id.2.1.314.1.0">LEMMA. </s> <s id="id.2.1.315.1.0">Siano quattro grandezze ABCD; & &longs;ia la A <lb/>maggiore della B, & C maggiore della D. </s> <s id="id.2.1.315.2.0">Dico, <lb/>che A ver&longs;o D hà proportione maggiore di quello <lb/>che hà B ver&longs;o C. </s> <s id="id.2.1.316.1.0"><emph type="italics"/>Hor percioche A ver&longs;o C hà proportion maggio<lb/>re, che B ver&longs;o C; & A parimente ver&longs;o D <emph.end type="italics"/><arrow.to.target n="note98"></arrow.to.target><lb/><emph type="italics"/>hà proportion maggiore di quel che ha ver&longs;o C: <lb/>Dunque A ver&longs;o D l'hauerà maggiore, che B <lb/>ver&longs;o C, Che bi&longs;ognaua mostrare. <emph.end type="italics"/></s> <s id="id.2.1.317.1.0"><margin.target id="note98"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <figure id="id.037.01.085.1.jpg" xlink:href="037/01/085/1.jpg"> </figure> <s id="id.2.1.319.1.0">PROPOSITIONE I. </s> <s id="id.2.1.320.1.0">La po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o attaccato alla Leua, ha la proportio<lb/>ne mede&longs;ima al detto pe&longs;o, che ha la di&longs;tanza della Leua fra il &longs;o&longs;te<lb/>gno po&longs;ta, & lo attaccamento del pe&longs;o, alla di&longs;tanza, che è dal &longs;o&longs;te<lb/>gno alla po&longs;&longs;anza. </s> <s id="id.2.1.321.1.0"><emph type="italics"/>Sia la leua AB, il cui &longs;oftegno &longs;ia C; & &longs;ia il pe&longs;o D pendente da A con AH, <lb/>&longs;i che AH &longs;ia &longs;empre à piombo dell'orizonte: & &longs;ia la po&longs;&longs;anza &longs;oftenente il pe<emph.end type="italics"/><lb/><figure id="id.037.01.085.2.jpg" xlink:href="037/01/085/2.jpg"></figure><lb/><emph type="italics"/>&longs;o in B. </s> <s id="id.2.1.321.2.0">Dico che la po&longs;&longs;anza posta in B ver&longs;o il pe&longs;o D &longs;ta co&longs;i, come la CA <emph.end type="italics"/><pb xlink:href="037/01/086.jpg"/><arrow.to.target n="note99"></arrow.to.target><emph type="italics"/>ver&longs;o la CB. </s> <s id="id.2.1.321.3.0">Faccia&longs;i come la BC alla CA, co&longs;i il pe&longs;o D ad vn'altro pe&longs;o <lb/>E, talche &longs;e egli in B &longs;arà appiccato, pe&longs;erà <expan abbr="egualm&etilde;te">egualmente</expan> con D, per e&longs;&longs;er il C cen<lb/>tro della grauezza di ambidue. </s> <s id="id.2.1.321.4.0">Per laqual co&longs;a vna po&longs;&longs;anza eguale ad e&longs;&longs;o E po<lb/>&longs;ta nel <lb/>mede&longs;i<lb/>mo lo <lb/>go pe<lb/>&longs;erà e<lb/>gual<lb/>mente <lb/>con e&longs;<emph.end type="italics"/><lb/><figure id="id.037.01.086.1.jpg" xlink:href="037/01/086/1.jpg"></figure><lb/><emph type="italics"/>&longs;o D, nella leua AB, collocando il &longs;o&longs;tegno &longs;uo in C, cioè impedirà, che il pe<emph.end type="italics"/><lb/><arrow.to.target n="note100"></arrow.to.target><emph type="italics"/>&longs;o D non inchini in giu&longs;o, &longs;i come impedi&longs;ce il pe&longs;o E. </s> <s id="id.2.1.321.5.0">Ma la po&longs;&longs;anza di B al <lb/>pe&longs;o D hàla mede&longs;ima proportione, che il pe&longs;o E ha all'iste&longs;&longs;o D: adunque la <lb/>po&longs;&longs;anza di B ver&longs;o il pe&longs;o D &longs;arà come CA ver&longs;o CB; cioè la di&longs;tanza del<lb/>la leua dal &longs;ostegno al &longs;o&longs;tenimento del pe&longs;o, alla di&longs;tanza dal &longs;ostegno alla po&longs;&longs;an<lb/>za, che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.324.1.0"><margin.target id="note99"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>del<emph.end type="italics"/> 1. <emph type="italics"/>di Archimede delle co&longs;a che <expan abbr="egualm&etilde;te">egualmente</expan> pe&longs;ano. <emph.end type="italics"/></s> <s id="id.2.1.325.1.0"><margin.target id="note100"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.326.1.0">Di quì ageuolmente &longs;i puote mo&longs;trare, che <expan abbr="quãto">quanto</expan> il &longs;o&longs;tegno &longs;arà piu <lb/>vicino al pe&longs;o, tanto minor po&longs;&longs;anza &longs;i ricerca à &longs;o&longs;tenere il detto <lb/>pe&longs;o. </s> <s id="id.2.1.327.1.0"><emph type="italics"/>Poste le co&longs;e mede&longs;ime &longs;ia il &longs;o&longs;tegno in F piu da pre&longs;&longs;o ad A, che C; & faccia&longs;i <lb/>come BF ad FA, co&longs;i il pe&longs;o D ad vn'altro pe&longs;o G, ilquale &longs;e in B &longs;ia ap<emph.end type="italics"/><lb/><arrow.to.target n="note101"></arrow.to.target><emph type="italics"/>piccato; i pe&longs;i DG dal &longs;o&longs;tegno F pe&longs;eranno egualmente. </s> <s id="id.2.1.327.2.0">Hor percioche BF <lb/>è mag<lb/>giore di <lb/>BC, & <lb/>CA <emph.end type="italics"/><lb/><arrow.to.target n="note102"></arrow.to.target><emph type="italics"/>maggio <lb/>re di AF; <lb/>la <lb/>propor<lb/>tione di <emph.end type="italics"/><lb/><figure id="id.037.01.086.2.jpg" xlink:href="037/01/086/2.jpg"></figure><lb/><emph type="italics"/>BF ver&longs;o FA &longs;arà maggiore, che di BC ver&longs;o CA: & perciò maggiore anco <lb/>&longs;arà la proportione del pe&longs;o D al pe&longs;o G, che de l'iste&longs;&longs;o D ad E: Dunque il <emph.end type="italics"/><lb/><arrow.to.target n="note103"></arrow.to.target><emph type="italics"/>pe&longs;o G &longs;arà minore del pe&longs;o E. </s> <s id="id.2.1.327.3.0">& concio&longs;ia che la po&longs;&longs;anza po&longs;ta in B eguale à <lb/>G pe&longs;i egualmente con D, auerrà, che minore po&longs;&longs;anza di quella, laquale è eguale <lb/>al pe&longs;o E &longs;o&longs;tenterà il pe&longs;o D; e&longs;&longs;endo la leua AB, & il &longs;o&longs;tegno &longs;uo doue è F, <lb/>che &longs;e egli fo&longs;&longs;e doue è C. </s> <s id="id.2.1.327.4.0">Similmente anche mo&longs;trera&longs;&longs;i, che quanto piu dapre&longs;&longs;o &longs;a <lb/>rà il &longs;o&longs;tegno al pe&longs;o D, &longs;empre vi &longs;i ricercherà anco po&longs;&longs;anza minore per &longs;o&longs;tentare <lb/>il detto pe&longs;o D. <emph.end type="italics"/></s> <s id="id.2.1.330.1.0"><margin.target id="note101"></margin.target><emph type="italics"/>Per la mede&longs;ima &longs;esta. <emph.end type="italics"/></s> <s id="id.2.1.331.1.0"><margin.target id="note102"></margin.target><emph type="italics"/>Per lo Lemma. <emph.end type="italics"/></s> <s id="id.2.1.332.1.0"><margin.target id="note103"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto<emph.end type="italics"/></s> <pb pagenum="36" xlink:href="037/01/087.jpg"/> <s id="id.2.1.333.1.0">COROLLARIO. </s> <s id="id.2.1.334.1.0">Onde &longs;i puote raccogliere chiaramente, che e&longs;&longs;endo AF minore di <lb/>FB, minor po&longs;&longs;anza anco &longs;i ricerca in B per &longs;o&longs;tenere il pe&longs;o D. </s> <s><lb/>& e&longs;&longs;endo eguale, eguale: & maggiore, maggiore. </s> <s id="id.2.1.335.1.0">PROPOSITIONE II. </s> <s id="id.2.1.336.1.0">In altra maniera po&longs;&longs;iamo v&longs;are la Leua. </s> <s id="id.2.1.337.1.0"><emph type="italics"/>Sia la leua AB, il cui &longs;o&longs;tegno &longs;ia B, & il pe&longs;o C &longs;ia attaccato, come &longs;i vuole, in D <emph.end type="italics"/><arrow.to.target n="note104"></arrow.to.target><lb/><emph type="italics"/>fra AB; & &longs;ia la po&longs;&longs;anza in A che &longs;ostiene il pe&longs;o C. </s> <s id="id.2.1.337.2.0">Dico, che &longs;i come <lb/>BD à BA; co&longs;i è la po&longs;&longs;anza di A' al pe&longs;o C. </s> <s id="id.2.1.337.3.0">Appicchi&longs;i in A il pe&longs;o E <lb/>eguale al C; & come AB ver&longs;o BD, co&longs;i faccia&longs;i il pe&longs;o E ver&longs;o vn'altro pe&longs;o, <emph.end type="italics"/><arrow.to.target n="note105"></arrow.to.target><lb/><emph type="italics"/>come F. </s> <s id="id.2.1.337.4.0">Et percioche i pe&longs;i CE &longs;ono tra &longs;e eguali, &longs;arà il pe&longs;o C ver&longs;o il pe&longs;o F <lb/>come AB ver&longs;o BD. </s> <s id="id.2.1.337.5.0">Attacchi&longs;i parimente il pe&longs;o F in A. </s> <s>& percioche il <emph.end type="italics"/><arrow.to.target n="note106"></arrow.to.target><lb/><emph type="italics"/>pe&longs;o E al pe&longs;o F <lb/>è come la grauez<lb/>za del pe&longs;o di E <lb/>alla grauezza di <lb/>F; & il pe&longs;o E <lb/>ad F è come AB <lb/>à BD; come <expan abbr="d&utilde;que">dun<lb/>que</expan> la grauezza <lb/>del pe&longs;o E alla <emph.end type="italics"/><arrow.to.target n="note107"></arrow.to.target><lb/><figure id="id.037.01.087.1.jpg" xlink:href="037/01/087/1.jpg"></figure><lb/><emph type="italics"/>grauezza del pe&longs;o F, co&longs;i è AB ver&longs;o BD. </s> <s id="id.2.1.337.6.0">ma come AB à BD, co&longs;i è la <lb/>grauezza del pe&longs;o E alla grauezza del pe&longs;o C: Per laqual co&longs;a la grauezza del <lb/>pe&longs;o E alla grauezza del pe&longs;o F co&longs;i &longs;arà, come la grauezza del pe&longs;o E alla gra<lb/>uezza del pe&longs;o C. </s> <s id="id.2.1.337.7.0">I pe&longs;i dunque CF hanno la mede&longs;ima grauezza: &longs;i che pon<lb/>ga&longs;i la po&longs;&longs;anza di A che &longs;o&longs;tenga il pe&longs;o F, &longs;arà la po&longs;&longs;anza di A eguale al pe&longs;o <lb/>F. </s> <s>& percioche il pe&longs;o E attaccato in A è graue egualmente, come il C appicca<emph.end type="italics"/><arrow.to.target n="note108"></arrow.to.target><lb/><emph type="italics"/>to in D; hauerà la proportione i&longs;te&longs;&longs;a la po&longs;&longs;anza di A ver&longs;o la grauezza del pe&longs;o <lb/>F appiccato in A, che ha alla grauezza del pe&longs;o C appiccato in D. </s> <s id="id.2.1.337.8.0">Ma la po&longs;&longs;an<lb/>za di A eguale ad F &longs;o&longs;tiene il pe&longs;o F; dunque la po&longs;&longs;anza di A &longs;o&longs;tenterà anco <lb/>il pe&longs;o C. </s> <s id="id.2.1.337.9.0">Et co&longs;i per e&longs;&longs;ere la po&longs;&longs;anza di A eguale al pe&longs;o F, & il pe&longs;o C ver&longs;o <lb/>il pe&longs;o F &longs;ia come AB à BD; &longs;arà il pe&longs;o C ver&longs;o la po&longs;&longs;anza po&longs;ta in A come <lb/>AB à BD. </s> <s>& conuertendo, come BD à BA, co&longs;i la po&longs;&longs;anza po&longs;ta in A ver<lb/>&longs;o il pe&longs;o C. </s> <s id="id.2.1.337.10.0">Dunque la po&longs;&longs;anza ver&longs;o il pe&longs;o co&longs;i &longs;arà, come la di&longs;tanza, che è fra <emph.end type="italics"/><arrow.to.target n="note109"></arrow.to.target><lb/><emph type="italics"/>il &longs;o&longs;tegno, & l'appiccamento del pe&longs;o alla di&longs;tanza, che è dal &longs;o&longs;tegno alla po&longs;&longs;an<lb/>za, che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.340.1.0"><margin.target id="note104"></margin.target><emph type="italics"/>Nella &longs;esta di questo de la bilancia. <emph.end type="italics"/></s> <s id="id.2.1.341.1.0"><margin.target id="note105"></margin.target><emph type="italics"/>Dalla<emph.end type="italics"/> 11. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.342.1.0"><margin.target id="note106"></margin.target><emph type="italics"/>Per la &longs;esta della <expan abbr="bilãcia">bilancia</expan><emph.end type="italics"/></s> <s id="id.2.1.343.1.0"><margin.target id="note107"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.344.1.0"><margin.target id="note108"></margin.target><emph type="italics"/>Per la &longs;ettima del<emph.end type="italics"/> 5. </s> <s id="id.2.1.345.1.0"><margin.target id="note109"></margin.target><emph type="italics"/>Per lo Corollario della<emph.end type="italics"/> 4 <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <pb xlink:href="037/01/088.jpg"/> <s id="id.2.1.346.1.0">Altramente. </s> <s id="id.2.1.347.1.0"><emph type="italics"/>Sia la leua AB, il cui &longs;o&longs;tegno &longs;ia B, & il pe&longs;o E &longs;ia pendente dal punto C, & <lb/>&longs;ia in A la forza, che &longs;ostiene l pe&longs;o E. </s> <s id="id.2.1.347.2.0">Dico, che &longs;i come BC à BA, co&longs;i è<emph.end type="italics"/><lb/><figure id="id.037.01.088.1.jpg" xlink:href="037/01/088/1.jpg"></figure><lb/><emph type="italics"/>anco la po&longs;&longs;anza di A ver&longs;o il pe&longs;o E. </s> <s id="id.2.1.347.3.0">Allunghi&longs;i AB in D, & faccia&longs;i <lb/>BD eguale à BC; & appicchi&longs;i il pe&longs;o F al punto D, che &longs;ia eguale al pe&longs;o E; <lb/>& parimente dal punto A &longs;i faccia pendere il punto G in modo, che il pe&longs;o F hab<lb/>bia la proportione i&longs;te&longs;&longs;a ver&longs;o il pe&longs;o G, che ha AB à BD. </s> <s id="id.2.1.347.4.0">i pe&longs;i FG verranno <lb/>à pe&longs;ar egualmente: & concio&longs;ia che CB &longs;ia eguale à BD, anco i pe&longs;i FE egua<lb/>li pe&longs;eranno egualmente. </s> <s id="id.2.1.347.5.0">Ma i pe&longs;i FEG nella bilancia, ouero nella leua DBA <lb/>appiccati, il cui &longs;o&longs;tegno è B, non pe&longs;eranno egualmente, ma inchineranno à ba&longs;&longs;o <lb/>dalla parte di A. </s> <s id="id.2.1.347.6.0">Per laqual co&longs;a ponga&longs;i in A tanta forza, che i pe&longs;i FEG pe&longs;i<lb/>no egualmente, &longs;arà la po&longs;&longs;anza in A eguale al pe&longs;o G; peroche i pe&longs;i FE pe&longs;a<lb/>no egualmente, & la forza in A niente altro deue fare, che &longs;o&longs;tenere il pe&longs;o G, ac<lb/>cioche non de&longs;cenda. </s> <s id="id.2.1.347.7.0">Et percio che i pe&longs;i FEG, & la po&longs;&longs;anza in A pe&longs;ano egual<lb/>mente, leuati dunque via i pe&longs;i FG, i quali pe&longs;ano egualmente, i re&longs;tanti pe&longs;eran<lb/>no pur egualmente, cioè la po&longs;&longs;anza in A co'l pe&longs;o E, cioè la po&longs;&longs;anza in A &longs;o<lb/>sterra il pe&longs;o E, &longs;i che la leua AB rimanga, come era prima. </s> <s id="id.2.1.347.8.0">Et per e&longs;&longs;ere la <lb/>po&longs;&longs;anza in A eguale al pe&longs;o G, & il pe&longs;o E eguale al pe&longs;o F, haurà la po&longs;&longs;anza <lb/>in A la proportione iste&longs;&longs;a al pe&longs;o E, che hà BD, cioè BC à BA, che bi&longs;ogna <lb/>ua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.349.1.0">COROLLARIO I. </s> <s id="id.2.1.350.1.0">Da que&longs;to etiandio, come prima, puote e&longs;&longs;ere manife&longs;to, che &longs;e il pe&longs;o <lb/>E &longs;arà po&longs;to piu vicino al &longs;o&longs;tegno B, come in H, minore <lb/>po&longs;&longs;anza po&longs;ta in A puote &longs;o&longs;tener il detto pe&longs;o. </s> <s id="id.2.1.351.1.0"><emph type="italics"/>Percioche minor proportione ha HB à BA, che CB à BA. </s> <s>& quanto piu da <emph.end type="italics"/><lb/><arrow.to.target n="note110"></arrow.to.target><emph type="italics"/>vicino il pe&longs;o &longs;arà al &longs;o&longs;tegno, &longs;empre anco &longs;i mo&longs;trerà &longs;imilmente minor po&longs;&longs;anza <lb/>poter &longs;o&longs;tener il pe&longs;o E. <emph.end type="italics"/></s> <s id="id.2.1.352.1.0"><margin.target id="note110"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.353.1.0">COROLLARIO II. </s> <s id="id.2.1.354.1.0">Segue etiandio, che la po&longs;&longs;anza in A &longs;empre è minore del pe&longs;o E: </s> <pb pagenum="37" xlink:href="037/01/089.jpg"/> <s id="id.2.1.356.1.0"><emph type="italics"/>Percioche pigli&longs;i tra A & B qual punto &longs;i voglia, come C, &longs;empre BC &longs;arà <lb/>minore di BA. <emph.end type="italics"/></s> <s id="id.2.1.357.1.0">COROLLARIO III. </s> <s id="id.2.1.358.1.0">Da que&longs;to parimente &longs;i puote cauare, che &longs;e due &longs;aranno le po&longs;&longs;anze, <lb/>l'vna in A, & l'altra in B, & ambedue &longs;o&longs;tentino il pe&longs;o E, la po&longs;<lb/>&longs;anza in A ver&longs;o la po&longs;&longs;anza in B è come BC ver&longs;o CA. </s> <s id="id.2.1.359.1.0"><emph type="italics"/>Percioche la leua BA fa l'officio di due leue, & AB &longs;ono come due &longs;o&longs;tegni, cioè <lb/>quando AB è leua, & la forza che &longs;o&longs;tiene è in A, &longs;arà il &longs;uo &longs;o&longs;tegno B. </s> <s id="id.2.1.359.2.0">Ma <lb/>quando BA è leua, & la po&longs;&longs;anza &longs;ta in B, il &longs;o&longs;tegno &longs;arà A, & il pe&longs;o <lb/>&longs;empre rimane appicca<lb/>to in C. </s> <s id="id.2.1.359.3.0">Et percio che la <lb/>po&longs;&longs;anza in A ver&longs;o il <lb/>pe&longs;o E è come BC à <emph.end type="italics"/><arrow.to.target n="note111"></arrow.to.target><lb/><emph type="italics"/>BA, & come il pe&longs;o <lb/>E alla po&longs;&longs;anza, che è <lb/>in B, co&longs;i è BA ad <lb/>AC, &longs;arà per la propor<emph.end type="italics"/><lb/><figure id="id.037.01.089.1.jpg" xlink:href="037/01/089/1.jpg"></figure><lb/><emph type="italics"/>tion eguale la po&longs;&longs;anza in A alla po&longs;&longs;anza in B come BC à CA, & à que<lb/>&longs;to modo facilmente ancora potremo cono&longs;cere la proportione, laquale è po&longs;ta de <lb/>Ari&longs;totele nelle que&longs;tioni Mecaniche alla que&longs;tione 29. <emph.end type="italics"/></s> <s id="id.2.1.361.1.0"><margin.target id="note111"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 22. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.362.1.0">COROLLARIO IIII. </s> <s id="id.2.1.363.1.0">E manife&longs;to etiandio, che ambedue le po&longs;&longs;anze in A, & in B <lb/>pre&longs;e in&longs;ieme, &longs;ono eguali al pe&longs;o E. </s> <s id="id.2.1.364.1.0"><emph type="italics"/>Percioche il pe&longs;o E alla po&longs;&longs;anza in A è come BA à BC, & l'i&longs;te&longs;&longs;o pe&longs;o E <lb/>ver&longs;o la po&longs;&longs;anza in B è come BA ad AC; Per laqual co&longs;a il pe&longs;o E ver<lb/>&longs;o l'vna, & l'altra po&longs;&longs;anza in A, & in B pre&longs;e in&longs;ieme, è come AB ver&longs;o <lb/>BC, & CA in&longs;ieme, cioè ver&longs;o BA. </s> <s id="id.2.1.364.2.0">il pe&longs;o dunque E è eguale ad amen<lb/>due le po&longs;&longs;anze pre&longs;e in&longs;ieme. <emph.end type="italics"/></s> <s id="id.2.1.365.1.0">PROPOSITIONE III. </s> <s id="id.2.1.366.1.0">In altro modo ancora po&longs;&longs;iamo v&longs;are la Leua. </s> <pb xlink:href="037/01/090.jpg"/> <s id="id.2.1.368.1.0"><emph type="italics"/>Sia la leua AB, il cui &longs;o&longs;tegno &longs;ia B. </s> <s>& &longs;ia il pe&longs;o C appiccato al punto A, <lb/>& &longs;ia la po&longs;&longs;anza in D, comunque &longs;i voglia tra AB, &longs;o&longs;tenente il pe&longs;o C. </s> <s id="id.2.1.368.2.0">Di<lb/>co che come AB à BD, co&longs;i è la po&longs;&longs;anza in D al pe&longs;o C. </s> <s id="id.2.1.368.3.0">Appicchi&longs;i al <lb/>punto D il pe&longs;o E eguale à C; & come BD à BA, co&longs;i faccia&longs;i il pe&longs;o <lb/>E ad vn'altro pe&longs;o, come F: & per e&longs;&longs;ere i pe&longs;i CE tra loro eguali, &longs;arà an<lb/>co il pe&longs;o C al <lb/>pe&longs;o F, come <lb/>BD à BA. <lb/></s> <s id="id.2.1.368.4.0">Appicchi&longs;i &longs;imil<lb/>mente il pe&longs;o F <lb/>in D. </s> <s>& per<lb/>che il pe&longs;o E ad <lb/>F è come la gra<lb/>uezza del pe&longs;o <lb/>E alla grauez<lb/>za del pe&longs;o F; <lb/>& il pe&longs;o E al <emph.end type="italics"/><lb/><figure id="id.037.01.090.1.jpg" xlink:href="037/01/090/1.jpg"></figure><lb/><arrow.to.target n="note112"></arrow.to.target><emph type="italics"/>pe&longs;o F è come BD à BA. </s> <s id="id.2.1.368.5.0">Come dunque la grauezza del pe&longs;o E alla gra<lb/>uezza del pe&longs;o F, co&longs;i è BD à BA. </s> <s id="id.2.1.368.6.0">Ma come BD à BA, co&longs;i è la gra<lb/>uezza del pe&longs;o E alla grauezza del pe&longs;o C. </s> <s id="id.2.1.368.7.0">Per laqual co&longs;a la grauezza del <emph.end type="italics"/><lb/><arrow.to.target n="note113"></arrow.to.target><emph type="italics"/>pe&longs;o E alla grauezza del pe&longs;o F ha la proportione mede&longs;ima, che ha alla gra<lb/>uezza del pe&longs;o C. </s> <s id="id.2.1.368.8.0">i pe&longs;i dunque CF hanno la grauezza mede&longs;ma. </s> <s id="id.2.1.368.9.0">Sia dunque <emph.end type="italics"/><lb/><arrow.to.target n="note114"></arrow.to.target><emph type="italics"/>la po&longs;&longs;anza in D &longs;o&longs;tenente il pe&longs;o F, che verrà ad e&longs;&longs;ere la detta po&longs;&longs;anza in <lb/>D eguale al pe&longs;o F. </s> <s>& percioche il pe&longs;o F posto in D è graue egualmente <lb/>come il pe&longs;o C po&longs;to in A; haurà la po&longs;&longs;anza in D la proportione mede&longs;ima <lb/>ver&longs;o la grauezza del pe&longs;o F, che ha alla grauezza del pe&longs;o C. </s> <s id="id.2.1.368.10.0">Ma la po&longs;&longs;anza <lb/>in D &longs;o&longs;tiene il pe&longs;o F, dunque la po&longs;&longs;anza in D &longs;o&longs;tenterà anco il pe&longs;o C; & <lb/>il pe&longs;o C alla po&longs;&longs;anza in D &longs;arà co&longs;i come il pe&longs;o C al pe&longs;o F; & C ad F <emph.end type="italics"/><lb/><arrow.to.target n="note115"></arrow.to.target><emph type="italics"/>è come BD à BA, &longs;arà dunque il pe&longs;o C alla po&longs;&longs;anza in D, come BD à <lb/>BA: & conuertendo come AB à BD, co&longs;i la po&longs;&longs;anza in D al pe&longs;o C. </s> <s id="id.2.1.368.11.0">La <lb/>po&longs;&longs;anza dunque al pe&longs;o, è come la di&longs;tanza dal &longs;ostegno allo appiccamento del pe<lb/>&longs;o alla distanza dal &longs;o&longs;tegno alla po&longs;&longs;anza. </s> <s id="id.2.1.368.12.0">che bi&longs;ognaua mostrare. <emph.end type="italics"/></s> <s id="id.2.1.370.1.0"><margin.target id="note112"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo della bilancia. <emph.end type="italics"/></s> <s id="id.2.1.371.1.0"><margin.target id="note113"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo della bilancia. <emph.end type="italics"/></s> <s id="id.2.1.372.1.0"><margin.target id="note114"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.373.1.0"><margin.target id="note115"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.374.1.0">Altramente. </s> <s id="id.2.1.375.1.0"><emph type="italics"/>Sia la leua AB, il cui &longs;o&longs;tegno &longs;ia B. </s> <s>& dal punto A &longs;ia fatto pendente il pe&longs;o <lb/>C, & &longs;ia la po&longs;&longs;anza in D &longs;o&longs;tenente il pe&longs;o C. </s> <s id="id.2.1.375.2.0">Dico, che come AB à BD, <lb/>co&longs;i è la po&longs;&longs;anza in D al pe&longs;o C. </s> <s id="id.2.1.375.3.0">allunghi&longs;ila AB in E, & faccia&longs;i BE egua<lb/>le à BA, & al punto E &longs;ia appiccato il pe&longs;o F eguale al pe&longs;o C; & come BD à <lb/>BE co&longs;i faccia&longs;i il pe&longs;o F ad vn'altro pe&longs;o G, ilquale &longs;ia appiccato al punto D, <lb/>i pe&longs;i FG pe&longs;eranno egualmente. </s> <s id="id.2.1.375.4.0">& percioche AB è eguale à BE, & i pe&longs;i <emph.end type="italics"/><pb pagenum="38" xlink:href="037/01/091.jpg"/><emph type="italics"/>FC &longs;ono eguali, &longs;imilmente i pe&longs;i FC pe&longs;eranno egualmente, ma i pe&longs;i FGC ap<lb/>piccati nella leua EBA, il cui &longs;o&longs;tegno è in B non pe&longs;eranno egualmente; ma in<lb/>chineranno in giu&longs;o dalla parte di A. </s> <s id="id.2.1.469.2.0">Ponga&longs;i dunque in D tanta forza, che i <lb/>pe&longs;i FGC pe&longs;ino egualmente; &longs;arà la po&longs;&longs;anza in D eguale al pe&longs;o G; peroche <emph.end type="italics"/><lb/><figure id="id.037.01.091.1.jpg" xlink:href="037/01/091/1.jpg"></figure><lb/><emph type="italics"/>i pe&longs;i FG pe&longs;ano egualmente, & la po&longs;&longs;anza in D niente altro deue fare, che <lb/>&longs;o&longs;tenere il pe&longs;o G che non di&longs;cenda. </s> <s id="id.2.1.469.3.0">& percioche i pe&longs;i FGC, & la po&longs;&longs;anza <lb/>in D pe&longs;ano egualmente, leuati via dunque i pe&longs;i FG, i quali pe&longs;ano egualmente, <lb/>i re&longs;tanti pe&longs;eranno egualmente, cioè la po&longs;&longs;anza in D co'l pe&longs;o C, cioè la po&longs;&longs;an<lb/>za in D &longs;o&longs;terrà il pe&longs;o C, talche la leua AB stia come prima. </s> <s id="id.2.1.469.4.0">& per e&longs;&longs;ere la <lb/>po&longs;&longs;anza in D eguale al pe&longs;o G, & il pe&longs;o C eguale al pe&longs;o, hauerà la po&longs;&longs;an<lb/>za posta in D la proportione mede&longs;ima al pe&longs;o C, che EB, cioè AB à BD. </s> <s><lb/>che bi&longs;ognaua mostrare. <emph.end type="italics"/></s> <s id="id.2.1.471.1.0">COROLLARIO I. </s> <s id="id.2.1.472.1.0">Da que&longs;to è chiaro ancora, come prima, che &longs;e &longs;arà po&longs;to il pe<lb/>&longs;o più vicino al &longs;o&longs;tegno B, come in H, il pe&longs;o douer&longs;i &longs;o<lb/>&longs;tenere da forza minore. </s> <s id="id.2.1.473.1.0"><emph type="italics"/>Percioche HB ha proportione minore à BD, che AB à BD. </s> <s>& quanto più <emph.end type="italics"/><arrow.to.target n="note136"></arrow.to.target><lb/><emph type="italics"/>da vicino &longs;arà al &longs;o&longs;tegno, &longs;empre anco minore forza vi &longs;i ricercherà. <emph.end type="italics"/></s> <s id="id.2.1.474.1.0"><margin.target id="note136"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.475.1.0">COROLLARIO II. </s> <s id="id.2.1.476.1.0">Egli è parimente manife&longs;to, che la po&longs;&longs;anza in D è &longs;empre <lb/>maggiore del pe&longs;o C. </s> <s id="id.2.1.477.1.0"><emph type="italics"/>Perche &longs;e tra AB &longs;i piglia qual &longs;i voglia punto, come D, &longs;empre AB &longs;arà mag<lb/>giore di BD. <emph.end type="italics"/></s> <s id="id.2.1.478.1.0"><emph type="italics"/>El è da auertire, che que&longs;te dimo&longs;trationi lequali habbiamo prodotte in mezo, &longs;i po&longs;&longs;o<lb/>no à tutte que&longs;te co&longs;e commodamente adattare non &longs;olamente e&longs;&longs;endo le leue egual<lb/>mente distanti dall'orizonte, ma anche inchinate le dette leue all'orizonte. </s> <s id="id.2.1.478.2.0">ilche è <lb/>chiaro da quel che nella bilancia &longs;i è diui&longs;ato. <emph.end type="italics"/></s> <pb xlink:href="037/01/092.jpg"/> <s id="id.2.1.480.1.0">PROPOSITIONE IIII. </s> <s id="id.2.1.481.1.0">Se la po&longs;&longs;anza mouerà il pe&longs;o appiccato nella leua, &longs;arà lo &longs;patio <lb/>della po&longs;&longs;anza mo&longs;&longs;a allo &longs;patio del pe&longs;o mo&longs;&longs;o, come la di&longs;tan<lb/>za dal &longs;o&longs;tegno alla po&longs;&longs;anza, alla di&longs;tanza dall'i&longs;te&longs;&longs;o &longs;o&longs;tegno <lb/>fin allo appiccamento del pe&longs;o. </s> <s id="id.2.1.482.1.0"><emph type="italics"/>Sia la leua AB, il cui &longs;o&longs;tegno C, & &longs;ia il pe&longs;o D attaccato al punto B, & &longs;ia <lb/>la po&longs;&longs;anza in A mouente il pe&longs;o D con la leua AB. </s> <s id="id.2.1.482.2.0">Dico lo &longs;patio della po&longs;<lb/>&longs;anza in A allo &longs;patio del pe&longs;o e&longs;&longs;ere co&longs;i come CA à CB. </s> <s id="id.2.1.482.3.0">Moua&longs;i la leua <lb/>AB, & affine che il pe&longs;o D &longs;i moua in sù, bi&longs;ogna che B &longs;i moua in sù, & A in <lb/>giù. </s> <s id="id.2.1.482.4.0">& percioche C è punto immobile; però mentre A, & B &longs;i mouono, de<lb/>&longs;criueranno circonferenze di cerchi. </s> <s id="id.2.1.482.5.0">Moua&longs;i dunque AB in EF; &longs;aranno AEBF <lb/>circonferenze di cerchi, i me<lb/>zi diametri de' quali &longs;ono CA <lb/>CB. </s> <s id="id.2.1.482.6.0">compi&longs;ca&longs;i tutta la cir<lb/>conferenza AGE, & tut<lb/>ta la BHF, & &longs;iano KH <lb/>i punti doue AB, & EF ta<lb/>gliano il cerchio BHF. </s> <s id="id.2.1.482.7.0">Hor <lb/>percioche l'angolo BCF è <emph.end type="italics"/><lb/><arrow.to.target n="note137"></arrow.to.target><emph type="italics"/>eguale all'angolo HCK, &longs;a<lb/>rà la circonferenza KH egua<emph.end type="italics"/><lb/><arrow.to.target n="note138"></arrow.to.target><emph type="italics"/>le alla circonferenza BF, & <lb/>concio&longs;ia, che le circonferen<lb/>ze AEKH &longs;iano &longs;otto l'i<lb/>&longs;te&longs;&longs;o angolo ACE, & la <lb/>circonferenza AE à tutta <lb/>la circonferenza AGE &longs;ia <lb/>come l'angolo ACE à quat<lb/>tro retti, & come l'i&longs;te&longs;&longs;o an<lb/>golo HCK à quattro retti, <lb/>co&longs;i anche è la circonferenza <lb/>HK à tutta la circonferentia <lb/>HBK, &longs;arà la circonferentia <emph.end type="italics"/><lb/><figure id="id.037.01.092.1.jpg" xlink:href="037/01/092/1.jpg"></figure><lb/><emph type="italics"/>AE à tutta la circonferentia AGE, come la circonferentia KH à tutta la <emph.end type="italics"/><lb/><arrow.to.target n="note139"></arrow.to.target><emph type="italics"/>KFH. </s> <s>& permutando come la circonferentia AE alla circonferenza KH, cioè <lb/>BF, co&longs;i tutta la circonferenza AGE à tutta la circonferenza BHF; ma tut<emph.end type="italics"/><lb/><arrow.to.target n="note140"></arrow.to.target><emph type="italics"/>ta la circonferenza AGE co&longs;i &longs;i ha à tutta la BHF, come il diametro del cer<lb/>chio AEG al diametro del cerchio BHF. </s> <s id="id.2.1.482.8.0">Come dunque la circonferenza AE <emph.end type="italics"/><pb pagenum="39" xlink:href="037/01/093.jpg"/><emph type="italics"/>ver&longs;o la circonferenza BF, co&longs;i è il diametro del cerchio AGE al diametro del <emph.end type="italics"/><arrow.to.target n="note141"></arrow.to.target><lb/><emph type="italics"/>cerchio BHF: ma come il diametro al diametro, co&longs;i è anche il mezo diametro al <lb/>mezo diametro, cioè CA à CB. </s> <s id="id.2.1.482.9.0">Per laqual co&longs;a come la circonferenza AE <lb/>alla circonferenza BF, co&longs;i CA à CB: ma la circonferenza AE è lo <lb/>&longs;patio della po&longs;&longs;anza mo&longs;&longs;a, & la circonferenza BF è eguale allo &longs;patio di D pe<lb/>&longs;o mo&longs;&longs;o, peroche lo &longs;patio del mouimento del pe&longs;o D &longs;empre è eguale allo &longs;patio <lb/>del mouimento del punto B, per e&longs;&longs;ere attaccato in B. </s> <s id="id.2.1.482.10.0">Lo &longs;patio dunque della po&longs;<lb/>&longs;anza mo&longs;&longs;a allo &longs;patio del pe&longs;o mo&longs;&longs;o è come CA à CB; cioè come la di&longs;tan<lb/>za dal &longs;o&longs;tegno alla po&longs;&longs;anza, alla distanza dall'i&longs;te&longs;&longs;o all'appiccamento del pe&longs;o. <lb/></s> <s id="id.2.1.482.11.0">che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.484.1.0"><margin.target id="note137"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.485.1.0"><margin.target id="note138"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 26. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <s id="id.2.1.486.1.0"><margin.target id="note139"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del<emph.end type="italics"/> 15. </s> <s id="id.2.1.487.1.0"><margin.target id="note140"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 23. <emph type="italics"/>del<emph.end type="italics"/> 8. <emph type="italics"/>di Pappo. <emph.end type="italics"/></s> <s id="id.2.1.488.1.0"><margin.target id="note141"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.489.1.0"><emph type="italics"/>Ma &longs;ia la leua AB, il cui &longs;o&longs;tegno B, & la po&longs;&longs;anza mouente in A, & il pe&longs;o <lb/>in C. </s> <s id="id.2.1.489.2.0">Dico lo &longs;patio della po&longs;&longs;anza mo&longs;&longs;a allo &longs;patio del pe&longs;o tra&longs;portato co&longs;i e&longs;<lb/>&longs;ere, come BA à BC. <lb/></s> <s id="id.2.1.489.3.0">Moua&longs;i la leua, & accioche <lb/>il pe&longs;o &longs;ia alzato in sù, egli <lb/>è nece&longs;&longs;ario, che anche i pun<lb/>ti CA &longs;i mouano in sù. <lb/></s> <s id="id.2.1.489.4.0">Moua&longs;i dunque A in sù <lb/>fin'in D; & &longs;ia il mouimen<lb/>to della leua BD. </s> <s id="id.2.1.489.5.0">mo&longs;tre<lb/>remo nel modo i&longs;te&longs;&longs;o, come <lb/>prima è detto, che i punti <lb/>CA de&longs;criuono circonferen<lb/>ze di cerchi, i cui mezi dia<lb/>metri &longs;ono BA BC. </s> <s>& di<lb/>mo&longs;treremo &longs;imilmente co&longs;i <lb/>e&longs;&longs;ere AD à CE, come il <lb/>mezo diametro AB al me<lb/>zo diametro BC. <emph.end type="italics"/></s> <s id="id.2.1.490.1.0"><emph type="italics"/>Et per la ragione i&longs;te&longs;&longs;a, &longs;e la <lb/>po&longs;&longs;anza fo&longs;&longs;e in C, & il <lb/>pe&longs;o in A &longs;i prouerà co&longs;i <lb/>e&longs;&longs;ere CE ver&longs;o AD, co<emph.end type="italics"/><lb/><figure id="id.037.01.093.1.jpg" xlink:href="037/01/093/1.jpg"></figure><lb/><emph type="italics"/>me BC à BA, cioè la di&longs;tanza dal &longs;o&longs;tegno alla po&longs;&longs;anza; alla di&longs;tanza dal<lb/>l'iste&longs;&longs;o allo attaccamento del pe&longs;o. </s> <s id="id.2.1.490.2.0">che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.492.1.0">COROLLARIO. </s> <s id="id.2.1.493.1.0">Da que&longs;te co&longs;e è manife&longs;to, che maggiore proportione ha lo &longs;pa<lb/>tio della po&longs;&longs;anza, che moue allo &longs;patio del pe&longs;o mo&longs;&longs;o, che il <lb/>pe&longs;o alla mede&longs;ima po&longs;&longs;anza. </s> <pb xlink:href="037/01/094.jpg"/> <s id="id.2.1.494.1.0"><emph type="italics"/>Percioche lo &longs;patio della po&longs;&longs;anza allo &longs;patio del pe&longs;o ha la mede&longs;ima proportione, <lb/>che il pe&longs;o alla po&longs;&longs;anza, che &longs;o&longs;tiene il detto pe&longs;o. </s> <s id="id.2.1.494.2.0">Ma la po&longs;&longs;anza, che &longs;ostie<lb/>ne è minore della po&longs;&longs;anza che moue, però il pe&longs;o haurà proportione minore alla <lb/>po&longs;&longs;anza che lo moue, che alla po&longs;&longs;anza, che lo &longs;ostiene. </s> <s id="id.2.1.494.3.0">Lo &longs;patio dunque della <emph.end type="italics"/><lb/><arrow.to.target n="note142"></arrow.to.target><emph type="italics"/>po&longs;&longs;anza che moue allo &longs;patio del pe&longs;o haurà proportione maggiore, che il pe&longs;o al<lb/>l'i&longs;te&longs;&longs;a po&longs;&longs;anza. <emph.end type="italics"/></s> <s id="id.2.1.495.1.0"><margin.target id="note142"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.496.1.0">PROPOSITIONE V. </s> <s id="id.2.1.497.1.0">La po&longs;&longs;anza che in qual &longs;i voglia modo &longs;o&longs;tenga il pe&longs;o con la le<lb/>ua hauerà la proportione mede&longs;ima ad e&longs;&longs;o pe&longs;o, che la di&longs;tan<lb/>za fra po&longs;ta dal &longs;o&longs;tegno al punto, doue dal centro della gra<lb/>uezza del pe&longs;o tirata vna linea à piombo all'orizonte tagli la <lb/>leua, alla di&longs;tanza che è fra il &longs;o&longs;tegno, & la po&longs;&longs;anza. </s> <s id="id.2.1.498.1.0"><emph type="italics"/>Sia la leua AB egualmente di&longs;tante dall'orizonte, col &longs;uo &longs;o&longs;tegno N. </s> <s id="id.2.1.498.2.0">&longs;ia dopo il pe<lb/>&longs;o AC, il cui centro della grauezza &longs;ia D, ilquale &longs;ia prima &longs;otto la leua: ma <lb/>il pe&longs;o &longs;ia appiccato à i punti AO. </s> <s>& dal punto D &longs;ia tirata la linea DE à <lb/>piombo dell' orizonte, & di AB. </s> <s id="id.2.1.498.3.0">Che &longs;e vi &longs;aranno altre leue ancora AF AG, <lb/>i cui &longs;o<lb/>stegni, <lb/>&longs;iano H <lb/>K, & il <lb/>pe&longs;o A <lb/>C &longs;ia ap<lb/>piccato <lb/>nella le<lb/>ua AG <lb/>ne i pun<lb/>ti AQ, <lb/>& nella <lb/>leua A <lb/>F ne' <expan abbr="p&utilde;ti">pun<lb/>ti</expan> AP: <lb/>& la li<lb/>nea DE <lb/>allunga<emph.end type="italics"/><lb/><figure id="id.037.01.094.1.jpg" xlink:href="037/01/094/1.jpg"></figure><lb/><emph type="italics"/>ta tagli AF in L, & AC in M. </s> <s id="id.2.1.498.4.0">Dico che la po&longs;&longs;anza in F &longs;o&longs;tenente il pe&longs;o AC <lb/>ha quella proportione ad e&longs;&longs;o pe&longs;o, che ha KL à KF; & la po&longs;&longs;anza in D ha quella <lb/>proportione al pe&longs;o, che ha NE ad NB; & la po&longs;&longs;anza in G al pe&longs;o quella, che ha <lb/>HM ad HG. </s> <s id="id.2.1.498.5.0">Hor percioche DL &longs;tà à piombo dell' orizonte, il pe&longs;o AC venga ap<emph.end type="italics"/><pb pagenum="40" xlink:href="037/01/095.jpg"/><emph type="italics"/>piccato doue &longs;i voglia nella linea DL, rimarrà nel modo i&longs;te&longs;&longs;o che &longs;i troua. </s> <s id="id.2.1.498.6.0">Per la<lb/>qual co&longs;a &longs;e nella leua AB &longs;i &longs;cioglieranno gli appiccamenti, che &longs;ono ad AO, il <lb/>pe&longs;o AC appiccato in E rimarrà nell'i&longs;te&longs;&longs;o modo, come hora rimane, cioè leuato via <lb/>il punto A, & la linea QO, nell'i&longs;te&longs;&longs;o modo il pe&longs;o appiccato in E rimarrà, come <lb/>era &longs;o&longs;tenuto da punti i&longs;te&longs;&longs;i AO, come &longs;i proua per lo commentario di Federico <lb/>Commandino nella &longs;esta propo&longs;itione di Archimede della quadratura della parabo<lb/>la, & dalla prima di que&longs;to della bilancia. </s> <s id="id.2.1.498.7.0">Co&longs;i percio che il pe&longs;o AC ha &longs;empre la <lb/>i&longs;te&longs;&longs;a di&longs;po&longs;itione ver&longs;o la bilancia, &longs;ia pur in AO &longs;ostentato, ouero pendente dal <lb/>punto E; la po&longs;&longs;anza mede&longs;ima in B &longs;o&longs;tenterà il pe&longs;o i&longs;te&longs;&longs;o AC pendente, ouero <lb/>da E, ouero da AO. </s> <s>ma la po&longs;&longs;anza in B &longs;o&longs;tenente il pe&longs;o AC appiccato in E co&longs;i <lb/>&longs;i hà ad e&longs;&longs;o pe&longs;o, come NE ad NB; La po&longs;&longs;anza dunque in B &longs;o&longs;tenente il pe&longs;o <lb/>AC da punti AO pendente &longs;arà co&longs;i ad e&longs;&longs;o pe&longs;o, come NE ad NB. </s> <s id="id.2.1.498.8.0">Non altra<emph.end type="italics"/><arrow.to.target n="note143"></arrow.to.target><lb/><emph type="italics"/>mente &longs;i mo&longs;trerà, che il pe&longs;o AC pendente dal punto L rimane, come &longs;e fo&longs;&longs;e &longs;oste <lb/>nuto da punti AP; & la po&longs;&longs;anza in F ad e&longs;&longs;o pe&longs;o e&longs;&longs;ere co&longs;i come<emph.end type="italics"/> KL <emph type="italics"/>à KF. </s> <s id="id.2.1.498.9.0">Ma <lb/>nella leua AG il pe&longs;o AC appiccato in M co&longs;i rimanere, come egli è &longs;o&longs;tenuto da <lb/>punti AQ & la po&longs;&longs;anza di G co&longs;i e&longs;&longs;ere al pe&longs;o AC, come HM ad HG, cioè co<lb/>me la di&longs;tanza dal &longs;o&longs;tegno al punto, doue la linea tirata à piombo dell' orizonte <lb/>dal centro della grauezza del pe&longs;o taglia la leua, alla di&longs;tanza dal &longs;o&longs;tegno alla po&longs;<lb/>&longs;anza. </s> <s id="id.2.1.498.10.0">che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.500.1.0"><margin.target id="note143"></margin.target><emph type="italics"/>Per la prima di questo. <emph.end type="italics"/></s> <s id="id.2.1.501.1.0"><emph type="italics"/>Che &longs;e FBG fo&longs;&longs;ero i &longs;o&longs;tegni delle leue, & le po&longs;&longs;anze fo&longs;&longs;ero in KNH &longs;o&longs;tenenti il pe<lb/>&longs;o, con &longs;imile modo &longs;i mo&longs;trerà la po&longs;&longs;anza in H, co&longs;i e&longs;&longs;ere al pe&longs;o, come GM à GH, <lb/>et la <expan abbr="po&longs;s&amacr;za">po&longs;sanza</expan> <expan abbr="&imacr;">in</expan> N al pe&longs;o, come BE à BN, et la <expan abbr="po&longs;s&amacr;za">po&longs;sanza</expan> <expan abbr="&itilde;">in</expan> K al pe&longs;o come FL ad FK. <emph.end type="italics"/></s> <s id="id.2.1.502.1.0"><emph type="italics"/>Et &longs;e le leue AB AF AG haue&longs;&longs;ero i &longs;o&longs;tegni in A, & il pe&longs;o fo&longs;&longs;e NO; poi dal <lb/>centro D del<lb/>la &longs;ua gra<lb/>uezza fo&longs;&longs;e <lb/>tirata la li<lb/>nea DME <lb/>L à piombo <lb/>di AB, & <lb/>dell' orizon<lb/>te, & fo&longs;&longs;e<lb/>ro le po&longs;&longs;an<lb/>ze in FB <lb/>G; <expan abbr="&longs;imilm&etilde;te">&longs;imilmen<lb/>te</expan> mo&longs;tre<lb/>ra&longs;&longs;i la po&longs;<lb/>&longs;anza di G <lb/>&longs;o&longs;tenente <lb/>il pe&longs;o N<emph.end type="italics"/><lb/><figure id="id.037.01.095.1.jpg" xlink:href="037/01/095/1.jpg"></figure><lb/><emph type="italics"/>O co&longs;i e&longs;&longs;ere ad e&longs;&longs;o pe&longs;o, come AM ad AG, & la po&longs;&longs;anza in B come AE ad <lb/>AB; & la po&longs;&longs;anza in F come AL ad AF. <emph.end type="italics"/></s> <pb xlink:href="037/01/096.jpg"/> <s id="id.2.1.504.1.0"><emph type="italics"/>Sia dapoi la leua AB egualmente di&longs;tante dall'orizonte, il cui &longs;o&longs;tegno &longs;ia D, & &longs;ia <lb/>BE il pe&longs;o, il cui centro della grauezza &longs;ia F &longs;opra la leua; & dal punto F tiri&longs;i la <lb/>linea FH à piombo, & dell' orizonte, & di e&longs;&longs;a AB; & &longs;ia &longs;o&longs;tenuto il pe&longs;o dal <lb/>punto B, & da PQ. </s> <s>&longs;iano po&longs;cia altre leue BLBM, i cui &longs;o&longs;tegni &longs;iano NO; <lb/>& la linea FH allungata tagli BM in K, & BL in G; & venga &longs;o&longs;tenuto il pe&longs;o <emph.end type="italics"/><lb/><figure id="id.037.01.096.1.jpg" xlink:href="037/01/096/1.jpg"></figure><lb/><emph type="italics"/>nella leua BL ne'punti BP; & nella leua BM dal punto B, & PR. </s> <s id="id.2.1.504.2.0">Dico, che <lb/>la po&longs;&longs;anza in L &longs;o&longs;tenente il pe&longs;o BE nella leua BL ha quella proportione ad <lb/>e&longs;&longs;o pe&longs;o, che NG ad NL; & la po&longs;&longs;anza in A al pe&longs;o ha quella proportio<lb/>ne, che DH à DA; & la po&longs;&longs;anza di M al pe&longs;o ha quella proportione, che <lb/>OK ad OM. </s> <s id="id.2.1.504.3.0">Hor percioche la linea KF tirata dal centro della grauezza F è <emph.end type="italics"/><lb/><arrow.to.target n="note144"></arrow.to.target><emph type="italics"/>à piombo dell' orizonte, &longs;ia pur &longs;ostenuto il pe&longs;o da qual &longs;i voglia punto della linea <lb/>KF, egli rimarrà, come hora &longs;i troua. </s> <s id="id.2.1.504.4.0">Se dunque &longs;arà &longs;ostenuto in H, rimarrà co <lb/>me prima, cioè leuato via il punto B, & PQ, i quali &longs;o&longs;tengono il pe&longs;o, rimarrà <lb/>il pe&longs;o BE nel modo che da e&longs;&longs;i era &longs;o&longs;tenuto. </s> <s id="id.2.1.504.5.0">Per la qual co&longs;a grauerà nella le<lb/>ua AB in H, & haurà alla leua quella di&longs;po&longs;itione mede&longs;ima, che prima, & per<lb/>ciò &longs;arà come &longs;e fo&longs;&longs;e appiccato in H. </s> <s id="id.2.1.504.6.0">La mede&longs;ma po&longs;&longs;anza dunque &longs;o&longs;terrà il me<lb/>de&longs;imo pe&longs;o BE &longs;o&longs;tentato ouero in H, ouero in B & Q. </s> <s>Ma la po&longs;&longs;anza in A <emph.end type="italics"/><lb/><arrow.to.target n="note145"></arrow.to.target><emph type="italics"/>&longs;o&longs;tenente il pe&longs;o BE appiccato in H con la leua AB ha l'i&longs;te&longs;&longs;a proportione ad e&longs;<lb/>&longs;o pe&longs;o, che DH à DA; l'i&longs;te&longs;&longs;a po&longs;&longs;anza dunque in A &longs;o&longs;tenente il pe&longs;o BE ne' <lb/>punti BQ &longs;o&longs;tentato, &longs;arà ad e&longs;&longs;o pe&longs;o come DH à DA. </s> <s id="id.2.1.504.7.0">Similmente &longs;i mo&longs;trerà <lb/>il pe&longs;o BE, &longs;e in G &longs;arà &longs;o&longs;tenuto, rimanere come egli era &longs;o&longs;tenuto da punti BP: <lb/>& nel punto K, come da punti BR. </s> <s id="id.2.1.504.8.0">Per la qual co&longs;a la po&longs;&longs;anza in L &longs;o&longs;tenente <emph.end type="italics"/><pb pagenum="41" xlink:href="037/01/097.jpg"/><emph type="italics"/>il pe&longs;o BE ad e&longs;&longs;o pe&longs;o co&longs;i &longs;arà come NG ad NL. </s> <s>ma la po&longs;&longs;anza in M al pe<lb/>&longs;o, come OK ad OM; cioè come la di&longs;tanza dal &longs;o&longs;tegno al punto, doue dal cen<lb/>tro della grauezza del pe&longs;o la linea tirata à piombo dell' orizonte taglia la leua, alla <lb/>di&longs;tanza dal &longs;o&longs;tegno alla po&longs;&longs;anza. </s> <s id="id.2.1.504.9.0">che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.506.1.0"><margin.target id="note144"></margin.target><emph type="italics"/>Per la prima di questo della bilancia. <emph.end type="italics"/></s> <s id="id.2.1.507.1.0"><margin.target id="note145"></margin.target><emph type="italics"/>Per la prima di questo. <emph.end type="italics"/></s> <s id="id.2.1.508.1.0"><emph type="italics"/>Che &longs;e LAM fo&longs;&longs;ero i &longs;o&longs;tegni, & le po&longs;&longs;anze in NDO; &longs;imilmente mo&longs;trera&longs;&longs;i <lb/>la po&longs;&longs;anza in N co&longs;i e&longs;&longs;ere al pe&longs;o, come LG ad LN; & la po&longs;&longs;anza in D, <lb/>come AH ad AD, & la po&longs;&longs;anza in O come MK ad MO. <emph.end type="italics"/></s> <s id="id.2.1.509.1.0"><emph type="italics"/>Et &longs;e le leue BA BL BM haue&longs;&longs;ero i &longs;o&longs;tegni in B, & il pe&longs;o fo&longs;&longs;e NO &longs;opra <lb/>la leua, & dal centro F della grauezza fo&longs;&longs;e tirata la linea FD EG à piombo <lb/>di AB, & dell' orizonte; & fo&longs;&longs;ero le po&longs;&longs;anze in LAM, &longs;imilmente proue<emph.end type="italics"/><lb/><figure id="id.037.01.097.1.jpg" xlink:href="037/01/097/1.jpg"></figure><lb/><emph type="italics"/>ra&longs;&longs;i la po&longs;&longs;anza in L &longs;o&longs;tenente il pe&longs;o co&longs;i e&longs;&longs;ere ad e&longs;&longs;o pe&longs;o, come BD à BL; <lb/>& la po&longs;&longs;anza in A al pe&longs;o come BE à BA, & la po&longs;&longs;anza in M come BG <lb/>à BM. <emph.end type="italics"/></s> <pb xlink:href="037/01/098.jpg"/> <s id="id.2.1.512.1.0"><emph type="italics"/>Sia vltimamente la leua AB egualmente di&longs;tante dall'orizonte, il cui &longs;ostegno &longs;ia <lb/>C, & il pe&longs;o DE habbia il centro della graueza F nella leua AB; & &longs;iano <lb/>alla fine altre leue GHKL, co i &longs;o&longs;tegni &longs;uoi MN; & il pe&longs;o nella leua GH <lb/>&longs;ia &longs;o&longs;tentato da i punti GO, & nella leua AB da punti AT, & nella leua <lb/>KL da punti KQ, & il centro F della grauezza &longs;ia parimente in amendue le le<emph.end type="italics"/><lb/><figure id="id.037.01.098.1.jpg" xlink:href="037/01/098/1.jpg"></figure><lb/><emph type="italics"/>ue GH<emph.end type="italics"/> K<emph type="italics"/>L, & &longs;iano le po&longs;&longs;anze in HBL. </s> <s id="id.2.1.512.3.0">Dico la po&longs;&longs;anza in H co&longs;i e&longs;&longs;ere al <lb/>pe&longs;o, come N<emph.end type="italics"/>F <emph type="italics"/>ad NH; & la po&longs;&longs;anza in B al pe&longs;o, come C<emph.end type="italics"/>F <emph type="italics"/>à CB, & la po&longs;<lb/>&longs;anza in L al pe&longs;o, come M<emph.end type="italics"/>F <emph type="italics"/>ad ML. </s> <s id="id.2.1.512.4.0">Hor percioche F è il centro della grauez<lb/>za del pe&longs;o DE, &longs;e dunque in<emph.end type="italics"/> F <emph type="italics"/>&longs;arà &longs;o&longs;tenuto, &longs;tarà il pe&longs;o DE come prima, per <lb/>la diffinitione del centro della grauezza; & &longs;arà come &longs;e egli fo&longs;&longs;e appiccato in<emph.end type="italics"/> F; <lb/><emph type="italics"/>& &longs;tarà nella leua in quel modo i&longs;te&longs;&longs;o, &longs;o&longs;tenga&longs;i pure ò da punti AP, ouero dal <lb/>punto<emph.end type="italics"/> F. </s> <s><emph type="italics"/>ilche parimente auerrà nelle leue GH KL, cioè che il pe&longs;o re&longs;terà nel mo<lb/>do i&longs;te&longs;&longs;o, &longs;o&longs;tenti&longs;i pur ò in<emph.end type="italics"/> F, <emph type="italics"/>ouero in GO ouero in KQ. </s> <s>La mede&longs;ma po&longs;&longs;anza <lb/>dunque in B &longs;o&longs;tenterà il pe&longs;o i&longs;te&longs;&longs;o DE appiccato, ouero in<emph.end type="italics"/> F, <emph type="italics"/>ouero in AP: & <lb/>quando egli è appiccato in<emph.end type="italics"/> F, <emph type="italics"/>è ad e&longs;&longs;o pe&longs;o come CF à CB, dunque la po&longs;&longs;anza &longs;o<lb/>&longs;tenente il pe&longs;o DE appiccato ad AP &longs;arà ad e&longs;&longs;o pe&longs;o come C<emph.end type="italics"/>F <emph type="italics"/>à CB. </s> <s>& nel mo<lb/>do i&longs;te&longs;&longs;o la po&longs;&longs;anza in H &longs;arà al pe&longs;o appiccato in OG co&longs;i, come N<emph.end type="italics"/>F <emph type="italics"/>ad NH. </s> <s>& <lb/>la po&longs;&longs;anza in L &longs;arà al pe&longs;o appiccato in KQ, come M<emph.end type="italics"/>F <emph type="italics"/>ad ML. </s> <s>ilche anco bi&longs;o<lb/>gnaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.514.1.0"><emph type="italics"/>Ma &longs;e li &longs;o&longs;tegni fo&longs;&longs;ero HBL, & le po&longs;&longs;anze fo&longs;&longs;ero in NCM; &longs;imilmente prouera&longs;&longs;i <lb/>la po&longs;&longs;anza in N co&longs;i e&longs;&longs;ere al pe&longs;o, come HF ad HN & la po&longs;&longs;anza in C come <lb/>BF à BC; & la po&longs;&longs;anza in M come LF ad LM. <emph.end type="italics"/></s> <pb pagenum="42" xlink:href="037/01/099.jpg"/> <s id="id.2.1.516.1.0"><emph type="italics"/>Et &longs;e le leue BA BC BD haue&longs;&longs;ero i &longs;o&longs;tegni in B, & fo&longs;&longs;ero i pe&longs;i in EF GH <lb/>KL, di modo che i loro centri della grauezza MNO fo&longs;&longs;ero nelle leue, & le <emph.end type="italics"/><lb/><figure id="id.037.01.099.1.jpg" xlink:href="037/01/099/1.jpg"></figure><lb/><emph type="italics"/>po&longs;&longs;anze fo&longs;&longs;ero in CAD. </s> <s id="id.2.1.516.2.0">Similmente prouera&longs;&longs;i, che la po&longs;&longs;anza in C co&longs;i è <lb/>al pe&longs;o EF, come BM à BC, & la po&longs;&longs;anza in A al pe&longs;o GH, come <lb/>BN à BA, & la po&longs;&longs;anza in D al pe&longs;o KL, come BO à BD. <emph.end type="italics"/></s> <s id="id.2.1.518.1.0">PROPOSITIONE VI. </s> <s id="id.2.1.519.1.0">Sia AB linea retta, ad angoli retti, dellaquale &longs;tia AD, la<lb/>quale dalla parte di D &longs;ia allungata come &longs;i vuole fin'al C, <lb/>& &longs;ia congiunta la CB, laquale parimente allunghi&longs;i dalla <lb/>parte di B fin ad E. </s> <s id="id.2.1.519.2.0">Dapoi &longs;iano dal punto B tirate altre <lb/>linee, come &longs;i vuole BF BG eguali ad AB tra AB BE; <lb/>& da i punti FG &longs;iano tirate le linee FH GK à piombo <lb/>delle &longs;udette, lequali &longs;i facciano eguali fra loro, & ad e&longs;&longs;a A <lb/>D come &longs;e BA AD fo&longs;&longs;ero mo&longs;&longs;e in BF FH, & in BG <lb/>GH; & congiungan&longs;i CH CK, lequali taglino le linee BF <lb/>BG ne'punti MN. </s> <s id="id.2.1.519.3.0">Dico che BN è minore di BM, & <lb/>BM di e&longs;&longs;a BA. </s> <pb xlink:href="037/01/100.jpg"/> <s id="id.2.1.521.1.0"><emph type="italics"/>Congiungan&longs;i BD BH B<emph.end type="italics"/>K, <emph type="italics"/>& percioche due linee DA AB &longs;ono eguali à due <emph.end type="italics"/><lb/><arrow.to.target n="note146"></arrow.to.target><emph type="italics"/>HF FB, & l'angolo DAB retto è anco eguale al retto HFB; &longs;aranno i <lb/>re&longs;tanti angoli eguali à i re&longs;tanti angoli, & HB eguale ad e&longs;&longs;a DB. </s> <s id="id.2.1.521.2.0">Similmen<lb/>te mo&longs;trera&longs;&longs;i il triangolo BKG e&longs;&longs;ere eguale al triangolo BHF. </s> <s id="id.2.1.521.3.0">Per laqual co<lb/>&longs;a co'l centro B, & con l'in<lb/>teruallo di vna di e&longs;&longs;e de&longs;cri<lb/>ua&longs;i il cerchio DH<emph.end type="italics"/> K<emph type="italics"/>E, il <lb/>quale tagli le linee CH CK <lb/>ne' punti OP; & congiun<lb/>gan&longs;i OB PB. </s> <s id="id.2.1.521.4.0">Percioche <lb/>dunque il punto K è più vi<lb/>cino ad E, che H; &longs;arà la <emph.end type="italics"/><lb/><arrow.to.target n="note147"></arrow.to.target><emph type="italics"/>linea CK maggiore di CH, <lb/>& CP minore di CO: dun<lb/>que PK &longs;arà maggiore di <lb/>OH. </s> <s id="id.2.1.521.5.0">Ma perche il triangolo <lb/>BKP di due lati eguali ha i <lb/>&longs;uoi lati BK BP eguali à i <lb/>lati BH BO del triangolo <lb/>BHO di due lati eguali, ma <lb/>ben la ba&longs;e KP maggiore <lb/>della ba&longs;e HO, &longs;arà l'ango<lb/>lo KBP maggiore dell' an<emph.end type="italics"/><lb/><arrow.to.target n="note148"></arrow.to.target><emph type="italics"/>golo HBO. </s> <s>dunque i restan<lb/>ti angoli alla ba&longs;e, cioè KPB <lb/>PKB pre&longs;i in&longs;ieme, i quali <lb/>tra loro &longs;ono eguali, &longs;aranno <lb/>minori de i re&longs;tanti angoli al<lb/>la ba&longs;e po&longs;ti, cioè OHB <lb/>HOB, iquali etiandio tra lo<lb/>ro &longs;ono eguali e&longs;&longs;endo che tut<lb/>ti gli angoli di cia&longs;cuno trian<emph.end type="italics"/><lb/><arrow.to.target n="note149"></arrow.to.target><emph type="italics"/>golo &longs;iano eguali à due angoli <lb/>retti. </s> <s id="id.2.1.521.6.0">Per laqual co&longs;a anche <lb/>le metà di que&longs;ti, cioè NKB <lb/>&longs;arà minore di MHB. </s> <s id="id.2.1.521.7.0">Et <emph.end type="italics"/><lb/><arrow.to.target n="note150"></arrow.to.target><emph type="italics"/>concio&longs;ia, che l'angolo BKG<emph.end type="italics"/><lb/><figure id="id.037.01.100.1.jpg" xlink:href="037/01/100/1.jpg"></figure><lb/><emph type="italics"/>&longs;ia eguale all'angolo BHF, &longs;arà NKG maggiore di MHF. </s> <s id="id.2.1.521.8.0">Se dunque nel <lb/>punto K &longs;i faccia l'angolo GKQ eguale ad FHM &longs;i &longs;arà il triangolo GKQ <lb/>eguale al triangolo FHM; Imperoche due angoli in FH di vno &longs;ono eguali à <lb/>due in GK d'vn'altro, & il lato FH è eguale al lato GK, &longs;arà GQ eguale <lb/>ad FM. </s> <s id="id.2.1.521.9.0">Adunque GN &longs;arà maggiore di FM. </s> <s>& co&longs;i per e&longs;&longs;ere BG egua<emph.end type="italics"/><pb pagenum="43" xlink:href="037/01/101.jpg"/><emph type="italics"/>le à BF, &longs;arà BN minore di e&longs;&longs;a BM. </s> <s>ma che BM &longs;ia minore di e&longs;&longs;a BA <lb/>è manife&longs;to, percioche BM, è minore di e&longs;&longs;a BF, laquale è eguale à BA. </s> <s>che <lb/>bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.523.1.0"><margin.target id="note146"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.524.1.0"><margin.target id="note147"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <s id="id.2.1.525.1.0"><margin.target id="note148"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 25. <emph type="italics"/>del<emph.end type="italics"/> 5. </s> <s id="id.2.1.526.1.0"><margin.target id="note149"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.527.1.0"><margin.target id="note150"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 26. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.528.1.0"><emph type="italics"/>Di più &longs;e tra BG BE &longs;i tiri à piacere vn'altra linea eguale à BG; & faccia&longs;i l'ope<lb/>ratione, come di &longs;opra è stato detto, prouera&longs;&longs;i &longs;imilmente la linea BR e&longs;&longs;er mi<lb/>nore di BN. </s> <s>& quanto più da vicino &longs;arà ad e&longs;&longs;a BE, &longs;arà anche &longs;empre minore. <emph.end type="italics"/></s> <s id="id.2.1.529.1.0">Che &longs;e i triangoli eguali BFH BGK fo&longs;&longs;ero di &longs;otto fra BC <lb/>BA collocati; & fo&longs;&longs;ero congiunte le linee HC KC, le<lb/>quali taglia&longs;&longs;ero le linee BF BG allungate dalla parte di FG <lb/>ne' punti MN, &longs;arà <lb/>la BN maggiore del<lb/>la BM, & la BM di <lb/>e&longs;&longs;a BA. </s> <s id="id.2.1.530.1.0"><emph type="italics"/>Imperoche allunghi&longs;i CH CK <lb/>fin alla circonferenza in OP, <lb/>& congiungan&longs;i BO BP; <lb/>con &longs;imile modo mo&longs;trera&longs;&longs;i <lb/>la linea PK e&longs;&longs;ere maggiore <lb/>ai OH, & l'angolo PKB e&longs;<lb/>&longs;ere minore dell <expan abbr="ãgolo">angolo</expan> OHB. <lb/></s> <s id="id.2.1.530.2.0">& percioche l'angolo BHF <lb/>è eguale dell' angolo BKG, &longs;a <lb/>rà tutto l'angolo PKG mi<lb/>nore dell' angolo OHF. </s> <s id="id.2.1.530.3.0">Per <lb/>laqual co&longs;a il re&longs;tante GKN <lb/>&longs;arà maggiore del re&longs;tante <lb/>FHM. </s> <s id="id.2.1.530.4.0">Se <expan abbr="d&utilde;que">dunque</expan> fara&longs;&longs;i l'an<lb/>golo GKQ eguale ad FHM <lb/>la linea KQ taglierà in modo <lb/>la GN, che GQ diuenterà <lb/>eguale ad FM. </s> <s id="id.2.1.530.5.0">Per laqual <lb/>co&longs;a maggiore &longs;arà GN, che <lb/>FM; allequali &longs;e &longs;aranno ag<lb/>giunte le eguali BF BG, &longs;a<lb/>rà BN maggiore di BM. </s> <s>& <lb/>per e&longs;&longs;ere BM maggiore di <lb/>FB, &longs;arà anco maggiore di <lb/>BA. </s> <s id="id.2.1.530.6.0">&longs;imilmente prouera&longs;&longs;i <lb/>che <expan abbr="quãto">quanto</expan> più da vicino &longs;arà <lb/>BG à BC, la linea BN &longs;em<lb/>pre &longs;arà maggiore. <emph.end type="italics"/></s> <figure id="id.037.01.101.1.jpg" xlink:href="037/01/101/1.jpg"> </figure> <pb xlink:href="037/01/102.jpg"/> <s id="id.2.1.532.1.0">PROPOSITIONE VII. </s> <s id="id.2.1.533.1.0">Sia la linea retta AB, à cui &longs;tia à piombo AD, laquale allun<lb/>ghi&longs;i dalla parte di D come pare &longs;in'à C, & congiunga&longs;i C <lb/>B, laquale etiandio &longs;i allunghi fin'ad E; & &longs;imilmente tra <lb/>AB BE &longs;iano, come pare, tirate BF BG eguali ad e&longs;&longs;a AB, <lb/>& da punti FG &longs;iano <lb/>tirate le linee FH GK <lb/>pur eguali ad e&longs;&longs;a AD, <lb/>& à piombo di BF BG, <lb/>come &longs;e BA AD fo&longs;<lb/>&longs;ero mo&longs;&longs;e in BF FH <lb/>BG GK: & congiun<lb/>gan&longs;i CH CK, lequali <lb/>taglino le linee allunga<lb/>te BF BG ne' punti <lb/>MN. </s> <s id="id.2.1.533.2.0">Dico che BN è <lb/>maggiore di BM, & <lb/>BM di e&longs;&longs;a BA. </s> <figure id="id.037.01.102.1.jpg" xlink:href="037/01/102/1.jpg"> </figure> <s id="id.2.1.535.1.0"><emph type="italics"/>Congiungan&longs;i BD BH BK, & <lb/>co'l centro B, & con lo &longs;patio <lb/>BD de&longs;criua&longs;i il cerchio. </s> <s id="id.2.1.535.2.0">&longs;imil<lb/>mente come nella precedente, di<lb/>mo&longs;treremo i punti KHDOP <lb/>e&longs;&longs;ere nella circonferenza del cer<lb/>chio; & i <expan abbr="tri&amacr;goli">triangoli</expan> ABD FBH <lb/>GBK e&longs;&longs;ere tra loro eguali, & <lb/>la linea PK e&longs;&longs;ere maggiore <lb/>della OH, & l'angolo PKB <lb/>e&longs;&longs;ere minore dell'angolo OHB. <lb/></s> <s id="id.2.1.535.3.0">Percioche <expan abbr="d&utilde;que">dunque</expan> l'angolo BHF <lb/>è eguale all'angolo BKG, &longs;arà <lb/>tutto l'angolo PKG minore <lb/>dell'angolo OHF. </s> <s id="id.2.1.535.4.0">Per laqual <lb/>co&longs;a il re&longs;tante G<emph.end type="italics"/>K<emph type="italics"/>N &longs;arà <lb/>maggiore del re&longs;tante FHM. <lb/></s> <s id="id.2.1.535.5.0">Se <expan abbr="d&utilde;que">dunque</expan> &longs;i &longs;arà l'angolo G<emph.end type="italics"/>K<emph type="italics"/>Q <emph.end type="italics"/><pb pagenum="44" xlink:href="037/01/103.jpg"/><emph type="italics"/>eguale ad e&longs;&longs;o FHM, &longs;arà il triangolo GKQ eguale al triangolo FHM, & <lb/>illato GQ al lato FM eguale; &longs;arà dunque maggiore GN di e&longs;&longs;a FM; & <lb/>perciò BN maggiore &longs;arà di BM. </s> <s>& BM &longs;arà maggiore di BA; impe<lb/>roche BM è maggiore di e&longs;&longs;a BF. </s> <s id="id.2.1.535.6.0">Che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.536.1.0"><emph type="italics"/>Et nel modo i&longs;te&longs;&longs;o in tutto, quanto più da pre&longs;&longs;o &longs;arà BG ad e&longs;&longs;a BE, &longs;empre la li<lb/>nea BN &longs;i dimo&longs;trerà e&longs;&longs;er maggiore. <emph.end type="italics"/></s> <s id="id.2.1.537.1.0">Che &longs;e &longs;aranno po&longs;ti di <lb/>&longs;otto i triangoli BF <lb/>HB GK tra AB <lb/>BC, & &longs;iano tiratele <lb/>linee CHO GKP, <lb/>lequali taglino le li<lb/>nee BF BG ne' pun<lb/>ti MN: &longs;arà la linea <lb/>BN minore di e&longs;&longs;a <lb/>BM, & BM di e&longs;sa <lb/>BA. </s> <figure id="id.037.01.103.1.jpg" xlink:href="037/01/103/1.jpg"> </figure> <s id="id.2.1.539.1.0"><emph type="italics"/>Congiungan&longs;i BO BP. </s> <s id="id.2.1.539.2.0">&longs;imilmen<lb/>te prouera&longs;&longs;i, che l'angolo P <lb/>KB è minore dell' angolo OH <lb/>B. </s> <s id="id.2.1.539.3.0">Hor percioche l'angolo F <lb/>HB è eguale all'angolo GKB; <lb/>&longs;arà l'angolo GKN maggio<lb/>re dell'angolo FHM: per la <lb/>qual co&longs;a la linea GN &longs;arà <lb/>maggiore di e&longs;&longs;a FM. </s> <s>& per<lb/>ciò la linea BN &longs;arà minore <lb/>della linea BM. </s> <s>& concio<lb/>&longs;ia che maggiore &longs;ia BF di <lb/>BM; &longs;arà BM minore di <lb/>BA. </s> <s id="id.2.1.539.4.0">& con &longs;imile modo <lb/>proueraßi, che quanto più B <lb/>G &longs;arà dapre&longs;&longs;o ad e&longs;&longs;a BC, <lb/>la linea BN &longs;empre &longs;arà <lb/>minore. <emph.end type="italics"/></s> <pb xlink:href="037/01/104.jpg"/> <s id="id.2.1.540.1.0">PROPOSITIONE VIII. </s> <s id="id.2.1.541.1.0">La po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o che habbia il centro della grauez<lb/>za &longs;opra la leuà egualmente di&longs;tante dall'orizonte, quanto <lb/>più il pe&longs;o &longs;i inalzerà da que&longs;to &longs;ito con la leua &longs;empre haurà <lb/>bi&longs;ogno di po&longs;sanza minore per e&longs;sere &longs;o&longs;tenuto: ma &longs;e &longs;arà <lb/>abba&longs;sato di maggiore. </s> <s id="id.2.1.542.1.0"><emph type="italics"/>Sia la leua AB egualmente di&longs;tante dall'orizonte, il cui &longs;o&longs;tegno &longs;ia C, & il pe&longs;o <lb/>BD il centro della grauezza delquale &longs;ia doue è H &longs;opra la leua; & &longs;ia la po&longs;&longs;an<emph.end type="italics"/><lb/><figure id="id.037.01.104.1.jpg" xlink:href="037/01/104/1.jpg"></figure><lb/><emph type="italics"/>za &longs;o&longs;tenente in A. </s> <s id="id.2.1.542.2.0">Moua&longs;i dapoi la leua AB in EF, & &longs;ia il pe&longs;o mo&longs;&longs;o <lb/>in FG. </s> <s id="id.2.1.542.3.0">Dico primieramente che minore po&longs;&longs;anza po&longs;ta in E &longs;o&longs;tenirà il pe&longs;o <lb/>FG con la leua EF, che la po&longs;&longs;anza in A il pe&longs;o BD con la leua AB. </s> <s id="id.2.1.542.4.0">&longs;ia <lb/>il K il centro della grauezza del pe&longs;o FG. </s> <s id="id.2.1.542.5.0">Dapoi &longs;iano tirate sì da H, come <emph.end type="italics"/><pb pagenum="45" xlink:href="037/01/105.jpg"/><emph type="italics"/>da K le linee HL KM à piombo de'loro orizonti, lequali &longs;i andaranno à tro<lb/>uare nel centro del mondo, & &longs;ia HL à piombo anche di e&longs;&longs;a AB. </s> <s id="id.2.1.542.6.0">Dapoi &longs;ia <lb/>tirata la linea KN à piombo di EF, laquale &longs;arà eguale ad HL, & la CN <lb/>eguale ad e&longs;&longs;a CL. </s> <s id="id.2.1.542.7.0">Hor percioche HL è à piombo dell'orizonte, la po&longs;&longs;anza <lb/>in A &longs;o&longs;tenente il pe&longs;o BD haurà quella proportione ad e&longs;&longs;o pe&longs;o, che CL à <emph.end type="italics"/><arrow.to.target n="note151"></arrow.to.target><lb/><emph type="italics"/>CA. </s> <s id="id.2.1.542.8.0">Di nuouo, percioche KM è à piombo dell'orizonte, la po&longs;&longs;anza in E &longs;o<lb/>&longs;tenente il pe&longs;o FG co&longs;i &longs;arà al pe&longs;o come CM à CE. </s> <s>& per e&longs;&longs;ere CN NK <lb/>eguali ad e&longs;&longs;e CL LH, & contenere angoli retti, &longs;arà CM minore di e&longs;&longs;a CL; <emph.end type="italics"/><arrow.to.target n="note152"></arrow.to.target><lb/><emph type="italics"/>Dunque CM à CA haurà proportione minore, che CL à CA; & CA <lb/>è eguale à CE, dunque haurà CM proportione minore à CE, che CL à <emph.end type="italics"/><arrow.to.target n="note153"></arrow.to.target><lb/><emph type="italics"/>CA: & per e&longs;&longs;erei pe&longs;i BD FG eguali, però che è il pe&longs;o mede&longs;imo. </s> <s id="id.2.1.542.9.0">Dun<lb/>que &longs;arà minore proportione della po&longs;&longs;anza in E &longs;o&longs;tenente il pe&longs;o FG ad e&longs;&longs;o <lb/>pe&longs;o, che della po&longs;&longs;anza in A &longs;o&longs;tenente il pe&longs;o BD ad e&longs;&longs;o pe&longs;o. </s> <s id="id.2.1.542.10.0">Per laqual <lb/>co&longs;a minore po&longs;&longs;anza po&longs;ta in E &longs;o&longs;tenterà il pe&longs;o FG, che la po&longs;&longs;anza in A <emph.end type="italics"/><arrow.to.target n="note154"></arrow.to.target><lb/><emph type="italics"/>il pe&longs;o BD. </s> <s>& quanto più &longs;arà inalzato il pe&longs;o, &longs;empre &longs;i mo&longs;trerà po&longs;&longs;anza <emph.end type="italics"/><arrow.to.target n="note155"></arrow.to.target><lb/><emph type="italics"/>anche minore douer &longs;o&longs;tenere il pe&longs;o, per e&longs;&longs;ere la linea PC minore della CM. <lb/></s> <s id="id.2.1.542.11.0">Sia dapoi la leua in QR, & il pe&longs;o in QS, il cui centro della grauezza &longs;ia O. <lb/></s> <s id="id.2.1.542.12.0">Dico che po&longs;&longs;anza maggiore &longs;i richiede in R per &longs;o&longs;tenere il pe&longs;o QS, che in <lb/>A per &longs;ostentare il pe&longs;o BD. </s> <s id="id.2.1.542.13.0">Tiri&longs;i dal centro O della grauezza la linea OT <lb/>a piombo dell'orizonte. </s> <s id="id.2.1.542.14.0">& percioche le linee HL OT &longs;e &longs;aranno allungate dal<lb/>la parte di L, & di T &longs;i andranno à ritrouare nel centro del mondo, &longs;arà la CT mag<lb/>giore della CL: & è la CA eguale ad e&longs;&longs;a CR, dunque la TC haurà pro<emph.end type="italics"/><arrow.to.target n="note156"></arrow.to.target><lb/><emph type="italics"/>portione maggiore à CR, che LC à CA. </s> <s id="id.2.1.542.15.0">Maggiore dunque &longs;arà la po&longs;&longs;an<emph.end type="italics"/><arrow.to.target n="note157"></arrow.to.target><lb/><emph type="italics"/>za in R &longs;o&longs;tenente il pe&longs;o QS, che in A &longs;o&longs;tenente il BD. </s> <s id="id.2.1.542.16.0">Similmente mo<lb/>&longs;trera&longs;&longs;i, che quanto la leua RQ abba&longs;&longs;ando&longs;i, &longs;arà più di&longs;tante dalla leua AB, <emph.end type="italics"/><arrow.to.target n="note158"></arrow.to.target><lb/><emph type="italics"/>&longs;empre più &longs;i ricercherà po&longs;&longs;anza maggiore à &longs;o&longs;tenere il pe&longs;o: peroche la di&longs;tanza <emph.end type="italics"/><arrow.to.target n="note159"></arrow.to.target><lb/><emph type="italics"/>CV è più lunga di CT. </s> <s id="id.2.1.542.17.0">Quanto dunque il pe&longs;o &longs;i alzerà più dal &longs;ito egualmente <lb/>di&longs;tante dall'orizonte, &longs;arà &longs;empre &longs;o&longs;tenuto da po&longs;&longs;anza minore; & quanto più &longs;i <lb/>abba&longs;&longs;erà, di po&longs;&longs;anza maggiore haurà me&longs;tieri per e&longs;&longs;er &longs;o&longs;tentato. </s> <s id="id.2.1.542.18.0">che bi&longs;ogna<lb/>ua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.544.1.0"><margin.target id="note151"></margin.target><emph type="italics"/>Per la quinta di questo. <emph.end type="italics"/></s> <s id="id.2.1.545.1.0"><margin.target id="note152"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.546.1.0"><margin.target id="note153"></margin.target><emph type="italics"/>Per la ottaua del quinto. <emph.end type="italics"/></s> <s id="id.2.1.547.1.0"><margin.target id="note154"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.548.1.0"><margin.target id="note155"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.549.1.0"><margin.target id="note156"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.550.1.0"><margin.target id="note157"></margin.target><emph type="italics"/>Per la ottaua del<emph.end type="italics"/> 5. </s> <s id="id.2.1.551.1.0"><margin.target id="note158"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.552.1.0"><margin.target id="note159"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.553.1.0">Quinci facilmente &longs;i caua, che la pos&longs;anza in A alla po&longs;sanza <lb/>in E co&longs;i è, come CL à CM. </s> <s id="id.2.1.554.1.0"><emph type="italics"/>Imperoche co&longs;iè LC à CA, come la po&longs;&longs;anza in A al pe&longs;o; & come CA, <lb/>cioè CE à CM, co&longs;i è il pe&longs;o alla po&longs;&longs;anza in E; Per laqual co&longs;a per la pro<emph.end type="italics"/><arrow.to.target n="note160"></arrow.to.target><lb/><emph type="italics"/>portion eguale, la po&longs;&longs;anza in A alla po&longs;&longs;anza in E &longs;arà come CL à CM. <emph.end type="italics"/></s> <s id="id.2.1.555.1.0"><margin.target id="note160"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 22. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.556.1.0"><emph type="italics"/>Con &longs;imile ragione mo&longs;treraßi non &longs;olamente che la po&longs;&longs;anza in A co&longs;i è alla po&longs;<lb/>&longs;anza in R, come CL à CT, ma che la po&longs;&longs;anza in E ancora alla po&longs;&longs;anza <lb/>in R è co&longs;i, come CM à CT, & co&longs;i nel re&longs;to. <emph.end type="italics"/></s> <pb xlink:href="037/01/106.jpg"/> <s id="id.2.1.558.1.0"><emph type="italics"/>Sia poi la leua AB egualmente di&longs;tante dall'orizonte, il cui &longs;o&longs;tegno &longs;ia B, & il <lb/>centro H della grauezza del pe&longs;o CD &longs;ia &longs;opra la leua; & moua&longs;i la leua in <lb/>BE, & il pe&longs;o in FG. </s> <s id="id.2.1.558.2.0">Dico che minore po&longs;&longs;anza po&longs;ta in E &longs;o&longs;tiene il pe&longs;o FG <lb/>con la leua EB, che la po&longs;&longs;anza in A il pe&longs;o CD con la leua AB. </s> <s id="id.2.1.558.3.0">Sia K <lb/>il centro della grauezza del pe&longs;o FG, & da i centri delle grauezze HK &longs;iano <emph.end type="italics"/><lb/><figure id="id.037.01.106.1.jpg" xlink:href="037/01/106/1.jpg"></figure><lb/><arrow.to.target n="note161"></arrow.to.target><emph type="italics"/>tirate le linee HL<emph.end type="italics"/> K<emph type="italics"/>M à piombo de'loro orizonti. </s> <s id="id.2.1.558.4.0">Hor percioche dalle co&longs;e <lb/>di &longs;opra mo&longs;trate BM è minore di BL, & BE è eguale à BA, haurà pro<emph.end type="italics"/><lb/><arrow.to.target n="note162"></arrow.to.target><emph type="italics"/>portione minore BM à BE, che BL à BA: ma come BM à BE, co&longs;i <emph.end type="italics"/><lb/><arrow.to.target n="note163"></arrow.to.target><emph type="italics"/>è la po&longs;&longs;anza in E &longs;o&longs;tenente il pe&longs;o FG ad e&longs;&longs;o pe&longs;o, & come BL a BA, <lb/>co&longs;i la po&longs;&longs;anza in A al pe&longs;o CD; la po&longs;&longs;anza in E al pe&longs;o FG haurà propor<emph.end type="italics"/><lb/><arrow.to.target n="note164"></arrow.to.target><emph type="italics"/>tione minore, che la po&longs;&longs;anza in A al pe&longs;o CD. </s> <s id="id.2.1.558.5.0">Dunque la po&longs;&longs;anza in E &longs;a<lb/>rà minore della po&longs;&longs;anza in A. </s> <s id="id.2.1.558.6.0">Similmente mo&longs;trera&longs;&longs;i quanto più il pe&longs;o &longs;i alze<lb/>rà, &longs;empre minore po&longs;&longs;anza &longs;o&longs;tenere il pe&longs;o, ma &longs;ia la leua in BO, & il pe&longs;o in <lb/>BQ, il cui centro della grauezza &longs;ia R. </s> <s id="id.2.1.558.7.0">Dico, che maggior po&longs;&longs;anza &longs;i ricerca <lb/>in O per &longs;o&longs;tenere il pe&longs;o PQ con la leua BO, che per &longs;o&longs;tenere il pe&longs;o CD con <lb/>la leua BA. </s> <s id="id.2.1.558.8.0">Sia tirata dal punto R la linea RS à piombo dell'orizonte. </s> <s id="id.2.1.558.9.0">& <emph.end type="italics"/><lb/><arrow.to.target n="note165"></arrow.to.target><emph type="italics"/>percioche BS è maggiore di BL, haurà BS proportione maggiore à BO, che <lb/>BL à BA; Per laqual co&longs;a la po&longs;&longs;anza in O &longs;o&longs;tenente il pe&longs;o PQ &longs;arà maggio <lb/>re della po&longs;&longs;anza in A &longs;o&longs;tenente il pe&longs;o CD. </s> <s>& à que&longs;to modo &longs;i mo&longs;trerà an<lb/>cora che quanto la leua BO abba&longs;&longs;ando&longs;i, &longs;arà più di&longs;tante dalla leua AB &longs;em<lb/>pre vi vorrà po&longs;&longs;anza maggiore à &longs;o&longs;tener il pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.560.1.0"><margin.target id="note161"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.561.1.0"><margin.target id="note162"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.562.1.0"><margin.target id="note163"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.563.1.0"><margin.target id="note164"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.564.1.0"><margin.target id="note165"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.565.1.0"><emph type="italics"/>Di qui parimente, come di &longs;opra è manife&longs;to, che la po&longs;&longs;anza in A è alla po&longs;&longs;anza in <emph.end type="italics"/><pb pagenum="46" xlink:href="037/01/107.jpg"/><emph type="italics"/>B, come BL à BM: & la po&longs;&longs;anza in A alla po&longs;&longs;anza in O, come BL à BS. </s> <s>& <lb/>la po&longs;&longs;anza in E alla po&longs;&longs;anza in O, come BM à BS. <emph.end type="italics"/></s> <s id="id.2.1.566.1.0"><emph type="italics"/>Oltre à ciò &longs;e &longs;i intenderà vn'altra po&longs;&longs;anza in B, per modo che due &longs;iano le po&longs;&longs;an<lb/>ze, che &longs;o&longs;tentino il pe&longs;o, minore &longs;arà la po&longs;&longs;anza in B, che &longs;o&longs;tiene il pe&longs;o PQ <lb/>con la leua BO, che il pe&longs;o CD con la leua BA. </s> <s>ma per lo contrario &longs;i ri<lb/>cerca po&longs;&longs;anza maggiore in B per &longs;o&longs;tenere il pe&longs;o FG con la leua BE, che <lb/>il pe&longs;o CD con la leua AB: percioche tirata la linea KN à piombo di EB, <lb/>&longs;arà EN eguale ad AL: Per laqual co&longs;a EM &longs;arà maggiore di LA. </s> <s id="id.2.1.566.2.0">Dun<emph.end type="italics"/><arrow.to.target n="note166"></arrow.to.target><lb/><emph type="italics"/>que EM haurà proportione maggiore ad EB, che LA ad AB, & LA <lb/>maggiore ad AB, che SO ad OB, lequali &longs;ono proportioni della po&longs;&longs;anza <emph.end type="italics"/><arrow.to.target n="note167"></arrow.to.target><lb/><emph type="italics"/>al pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.567.1.0"><margin.target id="note166"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.568.1.0"><margin.target id="note167"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.569.1.0"><emph type="italics"/>Similmente prouera&longs;&longs;i, che la po&longs;&longs;anza in B &longs;o&longs;tenente il pe&longs;o con la leua AB è al<lb/>la po&longs;&longs;anza &longs;o&longs;tenente po&longs;ta nell'i&longs;te&longs;&longs;o punto B con la leua EB, come LA <lb/>ad EM; & co&longs;i e&longs;&longs;ere anche alla po&longs;&longs;anza di B &longs;o&longs;tenente il pe&longs;o con la leua OB, <lb/>come AL ad OS. </s> <s id="id.2.1.569.2.0">Ma quelle po&longs;&longs;anze che &longs;o&longs;tengono con le leue EB OB <lb/>&longs;ono co&longs;i tra loro come EM ad OS. <emph.end type="italics"/></s> <s id="id.2.1.570.1.0"><emph type="italics"/>Dapoi mo&longs;treremo come nelle co&longs;e che di &longs;opra &longs;ono &longs;tate dette, che la po&longs;&longs;anza in B <lb/>ha quella proportione alla po&longs;&longs;anza in E, che EM ad MB; & la po&longs;&longs;anza <emph.end type="italics"/><arrow.to.target n="note168"></arrow.to.target><lb/><emph type="italics"/>in B co&longs;i e&longs;&longs;ere alla po&longs;&longs;anza in A, come AL ad LB, & la po&longs;&longs;anza in B <lb/>alla po&longs;&longs;anza in O, come OS ad SB. <emph.end type="italics"/><arrow.to.target n="note169"></arrow.to.target></s> <s id="id.2.1.571.1.0"><margin.target id="note168"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 3. <emph type="italics"/>corollario. <emph.end type="italics"/></s> <s id="id.2.1.572.1.0"><margin.target id="note169"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.573.1.0"><emph type="italics"/>Ma &longs;ia la leua AB e<lb/>gualmente di&longs;tante <lb/>dall'orizonte, il cui <lb/>&longs;o&longs;tegno &longs;ia B, & <lb/>il centro H della <lb/>grauezza del pe&longs;o <lb/>AC &longs;ia &longs;opra la <lb/>leua: & moua&longs;i la <lb/>leua in BE, & il <lb/>pe&longs;o in EF, & la <lb/>po&longs;&longs;anza in G. </s> <s>di <lb/>mo&longs;trera&longs;&longs;i parimen<lb/>te, come di &longs;opra, che <lb/>la po&longs;&longs;anza in G &longs;o <lb/>&longs;tenente il pe&longs;o EF <lb/>è minore della po&longs;<lb/>&longs;anza in D &longs;o&longs;te<emph.end type="italics"/><lb/><figure id="id.037.01.108.1.jpg" xlink:href="037/01/108/1.jpg"></figure><lb/><emph type="italics"/>nente il pe&longs;o AC. </s> <s>percioche e&longs;&longs;endo minore BM di BL haurà minore pro<lb/>portione MB à BG, che LB à BD. </s> <s>& à que&longs;to modo proueraßi, che quan<lb/>to il pe&longs;o più &longs;i alzerà con la leua, &longs;empre minore po&longs;&longs;anza &longs;i ricerca à &longs;o&longs;tenere <emph.end type="italics"/><pb xlink:href="037/01/108.jpg"/><emph type="italics"/>il detto pe&longs;o. </s> <s id="id.2.1.573.2.0">&longs;imilmente &longs;e la leua &longs;i moue in BO, & la po&longs;&longs;anza &longs;o&longs;tenente &longs;ia <lb/>in N, &longs;i mo&longs;trerà <lb/>la po&longs;&longs;anza in N e&longs;<lb/>&longs;ere maggiore della <lb/>po&longs;&longs;anza in D. </s> <s>pe<lb/>roche SB ha pro<lb/>portione maggiore <lb/>à BN che LB <lb/>à BD. </s> <s id="id.2.1.573.3.0">Mo&longs;tre<lb/>raßi ancora, che <lb/>quanto il pe&longs;o più <lb/>s'abba&longs;&longs;erà, &longs;empre <lb/>ricercar&longs;i po&longs;&longs;anza <lb/>maggiore à &longs;o&longs;tene<lb/>re il pe&longs;o. </s> <s id="id.2.1.573.4.0">che bi&longs;o<lb/>gnaua mo&longs;trare. <emph.end type="italics"/></s> <figure id="id.037.01.108.2.jpg" xlink:href="037/01/108/2.jpg"> </figure> <s id="id.2.1.576.1.0"><emph type="italics"/>Di quì <expan abbr="parim&emacr;te">parimente</expan> è chia<lb/>ro, che le po&longs;&longs;anze <lb/>in GDN co&longs;i tra loro &longs;ono, come BM à BL, & come BL à BS, & vlti<lb/>mamente come BM à BS. <emph.end type="italics"/></s> <s id="id.2.1.577.1.0">COROLLARIO</s> <s id="id.2.1.578.1.0">Da que&longs;te co&longs;e è manife&longs;to, che &longs;e la po&longs;sanza con la leua moue <lb/>rà in sù il pe&longs;o, il cui centro della grauezza &longs;ia &longs;opra la leua, <lb/>quanto più &longs;arà alzato il pe&longs;o, &longs;empre vi vorrà po&longs;sanza mi<lb/>nore per mouere il pe&longs;o. </s> <s id="id.2.1.579.1.0"><emph type="italics"/>Percioche doue la po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o è &longs;empre minore, &longs;arà parimente la po&longs;<lb/>&longs;anza, che lo moue &longs;empre minore. <emph.end type="italics"/></s> <s id="id.2.1.580.1.0"><emph type="italics"/>Da que&longs;te co&longs;e dimo&longs;treraßi etiandio, &longs;ia pur il centro della grauezza del pe&longs;o mede&longs;i<lb/>mo ò più da pre&longs;&longs;o, ò più da lunge della leua AB egualmente di&longs;tante dall' ori<lb/>zonte, che la po&longs;&longs;anza mede&longs;ima in A &longs;o&longs;terrà nondimeno il pe&longs;o: come &longs;e il cen<lb/>tro H della grauezza del pe&longs;o BD &longs;ia più da lunge dalla leua BA, che il cen<lb/>tro N della grauezza del pe&longs;o PV, pur che la linea HL tirata dal punto H <lb/>à piombo dell'orizonte, & della leua AB paßi per N, & &longs;ia il pe&longs;o PV <lb/>eguale al pe&longs;o BD; &longs;arà sì il pe&longs;o BD, & sì il pe&longs;o PV come &longs;e ambidue &longs;o&longs;<lb/>&longs;ero appiccati ad L; & &longs;ono eguali per e&longs;&longs;ere pre&longs;i in luogo di vn pe&longs;o &longs;olo, dun<lb/>que la i&longs;te&longs;&longs;a po&longs;&longs;anza in A &longs;o&longs;tenente il pe&longs;o BD &longs;o&longs;terrà anche il pe&longs;o PV. <emph.end type="italics"/></s> <pb pagenum="47" xlink:href="037/01/109.jpg"/> <s><emph type="italics"/>Ma nella leua EF quanto il centro della grauezza &longs;arà più da lunge dalla leua. <lb/></s> <s id="id.2.1.580.2.0">tanto più egualmente la po&longs;&longs;anza &longs;o&longs;tenterà il pe&longs;o mede&longs;imo, come &longs;e il centro K <lb/>della grauezza del pe&longs;o FG fo&longs;&longs;e più da lunge dalla leua EF, che il centro X <lb/>dalla grauezza del pe&longs;o <foreign lang="greek">*u</foreign>Z; in modo però, che la linea tirata dal punto<emph.end type="italics"/> K <emph type="italics"/>à <lb/>piombo della leua FE paßi per X; & &longs;ia il pe&longs;o FG eguale al pe&longs;o <foreign lang="greek">*u</foreign>Z; <lb/>& da punti KX &longs;iano tirate le linee KM X<36> à piombo de loro orizonti; &longs;a<emph.end type="italics"/><lb/><figure id="id.037.01.109.1.jpg" xlink:href="037/01/109/1.jpg"></figure><lb/><emph type="italics"/>rà la C<36> maggiore di CM; & perciò il pe&longs;o FG &longs;arà nella leua co&longs;i come <lb/>&longs;e fo&longs;&longs;e appiccato in M, & il pe&longs;o <foreign lang="greek">*u</foreign>Z come fo&longs;&longs;e appiccato in <36>. </s> <s id="id.2.1.580.3.0">Hor per<emph.end type="italics"/><arrow.to.target n="note170"></arrow.to.target><lb/><emph type="italics"/>cioche C<36> ha proportione maggiore à CE, che CM à CE, maggiore <lb/>&longs;arà la po&longs;&longs;anza po&longs;ta in E, che &longs;o&longs;terrà il pe&longs;o <foreign lang="greek">*u</foreign>Z, che FG. </s> <s id="id.2.1.580.4.0">Manella leua <lb/>QR per lo contrario &longs;i dimo&longs;trerà, cioè che quanto il centro della grauezza del pe<lb/>&longs;o mede&longs;imo è più da lunge dalla leua, tanto più anche maggiore è la po&longs;&longs;anza che <lb/>&longs;o&longs;tiene il pe&longs;o. </s> <s id="id.2.1.580.5.0">peroche maggiore è CT di CI, & perciò CT hauerà proportio<lb/>ne maggiore à CR, che CI à CR. </s> <s id="id.2.1.580.6.0">&longs;imilmente dimo&longs;treraßi, &longs;e il pe&longs;o &longs;arà col <lb/>locato fra la po&longs;&longs;anza, & il &longs;o&longs;tegno, ouero la po&longs;&longs;anza po&longs;ta fra il &longs;o&longs;tegno, & il <lb/>pe&longs;o, il che mede&longs;imamente auuenirà alla po&longs;&longs;anzà che moue peroche doue po&longs;&longs;anza <lb/>minore &longs;o&longs;tiene il pe&longs;o, iui po&longs;&longs;anza minore lo mouerà: & doue &longs;i ricerca po&longs;&longs;anza <lb/>maggiore in &longs;o&longs;tenere, iui anche maggiore vi vuole in mouere. <emph.end type="italics"/></s> <s id="id.2.1.582.1.0"><margin.target id="note170"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <pb xlink:href="037/01/110.jpg"/> <s id="id.2.1.583.1.0">PROPOSITIONE IX. </s> <s id="id.2.1.584.1.0">La po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o, che habbia il centro della &longs;ua gra<lb/>uezza &longs;otto la leua egualmente di&longs;tante dall'orizonte, quanto <lb/>più il pe&longs;o &longs;arà alzato da que&longs;to &longs;ito con la leua, haurà egli &longs;em<lb/>pre anco me&longs;tieri di po&longs;&longs;anza maggiore ad e&longs;&longs;ere &longs;o&longs;tenuto; <lb/>Ma &longs;e abba&longs;&longs;ato, di minore. </s> <s id="id.2.1.585.1.0"><emph type="italics"/>Sia la leua AB egualmente <expan abbr="di&longs;t&amacr;te">di&longs;tante</expan> dall'orizonte, il cui &longs;o&longs;tegno &longs;ia C, & &longs;ia il pe&longs;o AD, <lb/>il cui centro L della grauezza &longs;ia &longs;otto la leua, & &longs;ia in B la po&longs;&longs;anza &longs;o&longs;tenen<lb/>te il pe&longs;o AD: moua&longs;i dopo la leua in FG, & il pe&longs;o in FH. </s> <s id="id.2.1.585.2.0">Dico prima, <lb/>che po&longs;&longs;anza maggiore &longs;i ricerca in G per &longs;o&longs;tenere il pe&longs;o FH con la leua FG, <lb/>di quel che &longs;iala po&longs;&longs;anza in B e&longs;&longs;endo il pe&longs;o AD, ma con la leua AB. </s> <s id="id.2.1.585.3.0">&longs;ia <emph.end type="italics"/><lb/><figure id="id.037.01.110.1.jpg" xlink:href="037/01/110/1.jpg"></figure><lb/><emph type="italics"/>M il centro della grauezza del pe&longs;o FH, & da punti LM &longs;iano tirate le linee <emph.end type="italics"/><lb/><arrow.to.target n="note171"></arrow.to.target><emph type="italics"/>LK MN à piombo de'loro orizonti; & &longs;ia tirata la linea MS à piombo di FG, <lb/>che &longs;arà eguale ad LK, & CK &longs;arà etiandio eguale ad e&longs;&longs;a CS. </s> <s id="id.2.1.585.4.0">Percioche dun<emph.end type="italics"/><lb/><arrow.to.target n="note172"></arrow.to.target><emph type="italics"/>que CN è maggiore di CK haurà NC proportione maggiore à CG, che CK <lb/>à CB; & la po&longs;&longs;anza in B al pe&longs;o AD ha la mede&longs;ma proportione, che KC <emph.end type="italics"/><lb/><arrow.to.target n="note173"></arrow.to.target><emph type="italics"/>à CB: & come la po&longs;&longs;anza in G al pe&longs;o FH, co&longs;i è NC à CG; dunque la <emph.end type="italics"/><lb/><arrow.to.target n="note174"></arrow.to.target><emph type="italics"/>po&longs;&longs;anza in G hauerà maggiore proportione al pe&longs;o FH, che la po&longs;&longs;anza in B <lb/>al pe&longs;o AD. </s> <s id="id.2.1.585.5.0">Maggiore dunque è la po&longs;&longs;anza in G della po&longs;&longs;anza in B. </s> <s>che &longs;e <emph.end type="italics"/><pb pagenum="48" xlink:href="037/01/111.jpg"/><emph type="italics"/>la leua &longs;arà in OP, & il pe&longs;o in OQ: &longs;arà la po&longs;&longs;anza po&longs;ta in B maggiore, <lb/>che in P: percioche &longs;i dimo&longs;trerà nell'i&longs;te&longs;&longs;o modo CR e&longs;&longs;ere minore di CK, & <lb/>CR hauere proportione minore a CP, che CK a CB; & perciò la po&longs;&longs;anza <lb/>po&longs;ta in B e&longs;&longs;ere maggiore della po&longs;&longs;anza po&longs;ta in P. </s> <s id="id.2.1.585.6.0">& a que&longs;to modo mo&longs;tre<lb/>ra&longs;&longs;i che quanto più il pe&longs;o &longs;i alzerà dal &longs;ito AB, &longs;empre vi vorrà po&longs;&longs;anza mag<lb/>giore à &longs;o&longs;tenerlo. </s> <s id="id.2.1.585.7.0">ma per lo contrario accaderà &longs;e egli &longs;arà abba&longs;&longs;ato. </s> <s id="id.2.1.585.8.0">che bi&longs;o<lb/>gnaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.587.1.0"><margin.target id="note171"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.588.1.0"><margin.target id="note172"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.589.1.0"><margin.target id="note173"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.590.1.0"><margin.target id="note174"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.591.1.0"><emph type="italics"/>Di quà ancora &longs;i puote ageuolmente cauare, che le po&longs;&longs;anze po&longs;te in PBG &longs;ono in <lb/>modo di&longs;po&longs;te fra loro, come CR à CK; & come CK à CN, & come CN <lb/>à CR. <emph.end type="italics"/></s> <s id="id.2.1.592.1.0"><emph type="italics"/>Sia dopo la leua AB egualmente di&longs;tante dall'orizonte, co'l &longs;uo &longs;o&longs;tegno B; & il <lb/>pe&longs;o CD habbia il centro O della grauezza &longs;otto la leua, & &longs;ia in A la po&longs;<lb/>&longs;anza &longs;o&longs;tenente il pe&longs;o CD. </s> <s id="id.2.1.592.2.0">Moua&longs;i dapoi la leua in BE, & BF, & &longs;i tra<lb/>&longs;porti il pe&longs;o in GH KL. </s> <s id="id.2.1.592.3.0">Dico, che maggiore po&longs;&longs;anza per &longs;o&longs;tenere il pe&longs;o &longs;i <emph.end type="italics"/><lb/><figure id="id.037.01.111.1.jpg" xlink:href="037/01/111/1.jpg"></figure><lb/><emph type="italics"/>ricerca in E, che in A; & maggiore in A che in F &longs;iano tirate da i centri <lb/>delle grauezze le linee NM OP QR à piombo de gli orizonti, lequali allun<lb/>gate da la parte di NOQ &longs;i andranno à trouare nel centro del mondo. </s> <s id="id.2.1.592.4.0">Mo&longs;tre<lb/>ra&longs;&longs;i parimente come di &longs;opra, che BM è maggiore di BP, & BP maggio<emph.end type="italics"/><arrow.to.target n="note175"></arrow.to.target><lb/><emph type="italics"/>re di BR; & che BM ha proportione maggiore à BE, che BP à BA; & <lb/>BP à BA maggiore che BR à BF: & per que&longs;to la po&longs;&longs;anza in E mag<lb/>giore è della po&longs;&longs;anza in A; & la po&longs;&longs;anza in A maggiore della po&longs;&longs;anza in <emph.end type="italics"/><lb/>F. </s> <s><emph type="italics"/>& quanto la leua &longs;i alzerà più dal &longs;ito AB, mo&longs;trera&longs;&longs;i &longs;empre, che mag<pb xlink:href="037/01/112.jpg"/>giore po&longs;&longs;anza vi vuole à &longs;o&longs;tenere il pe&longs;o: ma &longs;e abba&longs;&longs;eraßi, minore. <emph.end type="italics"/></s> <s id="id.2.1.594.1.0"><margin.target id="note175"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.595.1.0"><emph type="italics"/>Di quì è chiaro etiandio che le po&longs;&longs;anze po&longs;te in EAF co&longs;i tra loro &longs;ono, come BM <lb/>à BP, & come BP à BR, & come BM à BR. <emph.end type="italics"/></s> <s id="id.2.1.596.1.0"><emph type="italics"/>Di più &longs;e in B &longs;arà vn'altra po&longs;&longs;anza, per modo, che due po&longs;&longs;anze &longs;iano quelle che <lb/>&longs;o&longs;tengano il pe&longs;o. </s> <s id="id.2.1.596.2.0">Di maggiore po&longs;&longs;anza è bi&longs;ogno in B per &longs;o&longs;tenere il pe&longs;o KL <lb/>con la leua BF, che per &longs;o&longs;tenere il pe&longs;o CD con la leua AB. </s> <s>& dauan<emph.end type="italics"/><lb/><figure id="id.037.01.112.1.jpg" xlink:href="037/01/112/1.jpg"></figure><lb/><emph type="italics"/>taggio anco maggiore con la leua AB, che con la leua BE: peroche RF ha <lb/>proportione maggiore ad FB, che PA ad AB; & PA ad AB mag<lb/>giore, che EM ad EB. <emph.end type="italics"/></s> <s id="id.2.1.598.1.0"><emph type="italics"/>Similmente mo&longs;treraßi, che le po&longs;&longs;anze in B &longs;o&longs;tenenti il pe&longs;o con le leue tra loro co&longs;i <lb/>e&longs;&longs;ere, come EM ad AP, & come AP ad FR, & come EM ad FR. <emph.end type="italics"/></s> <s id="id.2.1.599.1.0"><arrow.to.target n="note176"></arrow.to.target><emph type="italics"/>Oltre à ciò la po&longs;&longs;anza in B co&longs;i &longs;arà alla po&longs;&longs;anza in F, come RF ad RB; & <lb/>la po&longs;&longs;anza in B alla po&longs;&longs;anza in A come PA à PB, & la po&longs;&longs;anza in B al<emph.end type="italics"/><lb/><arrow.to.target n="note177"></arrow.to.target><emph type="italics"/>la po&longs;&longs;anza in E come EM ad MB. <emph.end type="italics"/></s> <s id="id.2.1.601.1.0"><margin.target id="note176"></margin.target><emph type="italics"/>Per lo<emph.end type="italics"/> 3. <emph type="italics"/>corrollario. <emph.end type="italics"/></s> <s id="id.2.1.602.1.0"><margin.target id="note177"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <pb pagenum="49" xlink:href="037/01/113.jpg"/> <s id="id.2.1.603.1.0"><emph type="italics"/>Ma &longs;ia la leua AB egualmente di&longs;tante <expan abbr="dall'orizõte">dall'orizonte</expan>, col &longs;uo &longs;o&longs;tegno B, & il pe&longs;o AC, <lb/>il cui centro della <lb/>grauezza &longs;ia &longs;ot<lb/>to la leua, & &longs;ia <lb/>la po&longs;&longs;anza <expan abbr="&longs;o&longs;ten&emacr;te">&longs;o&longs;te<lb/>nente</expan> il pe&longs;o in D, <lb/>& moua&longs;i la le<lb/>ua in BE BF, <lb/>& la po&longs;&longs;anza <lb/>in GH; &longs;imil<lb/>mente mo&longs;trera&longs;<lb/>&longs;i, che la po&longs;&longs;an<lb/>za in G è mag<lb/>giore della po&longs;&longs;an<lb/>za in D, & la <lb/>po&longs;&longs;anza in D <lb/>maggiore della <emph.end type="italics"/><lb/><figure id="id.037.01.113.1.jpg" xlink:href="037/01/113/1.jpg"></figure><lb/><emph type="italics"/>po&longs;&longs;anza in H. </s> <s>percioche KB ha proportione maggiore à BG, che BL à BD, <lb/>& BL à BD maggiore che MB à BH. </s> <s id="id.2.1.603.2.0">& à questa maniera mo&longs;trera&longs;&longs;i che <lb/>quanto la leua più &longs;i alzerà dal &longs;ito AB, dauantaggio douere &longs;empre e&longs;&longs;ere mag<lb/>gior la po&longs;&longs;anza per &longs;o&longs;tenere il pe&longs;o: & quanto più s'abba&longs;&longs;a, minore. </s> <s id="id.2.1.603.3.0">che dimo<lb/>&longs;trare era me&longs;tieri. <emph.end type="italics"/></s> <s id="id.2.1.605.1.0"><emph type="italics"/>Similmente in que&longs;te, le po&longs;&longs;anze poste in GDH co&longs;i tra loro &longs;aranno, come BK à <lb/>BL, & come BL à BM, & alla &longs;ine come BK à BM. <emph.end type="italics"/></s> <s id="id.2.1.606.1.0">COROLLARIO. </s> <s id="id.2.1.607.1.0">Da que&longs;te co&longs;e etiandio è pale&longs;e, che &longs;e la po&longs;&longs;anza mouerà con <lb/>la leua in sù vn pe&longs;o, che habbia il centro della grauezza &longs;otto <lb/>la leua; Quanto più il pe&longs;o &longs;arà alzato, &longs;empre vi vorrà po&longs;<lb/>&longs;anza maggiore per mouere il pe&longs;o. </s> <s id="id.2.1.608.1.0"><emph type="italics"/>Imperoche &longs;e la po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o è &longs;empre maggiore, &longs;arà parimente la <lb/>po&longs;&longs;anza che moue il pe&longs;o &longs;empre maggiore. <emph.end type="italics"/></s> <pb xlink:href="037/01/114.jpg"/> <s id="id.2.1.610.1.0"><emph type="italics"/>Da que&longs;te co&longs;e anco &longs;i cauerà facilmente &longs;e &longs;arà il centro della grauezza dell'i&longs;te&longs;&longs;o pe<lb/>&longs;o ò più da pre&longs;&longs;o, ò più da lunge dalla leua AB egualmente di&longs;tante dall'orizon<lb/>te, che la po&longs;&longs;anza mede&longs;ima po&longs;ta in B &longs;o&longs;terrà il pe&longs;o. </s> <s id="id.2.1.610.2.0">come &longs;e il centro L della <lb/>grauezza del pe&longs;o AD fo&longs;&longs;e più da lunge dalla leua BA, che il centro N <lb/>della grauezza del pe&longs;o PV, pur che la linea LK tirata dal punto L à piom<lb/>bo dell orizonte, & della leua AB pa&longs;&longs;i per N: &longs;imilmente come nella prece<emph.end type="italics"/><lb/><figure id="id.037.01.114.1.jpg" xlink:href="037/01/114/1.jpg"></figure><lb/><emph type="italics"/>dente &longs;i mo&longs;trerà, che la po&longs;&longs;anza mede&longs;ima in B &longs;ostiene & il pe&longs;o AD, & <lb/>il pe&longs;o PV. </s> <s id="id.2.1.610.3.0">Ma nella leua EF quanto il centro della grauezza &longs;arà più da lun<lb/>ge dalla leua, tanto haurà me&longs;tieri di po&longs;&longs;anza maggiore per &longs;ostenere il pe&longs;o. </s> <s id="id.2.1.610.4.0">co<lb/>me il centro M della grauezza del pe&longs;o FH &longs;ia più da lunge dalla leua EF, che <lb/>il centro S della grauezza del pe&longs;o XZ. </s> <s id="id.2.1.610.5.0">&longs;iano tirate da i punti MS le linee <lb/>MI SG à piombo de gli orizonti; &longs;arà CI maggiore di CG: & perciò la po&longs;&longs;an<lb/>za di E deue e&longs;&longs;ere maggiore &longs;o&longs;tenendo il pe&longs;o FH, che il pe&longs;o XZ. </s> <s id="id.2.1.610.6.0">Ma per <lb/>lo contrario &longs;i mo&longs;trerà nella leua OR, cioè che quanto il centro della grauezza <lb/>dell'i&longs;te&longs;&longs;o pe&longs;o è più da lunge dalla leua, il pe&longs;o viene &longs;o&longs;tentato da po&longs;&longs;anza mino <lb/>re. </s> <s id="id.2.1.610.7.0">peroche minore è C<foreign lang="greek">*u</foreign> de CT. </s> <s id="id.2.1.610.8.0">& in modo &longs;imile demo&longs;traraßi ancora &longs;tan<lb/>do il pe&longs;o fra la po&longs;&longs;anza, & il &longs;o&longs;tegno, ouero la po&longs;&longs;anza tra il &longs;ostegno, & il <emph.end type="italics"/><pb pagenum="50" xlink:href="037/01/115.jpg"/><emph type="italics"/>pe&longs;o, ilche parimente auerrà alla po&longs;&longs;anza che moue; peroche doue po&longs;&longs;anza mino<lb/>re &longs;o&longs;tien il pe&longs;o, iui minore po&longs;&longs;anza lo mouerà. </s> <s id="id.2.1.610.9.0">& doue vuole po&longs;&longs;anza maggio<lb/>re in &longs;o&longs;tentare, iui anco ella &longs;arà maggiore in mouere. <emph.end type="italics"/></s> <s id="id.2.1.612.1.0">PROPOSITIONE X. </s> <s id="id.2.1.613.1.0">La po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o che habbia il centro della grauez<lb/>za nella i&longs;te&longs;&longs;a leua, &longs;ia pure in qual &longs;i voglia modo tra&longs;porta<lb/>to il pe&longs;o con la leua; vi &longs;arà &longs;empre me&longs;tieri della po&longs;&longs;anza <lb/>i&longs;te&longs;&longs;a, acciò &longs;ia &longs;o&longs;tenuto. </s> <s id="id.2.1.614.1.0"><emph type="italics"/>Sia la leua AB egualmente di&longs;tante dall'orizonte, co'l &longs;uo &longs;o&longs;tegno C, & E cen<lb/>tro della grauezza del pe&longs;o &longs;ia in e&longs;&longs;a leua. </s> <s id="id.2.1.614.2.0">Moua&longs;i dapoi la leua in FG, & HK,<emph.end type="italics"/><lb/><figure id="id.037.01.115.1.jpg" xlink:href="037/01/115/1.jpg"></figure><lb/><emph type="italics"/>& il centro della grauezza in LM. </s> <s id="id.2.1.614.3.0">Dico che la mede&longs;ima po&longs;&longs;anza di KBG &longs;em<emph.end type="italics"/><arrow.to.target n="note178"></arrow.to.target><lb/><emph type="italics"/>pre &longs;o&longs;terrà l'iste&longs;&longs;o pe&longs;o. </s> <s id="id.2.1.614.4.0">Hor percioche il pe&longs;o nella leua AB è &longs;i fattamen<lb/>te di&longs;po&longs;to, come &longs;e egli fo&longs;&longs;e appiccato in E; & nella leua GF come &longs;e egli fo&longs;<lb/>&longs;e appiccato in L; & nella leua HK, come &longs;e egli fo&longs;&longs;e appiccato in M; & le <emph.end type="italics"/><pb xlink:href="037/01/116.jpg"/><emph type="italics"/>distanze CL CE CM &longs;ono tra loro eguali; & parimente CK CB CG pur <lb/>tra loro eguali; &longs;arà la po&longs;&longs;anza in B al pe&longs;o, come CE à CB; & la po&longs;&longs;an<lb/>za in K al pe&longs;o, come CM à CK, & la po&longs;&longs;anza in G al pe&longs;o, come CL<emph.end type="italics"/><lb/><figure id="id.037.01.116.1.jpg" xlink:href="037/01/116/1.jpg"></figure><lb/><emph type="italics"/>à CG. </s> <s id="id.2.1.614.5.0">La po&longs;&longs;anza mede&longs;ma dunque in KBG &longs;osterrà il pe&longs;o mede&longs;mo tra&longs;por<lb/>tato in vari &longs;iti. </s> <s id="id.2.1.614.6.0">che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.617.1.0"><margin.target id="note178"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.618.1.0"><emph type="italics"/>Similmente prouera&longs;&longs;i, &longs;e il pe&longs;o fo&longs;&longs;e tra la po&longs;&longs;anza, & il &longs;o&longs;tegno; ouero la po&longs;<lb/>&longs;anza tra il &longs;o&longs;tegno, & il pe&longs;o, che il mede&longs;imo auerrà alla po&longs;&longs;anza, che moue. <emph.end type="italics"/></s> <s id="id.2.1.619.1.0">PROPOSITIONE XI. </s> <s id="id.2.1.620.1.0">Se la di&longs;tanza della leua tra il &longs;o&longs;tegno, & la po&longs;&longs;anza haurà pro<lb/>portione maggiore alla di&longs;tanza trapo&longs;ta dal &longs;o&longs;tegno al pun<lb/>to, doue dal centro della grauezza del pe&longs;o tirata vna linea à <lb/>piombo dell'orizonte taglia la leua che non ha il pe&longs;o alla po&longs;<lb/>&longs;anza; il pe&longs;o veramente &longs;arà mo&longs;&longs;o dalla po&longs;&longs;anza. </s> <s id="id.2.1.621.1.0"><emph type="italics"/>Sia la leua AB, & dal punto A appicchi&longs;i il pe&longs;o C; cioè il punto A &longs;empre <lb/>&longs;ia quel punto, doue la linea tirata à piombo dal centro della grauezza del pe&longs;o ta<lb/>gli la leua; & &longs;ia la po&longs;&longs;anza in B, & il &longs;ostegno D; & DB habbia à DA <emph.end type="italics"/><pb pagenum="51" xlink:href="037/01/117.jpg"/><emph type="italics"/>proportione maggiore, che il pe&longs;o C alla po&longs;&longs;anza in B. </s> <s id="id.2.1.621.2.0">Dico che il pe&longs;o C &longs;a<lb/>rà mo&longs;&longs;o dalla po&longs;&longs;anza in B. </s> <s id="id.2.1.621.3.0">Faccia&longs;i come BD à DA, co&longs;i il pe&longs;o E alla <emph.end type="italics"/><arrow.to.target n="note179"></arrow.to.target><lb/><emph type="italics"/>po&longs;&longs;anza in B; & appicchi&longs;i parimente il pe&longs;o E in A: egliè chiaro che la po&longs;<lb/>&longs;anza in B pe<lb/>&longs;a <expan abbr="egualm&emacr;te">egualmente</expan> <expan abbr="cõ">con</expan> <lb/>e&longs;&longs;o E; cioè che <lb/>&longs;o&longs;tiene il detto <lb/>pe&longs;o E. </s> <s id="id.2.1.621.4.0">& per<lb/>cioche BD ha <lb/>proportion mag<lb/>giore à DA che <lb/>C alla po&longs;&longs;anza <lb/>in B. </s> <s>& come <lb/>BD à DA, co&longs;i<emph.end type="italics"/><lb/><figure id="id.037.01.117.1.jpg" xlink:href="037/01/117/1.jpg"></figure><lb/><emph type="italics"/>è il pe&longs;o F. </s> <s>alla po&longs;&longs;anza: adunque E haurà proportione maggiore alla po&longs;&longs;an<emph.end type="italics"/><arrow.to.target n="note180"></arrow.to.target><lb/><emph type="italics"/>za, che il pe&longs;o C alla po&longs;&longs;anza i&longs;te&longs;&longs;a. </s> <s id="id.2.1.621.5.0">Per laqual co&longs;a il pe&longs;o E &longs;arà maggiore <lb/>del pe&longs;o C. </s> <s>& perche la po&longs;&longs;anza pe&longs;a egualmente con e&longs;&longs;o E; dunque la po&longs;&longs;an<lb/>za non pe&longs;erà egualmente con e&longs;&longs;o C, ma per la forza &longs;ua inchinerà al ba&longs;&longs;o. </s> <s id="id.2.1.621.6.0">dun<lb/>que il pe&longs;o C &longs;arà mo&longs;&longs;o dalla po&longs;&longs;anza in B con la leua AB, il cui &longs;o&longs;tegno <lb/>è in D. <emph.end type="italics"/></s> <s id="id.2.1.623.1.0"><margin.target id="note179"></margin.target><emph type="italics"/>Per la prima di questo. <emph.end type="italics"/></s> <s id="id.2.1.624.1.0"><margin.target id="note180"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.625.1.0"><emph type="italics"/>Ma &longs;e la leua fo&longs;&longs;e AB, & il &longs;o&longs;tegno A, & il pe&longs;o C appiccato in D, & la <lb/>po&longs;&longs;anza in B, & BA haue&longs;&longs;e proportione maggiore ad AD, che il pe&longs;o C <lb/>alla po&longs;&longs;anza in B. </s> <s id="id.2.1.625.2.0">Dico che il pe&longs;o C mouera&longs;&longs;i dalla po&longs;&longs;anza in B. </s> <s id="id.2.1.625.3.0">faccia&longs;i co<emph.end type="italics"/><arrow.to.target n="note181"></arrow.to.target><lb/><emph type="italics"/>me BA ad AD, co&longs;i il pe<lb/>&longs;o E alla po&longs;&longs;anza in B: & <lb/>&longs;e E &longs;arà appiccato in D, la <lb/>po&longs;&longs;anza in B &longs;o&longs;tenterà il pe<lb/>&longs;o E. </s> <s id="id.2.1.625.4.0">Ma per hauere BA pro<lb/>portione maggiore ad AD, <lb/>che il pe&longs;o C alla po&longs;&longs;anza in <lb/>B; & come BA ad AD, <lb/>co&longs;i è il pe&longs;o E alla po&longs;&longs;anza in <lb/>B; dunque il pe&longs;o E haurà pro<lb/>portione maggiore alla po&longs;&longs;an<emph.end type="italics"/><arrow.to.target n="note182"></arrow.to.target><lb/><figure id="id.037.01.117.2.jpg" xlink:href="037/01/117/2.jpg"></figure><lb/><emph type="italics"/>za che è in B, che il pe&longs;o C all'i&longs;te&longs;&longs;a po&longs;&longs;anza: & perciò il pe&longs;o E &longs;arà maggio <lb/>re del pe&longs;o C; & la po&longs;&longs;anza in B &longs;o&longs;tiene il pe&longs;o E; dunque la po&longs;&longs;anza in B <lb/>con la leua AB mouerà il pe&longs;o C minore del pe&longs;o E appiccato in D, il cui &longs;o<lb/>stegno è A. <emph.end type="italics"/></s> <s id="id.2.1.627.1.0"><margin.target id="note181"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.628.1.0"><margin.target id="note182"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <pb xlink:href="037/01/118.jpg"/> <s id="id.2.1.629.1.0"><emph type="italics"/>Sia da capo la leua AB, & il &longs;uo &longs;o&longs;tegno A, & il pe&longs;o C &longs;ia appiccato in B, <lb/>& &longs;ia la po&longs;&longs;anza in D: & DA habbia proportione maggiore ad AB, che <lb/>il pe&longs;o C al<lb/>la po&longs;&longs;anza, <lb/>che è in D. </s> <s id="id.2.1.629.2.0">Di <lb/>co che il pe&longs;o C <lb/>&longs;arà mo&longs;&longs;o dal <lb/>la <expan abbr="poß&amacr;za">poßanza</expan> che <lb/>è in D. </s> <s id="id.2.1.629.3.0">Fac<lb/>cia&longs;i come D <lb/>A ad AB, <lb/>co&longs;i il pe&longs;o E <lb/>alla po&longs;&longs;anza,<emph.end type="italics"/><lb/><figure id="id.037.01.118.1.jpg" xlink:href="037/01/118/1.jpg"></figure><lb/><emph type="italics"/>che è in D; & &longs;ia il pe&longs;o E pendente dal punto B: la po&longs;&longs;anza in D &longs;o&longs;ter<lb/>rà il pe&longs;o E. </s> <s id="id.2.1.629.4.0">Ma DA tiene proportione maggiore ad AB, che C alla po&longs;<lb/>&longs;anza in D. </s> <s>& come DA ad AB, co&longs;i è il pe&longs;o E alla po&longs;&longs;anza in D; <lb/>dunque il pe&longs;o E haurà proportione maggiore alla po&longs;&longs;anza che è in D, che il <lb/>pe&longs;o C alla i&longs;te&longs;&longs;a po&longs;&longs;anza. </s> <s id="id.2.1.629.5.0">Per laqual co&longs;a il pe&longs;o E è maggiore del pe&longs;o C. <lb/></s> <s id="id.2.1.629.6.0">Et percioche la po&longs;&longs;anza in D &longs;o&longs;tiene il pe&longs;o E, dunque la detta po&longs;&longs;anza in <lb/>D mouerà il pe&longs;o C appiccato in B con la leua AB, il cui &longs;o&longs;tegno è A. </s> <s>che <lb/>bi&longs;ognaua prouare. <emph.end type="italics"/></s> <s id="id.2.1.631.1.0">Altramente. </s> <s id="id.2.1.632.1.0"><emph type="italics"/>Sia la leua AB, & il pe&longs;o C appiccato in A, & la po&longs;&longs;anza in B, & &longs;ia il <lb/>&longs;o&longs;tegno D; & DB habbia proportione maggiore à DA, che il pe&longs;o C alla <lb/>po&longs;&longs;anza in B. <lb/></s> <s id="id.2.1.632.2.0">Dico che il pe<lb/>&longs;o C &longs;arà mo&longs;<lb/>&longs;o dalla po&longs;&longs;an<lb/>za in B. </s> <s id="id.2.1.632.3.0">Fac<lb/>cia&longs;i BE ad <lb/>EA, come il <emph.end type="italics"/><lb/><figure id="id.037.01.118.2.jpg" xlink:href="037/01/118/2.jpg"></figure><lb/><emph type="italics"/>pe&longs;o C &longs;i ha inuer&longs;o la po&longs;&longs;anza, &longs;arà il punto E tra BD: percioche egli è me<emph.end type="italics"/><lb/><arrow.to.target n="note183"></arrow.to.target><emph type="italics"/>&longs;tieri che BE habbia proportione minore ad EA, che DB à DA; & però <lb/>BE &longs;arà minore di BD. </s> <s>& percioche la po&longs;&longs;anza in B &longs;o&longs;tiene il pe&longs;o C ap<lb/>piccato in A con la leua AB, che hà il &longs;o&longs;tegno E; dunque minore po&longs;&longs;an<lb/>za po&longs;ta in B, che la data &longs;o&longs;terrà il pe&longs;o mede&longs;imo nel &longs;o&longs;tegno D. </s> <s id="id.2.1.632.4.0">La po&longs;&longs;an<lb/>za data dunque po&longs;ta in B mouerà il pe&longs;o C con la leua AB, che ha il &longs;o&longs;te<lb/>gno in D. <emph.end type="italics"/></s> <s id="id.2.1.635.1.0"><margin.target id="note183"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <pb pagenum="52" xlink:href="037/01/119.jpg"/> <s id="id.2.1.636.1.0"><emph type="italics"/>Sia dapoi la leua AB, & il &longs;uo &longs;o&longs;tegno in A, & il pe&longs;o C appiccato in D, & <lb/>&longs;iala po&longs;&longs;anza in B; & AB habbia proportione maggiore ad AD, che il <lb/>pe&longs;o C alla po&longs;&longs;anza in B. </s> <s id="id.2.1.636.2.0">Di <lb/>co che il pe&longs;o C &longs;i mouerà dalla <lb/>po&longs;&longs;anza in B. </s> <s id="id.2.1.636.3.0">Faccia&longs;i AB ad <lb/>AE, come il pe&longs;o C alla po&longs;<lb/>&longs;anza; &longs;arà &longs;imilmente il punto E <lb/>tra BD, percioche egli è nece&longs;&longs;a<emph.end type="italics"/><arrow.to.target n="note184"></arrow.to.target><lb/><emph type="italics"/>rio che AE &longs;ia maggiore di A <lb/>D. </s> <s>& &longs;e il pe&longs;o C fo&longs;&longs;e appicca<emph.end type="italics"/><arrow.to.target n="note185"></arrow.to.target><lb/><figure id="id.037.01.119.1.jpg" xlink:href="037/01/119/1.jpg"></figure><lb/><emph type="italics"/>to in E, la po&longs;&longs;anza in B lo &longs;o&longs;tentarebbe. </s> <s id="id.2.1.636.4.0">ma po&longs;&longs;anza minore po&longs;ta in B, <lb/>che la data &longs;o&longs;tiene il pe&longs;o C appiccato in D; dunque la data po&longs;&longs;anza in B mo<emph.end type="italics"/><arrow.to.target n="note186"></arrow.to.target><lb/><emph type="italics"/>uerà il pe&longs;o C appiccato in D con la leua AB, che ha il &longs;uo &longs;o&longs;tegno A. <emph.end type="italics"/></s> <s id="id.2.1.638.1.0"><margin.target id="note184"></margin.target><emph type="italics"/>Per la ottaua del<emph.end type="italics"/> 5. </s> <s id="id.2.1.639.1.0"><margin.target id="note185"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.640.1.0"><margin.target id="note186"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 1. <emph type="italics"/>corollario del la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.641.1.0"><emph type="italics"/>Sia da capo la leua AB co'l &longs;o&longs;tegno &longs;uo A; & il pe&longs;o C &longs;ia appiccato in B, & <lb/>&longs;ia la po&longs;&longs;anza in D. </s> <s>& DA habbia proportione maggiore ad AB, che il pe<lb/>&longs;o C alla po&longs;&longs;anza in <lb/>D. </s> <s id="id.2.1.641.2.0">Dico che il pe&longs;o C <emph.end type="italics"/><arrow.to.target n="note187"></arrow.to.target><lb/><emph type="italics"/>&longs;arà mo&longs;&longs;o dalla po&longs;&longs;an<lb/>za in D. </s> <s id="id.2.1.641.3.0">faccia&longs;i come <lb/>il pe&longs;o C a'la po&longs;&longs;anza, <lb/>co&longs;i DA &longs;ia ad AE; <emph.end type="italics"/><arrow.to.target n="note188"></arrow.to.target><lb/><emph type="italics"/>&longs;arà AE maggiore di <emph.end type="italics"/><lb/><figure id="id.037.01.119.2.jpg" xlink:href="037/01/119/2.jpg"></figure><lb/><emph type="italics"/>AB; per e&longs;&longs;ere proportione maggiore da DA ad AB, che da DA ad AE. <emph.end type="italics"/><arrow.to.target n="note189"></arrow.to.target><lb/></s> <s><emph type="italics"/>Che &longs;e il pe&longs;o C &longs;arà appiccato in E, egli è chiaro, che la po&longs;&longs;anza in D &longs;o&longs;ter<lb/>rà il pe&longs;o C appiccato in E. </s> <s id="id.2.1.641.4.0">Ma po&longs;&longs;anza minore che la data &longs;o&longs;tiene l'i&longs;te&longs;&longs;o pe<lb/>&longs;o C in B; dunque la data po&longs;&longs;anza in D mouerà il pe&longs;o C appiccato in B, con <lb/>la leua AB che hà il &longs;o&longs;tegno &longs;uo A. </s> <s>come bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.643.1.0"><margin.target id="note187"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.644.1.0"><margin.target id="note188"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.645.1.0"><margin.target id="note189"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 1. <emph type="italics"/>corollario del la<emph.end type="italics"/> 3. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.646.1.0">PROPOSITIONE XII. </s> <s id="id.2.1.647.1.0">PROBLEMA. </s> <s id="id.2.1.648.1.0">Fare che vna data po&longs;&longs;anza, moua vn pe&longs;o dato con vna data le<lb/>ua. </s> <pb xlink:href="037/01/120.jpg"/> <s id="id.2.1.649.1.0"><emph type="italics"/>Sia il pe&longs;o A come cento, & la po&longs;&longs;anza che ha da mouere &longs;ia come diece; & &longs;ia <lb/>la data leua BC. </s> <s id="id.2.1.649.2.0">Egli è bi&longs;ogno che la po&longs;&longs;anza, che è diece moua il pe&longs;o A, che <lb/>è cento, con la leua BC. </s> <s id="id.2.1.649.3.0">Diuida&longs;i BC in D con &longs;i fatta maniera che CD hab<lb/>bia la proportione mede&longs;ima à DB, che ha cento à diece, cioè diece ad vno; per<emph.end type="italics"/><lb/><figure id="id.037.01.120.1.jpg" xlink:href="037/01/120/1.jpg"></figure><lb/><arrow.to.target n="note190"></arrow.to.target><emph type="italics"/>cioche &longs;e D &longs;i face&longs;&longs;e &longs;o&longs;tegno, egli è manife&longs;to, che la po&longs;&longs;anza in C come diece <lb/>pe&longs;erà egualmente co'l pe&longs;o A appiccato in B, cioè che &longs;o&longs;terrà il pe&longs;o A. </s> <s id="id.2.1.649.4.0">Pren<emph.end type="italics"/><lb/><arrow.to.target n="note191"></arrow.to.target><emph type="italics"/>da&longs;i tra BD qual &longs;i voglia punto, come E, & faccia&longs;i E il &longs;o&longs;tegno. </s> <s id="id.2.1.649.5.0">Hor per<lb/>cioche maggiore è la proportione di CE ad EB, che di CD à DB; CE haurà <lb/>proportione maggiore ad EB, che il pe&longs;o A alla po&longs;&longs;anza di diece po&longs;ta in C; <lb/>dunque la po&longs;&longs;anza di diece po&longs;ta in C mouerà il pe&longs;o A, che è cento, appiccato <emph.end type="italics"/><lb/><arrow.to.target n="note192"></arrow.to.target><emph type="italics"/>in B con la leua BC, che ha il &longs;uo &longs;o&longs;tegno E. <emph.end type="italics"/></s> <s id="id.2.1.651.1.0"><margin.target id="note190"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.652.1.0"><margin.target id="note191"></margin.target><emph type="italics"/>Per lo lemma di questo. <emph.end type="italics"/></s> <s id="id.2.1.653.1.0"><margin.target id="note192"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.654.1.0"><emph type="italics"/>Ma &longs;e la leua fo&longs;&longs;e BC, & il &longs;o&longs;tegno B. </s> <s>diuida&longs;i CB in D per &longs;i fatta maniera, <lb/>che CB habbia la proportione i&longs;te&longs;&longs;a à BD, che ha cento à diece: & &longs;e il pe&longs;o <lb/>A &longs;arà appic<lb/>cato in D, & <lb/>la po&longs;&longs;anza in <lb/>C, la po&longs;&longs;an<emph.end type="italics"/><lb/><arrow.to.target n="note193"></arrow.to.target><emph type="italics"/>za in C come <lb/>diece &longs;o&longs;terrà <lb/>anco il pe&longs;o <lb/>A appiccato <emph.end type="italics"/><lb/><figure id="id.037.01.120.2.jpg" xlink:href="037/01/120/2.jpg"></figure><lb/><emph type="italics"/>in D. </s> <s id="id.2.1.654.2.0">Prenda&longs;i qual &longs;i uoglia punto tra DB, come E, & ponga&longs;i il pe&longs;o A in <emph.end type="italics"/><lb/><arrow.to.target n="note194"></arrow.to.target><emph type="italics"/>E; & per e&longs;&longs;ere proportione maggiore da CB à BE, che da BC à BD; CB <lb/>haurà proportione maggiore à BE, che il pe&longs;o A di cento alla po&longs;&longs;anza di diece. <lb/></s> <s id="id.2.1.654.3.0">Dunque la po&longs;&longs;anza di diece po&longs;ta in C mouerà il pe&longs;o A di cento appiccato in E <emph.end type="italics"/><lb/><arrow.to.target n="note195"></arrow.to.target><emph type="italics"/>con la leua BC, che ha il &longs;o&longs;tegno &longs;uo B. </s> <s>che bi&longs;ognaua menar ad effetto. <emph.end type="italics"/></s> <s id="id.2.1.656.1.0"><margin.target id="note193"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.657.1.0"><margin.target id="note194"></margin.target><emph type="italics"/>Per la ottaua del quinto. <emph.end type="italics"/></s> <s id="id.2.1.658.1.0"><margin.target id="note195"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.659.1.0"><emph type="italics"/>Ma ciò non &longs;i puote mandar' ad e&longs;ecutione con la leua BC, che habbia il &longs;o&longs;tegno &longs;uo <lb/>in B, & il pe&longs;o A di cento &longs;ia appiccato in C. </s> <s id="id.2.1.659.2.0">Percioche ponga&longs;i la po&longs;&longs;anza <lb/>&longs;o&longs;tenente il pe&longs;o A comunque &longs;i &longs;ia tra BC, come in D; &longs;empre la po&longs;&longs;anza <lb/>&longs;arà maggiore del pe&longs;o A. </s> <s id="id.2.1.659.3.0">Per laqual co&longs;a egli è me&longs;tieri che &longs;empre la data po&longs;<emph.end type="italics"/><pb pagenum="53" xlink:href="037/01/121.jpg"/><emph type="italics"/>&longs;anza &longs;ia maggiore del pe&longs;o A. </s> <s id="id.2.1.659.4.0">Sia dunque la po&longs;&longs;anza data, come cento cin<emph.end type="italics"/><arrow.to.target n="note196"></arrow.to.target><lb/><emph type="italics"/>quanta. </s> <s id="id.2.1.659.5.0">Diuida&longs;i BC in D &longs;i fattamente che CB &longs;ia à BD come cento cin<lb/>quanta à cento, cioè tre à due: & &longs;e la po&longs;&longs;anza &longs;arà po&longs;ta in D, egli è chiaro, <lb/>che la po&longs;&longs;anza in D &longs;o&longs;ter<lb/>rà il pe&longs;o A appiccato in C. <emph.end type="italics"/><arrow.to.target n="note197"></arrow.to.target><lb/></s> <s><emph type="italics"/>& co&longs;i prenda&longs;i tra DC <lb/>qual &longs;i voglia punto, come <lb/>E, & ponga&longs;i la po&longs;&longs;anza <lb/>mouente in E, & per e&longs;&longs;ere <lb/>proportion maggiore da EB <lb/>à BC, che da DB à BC; <lb/>haurà EB proportione mag<lb/>giore à BC, che il pe&longs;o A <emph.end type="italics"/><arrow.to.target n="note198"></arrow.to.target><lb/><figure id="id.037.01.121.1.jpg" xlink:href="037/01/121/1.jpg"></figure><lb/><emph type="italics"/>alla po&longs;&longs;anza in E. </s> <s id="id.2.1.659.6.0">Dunque la po&longs;&longs;anza di cento cinquanta po&longs;ta in E mouerà il <lb/>pe&longs;o A di cento appiccato in C con la leua BC che hà il &longs;o&longs;tegno B. </s> <s>come bi<emph.end type="italics"/><arrow.to.target n="note199"></arrow.to.target><lb/><emph type="italics"/>&longs;ognaua oprare. <emph.end type="italics"/></s> <s id="id.2.1.661.1.0"><margin.target id="note196"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 2. <emph type="italics"/>corollario della<emph.end type="italics"/> 3. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.662.1.0"><margin.target id="note197"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.663.1.0"><margin.target id="note198"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto. <emph.end type="italics"/></s> <s id="id.2.1.664.1.0"><margin.target id="note199"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.665.1.0">COROLLARIO. </s> <s id="id.2.1.666.1.0">Di qui è manife&longs;to, &longs;e la data po&longs;&longs;anza &longs;arà maggiore del dato <lb/>pe&longs;o, que&longs;to poter&longs;i fare, ouero &longs;tando in maniera la leua, <lb/>che il &longs;o&longs;tegno &longs;uo &longs;ia fra il pe&longs;o, & la po&longs;&longs;anza; ouero che el<lb/>la habbia il pe&longs;o fra il &longs;o&longs;tegno, & la po&longs;&longs;anza; ouero alla fine <lb/>e&longs;&longs;endo po&longs;ta la po&longs;&longs;anza fra il pe&longs;o, & il &longs;o&longs;tegno. </s> <s id="id.2.1.667.1.0">Ma &longs;e la data po&longs;&longs;anza &longs;arà minore, ouero eguale al dato pe&longs;o, <lb/>egli è parimente chiaro, che il mede&longs;imo &longs;i puote mandare ad <lb/>e&longs;ecutione &longs;olamente &longs;tando la leua in maniera, che il &longs;o&longs;te<lb/>gno &longs;uo &longs;ia tra il pe&longs;o, & la po&longs;&longs;anza; ouero che ella habbia il <lb/>pe&longs;o fra il &longs;o&longs;tegno, & la po&longs;&longs;anza. </s> <s id="id.2.1.668.1.0">PROPOSITIONE XIII. </s> <s id="id.2.1.669.1.0">PROBLEMA. </s> <s id="id.2.1.670.1.0">Dati quanti &longs;i <expan abbr="vogliã">voglian</expan> pe&longs;i appiccati douunque &longs;i &longs;iano nella leua <lb/>il cui &longs;o&longs;tegno parimente &longs;ia dato, ritrouare vna po&longs;&longs;anza la <lb/>quale &longs;o&longs;tenga i dati pe&longs;i in vn punto dato. </s> <pb xlink:href="037/01/122.jpg"/> <s id="id.2.1.672.1.0"><emph type="italics"/>Siano i dati pe&longs;i ABC nella leua DE, & il &longs;o&longs;tegno &longs;uo F, douunque ne' pun<lb/>ti DGH &longs;iano appiccati, & habbia&longs;i à collocare la po&longs;&longs;anza nel punto E. </s> <s id="id.2.1.672.2.0">egli <lb/>è me&longs;tieri trouare la po&longs;&longs;anza, laquale &longs;o&longs;tenga in E i dati pe&longs;i ABC con la le <lb/>ua DE. </s> <s id="id.2.1.672.3.0">diuida&longs;i DG in K &longs;i fattamente, che DK &longs;ia à KG come il pe<lb/>&longs;o B al pe&longs;o A; dapoi diuida&longs;i KH in L &longs;i fattamente, che KL &longs;ia ad LH <lb/>come il pe&longs;o C à i pe&longs;i BA, & come FE ad FL, co&longs;i faccian&longs;i i pe&longs;i ABC <emph.end type="italics"/><lb/><figure id="id.037.01.122.1.jpg" xlink:href="037/01/122/1.jpg"></figure><lb/><emph type="italics"/>tutti in&longs;ieme alla po&longs;&longs;anza, laquale ponga&longs;i in E. </s> <s>dico, che la po&longs;&longs;anza in E. </s> <s>&longs;o<emph.end type="italics"/><lb/><arrow.to.target n="note200"></arrow.to.target><emph type="italics"/>&longs;tenterà i dati pe&longs;i ABC appiccati in DGH con la leua DE che ha il &longs;o&longs;te<emph.end type="italics"/><lb/><arrow.to.target n="note201"></arrow.to.target><emph type="italics"/>gno &longs;uo F. </s> <s id="id.2.1.672.4.0">Hor percioche &longs;e i pe&longs;i ABC fo&longs;&longs;ero appiccati in&longs;ieme in L, la po&longs;<lb/>&longs;anza in E &longs;o&longs;terrebbe i dati pe&longs;i appiccati in L; ma i pe&longs;i ABC pe&longs;ano tan<lb/>to in L, quanto &longs;e C in H, & BA in&longs;ieme fo&longs;&longs;ero appiccati in K; & AB <emph.end type="italics"/><lb/><arrow.to.target n="note202"></arrow.to.target><emph type="italics"/>nel K tanto pe&longs;ano, quanto &longs;e A in D, & B in G fo&longs;&longs;ero appiccati; dun<lb/>què la po&longs;&longs;anza in E &longs;o&longs;tenterà i dati pe&longs;i ABC appiccati in DGH con la <lb/>leua DE che ha il &longs;o&longs;tegno F. </s> <s id="id.2.1.672.5.0">Che &longs;e la po&longs;&longs;anza haue&longs;&longs;e ad e&longs;&longs;ere po&longs;ta in qual <lb/>&longs;i voglia altro punto dalla leua DE fuor che in F, come in K; faccia&longs;i come <lb/>FK ad FL, co&longs;i i pe&longs;i ABC &longs;iano alla po&longs;&longs;anza: &longs;imilmente dimo&longs;treremo, <lb/>che la po&longs;&longs;anza in K &longs;o&longs;terrà i pe&longs;i ABC ne' punti DGH appiccati. </s> <s id="id.2.1.672.6.0">come <lb/>bi&longs;ognaua fare. <emph.end type="italics"/></s> <s id="id.2.1.674.1.0"><margin.target id="note200"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.675.1.0"><margin.target id="note201"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo della bilancia. <emph.end type="italics"/></s> <s id="id.2.1.676.1.0"><margin.target id="note202"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.677.1.0"><emph type="italics"/>Da que&longs;ta, & dalla quinta di que&longs;to, &longs;e i pe&longs;i ABC &longs;aranno po&longs;ti in qual &longs;i voglia <lb/>modo nella leua DE, & che bi&longs;ogni ritrouare la po&longs;&longs;anza, la quale debba &longs;o&longs;te<lb/>nere in E i dati pe&longs;i &longs;iano tirate da i centri delle grauezze de i pe&longs;i le linee AB <lb/>C à piombo de gli orizonti, lequali taglino la leua DE ne' punti DGH; & <emph.end type="italics"/><pb pagenum="54" xlink:href="037/01/123.jpg"/><emph type="italics"/>&longs;i operino le altre co&longs;e nell'i&longs;te&longs;&longs;o modo: egli è manife&longs;to, che la po&longs;&longs;anza in E,<emph.end type="italics"/><lb/><figure id="id.037.01.123.1.jpg" xlink:href="037/01/123/1.jpg"></figure><lb/><emph type="italics"/>ouero in K &longs;o&longs;tenterà i dati pe&longs;i, percioche egli è l'i&longs;te&longs;&longs;o come &longs;e i pe&longs;i fo&longs;&longs;ero <lb/>appiccati in DGH. <emph.end type="italics"/></s> <s id="id.2.1.679.1.0">PROPOSITIONE XIIII. </s> <s id="id.2.1.680.1.0">PROBLEMA. </s> <s id="id.2.1.681.1.0">Fare che vna data po&longs;&longs;anza moua quanti pe&longs;i &longs;i vogliano, po&longs;ti <lb/>douunque, & in qualunque modo &longs;i &longs;ia in vna data leua. </s> <s id="id.2.1.682.1.0"><emph type="italics"/>Sia la data leua DE, & &longs;iano i dati pe&longs;i, come è po&longs;to nel precedente corollario, & <lb/>&longs;ia A come cento, B come cinquanta, & C come trenta; & la data po&longs;&longs;an<lb/>za &longs;ia come trenta. </s> <s id="id.2.1.682.2.0">&longs;iano po&longs;te le co&longs;e mede&longs;ime, & ritroui&longs;i il punto L; dapoi <emph.end type="italics"/><lb/><figure id="id.037.01.124.1.jpg" xlink:href="037/01/124/1.jpg"></figure><lb/><emph type="italics"/>diuida&longs;i LE in F, &longs;i fattamente che FE ad FL &longs;ia come cento ottanta à <lb/>trenta, cioè &longs;ei ad vno, & &longs;e F &longs;i face&longs;&longs;e &longs;o&longs;tegno, la po&longs;&longs;anza come trenta <emph.end type="italics"/><pb xlink:href="037/01/124.jpg"/><arrow.to.target n="note203"></arrow.to.target><emph type="italics"/>in E &longs;o&longs;terrebbe i pe&longs;i ABC. </s> <s id="id.2.1.682.3.0">pigli&longs;i dunque tra LF qualunque punto come <lb/>M, & faccia&longs;i M il &longs;o&longs;tegno: egli è manife&longs;to, che la po&longs;&longs;anza po&longs;ta in E co<emph.end type="italics"/><lb/><figure id="id.037.01.124.2.jpg" xlink:href="037/01/124/2.jpg"></figure><lb/><arrow.to.target n="note204"></arrow.to.target><emph type="italics"/>me trenta mouerài pe&longs;i ABC come cento ottanta con la leua DE. </s> <s>che bi&longs;o<lb/>gnaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.685.1.0"><margin.target id="note203"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 13. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.686.1.0"><margin.target id="note204"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.687.1.0"><emph type="italics"/>Ma ciò non potremo già vniuer&longs;almente menare ad effetto, &longs;e il &longs;o&longs;tegno fo&longs;&longs;e nelle <lb/>&longs;tremità della leua, come in D; peroche la proportione di DE à DL, cioè la <lb/>proportione de' pe&longs;i ABC alla po&longs;&longs;anza, laquale ha da &longs;o&longs;tenere i pe&longs;i &longs;empre <lb/>è data. </s> <s id="id.2.1.687.2.0">Laqual co&longs;a molto meno anco &longs;i potrebbe fare, &longs;e la po&longs;&longs;anza &longs;i haue&longs;&longs;e <lb/>à porre tra DL. <emph.end type="italics"/></s> <s id="id.2.1.688.1.0">PROPOSITIONE XV. </s> <s id="id.2.1.689.1.0">PROBLEMA. </s> <s id="id.2.1.690.1.0">Ma percioche mentre i pe&longs;i &longs;i mouono con la leua, ha la leua an<lb/>cora grauezza, della quale infin qui non &longs;i è fatto mentione <lb/>alcuna: però dimo&longs;triamo primieramente in che modo &longs;i tro<lb/>ui la po&longs;&longs;anza, la quale &longs;o&longs;tenga nel dato punto la leua data, il <lb/>cui &longs;o&longs;tegno &longs;ia parimente dato. </s> <s id="id.2.1.691.1.0"><emph type="italics"/>Sia la leua data AB, il cui &longs;o&longs;tegno C &longs;ia dato: & &longs;ia il punto D nelquale &longs;i hab<lb/>bia à collocare la po&longs;&longs;anza, che debba &longs;o&longs;tentare la leua AB, &longs;i fattamente che <lb/>re&longs;ti immobile. </s> <s id="id.2.1.691.2.0">&longs;ia dal punto C tirata la linea CE à piombo dell'orizonte la <lb/>quale diuida la leua AB in due parti AE EF; & della parte AE &longs;ia il <lb/>centro G della grauezza, & della parte EF il centro del'a grauezza &longs;ia H, <lb/>& dai punti GH &longs;iano tirate le linee GK HL à piombo de gli orizonti, le<emph.end type="italics"/><pb pagenum="55" xlink:href="037/01/125.jpg"/><emph type="italics"/>quali taglino la linea AF ne' punti KL. </s> <s id="id.2.1.691.3.0">Hor percioche la leua AB è diui<lb/>&longs;a dalla linea CE in due parti, cioè AE EF; però la leua AB, niente altro <lb/>&longs;arà, che due pe&longs;i AE EF nella leua, ouero bilancia AF po&longs;ti; il cui appicca <lb/>mento, ouero &longs;o&longs;tegno è C. </s> <s id="id.2.1.691.4.0">Per laqual co&longs;a i pe&longs;i AE EF &longs;aranno co&longs;i po&longs;ti,<emph.end type="italics"/><lb/><figure id="id.037.01.125.1.jpg" xlink:href="037/01/125/1.jpg"></figure><lb/><emph type="italics"/>come &longs;e fo&longs;&longs;ero appiccati in KL. </s> <s id="id.2.1.691.5.0">Diuida&longs;i dunque KL in M, &longs;i fattamente, <lb/>che KM &longs;ia ad ML come la grauezza della parte EF alla grauezza della <lb/>parte AE; & come CA à CM, co&longs;i faccia&longs;i la grauezza di tutta la leua <lb/>AB alla po&longs;&longs;anza, laquale &longs;e in D &longs;arà collocata (pur che DA &longs;ia à piombo <emph.end type="italics"/><arrow.to.target n="note205"></arrow.to.target><lb/><emph type="italics"/>dell'orizonte) pe&longs;erà egualmente con la leua; cioè &longs;o&longs;terrà la leua AB premendo <lb/>in giù, che bi&longs;ognaua trouare. <emph.end type="italics"/></s> <s id="id.2.1.693.1.0"><margin.target id="note205"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 13. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.694.1.0"><emph type="italics"/>Che &longs;e la po&longs;&longs;anza &longs;i haue&longs;&longs;e à porre nel punto B. </s> <s id="id.2.1.694.2.0">Faccia&longs;i come CF à CM, <lb/>co&longs;i il pe&longs;o AB alla po&longs;&longs;anza. </s> <s id="id.2.1.694.3.0">Con &longs;imile modo prouera&longs;&longs;i che la po&longs;&longs;anza in <lb/>B &longs;o&longs;terrà la leua AB. </s> <s>& l'i&longs;te&longs;&longs;o d mo&longs;trera&longs;&longs;i in qualunque altro &longs;ito s'haue&longs;<lb/>&longs;e à porre la po&longs;&longs;anza, (fuor che in E) come in N. </s> <s>peroche faccia&longs;i CO à <lb/>CM come AB alla po&longs;&longs;anza, laquale &longs;e &longs;i porrà in N &longs;o&longs;tenterà la leua AB. <emph.end type="italics"/></s> <s id="id.2.1.695.1.0">Ma aggiunga &longs;i il pe&longs;o appiccato, ouero po&longs;to nella leua; come, <lb/>po&longs;te le co&longs;e i&longs;te&longs;&longs;e, fia il pe&longs;o P appiccato in A; & la po&longs;<lb/>&longs;anza s'habbia à porre in B, &longs;i fattamente che &longs;o&longs;tenghi la le <lb/>ua AB in&longs;ieme col pe&longs;o P. </s> <s id="id.2.1.696.1.0"><emph type="italics"/>Diuida&longs;i AM in Q, &longs;i fattamente, che AQ &longs;ia à QM, come la grauezza <emph.end type="italics"/><arrow.to.target n="note206"></arrow.to.target><lb/><emph type="italics"/>della leua AB alla grauezza del pe&longs;o P; dapoi come CF à CQ, co&longs;i fac<emph.end type="italics"/><arrow.to.target n="note207"></arrow.to.target><lb/><emph type="italics"/>cia&longs;i la grauezza AB, & P in&longs;ieme alla po&longs;&longs;anza, la quale ponga&longs;i in B: egli <lb/>è manife&longs;to, che la po&longs;&longs;anza in B &longs;o&longs;terrà la leua AB in&longs;ieme co'l pe&longs;o P. </s> <s id="id.2.1.696.2.0">Che <lb/>&longs;e fo&longs;&longs;e CA à CM, come AB à P; &longs;arebbe il punto C il loro centro della <lb/>grauezza, & perciò la leua AB in&longs;ieme co'l pe&longs;o P &longs;enza la po&longs;&longs;anza po&longs;ta in <pb xlink:href="037/01/126.jpg"/>B &longs;tarà ferma. </s> <s id="id.2.1.696.3.0">Ma &longs;e il centro della grauezza de' pe&longs;i fo&longs;&longs;e tra CF, come in <lb/>O. </s> <s id="id.2.1.696.4.0">Faccia&longs;i come CF à CO, co&longs;i AB & P in&longs;ieme alla po&longs;&longs;anza, laqua<lb/>le in B &longs;o&longs;tenterà sì la leua AB come il pe&longs;o P. <emph.end type="italics"/></s> <s id="id.2.1.697.1.0"><margin.target id="note206"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 13 <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.698.1.0"><margin.target id="note207"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di Archimede delle cose che egualmente pe&longs;ano. <emph.end type="italics"/></s> <figure id="id.037.01.126.1.jpg" xlink:href="037/01/126/1.jpg"> </figure> <s id="id.2.1.700.1.0"><emph type="italics"/>Similmente mo&longs;trera&longs;&longs;i il mede&longs;imo &longs;e fo&longs;&longs;ero più pe&longs;i nella leua AB douunque, <lb/>& in qual modo &longs;i &longs;ia di&longs;po&longs;ti. <emph.end type="italics"/></s> <s id="id.2.1.701.1.0"><emph type="italics"/>Oltre à ciò da que&longs;te co&longs;e &longs;i puote cono&longs;cere, come nella decimaquarta propo&longs;itione di <lb/>que&longs;to habbiamo in&longs;egnato, in che modo cioè po&longs;&longs;iamo mouere i dati pe&longs;i po&longs;ti do<lb/>uunque &longs;i voglia nella leua, con vna data po&longs;&longs;anza, e con vna data leua, ilche po&longs;<lb/>&longs;iamo fare nell'i&longs;te&longs;&longs;o modo non &longs;olamente con&longs;iderando la grauezza della leua; ma <lb/>anco gli altri accidenti, iquali &longs;ono &longs;tati di &longs;opra mo&longs;trati &longs;enza la grauezza del<lb/>la leua; con &longs;imile modo con&longs;iderata la grauezza della leua in&longs;ieme co' pe&longs;i, ouero <lb/>&longs;enza pe&longs;i &longs;i mo&longs;treranno. <emph.end type="italics"/></s> <s id="id.2.1.702.1.0">IL FINE DELLA LEVA. </s> </chap> <pb pagenum="56" xlink:href="037/01/127.jpg"/> <chap> <s id="id.2.1.705.1.0">DELLA TAGLIA. </s> <s id="id.2.1.707.1.0">Con l'in&longs;trumento della Taglia &longs;i può mouere il pe<lb/>&longs;o in molti modi: ma percioche in tutti è la ragio<lb/>ne mede&longs;ima: però affine che la co&longs;a re&longs;ti più chia<lb/>ra, intenda&longs;i in quello che &longs;i ha da dire, che il pe<lb/>&longs;o &longs;empre &longs;i habbia da mouere all'insù ad angoli <lb/>retti al piano dell'orizonte in que&longs;to modo. </s> <pb xlink:href="037/01/128.jpg"/> <s id="id.2.1.708.1.0"><emph type="italics"/>Sia il pe&longs;o A ilquale &longs;i habbia ad alzare in sù ad angoli retti al piano dell'orizonte: <lb/>& come &longs;i co&longs;tuma di fare: &longs;ia <lb/>attaccata di &longs;opra vna taglia, <lb/>che habbia due girelle, gli a&longs;&longs;etti <lb/>dellequali &longs;iano in BC: & &longs;ia <lb/>anche legata vn'altra taglia al <lb/>pe&longs;o, laquale &longs;imilmente habbia <lb/>due girelle, gli a&longs;&longs;etti delle qua<lb/>li &longs;iano in DE: & per tutte <lb/>le girelle d'ambedue le taglie &longs;ia <lb/>condotta intorno la corda, la<lb/>quale in vno de i capi, come in <lb/>F deue e&longs;&longs;ere legata. </s> <s id="id.2.1.708.2.0">Ponga&longs;i <lb/>ancorala po&longs;&longs;anza che moue in <lb/>G, laquale mentre di&longs;cende, il <lb/>pe&longs;o A per lo contrario &longs;arà le<lb/>uato in &longs;u&longs;o, &longs;i come afferma Pa<lb/>po nell'ottauo libro delle rac<lb/>colte matematiche, & Vitruuio <lb/>nel decimo dell'architettura, & <lb/>altri. <emph.end type="italics"/></s> <figure id="id.037.01.128.1.jpg" xlink:href="037/01/128/1.jpg"> </figure> <s id="id.2.1.710.1.0">Hor in che modo que&longs;to <lb/>in&longs;trumento della ta<lb/>glia &longs;i riduca alla leua, <lb/>& perche vn pe&longs;o gran<lb/>de &longs;i moua da piccola <lb/>forza, & in qual modo, <lb/>& in quanto tempo; & <lb/>perche la corda debba <lb/>e&longs;&longs;ere legata da vn ca<lb/>po: & quale debba e&longs;<lb/>&longs;ere l'officio della ta<lb/>glia, che è po&longs;ta di &longs;ot<lb/>to, & quale di quella, <lb/>che &longs;tà di &longs;opra, & in <lb/>che modo &longs;i po&longs;&longs;a tro<lb/>uare ogni proportio<lb/>ne data ne i numeri tra la po&longs;&longs;anza, & il pe&longs;o, diciamo. </s> <pb pagenum="57" xlink:href="037/01/129.jpg"/> <s id="id.2.1.712.1.0">LEMMA. </s> <s id="id.2.1.713.1.0">Siano due linee rette AB CD egualmente di&longs;tanti, lequali <lb/>tocchino il cerchio ACE ne' punti AC, il centro delqual <lb/>cerchio &longs;ia F, & &longs;i congiunghino FA & FC. </s> <s id="id.2.1.713.2.0">dico che la <lb/>linea AFC è retta. </s> <s id="id.2.1.714.1.0"><emph type="italics"/>Tiri&longs;i la linea FE egualmente di&longs;tante dal<emph.end type="italics"/><arrow.to.target n="note208"></arrow.to.target><lb/><emph type="italics"/>le linee AB CD. </s> <s id="id.2.1.714.2.0">Et percioche AB <lb/>& FE &longs;ono egualmente di&longs;tanti, & <emph.end type="italics"/><arrow.to.target n="note209"></arrow.to.target><lb/><emph type="italics"/>l'angolo BAF èretto: &longs;arà anco A <lb/>FE retto, & all'i&longs;te&longs;&longs;o modo CFE &longs;a<emph.end type="italics"/><arrow.to.target n="note210"></arrow.to.target><lb/><emph type="italics"/>rà retto: adunque la linea AFC èret<lb/>ta, ilche s'hauea à dimo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.715.1.0"><margin.target id="note208"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <s id="id.2.1.716.1.0"><margin.target id="note209"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.717.1.0"><margin.target id="note210"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 14. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <figure id="id.037.01.129.1.jpg" xlink:href="037/01/129/1.jpg"> </figure> <s id="id.2.1.719.1.0">PROPOSITIONE I. </s> <s id="id.2.1.720.1.0">Se la corda &longs;i condurrà intorno alla girella della taglia, che &longs;ia <lb/>attaccata di &longs;opra, & che vno delli &longs;uoi capi &longs;i leghi al pe&longs;o, & <lb/>l'altro tratanto &longs;ia pre&longs;o dalla po&longs;sanza, che &longs;o&longs;tiene il detto <lb/>pe&longs;o; la po&longs;sanza &longs;arà eguale al pe&longs;o. </s> <pb xlink:href="037/01/130.jpg"/> <s id="id.2.1.722.1.0"><emph type="italics"/>Sia il pe&longs;o A alquale venga legata la corda à B: & la taglia, che habbia la girella <lb/>CEF il cui centro D appicchi&longs;i di &longs;opra: & &longs;ia parimente D il centro dell'a&longs;<lb/>&longs;etto, & d'intorno alla girella volga&longs;i la corda BCEFG: & &longs;ia in G la po&longs;<lb/>&longs;anza, che &longs;o&longs;tiene il pe&longs;o A. </s> <s id="id.2.1.722.2.0">Dico la po&longs;&longs;anza po&longs;ta in G e&longs;&longs;ere eguale al pe<lb/>&longs;o A. </s> <s id="id.2.1.722.3.0">&longs;ia FG <lb/>egualmente di<lb/>&longs;tante da CB. <lb/></s> <s id="id.2.1.722.4.0">Percioche <expan abbr="d&umacr;que">dun<lb/>que</expan> il pe&longs;o A <lb/>&longs;ta fermo, &longs;a<emph.end type="italics"/><lb/><arrow.to.target n="note211"></arrow.to.target><emph type="italics"/>rà CB à piom<lb/>bo del piano <lb/>dell'orizonte. <emph.end type="italics"/><lb/></s> <s><arrow.to.target n="note212"></arrow.to.target><emph type="italics"/>onde FG &longs;a<lb/>rà al piano <lb/>i&longs;te&longs;&longs;o à piom<lb/>bo. </s> <s id="id.2.1.722.5.0">Siano i <lb/>punti CF nel<lb/>la girella, da <lb/>quali le corde <lb/>CB FG &longs;cen<lb/>dano nel pia<lb/>no dell'orizon<lb/>te ad angoli <lb/>retti, tocche<lb/>ranno le dette <lb/>corde BC FG <lb/>la girella CE<emph.end type="italics"/><lb/><figure id="id.037.01.130.1.jpg" xlink:href="037/01/130/1.jpg"></figure><lb/><emph type="italics"/>F ne'punti CF peroche non po&longs;&longs;ono &longs;egare la girella. </s> <s id="id.2.1.722.6.0">Siano congiunte le li<emph.end type="italics"/><lb/><arrow.to.target n="note213"></arrow.to.target><emph type="italics"/>nee DC DF. </s> <s>&longs;arà retta la linea CF & &longs;aranno anche retti gli angoli DCB <lb/>DFG. </s> <s id="id.2.1.722.7.0">Ma percioche BC &longs;ta à piombo sì all'orizonte, come ad e&longs;&longs;a CF &longs;arà <emph.end type="italics"/><lb/><arrow.to.target n="note214"></arrow.to.target><emph type="italics"/>la detta CF egualmente di&longs;tante dall'orizonte. </s> <s id="id.2.1.722.8.0">& concio&longs;ia che il pe&longs;o &longs;ia attac<lb/>cato in CB & la po&longs;&longs;anza &longs;ia in G ch'è il mede&longs;imo, come &longs;e ella fo&longs;&longs;e in F: <emph.end type="italics"/><lb/><arrow.to.target n="note215"></arrow.to.target><emph type="italics"/>&longs;arà CF tanto quanto vna bilancia, ouero vna leua, il cui centro, ouero &longs;o&longs;tegno <lb/>&longs;arà D, imperoche la girella è &longs;o&longs;tenuta nell'a&longs;&longs;etto, & il punto D per e&longs;&longs;ere <lb/>centro dell'a&longs;&longs;etto, & della girella rimane immobile, &longs;e ben l'vno, & l'altro &longs;i vol<lb/>gono intorno. </s> <s id="id.2.1.722.9.0">Per laqual co&longs;a e&longs;&longs;endo la di&longs;tanza DC eguale alla di&longs;tanza DF, <lb/>& la po&longs;&longs;anza che è in F contrape&longs;i egualmente al pe&longs;o A attaccato in C &longs;o<lb/>&longs;tenendo il pe&longs;o in modo, che non cala al ba&longs;&longs;o, &longs;arà la po&longs;&longs;anza a&longs;&longs;egnata in F oue<lb/>ro in G che è tutt'vno, eguale al pe&longs;o A: percioche po&longs;ta in G fa l'i&longs;te&longs;&longs;o effet<lb/>to che &longs;e nel mede&longs;imo G fo&longs;&longs;e appiccato vn'altro pe&longs;o eguale al pe&longs;o A, liquali <lb/>pe&longs;i attaccati in CF contrape&longs;eranno egualmente. </s> <s id="id.2.1.722.10.0">Oltre à ciò non facendo&longs;i moto <emph.end type="italics"/><pb pagenum="58" xlink:href="037/01/131.jpg"/><emph type="italics"/>in niuna delle parti, &longs;arà l'i&longs;te&longs;&longs;o e&longs;&longs;endo circondata in que&longs;to modo la girella intor<lb/>no con vna corda &longs;ola BC e FG come &longs;e fu&longs;&longs;ero due corde BC FG legate <lb/>alla leua, ouero alla bilancia CF. <emph.end type="italics"/></s> <s id="id.2.1.724.1.0"><margin.target id="note211"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>di questo della bilancia. <emph.end type="italics"/></s> <s id="id.2.1.725.1.0"><margin.target id="note212"></margin.target><emph type="italics"/>Per la ottaua dell'vndecimo. <emph.end type="italics"/></s> <s id="id.2.1.726.1.0"><margin.target id="note213"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del terzo. <emph.end type="italics"/></s> <s id="id.2.1.727.1.0"><margin.target id="note214"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 28. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.728.1.0"><margin.target id="note215"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>del<emph.end type="italics"/> 1. <emph type="italics"/>d'Archimede delle co&longs;e che pe&longs;ano egualmente:. <emph.end type="italics"/></s> <s id="id.2.1.729.1.0">COROLLARIO. </s> <s id="id.2.1.730.1.0">Da que&longs;to può e&longs;&longs;ere manife&longs;to, che il mede&longs;imo pe&longs;o dalla i&longs;te&longs;<lb/>&longs;a po&longs;&longs;anza puote e&longs;&longs;ere tuttauia &longs;o&longs;tenuto &longs;enza anche alcu<lb/>no aiuto di que&longs;ta taglia. </s> <s id="id.2.1.731.1.0"><emph type="italics"/>Percioche &longs;ia il pe&longs;o H eguale al pe&longs;o A à cui &longs;ia legata la corda KL & &longs;iala <lb/>po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o H in L. </s> <s id="id.2.1.731.2.0">Hor concio&longs;ia che volendo &longs;o&longs;tenere al<lb/>cun pe&longs;o &longs;enza aiuto veruno vi bi&longs;ogni tanta forza, quanta &longs;ia eguale al pe&longs;o; la <lb/>po&longs;&longs;anza che è in L &longs;arà eguale al <lb/>pe&longs;o H, ma il pe&longs;o H è po&longs;to <lb/>eguale al pe&longs;o A, alquale è anco <lb/>eguale la po&longs;&longs;anza G. </s> <s id="id.2.1.731.3.0">&longs;arà dun<lb/>que la po&longs;&longs;anza in G eguale alla <lb/>po&longs;&longs;anza in L che è l'i&longs;te&longs;&longs;o, come &longs;e <lb/>la i&longs;te&longs;&longs;a po&longs;&longs;anza &longs;o&longs;tene&longs;&longs;e il pe&longs;o <lb/>mede&longs;imo. </s> <s id="id.2.1.731.4.0">Oltre à ciò &longs;e le po&longs;&longs;an<lb/>ze, lequali &longs;ono in G & in L &longs;o&longs;<lb/>&longs;ero eguali fra loro, & poi &longs;epara<lb/>tamente dai pe&longs;i minori, è co&longs;a chia<lb/>ra, che le dette po&longs;&longs;anze non &longs;areb<lb/>bono &longs;ufficienti à &longs;o&longs;tenere quei pe&longs;i <lb/>che &longs;e que&longs;te po&longs;&longs;anze &longs;aranno mag<lb/>giori, egli è manife&longs;to, che e&longs;&longs;e mo<lb/>ueranno i pe&longs;i. </s> <s id="id.2.1.731.5.0">& co&longs;i la po&longs;&longs;anza <lb/>in L col pe&longs;o H venirà ad e&longs;&longs;e<lb/>re nella proportione mede&longs;ima, co<lb/>me la po&longs;&longs;anza in G col pe&longs;o A. <emph.end type="italics"/></s> <figure id="id.037.01.131.1.jpg" xlink:href="037/01/131/1.jpg"> </figure> <s id="id.2.1.733.1.0"><emph type="italics"/>Ma perche nella dimo&longs;tratione è &longs;tato <lb/>pre&longs;uppo&longs;to che l'a&longs;&longs;etto &longs;i volga in <lb/>torno, ilquale il più delle volte &longs;tà <lb/>immobile, però &longs;tando anche immobile il detto a&longs;&longs;etto dimo&longs;tri&longs;i l'i&longs;te&longs;&longs;o. <emph.end type="italics"/></s> <pb xlink:href="037/01/132.jpg"/> <s id="id.2.1.735.1.0"><emph type="italics"/>Sia la girella della taglia CEF, il cui centro &longs;ia D, & &longs;ia l'a&longs;&longs;etto GHK, il cen<lb/>tro delquale &longs;ia mede&longs;imamente D: Tiri&longs;i il diametro CGDKF egualmente <lb/>di&longs;tante dall'orizonte. </s> <s id="id.2.1.735.2.0">et percioche <expan abbr="m&etilde;">men<lb/>tre</expan> la girella &longs;i volge, la circonferenza <lb/>del cerchio CEF &longs;empre va egual<lb/>mente di&longs;tante alla circonferenza del<lb/>l'a&longs;&longs;etto GHK: percioche ella &longs;i <lb/>volge intorno à l'a&longs;&longs;etto, & le circonfe<lb/>renze de' cerchi egualmente di&longs;tanti <lb/>hanno il centro mede&longs;imo, &longs;arà il pun<lb/>to D &longs;empre centro & della girella, <lb/>& dell'a&longs;&longs;etto. </s> <s id="id.2.1.735.3.0">Per laqual co&longs;a e&longs;&longs;en<lb/>do DC eguale à DF & DG ad <lb/>e&longs;&longs;o DK, &longs;arà GC ad e&longs;&longs;o KF egua<lb/>le. </s> <s id="id.2.1.735.4.0">Se dunque nella leua, ouero bilan<lb/>cia CF &longs;i attaccheranno pe&longs;i eguali, <lb/>contrape&longs;eranno egualmente, peroche <lb/>la di&longs;tanza CG è eguale alla di&longs;tan<lb/>za KF, & l'a&longs;&longs;etto GHK immobi<lb/>le &longs;erue per centro, ouero per &longs;o&longs;tegno. </s> <s id="id.2.1.735.5.0">Stando dunque immobile l'a&longs;&longs;etto, &longs;e la <lb/>po&longs;&longs;anza &longs;i metterà in F che &longs;o&longs;tenga il pe&longs;o appiccato in C, &longs;arà la po&longs;&longs;anza <lb/>in F ad e&longs;&longs;o pe&longs;o eguale, ilche era da mo&longs;trare. <emph.end type="italics"/></s> <figure id="id.037.01.132.1.jpg" xlink:href="037/01/132/1.jpg"> </figure> <s id="id.2.1.737.1.0"><emph type="italics"/>Et concio&longs;ia che del tutto &longs;ia il mede&longs;imo, che l'a&longs;&longs;etto ouero &longs;i volga intorno, ò non <lb/>&longs;i volga: però &longs;ia lecito nelle co&longs;e, che &longs;i hanno à dire, prendere in loco dello a&longs;&longs;etto <lb/>il centro &longs;olamente. <emph.end type="italics"/></s> <s id="id.2.1.738.1.0">PROPOSITIONE II. </s> <s id="id.2.1.739.1.0">Se la corda &longs;i condurrà intorno alla girella della taglia, che &longs;ia <lb/>legata al pe&longs;o, legando l'vn de' capi &longs;uoi in qualche loco, & <lb/>l'altro &longs;ia pre&longs;o dalla po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o, &longs;arà la po&longs;<lb/>&longs;anza la metà meno del pe&longs;o. </s> <s id="id.2.1.740.1.0"><emph type="italics"/>Sia il pe&longs;o A. </s> <s id="id.2.1.740.2.0">&longs;ia BCD la girella della taglia legata al pe&longs;o, il cui centro &longs;ia E, <lb/>&longs;ia dapoi inuolta d'intorno la girella la corda FBCDG, & legata in F, & &longs;ia <lb/>la po&longs;&longs;anza in G che &longs;o&longs;tiene il pe&longs;o A. </s> <s id="id.2.1.740.3.0">Dico che la po&longs;&longs;anza in G è la metà <lb/>meno del pe&longs;o A. </s> <s id="id.2.1.740.4.0">Siano le corde FB GD perpendicolari all' orizonte del pun<emph.end type="italics"/><lb/><arrow.to.target n="note216"></arrow.to.target><emph type="italics"/>to E, lequali &longs;aranno fra loro egualmente di&longs;tanti: & tocchino le dette corde <lb/>FBGD, il cerchio BCD ne i punti BD: congiunga&longs;i la linea BD ella pa&longs;<emph.end type="italics"/><pb pagenum="59" xlink:href="037/01/133.jpg"/><emph type="italics"/>&longs;erà per E centro, & &longs;arà egualmente di&longs;tante dall'orìzonte di e&longs;&longs;o centro, & <emph.end type="italics"/><arrow.to.target n="note217"></arrow.to.target><lb/><emph type="italics"/>concio&longs;ia che la G po&longs;&longs;anza debba &longs;o&longs;tenere il pe&longs;o A con la taglia; bi&longs;ogna, <lb/>che la corda &longs;ia legata dal'vno de' capi, come in<emph.end type="italics"/> F, <emph type="italics"/>&longs;i fattamente, che F fac<lb/>cia re&longs;i&longs;tenza egualmente almeno alla po&longs;&longs;anza, ch'è in G, altramente e&longs;&longs;a po&longs;&longs;an<lb/>za in G non potrebbe à modo alcuno &longs;o&longs;tenere il pe&longs;o. </s> <s id="id.2.1.740.5.0">Et perche la po&longs;&longs;anza <emph.end type="italics"/><lb/><figure id="id.037.01.133.1.jpg" xlink:href="037/01/133/1.jpg"></figure><lb/><emph type="italics"/>&longs;o&longs;tiene la girella mediante la corda, & la girella &longs;o&longs;tiene la parte re&longs;tante della <lb/>taglia mediante l'a&longs;&longs;etto, allaqual taglia il pe&longs;o è appiccato, pe&longs;erà que&longs;ta parte del<lb/>la taglia nell'a&longs;&longs;etto, cioè nel centro E: onde il pe&longs;o A pe&longs;erà &longs;imilmente nel me<lb/>de&longs;imo centro E, come &longs;e egli fo&longs;&longs;e appiccato in E. </s> <s id="id.2.1.740.6.0">Po&longs;ta dunque la po&longs;&longs;anza <lb/>che stà in G doue è D (perche egli è totalmente il mede&longs;imo) &longs;arà BD come <lb/>vna lèua, il cui &longs;o&longs;tegno &longs;arà B, & il pe&longs;o attaccato in E, & la po&longs;&longs;anza in D: <lb/>& e&longs;&longs;endo la corda FB immobile, conueneuolmente il B puote &longs;eruire per &longs;o<lb/>&longs;tegno. </s> <s id="id.2.1.740.7.0">Ma ciò più chiaramente apparerà dapoi. </s> <s id="id.2.1.740.8.0">Hora percioche la po&longs;&longs;anza al <emph.end type="italics"/><arrow.to.target n="note218"></arrow.to.target><lb/><emph type="italics"/>pe&longs;o ha la proportione mede&longs;ima, che hà BE à BD, & BE in proportione <lb/>è la metà manco di BD: dunque la po&longs;&longs;anza che è in G &longs;arà la metà meno del <lb/>pe&longs;o A. </s> <s id="id.2.1.740.9.0">Che bi&longs;ognaua dimo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.742.1.0"><margin.target id="note216"></margin.target><emph type="italics"/>Per la &longs;esta dell'vndecimo. <emph.end type="italics"/></s> <s id="id.2.1.743.1.0"><margin.target id="note217"></margin.target><emph type="italics"/>Per la procedense. <emph.end type="italics"/></s> <s id="id.2.1.744.1.0"><margin.target id="note218"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo nella leua. <emph.end type="italics"/></s> <pb xlink:href="037/01/134.jpg"/> <s id="id.2.1.745.1.0"><emph type="italics"/>Que&longs;to dunque &longs;tà nell'i&longs;te&longs;&longs;o modo con vna corda &longs;ola FBCDG condotta intor<lb/>no alla girella, come &longs;e fo&longs;&longs;ero due corde BF GD legate alla leua BD, il cui <emph.end type="italics"/><lb/><figure id="id.037.01.134.1.jpg" xlink:href="037/01/134/1.jpg"></figure><lb/><emph type="italics"/>&longs;o&longs;tegno &longs;arà B, & il pe&longs;o fo&longs;&longs;e attaccato in E & la po&longs;&longs;anza, che lo &longs;o&longs;tiene <lb/>fo&longs;&longs;e in D, ouero in G che è l'iste&longs;&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.747.1.0">COROLLARIO I. </s> <s id="id.2.1.748.1.0">Da que&longs;to dunque è manife&longs;to, che il pe&longs;o è &longs;o&longs;tenuto à que&longs;to <lb/>modo da po&longs;&longs;anza minore in proportlone della metà meno, <lb/>di quel che &longs;arebbe &longs;enza aiuto veruno di cotale taglia. </s> <pb pagenum="60" xlink:href="037/01/135.jpg"/> <s id="id.2.1.750.1.0"><emph type="italics"/>Come &longs;ia il pe&longs;o H eguale al <lb/>pe&longs;o A, alquale &longs;ia lega<lb/>tala corda KL, & la po&longs;<lb/>&longs;anza, che è in L &longs;o&longs;ten<lb/>ga il pe&longs;o H, &longs;arà la po&longs;<lb/>&longs;anza in L &longs;eparatamente <lb/>eguale al pe&longs;o H, & al <lb/>pe&longs;o A; ma la po&longs;&longs;an<lb/>za, che è in G in propor<lb/>tione è la metà manco del <lb/>pe&longs;o A. </s> <s id="id.2.1.750.2.0">Per laqual co&longs;a <lb/>la po&longs;&longs;anza che è in G &longs;a<lb/>rà la metà meno in propor<lb/>tione della po&longs;&longs;anza, che è <lb/>in L, & in que&longs;to modo <lb/>ne gli altri tutti di que&longs;ta <lb/>maniera &longs;i potrà ritrouare <lb/>la proportione. <emph.end type="italics"/></s> <figure id="id.037.01.135.1.jpg" xlink:href="037/01/135/1.jpg"> </figure> <s id="id.2.1.752.1.0">COROLLARIO II. </s> <s id="id.2.1.753.1.0">Egli è manife&longs;to ancora, &longs;e &longs;aranno due po&longs;&longs;anze l'vna in G & <lb/>l'altra in F, lequali &longs;o&longs;tengano il pe&longs;o A, che l'vna, & l'al<lb/>tra in&longs;ieme &longs;aranno eguali al pe&longs;o A, & cia&longs;cheduna di loro <lb/>&longs;o&longs;terrà la metà del pe&longs;o A. </s> <s id="id.2.1.754.1.0"><emph type="italics"/>Et que&longs;to è manife&longs;to dal terzo & dal quarto corollario del &longs;econdo di que&longs;to nel trat<lb/>tato della leua. <emph.end type="italics"/></s> <s id="id.2.1.755.1.0">COROLLARIO III. </s> <s id="id.2.1.756.1.0">Oltre à ciò que&longs;to parimente &longs;i fa noto, perche cioè la corda <lb/>debba e&longs;&longs;ere legata nell'vno de' capi. </s> <pb xlink:href="037/01/136.jpg"/> <s id="id.2.1.757.1.0">PROPOSITIONE III. </s> <s id="id.2.1.758.1.0">Se à cia&longs;cuna dell'vna, & l'altra girella delle due taglie, l'vna del<lb/>le quali &longs;ia po&longs;ta di &longs;opra, & l'altra di &longs;otto, & que&longs;ta &longs;ia lega<lb/>ta al pe&longs;o; &longs;arà condotta intorno la corda: legando l'vno de' <lb/>capi in qualche loco, & l'altro &longs;ia tenuto dalla po&longs;&longs;anza, che <lb/>&longs;o&longs;tiene il pe&longs;o, &longs;arà la po&longs;&longs;anza la metà meno del pe&longs;o. </s> <s id="id.2.1.759.1.0"><emph type="italics"/>Sia il pe&longs;o A, &longs;ia BCD la girella della <lb/>taglia, che &longs;ia legata al pe&longs;o A, il cui <lb/>centro &longs;ia K, & EFG &longs;ia la girella <lb/>della taglia appiccata di opra, il cui cen<lb/>tro &longs;ia H, dapoi &longs;ia condotta intorno <lb/>le girelle la corda LBCDMEFGN <lb/>laquale &longs;ia legata in L, & &longs;ia la po&longs;<lb/>&longs;anza, che &longs;o&longs;tiene il pe&longs;o A in N. <lb/></s> <s id="id.2.1.759.2.0">Dico la po&longs;&longs;anza, che &longs;ta in N e&longs;&longs;e<lb/>re la metà meno del pe&longs;o A. </s> <s id="id.2.1.759.3.0">Percio<lb/>che &longs;e la po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o <emph.end type="italics"/><lb/><arrow.to.target n="note219"></arrow.to.target><emph type="italics"/>A fo&longs;&longs;e collocata doue &longs;ta M, &longs;areb<lb/>be per certo la po&longs;&longs;anza in M la metà <emph.end type="italics"/><lb/><arrow.to.target n="note220"></arrow.to.target><emph type="italics"/>meno del pe&longs;o A: & alla po&longs;&longs;anza in <lb/>M è eguale la forza di N, percioche <lb/>egli è come &longs;e la po&longs;&longs;anza in M &longs;o&longs;te<lb/>ne&longs;&longs;e la metà del pe&longs;o A &longs;enza taglia, <lb/>alquale egualmente contrape&longs;a il pe&longs;o che <lb/>è in N per e&longs;&longs;ere eguale alla metà del <lb/>pe&longs;o A. </s> <s id="id.2.1.759.4.0">Per laqual co&longs;a la forza in N <lb/>che è alla metà del pe&longs;o A eguale, &longs;o<lb/>&longs;tenirà e&longs;&longs;o A. </s> <s id="id.2.1.759.5.0">La po&longs;&longs;anza dunque in <lb/>N che &longs;o&longs;tiene il pe&longs;o A, è la metà <lb/>meno di e&longs;&longs;o A. </s> <s>che bi&longs;ognaua mo<lb/>&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.760.1.0"><margin.target id="note219"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.761.1.0"><margin.target id="note220"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.136.1.jpg" xlink:href="037/01/136/1.jpg"> </figure> <pb pagenum="61" xlink:href="037/01/137.jpg"/> <s id="id.2.1.764.1.0"><emph type="italics"/>Ma &longs;e, come nella &longs;econda figura, la cor<lb/>da BCDEFGHKL &longs;arà inuolta <lb/>d'intorno à le girelle, & legata in <lb/>B: & la po&longs;&longs;anza in L &longs;o&longs;tenga il <lb/>pe&longs;o A, &longs;arà &longs;imilmente la po&longs;&longs;an<lb/>za in L la metà meno del pe&longs;o: <lb/>Peroche la girella della taglia di &longs;o<lb/>pra, & la taglia i&longs;te&longs;&longs;a &longs;ono del tut<lb/>to inutili: & è il mede&longs;imo, come &longs;e <lb/>la corda fo&longs;&longs;e legata in F, & che <lb/>la po&longs;&longs;anza in L &longs;o&longs;tene&longs;&longs;e il pe&longs;o <lb/>con la &longs;ola taglia legata al pe&longs;o, la <lb/>qual po&longs;&longs;anza è &longs;tata dimo&longs;trata e&longs;<lb/>&longs;ere la metà meno del pe&longs;o A. <emph.end type="italics"/></s> <figure id="id.037.01.137.1.jpg" xlink:href="037/01/137/1.jpg"> </figure> <s id="id.2.1.766.1.0">COROLLARIO. </s> <s id="id.2.1.767.1.0">Seguita da que&longs;te co&longs;e, che &longs;e &longs;aranno due po&longs;&longs;anze in BL, am<lb/>bedue tra loro &longs;aranno eguali. </s> <s id="id.2.1.768.1.0"><emph type="italics"/>Percioche ogn'vna di loro da per &longs;e è la metàmeno di e&longs;&longs;o A. <emph.end type="italics"/></s> <pb xlink:href="037/01/138.jpg"/> <s id="id.2.1.770.1.0">PROPOSITIONE IIII. </s> <s id="id.2.1.771.1.0">Sia la leua AB, il cui &longs;o&longs;tegno &longs;ia A, laqual leua &longs;ia diui&longs;a in <lb/>due parti eguali in D, & &longs;ia il pe&longs;o C appiccato in D, & <lb/>&longs;iano due po&longs;&longs;anze eguali in BD, che &longs;o&longs;tengano il pe&longs;o C. <lb/></s> <s id="id.2.1.771.2.0">Dico, che ogn'vna di que&longs;te po&longs;&longs;anze po&longs;te in BD è vn ter<lb/>zo del pe&longs;o C. </s> <s id="id.2.1.772.1.0"><emph type="italics"/>Hor percioche vna delle due po&longs;&longs;anze è collocata in D, & il pe&longs;o C &longs;tà appiccato <lb/>all'i&longs;te&longs;&longs;o punto D. </s> <s id="id.2.1.772.2.0">La po&longs;&longs;anza in D &longs;o&longs;ienirà la parte del pe&longs;o C, che &longs;arà <lb/>eguale ad e&longs;&longs;a po&longs;&longs;an<lb/>za D. </s> <s id="id.2.1.772.3.0">Per laqual co <lb/>&longs;ala po&longs;&longs;anza in B &longs;o<lb/>&longs;tenirà l'altra parte re <lb/>&longs;tante, laqual parte &longs;a <lb/>rà il doppio <expan abbr="t&amacr;to">tanto</expan>, quan<lb/>to è la po&longs;&longs;anza di B, <lb/>e&longs;&longs;endo che il pe&longs;o ver<lb/>&longs;o la po&longs;&longs;anza ha la <lb/>proportione i&longs;te&longs;&longs;a, che <emph.end type="italics"/><lb/><figure id="id.037.01.138.1.jpg" xlink:href="037/01/138/1.jpg"></figure><lb/><emph type="italics"/>ha AB ad AD: & le po&longs;&longs;anze po&longs;te in BD &longs;ono eguali, adunque la po&longs;<lb/>&longs;anza, che è in B &longs;o&longs;tenirà il doppio più di quello, che &longs;o&longs;tenirà la po&longs;&longs;anza, che è <lb/>in D. </s> <s id="id.2.1.772.4.0">Diuida&longs;i dunque il pe&longs;o C in due parti, l'vna delle quali &longs;ia il doppio del<lb/>l'altra: ilche &longs;i farà, &longs;e lo diuideremo in tre parti eguali EFG, & all'hora FG <lb/>&longs;arà il doppio di E. </s> <s id="id.2.1.772.5.0">Co&longs;i la po&longs;&longs;anza in D &longs;o&longs;tenirà la parte E, & la po&longs;&longs;anza <lb/>in B le altre due parti FG. </s> <s id="id.2.1.772.6.0">Ambedue dunque le po&longs;&longs;anze po&longs;te in BD tra <lb/>loro eguali <expan abbr="&longs;o&longs;terrãno">&longs;o&longs;terranno</expan> in&longs;ieme tutto il pe&longs;o C. </s> <s>& perche la po&longs;&longs;anza in D &longs;o&longs;tie<lb/>ne la parte E, laquale è la terza parte del pe&longs;o C, & ad e&longs;&longs;o è eguale, &longs;arà la po&longs;<lb/>&longs;anza in D vn terzo del pe&longs;o C: & concio&longs;ia che la po&longs;&longs;anza di B &longs;o&longs;tenga le <lb/>parti FG, la po&longs;&longs;anza dellequali po&longs;ta in B è la metà meno: &longs;arà la po&longs;&longs;anza <lb/>in B all'vna delle parti FG, come alla G eguale. </s> <s id="id.2.1.772.7.0">& il G è la terza parte <lb/>del pe&longs;o C. </s> <s id="id.2.1.772.8.0">La po&longs;&longs;anza dunque in B &longs;arà il terzo del pe&longs;o C. </s> <s id="id.2.1.772.9.0">Cia&longs;cuna delle <lb/>po&longs;&longs;anze dunque in BD è vn terzo del pe&longs;o C, che bi&longs;ognaua dimo&longs;trare. <emph.end type="italics"/></s> <pb pagenum="62" xlink:href="037/01/139.jpg"/> <s id="id.2.1.775.1.0"><emph type="italics"/>Et &longs;e fo&longs;&longs;ero due leue AB EF diui&longs;e in due parti eguali in GD, i &longs;o&longs;tegni delle<lb/>quali fo&longs;&longs;ero AF, & il pe&longs;o C fo&longs;&longs;e appiccato all'vna, & l'altra leua in DG<emph.end type="italics"/><lb/><figure id="id.037.01.139.1.jpg" xlink:href="037/01/139/1.jpg"></figure><lb/><emph type="italics"/>&longs;i fattamente, però che pe&longs;a&longs;&longs;e egualmente nell'vna, & l'altra: & fo&longs;&longs;ero due po&longs;<lb/>&longs;anze eguali in BG. </s> <s id="id.2.1.775.2.0">Si dimo&longs;trerà con ragione in tutto mede&longs;ima, che ogn'vna <lb/>delle po&longs;&longs;anze po&longs;te in B & G è vn terzo del pe&longs;o C. <emph.end type="italics"/></s> <s id="id.2.1.777.1.0">PROPOSITIONE V. </s> <s id="id.2.1.778.1.0">Se all'vna & l'altra, di cia&longs;cuna girella di due taglie, l'vna delle <lb/>quali &longs;ia po&longs;ta di &longs;opra, & l'altra di &longs;otto, & legata al pe&longs;o; &longs;a<lb/>rà condotta intornò la corda, legando vno de'&longs;uoi capi alla <lb/>taglia di &longs;otto, & l'altro &longs;ia tenuto dalla po&longs;&longs;anza, che &longs;o&longs;tie<lb/>ne il pe&longs;o: &longs;arà la po&longs;&longs;anza vn terzo del pe&longs;o. </s> <pb xlink:href="037/01/140.jpg"/> <s id="id.2.1.780.1.0"><emph type="italics"/>Sia il pe&longs;o A, &longs;ia BCD la girella della taglia legata al pe&longs;o A, il cui centro &longs;ia <lb/>E, & &longs;ia FGH l'altra girella della taglia appiccata di &longs;opra, il cui centro &longs;ia <lb/>K: &longs;ia condotta intorno alle girelle la corda LFGHBCDM, laquale &longs;ia lega<lb/>ta alla taglia di &longs;otto in L; & la po&longs;<lb/>&longs;anza, che &longs;o&longs;tiene il pe&longs;o A &longs;ia in <lb/>M. </s> <s id="id.2.1.780.2.0">Dico che la po&longs;&longs;anza in M è vn <lb/>terzo del pe&longs;o A. </s> <s id="id.2.1.780.3.0">Siano tirate le li<lb/>nee FH BD per li centri KE egual<lb/>mente di&longs;tanti dall'orizonte, &longs;i come <lb/>nelle precedenti è detto. </s> <s id="id.2.1.780.4.0">Hor percio<lb/>che la corda FL &longs;o&longs;tiene la taglia di <lb/>&longs;otto, laquale &longs;o&longs;tiene la girella nel &longs;uo <lb/>centro E: &longs;arà la corda di L come <lb/>po&longs;&longs;anza che &longs;o&longs;tiene la girella, tanto <lb/>quanto &longs;e fo&longs;&longs;e in e&longs;&longs;o E centro: & <lb/>la po&longs;&longs;anza di M è come &longs;e &longs;te&longs;&longs;e in <emph.end type="italics"/><lb/><arrow.to.target n="note221"></arrow.to.target><emph type="italics"/>D; &longs;i farà dunque DB come leua, il <lb/>cui &longs;o&longs;tegno &longs;arà B: ma il pe&longs;o A, <lb/>come di &longs;opra fù dimo&longs;trato, appicca<lb/>to in E viene &longs;o&longs;tenuto da due po&longs;<lb/>&longs;anze, l'vna po&longs;ta in D, & l'altra in <lb/>E. </s> <s id="id.2.1.780.5.0">& concio&longs;ia, che nel &longs;o&longs;tenere i <lb/>pe&longs;i &longs;tiano le leue FH BD immobi<lb/>li, &longs;e li pe&longs;i &longs;aranno appiccati alle cor<emph.end type="italics"/><lb/><arrow.to.target n="note222"></arrow.to.target><emph type="italics"/>de FL HB &longs;aranno que&longs;ti i&longs;te&longs;&longs;i egua<lb/>li, per hauere la leua FH il &longs;o&longs;tegno <lb/>nel mezo; altramente dall'vna delle <lb/>parti &longs;i farebbe il mouimento à ba&longs;&longs;o, <lb/>co&longs;a che tuttauia non accade; Adun<lb/>que tanto &longs;o&longs;tiene la corda FL, quan<lb/>to la HB. </s> <s id="id.2.1.780.6.0">Di più percioche dal me<lb/>zo della leua BD il pe&longs;o pende at<lb/>taccato, però &longs;e fo&longs;&longs;ero due po&longs;&longs;anze <lb/>in BD che &longs;o&longs;tene&longs;&longs;ero il pe&longs;o, &longs;areb<lb/>bon fra loro eguali: & benche la cor<emph.end type="italics"/><lb/><arrow.to.target n="note223"></arrow.to.target><emph type="italics"/>da FL &longs;o&longs;tenga e&longs;&longs;a ancora il pe&longs;o, <lb/>poiche ella &longs;ta in loco de la po&longs;&longs;anza <emph.end type="italics"/><lb/><arrow.to.target n="note224"></arrow.to.target><emph type="italics"/>E, nondimeno percioche &longs;o&longs;tiene da <lb/>quel mede&longs;imo punto, doue è appicca<lb/>to il pe&longs;o, non farà però che le po&longs;<emph.end type="italics"/><lb/><figure id="id.037.01.140.1.jpg" xlink:href="037/01/140/1.jpg"></figure><lb/><emph type="italics"/>&longs;anze, lequali &longs;ono in BD non &longs;iano tra loro eguali, peroche aiuta tanto all'v<lb/>na, quanto all'altra. </s> <s id="id.2.1.780.7.0">Ma le po&longs;&longs;anze che &longs;ono in BD &longs;ono le i&longs;te&longs;&longs;e, come &longs;e <emph.end type="italics"/><pb pagenum="63" xlink:href="037/01/141.jpg"/><emph type="italics"/>fu&longs;&longs;ero in HM. </s> <s id="id.2.1.780.8.0">Per laqual co&longs;a tanto &longs;o&longs;terrà la corda MD quanto la HB: ma <lb/>co&longs;i &longs;o&longs;tiene HB come FL; adunque la corda MD co&longs;i &longs;o&longs;tenirà, come FL, <lb/>cioè come &longs;e in D & in L fo&longs;&longs;ero appiccati pe&longs;i eguali. </s> <s id="id.2.1.780.9.0">Concio&longs;ia co&longs;a dunque, <lb/>che pe&longs;i eguali &longs;ian &longs;o&longs;tenuti da po&longs;&longs;anze vguali, le po&longs;&longs;anze in ML &longs;aranno egua<lb/>li, delle quali è in tutto vna ragione iste&longs;&longs;a, come &longs;e ambedue fo&longs;&longs;ero in DE. </s> <s><lb/>Onde, e&longs;&longs;endo che il pe&longs;o A &longs;tia attaccato nel mezo della leua BD, & che due <lb/>po&longs;&longs;anze po&longs;te in DE &longs;o&longs;tenente il pe&longs;o &longs;iano eguali: &longs;arà B il &longs;o&longs;tegno, & <lb/>cia&longs;cheduna po&longs;&longs;anza po&longs;ta in DE ouero in ML &longs;arà vn terzo del pe&longs;o A. <lb/></s> <s id="id.2.1.780.10.0">Adunque la po&longs;&longs;anza in M &longs;o&longs;tenente il pe&longs;o &longs;arà vn terzo del pe&longs;o A. </s> <s>che <emph.end type="italics"/><arrow.to.target n="note225"></arrow.to.target><lb/><emph type="italics"/>bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.782.1.0"><margin.target id="note221"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo<emph.end type="italics"/></s> <s id="id.2.1.783.1.0"><margin.target id="note222"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.784.1.0"><margin.target id="note223"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>corollario di questo. <emph.end type="italics"/></s> <s id="id.2.1.785.1.0"><margin.target id="note224"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo della leua. <emph.end type="italics"/></s> <s id="id.2.1.786.1.0"><margin.target id="note225"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.787.1.0">COROLLARIO. </s> <s id="id.2.1.788.1.0">Da que&longs;to è manife&longs;to, che ogn'vna delle corde MD FL HB <lb/>&longs;o&longs;tiene la terza parte del pe&longs;o A. </s> <pb xlink:href="037/01/142.jpg"/> <s id="id.2.1.789.1.0"><emph type="italics"/>Oltre à ciò &longs;e da M &longs;arà la corda portata intor<lb/>no ad vn'altra girella po&longs;ta più &longs;u nella ta<lb/>glia, che &longs;imilmente &longs;ia attaccata di &longs;opra, il <lb/>cui centro &longs;ia N &longs;i fattamente che peruen<lb/>ga in O, & iui &longs;ia tenuta dalla po&longs;&longs;anza; &longs;a <lb/>rà la po&longs;&longs;anza che in O &longs;o&longs;tiene il pe&longs;o A <lb/>parimente vn terzo del pe&longs;o. </s> <s id="id.2.1.789.2.0">Percioche la <lb/>corda MD &longs;o&longs;tiene tanto di pe&longs;o, come &longs;e in <lb/>D fo&longs;&longs;e appiccato il pe&longs;o eguale alla terza <emph.end type="italics"/><lb/><arrow.to.target n="note226"></arrow.to.target><emph type="italics"/>parte del pe&longs;o A, alla quale è pari la po&longs;<lb/>&longs;anza in O ad e&longs;&longs;a eguale, cioè vn terzo del <lb/>pe&longs;o A. </s> <s id="id.2.1.789.3.0">La po&longs;&longs;anza dunque in O è vn <lb/>terzo del pe&longs;o A. <emph.end type="italics"/></s> <s id="id.2.1.790.1.0"><margin.target id="note226"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.791.1.0"><emph type="italics"/>Et accioche non &longs;i ritorni à dire &longs;pe&longs;&longs;e volte il <lb/>mede&longs;imo, egli fà me&longs;tiero &longs;apere, che la po&longs;<lb/>&longs;anza in O è &longs;empre eguale à quella, che <lb/>&longs;ta in M. </s> <s>come &longs;arebbe à dire, &longs;e la po&longs;&longs;an<lb/>za in M fo&longs;&longs;e vn quarto, ouero vn quinto, <lb/>ò &longs;imile co&longs;a di e&longs;&longs;o pe&longs;o, la po&longs;&longs;anza parimen<lb/>te in O &longs;arà vn quarto, ouero vn quinto, <lb/>& co&longs;i di mano in mano dell'i&longs;te&longs;&longs;o pe&longs;o, nel <lb/>modo che è di&longs;po&longs;ta la po&longs;&longs;anza di M. <emph.end type="italics"/></s> <figure id="id.037.01.142.1.jpg" xlink:href="037/01/142/1.jpg"> </figure> <s id="id.2.1.793.1.0">Potrebbbe for&longs;e alcuno dubitare in alcune dimo&longs;trationi delle taglie come in que&longs;ta <lb/>quinta propo&longs;itione, tolta da me per e&longs;&longs;empio per e&longs;&longs;ere piu &longs;chietta delle altre, <lb/>che in fatto con la e&longs;perientia non riu&longs;ci&longs;&longs;ero in proportione le forze a' pe&longs;i, co<pb pagenum="64" xlink:href="037/01/143.jpg"/>mela ragione dimo&longs;tra; peroche pre&longs;upponendo &longs;i nelle dimo&longs;trationi matemati<lb/>che le linee &longs;enza larghezza, & profondità, & co&longs;i le altre co&longs;e imaginando &longs;i &longs;e<lb/>parate dalla materia, ageuolmente &longs;i per&longs;uadiamo e&longs;&longs;ere vere come dicono. </s> <s id="id.2.1.793.2.0">Ma <lb/>la e&longs;perientia poi molte volte mo&longs;tra diuer&longs;ità, & &longs;i trouiamo ingannati, facendo <lb/>la materia gran demente variare le co&longs;e. </s> <s id="id.2.1.793.3.0">In que&longs;ta propo&longs;itione &longs;i narra, che rauol<lb/>gendo d'intorno à due girelle di due taglie vna corda, & quel che &longs;egue, la forza <lb/>&longs;arà vn terzo del pe&longs;o, cioè &longs;e il pe&longs;o &longs;ara trecento, egli verrà &longs;o&longs;tenuto dalla po&longs;<lb/>&longs;anza di cento. </s> <s id="id.2.1.793.4.0">Direbbe alcuno ciò e&longs;&longs;ere dubbio&longs;o, peroche le girelle, gli a&longs;&longs;etti <lb/>&longs;uoi, le funi, & il pe&longs;o della taglia di &longs;otto fanno re&longs;i&longs;tenza alla forza, & grauano <lb/>sì, che ella non potrà &longs;o&longs;tenere il pe&longs;o. </s> <s id="id.2.1.793.5.0">Si ri&longs;ponde che que&longs;te co&longs;e ben farebbo<lb/>no re&longs;i&longs;tenza nel mouere il pe&longs;o, ma non già nel &longs;o&longs;tentarlo: & bi&longs;ogna notaro <lb/>con diligenza che l'autore in que&longs;te dimo&longs;trationi parla &longs;empre del &longs;o&longs;tenere &longs;o<lb/>lamente con le forze i pe&longs;i che non calino al ba&longs;&longs;o, non del mouere. </s> <s id="id.2.1.793.6.0">Però con<lb/>&longs;ideri&longs;i, che quando li pe&longs;i &longs;i hanno da far mouere con le po&longs;&longs;anze, allhora le gi<lb/>relle, & gli altri impedimenti faranno re&longs;i&longs;tenza; ma quando &longs;i ha da far &longs;olamen<lb/>te che il pe&longs;o &longs;tia fermo, & habbia il &longs;uo contrape&longs;o &longs;emplicemente &longs;enza porre <lb/>in con&longs;ideratione altri ri&longs;petti, che è officio della po&longs;&longs;anza &longs;o&longs;tenente; all'hora <lb/>nè le girelle, nè altro danno re&longs;i&longs;tenza veruna, & la proua fondata &longs;u la ragione <lb/>torna &longs;empre per eccellentia, anzi pare che quanto piu re&longs;i&longs;tenza vi &longs;ia, tanto piu <lb/>facilmente la forza &longs;o&longs;tenga. </s> <s id="id.2.1.793.7.0">Auertendo con tutto ciò, che nel fare la e&longs;perienza <lb/>bi&longs;ogna hauere riguardo alla taglia di &longs;otto, & alla corda, lequali hanno la &longs;ua <lb/>grauezza &longs;i fattamente, che &longs;e il pe&longs;o come nell'e&longs;&longs;empio propo&longs;to, &longs;arà trecento <lb/>libre, & la forza cento, & la taglia di &longs;otto con la &longs;ua fune quattordici, è me&longs;tieri <lb/>che alla po&longs;&longs;anza di M &longs;i aggiungano quattro libre, & due terzi di forza, ac<lb/>cioche po&longs;&longs;a &longs;o&longs;tenere tutto il pe&longs;o, & co&longs;i verrà ad e&longs;&longs;ere in M po&longs;&longs;anza vn ter<lb/>zo giu&longs;tamente del pe&longs;o. </s> <s id="id.2.1.793.8.0">Ma per &longs;apere quanta forza bi&longs;ogni aggiungere alla po&longs;<lb/>&longs;anza, accioche per ri&longs;petto alla taglia di &longs;otto, & alla fune, &longs;o&longs;tenghi il pe&longs;o tut<lb/>to, faccia&longs;i que&longs;ta ragione. </s> <s id="id.2.1.793.9.0">La taglia di &longs;otto con parte della fune, per gratia di <lb/>e&longs;&longs;empio, è quattordici libre, il pe&longs;o è trecento, & la po&longs;&longs;anza cento. </s> <s id="id.2.1.793.10.0">Hor per <lb/>la regola detta del tre. </s> <s id="id.2.1.793.11.0">Se trecento danno cento, che daranno quattordici? </s> <s id="id.2.1.793.12.0">Tro<lb/>ueran&longs;i quattro libre, & due terzi da e&longs;&longs;ere aggiunte alla po&longs;&longs;anza di M, per <lb/>&longs;o&longs;tenere il pe&longs;o A. </s> <s id="id.2.1.793.13.0">“Laqual co&longs;a tocca in &longs;o&longs;tanza l'auttore più à ba&longs;&longs;o <lb/>dicendo. </s> <s>& &longs;i come habbiamo ciò con&longs;iderato nella decimaquinta, & quel, <lb/>che &longs;egue. </s> <s id="id.2.1.793.14.0">ilqual loco bi&longs;ogna intendere in que&longs;ta maniera, che le taglie non <lb/>&longs;i deuono pigliare ad vn'i&longs;te&longs;&longs;o modo &longs;empre, ma diuer&longs;amente, come graua<lb/>no, ilche na&longs;ce dall'e&longs;&longs;ere in vari luoghi, & le po&longs;&longs;anze, & i pe&longs;i collocati, & fer<lb/>mate le taglie. </s> <s id="id.2.1.793.15.0">Hor nella &longs;econda propo&longs;itione di que&longs;to trattato has&longs;i da inten<lb/>dere la po&longs;&longs;anza e&longs;&longs;ere la meta meno del pe&longs;o, prendendo per lo pe&longs;o, & il pe&longs;o, <lb/>& la taglia di &longs;otto in&longs;ieme, à cui &longs;tà attaccato, come &longs;i vede chiaro nella dimo&longs;tra<lb/>tione della detta &longs;econda propo&longs;itione, doue &longs;i proua che la po&longs;&longs;anza &longs;o&longs;tiene la gi<lb/>rella, laquale &longs;o&longs;tiene anche il re&longs;to della taglia nell'a&longs;&longs;etto, alla qual taglia è attac<lb/>cato il pe&longs;o, oue &longs;i cono&longs;ce e&longs;pre&longs;&longs;o, che la taglia, & il pe&longs;o s'hanno à pigliare <lb/>per tutto il pe&longs;o. </s> <s id="id.2.1.793.16.0">Per la qual co&longs;a, &longs;e in quel ca&longs;o il pe&longs;o in&longs;ieme con la taglia pe<lb/>&longs;eranno vinti, la po&longs;&longs;anza che gli &longs;o&longs;tenterà &longs;arà dieci. </s> <s id="id.2.1.793.17.0">Et per vn'altro e&longs;&longs;empio <lb/>nella nona propo&longs;itione di que&longs;to nel primo ca&longs;o, &longs;e il pe&longs;o con la taglia di &longs;otto <lb/>pe&longs;eranno vinticinque, la po&longs;&longs;anza &longs;o&longs;tenente &longs;arà cinque. </s> <s id="id.2.1.793.18.0">& co&longs;i egli è me&longs;tieri <lb/>hauer con&longs;ideratione nelle altre, cioè di&longs;tinguere doue è la grauezza della taglia, <pb xlink:href="037/01/144.jpg"/>quando graua di &longs;otto &longs;olamente, come nelle allegate propo&longs;itioni, & &longs;imili: & <lb/>quando &longs;olamente di &longs;opra, come nelle propo&longs;itioni 17. & 18. & &longs;imili: & quan<lb/>do ambedue le taglie grauano di &longs;opra, & di &longs;otto, come nelle propo&longs;itioni 20. <lb/>22. & 23. & &longs;imili: & quando anche nel'vna taglia, ne l'altra grauano, come nella <lb/>prima propo&longs;itione & nella 19. anzi in e&longs;&longs;a 19. la taglia di &longs;otto aiuta la <expan abbr="po&longs;sãza">po&longs;sanza</expan> ad <lb/>e&longs;&longs;ere piu leggiera: & nel &longs;econdo ca&longs;o dopo il corollario della 16. propo&longs;itione, <lb/>& &longs;imili. </s> <s id="id.2.1.793.19.0">& oltre à ciò deue&longs;i por mente alle corde ancora, la grauezza delle qua<lb/>li non hà &longs;empre da e&longs;&longs;ere con&longs;iderata, peroche grauano nelle propo&longs;itioni 15. 17. <lb/>ma non grauano già nella 19. </s> <s id="id.2.1.794.1.0">Ne parmi etiandio che &longs;i habbia ad hauere punto di riguardo alla picciolezza, & <lb/>grandezza delle girelle po&longs;te nelle taglie, & de gli a&longs;&longs;etti &longs;uoi, credendo che per <lb/>neces&longs;ità habbiano da e&longs;&longs;ere lauorati con mi&longs;ura tale, & proportione co&longs;i accu<lb/>rata, che mancando da quella non rie&longs;cano le dimo&longs;trationi alla e&longs;perientia; per <lb/>roche, &longs;i come nota l'autore poco appre&longs;&longs;o, ba&longs;ta che con certa conueneuole mi&longs;u<lb/>ra, & proportione le girelle nelle taglie &longs;iano maggiori l'vna dell'altra &longs;i fattamen<lb/>te, che le corde non &longs;i tocchino, & freghino fra loro, & co&longs;i vengano ad impedi <lb/>re i mouimenti delle po&longs;&longs;anze, & de' pe&longs;i. </s> <s id="id.2.1.795.1.0">PROPOSITIONE VI. </s> <s id="id.2.1.796.1.0">Siano due leue AB CD diui&longs;e in due parti eguali in EF, li <lb/>&longs;o&longs;tegni delle quali &longs;iano in BD; & &longs;ia il pe&longs;o G in EF ap<lb/>piccato all'vna, & l'altra leua &longs;i fattamente, che pe&longs;i dall'vna, <lb/>& dall'altra egualmente: & &longs;iano due po&longs;&longs;anze in AC egua<lb/>li, che &longs;o&longs;tengano il pe&longs;o. </s> <s id="id.2.1.796.2.0">Dico, che ogn'vna delle po&longs;&longs;anze <lb/>in AC è vn quarto del pe&longs;o G. </s> <s id="id.2.1.797.1.0"><arrow.to.target n="note227"></arrow.to.target><emph type="italics"/>Concio&longs;ia che le po&longs;&longs;anze po<lb/>&longs;te in AC &longs;o&longs;tengano tut<lb/>to il pe&longs;o G, & la po&longs;&longs;an<lb/>za di A ver&longs;ola parte del <lb/>pe&longs;o, che &longs;o&longs;tiene, &longs;ia come <lb/>BE à BA, & la po&longs;&longs;an<lb/>za in C alla parte di e&longs;&longs;o <lb/>G pe&longs;o &longs;o&longs;tenuto da lei &longs;ia <lb/>co&longs;i, come DF à DC, & <lb/>come BE à BA, co&longs;i è <lb/>DF à DC: &longs;arà la po&longs;&longs;an<lb/>za po&longs;ta in A ver&longs;o la par<lb/>te del pe&longs;o, che &longs;o&longs;tiene, co<emph.end type="italics"/><lb/><figure id="id.037.01.145.1.jpg" xlink:href="037/01/145/1.jpg"></figure><lb/><emph type="italics"/>me la po&longs;&longs;anza di C ver&longs;o la parte di e&longs;&longs;o pe&longs;o, che &longs;o&longs;tiene: & le po&longs;&longs;anze po&longs;te <lb/>in AC &longs;ono eguali; &longs;aranno dunque le parti del pe&longs;o G eguali, lequali &longs;ono &longs;o<emph.end type="italics"/><pb pagenum="65" xlink:href="037/01/145.jpg"/><emph type="italics"/>&longs;tenute dalle po&longs;&longs;anze. </s> <s id="id.2.1.797.2.0">Per laqual co&longs;a cia&longs;cuna po&longs;&longs;anza po&longs;ta in AC &longs;o&longs;terrà <lb/>la metà del pe&longs;o G. </s> <s id="id.2.1.797.3.0">Mala po&longs;&longs;anza in A è la metà meno del pe&longs;o, che &longs;o&longs;tie<lb/>ne; adunque la po&longs;&longs;anza in A &longs;arà per lo mezo della metà, cioè eguale alla quar<lb/>ta portione del pe&longs;o G; & però &longs;arà il quarto del pe&longs;o G, nè altramente &longs;i di<lb/>mo&longs;trerà la po&longs;&longs;anza in C e&longs;&longs;ere vn quarto dell'i&longs;te&longs;&longs;o pe&longs;o G. </s> <s>che bi&longs;ognaua <lb/>mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.799.1.0"><margin.target id="note227"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo nella leua. <emph.end type="italics"/></s> <s id="id.2.1.800.1.0"><emph type="italics"/>Ma &longs;e &longs;aranno tre leue AB <lb/>CD EF diui&longs;e in due <lb/>parti eguali in GHK, li <lb/>&longs;o&longs;tegni delle quali &longs;iano <lb/>BDF, & il pe&longs;o L &longs;ia <lb/>nell'i&longs;te&longs;&longs;o modo appicca<lb/>to in GHK: & &longs;iano <lb/>tre po&longs;&longs;anze in ACE <lb/>eguali, che &longs;o&longs;tengano il <lb/>pe&longs;o: &longs;i mo&longs;trerà &longs;imil<lb/>mente cia&longs;cuna po&longs;&longs;anza <lb/>e&longs;&longs;ere vn &longs;e&longs;to del pe&longs;o <lb/>L: & con questo ordi<lb/>ne &longs;e fo&longs;&longs;ero quattro le<lb/>ue, & quattro po&longs;&longs;anze, <lb/>cia&longs;cuna po&longs;&longs;anza &longs;arà <lb/>la ottaua parte del pe&longs;o, & co&longs;i di mano in mano in infinito. <emph.end type="italics"/></s> <figure id="id.037.01.145.2.jpg" xlink:href="037/01/145/2.jpg"> </figure> <s id="id.2.1.802.1.0">PROPOSITIONE VII. </s> <s id="id.2.1.803.1.0">Se à tre girelle di due taglie, l'vna delle quali po&longs;ta di &longs;opra hab<lb/>bia vna &longs;ola girella, & l'altra di &longs;otto ne habbia due, & &longs;ia lega<lb/>ta al pe&longs;o; &longs;ia po&longs;ta d'intorno la corda; legando l'vn de' capi <lb/>&longs;uoi in qualche loco, & l'altro &longs;ia tenuto dalla po&longs;&longs;anza, che <lb/>&longs;o&longs;tiene il pe&longs;o. </s> <s id="id.2.1.803.2.0">La po&longs;&longs;anza &longs;arà vn quarto del pe&longs;o. </s> <pb xlink:href="037/01/146.jpg"/> <s id="id.2.1.805.1.0"><emph type="italics"/>Sia il pe&longs;o A: &longs;iano le tre girelle, il centro dellequali &longs;ia BCD: & la girella, il <lb/>cui centro è D, &longs;ia della taglia appiccata di <lb/>&longs;opra: ma quelle girelle, il cui centro è in B <lb/>C &longs;iano della taglia legata al pe&longs;o A: & <lb/>la corda EFGHKLNOP &longs;ia condotta <lb/>intorno à tutte le girelle, & legata in E: & <lb/>&longs;ia la forza che &longs;o&longs;tiene il pe&longs;o A in P. <lb/></s> <s id="id.2.1.805.2.0">Dicola po&longs;&longs;anza in P e&longs;&longs;ere vn quarto del <lb/>pe&longs;o A. </s> <s id="id.2.1.805.3.0">Siano tirate le linee KL GF ON <lb/>per li centri delle girelle, &longs;i che &longs;iano egual<lb/>mente di&longs;tanti dall'orizonte; le quali per le co <lb/>&longs;e, che già &longs;ono dette, &longs;aranno come leue. </s> <s id="id.2.1.805.4.0">& <lb/>percioche per cagione della leua, ouero bilan<lb/>cia KL, il cui &longs;o&longs;tegno, ouero centro è nel <emph.end type="italics"/><lb/><arrow.to.target n="note228"></arrow.to.target><emph type="italics"/>mezo, tanto &longs;o&longs;tiene la corda KG, quanto <lb/>la NL non &longs;i facendo mouimento in niu<lb/>na delle parti: Di più per cau&longs;a della leua <lb/>GF dal cui mezo, come &longs;o&longs;pe&longs;o dipende il <lb/>pe&longs;o; &longs;e fo&longs;&longs;ero due po&longs;&longs;anze in GF, oue<emph.end type="italics"/><lb/><arrow.to.target n="note229"></arrow.to.target><emph type="italics"/>ro in HE, (percioche &longs;i come è &longs;tato più <lb/>volte detto, la ragione dell'vno, & dell'al<lb/>tro &longs;ito è pari) &longs;arebbono per certo que&longs;te <lb/>tali po&longs;&longs;anze eguali fra loro. </s> <s id="id.2.1.805.5.0">Onde co&longs;i &longs;o<lb/>&longs;tiene la corda HG, come EF: &longs;imilmen<lb/>te &longs;imo&longs;trerà tanto &longs;o&longs;tenere la corda PO, <lb/>quanto la NL. </s> <s id="id.2.1.805.6.0">Per laqual co&longs;a le corde <lb/>PO KG EF LN &longs;o&longs;tengono egualmen<lb/>te. </s> <s id="id.2.1.805.7.0">Adunque &longs;o&longs;tiene egualmente sì la cor<lb/>da PO, come la KG. </s> <s id="id.2.1.805.8.0">Se dunque s'inten<lb/>de&longs;&longs;ero e&longs;&longs;ere due po&longs;&longs;anze in OG, ouero in <lb/>PH, che è il mede&longs;imo, lequali tuttauia &longs;o<lb/>&longs;tenghino il pe&longs;o, come &longs;o&longs;tengono le corde, <lb/>&longs;arebbono per certo eguali: & GF ON <lb/>baurebbono le forze di due leue, il &longs;o&longs;tegno <lb/>delle quali &longs;aranno FN & il pe&longs;o A &longs;a <lb/>rà appiccato in BC, che è il mezo delle le<lb/>ue. </s> <s id="id.2.1.805.9.0">& percioche tutte le corde &longs;o&longs;tengo<lb/>no egualmente, tanto &longs;o&longs;teniranno le due <emph.end type="italics"/><lb/><figure id="id.037.01.146.1.jpg" xlink:href="037/01/146/1.jpg"></figure><lb/><emph type="italics"/>PO LN quanto le due KG EF. </s> <s id="id.2.1.805.10.0">tanto dunque &longs;o&longs;terrà la leua ON, quan<emph.end type="italics"/><lb/><arrow.to.target n="note230"></arrow.to.target><emph type="italics"/>to la leua GF. </s> <s id="id.2.1.805.11.0">Onde nell'vna, & l'altra leua ON GF pe&longs;erà egualmente il <lb/>pe&longs;o. </s> <s id="id.2.1.805.12.0">&longs;arà dunque ogni po&longs;&longs;anza che è in PH vn quarto del pe&longs;o A. </s> <s>& e&longs;&longs;en<emph.end type="italics"/><pb pagenum="66" xlink:href="037/01/147.jpg"/><emph type="italics"/>do, che la corda KG &longs;i prenda in loco di po&longs;&longs;anza, come quella, che non &longs;o&longs;tiene <lb/>altramente di quel che faccia PO, &longs;arà la po&longs;&longs;anza di P, che &longs;o&longs;tiene il pe&longs;o A <lb/>vn quarto di e&longs;&longs;o pe&longs;o. </s> <s id="id.2.1.805.13.0">che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.807.1.0"><margin.target id="note228"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.808.1.0"><margin.target id="note229"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 2. <emph type="italics"/>corollario della<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.809.1.0"><margin.target id="note230"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.810.1.0">COROLLARIO I. </s> <s id="id.2.1.811.1.0">Di qui è manife&longs;to, che cia&longs;cuna corda EF GK LN OP &longs;o<lb/>&longs;tiene la quarta parte del pe&longs;o A. </s> <s id="id.2.1.812.1.0">COROLLARIO II. </s> <s id="id.2.1.813.1.0">E chiaro ancora, che non meno &longs;o&longs;tiene la girella il cui centro <lb/>è C, di quello che faccia la girella, il centro dellaquale è B. </s> <pb xlink:href="037/01/148.jpg"/> <s id="id.2.1.815.1.0">Altramente. </s> <s id="id.2.1.816.1.0"><emph type="italics"/>Po&longs;te ancora le co&longs;e mede&longs;ime, &longs;e fo&longs;&longs;ero <lb/>due po&longs;&longs;anze eguali, che &longs;o&longs;tene&longs;&longs;ero <emph.end type="italics"/><lb/><arrow.to.target n="note231"></arrow.to.target><emph type="italics"/>il pe&longs;o A, l'vna in O, & l'altra in <lb/>C: &longs;arebbe cia&longs;cuna delle dette po&longs;&longs;an<lb/>ze vn terzo del pe&longs;o A. </s> <s id="id.2.1.816.2.0">Ma perche <lb/>la leua GF, il cui &longs;o&longs;tegno è F, è <lb/>diui&longs;a in due parti eguali nel C. </s> <s>&longs;e dun<lb/>que &longs;i porrà la po&longs;&longs;anza in G che &longs;o<lb/>&longs;tenga l'i&longs;te&longs;&longs;o pe&longs;o, come la po&longs;&longs;anza <lb/>di C, &longs;arà la po&longs;&longs;anza di G la metà <lb/>della po&longs;&longs;anza, che fo&longs;&longs;e in C; per<lb/>cioche &longs;e la po&longs;&longs;anza di C per &longs;e &longs;te&longs;&longs;a <lb/>&longs;o&longs;tene&longs;&longs;e il pe&longs;o, che è appiccato in C, <lb/>&longs;arebbe per certo eguale ad e&longs;&longs;o pe&longs;o; et <lb/>&longs;e l'i&longs;te&longs;&longs;o pe&longs;o fo&longs;&longs;e &longs;o&longs;tenuto dalla po&longs;<lb/>&longs;anza di G, &longs;arebbe il doppio di e&longs;&longs;a G <lb/>po&longs;&longs;anza, & la po&longs;&longs;anza di C &longs;areb<emph.end type="italics"/><lb/><arrow.to.target n="note232"></arrow.to.target><emph type="italics"/>be vn terzo del pe&longs;o A; dunque la <lb/>po&longs;&longs;anza di G &longs;arebbe vn &longs;e&longs;to della <lb/>po&longs;&longs;anza del pe&longs;o A. </s> <s id="id.2.1.816.3.0">Per laqual co <lb/>&longs;a, e&longs;&longs;endo, che la po&longs;&longs;anza di O &longs;ia vn <lb/>terzo del pe&longs;o A, & la po&longs;&longs;anza di <lb/>G vn &longs;e&longs;to: &longs;ara l'vna, & l'altra po&longs;<lb/>&longs;anza in&longs;ieme po&longs;te in OG la metà <lb/>del pe&longs;o A, percioche la terza par<lb/>te con la &longs;e&longs;ta &longs;à la metà. </s> <s id="id.2.1.816.4.0">Ma per<lb/>cioche la po&longs;&longs;anza di OG, ouero di <lb/>PH, (come prima è detto) &longs;ono fra <lb/>loro eguali, & l'vna, & l'altra in&longs;ie<lb/>me &longs;ono la metà del pe&longs;o A, &longs;arà <lb/>ogn'vna delle po&longs;&longs;anze po&longs;te in PH <lb/>vn quarto di e&longs;&longs;o A. </s> <s id="id.2.1.816.5.0">Adunque la <lb/>po&longs;&longs;anza di P che &longs;o&longs;tiene il pe&longs;o A <lb/>&longs;arà vn quarto di e&longs;&longs;o pe&longs;o A. </s> <s>che era <lb/>da mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.817.1.0"><margin.target id="note231"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.818.1.0"><margin.target id="note232"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>di questo della leua. <emph.end type="italics"/></s> <figure id="id.037.01.148.1.jpg" xlink:href="037/01/148/1.jpg"> </figure> <pb pagenum="67" xlink:href="037/01/149.jpg"/> <s id="id.2.1.821.1.0"><emph type="italics"/>Ma &longs;e'la corda <lb/>&longs;arà legata in <lb/>E, & &longs;ia <lb/>dauantaggio <lb/>inuolta intor<lb/>no à quattro <lb/>girelle, et per<lb/>uenga in P, <lb/>&longs;imo&longs;trerà <expan abbr="&longs;imilm&emacr;te">&longs;i<lb/>milmente</expan>, che <lb/>la po&longs;&longs;anza <lb/>di P &longs;arà <lb/>vn quarto <lb/>del pe&longs;o A; <lb/>peroche egli <lb/>è il mede&longs;i<lb/>mo, come &longs;e <lb/>la corda fo&longs;<lb/>&longs;e legata in <lb/>L, & che la <lb/>po&longs;&longs;anza &longs;o<lb/>&longs;tene&longs;&longs;e il pe<lb/>&longs;o con la cor<lb/>da inuolta in<lb/>torno à tre gi<lb/>relle &longs;olamen<lb/>te, i centri <lb/>delle quali fo&longs;<lb/>&longs;ero BCQ, <lb/>percioche la <lb/>girella, il cui <lb/>centro è D, <lb/>del tutto è <lb/>inutile. <emph.end type="italics"/></s> <figure id="id.037.01.149.1.jpg" xlink:href="037/01/149/1.jpg"> </figure> <pb xlink:href="037/01/150.jpg"/> <s id="id.2.1.823.1.0">PROPOSITIONE VIII. </s> <s id="id.2.1.824.1.0">Siano due leue AB CD diui&longs;e in due parti eguali EF, i &longs;o<lb/>&longs;tegni delle quali &longs;iano AC, & &longs;ia appiccato il pe&longs;o G ne' <lb/>punti EF all'vna, & l'altra leua, &longs;i fattamente, che dall'vno, <lb/>& l'altro pe&longs;i egualmente: & &longs;iano tre po&longs;&longs;anze eguali in BD <lb/>E che &longs;o&longs;tenghino il pe&longs;o G. </s> <s id="id.2.1.824.2.0">Dico, che cia&longs;cuna delle det<lb/>te po&longs;&longs;anze &longs;eparatamente è vn quinto del pe&longs;o G. </s> <s id="id.2.1.825.1.0"><emph type="italics"/>Percioche il pe&longs;o G &longs;ta appiccato in EF, & &longs;ono le tre po&longs;&longs;anze in EBD egua<lb/>li: però la po&longs;&longs;anza di E &longs;o&longs;terrà la parte &longs;olamente del pe&longs;o G, che &longs;arà eguale <lb/>ad e&longs;&longs;a po&longs;&longs;anza di E, ma <lb/>le po&longs;&longs;anze di BD &longs;o&longs;terran<lb/>no la parte re&longs;tante, & la <emph.end type="italics"/><lb/><arrow.to.target n="note233"></arrow.to.target><emph type="italics"/>parte, che è da B &longs;o&longs;tenu<lb/>ta, &longs;arà il doppio di e&longs;&longs;o: ma <lb/>la parte &longs;o&longs;tenuta da D &longs;a<lb/>rà &longs;imilmente il doppio di e&longs;<lb/>&longs;o D per cau&longs;a della pro<lb/>portione di BA ver&longs;o AE, <lb/>& di DC ver&longs;o CF. </s> <s id="id.2.1.825.2.0">Con<lb/>cio&longs;ia dunque, che le po&longs;&longs;an<emph.end type="italics"/><lb/><arrow.to.target n="note234"></arrow.to.target><emph type="italics"/>ze di BD &longs;iano eguali, &longs;a<lb/>ranno anche (per quel che di <lb/>&longs;opra è detto) le parti del pe<emph.end type="italics"/><lb/><figure id="id.037.01.150.1.jpg" xlink:href="037/01/150/1.jpg"></figure><lb/><emph type="italics"/>&longs;o G, lequali &longs;ono &longs;o&longs;tenute dalle po&longs;&longs;anze di BD, fra loro eguali, & ogni vna <lb/>&longs;arà il doppio di quella tal parte, che è &longs;o&longs;tenuta dalla po&longs;&longs;anza di E. </s> <s id="id.2.1.825.3.0">Diuida&longs;i <lb/>dunque il pe&longs;o G in tre parti, delle quali due &longs;iano fra loro eguali, & di più ogni <lb/>vna di loro &longs;eparatamente &longs;ia il doppio dell'altra terza parte, ilche accaderà, &longs;e <lb/>in cinque parti eguali HKLMN &longs;arà diui&longs;o: percioche la parte compo&longs;ta di due <lb/>parti KL è il doppio della parte H, & la parte ancora di MN è &longs;imilmen<lb/>te il doppio della parte i&longs;te&longs;&longs;a H. </s> <s id="id.2.1.825.4.0">Per laqual co&longs;a anche la parte KL &longs;arà egua<lb/>le alla parte MN. </s> <s id="id.2.1.825.5.0">Ma &longs;o&longs;tenga la po&longs;&longs;anza di E la parte di H; & la po&longs;&longs;an<lb/>za di B le parti di KL: & la po&longs;&longs;anza di D le parti MN; adunque le tre <lb/>po&longs;&longs;anze eguali po&longs;te in BDE &longs;o&longs;terranno tutto il pe&longs;o G: & ogn'vna delle <lb/>po&longs;&longs;anze di BD &longs;o&longs;terrà il doppio di quel che &longs;o&longs;tiene la po&longs;&longs;anza di E. </s> <s id="id.2.1.825.6.0">Però <lb/>e&longs;&longs;endo che la po&longs;&longs;anza di E &longs;o&longs;tenga la parte di H, laquale è la quinta parte del <lb/>pe&longs;o G, & &longs;ia ad e&longs;&longs;o eguale, &longs;arà la po&longs;&longs;anza di E vn quinto del pe&longs;o G. </s> <s>& <lb/>percioche la po&longs;&longs;anza di B &longs;o&longs;tiene le parti di KL, lequali &longs;ono il doppio & del<emph.end type="italics"/><pb pagenum="68" xlink:href="037/01/151.jpg"/><emph type="italics"/>la po&longs;&longs;anza di B, & della parte di H, &longs;arà ancora la po&longs;&longs;anza di B ad e&longs;&longs;o H <lb/>eguale. </s> <s id="id.2.1.825.7.0">Per laqual co&longs;a &longs;arà vn quinto del pe&longs;o G. </s> <s id="id.2.1.825.8.0">Ne altrimente &longs;i dimo&longs;tre<lb/>rà, che la po&longs;&longs;anza di D è vn quinto del pe&longs;o G. </s> <s>cia&longs;cuna po&longs;&longs;anza dunque in <lb/>BDE è vn quinto del pe&longs;o G. </s> <s>che bi&longs;ognaua dimo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.827.1.0"><margin.target id="note233"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>di questa nella leua. <emph.end type="italics"/></s> <s id="id.2.1.828.1.0"><margin.target id="note234"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.151.1.jpg" xlink:href="037/01/151/1.jpg"> </figure> <s id="id.2.1.830.1.0"><emph type="italics"/>Che &longs;e &longs;aranno tre leue AB <lb/>CD EF diui&longs;e in due <lb/>parti eguali in GHK, i <lb/>&longs;o&longs;tegni dellequali &longs;iano A <lb/>CE, & il pe&longs;o L nel mo<lb/>do i&longs;te&longs;&longs;o &longs;ia appiccato in <lb/>GHK, & &longs;iano quattro <lb/>po&longs;&longs;anze eguali in BD <lb/>FG che &longs;o&longs;tengano il pe<lb/>&longs;o L; &longs;i mo&longs;trerà con &longs;imi<lb/>gliante modo, che cia&longs;cuna <lb/>po&longs;&longs;anza in BD FG &longs;a<lb/>rà vn &longs;ettimo del pe&longs;o L: <lb/>& &longs;e quattro fo&longs;&longs;ero le le<lb/>ue, & cinque le po&longs;&longs;anze <lb/>eguali &longs;o&longs;tenenti il pe&longs;o; con l'i&longs;te&longs;&longs;o modo ancora &longs;i mo&longs;trerebbe che ogni vna del<lb/>le po&longs;&longs;anze &longs;arebbe vn nono del pe&longs;o, & co&longs;i di mano in mano &longs;ucce&longs;&longs;iuamente. <emph.end type="italics"/></s> <s id="id.2.1.831.1.0">PROPOSITIONE IX. </s> <s id="id.2.1.832.1.0">Se à quattro girelle di due taglie, l'vna delle quali &longs;ia po&longs;ta di <lb/>&longs;opra, & l'altra di &longs;otto legata al pe&longs;o, &longs;ia condotta intorno <lb/>la corda, legando l'vno de'&longs;uoi capi alla taglia di &longs;otto, & l'al<lb/>tro &longs;ia ritenuto dalla po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o. </s> <s id="id.2.1.832.2.0">&longs;arà la po&longs;<lb/>&longs;anza vn quinto del pe&longs;o. </s> <pb xlink:href="037/01/152.jpg"/> <s id="id.2.1.833.1.0"><emph type="italics"/>Sia il pe&longs;o A, alquale &longs;ia legata <lb/>la taglia, che habbia due girel<lb/>le, i cui centri &longs;iano BC: & <lb/>&longs;ia la taglia appiccata di &longs;opra, <lb/>che habbia due altre girelle, i <lb/>cui centri &longs;iano DE, & la <lb/>corda &longs;ia tirata intorno à tutte <lb/>le girelle, laquale &longs;ia legata al<lb/>la taglia di &longs;otto in F: & &longs;ia <lb/>la po&longs;&longs;anza in G che &longs;o&longs;tiene <lb/>il pe&longs;o A. </s> <s id="id.2.1.833.2.0">Dico che la po&longs;&longs;an<lb/>za di G è vn quinto del pe&longs;o <lb/>A. </s> <s id="id.2.1.833.3.0">Siano tirate le linee HK <lb/>LM per li centri BC egual<lb/>mente di&longs;tanti dall'orizonte, le <lb/>quali nel modo i&longs;te&longs;&longs;o, che di <lb/>&longs;opra è &longs;tato detto, dimo&longs;trere<lb/>mo e&longs;&longs;ere come leue, i &longs;o&longs;tegni <lb/>delle quali &longs;ono KM, & il pe<lb/>&longs;o A pende attaccato nel me<lb/>zo BC dell'vna, & l'altra le<lb/>ua, & le tre po&longs;&longs;anze LHC, che <lb/>&longs;o&longs;tengono il pe&longs;o, lequali con <lb/>&longs;imile modo mo&longs;treremo e&longs;&longs;ere <lb/>eguali: percioche le corde fanno <lb/>l'i&longs;te&longs;&longs;o officio, come &longs;e fo&longs;&longs;ero <lb/>po&longs;&longs;anze: & percioche il pe&longs;o <lb/>dall'vna, & l'altra leua HK <lb/>LM pe&longs;a egualmente, ilche &longs;i <lb/>dimo&longs;trerà ancora, come nelle <lb/>precedenti è &longs;tato dimo&longs;trato: <emph.end type="italics"/><lb/><arrow.to.target n="note235"></arrow.to.target><emph type="italics"/>&longs;arà ogni po&longs;&longs;anza po&longs;ta sì in L <lb/>ouero in G, che è il mede&longs;imo; <lb/>& sì in H & in C, cioè in F <lb/>vn quinto del pe&longs;o A. </s> <s id="id.2.1.833.4.0">La po&longs;<lb/>&longs;anza dunque di G, che &longs;o&longs;tie<lb/>ne il pe&longs;o A. </s> <s>&longs;arà vn quinto <lb/>di e&longs;&longs;o pe&longs;o A. </s> <s>che bi&longs;ognaua <lb/>mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.834.1.0"><margin.target id="note235"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.152.1.jpg" xlink:href="037/01/152/1.jpg"> </figure> <pb pagenum="69" xlink:href="037/01/153.jpg"/> <s id="id.2.1.837.1.0"><emph type="italics"/>Che &longs;e dauantaggio &longs;i traporterà la cor<lb/>da in F d'intorno ad vn'altra girella, <lb/>il cui centro &longs;ia N, & &longs;ia legata <lb/>in O, &longs;i prouerà &longs;imilmente per due <lb/>ragioni, come nella &longs;ettima propo&longs;i<lb/>tione di que&longs;to, che la po&longs;&longs;anza di G <lb/>che &longs;o&longs;tiene il pe&longs;o A, è vn &longs;e&longs;to <emph.end type="italics"/><arrow.to.target n="note236"></arrow.to.target><lb/><emph type="italics"/>di e&longs;&longs;o pe&longs;o A. </s> <s id="id.2.1.837.2.0">Percioche prima dal<lb/>le treleue LM HK FP licui &longs;o<lb/>stegni &longs;ono in KP, & il pe&longs;o è ap<lb/>piccato nel mezo delle leue, & le tre <lb/>po&longs;&longs;anze po&longs;te in LHF che &longs;o&longs;ten<emph.end type="italics"/><arrow.to.target n="note237"></arrow.to.target><lb/><emph type="italics"/>gono il pe&longs;o &longs;ono eguali: poi dalle po&longs;<lb/>&longs;anze di LHN cia&longs;cuna delle quali <lb/>&longs;arebbe vn quinto del pe&longs;o A, per<lb/>cioche ambedue le po&longs;&longs;anze in&longs;ieme <lb/>po&longs;te in LH &longs;arebbono &longs;otto doppie &longs;e&longs;<lb/>quialtere al pe&longs;o, & la po&longs;&longs;anza di F <lb/>&longs;arebbe vn decimo, e&longs;&longs;endo la metà di <lb/>e&longs;&longs;a N. </s> <s id="id.2.1.837.3.0">Ma due quinte parti con <lb/>vna decima parte fanno la metà, la <lb/>qual metà &longs;e &longs;arà diui&longs;a per tre, ri<lb/>&longs;ponderà la &longs;e&longs;ta parte del pe&longs;o à cia<lb/>&longs;cuna delle po&longs;&longs;anze po&longs;te in LHF. <lb/></s> <s id="id.2.1.837.4.0">Dalle quali co&longs;e è manife&longs;to la po&longs;&longs;an<lb/>za di G e&longs;&longs;ere vn &longs;e&longs;to del pe&longs;o A; <lb/>& &longs;i dimo&longs;trerà &longs;imilmente che cia<lb/>&longs;cuna girella &longs;o&longs;tiene eguale portione <lb/>del pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.838.1.0"><margin.target id="note236"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo<emph.end type="italics"/></s> <s id="id.2.1.839.1.0"><margin.target id="note237"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.153.1.jpg" xlink:href="037/01/153/1.jpg"> </figure> <pb xlink:href="037/01/154.jpg"/> <s id="id.2.1.842.1.0">In que&longs;to trattato della taglia, &longs;i come in tutti gli altri ancora, l'autore pre&longs;uppone, <lb/>che qualunque per&longs;ona &longs;i mette à leggere il &longs;uo libro delle Mechaniche &longs;ia inten<lb/>dente di numeri, & di Geometria, & però ha &longs;empre mantenuto quello accurato <lb/>&longs;tile, & dimo&longs;tratiuo co&longs;tumato da buoni Matematici, v&longs;ando i vocaboli proprij <lb/>della &longs;cienza, alcuni de' quali io hò ben potuto volgarizare facilmente, &longs;i che <lb/>ogn'vno gli po&longs;&longs;a intendere, come per e&longs;&longs;empio, nelle proportioni duplum, tri<lb/>plum, quadruplum, & gli altri &longs;imili, ponendo in vece loro due volte tanto, tre <lb/>volte tanto, & quattro volte tanto: & co&longs;i per 'oppo&longs;ito &longs;ubduplum, <expan abbr="&longs;ubtripl&utilde;">&longs;ubtriplum</expan>, <lb/>& &longs;ubquadruplum, la metà, vn terzo, & vn quarto: & parimente &longs;e&longs;quialterum, <lb/>&longs;e&longs;quitertium, & &longs;e&longs;quiquartum, & gli altri &longs;imili, che vogliono dire vna volta & <lb/>meza, vna volta, & vn terzo, & vna volta & vn quarto. </s> <s id="id.2.1.842.2.0">Que&longs;ti dico s'hanno po<lb/>tuto ben dire, & facilmente nella no&longs;tra lingua. </s> <s id="id.2.1.842.3.0">Ma nell'ampiezza delle propor<lb/>tioni trouando&longs;i altri vocaboli a&longs;&longs;ai, i quali non è pos&longs;ibile co&longs;i adattare alla no<lb/>&longs;tra lingua, tra quali alcuni &longs;i trouano po&longs;ti dall'autore in que&longs;to trattato della ta<lb/>glia, & io &longs;ono &longs;tato sforzato à la&longs;ciargli co&longs;i, come erano, per mancamento di pa<lb/>role, che nella no&longs;tra fauella gli po&longs;&longs;ano e&longs;primere; hò giudicato douer e&longs;&longs;ere co <lb/>&longs;a vtile il dichiarare tutti i predetti vocaboli pertinenti alle proportioni, che ha il <lb/>pe&longs;o alla po&longs;&longs;anza, & la po&longs;&longs;anza al pe&longs;o &longs;critti dall'autore in que&longs;to trattato della <lb/>taglia, accioche quelle per&longs;one lequali non po&longs;&longs;edono que&longs;ti termini, non habbia <lb/>no fatica di andare &longs;tudiando i loro &longs;ignificati. </s> <s id="id.2.1.843.1.0">Dico dunque vna quantità poter&longs;i paragonare, & hauere proportione con vn'altra <lb/>in tre modi principali, la&longs;ciando hora le più &longs;ottili di&longs;tintioni. </s> <s id="id.2.1.843.2.0">Primieramente <lb/>come maggiore ver&longs;o la minore, dapoi come minore ver&longs;o la maggiore, & in fi<lb/>ne come eguale ver&longs;o la eguale. </s> <s id="id.2.1.843.3.0">Tutta la dottrina delle'proportioni, con&longs;i&longs;te in <lb/>que&longs;ti riguardi, cioè dal maggiore al minore, dal minore al maggiore, & dall'e<lb/>quale all'equale. </s> <s id="id.2.1.843.4.0">Hor quando vna quantità, che &longs;ia maggiore è paragonata con <lb/>vn'altra, che &longs;ia minore, che &longs;i dice proportione di maggiore di&longs;uguaglianza, na<lb/>&longs;cono cinque generi di proportioni, l'vno è il moltiplice &longs;chietto, il &longs;econdo è il <lb/>&longs;opraparticolare, il terzo il &longs;oprapartiente, il quarto il moltiplice &longs;opraparticola<lb/>re, & il quinto & vltimo il moltiplice &longs;oprapartiente. </s> <s id="id.2.1.843.5.0">Ma quando &longs;i fa compara<lb/>tione della minore quantità ver&longs;o la maggiore, all'hora &longs;i producono cinque altri <lb/>generi oppo&longs;ti apunto à i predetti cinque, & &longs;i dicono di minore di&longs;uguaglian<lb/>za, à i quali per fargli differenti da loro &longs;i aggiunge da Latini il &longs;ub, cioè &longs;otto, <lb/>&longs;criuendo &longs;i &longs;otto moltiplice, &longs;otto&longs;opra particolare, &longs;otto &longs;oprapartiente, &longs;otto <lb/>moltiplice &longs;opra particolare, & &longs;otto moltiplice &longs;oprapartiente. </s> <s id="id.2.1.843.6.0">Tutte le propor<lb/>tioni dunque &longs;ono compre&longs;e in vniuer&longs;ale da que&longs;ti diece generi oppo&longs;ti fra &longs;e <lb/>l'vn l'altro, cia&longs;cheduno de quali poi ha le &longs;ue &longs;petie differenti di proportioni. </s> <s id="id.2.1.843.7.0">Ma <lb/>io non hò qui intentione di numerarle, nè dichiarare diffu&longs;amente que&longs;ta materia <lb/>delle proportioni, ma &longs;olamente li vocaboli po&longs;ti dall'autore nel pre&longs;ente libro <lb/>della taglia, ba&longs;tando mi hauerne dato in generale vna rozza cognitione. </s> <s id="id.2.1.843.8.0">Ma chi <lb/>di ciò de&longs;idera hauere intero cono&longs;cimento legga tra i &longs;crittori della lingua Ita<lb/>liana Fra Luca dal Borgo, il Tartaglia ne i libri della Arithmetica, & il dottis&longs;imo <lb/>Zarlino nella prima parte delle In&longs;titutioni Harmoniche. </s> <s id="id.2.1.843.9.0">Dice l'autore in que&longs;to <lb/>loco. </s> <s id="id.2.1.843.10.0">Percio che &longs;arebbono ambedue le po&longs;&longs;anze in&longs;ieme in LH &longs;otto doppie <lb/>&longs;e&longs;quialtere di e&longs;&longs;o pe&longs;o. </s> <s id="id.2.1.843.11.0">Cioè le due po&longs;&longs;anze po&longs;te in LH haurebbono quella <lb/>proportione ver&longs;o il pe&longs;o, che ha 2. à 5. </s> <s>cioè &longs;e il pe&longs;o fo&longs;&longs;e come cinque, le po&longs;<lb/>&longs;anze larebbono come 2. che è la proportione &longs;otto doppia &longs;e&longs;quialtera. </s> <s id="id.2.1.843.12.0">Segue <pb pagenum="70" xlink:href="037/01/155.jpg"/>poi, Ma due quinte con vna decima fanno la me <lb/>tà, cioè à &longs;ommare in&longs;ieme due quinti, & vn <lb/>decimo fanno la metà di cinque, pero che li <lb/>due quinti &longs;ono due parti del cinque, & la deci <lb/>ma parte è la metà di vn quinto, tanto che met<lb/>tono in&longs;ieme due, & mezo, che &longs;ono la metà di <lb/>cinque. </s> <s id="id.2.1.843.13.0">Che &longs;e que&longs;ta metà poi &longs;arà diui&longs;a per <lb/>tre, ne riu&longs;cirà la &longs;e&longs;ta parte da e&longs;&longs;ere attribuita à <lb/>cia&longs;cheduna delle tre po&longs;&longs;anze po&longs;te in LHF. <lb/></s> <s id="id.2.1.843.14.0">Il modo del diuidere la metà per tre è facile, & <lb/>fas&longs;i in que&longs;ta maniera ponendo tre di &longs;opra, & <lb/>vno di &longs;otto; & vno di &longs;opra, & due di &longs;otto <expan abbr="cõ">con</expan> <lb/>la &longs;ua linea nel mezo, come &longs;i co&longs;tuma, & mol<lb/>tiplicando il tre intero co'l due denominatore <lb/>della metà, ne viene 6, alquale di &longs;opra &longs;i ag<lb/>giunge vno, & è vn &longs;e&longs;to. </s> <s id="id.2.1.844.1.0"><emph type="italics"/>Che &longs;e come nella terza figura la corda &longs;i allunghe<lb/>rà in O, & &longs;i condarrà intorno ad vn'altra gi<lb/>rella, il cui centro &longs;ia Q, la qual corda poi &longs;i <lb/>leghi in R alla taglia di &longs;otto; &longs;arà la po&longs;&longs;an<lb/>za di G vn &longs;ettimo del pe&longs;o. </s> <s id="id.2.1.844.2.0">& co&longs;i proceden<lb/>do in infinito, la proportione della po&longs;&longs;anza al pe<emph.end type="italics"/><arrow.to.target n="note238"></arrow.to.target><lb/><emph type="italics"/>&longs;o, quanto &longs;i voglia &longs;otto moltiplice ver&longs;o il pe<lb/>&longs;o &longs;i potrà trouare. </s> <s id="id.2.1.844.3.0">Dapoi &longs;i mo&longs;trerà &longs;empre, <lb/>come nelle precedenti, che &longs;e la po&longs;&longs;anza, la<lb/>quale &longs;o&longs;tiene il pe&longs;o &longs;arà vn quarto, ouero vn <lb/>quinto, ouero in qual &longs;i voglia altro modo &longs;arà <lb/>di&longs;po&longs;ta ver&longs;o il pe&longs;o, che &longs;imilmente cia&longs;cuna <lb/>corda &longs;o&longs;terrà la quarta, ò la quinta, ouero qual <lb/>&longs;i voglia altra parte del pe&longs;o, &longs;i come la i&longs;te&longs;&longs;a <lb/>po&longs;&longs;anza: peroche le corde fanno il mede&longs;imo, <lb/>come &longs;e fo&longs;&longs;ero tante po&longs;&longs;anze: & le girelle co<lb/>me &longs;e fo&longs;&longs;ero tante leue. <emph.end type="italics"/></s> <s id="id.2.1.845.1.0"><margin.target id="note238"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.846.1.0">Sotto moltiplice. </s> <s id="id.2.1.846.2.0">Que&longs;to è il primo genere delle <lb/>proportioni, che &longs;i riguardano dal minore al <lb/>maggiore, detto di minore di&longs;uguaglianza, il <lb/>quale &longs;otto di &longs;e tiene a&longs;&longs;ais&longs;ime &longs;petie, & è op<lb/>po&longs;to come ho ricordato, al moltiplice. </s> <s id="id.2.1.846.3.0">Dice <lb/>l'autore: & co&longs;i procedendo in infinito &longs;i potrà <lb/>ritrouare qual &longs;i voglia proportione &longs;otto mol<lb/>tiplice. </s> <s id="id.2.1.846.4.0">Percio che la po&longs;&longs;anza è minore del pe<lb/>&longs;o, & però ver&longs;o lui ha proportione &longs;otto mol<lb/>tiplice, come di vno ver&longs;o due, & di due ver<lb/>&longs;o quattro per darne e&longs;&longs;empio, & co&longs;i de gli al<lb/>tri numeri tali. </s> <figure id="id.037.01.155.1.jpg" xlink:href="037/01/155/1.jpg"> </figure> <pb xlink:href="037/01/156.jpg"/> <s id="id.2.1.849.1.0">COROLLARIO</s> <s id="id.2.1.850.1.0">Di qui è manife&longs;to, che le girelle della taglia, allaquale è legato <lb/>il pe&longs;o, fanno sì, che il pe&longs;o è &longs;o&longs;tenuto da po&longs;&longs;anza minore, <lb/>di quel che &longs;ia e&longs;&longs;o pe&longs;o, co&longs;a che veramente non fanno le gi<lb/>relle della taglia di &longs;opra. </s> <s id="id.2.1.851.1.0"><emph type="italics"/>Egli nondimeno conuiene &longs;apere, che come &longs;uole &longs;ar&longs;i, la girella della taglia di &longs;otto, <lb/>il cui centro è N, deue e&longs;&longs;ere minore di quella girella, il cui centro è C, & que<lb/>&longs;ta anche minore di quella, che ha il centro in B: & in &longs;omma &longs;e &longs;aranno più gi<lb/>relle nella taglia di &longs;otto legata al pe&longs;o, &longs;empre quella girella deue e&longs;&longs;ere maggiore <lb/>delle altre, che è più vicina al pe&longs;o attaccato: ma al contrario hanno à di&longs;por&longs;i le <lb/>girelle nella taglia di &longs;opra, ilche &longs;i co&longs;tuma di fare, acciò che le corde fra loro non <lb/>&longs;i intrichino; peroche in quanto alle girelle, &longs;iano ò grandi, ò picciole, non importa <lb/>nulla, &longs;eguendone &longs;empre l'i&longs;te&longs;&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.852.1.0"><emph type="italics"/>Di più è da notare, ilche etiandio dalle co&longs;e dette facilmente appare, che grandi&longs;&longs;ima <lb/>differenza na&longs;ce tra la po&longs;&longs;anza, & il pe&longs;o dal legare la corda ouero in R della ta<lb/>glia di &longs;otto, ouero in S, percioche &longs;e &longs;i legherà in S, la po&longs;&longs;anza di G &longs;arà vn <lb/>&longs;e&longs;to del pe&longs;o; ma &longs;e in R vn &longs;ettimo, co&longs;a che non accade alla taglia di &longs;opra: <lb/>percioche leghi&longs;i la corda, come nella precedente figura, ouero in T, ouero in O, <lb/>&longs;empre la po&longs;&longs;anza di G &longs;arà vn &longs;e&longs;to di e&longs;&longs;o pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.853.1.0"><emph type="italics"/>Dopo que&longs;te co&longs;e egli è da con&longs;iderare in che modo la forza moua il pe&longs;o, & di più lo <lb/>&longs;patio, & il tempo della po&longs;&longs;anza, che moue, & del pe&longs;o che è mo&longs;&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.854.1.0">“Di piu egli è da notare ilche etiandio è manife&longs;to dalle co&longs;e dette &c. </s> <s id="id.2.1.854.2.0">Qui potreb<lb/>be for&longs;e ad alcuno parere difficile in che modo po&longs;&longs;a e&longs;&longs;ere, che dal legare la cor<lb/>da in R, ouero in S, come &longs;i vede in que&longs;ta figura, na&longs;ca tanta differenza. </s> <s id="id.2.1.854.3.0">Onde <lb/>noti&longs;i che legando la corda in S, la girella Q re&longs;ta del tutto inutile, & è come <lb/>&longs;e ella non vi fo&longs;&longs;e; & la corda per non e&longs;&longs;ere attaccata in R alla taglia di &longs;otto, <lb/>ma in S fuori non &longs;o&longs;tiene la taglia, talche la forza di G viene ad e&longs;&longs;ere &longs;olamen<lb/>te vn &longs;e&longs;to del pe&longs;o. </s> <s id="id.2.1.854.4.0">&longs;oggiunge poi ilche non auiene alla taglia di &longs;opra.” Doue <lb/>auerta&longs;i che mentre &longs;i ha tenuto propo&longs;ito delle lettere S & R, ha bi&longs;ognato guar<lb/>dare nella qui &longs;opra&longs;critta figura, ma in parlando di TO, egli è me&longs;tieri per in<lb/>tendere que&longs;to loco mirare nella figura precedente, che è la &longs;econda della nona <lb/>propo&longs;itione, peroche iui &longs;ono le lettere TO. </s> <s id="id.2.1.854.5.0">La ragione per la quale non na&longs;ca <lb/>differenza nella po&longs;&longs;anza à legare la corda in T ouero in O, ma &longs;ia tutto vno, <lb/>è che la taglia di &longs;opra &longs;ta &longs;empre ferma, per modo, che non importa nulla il le<lb/>gare la corda in O nella taglia di &longs;opra, ouero in T fuori di e&longs;&longs;a, poiche am<lb/>bidue i luoghi &longs;ono immobili, & iui la corda &longs;ta ferma. </s> <s id="id.2.1.854.6.0">Lequali tutte co&longs;e l'auto<lb/>re hà toccato breuis&longs;imamente per e&longs;&longs;ere que&longs;to trattato della taglia lungo, la<lb/>&longs;ciando al lettore ancora qualche co&longs;a da &longs;peculare per &longs;e mede&longs;imo. </s> <pb pagenum="71" xlink:href="037/01/157.jpg"/> <s id="id.2.1.856.1.0">PROPOSITIONE X. </s> <s id="id.2.1.857.1.0">Se la corda &longs;arà inuolta intorno alla girella della taglia appicca<lb/>ta di &longs;opra, all'vno de'capi, dellaqual corda &longs;ia attaccato il pe<lb/>&longs;o, & all'altro po&longs;ta la po&longs;&longs;anza, che moue. </s> <s id="id.2.1.857.2.0">La detta po&longs;&longs;anza mo<lb/>uerà con la leua &longs;empre egualmente di&longs;tante dall'orizonte. </s> <s id="id.2.1.858.1.0"><emph type="italics"/>Sia il pe&longs;o A. </s> <s id="id.2.1.858.2.0">&longs;ia la girella della taglia appiccata di &longs;opra, che habbia il centro K. <lb/></s> <s id="id.2.1.858.3.0">Sia dapoi la corda HB CDEF legata al pe&longs;o A in H, & &longs;ia inuolta d'intor<lb/>no alla girella; & &longs;ia la taglia per modo appiccata <lb/>in L, che non habbia alcun altro mouimento fuor<lb/>che il volgimento libero della girella d'intorno al <lb/>&longs;uo a&longs;&longs;etto, & &longs;ia la po&longs;&longs;anza in F che moua il <lb/>pe&longs;o A. </s> <s id="id.2.1.858.4.0">Dico, che la po&longs;&longs;anza di F mouerà <lb/>&longs;empre il pe&longs;o A con la leua egualmente di&longs;tan<lb/>te dall'orizonte. </s> <s id="id.2.1.858.5.0">&longs;ia tirata la linea BKE egual<lb/>mente di&longs;tante dall'orizonte, & &longs;iano i punti BE <emph.end type="italics"/><arrow.to.target n="note239"></arrow.to.target><lb/><emph type="italics"/>doue le corde BH & EF toccano il cerchio: <lb/>&longs;arà BKE la leua, il &longs;o&longs;tegno dellaquale è nel <lb/>&longs;uo mezo, che è K, come di &longs;opra è detto. </s> <s id="id.2.1.858.6.0">Men<lb/>tre che dunque la forza di F inchina al ba&longs;&longs;o ver<lb/>&longs;o M, la leua EB &longs;i mouerà, mouendo&longs;i tut<lb/>ta la girella, cioè volgendo&longs;i attorno. </s> <s id="id.2.1.858.7.0">Mentre <lb/>che dunque F &longs;ta in M &longs;ia il punto E della <lb/>leua mo&longs;&longs;o fin ad I, & il B &longs;in'al C, di mo<lb/>do, che la leua &longs;ia in CI. </s> <s id="id.2.1.858.8.0">Dapoi &longs;i faccia la li<lb/>nea NM eguale ad e&longs;&longs;a FE: & quando il <lb/>punto E, &longs;arà in I all'hora il punto della cor<lb/>da, ilquale era in E &longs;arà in N, & quello,<emph.end type="italics"/><lb/><figure id="id.037.01.157.1.jpg" xlink:href="037/01/157/1.jpg"></figure><lb/><emph type="italics"/>che era in B &longs;arà in C di modo, che tirata la linea CI pa&longs;&longs;erà per lo centro <lb/>K. </s> <s id="id.2.1.858.9.0">Hor mentre il B &longs;ta in C &longs;ia il punto H in G, & &longs;arà BH al CBG <lb/>eguale, e&longs;&longs;endo la mede&longs;ima corda. </s> <s id="id.2.1.858.10.0">& percioche mentre EF inchina in MN <lb/>rimane pur &longs;empre EFM à piombo dell'orizonte, & tocca il cerchio nel punto <lb/>E di modo, che la linea tirata dal punto E per lo centro K &longs;ia &longs;empre egualmen<lb/>te di&longs;tante dall'orizonte, ilche mede&longs;imamente auiene alla corda BG & al pun<pb xlink:href="037/01/158.jpg"/>to B. </s> <s id="id.2.1.858.11.0">Mentre dunque il cerchio, ouero la girella &longs;i volge intorno, &longs;empre &longs;i mo<lb/>ue la leua EB, & &longs;em<lb/>pre ancora rimane vn'al<lb/>tra leua in EB, e&longs;&longs;endo <lb/>che per natura di e&longs;&longs;a gi<lb/>rella, nellaquale &longs;empre, <lb/>mentre &longs;i moue, re&longs;ti il <lb/>diametro da B in E, <lb/>(ilquale è in loco di le<lb/>ua) auuiene che parten<lb/>do&longs;ene vna, &longs;ucceda <lb/>l'altra &longs;empre, durando <lb/>però cotale aggiramen<lb/>to; & co&longs;i accade, che <lb/>la po&longs;&longs;anza moua il pe<lb/>&longs;o &longs;empre con la leua <lb/>EB egualmente di&longs;tan<lb/>te dall'orizonte, ilche <lb/>bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.860.1.0"><margin.target id="note239"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.158.1.jpg" xlink:href="037/01/158/1.jpg"> </figure> <s id="id.2.1.862.1.0">Po&longs;te le co&longs;e i&longs;te&longs;&longs;e, lo &longs;patio della po&longs;&longs;anza, che moue il pe&longs;o, è <lb/>eguale allo &longs;patio dello i&longs;te&longs;&longs;o pe&longs;o, che è mo&longs;&longs;o. </s> <s id="id.2.1.863.1.0"><emph type="italics"/>Percioche egli è &longs;tato dimo&longs;trato, che mentre F &longs;tà in M, il pe&longs;o A, cioè il punto <lb/>H è in G: & concio&longs;ia che la corda HBCDEF &longs;ia eguale alla GBCDEN <lb/>FM per e&longs;&longs;ere la corda i&longs;te&longs;&longs;a: leuata via dunque la commune GBCDENF <lb/>&longs;arà la HG alla FM eguale, & &longs;imilmente &longs;i mo&longs;trerà la di&longs;ce&longs;a di F e&longs;&longs;ere <lb/>&longs;empre eguale alla &longs;alita di H. </s> <s id="id.2.1.863.2.0">Adunque lo &longs;patio della po&longs;&longs;anza è eguale allo <lb/>&longs;patio del pe&longs;o. </s> <s id="id.2.1.863.3.0">che era da dimo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.864.1.0">Oltre à ciò la po&longs;&longs;anza moue il pe&longs;o i&longs;te&longs;&longs;o per i&longs;patio eguale in <lb/>tempo eguale, tanto con la corda inuolta intorno alla girella <lb/>della taglia appiccata di &longs;opra, quanto &longs;enza taglia, pur che li <lb/>mouimenti di e&longs;&longs;a po&longs;&longs;anza in velocità &longs;iano eguali. </s> <pb pagenum="72" xlink:href="037/01/159.jpg"/> <s id="id.2.1.866.1.0"><emph type="italics"/>Stando le co&longs;e i&longs;te&longs;&longs;e, &longs;ia vn'altro pe&longs;o P eguale al pe&longs;o A, alquale &longs;ia legata la cor<lb/>da TQ à piombo dell'orizonte: & &longs;ia TQ eguale ad e&longs;&longs;a HB: & muoua <lb/>la po&longs;&longs;anza di Q il <lb/>pe&longs;o P all'insù ad <lb/>angoli retti all'orizon<lb/>te, come &longs;i moue il pe<lb/>&longs;o A. </s> <s id="id.2.1.866.2.0">Dico, che per <lb/>eguale &longs;patio, & in <lb/>vno i&longs;te&longs;&longs;o tempo la <lb/>po&longs;&longs;anza di Q mo<lb/>ue il pe&longs;o P, & la <lb/>po&longs;&longs;anza di<emph.end type="italics"/> F <emph type="italics"/>il pe<lb/>&longs;o A: ilche è il me<lb/>de&longs;imo, come &longs;e l'i<lb/>&longs;te&longs;&longs;o pe&longs;o fo&longs;&longs;e mo&longs;<lb/>&longs;o in tempo eguale, <lb/>&longs;econdo che habbia<lb/>mo propo&longs;to. </s> <s id="id.2.1.866.3.0">Sia <lb/>allungata la EF in <lb/>S, & la TQ in R, <lb/>& &longs;iano le QRFS <lb/>fatte eguali non &longs;olo <lb/>fra &longs;e, ma etiandio <lb/>ad e&longs;&longs;a BH. </s> <s id="id.2.1.866.4.0">Hor <lb/>concio&longs;ia che le TQ <lb/>QR &longs;iano eguali ad <lb/>e&longs;&longs;e HB FS, & <lb/>la forza di Q mo<lb/>ua il pe&longs;o P per <lb/>la linea retta TQ <lb/>R: & dall'altro <emph.end type="italics"/><lb/><figure id="id.037.01.159.1.jpg" xlink:href="037/01/159/1.jpg"></figure><lb/><emph type="italics"/>canto la forza di F moua A per la retta HB, & le velocità de i mouimenti <lb/>dell'una, & l'altra po&longs;&longs;anza &longs;iano eguali, all'hor che nell'i&longs;te&longs;&longs;o tempo la po&longs;&longs;anza di <lb/>Q &longs;arà in R, & la po&longs;&longs;anza di F &longs;arà in S, e&longs;&longs;endo gli &longs;patij eguali: & men<lb/>tre la po&longs;&longs;anza di Q è in R, il pe&longs;o P, cioè il punto T &longs;arà in Q, per e&longs;&longs;e<lb/>rela TQ eguale ad e&longs;&longs;a QR, & mentre che la po&longs;&longs;anza di F &longs;ta in S, il pe<lb/>&longs;o A, cioè il punto H &longs;arà in B; ma lo &longs;patio TQ è eguale allo &longs;patio HB: <lb/>adunque le po&longs;&longs;anze di FQ mo&longs;&longs;e egualmente moueranno i pe&longs;i PA eguali <lb/>per eguali &longs;patij in tempo eguale. </s> <s id="id.2.1.866.5.0">che era da mo&longs;trare. <emph.end type="italics"/></s> <pb xlink:href="037/01/160.jpg"/> <s id="id.2.1.868.1.0">PROPOSITIONE XI. </s> <s id="id.2.1.869.1.0">Se la corda &longs;arà inuolta intorno alla girella della taglia legata al <lb/>pe&longs;o, laqual corda con vno de' &longs;uoi capi &longs;ia legata in qualche <lb/>luogo, & con l'altro pre&longs;a dalla po&longs;&longs;anza che moue il pe&longs;o; La <lb/>po&longs;&longs;anza mouerà &longs;empre con la leua egualmente di&longs;tante dal <lb/>l'orizonte. </s> <s id="id.2.1.870.1.0"><emph type="italics"/>Sia il pe&longs;o A: &longs;ia la girella CED <lb/>della taglia legata al pe&longs;o A, <lb/>da KH, & &longs;ia KH ad ango<lb/>li retti dell'orizonte, di modo <lb/>the il pe&longs;o &longs;egua &longs;empre il mo<lb/>uimento della taglia, &longs;ia pur fat<lb/>to all'insù, ouero all'ingiù, & <lb/>&longs;ia il centro della girella K, & <lb/>la corda inuolta intorno alla gi<lb/>rella &longs;ia BCDEF, la quale <lb/>&longs;ia legata in B, di modo che <lb/>&longs;tia immobile in B: & &longs;ia in F <lb/>la po&longs;&longs;anza, che moue il pe&longs;o A. <lb/></s> <s id="id.2.1.870.2.0">Dico che la po&longs;&longs;anza di F mo<lb/>ue &longs;empre il pe&longs;o A con la le<lb/>ua egualmente di&longs;tante dall'ori<lb/>zonte. </s> <s id="id.2.1.870.3.0">Siano BC EF egual<lb/>mente di&longs;tanti sì fra loro, come <lb/>ad e&longs;&longs;a KH, & à piombo al<emph.end type="italics"/><lb/><arrow.to.target n="note240"></arrow.to.target><emph type="italics"/>l'orizonte della i&longs;te&longs;&longs;a KH, & <lb/>toccanti il cerchio CED ne i <lb/>punti EC, & &longs;ia congiunta la <lb/>EC laquale pa&longs;&longs;erà per lo cen<lb/>tro K, & &longs;arà egualmente di <lb/>&longs;tante dall'orizonte, &longs;i come pri<lb/>ma è detto. </s> <s id="id.2.1.870.4.0">Hor percioche la <lb/>girella CED &longs;i volge d'intor<lb/>no K &longs;uo centro, però mentre <lb/>la forza di F tira sù il punto E <lb/>dourebbe di&longs;cendere il punto C <lb/>& tirare in giù B: mala cor<lb/>da po&longs;ta in B è immobile, on<lb/>de BC non può di&longs;cendere. <lb/></s> <s id="id.2.1.870.5.0">Per laqual co&longs;a mentre la po&longs;<emph.end type="italics"/><lb/><figure id="id.037.01.161.1.jpg" xlink:href="037/01/161/1.jpg"></figure><pb pagenum="73" xlink:href="037/01/161.jpg"/><emph type="italics"/>&longs;anza di F tira sùlo E, tutta la girella &longs;i mouerà in sù, & per con&longs;equenza tut<lb/>ta la taglia, & il pe&longs;o; & EKC &longs;arà come leua, il cui &longs;o&longs;tegno &longs;arà C: pero<emph.end type="italics"/><arrow.to.target n="note241"></arrow.to.target><lb/><emph type="italics"/>che il punto C per cau&longs;a di BC qua&longs;i è immobile, ma la po&longs;&longs;anza che moue la <lb/>leua è in F con la corda EF, & il pe&longs;o &longs;ta appiccato in K. </s> <s id="id.2.1.870.6.0">Che &longs;e il punto <lb/>C fo&longs;&longs;e del tutto immobile, & &longs;i moua la leua EC in NC, & &longs;i diuida NC <lb/>in due parti eguali in L: &longs;aranno CL LN eguali ad e&longs;&longs;e CK KE. </s> <s id="id.2.1.870.7.0">Per la <lb/>qual co&longs;a &longs;e la leua EC fo&longs;&longs;e in CN, il punto K &longs;arebbe in L: & &longs;e &longs;i con <lb/>duce&longs;&longs;e la linea LM à piombo dell'orizonte, laquale &longs;ia anche eguale alla KH, <lb/>&longs;arebbe il pe&longs;o A, cioè il punto H in M. </s> <s id="id.2.1.870.8.0">Ma percioche la po&longs;&longs;anza di F men<lb/>tre và in &longs;u&longs;o mouendo la girella &longs;empre &longs;i moue &longs;opra la linea retta EFG, laquale <lb/>è anco egualmente di&longs;tante &longs;empre da BC, &longs;arà nece&longs;&longs;ario, che la girella della ta<lb/>glia &longs;empre &longs;i troui tra le linee EG BC, & il centro K &longs;tando nel mezo, &longs;i mo<lb/>uerà &longs;empre &longs;opra la linea retta HKT. </s> <s id="id.2.1.870.9.0">Sia condotta adunque per L la linea <lb/>PT LQ egualmente di&longs;tante sì dall orizonte, come dalla EC, laquale &longs;eghi la <lb/>HK allungata in T, & co'l centro T, & lo &longs;patio TQ &longs;i formi il cerchio QR <lb/>PS, ilquale &longs;arà eguale al cerchio CED; & li punti PQ toccheranno le cor<lb/>de FE BC ne i punti PQ. </s> <s>Peroche il rettangolo PECQ & la PT & la <lb/>TQ &longs;ono eguali ad e&longs;&longs;e EK KC. </s> <s id="id.2.1.870.10.0">Dapoi per T &longs;ia tirato RTS diametro <lb/>del cerchio PQS egualmente di&longs;tante ad e&longs;&longs;a NC, & &longs;ia fatta TO eguale <emph.end type="italics"/><arrow.to.target n="note242"></arrow.to.target><lb/><emph type="italics"/>alla KH. </s> <s id="id.2.1.870.11.0">Hor mentre il centro K &longs;arà mo&longs;&longs;o fin alla linea PQ all'hora il cen<lb/>tro K &longs;arà in T. </s> <s id="id.2.1.870.12.0">Maegliè &longs;tato dimo&longs;trato, che il centro della girella &longs;i moue <lb/>&longs;empre per la linea retta HT. </s> <s id="id.2.1.870.13.0">Onde accioche il centro K &longs;ia nella linea PQ egual<lb/>mente di&longs;tante ad e&longs;&longs;a EC, egli è nece&longs;&longs;ario, che e&longs;&longs;o &longs;ia in T: & accioche an<lb/>chora la leua EC &longs;i alzi nell'angolo ECN egli è nece&longs;&longs;ario, che &longs;ia in RS & <lb/>non in CN percioche l'angolo RSE all'angolo NCE è eguale & co&longs;i il &longs;o<emph.end type="italics"/><arrow.to.target n="note243"></arrow.to.target><lb/><emph type="italics"/>&longs;tegno C non è del tutto immobile, mouendo&longs;i tutta la girella all'insù, & tutta <lb/>mutt'il luogo: nondimeno il C ha ragione di &longs;o&longs;tegno, peroche meno &longs;i moue C <lb/>di quel che fà K & E, percioche &longs;i moue il punto E fin ad R, & il K fin al T, <lb/>ma il punto C fin ad S &longs;olamente. </s> <s id="id.2.1.870.14.0">Per laqual co&longs;a mentre il centro K &longs;i troua <lb/>in T, il &longs;ito della girella &longs;arà QRPS: & il pe&longs;o A, cioè il punto H &longs;arà <lb/>in O, e&longs;&longs;endo TO eguale à KH; ma il &longs;ito di EC, cioè della leua mo&longs;&longs;a, &longs;arà <lb/>RS: & la po&longs;&longs;anza di F &longs;arà mo&longs;&longs;a in &longs;u&longs;o per la retta linea EFG: ma nel<lb/>l'i&longs;te&longs;&longs;o tempo, che K &longs;arà in T, &longs;ia la po&longs;&longs;anza in G; & mentre la leua EC in <lb/>que&longs;to modo &longs;i moue, rimangono pur &longs;empre GPBQ fra loro egualmente di<lb/>&longs;tanti, & à piombo dell'orizonte, talche doue toccano la girella, come ne' punti <lb/>PQ, &longs;empre la linea PQ &longs;arà il diametro della girella & come leua egualmen<lb/>te di&longs;tante dall'orizonte. </s> <s id="id.2.1.870.15.0">Mentre dunque la girella &longs;i moue, & và attorno, &longs;em<lb/>pre anche &longs;i moue la leua EC, & &longs;empre rimane vn'altra leua nella girella egual<lb/>mente di&longs;tante dall'orizonte, come PQ, per modo, che la po&longs;&longs;anza di F moua <lb/>il pe&longs;o, &longs;tando la leua egualmente di&longs;tante all'orizonte, il cui &longs;o&longs;tegno &longs;acà &longs;empre <lb/>nella linea CB, & il pe&longs;o nel mezo della leua appiccato: & la po&longs;&longs;anza nella li<lb/>nea EG, che era da mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.873.1.0"><margin.target id="note240"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.874.1.0"><margin.target id="note241"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.875.1.0"><margin.target id="note242"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 34. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <s id="id.2.1.876.1.0"><margin.target id="note243"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo. <emph.end type="italics"/></s> <pb xlink:href="037/01/162.jpg"/> <s id="id.2.1.877.1.0">Stando le co&longs;e i&longs;te&longs;&longs;e. </s> <s id="id.2.1.877.2.0">Lo &longs;patio della po&longs;&longs;anza, che moue il pe<lb/>&longs;o è il doppio dello &longs;patio dell'i&longs;te&longs;&longs;o pe&longs;o mo&longs;&longs;o. </s> <s id="id.2.1.878.1.0"><emph type="italics"/>E&longs;&longs;endo &longs;tato dimo&longs;trato, che mentre il K &longs;tà nel T, il pe&longs;o A cioè il punto H <lb/>e&longs;&longs;ere in O: & nell'i&longs;te&longs;&longs;o tempo ancora la po&longs;&longs;anza di F e&longs;&longs;ere in G: & per<lb/>cioche la corda BCDEF eguale è alla corda EQSPG, peroche è la mede&longs;ima <lb/>corda: & la corda che è inuolta intorno al mezo cerchio CDE eguale è alla cor<lb/>da, che &longs;ta d'intorno al mezo cerchio QSP: tolti via dunque li due pezzi di cor<lb/>da communi BQ, & FP: &longs;arà il re&longs;tante della corda FG eguale ad e&longs;&longs;i due <lb/>pezzi di corda rima&longs;i CQ & EP in&longs;ieme pre&longs;i. </s> <s id="id.2.1.878.2.0">Ma EP eguale è al TK, <lb/>& il CQ &longs;arà anche eguale ad e&longs;&longs;o TK, peroche &longs;ono PK & TC parallelo<lb/>grammi rettangoli. </s> <s id="id.2.1.878.3.0">Per laqual co&longs;a le linee EPCQ in&longs;ieme &longs;ono due volte tan<lb/>to, quanto è TK. </s> <s id="id.2.1.878.4.0">Adunque la corda FC &longs;arà due volte tanto quanto la TK. </s> <s><lb/>& percioche la KH è eguale alla TO, leuando via la corda commune KO &longs;a <lb/>rà la KT eguale ad e&longs;&longs;a KO. </s> <s id="id.2.1.878.5.0">Per laqual co&longs;a la corda FG &longs;arà due volte tan<lb/>to quanto e&longs;&longs;a HO: cioè lo &longs;patio della po&longs;&longs;anza due volte tanto quanto lo &longs;pa<lb/>tio del pe&longs;o, che era da mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.879.1.0">“Parallelogrammi rettangoli. </s> <s id="id.2.1.879.2.0">Vuol dire figure di linee egualmente diftanti fra loro, <lb/>lequali formino angoli retti à differenza di altre figure, che &longs;e ben &longs;ono di linee <lb/>egualmente di&longs;tanti, non formano tuttauia angoli retti. </s> <s id="id.2.1.880.1.0">Dapoi la po&longs;&longs;anza mouerà il pe&longs;o i&longs;te&longs;&longs;o in tempo eguale per la <lb/>metà dello &longs;patio, con la corda inuolta d'intorno alla girella <lb/>della taglia legata al pe&longs;o, che &longs;enza taglia; pur che le veloci<lb/>tà de' mouimenti di e&longs;&longs;a po&longs;&longs;anza &longs;iano eguali. </s> <pb pagenum="74" xlink:href="037/01/163.jpg"/> <s id="id.2.1.882.1.0"><emph type="italics"/>Peroche &longs;ia, &longs;tando le co&longs;e i&longs;te&longs;&longs;e, vn'altro pe&longs;o V eguale al pe&longs;o A al quale &longs;ia <lb/>legata la corda <36>X & &longs;ia in X la po&longs;&longs;anza, che moue il pe&longs;o V, Dico, &longs;e le ve<lb/>locità de' mouimenti dell'vna, & l'altra po&longs;&longs;anza &longs;aranno eguali, che la po&longs;&longs;anza <emph.end type="italics"/><lb/><figure id="id.037.01.163.1.jpg" xlink:href="037/01/163/1.jpg"></figure><lb/><emph type="italics"/>di<emph.end type="italics"/> F <emph type="italics"/>mouerà il pe&longs;o A nell'i&longs;te&longs;&longs;o tempo per la metà dello &longs;patio, per lo quale <lb/>il pe&longs;o V &longs;arà mo&longs;&longs;o dalla po&longs;&longs;anza di X, che è il mede&longs;imo, come &longs;el'i&longs;te&longs;&longs;o pe<lb/>&longs;o in tempo eguale fo&longs;&longs;e mo&longs;&longs;o. </s> <s id="id.2.1.882.2.0">Moua la po&longs;&longs;anza di X il pe&longs;o V, & la po&longs;&longs;an<lb/>za peruenga in <foreign lang="greek">*u</foreign>; & &longs;ia X<foreign lang="greek">*u</foreign> eguale ad e&longs;&longs;a FG: & &longs;i faccia <foreign lang="greek">*u</foreign>Z eguale <lb/>à X<36>, talche quando la po&longs;&longs;anza di X &longs;arà in <foreign lang="greek">*u</foreign>, &longs;ia il pe&longs;o V cioè il punto <36><emph.end type="italics"/><pb xlink:href="037/01/164.jpg"/><emph type="italics"/>in Z; ma <36>Z è eguale ad FG, e&longs;&longs;endo eguale ad X<foreign lang="greek">*u</foreign>: dunque <36>Z &longs;arà due <lb/>volte tanto, quanto OH. </s> <s id="id.2.1.882.3.0">Per laqual co&longs;a mentre le po&longs;&longs;anze &longs;aranno in G<foreign lang="greek">*u</foreign>, i <lb/>pe&longs;i AV &longs;aranno in OZ. </s> <s id="id.2.1.882.4.0">Hor nell'i&longs;te&longs;&longs;o tempo &longs;aranno le po&longs;&longs;anze in G<foreign lang="greek">*u</foreign>, <lb/>peroche le vetocità de mouimenti &longs;ono eguali: onde la forza di F mouerà il pe<lb/>&longs;o A nel mede&longs;imo tempo per la metà di quello &longs;patio, per loquale il pe&longs;o V &longs;a<emph.end type="italics"/><lb/><figure id="id.037.01.164.1.jpg" xlink:href="037/01/164/1.jpg"></figure><lb/><emph type="italics"/>rà mo&longs;&longs;o dalla po&longs;&longs;anza di X: & li pe&longs;i &longs;ono eguali, adunque la po&longs;&longs;anza moue<lb/>rà il pe&longs;o i&longs;l e&longs;&longs;o in tempo eguale per la metà dello &longs;patio, con la corda, & la taglia <lb/>legata in que&longs;to modo al pe&longs;o, che &longs;enza taglia; purche le velocità della po&longs;&longs;anza <lb/>de'mouimenti &longs;iano eguali, che era da mo&longs;trar&longs;i. <emph.end type="italics"/></s> <pb pagenum="75" xlink:href="037/01/165.jpg"/> <s id="id.2.1.886.1.0">PROPOSITIONE XII. </s> <s id="id.2.1.887.1.0">Se la corda &longs;arà riuolta d'intorno à più girelle, legando l'vno de' <lb/>capi &longs;uoi in qualche loco, & l'altro &longs;ia tenuto dalla po&longs;&longs;anza, <lb/>che moue il pe&longs;o: La po&longs;&longs;anza mouerà con le leue &longs;empre <lb/>egualmente di&longs;tanti dall'orizonte. </s> <s id="id.2.1.888.1.0"><emph type="italics"/>Sia il pe&longs;o A. </s> <s id="id.2.1.888.2.0">&longs;ia la girella CED della <lb/>taglia legata al pe&longs;o da KS ad angoli ret<lb/>ti all'orizonte; di modo, che il pe&longs;o &longs;egua <lb/>&longs;empre il &longs;uo mouimento ò &longs;u&longs;o, ò giu&longs;o, <lb/>che &longs;ia fatto. </s> <s id="id.2.1.888.3.0">Sia dapoi la girella intorno <lb/>al centro L della taglia appiccata di &longs;opra; <lb/>& &longs;iala corda BCDEHMNO riuol<lb/>ta d'intorno alle girelle, laquale &longs;ia legata <lb/>in B; & &longs;ia in O la forza mouente il <lb/>pe&longs;o A, mouendo&longs;i al ba&longs;&longs;o per OP. </s> <s id="id.2.1.888.4.0">Di<lb/>co che la po&longs;&longs;anza di O mouerà &longs;empre il <lb/>pe&longs;o A con le leue &longs;empre egualmente <lb/>di&longs;tanti dall'orizonte. </s> <s id="id.2.1.888.5.0">&longs;ia tirata la linea <lb/>NH per lo centro L egualmente di&longs;tan<lb/>te dall'orizonte, che &longs;arà la leua della girel<emph.end type="italics"/><arrow.to.target n="note244"></arrow.to.target><lb/><emph type="italics"/>la, il cui centro è L: &longs;ia tirata da poi la <lb/>EC per lo centro K, &longs;imilmente di&longs;tan<emph.end type="italics"/><arrow.to.target n="note245"></arrow.to.target><lb/><emph type="italics"/>te egualmente dall'orizonte, la quale &longs;arà <lb/>anche la leua della girella, il cui centro è <emph.end type="italics"/><arrow.to.target n="note246"></arrow.to.target><lb/><emph type="italics"/>K. </s> <s id="id.2.1.888.6.0">Moua&longs;i la po&longs;&longs;anza di O in giu&longs;o, la <lb/>quale mentre in giu&longs;o &longs;i moue, mouerà la <lb/>leua NH, & mentre la leua &longs;i moue, la <lb/>N &longs;i mouerà in giu&longs;o, & la H in &longs;u&longs;o, <lb/>come è detto di &longs;opra. </s> <s id="id.2.1.888.7.0">Ma mentre la H <lb/>&longs;i moue in &longs;u&longs;o, moue etiandio in &longs;u&longs;o la E, <lb/>& la leua EC, il cui &longs;o&longs;tegno è C, ma <lb/>il &longs;o&longs;tegno C non puote mouere in giu&longs;o <lb/>il B; però la girella il cui centro è K mo<lb/>uera&longs;&longs;i in &longs;u&longs;o, & per con&longs;equenza la ta<lb/>glia, & il pe&longs;o A, come nella preceden<lb/>te è stato detto. </s> <s id="id.2.1.888.8.0">& perche per la mede&longs;i<lb/>ma cau&longs;a, che è stata a&longs;&longs;egnata nelle pre<lb/>cedenti, rimangono &longs;empre le leue egual<lb/>mente distanti dall'orizonte in HN, & <emph.end type="italics"/><figure id="id.037.01.166.1.jpg" xlink:href="037/01/166/1.jpg"></figure><pb xlink:href="037/01/166.jpg"/><emph type="italics"/>in EC, la po&longs;&longs;anza dun<lb/>que mouente il pe&longs;o A <lb/>lo mouerà &longs;empre &longs;tando <lb/>le leue egualmente distan<lb/>ti dall'orizonte; che era da <lb/>mo&longs;trar&longs;i. <emph.end type="italics"/></s> <s id="id.2.1.890.1.0"><margin.target id="note244"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>&<emph.end type="italics"/> 10. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.891.1.0"><margin.target id="note245"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.892.1.0"><margin.target id="note246"></margin.target><emph type="italics"/>Par la<emph.end type="italics"/> 10. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.893.1.0"><emph type="italics"/>Et &longs;e la corda &longs;arà riuolta d'in<lb/>torno à più girelle; &longs;imil<lb/>mente &longs;i dimo&longs;trerà la po&longs;<lb/>&longs;anza mouere il pe&longs;o con <lb/>le leue &longs;empre egualmente <lb/>di&longs;tanti dall'orizonte: & <lb/>le leue delle girelle della ta<lb/>glia di &longs;opra &longs;empre e&longs;&longs;e<lb/>re come HN, i &longs;o&longs;tegni <lb/>delle quali &longs;aranno &longs;empre <lb/>nel mezo: ma le leue delle <lb/>girelle della taglia di &longs;otto <lb/>&longs;empre e&longs;&longs;ere, come EC; <lb/>li cui &longs;o&longs;tegni &longs;aranno nel<lb/>le &longs;tremità delle leue. <emph.end type="italics"/></s> <s id="id.2.1.894.1.0">Stando le co&longs;e i&longs;te&longs;&longs;e, <lb/>lo &longs;patio della po&longs;<lb/>&longs;anza, è il doppio <lb/>dello &longs;patio del pe<lb/>&longs;o. </s> <figure id="id.037.01.166.2.jpg" xlink:href="037/01/166/2.jpg"> </figure> <s id="id.2.1.896.1.0"><emph type="italics"/>Sia mo&longs;&longs;o il centro K fin al centro R; & &longs;ia la girella FTG: poi &longs;ia per lo cen<lb/>tro R condotta la linea GF egualmente di&longs;tante da e&longs;&longs;a EC: le corde EH <lb/>CB toccheranno la girella ne i punti GF. </s> <s id="id.2.1.896.2.0">Faccia&longs;i alla fine RQ eguale à <lb/>KS. </s> <s id="id.2.1.896.3.0">Mentre dunque K &longs;arà in R, il pe&longs;o A, cioè il punto S &longs;arà in Q,<emph.end type="italics"/><pb pagenum="76" xlink:href="037/01/167.jpg"/><emph type="italics"/>& mentre il centro della girella è in R, &longs;ia la po&longs;&longs;anza di O mo&longs;&longs;a in P. </s> <s id="id.2.1.896.4.0">& <lb/>percioche la corda BCDEHMNO eguale è alla corda BFTGHMNP <lb/>per e&longs;&longs;er la corda iste&longs;&longs;a, & FTG è eguale à CDE; leuate via dunque le com<lb/>muni BF & GHMNO, &longs;arà la re&longs;tante OP eguale ad e&longs;&longs;e FC EG pre<lb/>&longs;e in&longs;ieme: & per con&longs;equenza due volte tanto, quanto è KR, & QS. </s> <s id="id.2.1.896.5.0">& e&longs;<lb/>&longs;endo OP lo &longs;patio della po&longs;&longs;anza mo&longs;&longs;a, & SQ lo &longs;patio del pe&longs;o mo&longs;&longs;o; <lb/>&longs;arà lo &longs;patio della po&longs;&longs;anza due volte tanto quanto lo &longs;patio del pe&longs;o. </s> <s id="id.2.1.896.6.0">che era <lb/>da mostrar&longs;i. <emph.end type="italics"/></s> <s id="id.2.1.897.1.0">Oltre à ciò la po&longs;&longs;anza mouerà il pe&longs;o i&longs;te&longs;&longs;o in tempo eguale <lb/>per la metà dello &longs;patio, con vna corda riuolta d'intorno à <lb/>due girelle, l'una delle quali &longs;ia della taglia di &longs;opra, & l'altra <lb/>&longs;ia della taglia legata al pe&longs;o; che &longs;enza taglie: pur che i mo<lb/>uimenti di e&longs;&longs;a po&longs;&longs;anza &longs;iano egualmente veloci. </s> <pb xlink:href="037/01/168.jpg"/> <s id="id.2.1.898.1.0"><emph type="italics"/>Percioche &longs;tando le co<lb/>&longs;e i&longs;te&longs;&longs;e, &longs;ia il pe&longs;o <lb/>V eguale ad e&longs;&longs;o A, <lb/>alquale &longs;ia legata la <lb/>corda X<36>; & &longs;ia <lb/>la po&longs;&longs;anza in X che <lb/>moue il pe&longs;o V; la <lb/>quale mentre moue <lb/>il pe&longs;o, peruenga in <lb/><foreign lang="greek">*u</foreign>: & &longs;iano fatte <lb/>X<foreign lang="greek">*u</foreign> Z<36> eguali ad <lb/>e&longs;&longs;a OP; &longs;arà Z<36><lb/>due volte tanto <expan abbr="quãto">quan<lb/>to</expan> QS. </s> <s>& &longs;e le <lb/>velocità de' moui<lb/>menti dell'vna, & <lb/>l'altra po&longs;&longs;anza &longs;a<lb/>ranno eguali; egli è <lb/>manife&longs;to, che il pe<lb/>&longs;o V trapa&longs;&longs;a due <lb/>volte tanto &longs;patio <lb/>nell'i&longs;te&longs;&longs;o tempo, di <lb/>quel che trapa&longs;&longs;i il <lb/>pe&longs;o A: percioche <lb/>nel tempo mede&longs;imo <lb/>la po&longs;&longs;anza di X per<lb/>uiene ad <foreign lang="greek">*u</foreign>, & la <lb/>po&longs;&longs;anza di O à P; <lb/>& li pe&longs;i &longs;imilmen<lb/>te in ZQ. </s> <s>che era <lb/>da mo&longs;trar&longs;i. <emph.end type="italics"/></s> <figure id="id.037.01.168.1.jpg" xlink:href="037/01/168/1.jpg"> </figure> <s id="id.2.1.900.1.0">PROPOSITIONE XIII. </s> <s id="id.2.1.901.1.0">Riuolgendo la corda d'intorno à due girelle di due taglie, I'vna <lb/>dellequali &longs;ia di &longs;opra, & l'altra di &longs;otto, & legata al pe&longs;o; e&longs;<lb/>&longs;endo anche l'vno de' capi di detta corda legato alla taglia di <lb/>&longs;otto, & l'altro tenuto dalla po&longs;&longs;anza che moue; &longs;arà lo &longs;patio <lb/>cor&longs;o della po&longs;&longs;anza, che tira, tre volte tanto quanto lo &longs;pati <lb/>del pe&longs;o mo&longs;&longs;o. </s> <pb pagenum="77" xlink:href="037/01/169.jpg"/> <s id="id.2.1.903.1.0"><emph type="italics"/>Sia il pe&longs;o A; &longs;ia BCD la girel<lb/>la della taglia legata al pe&longs;o A, <lb/>attaccato da EQ, & &longs;ia E il <lb/>centro della girella; &longs;ia dapoi F <lb/>GH la girella della taglia appic<lb/>cata di &longs;opra, il cui centro K; & <lb/>&longs;ia la corda LFGHDBCM ri<lb/>uolta intorno à tutte le girelle, & <lb/>legata alla taglia di &longs;otto in L: <lb/>& &longs;ia in M la po&longs;&longs;anza, che <lb/>moue. </s> <s id="id.2.1.903.2.0">Dico lo &longs;patio cor&longs;o dalla <lb/>po&longs;&longs;anza di M, mentre moue il <lb/>pe&longs;o, e&longs;&longs;ere triplo dello &longs;patio del <lb/>pe&longs;o mo&longs;&longs;o A. </s> <s id="id.2.1.903.3.0">Moua&longs;i la po&longs;&longs;an<lb/>za di M fin ad N; & il centro <lb/>E &longs;ia mo&longs;&longs;o fin ad O; & L fin <lb/>à P; & il pe&longs;o A, cioè il pun<lb/>to Q fin ad R; & la girella <lb/>mo&longs;&longs;a &longs;ia TSV. </s> <s id="id.2.1.903.4.0">Siano condot<lb/>te per EO le linee ST BD <lb/>egualmente di&longs;tanti dall'orizonte, <lb/>lequali &longs;aranno anche tra loro e<lb/>gualmente di&longs;tanti. </s> <s id="id.2.1.903.5.0">Ma percio<lb/>che mentre E &longs;ta in O, il pun<lb/>to Q &longs;ta in R; &longs;arà EQ egua<lb/>le ad OR, & EO ade&longs;&longs;o QR <lb/>eguale; &longs;imilmente LQ &longs;arà <lb/>eguale à PR, & LP ad e&longs;&longs;o <lb/>QR eguale. </s> <s id="id.2.1.903.6.0">Adunque le tre <lb/>QR EO LP fra loro &longs;aranno <lb/>eguali; à cui &longs;ono etiandio eguali <lb/>BS DT. </s> <s id="id.2.1.903.7.0">Et percioche la corda <lb/>LFGHDCBM è eguale alla <lb/>corda PFGHTVSN e&longs;&longs;en<lb/>do vna corda i&longs;te&longs;&longs;a, & la corda, <lb/>che è intorno al mezo cerchio <lb/>TVS è eguale alla corda, che è <lb/>intorno al mezo cerchio BCD; <lb/>tolte via dunque le communi PF <lb/>GHT, & SM; &longs;arà la re&longs;tan<lb/>te MN eguale alle tre BS <lb/>LP DT pre&longs;e in&longs;ieme. </s> <s id="id.2.1.903.8.0">ma BS LP DT in&longs;ieme &longs;ono tre volte tanto, quanto <emph.end type="italics"/><lb/><figure id="id.037.01.170.1.jpg" xlink:href="037/01/170/1.jpg"></figure><pb xlink:href="037/01/170.jpg"/><emph type="italics"/>EQ, & per con&longs;e<lb/>quenza QR. </s> <s id="id.2.1.903.9.0">Lo <lb/>&longs;patio dunque MN <lb/>della traportata po&longs;<lb/>&longs;anza è tre volte <lb/>tanto, quanto lo &longs;pa<lb/>tio QR del pe&longs;o <lb/>mo&longs;&longs;o. </s> <s id="id.2.1.903.10.0">che era da <lb/>mo&longs;trar&longs;i. <emph.end type="italics"/></s> <s id="id.2.1.905.1.0"><emph type="italics"/>Il tempo ancora di que<lb/>sto mouimento è <lb/>manife&longs;to, percio<lb/>che la po&longs;&longs;anza i&longs;te&longs;<lb/>&longs;a in tempo eguale <lb/>mouerà l'i&longs;te&longs;&longs;o pe<lb/>&longs;o in i&longs;patio tre co<lb/>tanto maggiore &longs;en<lb/>za tali taglie, di <lb/>quel che &longs;arebbe <lb/>con e&longs;&longs;e taglie à que<lb/>&longs;to modo commoda<lb/>te. </s> <s id="id.2.1.905.2.0">Lo &longs;patio del <lb/>pe&longs;o mo&longs;&longs;o &longs;enza le <lb/>taglie è eguale allo <lb/>&longs;patio della po&longs;&longs;an<lb/>za. </s> <s id="id.2.1.905.3.0">& in que&longs;to <lb/>modo ritrouaremo <lb/>in tutte il tempo. <emph.end type="italics"/></s> <figure id="id.037.01.170.2.jpg" xlink:href="037/01/170/2.jpg"> </figure> <pb pagenum="78" xlink:href="037/01/171.jpg"/> <s id="id.2.1.908.1.0">PROPOSITIONE XIIII. </s> <s id="id.2.1.909.1.0">Legando la corda d'intorno à tre girelle di due taglie, l'vna del<lb/>lequali &longs;ia di &longs;opra, & habbia vna &longs;ola girella, & l'altra di &longs;ot<lb/>to, & ne habbia due, & &longs;ia lega<lb/>ta al pe&longs;o; laqual corda &longs;ia le<lb/>gata con l'vno de' capi &longs;uoi in <lb/>qualche loco, & l'altro tenu<lb/>to dalla po&longs;&longs;anza, che moue il <lb/>pe&longs;o: &longs;arà lo &longs;patio cor&longs;o dal<lb/>la po&longs;&longs;anza, che tira, quattro <lb/>volte tanto, quanto è lo &longs;patio <lb/>del pe&longs;o mo&longs;&longs;o. </s> <s id="id.2.1.910.1.0"><emph type="italics"/>Sia il pe&longs;o A, &longs;iano le due girelle, i cui cen<lb/>tri <emph.end type="italics"/>K <emph type="italics"/>I della taglia legata al pe&longs;o con <lb/>K<foreign lang="greek">a</foreign>; di modo, che il pe&longs;o &longs;empre &longs;egua il <lb/>mouimento della taglia in &longs;u&longs;o, ouero in <lb/>giu&longs;o: &longs;ia dapoi la girella il cui centro L <lb/>della taglia appe&longs;a di &longs;opra in <foreign lang="greek">d</foreign>; & &longs;ia la <lb/>corda BCDEFGHZMNO riuolta <lb/>intorno à tutte le girelle, & legata in B; <lb/>& &longs;ia in O la po&longs;&longs;anza, che moue il pe<lb/>&longs;o A. </s> <s id="id.2.1.910.2.0">Dico lo &longs;patio, ilquale la po&longs;&longs;an<lb/>za di O mouendo trapa&longs;&longs;a, e&longs;&longs;ere quat<lb/>tro volte tanto, quanto lo &longs;patio del pe<lb/>&longs;o A mo&longs;&longs;o. </s> <s id="id.2.1.910.3.0">Mouan&longs;i le girelle della <lb/>taglia legata al pe&longs;o; & mentre il centro <lb/>K è in R, il centro I &longs;ia in S, & il <lb/>pe&longs;o A, cioè il punto <foreign lang="greek">a</foreign> in <foreign lang="greek">b</foreign>: &longs;aranno <lb/>IS KR <foreign lang="greek">ab</foreign> tra &longs;e eguali, & parimen<lb/>te KI ad e&longs;&longs;a RS eguale: percioche le <lb/>girelle mantengono fra &longs;e la di&longs;tanza me<lb/>de&longs;ima &longs;empre; & K<foreign lang="greek">a</foreign> &longs;arà eguale ad e&longs;<lb/>&longs;a R<foreign lang="greek">b </foreign>.</s> <s id="id.2.1.910.4.0">&longs;iano condotte per li centri delle <lb/>girelle le linee FHQTECVXNZ <lb/>egualmente distanti dall orizonte, lequa<lb/>li tocchino le corde ne i punti FH QT<emph.end type="italics"/><lb/><figure id="id.037.01.172.1.jpg" xlink:href="037/01/172/1.jpg"></figure><pb xlink:href="037/01/172.jpg"/><emph type="italics"/>EC VX NZ che parimente &longs;aranno <lb/>fra loro egualmente di&longs;tanti: & EQ CT <lb/>VN XZ non &longs;olamente fra &longs;e, ma <lb/>ancora ad e&longs;&longs;e IS KR <foreign lang="greek">ab</foreign> &longs;aranno e<lb/>guali: & mentre li centri KI &longs;ono in <lb/>RS, la po&longs;&longs;anza di O &longs;ia mo&longs;&longs;a in P. <lb/></s> <s id="id.2.1.910.5.0">Et percioche la corda BCDEFGHZ <lb/>MNO è eguale alla corda BT<36>QF <lb/>GHX<foreign lang="greek">*u</foreign>VP e&longs;&longs;endo vna corda mede<lb/>&longs;ima, & le corde d'intorno à mezi cerchi <lb/>T<36>Q X<foreign lang="greek">*u</foreign>V &longs;ono eguali alle corde, che <lb/>&longs;ono d'intorno à CDE ZMN; tolte <lb/>via dunque le communi BT, QFGHX, <lb/>& VO; &longs;arà OP eguale ad e&longs;&longs;e VN <lb/>XZ CT QE pre&longs;e tutte in&longs;ieme. </s> <s id="id.2.1.910.6.0">ma le <lb/>quattro VN ZX CT QE &longs;ono tra &longs;e <lb/>eguali, & in&longs;ieme quattro volte tanto <lb/>quanto KR & <foreign lang="greek">ab</foreign>. </s> <s id="id.2.1.910.7.0">Per laqual co&longs;a OP <lb/>&longs;arà quattro volte tanto quanto è e&longs;&longs;a <lb/><foreign lang="greek">ab</foreign>. </s> <s id="id.2.1.910.8.0">Adunque lo &longs;patio della po&longs;&longs;anza <lb/>è quattro volte tanto quanto è lo &longs;patio <lb/>del pe&longs;o. </s> <s id="id.2.1.910.9.0">che era da mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.912.1.0"><emph type="italics"/>Et &longs;e la corda in P &longs;arà dauantaggio ri<lb/>uolta d'intorno ad vn'altra girella ver&longs;o il <lb/><foreign lang="greek">d</foreign>, & la po&longs;&longs;anza mouendo&longs;i in giù mo<lb/>ua in sù il pe&longs;o: &longs;imilmente &longs;i mo&longs;trerà <lb/>lo &longs;patio della po&longs;&longs;anza e&longs;&longs;ere quattro <lb/>volte tanto quanto lo &longs;patio del pe&longs;o. <emph.end type="italics"/></s> <figure id="id.037.01.172.2.jpg" xlink:href="037/01/172/2.jpg"> </figure> <s id="id.2.1.914.1.0"><emph type="italics"/>Ma &longs;e la corda in B &longs;i riuolgerà d'intorno ad <lb/>vn'altra girella, laqual corda &longs;i leghi da<emph.end type="italics"/><lb/><arrow.to.target n="note247"></arrow.to.target><emph type="italics"/>poi alla taglia di &longs;otto; &longs;arà la po&longs;&longs;anza <lb/>di O, che &longs;o&longs;tiene il pe&longs;o A vn quinto <lb/>dal pe&longs;o. </s> <s id="id.2.1.914.2.0">& &longs;e in O &longs;arà la po&longs;&longs;anza, <lb/>che moua il pe&longs;o A; &longs;imilmète &longs;i dimo&longs;tre <lb/>rà lo &longs;patio della po&longs;&longs;anza po&longs;ta in O e&longs;<lb/>&longs;ere cinque volte tanto quanto lo &longs;patio del pe&longs;o A. <emph.end type="italics"/></s> <s id="id.2.1.915.1.0"><margin.target id="note247"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.916.1.0"><emph type="italics"/>Et &longs;e la corda &longs;i adatterà in modo d'intorno alle girelle, che la po&longs;&longs;anza di O &longs;o&longs;tenen<lb/>te il pe&longs;o &longs;ia vn &longs;e&longs;to del pe&longs;o; & in loco della po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o, &longs;i met<lb/>ta in O la po&longs;&longs;anza, che lo moua; nell'i&longs;te&longs;&longs;o modo &longs;i mo&longs;trerà lo &longs;patio della po&longs;<lb/>&longs;anza e&longs;&longs;ere &longs;ei volte tanto quanto lo &longs;patio del pe&longs;o mo&longs;&longs;o. </s> <s id="id.2.1.916.2.0">& co&longs;i procedendo in <emph.end type="italics"/><pb pagenum="79" xlink:href="037/01/173.jpg"/><emph type="italics"/>infinito &longs;i troueranno le proportioni dello &longs;patio della po&longs;&longs;anza allo &longs;patio del pe<lb/>&longs;o mo&longs;&longs;o quanto &longs;i vogliano moltiplici. <emph.end type="italics"/></s> <s id="id.2.1.917.1.0">Et co&longs;i procedendo in infinito &longs;i troueranno le proportioni dello &longs;patio della po&longs;<lb/>&longs;anza allo &longs;patio del pe&longs;o mo&longs;&longs;o quanto &longs;i vorrà moltiplici. </s> <s id="id.2.1.917.2.0">Già è detto che mol<lb/>tiplice è il primo genere delle proportioni nelle quantità paragonate dal mag<lb/>giore al minore, però qui vuol dire, che con tale regola &longs;i ritroueranno le pro<lb/>portioni dello &longs;patio del pe&longs;o allo &longs;patio della po&longs;&longs;anza in infinito, <expan abbr="dou&etilde;do">douendo</expan> e&longs;&longs;ere <lb/>lo &longs;patio della po&longs;&longs;anza mouente moltiplice, cioè molte volte maggiore dello <lb/>&longs;patio del pe&longs;o mo&longs;&longs;o, come appare nel pre&longs;ente e&longs;&longs;empio, che è &longs;ei volte più, <lb/>come &longs;ei ad vno; & que&longs;to è il &longs;ignificato di moltiplice. </s> <s id="id.2.1.918.1.0">COROLLARIO I. </s> <s id="id.2.1.919.1.0">Da que&longs;te co&longs;e è manife&longs;to, co&longs;i hauer&longs;i il pe&longs;o ver&longs;o la po&longs;&longs;an<lb/>za, che lo &longs;o&longs;tiene, come lo &longs;patio della po&longs;&longs;anza che moue al<lb/>lo &longs;patio del pe&longs;o mo&longs;&longs;o. </s> <s id="id.2.1.920.1.0"><emph type="italics"/>Come &longs;e il pe&longs;o A &longs;arà cinque volte tanto quanto la po&longs;&longs;anza di O, che &longs;o&longs;tiene <lb/>il detto pe&longs;o A; &longs;arà anche lo &longs;patio OP della po&longs;&longs;anza mouente il pe&longs;o cin<lb/>que volte tanto quanto lo &longs;patio <foreign lang="greek">a b</foreign> del pe&longs;o mo&longs;&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.921.1.0">COROLLARIO II. </s> <s id="id.2.1.922.1.0">E manife&longs;to ancora per le co&longs;e dette, che le girelle della taglia, <lb/>laquale è legata al pe&longs;o, fanno sì, che minore &longs;patio è quello, <lb/>ilquale è de&longs;critto dal pe&longs;o mo&longs;&longs;o, che dalla po&longs;&longs;anza che tira; <lb/>& che in tempo maggiore &longs;i de&longs;criua vn dato &longs;patio eguale, <lb/>che &longs;enza loro: ilche veramente non fanno le girelle della ta<lb/>glia di &longs;opra. </s> <s id="id.2.1.923.1.0"><emph type="italics"/>Mo&longs;trata la proportione moltiplice, che ha il pe&longs;o ver&longs;o la po&longs;&longs;anza, hora &longs;i mo&longs;tri per <lb/>lo contrario la proportione moltiplice, che haue la po&longs;&longs;anza ver&longs;o il pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.924.1.0">PROPOSITIONE XV. </s> <s id="id.2.1.925.1.0">Se la corda &longs;arà inuolta d'intorno alla girella della taglia tenu<lb/>ta di &longs;opra dalla po&longs;&longs;anza; l'vn capo dellaquale &longs;ia legato in <lb/>qualche loco, ma all'altro &longs;ia appiccato il pe&longs;o, &longs;arà la po&longs;&longs;an<lb/>za due volte tanto quanto il pe&longs;o. </s> <pb xlink:href="037/01/174.jpg"/> <s id="id.2.1.926.1.0"><emph type="italics"/>Sia la taglia, che habbia la girella co'l &longs;uo centro A; & &longs;ia il pe&longs;o B legato alla <lb/>corda CDEFG, laquale &longs;ia in uol<lb/>ta d'intorno alla girella, & alla fine <lb/>legata in G; & &longs;ia la po&longs;&longs;anza, che <lb/>&longs;o&longs;tiene il pe&longs;o in H. </s> <s id="id.2.1.926.2.0">Dico, che la <lb/>po&longs;&longs;anza di H è due volte tanto quan<lb/>to il pe&longs;o B. </s> <s id="id.2.1.926.3.0">Sia condotta la linea <lb/>DF per lo centro A egualmente di <lb/>&longs;tante dall'orizonte. </s> <s id="id.2.1.926.4.0">Percioche dun<lb/>que la po&longs;&longs;anza di H &longs;o&longs;tiene la ta<lb/>glia, laquale &longs;o&longs;tiene la girella nel &longs;uo <lb/>centro A, laqual girella &longs;o&longs;tiene il pe<lb/>&longs;o; &longs;arà la po&longs;&longs;anza, che &longs;o&longs;tiene la gi<lb/>rella, come &longs;e fo&longs;&longs;e po&longs;ta in A; &longs;tan<lb/>do dunque e&longs;&longs;a in A, & il pe&longs;o ap<lb/>piccato in D, & legato alla corda <lb/>CD; &longs;arà la DF come leua, il cui <lb/>&longs;o&longs;tegno &longs;arà F, il pe&longs;o in D & la <emph.end type="italics"/><lb/><arrow.to.target n="note248"></arrow.to.target><emph type="italics"/>po&longs;&longs;anza in A. </s> <s id="id.2.1.926.5.0">Ma la po&longs;&longs;anza ver<lb/>&longs;o il pe&longs;o è come DF ad FA, & <lb/>DF è il doppio di FA: adunque la <lb/>po&longs;&longs;anza di A ouero di H, che è <lb/>l'i&longs;te&longs;&longs;o, &longs;arà due volte tanto, quanto il <lb/>pe&longs;o B. </s> <s>che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.927.1.0"><margin.target id="note248"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>di questo nella leua. <emph.end type="italics"/></s> <figure id="id.037.01.174.1.jpg" xlink:href="037/01/174/1.jpg"> </figure> <s id="id.2.1.929.1.0"><emph type="italics"/>Oltre à ciò occorre à con&longs;iderare, &longs;tando ferme tutte que&longs;te co&longs;e, che egli è l'i&longs;te&longs;&longs;o, e&longs;<lb/>&longs;endo vna corda &longs;ola CDEFG in que&longs;to modo inuolta d'intorno alla girella, co<lb/>me &longs;e fo&longs;&longs;ero due corde CDFG legate nella leua, ouero nella bilancia DF. <emph.end type="italics"/></s> <s id="id.2.1.930.1.0">Altramente. </s> <s id="id.2.1.931.1.0"><emph type="italics"/>Stando le mede&longs;ime co&longs;e, &longs;e in G fo&longs;&longs;e appiccato un pe&longs;o K eguale al pe&longs;o B, li pe&longs;i <lb/>BK pe&longs;erebbono egualmente nella bilancia DF, il cui centro A. </s> <s id="id.2.1.931.2.0">Ma la po&longs;<lb/>&longs;anza di H, laquale &longs;o&longs;tiene i pe&longs;i BK è eguale ad ambidue pre&longs;i in&longs;ieme, & i <lb/>pe&longs;i BK &longs;ono due volte tanto quanto è e&longs;&longs;o B. </s> <s id="id.2.1.931.3.0">Adunque la po&longs;&longs;anza di H &longs;a<lb/>rà due volte tanto quanto è il B. </s> <s>& percioche la corda legata in G non fa al<lb/>tro niente, &longs;e non che &longs;o&longs;tiene il pe&longs;o B, che non di&longs;cenda, laqual co&longs;a parimente <lb/>fà il pe&longs;o K appiccato in G: la po&longs;&longs;anza dunque di H, che &longs;o&longs;tiene il pe&longs;o B, <lb/>e&longs;&longs;endo la corda legata in G, è due volte tanto quanto il pe&longs;o B. </s> <s>che bi&longs;ognaua <lb/>mostrare. <emph.end type="italics"/></s> <pb pagenum="80" xlink:href="037/01/175.jpg"/> <s id="id.2.1.933.1.0">PROPOSITIONE XVI. </s> <s id="id.2.1.934.1.0">Po&longs;te le co&longs;e i&longs;te&longs;&longs;e, &longs;e in H &longs;arà la po&longs;&longs;anza che moue il pe&longs;o, <lb/>mouerà ella con la leua egualmente di&longs;tante dall'orizonte. </s> <s id="id.2.1.935.1.0"><emph type="italics"/>Que&longs;to etiandio &longs;i mo&longs;trerà, co<lb/>me è detto di &longs;opra. </s> <s id="id.2.1.935.2.0">Moua&longs;i <lb/>la girella in sù, & habbia il <lb/>&longs;ito di MNO, il cui centro <lb/>L: & per L &longs;ia condotta la <lb/>linea MLO egualmente di<lb/>&longs;tante da e&longs;&longs;a DF, & dall'o<lb/>rizonte. </s> <s id="id.2.1.935.3.0">& percioche le cor<lb/>de toccano il cerchio MON <lb/>ne i punti MO; però e&longs;&longs;en<lb/>do che la po&longs;&longs;anza di A, oue<lb/>ro di H, che è l'i&longs;te&longs;&longs;o, mo<lb/>ua il pe&longs;o B appiccato in D <lb/>con la leua DF, il cui &longs;o&longs;te<lb/>gno è F; &longs;empre rimarrà da <lb/>uantaggio vn'altra leua, co<lb/>me MO egualmente di&longs;tan<lb/>te dall'orizonte, di modo che <lb/>&longs;empre la po&longs;&longs;anza moua il pe<lb/>&longs;o, &longs;tando la leua egualmente <lb/>di&longs;tante dall'orizonte, il cui <lb/>&longs;o&longs;tegno &longs;empre è nella linea <lb/>OG, & il pe&longs;o in MC, & <lb/>la po&longs;&longs;anza nel centro della <lb/>girella. <emph.end type="italics"/></s> <figure id="id.037.01.175.1.jpg" xlink:href="037/01/175/1.jpg"> </figure> <s id="id.2.1.937.1.0">Po&longs;te le co&longs;e mede&longs;ime, lo &longs;patio del pe&longs;o mo&longs;&longs;o è due volte tan<lb/>to quanto lo &longs;patio della po&longs;&longs;anza, che moue. </s> <pb xlink:href="037/01/176.jpg"/> <s id="id.2.1.938.1.0"><emph type="italics"/>Sia mo&longs;&longs;a la girella dal centro A fin al centro L; & il pe&longs;o B, cioè il punto e, <lb/>nell'i&longs;te&longs;&longs;o tempo &longs;ia mo&longs;&longs;o nel P; & la po&longs;&longs;anza di H fin in K; farà AH <lb/>ad e&longs;&longs;a LK eguale, & <lb/>AL ad e&longs;&longs;a HK: & <lb/>percioche le corda CDE <lb/>FG eguale è alla corda <lb/>PMNOG, peroche è <lb/>vna corda i&longs;te&longs;&longs;a, & la <lb/>corda d'intorno al mezo <lb/>cerchio MNO eguale è <lb/>alla corda d'intorno al me<lb/>zo cerchio DEF: tolte <lb/>via dunque le communi <lb/>corde DP FG, &longs;arà <lb/>PC eguale à DM FO <lb/>pre&longs;e in&longs;ieme, lequali cor<lb/>de &longs;ono due volte tanto <lb/>quanto è e&longs;&longs;a AL & per <lb/>con&longs;eguenza e&longs;&longs;a HK. <lb/></s> <s id="id.2.1.938.2.0">Lo &longs;patio dunque del pe<lb/>&longs;o mo&longs;&longs;o CP è due vol<lb/>te tanto, quanto è lo &longs;pa<lb/>tio della po&longs;&longs;anza HK. </s> <s><lb/>che bi&longs;ognaua mostrare. <emph.end type="italics"/></s> <figure id="id.037.01.176.1.jpg" xlink:href="037/01/176/1.jpg"> </figure> <s id="id.2.1.940.1.0">COROLLARIO</s> <s id="id.2.1.941.1.0">Da que&longs;to è manife&longs;to, l'i&longs;te&longs;&longs;o pe&longs;o e&longs;&longs;ere tirato dalla i&longs;te&longs;&longs;a po&longs;<lb/>&longs;anza in tempo eguale per due volte tanto &longs;patio con la taglia <lb/>in que&longs;to modo accommodata, che &longs;enza taglia; pur che i <lb/>mouimenti di e&longs;&longs;a po&longs;&longs;anza &longs;iano eguali in velocità. </s> <s id="id.2.1.942.1.0"><emph type="italics"/>Percioche lo &longs;patio del pe&longs;o mo&longs;&longs;o &longs;enza taglia è vguale allo &longs;patio della po&longs;&longs;anza. <emph.end type="italics"/></s> <pb pagenum="81" xlink:href="037/01/177.jpg"/> <s id="id.2.1.944.1.0"><emph type="italics"/>Che &longs;e la corda &longs;arà in G riuolta d'intorno ad vn'altra girella, il cui centro K; & <lb/>&longs;ia la taglia di cotale girella attaccata di &longs;otto, laquale non habbia alcuno altro <lb/>mouimento, &longs;e non il libero riuolgimento della girella d'intorno all'a&longs;&longs;etto &longs;uo; & <lb/>la corda &longs;i leghi in M; &longs;arà <lb/>la po&longs;&longs;anza di H che &longs;o&longs;tiene <lb/>il pe&longs;o B. </s> <s>&longs;imilmente due vol<lb/>te tanto, quanto è e&longs;&longs;o pe&longs;o. </s> <s id="id.2.1.944.2.0">il <lb/>che per certo è manife&longs;to, con<lb/>cio&longs;ia, che egli &longs;ia in tutto vna <lb/>co&longs;a i&longs;te&longs;&longs;a, &longs;e ouero la corda &longs;ia <lb/>in M ouero in G legata, per<lb/>cioche la girella del centro K <lb/>non fà nulla, & è totalmente <lb/>inutile. <emph.end type="italics"/></s> <s id="id.2.1.945.1.0"><emph type="italics"/>Ma &longs;e la po&longs;&longs;anza che &longs;o&longs;tiene il <lb/>pe&longs;o B &longs;arà in M, & la ta<lb/>glia di &longs;opra &longs;ia appiccata in <lb/>sù; &longs;arà la po&longs;&longs;anza di M e<lb/>guale al pe&longs;o B. <emph.end type="italics"/></s> <s id="id.2.1.946.1.0"><emph type="italics"/>Percioche la po&longs;&longs;anza di G, che <emph.end type="italics"/><arrow.to.target n="note249"></arrow.to.target><lb/><emph type="italics"/>&longs;o&longs;tiene il pe&longs;o B è eguale al <lb/>pe&longs;o B; & ad e&longs;&longs;a po&longs;&longs;anza <lb/>di G è eguale la po&longs;&longs;anza di <lb/>L; percioche GL è leua, il <lb/>cui &longs;o&longs;tegno è K; & la di<lb/>&longs;tanza GK è eguale alla di&longs;tan<lb/>za KL; &longs;arà dunque la po&longs;<lb/>&longs;anza di L, ouero (che è il me<lb/>de&longs;imo,) di M eguale al pe&longs;o B. <emph.end type="italics"/></s> <s id="id.2.1.947.1.0"><margin.target id="note249"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.177.1.jpg" xlink:href="037/01/177/1.jpg"> </figure> <s id="id.2.1.949.1.0"><emph type="italics"/>Que&longs;to tale mouimento &longs;i fà nel<lb/>le leue DF LG i cui &longs;o&longs;tegni <lb/>&longs;ono KA, & il pe&longs;o in D, & la po&longs;&longs;anza in F; ma nella leua LG la po&longs;&longs;an<lb/>za &longs;tà in L, & il pe&longs;o come &longs;e fu&longs;&longs;e in G. <emph.end type="italics"/></s> <s id="id.2.1.950.1.0"><emph type="italics"/>Se poi &longs;arà in M la po&longs;&longs;anza, che moue il pe&longs;o, & &longs;i tra&longs;porti, la po&longs;&longs;anza in N <lb/>& il pe&longs;o &longs;ia mo&longs;&longs;o fin ad O; &longs;arà lo &longs;patio MN della po&longs;&longs;anza eguale allo &longs;patio <lb/>di CO pe&longs;o; percioche e&longs;&longs;endo la corda MLGFDC eguale alla corda NLG <lb/>FDO, peroche è vna i&longs;te&longs;&longs;a corda; leuata via la commune MLGFDO, &longs;arà lo <lb/>&longs;patio MN della po&longs;&longs;anza eguale allo &longs;patio CO del pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.951.1.0"><emph type="italics"/>Et &longs;e la corda in M &longs;arà inuolta intorno à più girelle, &longs;empre la po&longs;&longs;anza, che in vno <lb/>delli &longs;uoi e&longs;tremi &longs;o&longs;terrà il pe&longs;o &longs;arà eguale ad e&longs;&longs;o pe&longs;o: & gli &longs;patij del pe&longs;o, & <lb/>della po&longs;&longs;anza che moue &longs;empre &longs;i mo&longs;treranno e&longs;&longs;ere eguali. <emph.end type="italics"/></s> <pb xlink:href="037/01/178.jpg"/> <s id="id.2.1.953.1.0">PROPOSITIONE XVII. </s> <s id="id.2.1.954.1.0">Se à cia&longs;cuna delle due girelle di due taglie, l'vna delle quali &longs;ia <lb/>&longs;o &longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & l'altra &longs;ia po&longs;ta di &longs;otto, & <lb/>iui attaccata, &longs;i condurrà intorno la corda; con l'vno de' &longs;uoi <lb/>capi legato alla taglia di &longs;opra, & l'altro appiccato al pe&longs;o; la <lb/>po&longs;&longs;anza &longs;arà tre volte tanto quanto il pe&longs;o. </s> <s id="id.2.1.955.1.0"><emph type="italics"/>Sia la girella co'l centro A della <lb/>taglia attaccata di &longs;otto; & &longs;ia la <lb/>corda BCDEFG inuolta intor<lb/>no non &longs;olamente à cote&longs;ta girel<lb/>la, ma etiandio alla girella della <lb/>taglia di &longs;opra, che ha il centro K; <lb/>& &longs;ia la corda legata in B della <lb/>taglia di &longs;opra; & in G &longs;ia at<lb/>taccato il pe&longs;o H; & la po&longs;&longs;an<lb/>za in L &longs;o&longs;tenga il pe&longs;o H. </s> <s id="id.2.1.955.2.0">Di<lb/>co che la po&longs;&longs;anza in L ètre vol<lb/>te tanto quanto il pe&longs;o H, per<lb/>cioche &longs;e fo&longs;&longs;ero due po&longs;&longs;anze, che <lb/>&longs;o&longs;tenne&longs;&longs;ero il pe&longs;o H vna in K, <lb/>& l'altra in B, &longs;arebbono ambe<lb/>due in&longs;ieme tre volte tanto quan<emph.end type="italics"/><lb/><arrow.to.target n="note250"></arrow.to.target><emph type="italics"/>to il pe&longs;o H: percioche la po&longs;&longs;an<lb/>za in K è due volte tanto quan<lb/>to il pe&longs;o H, & la po&longs;&longs;anza in <lb/>B è eguale ad e&longs;&longs;o pe&longs;o. </s> <s id="id.2.1.955.3.0">& per <lb/>cioche la &longs;ola po&longs;&longs;anza in L è <lb/>eguale ad ambedue le po&longs;&longs;anze in <lb/>KB, peroche la po&longs;&longs;anza in L &longs;o<lb/>&longs;tiene sì la po&longs;&longs;anza po&longs;ta in K, <lb/>come la po&longs;&longs;anza po&longs;ta in B; & <lb/>la detta po&longs;&longs;anza in L fa l'i&longs;te&longs;&longs;o, <lb/>come &longs;e fu&longs;&longs;ero due po&longs;&longs;anze, l'v<lb/>na in K & l'altra in B. </s> <s id="id.2.1.955.4.0">Sarà <lb/>dunque tre volte tanto la po&longs;&longs;an<lb/>za in L quanto il pe&longs;o H. </s> <s id="id.2.1.955.5.0">Che <lb/>bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.956.1.0"><margin.target id="note250"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo. </s> <s id="id.2.1.956.2.0">Nella prece dente. <emph.end type="italics"/></s> <figure id="id.037.01.178.1.jpg" xlink:href="037/01/178/1.jpg"> </figure> <pb pagenum="82" xlink:href="037/01/179.jpg"/> <s id="id.2.1.959.1.0">Ma &longs;e in L &longs;arà la po&longs;&longs;anza, che moue il pe&longs;o. </s> <s id="id.2.1.959.2.0">Dico lo &longs;patio <lb/>del pe&longs;o mo&longs;&longs;o e&longs;&longs;ere tre volte tanto, quanto lo &longs;patio della <lb/>po&longs;&longs;anza mo&longs;&longs;a. </s> <s id="id.2.1.960.1.0"><emph type="italics"/>Moua&longs;i il centro della girella <lb/>K fin ad M, lo &longs;patio <lb/>delquale mouimento è ve<lb/>ramente eguale allo &longs;patio <lb/>della po&longs;&longs;anza mo&longs;&longs;a, co<lb/>me è detto di &longs;opra: & <emph.end type="italics"/><arrow.to.target n="note251"></arrow.to.target><lb/><emph type="italics"/>quando K &longs;arà in M, B <lb/>&longs;arà in N, & NB &longs;a <lb/>rà eguale ad MK; & <lb/>mentre K è in M, fia il <lb/>pe&longs;o H, cioè il punto G <lb/>mo&longs;&longs;o in O; & per MK <lb/>&longs;iano condotte le linee EF <lb/>PQ egualmente di&longs;tanti <lb/>dall'orizonte; &longs;arà cia&longs;cu<lb/>na delle EP BN FQ <lb/>eguale ad e&longs;&longs;a KM. </s> <s id="id.2.1.960.2.0">Et <lb/>percioche la coda BCD <lb/>EFG eguale è alla corda <lb/>NCDPQO; e&longs;&longs;endo <lb/>vna mede&longs;ima corda; & <lb/>la corda po&longs;ta intorno al <lb/>mezo cerchio ERF e<lb/>guale è alla corda po&longs;ta in <lb/>torno al mezo cerchio <lb/>PSQ; tolte via dunque <lb/>le corde communi BC <lb/>DE, & FO, &longs;arà OG <lb/>eguale alle tre corde QF <lb/>NB PE pre&longs;e in&longs;ieme. <lb/></s> <s id="id.2.1.960.3.0">ma QF NB PE in<lb/>&longs;ieme &longs;ono tre volte tanto <lb/>quanto MK, cioè lo &longs;pa<lb/>tio della po&longs;&longs;anza mo&longs;&longs;a; <lb/>lo &longs;patio dunque GO del <lb/>pe&longs;o H mo&longs;&longs;o, è tre vol<lb/>te tanto quanto è lo &longs;pa<lb/>tio della po&longs;&longs;anza mo&longs;&longs;a. <lb/></s> <s id="id.2.1.960.4.0">che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.961.1.0"><margin.target id="note251"></margin.target><emph type="italics"/>Nella precedente. <emph.end type="italics"/></s> <figure id="id.037.01.179.1.jpg" xlink:href="037/01/179/1.jpg"> </figure> <pb xlink:href="037/01/180.jpg"/> <s id="id.2.1.964.1.0">PROPOSITIONE XVIII. </s> <s id="id.2.1.965.1.0">Se ad ambedue le girelle delle due taglie: l'vna delle quali &longs;ia &longs;o <lb/>&longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & l'altra &longs;ia po&longs;ta di &longs;otto, & <lb/>iui attaccata, &longs;arà inuolta intorno la corda; con l'vno de' ca<lb/>pi &longs;uoi in qualche luogo legato, ma non già nella taglia di &longs;o<lb/>pra, & all'altro &longs;ia appiccato il pe&longs;o; la po&longs;&longs;anza &longs;arà quattro <lb/>volte tanto quanto il pe&longs;o. </s> <s id="id.2.1.966.1.0"><emph type="italics"/>Sia la taglia di &longs;otto, che habbia due gi<lb/>relle con li centri &longs;uoi AB; & &longs;ia <lb/>la taglia di &longs;opra, che &longs;imilmente hab<lb/>bia due girelle con li centri &longs;uoi CD: <lb/>& &longs;ia la corda EFGHKLMNOP <lb/>riuolta d'intorno à tutte le girelle, che <lb/>&longs;ia legata poi in E, & &longs;ia appicca<lb/>to in P il pe&longs;o Q: & &longs;ia la po&longs;&longs;an<lb/>za in R. </s> <s id="id.2.1.966.2.0">Dico la po&longs;&longs;anza di R e&longs;<lb/>&longs;ere quattro volte tanto quanto il pe<lb/>&longs;o Q: concio&longs;ia che &longs;e &longs;i intenderan<emph.end type="italics"/><lb/><arrow.to.target n="note252"></arrow.to.target><emph type="italics"/>no due po&longs;&longs;anze, l'vna in K & l'al<lb/>tra in D, la po&longs;&longs;anza in K che &longs;o<lb/>&longs;tiene il pe&longs;o Q con la corda KLM <lb/>NOP &longs;arà eguale al pe&longs;o; & &longs;aran<lb/>no le due po&longs;&longs;anze in&longs;ieme l'vna in D <lb/>& l'altra in K &longs;ostenenti il pe&longs;o Q <lb/>tre volte tanto quanto l'i&longs;te&longs;&longs;o pe&longs;o. <lb/></s> <s id="id.2.1.966.3.0">Ma la po&longs;&longs;anza di C è due volte tan<lb/>to quanto la po&longs;&longs;anza di K, & per con <emph.end type="italics"/><lb/><arrow.to.target n="note253"></arrow.to.target><emph type="italics"/>&longs;equenza del pe&longs;o Q. </s> <s>peroche egli è <lb/>la mede&longs;ima co&longs;a, come &longs;e in K fo&longs;&longs;e <lb/>appiccato vn pe&longs;o eguale al pe&longs;o Q, <lb/>delquale è due volte tanto la po&longs;&longs;anza <lb/>di C. </s> <s id="id.2.1.966.4.0">Adunque due po&longs;&longs;anze po&longs;te <lb/>in DC &longs;ono quattro volte tanto quan<lb/>to è il pe&longs;o Q. </s> <s>& concio&longs;ia, che la <lb/>po&longs;&longs;anza di R &longs;o&longs;tenga con le girelle <lb/>il pe&longs;o Q, &longs;arà la po&longs;&longs;anza di R co<lb/>me &longs;e fo&longs;&longs;ero due po&longs;&longs;anze l'vna in D<emph.end type="italics"/><lb/><figure id="id.037.01.181.1.jpg" xlink:href="037/01/181/1.jpg"></figure><pb pagenum="83" xlink:href="037/01/181.jpg"/><emph type="italics"/>& l'altra in C: & l'vna, & l'altra in&longs;ieme &longs;o&longs;tene&longs;&longs;e il pe&longs;o Q. </s> <s>La po&longs;&longs;anza <lb/>dunque di R è quattro volte tanto quanto il pe&longs;o Q. </s> <s>che bi&longs;ognaua dimo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.968.1.0"><margin.target id="note252"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.969.1.0"><margin.target id="note253"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.970.1.0">COROLLARIO</s> <s id="id.2.1.971.1.0">Dalla qual co&longs;a è manife&longs;to, che &longs;e la corda &longs;arà legata in G, & <lb/>riuolta d'intorno alle girelle, i cui centri &longs;ono BCD; &longs;arà <lb/>la po&longs;&longs;anza di R che &longs;o&longs;tiene quat<lb/>tro volte tanto, &longs;imilmente quan<lb/>to il pe&longs;o Q. </s> <s>Percioche la girel<lb/>la il cui centro è A non fà nulla. </s> <s id="id.2.1.972.1.0">Che &longs;e la po&longs;&longs;anza mouènte il pe&longs;o &longs;a <lb/>rà in R. </s> <s id="id.2.1.972.2.0">Dico lo &longs;patio del pe&longs;o <lb/>mo&longs;&longs;o e&longs;&longs;ere quattro volte tanto <lb/>quanto lo &longs;patio della po&longs;&longs;anza. </s> <figure id="id.037.01.181.2.jpg" xlink:href="037/01/181/2.jpg"> </figure> <s id="id.2.1.974.1.0"><emph type="italics"/>Siano mo&longs;&longs;i i centri CD delle girelle fin ad ST; <lb/>&longs;aranno per le co&longs;e di &longs;opra dette CS DT <lb/>eguali allo &longs;patio della po&longs;&longs;anza; & per SDT <lb/>&longs;iano condotte le linee HK VX NO <foreign lang="greek">*u</foreign>Z <lb/>egualmente di&longs;tanti dall'orizonte; & mentre <lb/>li centri CD &longs;ono in ST, &longs;ia il pe&longs;o Q, <lb/>cioè il punto P mo&longs;&longs;o in <36>. </s> <s id="id.2.1.974.2.0">& percioche <lb/>la corda EFGHKLMNOP eguale è al <lb/>la corda EFGVXLM<foreign lang="greek">*u</foreign>Z<36>; e&longs;&longs;endo vna <lb/>mede&longs;ima corda: & le corde po&longs;te d'intorno à <lb/>mezi cerchi NIOH<foreign lang="greek">a</foreign>K &longs;iano eguali alle cor<lb/>de, lequali &longs;ono intorno à i mezi cerchi <foreign lang="greek">*ud</foreign>Z <lb/>V<foreign lang="greek">b</foreign>X; tolte via dunque le communi EFGH <lb/>KLMN & O<36>; &longs;arà P<36> eguale ad e&longs;&longs;e <lb/>N<foreign lang="greek">*u</foreign> ZO VH XK in&longs;ieme pre&longs;e, ma le quat<lb/>tro N<foreign lang="greek">*u</foreign> ZO VH XK tutte in&longs;ieme &longs;ono <lb/>quattro volte tanto quanto DT cioè lo &longs;pa<lb/>tio della po&longs;&longs;anza. </s> <s id="id.2.1.974.3.0">Lo &longs;patio dunque PQ del <lb/>pe&longs;o è quattro volte tanto quanto lo &longs;patio <lb/>della po&longs;&longs;anza. </s> <s id="id.2.1.974.4.0">che era da mo&longs;trar&longs;i. <emph.end type="italics"/></s> <pb xlink:href="037/01/182.jpg"/> <s id="id.2.1.975.1.0"><emph type="italics"/>Ma &longs;e la corda &longs;a <lb/>rà rilegata in <lb/>E della taglia <lb/>di &longs;opra, & la <lb/>po&longs;&longs;anza di R <lb/>&longs;o&longs;tenga il pe<lb/>&longs;o Q. </s> <s>&longs;arà la <lb/>po&longs;&longs;anza di R <lb/>cinque volte <lb/>tanto quanto <lb/>il pe&longs;o Q. </s> <s>& <lb/>&longs;e in R &longs;arà <lb/>la po&longs;&longs;anza, <lb/>che moue il pe<lb/>&longs;o &longs;arà lo &longs;pa<lb/>tio del pe&longs;o <lb/>mo&longs;&longs;o cinque <lb/>volte tanto, <lb/>quanto lo &longs;pa<lb/>tio della po&longs;<lb/>&longs;anza. </s> <s id="id.2.1.975.2.0">Lequa<lb/>li co&longs;e tutte &longs;i <lb/>dimo&longs;treranno <lb/>con modo &longs;imi<lb/>le, come nelle <lb/>precedenti è <lb/>&longs;tato fatto. <emph.end type="italics"/></s> <figure id="id.037.01.182.1.jpg" xlink:href="037/01/182/1.jpg"> </figure> <pb pagenum="84" xlink:href="037/01/183.jpg"/> <s id="id.2.1.978.1.0"><emph type="italics"/>Ma &longs;e la po&longs;&longs;anza di R &longs;o<lb/>&longs;tene&longs;&longs;e il pe&longs;o Q hauen<lb/>do la taglia tre girelle, i <lb/>cui centri &longs;iano ABC; & <lb/>&longs;ia vn'altra taglia di &longs;otto, <lb/>che habbia due, ò tre girel<lb/>le, i cui centri &longs;iano DEF; <lb/>& &longs;ia la corda riuolta d'in <lb/>torno à tutte le girelle, & <lb/>&longs;ia legata in G ouero in H; <lb/>&longs;imilmente mo&longs;trera&longs;&longs;i la <lb/>po&longs;&longs;anza di R e&longs;&longs;ere &longs;ei <lb/>volte tanto quanto il pe&longs;o <lb/>Q. </s> <s>& &longs;e in R &longs;arà la <lb/>forza mouente il pe&longs;o, &longs;i <lb/>mo&longs;trerà lo &longs;patio del pe&longs;o <lb/>mo&longs;&longs;o e&longs;&longs;ere &longs;ei volte tan<lb/>to quanto lo &longs;patio della <lb/>po&longs;&longs;anza. <emph.end type="italics"/></s> <figure id="id.037.01.183.1.jpg" xlink:href="037/01/183/1.jpg"> </figure> <s id="id.2.1.980.1.0"><emph type="italics"/>Et &longs;e la corda &longs;arà legata in <lb/>K della taglia di &longs;opra, & <lb/>in R &longs;ia la po&longs;&longs;anza che <lb/>&longs;o&longs;tiene il pe&longs;o; con modo <lb/>&longs;imile &longs;i prouerà la po&longs;&longs;an<lb/>za di R e&longs;&longs;ere &longs;ette volte <lb/>tanto quanto il pe&longs;o Q. <emph.end type="italics"/></s> <s id="id.2.1.981.1.0"><emph type="italics"/>Et &longs;e in R &longs;arà la po&longs;&longs;anza <lb/>che moue, &longs;i mo&longs;trerà lo &longs;pa<lb/>tio del pe&longs;o Q e&longs;&longs;ere &longs;ette <lb/>volte tanto quanto lo &longs;pa<lb/>tio della po&longs;&longs;anza. </s> <s id="id.2.1.981.2.0">& co&longs;i <lb/>in infinito ogni proportio<lb/>ne molteplice della po&longs;&longs;an<lb/>za ver&longs;o il pe&longs;o potra&longs;&longs;i <lb/>trouare. </s> <s id="id.2.1.981.3.0">& &longs;i mo&longs;trerà <lb/>&longs;empre, co&longs;i e&longs;&longs;ere il pe&longs;o <lb/>ver&longs;o la po&longs;&longs;anza che lo &longs;o<lb/>&longs;tiene, come lo &longs;patio della <lb/>po&longs;&longs;anza che moue il pe&longs;o, <lb/>allo &longs;patio del pe&longs;o mo&longs;&longs;o. <emph.end type="italics"/></s> <pb xlink:href="037/01/184.jpg"/> <s id="id.2.1.982.1.0"><emph type="italics"/>Hor il mouimento <lb/>delle leue delle gi<lb/>relle in que&longs;te &longs;i fà <lb/>in cotal modo, <lb/>cioè le leue delle <lb/>girelle della taglia <lb/>di &longs;opra &longs;i mouo<lb/>no, come è detto, <lb/>nella decima&longs;e&longs;ta <lb/>di que&longs;to; cioè han<lb/>no il &longs;o&longs;tegno nel<lb/>le &longs;tremità, la po&longs;<lb/>&longs;anza nel mezo, <lb/>& il pe&longs;o nell'al<lb/>tra &longs;tremità ap<lb/>piccato. </s> <s id="id.2.1.982.2.0">Ma le <lb/>leue della taglia di <lb/>&longs;otto hanno il &longs;o<lb/>&longs;tegno nel mezo, <lb/>& il pe&longs;o, & la <lb/>po&longs;&longs;anza nelle &longs;tro<lb/>mità. <emph.end type="italics"/></s> <figure id="id.037.01.184.1.jpg" xlink:href="037/01/184/1.jpg"> </figure> <pb pagenum="85" xlink:href="037/01/185.jpg"/> <s id="id.2.1.985.1.0">COROLLARIO</s> <s id="id.2.1.986.1.0">In que&longs;te co&longs;e è manife&longs;to, che le girelle della taglia di &longs;opra &longs;o<lb/>no cagione, che il pe&longs;o &longs;i moua da po&longs;&longs;anza maggio re di e&longs;&longs;o <lb/>pe&longs;o, & per maggiore &longs;patio di quel che è lo &longs;patio di e&longs;&longs;a po&longs;<lb/>&longs;anza, & per eguale in manco tempo: co&longs;a che veramente <lb/>non fanno le girelle della taglia di &longs;otto. </s> <s id="id.2.1.987.1.0"><emph type="italics"/>In altro modo ancora po&longs;&longs;iamo ritrouare que&longs;ta proportione moltiplice della po&longs;&longs;an<lb/>za ver&longs;o il pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.988.1.0">PROPOSITIONE XIX. </s> <s id="id.2.1.989.1.0">Se à cia&longs;cuna delle girelle dell'vna, & l'altra delle due taglio, l'v<lb/>na delle quali &longs;ia appiccata di &longs;opra, & l'altra di &longs;otto ritenu<lb/>ta dalla po&longs;&longs;anza, che &longs;o&longs;tiene, &longs;i riuolga intorno la corda; con <lb/>l'vno de' capi &longs;uoi legato in qualche loco, & con l'altro attac<lb/>cato al pe&longs;o: la po&longs;&longs;anza &longs;arà due volte tanto quanto il pe&longs;o. </s> <pb xlink:href="037/01/186.jpg"/> <s id="id.2.1.991.1.0"><emph type="italics"/>Sia la girella della taglia appiccata di &longs;opra, il cui centro &longs;ia A; & BCD &longs;ia del<lb/>la taglia di &longs;otto; &longs;ia dapoi la corda EBCDFGHL rilegata in E; & in L &longs;ia <lb/>appiccato il pe&longs;o M; & &longs;ia la po&longs;<lb/>&longs;anza che &longs;ostiene il pe&longs;o M po&longs;ta in <lb/>N. </s> <s id="id.2.1.991.2.0">Dico la po&longs;&longs;anza di N e&longs;&longs;ere <emph.end type="italics"/><lb/><arrow.to.target n="note254"></arrow.to.target><emph type="italics"/>due volte tanto quanto il pe&longs;o M. </s> <s id="id.2.1.991.3.0">Per <lb/>cioche e&longs;&longs;endo &longs;tato di &longs;opra mo&longs;trato <lb/>la po&longs;&longs;anza di L, laquale per gratia <lb/>di e&longs;&longs;empio, &longs;o&longs;tenga il pe&longs;o O ap<lb/>piccato in N, e&longs;&longs;ere la metà meno di <lb/>e&longs;&longs;o pe&longs;o; adunque la po&longs;&longs;anza di N, <lb/>che è eguale al pe&longs;o O &longs;o&longs;tenirà il pe<lb/>&longs;o M, che è eguale alla po&longs;&longs;anza di L; <lb/>& &longs;arà detta po&longs;&longs;anza due volte tan<lb/>to quanto il pe&longs;o M. </s> <s>che bi&longs;ognaua <lb/>mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.992.1.0"><margin.target id="note254"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.186.1.jpg" xlink:href="037/01/186/1.jpg"> </figure> <s id="id.2.1.994.1.0">Altramente. </s> <s id="id.2.1.995.1.0"><emph type="italics"/>Po&longs;te le co&longs;e i&longs;te&longs;&longs;e. </s> <s id="id.2.1.995.2.0">Percioche la po&longs;&longs;an<lb/>za di F, ouero di D, che è l'i&longs;te&longs;&longs;o, <emph.end type="italics"/><lb/><arrow.to.target n="note255"></arrow.to.target><emph type="italics"/>è eguale al pe&longs;o M: & BD è vna <lb/>leua, il cui &longs;o&longs;tegno è B, & la po&longs;<lb/>&longs;anza di N è come &longs;e ella fo&longs;&longs;e nel <lb/>mezo della leua, & il pe&longs;o eguale ad <lb/>e&longs;&longs;o M &longs;tà come &longs;e egli fu&longs;&longs;e in D <lb/>per cau&longs;a della corda FD, che è l'i<lb/>&longs;te&longs;&longs;o, come &longs;e BCD fo&longs;&longs;e la girella <lb/>della taglia di &longs;opra, & il pe&longs;o fo&longs;&longs;e <lb/>appiccato nella corda DF, &longs;i come <lb/>nella decimaquinta, & nella decima<lb/>&longs;e&longs;ta è detto. </s> <s id="id.2.1.995.3.0">La po&longs;&longs;anza dunque di <lb/>N è due volte tanto, quanto il pe&longs;o <lb/>M. </s> <s>che era da mo&longs;trar&longs;i. <emph.end type="italics"/></s> <s id="id.2.1.996.1.0"><margin.target id="note255"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.997.1.0"><emph type="italics"/>Ma &longs;e in N &longs;arà la po&longs;&longs;anza, che moue <lb/>il pe&longs;o M, &longs;arà lo &longs;patio del pe&longs;o M <lb/>due volte tanto quanto la po&longs;&longs;anza po&longs;ta in N, ilche è manife&longs;to dalla duodecima <lb/>di que&longs;to; percioche lo &longs;patio del punto L che inchina in giu&longs;o, è due volte tanto <lb/>quanto lo &longs;patio di N che và in &longs;u&longs;o; &longs;arà dunque per lo contrario lo &longs;patio del<lb/>la po&longs;&longs;anza di N che inchina in giù la metà meno dello &longs;patio del pe&longs;o M mo&longs;<lb/>&longs;a all'in sù. <emph.end type="italics"/></s> <s id="id.2.1.998.1.0"><emph type="italics"/>Hor &longs;i come dalla terza, dalla quinta, & dalla &longs;ettima di que&longs;to &c. </s> <s id="id.2.1.998.2.0">&longs;i po&longs;&longs;ono rac<emph.end type="italics"/><pb pagenum="86" xlink:href="037/01/187.jpg"/><emph type="italics"/>cogliere le ragioni del pe&longs;o O, &longs;iano quanto &longs;i voglia molteplici ad e&longs;&longs;a po&longs;&longs;anza <lb/>po&longs;ta in L, con l'i&longs;te&longs;&longs;o modo parimente &longs;i potranno mo&longs;trare le ragioni quanto <lb/>&longs;i voglia molteplici della po&longs;&longs;anza po&longs;ta in N, che &longs;o&longs;tiene il pe&longs;o M. </s> <s>& co&longs;i <lb/>dalla decimaterza, & dalla decimaquarta &longs;i mo&longs;treranno le ragioni quanto &longs;i voglia <lb/>molteplici allo &longs;patio del pe&longs;o M, allo &longs;patio della po&longs;&longs;anza po&longs;ta in N. <emph.end type="italics"/></s> <s id="id.2.1.999.1.0"><emph type="italics"/>Si potrà ancora dalla decima&longs;ettima, & dalla decimaottaua di que&longs;to ritrouare la <lb/>proportione molteplice, laquale ha la po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o ver&longs;o l'i&longs;te&longs;&longs;o <lb/>pe&longs;o, &longs;i come la proportione della po&longs;&longs;anza di N al pe&longs;o M &longs;i dimo&longs;traua nel<lb/>la propo&longs;itione decimaquinta, & decima&longs;e&longs;ta: & &longs;i trouerà co&longs;i e&longs;&longs;ere il pe&longs;o <lb/>alla po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o; come lo &longs;patio della po&longs;&longs;anza, che moue allo <lb/>&longs;patio del pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.1000.1.0"><emph type="italics"/>Li mouimenti delle leue in que&longs;te &longs;i fà in cotal modo, cioè le leue delle girelle della ta<lb/>glia di &longs;otto &longs;i mouono, come della leua BD, laquale &longs;i moue, come &longs;e B fo&longs;&longs;e il <lb/>&longs;o&longs;tegno, & il pe&longs;o &longs;te&longs;&longs;e in D, & la po&longs;&longs;anza nel mezo. </s> <s id="id.2.1.1000.2.0">Ma le leue delle girel<lb/>le della taglia di &longs;opra &longs;i mouono, come FH, il cui &longs;o&longs;tegno è nel mezo, il pe&longs;o in <lb/>H & la po&longs;&longs;anza in F. <emph.end type="italics"/></s> <s id="id.2.1.1001.1.0">COROLLARIO. </s> <s id="id.2.1.1002.1.0">Da que&longs;to è manife&longs;to, che le girelle della taglia di &longs;otto in que<lb/>&longs;te fanno effetto tale, che il pe&longs;o vien mo&longs;&longs;o da po&longs;&longs;anza mag<lb/>giore, di quel che &longs;ia e&longs;&longs;o pe&longs;o, & per maggiore &longs;patio dello <lb/>&longs;patio di e&longs;&longs;a po&longs;&longs;anza, & per eguale in manco tempo. </s> <s id="id.2.1.1002.2.0">Co&longs;a <lb/>che non fanno già le girelle della taglia di &longs;opra. </s> <s id="id.2.1.1003.1.0"><emph type="italics"/>Cono&longs;ciute le proportioni molteplici, hor egli è da accostar&longs;i alle &longs;opra particolari. <emph.end type="italics"/></s> <s id="id.2.1.1004.1.0">Cono&longs;ciute le proportioni molteplici, già egli è da venire alle &longs;opraparticolari. </s> <s id="id.2.1.1004.2.0">Il <lb/>genere &longs;opraparticolare è il &longs;econdo propo&longs;to di &longs;opra, quando cio è &longs;i paragona <lb/>vna quantità maggiore ver&longs;o vna minore &longs;i fattamente, che e&longs;&longs;a maggiore con<lb/>tenga la minore vna ò piu volte, & di piu parte di e&longs;&longs;a, che la pos&longs;i numerare in<lb/>teramente: come per e&longs;&longs;empio, il tre contiene il due vna volta, & più la metà di <lb/>e&longs;&longs;o due, cioè vno, ilquale puote numerare il tre. </s> <s id="id.2.1.1004.3.0">Intende dunque l'autore d'in<lb/>ue&longs;tigare la proportione &longs;opraparticolare, che hà il pe&longs;o alla po&longs;&longs;anza. </s> <pb xlink:href="037/01/188.jpg"/> <s id="id.2.1.1006.1.0">PROPOSITIONE XX. </s> <s id="id.2.1.1007.1.0">Se à cia&longs;cuna delle girelle dell'vna & l'altra delle due taglie, l'v<lb/>na delle quali &longs;ia &longs;o&longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & di &longs;otto <lb/>&longs;ia po&longs;ta, & legata al pe&longs;o, &longs;arà inuolta d'intorno la corda; <lb/>con l'vno de' &longs;uoi capi legato in qualche loco, & l'altro attac<lb/>cato alla taglia di &longs;otto; il pe&longs;o &longs;arà vna volta & meza tanto <lb/>quanto la po&longs;&longs;anza. </s> <s id="id.2.1.1008.1.0"><emph type="italics"/>Sia ABC la girella della taglia di &longs;opra, & DEF <lb/>quella della taglia di &longs;otto legata al pe&longs;o G; & <lb/>&longs;ia la corda HABCDEFK inuolta d'intorno <lb/>alle, girelle laqual corda &longs;ia legata in K, & in H <lb/>alla taglia di &longs;otto; & &longs;ia in L la po&longs;&longs;anza che <lb/>&longs;o&longs;tiene il pe&longs;o G. </s> <s id="id.2.1.1008.2.0">Dico, che il pe&longs;o è vna volta <lb/>& meza tanto quanto la po&longs;&longs;anza. </s> <s id="id.2.1.1008.3.0">Hor percio<emph.end type="italics"/><lb/><arrow.to.target n="note256"></arrow.to.target><emph type="italics"/>che l'vna, & l'altra corda CD AH &longs;o&longs;tiene la <lb/>terza parte del pe&longs;o G; &longs;arà ogn'vna delle po&longs;<lb/>&longs;anze po&longs;te in DH vn terzo del pe&longs;o G; alle <lb/>quali tutte pre&longs;e in&longs;ieme è eguale la po&longs;&longs;anza di <emph.end type="italics"/><lb/><arrow.to.target n="note257"></arrow.to.target><emph type="italics"/>L: peroche la detta po&longs;&longs;anza di L è due volte <lb/>tanto quanto è la po&longs;&longs;anza di D, & di quella <lb/>che &longs;ta in H. </s> <s id="id.2.1.1008.4.0">Per laqual co&longs;a la po&longs;&longs;anza di L <lb/>viene ad e&longs;&longs;ere &longs;otto &longs;e&longs;quialtera del pe&longs;o G. <lb/></s> <s id="id.2.1.1008.5.0">Adunque il pe&longs;o G ver&longs;o la po&longs;&longs;anza di L è co<lb/>me tre à due. </s> <s id="id.2.1.1008.6.0">cioè vna volta & meza. </s> <s id="id.2.1.1008.7.0">che bi&longs;o<lb/>gnaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.1009.1.0"><margin.target id="note256"></margin.target><emph type="italics"/>Per il corollario della<emph.end type="italics"/> 5. <emph type="italics"/>di que&longs;to. <emph.end type="italics"/></s> <s id="id.2.1.1010.1.0"><margin.target id="note257"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.188.1.jpg" xlink:href="037/01/188/1.jpg"> </figure> <s id="id.2.1.1012.1.0">“Per laqual co&longs;a la po&longs;&longs;anza di L è &longs;otto &longs;e&longs;quialtera del pe&longs;o G. </s> <s id="id.2.1.1012.2.0">Hò detto, che il <lb/>&longs;opraparticolare è il &longs;econdo genere de'moltiplici, la prima &longs;petie del quale è <lb/>tre à due, che è &longs;e&longs;quialtera, cioè vna volta & meza. </s> <s id="id.2.1.1012.3.0">Hor chi fà comparatione <lb/>al contrario di due à tre na&longs;ce la &longs;otto &longs;e&longs;quialtera, hauendo forza quella voce <lb/>&longs;otto di paragonare la minore quantita con la maggiore. </s> <s id="id.2.1.1012.4.0">La po&longs;&longs;anza dunque di <lb/>L &longs;arà in proportione co'l pe&longs;o G come dueà tre, & in que&longs;ta gui&longs;a deue&longs;i in<lb/>tendere &longs;empre tale vocabolo. </s> <pb pagenum="87" xlink:href="037/01/189.jpg"/> <s id="id.2.1.1014.1.0">Ma &longs;e la po&longs;&longs;anza che moue il pe&longs;o &longs;arà in L: Dico lo &longs;patio <lb/>della po&longs;&longs;anza e&longs;&longs;ere vna volta & meza tanto, quanto lo &longs;pa<lb/>tio del pe&longs;o. </s> <s id="id.2.1.1015.1.0"><emph type="italics"/>Stando le co&longs;e i&longs;te&longs;&longs;e, peruenga la girella <lb/>ABC fin ad MNO, & la girella <lb/>DEF fin à PQR; & H in S; <lb/>& il pe&longs;o G fin in T. </s> <s id="id.2.1.1015.2.0">Et perche la <lb/>corda HABCDEFK è eguale alla <lb/>corda SMNOPQRK e&longs;&longs;endo la <lb/>corda i&longs;te&longs;&longs;a; & le corde che &longs;ono d'in<lb/>torno à mezi cerchi ABCMNO &longs;o <lb/>no tra loro eguali, & quelle, che &longs;ono <lb/>d'intorno alli mezi cerchi DEF PQR <lb/>&longs;imilmente &longs;ono tra loro eguali; tolte <lb/>via dunque le corde AS CP RK <lb/>communi, &longs;aranno le due CO MA e<lb/>guali alle tre DP HS FR. </s> <s>ma l'v<lb/>na, & l'altra di CO AM &longs;eparata<lb/>mente è eguale allo &longs;patio della po&longs;&longs;an<lb/>za mo&longs;&longs;a. </s> <s id="id.2.1.1015.3.0">Per laqual co&longs;a le due CO <lb/>MA in&longs;ieme &longs;aranno due volte tanto <lb/>quanto lo &longs;patio della po&longs;&longs;anza; & le <lb/>tre DP HS FR in&longs;ieme con &longs;imile <lb/>modo &longs;aranno tre volte tanto quanto <lb/>lo &longs;patio del pe&longs;o mo&longs;&longs;o. </s> <s id="id.2.1.1015.4.0">Ma la metà, <lb/>cioè lo &longs;patio della po&longs;&longs;anza mo&longs;&longs;a, al<lb/>la terza parte, cioè allo &longs;patio del pe&longs;o <lb/>mo&longs;&longs;o, ha proportione tale quale è dal <lb/>doppio della metà al doppio del terzo, <lb/>cioè come il tutto à duo terzi, che è come <lb/>tre à due. </s> <s id="id.2.1.1015.5.0">Lo &longs;patio dunque della po&longs;&longs;an<lb/>za po&longs;ta in L è vna volta & meza tan<lb/>to quanto lo &longs;patio del pe&longs;o G mo&longs;&longs;o. <lb/></s> <s id="id.2.1.1015.6.0">che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <figure id="id.037.01.189.1.jpg" xlink:href="037/01/189/1.jpg"> </figure> <pb xlink:href="037/01/190.jpg"/> <s id="id.2.1.1017.1.0">PROPOSITIONE XXI. </s> <s id="id.2.1.1018.1.0">Se à tre girelle di due taglie, l vna delle quali &longs;ia &longs;o&longs;tenuta dalla <lb/>po&longs;&longs;anza di &longs;opra con vna &longs;ola girella, & l'altra con due girel<lb/>le &longs;ia po&longs;ta di &longs;otto, & legata al pe&longs;o, &longs;arà inuolta d'intorno <lb/>la corda, con l'vno de' &longs;uoi capi legato in qualche luogo, & <lb/>l'altro legato nella taglia di &longs;opra; il pe&longs;o &longs;arà vna volta, & vn <lb/>terzo tanto quanto la po&longs;&longs;anza. </s> <s id="id.2.1.1019.1.0"><emph type="italics"/>Sia il pe&longs;o A legato alla taglia di <lb/>&longs;otto, laquale habbia due girelle, i <lb/>cui centri &longs;iano BC, & la taglia <lb/>di &longs;opra habbia la girella co'l centro <lb/>D; & &longs;ia la corda EFGHKL <lb/>MN riuolta d'intorno à tutte le gi<lb/>relle, laquale &longs;ia legata in N, & <lb/>in E dalla taglia di &longs;opra; & &longs;ia <lb/>la po&longs;&longs;anza in O, che &longs;o&longs;tenga il pe<lb/>&longs;o A. </s> <s id="id.2.1.1019.2.0">Dico che il pe&longs;o è vna volta <lb/>& vn terzo tanto quanto è la po&longs;&longs;an<lb/>za. </s> <s id="id.2.1.1019.3.0">Et percioche cia&longs;cheduna delle <emph.end type="italics"/><lb/><arrow.to.target n="note258"></arrow.to.target><emph type="italics"/>corde NM HG EF KL &longs;o&longs;tie<lb/>ne la quarta parte del pe&longs;o A; & <lb/>tutte in&longs;ieme &longs;o&longs;tengono tutto il pe<lb/>&longs;o; le tre HG EF KL in&longs;ieme <lb/>&longs;o&longs;terranno le tre parti del pe&longs;o A. <lb/></s> <s id="id.2.1.1019.4.0">Per laqual co&longs;a il pe&longs;o A ver&longs;o tut<lb/>te que&longs;te in&longs;ieme &longs;arà come quattro <lb/>à tre: & concio&longs;ia che la po&longs;&longs;anza di <lb/>O faccia il mede&longs;imo, che &longs;anno le <lb/>corde HG EF KL tutte in&longs;ie<lb/>me; peroche le &longs;o&longs;tiene tutte; &longs;arà la <lb/>po&longs;&longs;anza di O eguale à le tre HG <lb/>EF KL in&longs;ieme; & perciò il pe&longs;o <lb/>A ver&longs;o la po&longs;&longs;anza di O &longs;arà co<lb/>me quattro à tre, cioè vna volta, & <lb/>vn terzo. </s> <s id="id.2.1.1019.5.0">che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.1020.1.0"><margin.target id="note258"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 1. <emph type="italics"/>corolario della<emph.end type="italics"/> V. <emph type="italics"/>di que&longs;to. <emph.end type="italics"/></s> <figure id="id.037.01.190.1.jpg" xlink:href="037/01/190/1.jpg"> </figure> <pb pagenum="88" xlink:href="037/01/191.jpg"/> <s id="id.2.1.1023.1.0">Ma &longs;e in O &longs;arà la po&longs;&longs;an<lb/>za che moua il pe&longs;o A. <lb/></s> <s id="id.2.1.1023.2.0">Dico lo &longs;patio cor&longs;o dal<lb/>la po&longs;&longs;anza di O e&longs;&longs;ere <lb/>vna volta & vn terzo tan<lb/>to quanto è lo &longs;patio del <lb/>pe&longs;o A mo&longs;&longs;o. </s> <s id="id.2.1.1024.1.0"><emph type="italics"/>Stando le co&longs;e mede&longs;ime, &longs;ia il centro <lb/>B mo&longs;&longs;o in P; & C fin in Q; <lb/>& D in R; & E in S nel<lb/>l'i&longs;te&longs;&longs;o tempo: & &longs;iano per li cen<lb/>tri condotte le linee ML<36>ZFG <lb/>TV HK X<foreign lang="greek">*u</foreign> egualmente di&longs;tan<lb/>ti, & dall' orizonte, & fra &longs;e &longs;te&longs;<lb/>&longs;e: &longs;imilmente, come nella prece<lb/>dente &longs;i dimo&longs;trerà, le tre corde <lb/>XH SE <foreign lang="greek">*u</foreign>K e&longs;&longs;ere eguali alle <lb/>quattro TG VF ZL <36>M. </s> <s>& <lb/>percioche le tre XH SE <foreign lang="greek">*u</foreign>K &longs;o<lb/>no in&longs;ieme tre volte tanto quanto <lb/>lo &longs;patio della po&longs;&longs;anza: ma le <lb/>quattro TG VF ZL <36>M in<lb/>&longs;ieme &longs;ono quattro volte <expan abbr="tãto">tanto</expan> quan<lb/>to lo &longs;patio del pe&longs;o mo&longs;&longs;o; &longs;arà lo <lb/>&longs;patio della po&longs;&longs;anza ver&longs;o lo &longs;pa<lb/>tio del pe&longs;o, come la terza parte <lb/>alla quarta parte. </s> <s id="id.2.1.1024.2.0">Ma la terza <lb/>parte ver&longs;o la quarta parte è come <lb/>tre terzi à tre quarti, cioè come il <lb/>tutto ver&longs;o tre quarti, che è come <lb/>quattro ver&longs;o tre. </s> <s id="id.2.1.1024.3.0">Lo &longs;patio dun<lb/>que della po&longs;&longs;anza allo &longs;patio del <lb/>pe&longs;o mo&longs;&longs;o hà proportione di vna <lb/>volta & vn terzo. </s> <s id="id.2.1.1024.4.0">che era damo<lb/>&longs;trar&longs;i. <emph.end type="italics"/></s> <figure id="id.037.01.191.1.jpg" xlink:href="037/01/191/1.jpg"> </figure> <s id="id.2.1.1026.1.0"><emph type="italics"/>Ma &longs;e la corda in E &longs;arà inuolta d'in <lb/>torno vn'altra girella, laqual cor<emph.end type="italics"/><pb xlink:href="037/01/192.jpg"/><emph type="italics"/>da poi &longs;ia legata alla taglia di &longs;ot<lb/>to; &longs;imilmente si mo&longs;trerà la pro<lb/>portione del pe&longs;o alla po&longs;&longs;anza di <lb/>O, che lo &longs;o&longs;tiene e&longs;&longs;ere vna vol<lb/>ta & vn quarto; che &longs;e la po&longs;&longs;an<lb/>za e&longs;&longs;ere vna volta, & vn quar<lb/>to ver&longs;o lo &longs;patio del pe&longs;o. </s> <s>& <lb/>co&longs;i in infinito procedendo ritro<lb/>ueremo qual &longs;i voglia proportione <lb/>&longs;opraparticolare del pe&longs;o ver&longs;o la po&longs;&longs;an<lb/>za, che &longs;o&longs;tiene il pe&longs;o, come lo <lb/>&longs;patio della po&longs;&longs;anza mouemte al<lb/>lo &longs;patio del pe&longs;o mo&longs;&longs;o. </s> <figure id="id.037.01.192.1.jpg" xlink:href="037/01/192/1.jpg"> </figure> <s id="id.2.1.1027.1.0">Il mouimento po&longs;cia delle leue &longs;i fà <lb/>in que&longs;to modo, cioè della leua <lb/>ML è il &longs;o&longs;tegno M, e&longs;&longs;endo <lb/>la corda legata in N, & la po&longs;&longs;anza in L. <lb/></s> <s>ma percioche il punto L và in <lb/>sù, il quale è mo&longs;&longs;o dalla corda KL, <lb/>però K &longs;i mouerà in sù, & <lb/>la po&longs;&longs;anza nel mezo;</s> <s id="id.2.1.1027.2.0">Ma la le<lb/>ua FG haurà per &longs;o&longs;tegno G,<lb/> il pe&longs;o nel mezo, & la po&longs;&longs;an<lb/>za in F; peroche il punto F &longs;i <lb/>moue in sù dalla corda EF. </s> <s id="id.2.1.1027.3.0">Ol<lb/>tre à ciò il G china in giù nella <lb/>girella; peroche la H anchora <lb/>nella &longs;ua girella &longs;i moue all'ingiù. </s> <pb pagenum="89" xlink:href="037/01/193.jpg"/> <s id="id.2.1.1028.1.0">PROPOSITIONE XXII. </s> <s id="id.2.1.1029.1.0">Se all'vna & l'altra di cia&longs;cuna girella delle due taglie, l'vna del<lb/>le quali &longs;ia &longs;o&longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & l'altra po&longs;ta di <lb/>&longs;otto, & legata al pe&longs;o, &longs;arà condotta d'intorno la corda; con <lb/>l'vno de &longs;uoi capi legato in qualche luogo, & l'altro attaccato <lb/>alla taglia di &longs;opra. </s> <s id="id.2.1.1029.2.0">&longs;arà la po&longs;&longs;anza vna volta & meza tanto <lb/>quanto il pe&longs;o. </s> <s id="id.2.1.1030.1.0"><emph type="italics"/>Sia la girella ABC della taglia legata <lb/>al pe&longs;o D; & EFG la girella del<lb/>la taglia di &longs;opra, il cui centro &longs;ia H; <lb/>&longs;ia dapoi la corda KABCEFGL ri<lb/>uolta d'intorno alle girelle, & legata <lb/>in L & in K alla taglia di &longs;opra; & <lb/>&longs;ia in M la po&longs;&longs;anza, che &longs;o&longs;tiene il <emph.end type="italics"/><arrow.to.target n="note259"></arrow.to.target><lb/><emph type="italics"/>pe&longs;o D. </s> <s id="id.2.1.1030.2.0">Dico che la po&longs;&longs;anza è vna <lb/>volta & meza quanto è il pe&longs;o. </s> <s id="id.2.1.1030.3.0">Hor <emph.end type="italics"/><arrow.to.target n="note260"></arrow.to.target><lb/><emph type="italics"/>percioche la po&longs;&longs;anza di E &longs;o&longs;tenente <lb/>il pe&longs;o D è la metà meno del pe&longs;o D; <emph.end type="italics"/><arrow.to.target n="note261"></arrow.to.target><lb/><emph type="italics"/>& la po&longs;&longs;anza di H è due volte quan<lb/>to la po&longs;&longs;anza po&longs;ta in E; &longs;arà la po&longs;<lb/>&longs;anza di H eguale al pe&longs;o D; & con<lb/>cio&longs;ia, che la po&longs;&longs;anza di K &longs;ia la me<lb/>tà meno del pe&longs;o D; &longs;aranno ambe<lb/>due le po&longs;&longs;anze in&longs;ieme po&longs;te in HK <lb/>vna volta & meza quanto il pe&longs;o D. <lb/></s> <s id="id.2.1.1030.4.0">e&longs;&longs;endo adunque la po&longs;&longs;anza di M egua<lb/>le à due po&longs;&longs;anze in HK pre&longs;e in&longs;ie<lb/>me, &longs;i come di &longs;opra è &longs;tato dichiarato; <lb/>&longs;arà la po&longs;&longs;anza di M vna volta & <lb/>meza quanto il pe&longs;o D. </s> <s>che bi&longs;ogna<lb/>ua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.1031.1.0"><margin.target id="note259"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1032.1.0"><margin.target id="note260"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1033.1.0"><margin.target id="note261"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 2. <emph type="italics"/>corollario del la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.193.1.jpg" xlink:href="037/01/193/1.jpg"> </figure> <s id="id.2.1.1035.1.0"><emph type="italics"/>Ma &longs;e la po&longs;&longs;anza che moue il pe&longs;o &longs;arà in <lb/>M, &longs;i mo&longs;trerà &longs;imilmente, come nelle <lb/>precedenti, lo &longs;patio del pe&longs;o e&longs;&longs;ere vna <lb/>volta & meza tanto quanto lo &longs;patio <lb/>della po&longs;&longs;anza. <emph.end type="italics"/></s> <pb xlink:href="037/01/194.jpg"/> <s id="id.2.1.1037.1.0"><emph type="italics"/>Et &longs;e la corda in K &longs;arà inuolta d'interno ad vn'altra girella, il cui centro &longs;ia N; <lb/>laquale dapoi &longs;ta rilegata alla taglia di &longs;otto in O; & la po&longs;&longs;anza di M &longs;o&longs;ten<lb/>ga il pe&longs;o D. </s> <s id="id.2.1.1037.2.0">Dico la proportione <lb/>della po&longs;&longs;anza al pe&longs;o e&longs;&longs;ere vna <lb/>volta, & vn terzo. <emph.end type="italics"/></s> <s id="id.2.1.1038.1.0"><emph type="italics"/>Hor percioche la po&longs;&longs;anza di E che <emph.end type="italics"/><lb/><arrow.to.target n="note262"></arrow.to.target><emph type="italics"/>&longs;o&longs;tiene il pe&longs;o D con la corda EC <lb/>BAKPO è vn terzo di e&longs;&longs;o D, <emph.end type="italics"/><lb/><arrow.to.target n="note263"></arrow.to.target><emph type="italics"/>& la po&longs;&longs;anza di H è due volte <lb/>tanto quanto e&longs;&longs;o E; &longs;arà la po&longs;<lb/>&longs;anza di H &longs;otto &longs;e&longs;quialtera al pe<lb/>&longs;o D. </s> <s>& nel modo iste&longs;&longs;o, per<lb/>cioche la po&longs;&longs;anza di O, laquale <lb/>è come &longs;e fo&longs;&longs;e nel centro della gi<lb/>rella ABC è vn terzo del pe&longs;o <emph.end type="italics"/><lb/><arrow.to.target n="note264"></arrow.to.target><emph type="italics"/>D, & la po&longs;&longs;anza di N è due <lb/>volte tanto quanto è e&longs;&longs;o O. </s> <s>&longs;arà <lb/>parimente la po&longs;&longs;anza di N &longs;otto <lb/>&longs;e&longs;quialtera al pe&longs;o D. </s> <s id="id.2.1.1038.2.0">Per laqual <lb/>co&longs;a due po&longs;&longs;anze in&longs;ieme po&longs;te in <lb/>HN &longs;uperano il pe&longs;o D d'vna <lb/>terza parte, & &longs;ono ver&longs;o il detto <lb/>D in ragione di vna volta & vn <lb/>terzo. </s> <s id="id.2.1.1038.3.0">& concio&longs;ia, che la po&longs;&longs;an<lb/>za di M &longs;ia eguale alle due po&longs;&longs;an<lb/>ze di HN pre&longs;e in&longs;ieme, &longs;upere<lb/>ra mede&longs;imamente la detta po&longs;&longs;an<lb/>za di M il pe&longs;o D di vn terzo. <lb/></s> <s id="id.2.1.1038.4.0">Adunque la proportione della po&longs;<lb/>&longs;anza po&longs;ta in M ver&longs;o il pe&longs;o D <lb/>è vna volta, & vn terzo. </s> <s id="id.2.1.1038.5.0">che bi<lb/>&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.1039.1.0"><margin.target id="note262"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1040.1.0"><margin.target id="note263"></margin.target><emph type="italics"/>Dalla<emph.end type="italics"/> 15. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1041.1.0"><margin.target id="note264"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>&<emph.end type="italics"/> 15. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.194.1.jpg" xlink:href="037/01/194/1.jpg"> </figure> <s id="id.2.1.1043.1.0"><emph type="italics"/>Che &longs;e la po&longs;&longs;anza mouente il pe&longs;o &longs;a<lb/>rà in M, con modo &longs;imile proue<lb/>ra&longs;&longs;i lo &longs;patio del pe&longs;o D e&longs;&longs;ere vna <lb/>volta & vn terzo tanto quanto la <lb/>po&longs;&longs;anza di M. <emph.end type="italics"/></s> <s id="id.2.1.1044.1.0"><emph type="italics"/>Et &longs;e la corda in O &longs;arà inuolta d'in<lb/>torno ad vn'altra girella, laquale dapoi &longs;ia legata alla taglia di &longs;opra; nell'i&longs;te&longs;&longs;o <lb/>modo dimo&longs;treremo la proportione della po&longs;&longs;anza M, che &longs;o&longs;tiene il pe&longs;o e&longs;&longs;ere <lb/>vna volta & vn quarto tanto quanto il pe&longs;o. </s> <s id="id.2.1.1044.2.0">& &longs;e in M &longs;arà la po&longs;&longs;anza che <lb/>moue, &longs;imilmente mo&longs;treraßilo &longs;patio del pe&longs;o e&longs;&longs;ere vna volta & vn quarto tan<emph.end type="italics"/><pb pagenum="90" xlink:href="037/01/195.jpg"/><emph type="italics"/>to quanto lo &longs;patio della po&longs;&longs;anza. </s> <s id="id.2.1.1044.3.0">& co&longs;i procedendo in infinito ritrouereme <lb/>qual &longs;i voglia proportione &longs;opraparticolare della po&longs;&longs;anza al pe&longs;o, & &longs;empre <lb/>mostreremo la po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o co&longs;i e&longs;&longs;ere ver&longs;o il pe&longs;o, come lo &longs;pa<lb/>tio del pe&longs;o allo &longs;patio della po&longs;&longs;anza, che moue il pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.1045.1.0"><emph type="italics"/>Ma il mouimento della leua EG è come &longs;e G fo&longs;&longs;e il &longs;o&longs;tegno, e&longs;&longs;endo la corda legata <lb/>in L, & il pe&longs;o, come &longs;e fo&longs;&longs;e appiccato in E, & la po&longs;&longs;anza nel mezo. </s> <s id="id.2.1.1045.2.0">Ma <lb/>della leua CA il &longs;o&longs;tegno è A, il pe&longs;o nel mezo, & la po&longs;&longs;anza in C. </s> <s>& il <lb/>K è il &longs;o&longs;tegno della leua PK, il pe&longs;o in P, & la po&longs;&longs;anza nel mezo. </s> <s id="id.2.1.1045.3.0">Le qua<lb/>li co&longs;e tutte &longs;i dimo&longs;treranno, come nelle precedenti. <emph.end type="italics"/></s> <s id="id.2.1.1046.1.0">PROPOSITIONE XXIII. </s> <s id="id.2.1.1047.1.0">Se all'vna, & l'altra delle due girelle di due taglie, l'vna dellequa<lb/>li &longs;ia &longs;o&longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & l'altra po&longs;ta à ba&longs;&longs;o, <lb/>& legata al pe&longs;o, &longs;ia menata intorno la corda, legando am<lb/>bidue li &longs;uoi capi in qualche luogo, non già nelle taglie; la <lb/>po&longs;&longs;anza &longs;arà eguale al pe&longs;o. </s> <pb xlink:href="037/01/196.jpg"/> <s id="id.2.1.1049.1.0"><emph type="italics"/>Sia la girella della taglia di &longs;o <lb/>pra ABC, il cui centro D; <lb/>& la girella della taglia le<lb/>gata al pe&longs;o H &longs;ia EFG; il <lb/>cui centro K; & &longs;ia la cor<lb/>da LEFGABCM ri<lb/>uolta d'intorno alle girelle <lb/>& legata in LM; & &longs;ia <lb/>in N la po&longs;&longs;anza che &longs;o<lb/>&longs;tiene il pe&longs;o H. </s> <s id="id.2.1.1049.2.0">Dico che <lb/>la po&longs;&longs;anza di N è egua<lb/>le al pe&longs;o H. </s> <s id="id.2.1.1049.3.0">Prenda&longs;i <lb/>il punto O douunque &longs;i <lb/>&longs;ia nella corda AG. </s> <s id="id.2.1.1049.4.0">Hor <lb/>percioche &longs;e la po&longs;&longs;anza, <lb/>che &longs;o&longs;tiene il pe&longs;o H fo&longs;<emph.end type="italics"/><lb/><arrow.to.target n="note265"></arrow.to.target><emph type="italics"/>&longs;e in O, &longs;arebbe la metà <lb/>meno del pe&longs;o H, & la <emph.end type="italics"/><lb/><arrow.to.target n="note266"></arrow.to.target><emph type="italics"/>po&longs;&longs;anza po&longs;ta in D è due <lb/>volte quanto è quella di O, <lb/>ouero (che è l'i&longs;te&longs;&longs;o) di N; <lb/>&longs;arà la po&longs;&longs;anza di N e<lb/>guale al pe&longs;o H. </s> <s>che bi<lb/>&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.1050.1.0"><margin.target id="note265"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1051.1.0"><margin.target id="note266"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.196.1.jpg" xlink:href="037/01/196/1.jpg"> </figure> <s id="id.2.1.1053.1.0">Et &longs;e in N &longs;arà la po&longs;<lb/>&longs;anza, che moue il <lb/>pe&longs;o. </s> <s id="id.2.1.1053.2.0">Dico, che lo <lb/>&longs;patio della po&longs;&longs;an<lb/>za po&longs;ta in N è e<lb/>guale allo &longs;patio del <lb/>pe&longs;o H mo&longs;&longs;o. </s> <s id="id.2.1.1054.1.0"><arrow.to.target n="note267"></arrow.to.target><emph type="italics"/>Percio che lo &longs;patio del punto O mo&longs;&longs;o è due volte tanto quanto è lo &longs;patio sì del pe<emph.end type="italics"/><lb/><arrow.to.target n="note268"></arrow.to.target><emph type="italics"/>&longs;o H mo&longs;&longs;o, come della po&longs;&longs;anza N mo&longs;&longs;a; &longs;arà lo &longs;patio della po&longs;&longs;anza N <lb/>allo &longs;patio del pe&longs;o H eguale. <emph.end type="italics"/></s> <s id="id.2.1.1056.1.0"><margin.target id="note267"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1057.1.0"><margin.target id="note268"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <pb pagenum="91" xlink:href="037/01/197.jpg"/> <s id="id.2.1.1058.1.0">Altramente. </s> <s id="id.2.1.1059.1.0"><emph type="italics"/>Stando le co&longs;e i&longs;te&longs;&longs;e. </s> <s id="id.2.1.1059.2.0">&longs;ia tra<lb/>portato il centro della gi<lb/>rella ABC fin à P; & <lb/>la girella habbia il &longs;ito in <lb/>QRS. </s> <s id="id.2.1.1059.3.0">Dapoi nell'i&longs;te&longs;&longs;o <lb/>tempo la girella EFG &longs;ia <lb/>in TVX, il cui centro <lb/>&longs;ia <foreign lang="greek">*u</foreign>, & il pe&longs;o &longs;ia per<lb/>uenuto in Z. </s> <s>&longs;iano tira<lb/>te per i centri delle girel<lb/>le le linee GETX AC <lb/>QS egualmente di&longs;tanti <lb/>dall' orizonte. </s> <s id="id.2.1.1059.4.0">& &longs;i come <lb/>nelle altre fu dimo&longs;trato, <lb/>le due corde AQ CS &longs;a <lb/>ranno eguali alle due cor<lb/>de XG TE; ma AQ <lb/>CS in&longs;ieme &longs;ono due vol<lb/>te tanto quanto lo &longs;patio <lb/>della po&longs;&longs;anza mo&longs;&longs;a; & <lb/>le due XG TE in&longs;ieme <lb/>&longs;imilmente &longs;ono due vol<lb/>te tanto quanto lo &longs;patio <lb/>del pe&longs;o; &longs;arà dunque lo <lb/>&longs;patio della po&longs;&longs;anza egua<lb/>le allo &longs;patio del pe&longs;o. </s> <s id="id.2.1.1059.5.0">che <lb/>bi&longs;ognaua mostrare. <emph.end type="italics"/></s> <figure id="id.037.01.197.1.jpg" xlink:href="037/01/197/1.jpg"> </figure> <pb xlink:href="037/01/198.jpg"/> <s id="id.2.1.1061.1.0"><emph type="italics"/>Che &longs;e l'vna, & l'altra taglia haurà etiandio due girelle, i cui centri &longs;iano ABCD, <lb/>& la corda &longs;ia inuolta d'intorno à tutte, la quale &longs;ia rilegata in LM; &longs;imilmen<lb/>te &longs;i mo&longs;trerà, che la po&longs;&longs;an<lb/>za di N è eguale al pe&longs;o H. <lb/></s> <s id="id.2.1.1061.2.0">Peroche cia&longs;cuna po&longs;&longs;anza <lb/>po&longs;ta in EF &longs;o&longs;tenente il <lb/>pe&longs;o è vn quarto del pe&longs;o; & <lb/>le po&longs;&longs;anze di CD &longs;ono due <lb/>volte tanto quanto quelle, <lb/>che &longs;ono in EF; &longs;arà cia<lb/>&longs;cuna po&longs;&longs;anza di CD la <lb/>metà del pe&longs;o H. </s> <s id="id.2.1.1061.3.0">Per la<lb/>qual co&longs;a le po&longs;&longs;anze di CD <lb/>pre&longs;e in&longs;ieme &longs;aranno eguali <lb/>al pe&longs;o H. </s> <s id="id.2.1.1061.4.0">Et percioche la <lb/>po&longs;&longs;anza di N è eguale à <lb/>due po&longs;&longs;anze po&longs;te in CD; <lb/>&longs;arà la po&longs;&longs;anza di N egua<lb/>le al pe&longs;o H. <emph.end type="italics"/></s> <figure id="id.037.01.198.1.jpg" xlink:href="037/01/198/1.jpg"> </figure> <s id="id.2.1.1063.1.0"><emph type="italics"/>Et &longs;e la po&longs;&longs;anza che moue &longs;a<lb/>rà in N, con modo &longs;imile <lb/>&longs;i mo&longs;trerà lo &longs;patio della po&longs;<lb/>&longs;anza e&longs;&longs;ere eguale allo &longs;pa<lb/>tio del pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.1064.1.0"><emph type="italics"/>Ma &longs;e l'vna & l'altra taglia ha<lb/>uerà tre, ò quattro, oue<lb/>ro quante &longs;i voglia girelle, <lb/>&longs;empre &longs;i dimostrerà la po&longs;<lb/>&longs;anza di N e&longs;&longs;ere eguale al <lb/>pe&longs;o H; & lo &longs;patio della <lb/>po&longs;&longs;anza mouente il pe&longs;o e&longs;<lb/>&longs;ere eguale allo &longs;patio del pe<lb/>&longs;o mo&longs;&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.1065.1.0"><emph type="italics"/>Ma i mouimenti delle leue in <lb/>que&longs;ta maniera &longs;ono di&longs;po&longs;ti, <lb/>che il &longs;o&longs;tegno delle girelle <lb/>della taglia di &longs;opra, come <lb/>AC della figura preceden<lb/>te è in C, il pe&longs;o appiccato in A, & la po&longs;&longs;anza nel mezo in D. </s> <s>ma le leue <lb/>delle girelle della taglia di &longs;otto co&longs;i &longs;i mouono, che di e&longs;&longs;o GE il &longs;o&longs;tegno &longs;ia E, <lb/>il pe&longs;o appiccato nel mezo, & la po&longs;&longs;anza in G. <emph.end type="italics"/></s> <pb pagenum="92" xlink:href="037/01/199.jpg"/> <s id="id.2.1.1067.1.0">PROPOSITIONE XXIIII. </s> <s id="id.2.1.1068.1.0">Se à tre girelle di due taglie, l'vna delle quali, che habbia vna gi<lb/>rella &longs;olamente &longs;ia &longs;o&longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & l'altra <lb/>po&longs;ta di &longs;otto con due girelle, & legata al pe&longs;o, &longs;arà girata in<lb/>torno la corda: e&longs;&longs;endo li due &longs;uoi capi <expan abbr="legat">legati</expan> in qualche luo<lb/>go, ma non già nella taglia di &longs;opra: il pe&longs;o &longs;arà il doppio del<lb/>la po&longs;&longs;anza. </s> <s id="id.2.1.1069.1.0"><emph type="italics"/>Siano AB i centri delle girelle della <lb/>taglia legata al pe&longs;o C: & il D <lb/>&longs;ia il centro della girella di &longs;opra; <lb/>&longs;ia dapoi la corda riuolta d'intorno <lb/>à tutte le girelle, & rilegata in EF; <lb/>& &longs;ia in G la po&longs;&longs;anza, che &longs;o<lb/>&longs;tiene il pe&longs;o C. </s> <s id="id.2.1.1069.2.0">Dico, che il pe&longs;o <lb/>C è due volte tanto quanto la <lb/>po&longs;&longs;anza G. </s> <s id="id.2.1.1069.3.0">Hor percioche &longs;e in <lb/>HK fo&longs;&longs;ero due po&longs;&longs;anze, che &longs;o<lb/>&longs;tene&longs;&longs;ero il pe&longs;o con due corde ri<lb/>uolte d'intorno alle girelle &longs;olamen<lb/>te della taglia di &longs;otto, &longs;arebbe per <lb/>certo l'vna & l'altra po&longs;&longs;anza po<emph.end type="italics"/><arrow.to.target n="note269"></arrow.to.target><lb/><emph type="italics"/>&longs;ta in KH vn quarto del pe&longs;o C; <lb/>Mala po&longs;&longs;anza di G è eguale alle <lb/>po&longs;&longs;anze di HK pre&longs;e in&longs;ieme: <emph.end type="italics"/><arrow.to.target n="note270"></arrow.to.target><lb/><emph type="italics"/>percioche è due volte tanto quan<lb/>to cia&longs;cuna delle po&longs;&longs;anze di H, <lb/>& K; &longs;arà la po&longs;&longs;anza di G la <lb/>metà del pe&longs;o C. </s> <s>il pe&longs;o dunque <lb/>&longs;arà il doppio della po&longs;&longs;anza. </s> <s id="id.2.1.1069.4.0">che <lb/>bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.1070.1.0"><margin.target id="note269"></margin.target><emph type="italics"/>Dalla<emph.end type="italics"/> 7. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1071.1.0"><margin.target id="note270"></margin.target><emph type="italics"/>Dalla<emph.end type="italics"/> 15. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.199.1.jpg" xlink:href="037/01/199/1.jpg"> </figure> <pb xlink:href="037/01/200.jpg"/> <s id="id.2.1.1073.1.0">Et &longs;e in G &longs;arà la po&longs;&longs;anza <lb/>mouente il pe&longs;o. </s> <s id="id.2.1.1073.2.0">Dico <lb/>che lo &longs;patio della po&longs;&longs;an<lb/>za è il doppio dello &longs;pa<lb/>tio del pe&longs;o. </s> <figure id="id.037.01.200.1.jpg" xlink:href="037/01/200/1.jpg"> </figure> <s id="id.2.1.1075.1.0"><emph type="italics"/>Stando le co&longs;e i&longs;te&longs;&longs;e. </s> <s id="id.2.1.1075.2.0">&longs;iano mo&longs;&longs;e le gi<lb/>relle; &longs;i dimo&longs;trerà &longs;imilmente am<lb/>bedue quelle corde LM NO e&longs;<lb/>&longs;ere eguali alle quattro PQ RS <lb/>TV X<foreign lang="greek">*u</foreign>. </s> <s id="id.2.1.1075.3.0">Ma LM NO in<lb/>&longs;ieme &longs;ono il doppio dello &longs;patio <lb/>della po&longs;&longs;anza di G mo&longs;&longs;a; & <lb/>le quattro PQ RS TV X<foreign lang="greek">*u</foreign> <lb/>in&longs;ieme &longs;ono quattro volte tanto <lb/>quanto lo &longs;patio del pe&longs;o mo&longs;&longs;o. <lb/></s> <s id="id.2.1.1075.4.0">Lo &longs;patio dunque della po&longs;&longs;anza <lb/>ver&longs;o lo &longs;patio del pe&longs;o è come la <lb/>metà ad vn quarto. </s> <s id="id.2.1.1075.5.0">Sarà dunque <lb/>lo &longs;patio della po&longs;&longs;anza allo &longs;patio <lb/>del pe&longs;o il doppiò. <emph.end type="italics"/></s> <pb pagenum="93" xlink:href="037/01/201.jpg"/> <s id="id.2.1.1077.1.0"><emph type="italics"/>Di qui egli è da con&longs;iderare <lb/>in che modo &longs;i faccia il mo<lb/>uimento; percioche e&longs;&longs;en<lb/>do legata la corda in F, <lb/>la leua NO nella prima <lb/>figura haurà il &longs;o&longs;tegno in <lb/>O, il pe&longs;o nel mezo, & la <lb/>po&longs;&longs;anza in N. </s> <s><lb/></s> <s id="id.2.1.1077.2.0">&longs;imilmente percioche la corda <lb/>è rilegata in E, la leua <lb/>PQ haurà il &longs;o&longs;tegno in <lb/>P, & il pe&longs;o ne, nezo, <lb/>& la po&longs;&longs;anza in Q. </s> <s>On<lb/>de le parti delle girelle di <lb/>N & Q &longs;i moueranno <lb/>in sù; adunque le girelle <lb/>&longs;i moueranno non ad vna <lb/>parte, ma in contrarie par <lb/>ti, cioè vna alla de&longs;tra, & <lb/>l'altra alla &longs;ini&longs;tra. </s> <s id="id.2.1.1077.3.0">& <lb/>percioche le po&longs;&longs;anze di <lb/>NQ &longs;ono le i&longs;te&longs;&longs;e, che <lb/>&longs;ono in LM; le po&longs;&longs;an<lb/>ze dunque di LM e&longs;&longs;en<lb/>do eguali &longs;i moueranno <lb/>in sù. </s> <s id="id.2.1.1077.4.0">La leua dunque <lb/>LM non &longs;i mouerà in <lb/>niuna delle parti. </s> <s id="id.2.1.1077.5.0">Per la <lb/>qual co&longs;a no anche la gi<lb/>rella &longs;i girerà intorno. <lb/></s> <s id="id.2.1.1077.6.0">Co&longs;i LM &longs;arà come bi<lb/>lancia, il <expan abbr="cni">cui</expan> centro D, <lb/>& li pe&longs;i appiccati in LM <lb/>&longs;aranno eguali alla quar<emph.end type="italics"/><lb/><figure id="id.037.01.201.1.jpg" xlink:href="037/01/201/1.jpg"></figure><lb/><emph type="italics"/>ta parte del pe&longs;o C; peroche cia&longs;cheduna corda in LN MQ &longs;o&longs;tiene la quar<lb/>ta parte del pe&longs;o C; &longs;i mouerà dunque tutta la girella, il cui centro è D in sù, <lb/>ma non già voltera&longs;&longs;i intorno. <emph.end type="italics"/></s> <pb xlink:href="037/01/202.jpg"/> <s id="id.2.1.1080.1.0"><emph type="italics"/>Et &longs;e la corda po&longs;ta in F &longs;iriuolgerà <lb/>d'intorno à due altre girelle, i cui <lb/>centri &longs;iano HK laquale dapoi &longs;ia <lb/>rilegata in L; &longs;arà la proportione <lb/>del pe&longs;o alla po&longs;&longs;anza vna volta & <lb/>meza. <emph.end type="italics"/></s> <s id="id.2.1.1081.1.0"><arrow.to.target n="note271"></arrow.to.target><emph type="italics"/>Percioche &longs;e fo&longs;&longs;ero quattro po&longs;&longs;anze in <lb/>MNOI, cia&longs;cheduna di loro &longs;areb<lb/>be vn &longs;e&longs;to del pe&longs;o C. </s> <s id="id.2.1.1081.2.0">Per laqual <lb/>co&longs;a quattro po&longs;&longs;anze in&longs;ieme in MN <lb/>OI &longs;aranno quattro &longs;e&longs;ti del pe&longs;o C. </s> <s><lb/>& percioche due po&longs;&longs;anze in&longs;ieme po<lb/>&longs;te in HD &longs;ono eguali à quattro po&longs;<lb/>&longs;anze po&longs;te in MNOI; & la po&longs;<lb/>&longs;anza di G è eguale alle po&longs;&longs;anze di <lb/>DH; &longs;arà la po&longs;&longs;anza di G egua<lb/>le à quattro po&longs;&longs;anze in&longs;ieme po&longs;te <lb/>in MNOI; & perciò &longs;arà quat<lb/>tro &longs;e&longs;ti del pe&longs;o C. </s> <s id="id.2.1.1081.3.0">La proportio<lb/>ne dunque del pe&longs;o C alla po&longs;&longs;anza <lb/>di G è vna volta & meza. <emph.end type="italics"/></s> <s id="id.2.1.1082.1.0"><margin.target id="note271"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.202.1.jpg" xlink:href="037/01/202/1.jpg"> </figure> <s id="id.2.1.1084.1.0"><emph type="italics"/>Et &longs;e in G &longs;arà la po&longs;&longs;anza, che moue, <lb/>con modo &longs;imile &longs;i mo&longs;trerà lo &longs;patio <lb/>della po&longs;&longs;anza e&longs;&longs;ere vna volta & <lb/>meza tanto quanto lo &longs;patio del pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.1085.1.0"><emph type="italics"/>Et &longs;e la corda di L &longs;arà dauantaggio <lb/>riuolta d'intorno due altre girelle, &longs;i<lb/>milmente &longs;i dimo&longs;trerà la proportio<lb/>ne del pe&longs;o alla po&longs;&longs;anza e&longs;&longs;ere vna <lb/>volta, & vn terzo. </s> <s id="id.2.1.1085.2.0">Che &longs;e in G <lb/>&longs;arà la po&longs;&longs;anza che moue, &longs;i mo&longs;tre<lb/>rà lo &longs;patio della po&longs;&longs;anza e&longs;&longs;ere vna <lb/>volta, & vn terzo quanto lo &longs;patio <lb/>del pe&longs;o, & co&longs;i di mano in mano <lb/>procedendo in infinito ritroueremo <lb/>qual &longs;i voglia proportione &longs;oprapar<lb/>ticolare del pe&longs;o alla po&longs;&longs;anza. </s> <s id="id.2.1.1085.3.0">& <lb/>&longs;empre ritroueremo co&longs;i e&longs;&longs;ere il pe&longs;o <lb/>ver&longs;o la po&longs;&longs;anza che lo &longs;o&longs;tiene, co<lb/>me lo &longs;patio della po&longs;&longs;anza che moue <lb/>allo &longs;patio del pe&longs;o mo&longs;&longs;o dalla po&longs;<lb/>&longs;anza. <emph.end type="italics"/></s> <pb pagenum="94" xlink:href="037/01/203.jpg"/> <s id="id.2.1.1087.1.0"><emph type="italics"/>Il mouimento delle leue &longs;i fa in que&longs;to modo, la leua <foreign lang="greek">*u</foreign> Z, e&longs;&longs;endo la corda legata in <lb/>E ha il &longs;o&longs;tegno in <foreign lang="greek">*u</foreign>, il pe&longs;o attaccato in B nel mezo, & la po&longs;&longs;anza in Z. </s> <s><lb/>& la leua PQ ha il &longs;ostegno in P, la po&longs;&longs;anza nel mezo, & il pe&longs;o in Q. <lb/></s> <s>Percioche bi&longs;ogna, che le girelle, i cui centri &longs;ono BD, &longs;i mouano nella parte <lb/>i&longs;te&longs;&longs;a, cioè che QZ &longs;i mouano all'insu. </s> <s id="id.2.1.1087.2.0">& percioche la corda è rilegata in L, <lb/>&longs;ara il T il &longs;o&longs;tegno della leua ST, che ha il pe&longs;o nel mezo, & la po&longs;&longs;anza in <lb/>S; & percioche S &longs;i moue all'insù, è co&longs;a nece&longs;&longs;aria, che R anchora &longs;i moua <lb/>all'insù; & però, F &longs;arà il &longs;o&longs;tegno della leua FR, & il pe&longs;o &longs;arà in R, & <lb/>la po&longs;&longs;anza nel mezo. </s> <s id="id.2.1.1087.3.0">Le girelle dunque, i cui centri &longs;ono HK &longs;i mouono in <lb/>parti contrarie di quelle, le quali hanno i centri BD; Per laqual co&longs;a le parti del<lb/>le girelle PF nelle girelle inchineranno al ba&longs;&longs;o, cioè ver&longs;o XV. </s> <s id="id.2.1.1087.4.0">La leua dun<lb/>que VX non &longs;i mouerà nè in vna, nè in altra parte, mouendo&longs;i P & F al <lb/>ba&longs;&longs;o; & VX &longs;arà come leua, nel cui mezo &longs;ia appiccato il pe&longs;o, & in VX <lb/>due po&longs;&longs;anze eguali alla &longs;e&longs;ta parte del pe&longs;o C. </s> <s id="id.2.1.1087.5.0">Percioche le po&longs;&longs;anze di MO, <lb/>cioè le corde PV FX &longs;o&longs;tengono la &longs;e&longs;ta parte del pe&longs;o C. </s> <s id="id.2.1.1087.6.0">Adunque tutta <lb/>la girella, il cui centro è A &longs;i mouerà in sù in&longs;ieme con la taglia, ma non già &longs;i <lb/>volgerà intorno. <emph.end type="italics"/></s> <s id="id.2.1.1088.1.0">PROPOSITIONE XXV. </s> <s id="id.2.1.1089.1.0">Se à tre girelle di due taglie, l'vna delle quali habbia due girel<lb/>le, & &longs;ia tenuta di &longs;opra dalla po&longs;&longs;anza; & l'altra habbia vna &longs;o <lb/>la girella, & &longs;ia po&longs;ta di &longs;otto, & legata al pe&longs;o, &longs;arà inuolta in <lb/>torno la corda: e&longs;&longs;endo legato l'vn & l'altro de'&longs;uoi capi in <lb/>qualche luogo, ma non già nella taglia di &longs;otto. </s> <s id="id.2.1.1089.2.0">La pos&longs;anza <lb/>&longs;arà due volte tanto quanto il pe&longs;o. </s> <pb xlink:href="037/01/204.jpg"/> <s id="id.2.1.1091.1.0"><emph type="italics"/>Sia il pe&longs;o A legato alla taglia di &longs;otto, laquale habbia la girella &longs;ua co'l centro B; <lb/>ma la taglia di &longs;opra habbia due girelle, i cui centri &longs;iano CD, & &longs;ia la corda <lb/>inuolta d'intorno à tutte le girelle, & rilegata in EF; & la po&longs;&longs;anza che fo&longs;tie<lb/>ne il pe&longs;o &longs;ia in G. </s> <s id="id.2.1.1091.2.0">Di<lb/>co la po&longs;&longs;anza di G e&longs;&longs;e<lb/>re due volte tanto quanto <emph.end type="italics"/><lb/><arrow.to.target n="note272"></arrow.to.target><emph type="italics"/>il pe&longs;o A. </s> <s id="id.2.1.1091.4.0">Percioche &longs;e <lb/>in HK fo&longs;&longs;ero due po&longs;&longs;an<lb/>ze, che &longs;o&longs;tene&longs;&longs;ero il pe<lb/>&longs;o, l'vna & l'altra &longs;areb<emph.end type="italics"/><lb/><arrow.to.target n="note273"></arrow.to.target><emph type="italics"/>be la metà del pe&longs;o A: <lb/>ma la po&longs;&longs;anza di D è due <lb/>volte tanto quanto la po&longs;<lb/>&longs;anza di H, & la po&longs;&longs;an<lb/>za di C è due volte tanto <lb/>quanto la po&longs;&longs;anza di K; <lb/>Per laqual co&longs;a due po&longs;<lb/>&longs;anze in&longs;ieme po&longs;te in CD <lb/>&longs;aranno il doppio di ambe<lb/>due le po&longs;&longs;anze di HK pre<lb/>&longs;e in&longs;ieme. </s> <s id="id.2.1.1091.5.0">Ma le po&longs;&longs;an<lb/>ze di HK &longs;ono eguali al <lb/>pe&longs;o A & le po&longs;&longs;anze di <lb/>CD &longs;ono etiandio eguali <lb/>ad e&longs;&longs;a po&longs;&longs;anza di G; la <lb/>po&longs;&longs;anza dunque di G &longs;a<lb/>rà il doppio del pe&longs;o A, <lb/>che bi&longs;ognaua mo&longs;trare. <emph.end type="italics"/></s> <s id="id.2.1.1092.1.0"><margin.target id="note272"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 2. <emph type="italics"/>corollario del la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1093.1.0"><margin.target id="note273"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.204.1.jpg" xlink:href="037/01/204/1.jpg"> </figure> <s id="id.2.1.1095.1.0"><emph type="italics"/>Ma &longs;e in G &longs;arà la po&longs;&longs;anza <lb/>mouente il pe&longs;o, &longs;imilmen<lb/>te &longs;i mo&longs;trerà, come nella <lb/>precedente lo &longs;patio del <lb/>pe&longs;o e&longs;&longs;ere il doppio dello <lb/>&longs;patio della po&longs;&longs;anza. <emph.end type="italics"/></s> <s id="id.2.1.1096.1.0"><emph type="italics"/>Qui parimente è da con&longs;ide<lb/>rare, che la leua PQ non &longs;i moue, peroche la leua LM hà il &longs;o&longs;tegno in L, la <lb/>po&longs;&longs;anza nel mezo, & il pe&longs;o in M. </s> <s id="id.2.1.1096.2.0">Mala leua NO hà il &longs;o&longs;tegno in O, la po&longs;<lb/>&longs;anza nel mezo, & il pe&longs;o in N. </s> <s id="id.2.1.1096.3.0">Per laqual co&longs;a M, & N &longs;i moueranno all'in <lb/>sù. </s> <s id="id.2.1.1096.4.0">Le girelle dunque, lequali hanno i centri CD &longs;i mouono in parti contrarie. <lb/></s> <s id="id.2.1.1096.5.0">Onde la leua PQ non &longs;i mouerà nè all'vna, nè all'altra parte; & &longs;arà come &longs;e <lb/>fo&longs;&longs;e appiccato il pe&longs;o nel mezo, & in PQ due po&longs;&longs;anze fu&longs;&longs;ero eguali alla metà <lb/>del pe&longs;o A. </s> <s id="id.2.1.1096.6.0">Peroche l'vna & l'altra po&longs;&longs;anza di HK è la metà del pe&longs;o A. <emph.end type="italics"/></s> <pb pagenum="95" xlink:href="037/01/205.jpg"/> <s><emph type="italics"/>Tutta la girella dunque il cui centro è B &longs;i mouerà all'insù, ma non già &longs;i volge<lb/>rà intorno. <emph.end type="italics"/></s> <s id="id.2.1.1097.1.0">Et &longs;e la corda di F &longs;i volge&longs;&longs;e <lb/>ancora d'intorno à due al<lb/>tre girelle, i cui centri &longs;o&longs;<lb/>&longs;ero HK, laqual corda <lb/>poi &longs;ia legata in L; &longs;arà la <lb/>proportione della po&longs;&longs;an<lb/>za po&longs;ta in G vna volta <lb/>& meza quanto il pe&longs;o A. </s> <figure id="id.037.01.205.1.jpg" xlink:href="037/01/205/1.jpg"> </figure> <s id="id.2.1.1099.1.0"><emph type="italics"/>Percioche &longs;e in MN OP fo&longs;&longs;ero quat<lb/>tro po&longs;&longs;anze &longs;o&longs;tenenti il pe&longs;o, cia<emph.end type="italics"/><arrow.to.target n="note274"></arrow.to.target><lb/><emph type="italics"/>&longs;cheduna di loro &longs;arebbe il quarto del <lb/>pe&longs;o A: ma concio&longs;ia che la po&longs;&longs;an<emph.end type="italics"/><arrow.to.target n="note275"></arrow.to.target><lb/><emph type="italics"/>za di K &longs;ia il doppio della po&longs;&longs;anza <lb/>di N; &longs;arà la po&longs;&longs;anza di K vn <lb/>quarto del pe&longs;o A. </s> <s>& percioche <lb/>la po&longs;&longs;anza po&longs;ta in D è eguale al <lb/>le due po&longs;&longs;anze MO; &longs;arà ancho<lb/>ra la po&longs;&longs;anza di D vn quarto del <lb/>pe&longs;o A. </s> <s id="id.2.1.1099.2.0">Et di più e&longs;&longs;endo la po&longs;<lb/>&longs;anza di C vn quarto della po&longs;&longs;an<lb/>za di P, &longs;arà &longs;imilmente la po&longs;&longs;an<lb/>za di C vn quarto del pe&longs;o A. <lb/></s> <s id="id.2.1.1099.3.0">Tre po&longs;&longs;anze dunque po&longs;te in CDK <lb/>&longs;ono eguali à tre metà del pe&longs;o A. <lb/></s> <s id="id.2.1.1099.4.0">Ma percioche la po&longs;&longs;anza di G è <lb/>eguale alle po&longs;&longs;anze di CDK, &longs;a<lb/>rà la po&longs;&longs;anza di G eguale alle tre <lb/>metà del pe&longs;o A. </s> <s id="id.2.1.1099.5.0">La proportio<lb/>ne dunque della po&longs;&longs;anza al pe&longs;o è <lb/>vna volta, & meza. <emph.end type="italics"/></s> <s id="id.2.1.1100.1.0"><margin.target id="note274"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1101.1.0"><margin.target id="note275"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1102.1.0"><emph type="italics"/>Che &longs;e in G &longs;arà la po&longs;&longs;anza, che mo<lb/>ue, &longs;arà lo &longs;patio del pe&longs;o vna volta <lb/>& meza tanto quanto lo &longs;patio del<lb/>la po&longs;&longs;anza. <emph.end type="italics"/></s> <s id="id.2.1.1103.1.0"><emph type="italics"/>Et &longs;e la corda in L &longs;arà inuolta dauan<pb xlink:href="037/01/206.jpg"/>taggio d'intorno à due altre girelle, <lb/>&longs;imilmente &longs;i mostrerà la proportio<lb/>ne della po&longs;&longs;anza al pe&longs;o e&longs;&longs;ere vna <lb/>volta & vn terzo. </s> <s id="id.2.1.1103.2.0">& co&longs;i in infini<lb/>to ritroueremo tutte le proportioni <lb/>&longs;opraparticolari della po&longs;&longs;anza al pe<lb/>&longs;o. </s> <s id="id.2.1.1103.3.0">& mo&longs;treremo la po&longs;&longs;anza che <lb/>&longs;o&longs;tiene il pe&longs;o e&longs;&longs;ere co&longs;i ver&longs;o il pe<lb/>&longs;o, come lo &longs;patio d l pe&longs;o mo&longs;&longs;o <lb/>allo &longs;patio della po&longs;&longs;anza che mo<lb/>ue il pe&longs;o. <emph.end type="italics"/></s> <figure id="id.037.01.206.1.jpg" xlink:href="037/01/206/1.jpg"> </figure> <s id="id.2.1.1105.1.0"><emph type="italics"/>Il mouimento delle leue &longs;i farà in que<lb/>&longs;to modo, cioè il Q &longs;arà il &longs;o&longs;tegno <lb/>della leua QR, la po&longs;&longs;anza nel me<lb/>zo, il pe&longs;o in R; & della leua Z <36><lb/>il &longs;o&longs;tegno &longs;arà il Z, il pe&longs;o nel <lb/>mezo, & la po&longs;&longs;anza in <36>. </s> <s id="id.2.1.1105.2.0">&longs;imil<lb/>mente lo X &longs;arà il &longs;o&longs;tegno della le <lb/>ua VX, la po&longs;&longs;anza nel mezo, & <lb/>il pe&longs;o in V. </s> <s>& percioche lo V <lb/>&longs;i moue all'insù, &longs;i mouerà all in sù lo <lb/><foreign lang="greek">*u</foreign> ancora, & della leua <foreign lang="greek">*u</foreign> F il &longs;o<lb/>&longs;tegno &longs;arà F. </s> <s id="id.2.1.1105.3.0">Per laqual co&longs;a F <lb/>& Z nelle girelle &longs;i moueranno in <lb/>giù. </s> <s id="id.2.1.1105.4.0">& perciò la leua ST non &longs;i <lb/>mouerà nè in vna, nè in altra par<lb/>te; & ST &longs;arà come bilancia, il <lb/>cui centro &longs;arà D, & i pe&longs;i po&longs;ti <lb/>in ST &longs;aranno eguali alla quarta <lb/>parte del pe&longs;o A. </s> <s id="id.2.1.1105.5.0">Peroche cia&longs;cu<lb/>na corda SZ TF &longs;o&longs;tiene la quar<lb/>ta parte del pe&longs;o A. </s> <s id="id.2.1.1105.6.0">La girellá <lb/>dunque del centro D &longs;i monerà al<lb/>l'insù, ma non &longs;i volgerà intorno. <emph.end type="italics"/></s> <s id="id.2.1.1106.1.0"><emph type="italics"/>Fin qui, &longs;ono &longs;tate dichiarate le propor<lb/>tioni molteplici, & &longs;otto molteplici <lb/>che ha il pe&longs;o alla po&longs;&longs;anza; & da<lb/>poi le proportioni &longs;opraparticolari, <lb/>& &longs;otto &longs;opraparticolari. </s> <s id="id.2.1.1106.2.0">Hora re<lb/>&longs;ta, che &longs;i manife&longs;tino le proportio<emph.end type="italics"/><pb pagenum="96" xlink:href="037/01/207.jpg"/><emph type="italics"/>ni tra il pe&longs;o, & la po&longs;&longs;anza &longs;oprapartienti, & molteplici &longs;opraparticolari, & <lb/>molteplici &longs;oprapartienti. <emph.end type="italics"/></s> <s id="id.2.1.1107.1.0">“Et dapoi le &longs;opraparticolari, & le &longs;otto &longs;opraparticolari furono dichiarate. </s> <s id="id.2.1.1107.2.0">Dal co<lb/>no&longs;cimento del &longs;opraparticolare &longs;i intende ageuolmente il &longs;otto &longs;opraparticolare <lb/>che gli è oppo&longs;to; pero che paragonando come è detto il 3. co'l 2. na&longs;ce il &longs;o<lb/>praparticolare, & per lo contrario il 2. co'l 3. &longs;i produce il &longs;otto &longs;opraparticolare <lb/>per la forza di quella voce &longs;otto. </s> <s id="id.2.1.1108.1.0">“Hora re&longs;ta &c. </s> <s id="id.2.1.1108.2.0">Qui propone di trattare delle proportioni, che il pe&longs;o hà con la po&longs;<lb/>&longs;anza nel genere &longs;oprapartiente, & nel genere compo&longs;to del molteplice &longs;oprapar<lb/>ticolare, & del molteplice &longs;oprapartiente. </s> <s id="id.2.1.1108.3.0">il genere &longs;o prapartiente è diuer&longs;o dal <lb/>&longs;opraparticolare, che doue nel &longs;opraparticolare vna quantità contiene l'altra vna <lb/>ò più volte, & più parte, che può interamente numerare & l'vna, & l'altra: nel <lb/>&longs;oprapartiente contiene vna, ò più volte, & dauantaggio parte che non le puo<lb/>te numerare, & mi&longs;urare perfettamente, come il cinque contiene il 3. vna volta, <lb/>& piu parte di e&longs;&longs;o, che è il 2. il quale non è mi&longs;ura commune di ambidue loro, <lb/>& &longs;i denomina &longs;oprabipartiente terze, pero che contiene vna volta, & piu due <lb/>terze parti del contenuto. </s> <s id="id.2.1.1109.1.0">“Segue poi. </s> <s id="id.2.1.1109.2.0">Et le molteplici &longs;opraparticolari, che hò di &longs;opra mo&longs;trato. </s> <s id="id.2.1.1109.3.0">Componen<lb/>do due generi in&longs;ieme il molteplice, & il &longs;opraparticolare na&longs;ce que&longs;to moltepli<lb/>ce &longs;opraparticolare, nelquale vna quantità contiene l'altra molte volte, & più par<lb/>te di e&longs;&longs;a, che è mi&longs;ura commune di ambedue. </s> <s id="id.2.1.1109.4.0">La primiera &longs;ua &longs;petie è il 5. pa<lb/>ragonato co'l due, che lo contiene due volte, & piu la metà di lui, cioè vno, mi<lb/>&longs;ura di ambedue. </s> <s id="id.2.1.1109.5.0">Chiama&longs;i que&longs;ta proportione doppia &longs;e&longs;quialtera. </s> <s id="id.2.1.1109.6.0">Mettendo <lb/>parimente in&longs;ieme il genere molteplice co'l &longs;oprapartiente, &longs;i fa il molteplice &longs;o<lb/>prapartiente, il quale è differente dal &longs;opradetto per ri&longs;petto che in lui la maggior <lb/>quantità contiene la minore molte volte, & piu parte di e&longs;&longs;a, che non puote e&longs;&longs;e<lb/>re loro mi&longs;ura commune; la prima &longs;petie del qual genere è come 8. à 3. peroche <lb/>l'otto contiene il 3. due volte, & piu parte di e&longs;&longs;o 3. cioè 2. che non gli puo mi&longs;u<lb/>rare ambidue, concio&longs;ia che il 2. non puo mi&longs;urare il 3. come fà l'otto per e&longs;&longs;ere <lb/>que&longs;ti due numeri 8. & 3. tra &longs;e primi. </s> <s id="id.2.1.1109.7.0">& chiama&longs;i proportione doppia &longs;oprabi<lb/>partiente. </s> <s id="id.2.1.1109.8.0">Vuole dunque l'autore andar inue&longs;tigando le proportioni fra il pe&longs;o, <lb/>& la po&longs;&longs;anza ne i predetti generi ancora, come hà fatto ne gli altri. </s> <s id="id.2.1.1110.1.0">Da que&longs;te poche co&longs;e, lequali hò qui narrato per ageuolare l'intédimento de i voca<lb/>boli pertinenti alle proportioni po&longs;te da l'autore, &longs;i potrà facilmente con qual<lb/>che &longs;tudio comprendere tutta la &longs;omma delle vltime dimo&longs;trationi della taglia, <lb/>nelle quali &longs;ono que&longs;ti vocaboli di proportioni, quantunque in ogni loco qua&longs;i <lb/>con gli e&longs;&longs;empi &longs;tes&longs;i de' numeri &longs;iano dall'autore manife&longs;tate. </s> <s id="id.2.1.1111.1.0">PROPOSITIONE XXVI. </s> <s id="id.2.1.1112.1.0">PROBLEMA. </s> <s id="id.2.1.1113.1.0">Se vogliamo trouare la proportione &longs;oprapartiente, come &longs;e la <lb/>proportione, laquale hà il pe&longs;o alla po&longs;&longs;anza che &longs;o&longs;tiene il pe<lb/>&longs;o &longs;arà &longs;oprabipartiente, come il cinque à tre. </s> <pb xlink:href="037/01/208.jpg"/> <s id="id.2.1.1114.1.0"><arrow.to.target n="note276"></arrow.to.target><emph type="italics"/>Ponga&longs;i la po&longs;&longs;anza in A, the &longs;o<lb/>&longs;tenga il pe&longs;o B, & il pe&longs;o B <lb/>habbia proportione alla po&longs;&longs;an<lb/>za A, come cinque ad vno; <lb/>cioè &longs;ia la po&longs;&longs;anza di A vn quin<lb/>to del pe&longs;o B: dapoi riuolgendo <lb/>la corda i&longs;te&longs;&longs;a d'intorno ad altre <emph.end type="italics"/><lb/><arrow.to.target n="note277"></arrow.to.target><emph type="italics"/>girelle, ritroui&longs;i la po&longs;&longs;anza di C, <lb/>laquale &longs;ia tre volte tanto quan<lb/>to la po&longs;&longs;anza di A. </s> <s id="id.2.1.1114.2.0">Et percio <lb/>che il pe&longs;o B alla po&longs;&longs;anza po<lb/>&longs;ta in A è come cinque ad vno; <lb/>& la po&longs;&longs;anza di A alla po&longs;&longs;an<lb/>za di C è come vno ver&longs;o tre, &longs;a <lb/>rà il pe&longs;o B ver&longs;o la po&longs;&longs;anza <lb/>di C come cinque à tre, cioè &longs;o<lb/>prabipartiente. <emph.end type="italics"/></s> <s id="id.2.1.1115.1.0"><margin.target id="note276"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1116.1.0"><margin.target id="note277"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.208.1.jpg" xlink:href="037/01/208/1.jpg"> </figure> <s id="id.2.1.1118.1.0"><emph type="italics"/>Et à que&longs;to modo tutte le proportio<lb/>ni &longs;oprapartienti del pe&longs;o alla po&longs;<lb/>&longs;anza &longs;i troueranno; come &longs;e la <lb/>proportione &longs;opratrepartiente vor<lb/>rà alcuno trouare, proceda con <lb/>l'ordine iste&longs;&longs;o: cioè faccia&longs;i che la <lb/>po&longs;&longs;anza di A &longs;ostenente il pe<lb/>&longs;o B &longs;ia vn &longs;ettimo del pe&longs;o B; <lb/>Dapoi &longs;i faccia, che la po&longs;&longs;anza <lb/>di C &longs;ia quattro volte tanto quan<lb/>to è quella di A; &longs;arà il pe&longs;o B <lb/>ver&longs;o la po&longs;&longs;anza di C, come &longs;et<lb/>te à quattro; cioè &longs;opratrepar<lb/>tiente. <emph.end type="italics"/></s> <s id="id.2.1.1119.1.0">Ma &longs;e in C &longs;arà la po&longs;&longs;an<lb/>za mouente il pe&longs;o, &longs;arà <lb/>lo &longs;patio della po&longs;&longs;anza <lb/>&longs;oprabipartiente allo &longs;pa<lb/>tio del pe&longs;o. </s> <s id="id.2.1.1120.1.0"><arrow.to.target n="note278"></arrow.to.target><emph type="italics"/>Per cioche lo &longs;patio della po&longs;&longs;anza <lb/>po&longs;ta in C è la terza parte della <lb/>&longs;patio della po&longs;&longs;anza po&longs;ta in A,<emph.end type="italics"/><pb pagenum="97" xlink:href="037/01/209.jpg"/><emph type="italics"/>cioè, che co&longs;i &longs;ono tra loro, come il cinque al quindici: & lo &longs;patio della po&longs;&longs;anza <emph.end type="italics"/><arrow.to.target n="note279"></arrow.to.target><lb/><emph type="italics"/>di A è cinque volte tanto quanto lo &longs;patio del pe&longs;o B, cioè come quindici à tre. </s> <s><lb/>&longs;arà dunque lo &longs;patio della po&longs;&longs;anza posta in C ver&longs;o lo &longs;patio del pe&longs;o B come <lb/>cinque à tre; cioè &longs;oprabipartiente: & &longs;empre dimostreremo, co&longs;i e&longs;&longs;ere lo &longs;patio <lb/>della po&longs;&longs;anza che moue allo &longs;patio del pe&longs;o; come il pe&longs;o alla po&longs;&longs;anza che lo &longs;o<lb/>&longs;tiene. <emph.end type="italics"/></s> <s id="id.2.1.1121.1.0"><margin.target id="note278"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1122.1.0"><margin.target id="note279"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 14. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1123.1.0"><emph type="italics"/>Et con ragione del tutto &longs;imile ritroueremo la proportione &longs;oprapartiente della po&longs;&longs;an<lb/>za al pe&longs;o. </s> <s id="id.2.1.1123.2.0">Peroche &longs;e C fo&longs;&longs;e di &longs;otto, & in e&longs;&longs;o fo&longs;&longs;e appiccato il pe&longs;o; & il <lb/>B di &longs;opra, nelquale fo&longs;&longs;e la po&longs;&longs;anza che in C &longs;o&longs;tiene il pe&longs;o, &longs;arebbe la po&longs;<lb/>&longs;anza di B &longs;oprabipartiente al pe&longs;o appiccato in C: e&longs;&longs;endo il B allo A come <emph.end type="italics"/><arrow.to.target n="note280"></arrow.to.target><lb/><emph type="italics"/>cinque ad vno; ma A al C come l'vno al tre. <emph.end type="italics"/></s> <s id="id.2.1.1124.1.0"><margin.target id="note280"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>& per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1125.1.0">Ma &longs;e vorremo trouare la proportione molteplice &longs;oprapartico<lb/>lare; come &longs;e la proportione, laquale ha il pe&longs;o alla po&longs;&longs;anza, <lb/>che lo &longs;o&longs;tiene &longs;ia doppia &longs;e&longs;quialtera, come cinque à due. </s> <s id="id.2.1.1126.1.0"><emph type="italics"/>Nell'i&longs;te&longs;&longs;o modo, co'l quale ritrouiamo le &longs;oprapartienti, ritroueremo ancora tutte que<lb/>ste molteplici &longs;opraparticolari. </s> <s id="id.2.1.1126.2.0">Come faccia&longs;i il pe&longs;o po&longs;to in B alla po&longs;&longs;anza di A, <emph.end type="italics"/><arrow.to.target n="note281"></arrow.to.target><lb/><emph type="italics"/>come il cinque all'vno; & la po&longs;&longs;anza di C alla po&longs;&longs;anza di A come il due all'vno; <lb/>co&longs;a che &longs;i farà, &longs;e la corda &longs;arà rilegata in D, ouero in E; ma non già alla ta<emph.end type="italics"/><arrow.to.target n="note282"></arrow.to.target><lb/><emph type="italics"/>glia di &longs;opra; &longs;arà il pe&longs;o B alla po&longs;&longs;anza di C, come il cinque al due, cioè dop<lb/>pio &longs;e&longs;quialtero. <emph.end type="italics"/></s> <s id="id.2.1.1127.1.0"><margin.target id="note281"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1128.1.0"><margin.target id="note282"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>&<emph.end type="italics"/> 16. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1129.1.0"><emph type="italics"/>Et per lo contrario ritrouaremo la proportione molteplice &longs;opraparticolare della po&longs;<lb/>&longs;anza al pe&longs;o; & come nelle altre &longs;i mo&longs;trerà co&longs;i e&longs;&longs;ere lo &longs;patio della po&longs;&longs;anza <lb/>che moue allo &longs;patio del pe&longs;o, come il pe&longs;o alla po&longs;&longs;anza, che lo &longs;o&sacute;tiene. <emph.end type="italics"/></s> <s id="id.2.1.1130.1.0">Con l'i&longs;te&longs;&longs;o modo ritrouaremo ancora ogni proportione &longs;opra<lb/>partiente; come &longs;e la proportione, laquale ha la po&longs;&longs;anza co'l <lb/>pe&longs;o, &longs;arà doppia &longs;oprabipartiente, come l'otto al tre. </s> <s id="id.2.1.1131.1.0"><emph type="italics"/>Faccia&longs;i la po&longs;&longs;anza po&longs;ta in A &longs;o&longs;tenente il pe&longs;o B vn'ottauo del pe&longs;o B, & <emph.end type="italics"/><arrow.to.target n="note283"></arrow.to.target><lb/><emph type="italics"/>la po&longs;&longs;anza di C &longs;ia vn terzo della po&longs;&longs;anza di A; &longs;arà il pe&longs;o B alla po&longs;&longs;an<lb/>za di C, come l'otto al tre. </s> <s id="id.2.1.1131.2.0">& per lo contrario ritroueremo ogni proportione mol<emph.end type="italics"/><arrow.to.target n="note284"></arrow.to.target><pb xlink:href="037/01/210.jpg"/><emph type="italics"/>teplice &longs;oprapartiente della po&longs;&longs;anza al pe&longs;o. </s> <s id="id.2.1.1131.3.0">& come nelle altre ritrouaremo co&longs;i <lb/>e&longs;&longs;ere il pe&longs;o alla po&longs;&longs;anza che lo &longs;o&longs;tiene, come lo &longs;patio della po&longs;&longs;anza che moue <lb/>allo &longs;patio del pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.1132.1.0"><margin.target id="note283"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1133.1.0"><margin.target id="note284"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1134.1.0"><emph type="italics"/>Ma egli è da notare, che benche più volte &longs;ia &longs;tato detto nelle demo&longs;irationi prece<lb/>denti, la po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o e&longs;&longs;ere due volte tanto quanto e&longs;&longs;o pe&longs;o, ò tre, <lb/>& co&longs;i di mano in mano, come nella decimaquinta di questo è &longs;tato mo&longs;trato; non<lb/>dimeno percioche la po&longs;&longs;anza &longs;o&longs;tiene non &longs;olamente il pe&longs;o, ma la taglia ancora, <lb/>però egli pare, che &longs;ia me&longs;tieri porre la po&longs;&longs;anza di molto maggiore virtù, & di pro<lb/>portione maggiore ver&longs;o il pe&longs;o. </s> <s id="id.2.1.1134.2.0">ilche è vero, &longs;e vogliamo con&longs;iderare etiandio la <lb/>grauezza della taglia. </s> <s id="id.2.1.1134.3.0">Ma percioche cerchiamo la proportione che è fra la po&longs;<lb/>&longs;anza & il pe&longs;o, però habbiamo trala&longs;ciato cote&longs;ta grauezza della taglia, laquale <lb/>&longs;e alcuno vorrà anche con&longs;iderare, alla po&longs;&longs;anza potrà aggiungere forza che &longs;ia <lb/>eguale alla taglia. </s> <s id="id.2.1.1134.4.0">ilche mede&longs;imamente &longs;i potrà o&longs;&longs;eruare nella corda. </s> <s id="id.2.1.1134.5.0">& &longs;i co<lb/>me habbiamo ciò con&longs;iderato nella decimaquinta, l'i&longs;te&longs;&longs;o parimente nelle altre po<lb/>tremo con&longs;iderare. <emph.end type="italics"/></s> <pb pagenum="98" xlink:href="037/01/211.jpg"/> <s id="id.2.1.1136.1.0"><emph type="italics"/>Egli è me&longs;tieri &longs;apere etiandio, che &longs;i come tut<lb/>te le proportioni tra la po&longs;&longs;anza, & il pe&longs;o <lb/>&longs;ono &longs;tate ritrouate con vna &longs;ola corda: co&longs;i <lb/>ancora potranno&longs;i le i&longs;te&longs;&longs;e ritrouare con più <lb/>corde, & con più taglie. </s> <s id="id.2.1.1136.2.0">come &longs;e vorremo <lb/>ritrouare la proportione molteplice &longs;oprapar<lb/>ticolare con più corde, cioè &longs;e la proportio<lb/>ne, laquale hà il pe&longs;o alla po&longs;&longs;anza che lo &longs;o<lb/>&longs;tiene &longs;arà doppia &longs;e&longs;quialtera, come cinque <lb/>à due; bi&longs;ogna comporre que&longs;ta proportione <lb/>da più proportioni come per gratia di e&longs;&longs;em<lb/>pio dalla proportione &longs;e&longs;quiquarta, che è il <lb/>cinque al quattro, & dalla doppia, che è il <lb/>quattro al due. </s> <s id="id.2.1.1136.3.0">Ponga&longs;i dunque la po&longs;&longs;an<emph.end type="italics"/><arrow.to.target n="note285"></arrow.to.target><lb/><emph type="italics"/>za di A che &longs;o&longs;tenga il pe&longs;o B, alla qua<lb/>le il pe&longs;o habbia la proportione di vna volta <lb/>& vn quarto, come cinque à quattro: da<emph.end type="italics"/><arrow.to.target n="note286"></arrow.to.target><lb/><emph type="italics"/>poi con vn'altra corda &longs;i troui la po&longs;&longs;anza <lb/>di C, della quale &longs;ia doppia la po&longs;&longs;anza di <lb/>A. </s> <s>& percioche il B all' A è come cin<lb/>que à quattro: & l' A al C come il quat<lb/>tro al due: &longs;arà la po&longs;&longs;anza di B alla po&longs;<lb/>&longs;anza di C come il cinque al due; cioè ha<lb/>urà la proportione doppia &longs;e&longs;quialtera. <emph.end type="italics"/></s> <s id="id.2.1.1137.1.0"><margin.target id="note285"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 21. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1138.1.0"><margin.target id="note286"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.211.1.jpg" xlink:href="037/01/211/1.jpg"> </figure> <s id="id.2.1.1140.1.0"><emph type="italics"/>Et è da notare poter&longs;i trouar' anche que&longs;ta pro<lb/>portione, &longs;e comporremo la proportione di <lb/>cinque à due da più, come cinque à quindici, <lb/>& il quindici al venti, & il venti al due. </s> <s id="id.2.1.1140.2.0">Et <lb/>in que&longs;to modo ritroueremo non &longs;olo ogni al<lb/>tra proportione, ma qualunque &longs;i &longs;ia in mol<lb/>ti, & infiniti modi ritroueremo. </s> <s id="id.2.1.1140.3.0">percioche <lb/>ogni proportione &longs;i può comporre di propor<lb/>tioni infinite. </s> <s id="id.2.1.1140.4.0">come è manife&longs;to nel commen<lb/>tario di Eutocio nella quarta propo&longs;itione del <lb/>&longs;econdo libro di Archimede della sfera, & <lb/>Cilindro. <emph.end type="italics"/></s> <s id="id.2.1.1141.1.0">Po&longs;&longs;iamo ancora v&longs;are più corde: & <lb/>adoperare le taglie di &longs;otto &longs;ola<lb/>mente, ouero quelle di &longs;opra. </s> <pb xlink:href="037/01/212.jpg"/> <s id="id.2.1.1142.1.0"><emph type="italics"/>Sia il pe&longs;o A alquale &longs;ia legata la ta<lb/>glia, che habbia la girella col centro <lb/>B; &longs;ia rilegata la corda in C, la <lb/>quale &longs;ia inuolta d'intorno alla gi<lb/><emph.end type="italics"/><arrow.to.target n="note286b"></arrow.to.target><emph type="italics"/>rella, & peruenga la corda in D: <lb/>&longs;arà la po&longs;&longs;anza di D &longs;o&longs;tenente il <lb/>pe&longs;o A la metà del pe&longs;o A. </s> <s id="id.2.1.1142.2.0">Da<lb/><emph.end type="italics"/><arrow.to.target n="note286c"></arrow.to.target><emph type="italics"/>poi la corda in D &longs;ia rilegata ad <lb/>vn'altra corda, laquale &longs;ia legata <lb/>in E, & peruenga in F. </s> <s>&longs;arà <lb/>la po&longs;&longs;anza di F la metà di quel<lb/>lo, che &longs;o&longs;tiene la po&longs;&longs;anza in D: <lb/>percioche egli è come &longs;e il D &longs;o&longs;te<lb/>ne&longs;&longs;e la metà del pe&longs;o A &longs;enza ta<lb/>glia: per laqual co&longs;a la po&longs;&longs;anza di <lb/>F &longs;arà vn quarto del pe&longs;o A. </s> <s>& <lb/>&longs;e dauantaggio la corda di F &longs;i ri<lb/>legherà ad vn'altra traglia, & &longs;i ri<lb/>uolga intorno alla &longs;ua girella vn'al<lb/>tra corda, laquale &longs;ia legata in G, <lb/>& peruenga in H: &longs;arà la po&longs;&longs;an<lb/>za di H la metà della po&longs;&longs;anza di <lb/>F. </s> <s id="id.2.1.1142.3.0">Adunque la po&longs;&longs;anza di N è <lb/>vn'ottauo del pe&longs;o A. </s> <s>& co&longs;i in <lb/>infinito ritroueremo &longs;empre la po&longs;<lb/>&longs;anza in proportione &longs;otto doppia ver&longs;o la precedente po&longs;&longs;anza. </s> <figure id="id.037.01.212.1.jpg" xlink:href="037/01/212/1.jpg"> </figure> <s id="id.2.1.1142b.1.0">Et &longs;e in H &longs;arà la po&longs;&longs;anza che mo<lb/>ue, &longs;arà lo &longs;patio della po&longs;&longs;anza ot<lb/>to volte tanto quanto lo &longs;patio del <lb/>pe&longs;o: percioche lo &longs;patio di D è <lb/><emph.end type="italics"/><arrow.to.target n="note286d"></arrow.to.target><emph type="italics"/>due volte tanto quanto lo &longs;patio del <lb/>pe&longs;o A, & lo &longs;patio di F è due <lb/>volte tanto quanto lo &longs;patio di D: <lb/>&longs;arà lo &longs;patio di F quattro volte <lb/>tanto quanto lo &longs;patio di A pe&longs;o. </s> <s><lb/>&longs;imilmente percioche lo &longs;patio della <lb/>po&longs;&longs;anza di N è il doppio dello &longs;pa<lb/>tio di F, &longs;arà lo &longs;patio della po&longs;&longs;an<lb/>za di N otto volte tanto quanto il pe&longs;o A. <emph.end type="italics"/></s> <s id="id.2.1.1137b.1.0"><margin.target id="note286b"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1137c.1.0"><margin.target id="note286c"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1137d.1.0"><margin.target id="note286d"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo.<emph.end type="italics"/> </s> <pb pagenum="99" xlink:href="037/01/213.jpg"/> <s id="id.2.1.1143.1.0"><emph type="italics"/>Sia poi il pe&longs;o A legato alla fune, la<lb/>quale &longs;ia inuolta d'intorno alla girel<lb/>la della taglia di &longs;opra, & rilegata in <lb/>B, & &longs;ia la po&longs;&longs;anza di C che &longs;o <lb/>&longs;tenga il pe&longs;o A; &longs;arà la po&longs;&longs;anza <lb/>di C due volte tanto quanto il pe&longs;o <emph.end type="italics"/><arrow.to.target n="note287"></arrow.to.target><lb/><emph type="italics"/>A: dapoi C &longs;ia rilegata ad vn'al<lb/>tra fune, laquale &longs;ia rinuolta d'intor<lb/>no la girella d'vn'altra taglia, & ri<lb/>legata in D; &longs;arà la po&longs;&longs;anza di E <emph.end type="italics"/><arrow.to.target n="note288"></arrow.to.target><lb/><emph type="italics"/>due volte tanto quanto la po&longs;&longs;anza <lb/>di C. </s> <s id="id.2.1.1143.2.0">Per laqual co&longs;a la po&longs;&longs;anza <lb/>di E &longs;arà quattro volte tanto quan<lb/>to il pe&longs;o A. </s> <s id="id.2.1.1143.3.0">Et &longs;e dauantaggio <lb/>lo E &longs;i rilegherà ad vn'altra fune, <lb/>laquale &longs;ia inuolta dintorno' alla gi<lb/>rella d'vn'altra taglia ancora, & &longs;ia <lb/>rilegata in F; &longs;arà la po&longs;&longs;anza di G <lb/>due volte tanto quanto la po&longs;&longs;anza <lb/>di E. </s> <s id="id.2.1.1143.4.0">Adunque la po&longs;&longs;anza po&longs;ta <lb/>in G è otto volte tanto quanto il pe<lb/>&longs;o A; & co&longs;i in infinito ritrouere<lb/>mo &longs;empre la po&longs;&longs;anza e&longs;&longs;ere due vol<lb/>te tanto quanto la po&longs;&longs;anza prece<lb/>dente. <emph.end type="italics"/></s> <s id="id.2.1.1144.1.0"><margin.target id="note287"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1145.1.0"><margin.target id="note288"></margin.target><emph type="italics"/>Per la iste&longs;&longs;a. <emph.end type="italics"/></s> <figure id="id.037.01.213.1.jpg" xlink:href="037/01/213/1.jpg"> </figure> <s id="id.2.1.1147.1.0"><emph type="italics"/>Ma &longs;e in G fo&longs;&longs;e la po&longs;&longs;anza che moue, <lb/>&longs;arà lo &longs;patio del pe&longs;o otto volte tan<lb/>to quanto lo &longs;patio della po&longs;&longs;anza po<lb/>&longs;ta in G: percioche lo &longs;patio del pe<emph.end type="italics"/><arrow.to.target n="note289"></arrow.to.target><lb/><emph type="italics"/>&longs;o A è due volte tanto quanto lo <lb/>&longs;patio della po&longs;&longs;anza posta in C, & <lb/>il C è due volte tanto quanto è lo <lb/>&longs;patio di e&longs;&longs;o E. </s> <s id="id.2.1.1147.2.0">Per laqual co&longs;a lo <lb/>&longs;patio del pe&longs;o A &longs;arà quattro vol<lb/>te tanto quanto lo &longs;patio della po&longs;&longs;an<lb/>za di E. </s> <s>&longs;imilmente percioche lo <lb/>&longs;patio di E è due volte tanto quan<lb/>to è lo &longs;patio della po&longs;&longs;anza po&longs;ta in <lb/>G; &longs;arà dunque lo &longs;patio del pe&longs;o A <lb/>otto volte tanto quanto lo &longs;patio della <lb/>po&longs;&longs;anza po&longs;ta in G. <emph.end type="italics"/></s> <s id="id.2.1.1148.1.0"><margin.target id="note289"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <pb xlink:href="037/01/214.jpg"/> <s id="id.2.1.1149.1.0">COROLLARIO. </s> <s id="id.2.1.1150.1.0">Da que&longs;te co&longs;e è manife&longs;to, che &longs;empre lo &longs;patio della po&longs;&longs;anza <lb/>che moue ha proportione maggiore ver&longs;o lo &longs;patio del pe&longs;o <lb/>mo&longs;&longs;o, di quel che ha il pe&longs;o ver&longs;o la mede&longs;ima po&longs;&longs;anza. </s> <s id="id.2.1.1151.1.0"><emph type="italics"/>Que&longs;to è chiaro da quelle co&longs;e lequali &longs;ono &longs;tate dette nel corollario della quarta pro<lb/>po&longs;itione di que&longs;to nella leua. <emph.end type="italics"/></s> <s id="id.2.1.1152.1.0">PROPOSITIONE XXVII. </s> <s id="id.2.1.1153.1.0">PROBLEMA. </s> <s id="id.2.1.1154.1.0">Che &longs;i moua vn pe&longs;o dato da vna po&longs;&longs;anza data con le taglie. </s> <s id="id.2.1.1155.1.0"><emph type="italics"/>La po&longs;&longs;anza data ò che ella è maggiore, ouero eguale, ò pure minore del pe&longs;o dato. <emph.end type="italics"/></s> <s id="id.2.1.1156.1.0"><emph type="italics"/>Se è maggiore, all'hora la po&longs;<lb/>&longs;anza, &longs;enza altro &longs;tromen<lb/>to, ò fune inuolta d'intor<lb/>no alla girella della taglia <emph.end type="italics"/><lb/><arrow.to.target n="note290"></arrow.to.target><emph type="italics"/>appiccata di &longs;opra, mouerà <lb/>il pe&longs;o dato. </s> <s id="id.2.1.1156.2.0">percio che po&longs;<lb/>&longs;anza minore della data pe<lb/>&longs;a tanto quanto il pe&longs;o, <lb/>adunque la data, che è mag<lb/>giore mouerà. </s> <s id="id.2.1.1156.3.0">L'i&longs;te&longs;&longs;o &longs;i <lb/>può fare in tutte le propo<lb/>&longs;itioni nelle quali la po&longs;&longs;an<lb/>za, che &longs;o&longs;tiene il pe&longs;o è &longs;ta<lb/>ta dimo&longs;trata ò eguale, ò <lb/>minore del pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.1157.1.0"><margin.target id="note290"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.214.1.jpg" xlink:href="037/01/214/1.jpg"> </figure> <pb pagenum="100" xlink:href="037/01/215.jpg"/> <s id="id.2.1.1160.1.0"><emph type="italics"/>Ma &longs;e eguale mouerà il <lb/>pe&longs;o e&longs;&longs;endo la fune <lb/>inuolta d'intorno al <lb/>la girella della ta<lb/>glia legata al pe&longs;o, <emph.end type="italics"/><arrow.to.target n="note291"></arrow.to.target><lb/><emph type="italics"/>percio che la po&longs;&longs;an<lb/>za che &longs;o&longs;tiene il pe<lb/>&longs;o è la metà del pe<lb/>&longs;o. </s> <s id="id.2.1.1160.2.0">la po&longs;&longs;anza dun<lb/>que eguale al. </s> <s id="id.2.1.1160.3.0">pe&longs;o <lb/>mouerà il pe&longs;o da<lb/>to. </s> <s id="id.2.1.1160.4.0">ilche parimen<lb/>te &longs;i puote fare &longs;e<lb/>condo le propo&longs;itio<lb/>ni, nellequali &longs;i è <lb/>mo&longs;trato la po&longs;&longs;an<lb/>za e&longs;&longs;ere minore del <lb/>pe&longs;o. <emph.end type="italics"/></s> <s id="id.2.1.1161.1.0"><margin.target id="note291"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.215.1.jpg" xlink:href="037/01/215/1.jpg"> </figure> <pb xlink:href="037/01/216.jpg"/> <s id="id.2.1.1163.1.0"><emph type="italics"/>Che &longs;e è minore, &longs;ia il pe&longs;o <lb/>dato come &longs;e&longs;&longs;anta, & la <lb/>po&longs;&longs;anza che moue &longs;ia da<lb/>ta come tredici. </s> <s id="id.2.1.1163.2.0">Troui&longs;i <lb/>la po&longs;&longs;anza di A, che &longs;o <lb/>&longs;tenga il pe&longs;o B, laquale <emph.end type="italics"/><lb/><arrow.to.target n="note292"></arrow.to.target><emph type="italics"/>&longs;ia vn quinto del pe&longs;o </s> <s id="id.2.1.1163.3.0">& percioche la po&longs;&longs;anza <lb/>di A che &longs;o&longs;tiene il pe&longs;o <lb/>è come dodici; adunque <lb/>po&longs;&longs;anza maggiore di do<lb/>dici po&longs;ta in A mouerà <lb/>il pe&longs;o B. </s> <s id="id.2.1.1163.4.0">Per laqual co<lb/>&longs;a la po&longs;&longs;anza come tredi<lb/>ci po&longs;ta in A mouerà il <lb/>pe&longs;o B. </s> <s>che bi&longs;ognaua <lb/>fare. <emph.end type="italics"/></s> <s id="id.2.1.1164.1.0"><margin.target id="note292"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <figure id="id.037.01.216.1.jpg" xlink:href="037/01/216/1.jpg"> </figure> <s id="id.2.1.1166.1.0"><emph type="italics"/>Egli è parimente da auerti<lb/>re nel mouere i pe&longs;i, che <lb/>la po&longs;&longs;anza alcuna volta <lb/>meglio for&longs;e moue mouen<lb/>do&longs;i in giù, che mouendo&longs;i <lb/>in sù. </s> <s id="id.2.1.1166.2.0">come volga&longs;i dauan<lb/>taggio la fune d'intorno ad <lb/>vn'altra girella della ta<lb/>glia di &longs;opra, il cui centro <lb/>&longs;ia C, & la fune per<lb/>uenga in D; &longs;arà la po&longs;<emph.end type="italics"/><lb/><arrow.to.target n="note293"></arrow.to.target><emph type="italics"/>&longs;anza di D &longs;o&longs;tenente il <lb/>pe&longs;o B &longs;imilmente dodi<lb/>ci, &longs;i come ella era in A. <lb/></s> <s id="id.2.1.1166.3.0">Però la po&longs;&longs;anza di tredici <lb/>po&longs;ta in D mouerà il pe<lb/>&longs;o B. </s> <s id="id.2.1.1166.4.0">& percioche &longs;i <lb/>moue in giù, for&longs;e tirerà <lb/>più facilmente, che &longs;e fo&longs;<lb/>&longs;e po&longs;ta in A, ma il tem<lb/>po è l'i&longs;te&longs;&longs;o, &longs;i come egli <lb/>era etiandio in A. <emph.end type="italics"/></s> <s id="id.2.1.1168.1.0"><margin.target id="note293"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <pb pagenum="101" xlink:href="037/01/217.jpg"/> <s id="id.2.1.1169.1.0">PROPOSITIONE XXVIII. </s> <s id="id.2.1.1170.1.0">PROBLEMA. </s> <s id="id.2.1.1171.1.0">Sia propo&longs;to à noi il fare, che la po&longs;&longs;anza mouente il pe&longs;o, & il <lb/>pe&longs;o &longs;i mouano per gli <lb/>&longs;patij dati, i quali &longs;ia<lb/>no fra loro commen<lb/>&longs;urabili. </s> <s id="id.2.1.1172.1.0"><emph type="italics"/>Sia dato lo &longs;patio della po&longs;&longs;anza <emph.end type="italics"/><arrow.to.target n="note294"></arrow.to.target><lb/><emph type="italics"/>come tre, & del pe&longs;o come <lb/>quattro. </s> <s id="id.2.1.1172.2.0">ritroui&longs;i la po&longs;&longs;anza <lb/>di A &longs;o&longs;tenente il pe&longs;o B, la <lb/>quale &longs;ia vna volta, & vn ter<lb/>zo quanto il pe&longs;o, come quat<lb/>tro à tre. </s> <s id="id.2.1.1172.3.0">Se dunque in A <emph.end type="italics"/><arrow.to.target n="note295"></arrow.to.target><lb/><emph type="italics"/>fo&longs;&longs;e la po&longs;&longs;anza mouente il <lb/>pe&longs;o; &longs;arebbe lo &longs;patio del pe<lb/>&longs;o vna volta, & vn terzo <lb/>quanto lo &longs;patio della po&longs;&longs;an<lb/>za, cioè come quattro à tre; <lb/>che bi&longs;ognaua fare. <emph.end type="italics"/></s> <s id="id.2.1.1173.1.0"><margin.target id="note294"></margin.target><emph type="italics"/>Per la.<emph.end type="italics"/> 22. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <s id="id.2.1.1174.1.0"><margin.target id="note295"></margin.target><emph type="italics"/>Per l'iste&longs;&longs;a. <emph.end type="italics"/></s> <figure id="id.037.01.217.1.jpg" xlink:href="037/01/217/1.jpg"> </figure> <s id="id.2.1.1176.1.0"><emph type="italics"/>Ciò po&longs;&longs;iamo menar ad effetto con <lb/>vna &longs;ola fune per le co&longs;e det<lb/>te nella vige&longs;ima &longs;econda, & <lb/>nella vige&longs;ima quinta di que<lb/>&longs;to. </s> <s id="id.2.1.1176.2.0">che &longs;e ciò vorremo fare <lb/>con più funi, potremo porlo <lb/>in opra non &longs;olo con molti, <lb/>ma con modi infiniti, come di <lb/>&longs;opra è detto. </s> <s id="id.2.1.1176.3.0">Per laqual co<lb/>&longs;a ciò ben po&longs;siamo affermare, <emph.end type="italics"/><arrow.to.target n="note296"></arrow.to.target><lb/><emph type="italics"/>che pare co&longs;a marauiglio&longs;a, <lb/>cioè. <emph.end type="italics"/></s> <s id="id.2.1.1177.1.0"><margin.target id="note296"></margin.target><emph type="italics"/>Nella<emph.end type="italics"/> 26. <emph type="italics"/>di questo. <emph.end type="italics"/></s> <pb xlink:href="037/01/218.jpg"/> <s id="id.2.1.1178.1.0">COROLLARIO I. </s> <s id="id.2.1.1179.1.0">Da que&longs;te co&longs;e e&longs;&longs;ere manife&longs;to, Qualunque data proportione <lb/>nei numeri tra il pe&longs;o, & la po&longs;&longs;anza; & tra lo &longs;patio del pe&longs;o <lb/>mo&longs;&longs;o, & lo &longs;patio della po&longs;&longs;anza mo&longs;&longs;a; poter&longs;i trouare con <lb/>le taglie in modi infiniti. </s> <s id="id.2.1.1180.1.0">COROLLARIO II. </s> <s id="id.2.1.1181.1.0">Dalle co&longs;e dette è manife&longs;to etiandio che quanto più facilmen<lb/>te &longs;i moue il pe&longs;o, tanto maggiore e&longs;&longs;ere etiandio il tempo; <lb/>ma quanto più difficilmente, tanto minore e&longs;&longs;ere: & co&longs;i per <lb/>lo contrario. </s> <s id="id.2.1.1182.1.0">IL FINE DELLA TAGLIA. </s> </chap> <pb pagenum="102" xlink:href="037/01/219.jpg"/> <chap> <s id="id.2.1.1184.1.0">DELL'ASSE <lb/>NELLA ROTA. </s> <figure id="id.037.01.219.1.jpg" xlink:href="037/01/219/1.jpg"> </figure> <s id="id.2.1.1187.1.0">La fabrica, & compo&longs;itione di que&longs;to i&longs;trumento <lb/>in&longs;egna Pappo nell'ottauo libro delle raccolte ma<lb/>tematiche: & chiama a&longs;&longs;e AB, & timpano CD <lb/>d'intorno al centro mede&longs;imo (che noi diremo ro<lb/>ta) & noma&longs;citale quei ba&longs;toni i quali &longs;ono fic<lb/>cati ne'buchi della rota notate per EFGH, & le altre &longs;uc<lb/>ce&longs;&longs;iuamente, che noi pur diremo raggi. </s> <s id="id.2.1.1187.2.0">talche la po&longs;&longs;anza, <pb xlink:href="037/01/220.jpg"/>laquale è &longs;empre ne i raggi, come in F, mentre ella volge <lb/>intorno la rota, & l'a&longs;&longs;e, moua anco in sù il pe&longs;o K appicca<lb/>to all'a&longs;&longs;e con la corda LM riuolta d'intorno all'a&longs;&longs;e. </s> <s id="id.2.1.1187.3.0">A noi <lb/>re&longs;ta dunque, di mo&longs;trare, perche i gran pe&longs;i da piccola forza, <lb/><figure id="id.037.01.220.1.jpg" xlink:href="037/01/220/1.jpg"></figure><lb/>& in che modo etiandio &longs;i mouano con que&longs;to i&longs;trumento: <lb/>& di più manife&longs;tare la ragione del tempo, & dello &longs;patio del<lb/>la po&longs;&longs;anza mouente, & del pe&longs;o mo&longs;&longs;o fra loro, & ridurre l'v<lb/>&longs;o di cote&longs;to i&longs;trumento alla leua. </s> <pb pagenum="103" xlink:href="037/01/221.jpg"/> <s id="id.2.1.1190.1.0">PROPOSITIONE I. </s> <s id="id.2.1.1191.1.0">La po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o con l'a&longs;&longs;e nella rota, ha la pro