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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" > <archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info> <author>Stelliola, Niccolà Antonio</author> <title>De gli elementi mechanici</title> <date>1597</date> <place>Naples</place> <translator/> <lang>it</lang> <cvs_file>stell_mecha_01_it_1597</cvs_file> <cvs_version/> <locator>041.xml</locator> </info> <text> <front> </front> <body> <chap> <pb id="p.0001"/><p type="head"> <s>DE GLI <lb/>ELEMENTI <lb/>MECHANICI<lb/><figure id="fig1"/><lb/>La &longs;tatera. Leua. <lb/><!-- KEEP S--></s> <s>Raggi nell'a&longs;&longs;e.</s><s> Rote vettiue <lb/></s><s>Taglia. </s><s>Rote motiue. <lb/></s> <s>Cugno.<!-- KEEP S--></s><s> Vite. <!-- KEEP S--></s></p><p type="head"> <s><emph type="italics"/>DI C. ANTONIO STELLIOLA.<emph.end type="italics"/></s></p><figure/><p type="head"> <s>IN NAPOLI, Nella Stamparia à Porta Regale <lb/>M. D. XCVII.<!-- KEEP S--></s></p><pb pagenum="1"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE <lb/>di tutta l'opera.<emph.end type="italics"/></s></p><p type="main"> <s>Cerchiamo come po&longs;&longs;a la potenza <lb/>minore vincer di forza la maggiore: <lb/>e la potenza piu tarda, vincer di mo­<lb/>uimento la piu veloce. </s> <s>e que&longs;to con <lb/>Leue, Taglie, Viti, Rote, e tutti in&longs;trumenti che <lb/>moltiplicar po&longs;&longs;ono il momento, o della forza, <lb/>o della velocità. </s> <s>Qual &longs;oggetto communemente <lb/>gli antichi chiamarono Mechaniche. <!-- KEEP S--></s> <s>Il che tut­<lb/>to &longs;i tratterà &longs;econdo le &longs;uppo&longs;itioni fatte de mo­<lb/>menti, o per linee parallelle, o per linee con­<lb/>correnti ad vn ponto, o per circonferenze d'in­<lb/>torno vn centro i&longs;te&longs;&longs;o: e &longs;econdo il &longs;olito v&longs;o de <lb/>mathematici deducendo le dimo&longs;trationi, e cau­ <lb/>&longs;e de gli effetti, dalli primi e proprij principij. </s></p><figure/><pb pagenum="2"/><p type="head"> <s><emph type="italics"/>DEFINITIONI.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Centro di pe&longs;o diciamo il ponto, per cui il corpo co­<lb/>munque &longs;o&longs;pe&longs;o, non muta po&longs;itione. </s></p><p type="head"> <s>II.<!-- KEEP S--></s></p><p type="main"> <s>Corpo egualmente di&longs;te&longs;o diciamo, che comunque <lb/>tagliato con pianezze parallele, fa figure &longs;uperficiali <lb/>eguali e &longs;imili. </s></p><p type="head"> <s>III.<!-- KEEP S--></s></p><p type="main"> <s>Applicar&longs;i diciamo vn corpo ad vna linea, quando <lb/>detto corpo vgualmente di&longs;te&longs;o occupi la lunghezza di <lb/>detta linea. </s></p><p type="head"> <s>IIII.<!-- KEEP S--></s></p><p type="main"> <s>Linea di momento diciamo, per cui il centro di pe­<lb/>&longs;o della grauezza da impedimento libera &longs;i moue. </s></p><p type="head"> <s>V.<!-- KEEP S--></s></p><p type="main"> <s>Libra ò &longs;tatera diciamo la linea a cui &longs;i applicano, ò <lb/>appendono le grauezze: e che &longs;ia &longs;u&longs;pe&longs;a da vn &longs;ol <lb/>ponto. </s></p><p type="head"> <s>VI.<!-- KEEP S--></s></p><p type="main"> <s>E leua diciamo la linea &longs;o&longs;tenuta da due ponti, o &longs;o­<lb/>&longs;tenuta da vn ponto e mo&longs;&longs;a da vna po&longs;&longs;anza. </s></p><p type="head"> <s>VII.<!-- KEEP S--></s></p><p type="main"> <s>Ponto di momento diciamo nella &longs;tatera e leua, il <lb/>ponto, nel quale s'incontra la linea del momento, con <lb/>la linea della &longs;tatera. </s></p><p type="head"> <s>VIII.<!-- KEEP S--></s></p><p type="main"> <s>E ponto di appen&longs;ione: il ponto, onde perde la gra-<pb pagenum="3"/>uezza &longs;taccata dalla &longs;tatera, o leua, nelquale i&longs;te&longs;&longs;o pon­<lb/>to s'intende hauer il &longs;uo momento. </s></p><p type="head"> <s>IX.<!-- KEEP S--></s></p><p type="main"> <s>Et Horizonte de pe&longs;i la &longs;uperficie in cui le linee de <lb/>momenti tutte vanno perpendicolarmente. </s></p><p type="head"> <s><emph type="italics"/>Appendice.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Dalche è manife&longs;to, che l'Horizonte de' momenti pa­<lb/>ralleli, &longs;ia &longs;uperficie piana: e delli concorrenti &longs;ia &longs;uper­<lb/>cie sferica. </s></p><p type="head"> <s><emph type="italics"/>POSITIONI.<emph.end type="italics"/><lb/>I.</s></p><p type="main"> <s>Pigliamo nelli corpi egualmente di&longs;te&longs;i il centro del <lb/>pe&longs;o e&longs;&longs;er nella &longs;uperficie, che diuide egualmente la <lb/>lunghezza di detto corpo. <!-- KEEP S--></s></p><p type="head"> <s>II.<!-- KEEP S--></s></p><p type="main"> <s>Che grauezze eguali appe&longs;e o nell'i&longs;te&longs;&longs;o ponto, o in <lb/>ponti della libra egualmente di&longs;tanti dalla &longs;u&longs;pen&longs;ione <lb/>della &longs;tatera, habbiano momento eguale. </s></p><p type="head"> <s>III.<!-- KEEP S--></s></p><p type="main"> <s>Che nelli corpi di vna i&longs;te&longs;&longs;a natura &longs;ia proportionale <lb/>il pe&longs;o alla quantità delli corpi. </s></p><p type="head"> <s>IIII.<!-- KEEP S--></s></p><p type="main"> <s>E, che la grauezza appe&longs;a non &longs;i fermi, &longs;in che il <expan abbr="c&etilde;-tro">cen­<lb/>tro</expan> del pe&longs;o non &longs;ia nella perpendicolare del ponto del <lb/>&longs;o&longs;tenimento. </s></p><pb pagenum="4"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Se &longs;i togliono due quantità da due altre, che &longs;iano <lb/>eguali, e tra di loro, & alla compo&longs;ta delle due tolte: di­<lb/>co che le re&longs;tanti alle tolte &longs;cambieuolmente &longs;ono egua­ <lb/>li. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Siano le due quantità, A & B, & alla compo&longs;ta di ambe &longs;iano egua <lb/>li, la C D, & la E F; e dalla C D, toglia&longs;i eguale ad A, che &longs;ia, <lb/>C G, e dalla E F toglia&longs;i eguale a B, che &longs;ia E H. <!-- KEEP S--></s> <s>dico che la re&longs;tan­<lb/>te H F, è vguale ad A; e la G D, eguale a B. </s> <s>Si mo&longs;tra perciò <lb/>che e&longs;&longs;endo C D, eguale ad A e B in&longs;ieme: tolti dall'vna e l'altra &longs;um­<lb/>male A, e C G eguali: le re&longs;tanti, B, e G D di con&longs;eguenza &longs;o­<lb/>no eguali. </s> <s>Similmente perche la E F &longs;i pone vguale alle A, & B <lb/>gionte in&longs;ieme; tolte la E H, & B vguali: le re&longs;tanti, H F, e A &longs;o­<lb/>no di con&longs;eguenza eguali. </s> <s>è adunque la H F eguale a C G: e la G D <lb/>eguale ad E H. <!-- KEEP S--></s> <s>il che hauea da mo&longs;trar&longs;i.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Dalche è manife&longs;to, che le i&longs;te&longs;&longs;e re&longs;tanti &longs;cambieuol­ <lb/>mente &longs;ono proportionali alle tolte. </s></p><pb pagenum="5"/><p type="main"> <s>Percioche e&longs;&longs;endo le C G H F eguali. </s> <s>e le G D E H <lb/>anco eguali: ma le eguali &longs;ono proportionali: &longs;ono <expan abbr="dũque">dunque</expan> <lb/>come C G ad E H, co&longs;i H F ad G D: ilche hauea da mo­<lb/>&longs;trar&longs;i. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>II.<!-- KEEP S--></s></p><p type="main"> <s>Se alla linea della &longs;tatera &longs;i applicano continuatamen­<lb/>te due corpi: li centri delli corpi applicati, &longs;ono di&longs;tanti <lb/>dal centro ditutto il compo&longs;to, di di&longs;tanze proportio­<lb/>nali alli pe&longs;i, pigliati reciprocamente. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la linea della &longs;tatera, A B, l'vn delli corpi applicati &longs;ia <lb/>B C, l'altro &longs;ia C A, e l'applicatione del corpo B C occupi la parte di li­<lb/>nea B D, e del corpo C A, la parte A D, e diuida&longs;i B D, in par­<lb/>ti vguali al ponto E: & A D in parti eguali il ponto F: è manifesto <lb/>che del corpo applicato à B D, il ponto del momento &longs;ia E, e del cor­<lb/>po applicato a D A, il ponto del momento &longs;ia F, dico che diui&longs;a B A <lb/>tutta per metà nel ponto G, che è ponto di <expan abbr="mom&etilde;to">momento</expan> <lb/>della grauezza tut­ <lb/>ta compo&longs;ta di ambedue: c'habbia la di&longs;tanza F G a G E la ragione <lb/>che'lpe&longs;o di B C al pe&longs;o di C A. <!-- KEEP S--></s> <s>Si mo&longs;tra percioche la ragione del <lb/>pe&longs;o di B C, al pe&longs;o di C A, e l'i&longs;te&longs;&longs;a che delli corpi: e delli corpi<emph.end type="italics"/><pb pagenum="6"/><emph type="italics"/>vgualmente di&longs;te&longs;i, e l'i&longs;te&longs;&longs;a che delle linee: qual è della linea B D <lb/>a D A. <!-- KEEP S--></s> <s>e delle loro metà di E D a D F cioè di F G, a G E: <lb/>& perche &longs;e due quantità compongono quantità, e le metà del­<lb/>le componenti, compongono la metà della tutta: ma le metà delle li­<lb/>nee componenti &longs;ono A F, e B E, la metà della tutta, e co&longs;i la B G <lb/>come la A G. <!-- KEEP S--></s> <s>perciò togliendo due quantità A F B E dalle due, <lb/>A G, G E eguali tra di loro, & alla compo&longs;ta di A F, B E. <!-- KEEP S--></s><lb/> <s>le re&longs;tanti &longs;cambieuolmente &longs;ono proportionali, e perciò F G, a G E <lb/>&longs;arà nell'i&longs;te&longs;&longs;a ragione di B E, ad A F, cioè della doppia, B D, a D <lb/>A: qual è l'i&longs;te&longs;&longs;a del corpo, B C, a C A: e della grauezza di B C, a <lb/>C A. <!-- KEEP S--></s> <s>la di&longs;tanza dunque F G, alla di&longs;tanza E G, ha la ragione che'l <lb/>pe&longs;o di B C, al pe&longs;o di C A, il che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>III.<!-- KEEP S--></s></p><p type="main"> <s>Se ad vn vna &longs;tatera &longs;iano appe&longs;e due grauezze, e l' <lb/>interuallo delli ponti della &longs;o&longs;pen&longs;ione &longs;i diuida nella <lb/>ragione delle grauezze: &longs;o&longs;pe&longs;a la &longs;tateradal ponto del­<lb/>la diui&longs;ione, &longs;ta in equilibrio. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la &longs;tatera A B: le grauezze in e&longs;&longs;a &longs;o&longs;pe&longs;e: C, & D: la C, dal <lb/>ponto, A, & la D, dal ponto, ‘B, & in quellaragione che ha la gra­<lb/>uezza D, alla grauezza C, &longs;i diuida A B nelponto E dico che &longs;o&longs;pe&longs;a<emph.end type="italics"/><pb pagenum="7"/><emph type="italics"/>la &longs;tatera nel ponto E, &longs;ta in equilibrio. </s> <s>Si mo&longs;tra alla linea, B E, ta­<lb/>gli&longs;i eguale la linea A F, dunque giunta communemente, F E, &longs;arà B <lb/>F, vguale ad A E, e perciò haurà B F, ad F A, l'i&longs;te&longs;&longs;a ragione, che <lb/>D, a C. <!-- KEEP S--></s> <s>faccia&longs;i alla B F, vguale, B G, & alla A F, vguale A H, <lb/>dunque &longs;e alla linea, G F, s'intenda applicato un corpo eguale di pe&longs;o <lb/>alla grauezza D, e tal corpo &longs;i allunghi nella iste&longs;&longs;a gro&longs;&longs;ezza fin ad H, <lb/>&longs;arà il corpo applicato ad F H, uguale di pe&longs;o a C: percio che hauendo <lb/>G F, ad F H, l'i&longs;te&longs;&longs;a ragione che D, a C, e li corpi applicati l'i&longs;te&longs;­<lb/>&longs;a delle linee: &longs;ono perciò come la grauezza D alla C, co&longs;i il cor­<lb/>po applicato ad F G, al corpo applicato ad H F: dunque mutando, &longs;o­<lb/>no anco proportionali: ma il corpo applicato a G F, è di pe&longs;o uguale al <lb/>D, dunque l'applicato ad F H è vguale di pe&longs;o a C: & è delli due ap­<lb/>plicati, il commune punto di momento in E. <!-- KEEP S--></s> <s>Dunque delli D C in&longs;ie­<lb/>me pigliati il commun momento è nel ponto i&longs;te&longs;&longs;o: & percio la &longs;tatera <lb/>&longs;o&longs;tenuta in E, &longs;ta in equilibrio, ilche &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice.<emph.end type="italics"/> I.<!-- KEEP S--></s></p><p type="main"> <s>Dal che è manife&longs;to che'l centro commune di due <lb/>pe&longs;i è il ponto che diuide l'interuallo de'centri loro, re­<lb/>ciprocamente. </s></p><p type="head"> <s><emph type="italics"/>Appendice.<emph.end type="italics"/> II.<!-- KEEP S--></s></p><p type="main"> <s>E &longs;e due grauezze diui&longs;amente &longs;i appendono: che di­<lb/>ui&longs;o l'interuallo nella ragione delle grauezze recipro­<lb/>camente: dette grauezze, fanno l'i&longs;te&longs;&longs;o effetto nel mo­ <lb/><expan abbr="m&etilde;to">mento</expan>, che &longs;e in detto ponto <expan abbr="giuntam&etilde;te">giuntamente</expan> fu&longs;&longs;ero appe&longs;e. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>IIII.<!-- KEEP S--></s></p><p type="main"> <s>Se due grauezze appe&longs;e in due ponti facciano equi­<lb/>pondio: e di nuouo appe&longs;e in due altri ponti facciano <pb pagenum="8"/>equipondio; l'interualli delle &longs;o&longs;pen&longs;ioni mutate, &longs;ono <lb/>proportionali con li pe&longs;i reciprocamente. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la &longs;tatera A B: il ponto della &longs;o&longs;pen&longs;ione C, li ponti onde &longs;one <lb/>appe&longs;e le grauezze che fanno equipondio A & B. le grauezze appe&longs;e <lb/>D & E. <!-- KEEP S--></s> <s>Quali di nuouo appe&longs;e nelli ponti F & G faccino equipondio: <lb/>dico che la F A interuallo delle due &longs;o&longs;pen&longs;ioni di D, a B G, inter­<lb/>uallo delle <expan abbr="&longs;u&longs;p&etilde;&longs;ioni">&longs;u&longs;pen&longs;ioni</expan> di E; ha quella ragione che la grauezza c alla gra­<lb/>uezza D. <!-- KEEP S--></s> <s>Si mo&longs;tra perche D et E grauezze nella <expan abbr="&longs;u&longs;p&etilde;&longs;ion">&longs;u&longs;pen&longs;ion</expan> prima han­<lb/>no equipondio: dunque la ragione della grauezza D ad E, è l'i&longs;te&longs;&longs;a che <lb/>di B C a C A: e nella &longs;econda &longs;u&longs;pen&longs;ione la ragione di D ad E e l'i&longs;te&longs;­<lb/>&longs;a che di G C a C F. <!-- KEEP S--></s> <s>e perciò come B C à C A, co&longs;i G C à C F, e per che <lb/>da due &longs;i togliono due altre nell'i&longs;te&longs;&longs;a ragione, le re&longs;tanti anco &longs;ono nel­<lb/>l'i&longs;te&longs;&longs;a ragione. </s> <s>è dunque B G ad F A, come D ad E, ilche hauea da <lb/>mo&longs;trar&longs;i.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>V.<!-- KEEP S--></s></p><p type="main"> <s>Se due grauezze facciano equipondio, e gionte ò tol­<lb/>te due altre grauezze facciano anco equipondio: le gion­ <lb/>te ancora ò le tolte &longs;ono nell'i&longs;te&longs;&longs;a raggione. </s></p><pb pagenum="9"/><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la &longs;tatera A B: il ponto della &longs;u&longs;pen&longs;ione C: le grauezze appe­<lb/>&longs;e D et E: che facciano equipondio: e di nouo aggiuntoui due altre F e G <lb/>facciano anco equipondio. </s> <s>dico che la grauezza F a G, ha la ragione <lb/>che D ad E: qual'è l'i&longs;te&longs;&longs;a che di B C a C. A. <!-- KEEP S--></s> <s>&longs;i mo&longs;tra perche D & E, <lb/>fanno equipondio. </s> <s>& F e G fanno equipondio: perciò &longs;arà, come B C <lb/>à C A co&longs;i D F ad E G e nell'i&longs;te&longs;&longs;a era D ad E dunque le re&longs;tanti F e G <lb/>&longs;ono anco nell'i&longs;te&longs;&longs;a ragione: e non altrimente che nella &longs;uppo&longs;ition del <lb/>la compo&longs;ta, &longs;i mo&longs;tra nella &longs;uppo&longs;ition delli re&longs;idui. </s> <s>Ha&longs;&longs;i dunque l'in­<lb/>tento.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>VI.<!-- KEEP S--></s></p><p type="main"> <s>Date quante &longs;i voglia grauezze appe&longs;e in vn'i&longs;te&longs;&longs;a <lb/>&longs;tatera, ritrouare il ponto del momento commune. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la &longs;tatera A B dalli cui ponti A e B &longs;iano &longs;o&longs;pe&longs;e le grauezze C D <lb/>e &longs;iano in altri ponti &longs;o&longs;pe&longs;i altri pe&longs;i, come E nel ponto F: &longs;i cerca il pon­ <lb/>to del momento commune. </s> <s>diuida&longs;i la B A nella ragione di C a D reci­<lb/>procamente &longs;e dunque il detto punto uiene in F e&longs;&longs;endo F il ponto del<emph.end type="italics"/><pb pagenum="10"/><figure id="fig2"/><lb/><emph type="italics"/>momento delle C D pigliate in&longs;ieme, &longs;arà ponto di momento commu <lb/>ne delle grauezze C E D, tutte. </s> <s>Et harra&longs;&longs;i l'intento.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Ma &longs;e'l dato ponto ca&longs;chi altroue come in H, perche le grauezze <lb/>D, & C appe&longs;e in A e B fanno l'i&longs;te&longs;&longs;o effetto che &longs;e giuntamente fu&longs;&longs;e­ <lb/>ro appe&longs;e in H: perciò &longs;e quella ragione che hà il compo&longs;to di C D <lb/>ad E habbia reciprocamente F G a G H, &longs;arà G ponto di momento <lb/>commune di tutti. </s> <s>con l'i&longs;te&longs;&longs;o ordine &longs;i ritrouerà il centro di quante <lb/>altre &longs;i uogliano, il che &longs;i hauea da trouare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE<emph.end type="italics"/><lb/>VII.</s></p><p type="main"> <s>Delle grauezze che fanno <expan abbr="equipõdio">equipondio</expan>, compo&longs;te le ra­ <lb/>gioni delle grauezze e delle di&longs;tanze, li e&longs;tremi termini <lb/>&longs;ono eguali. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la &longs;tatera A B il ponto del &longs;ostenimento C le due grauezze che <lb/>fanno <expan abbr="equipõdio">equipondio</expan> D & E: de quali la D &longs;ia &longs;o&longs;pe&longs;a dal ponto A la E dal <expan abbr="põ">pom</expan> <lb/>to B: dico che compo&longs;tala ragione della grauezza D ad E: e della <expan abbr="di&longs;tãza">di&longs;tanza</expan> <lb/>A C a C B: cioè fatto che la <expan abbr="quãtità">quantità</expan> F a G &longs;ia come la grauezza D ad E e <lb/>la quantità G ad H come la di&longs;tanza A C alla C B, che F & H<emph.end type="italics"/><pb pagenum="11"/><figure id="fig3"/><lb/><emph type="italics"/>e&longs;tremi termini &longs;iano uguali. </s> <s>&longs;i mo&longs;tra: perche A C a C B &longs;i è po&longs;ta co­<lb/>me G ad H: dunque riuoltando H à G, è come B C à C A. e per l'e­<lb/>quipondio, come la di&longs;tanza B C a C A co&longs;i la grauezza D ad E, & <lb/>come D ad E co&longs;i &longs;i è pigliato F a G: dnnque F a G e come B C a C A, <lb/>e nell'i&longs;te&longs;&longs;a ragione era H a G. <!-- KEEP S--></s> <s>hanno dunque li due termini F et H l'i­<lb/>&longs;te&longs;&longs;a ragione al termine G. e perciò li F & H &longs;ono eguali tra di loro: <lb/>il che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE<emph.end type="italics"/><lb/>VIII.</s></p><p type="main"> <s>Li momenti delle grauezze uguali, appe&longs;e in di&longs;tan­ <lb/>ze ineguali, hanno fra di loro la proportione che le di­<lb/>&longs;tanze. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la &longs;tatera A B, il ponto del &longs;o&longs;tenimento C, le grauezze uguali D <lb/>& E. <!-- KEEP S--></s> <s>de quali il D &longs;ia appe&longs;o in A, & l'c in F. <!-- KEEP S--></s> <s>dico che il momento <lb/>di D al momento di E, hà quella ragione che l'interuallo di A C all'­<lb/>interuallo di F C. <!-- KEEP S--></s> <s>&longs;i mo&longs;tra, pigliato dall' altra parte del &longs;e&longs;tem­<lb/>mento C, qual &longs;i uoglia ponto B: intenda&longs;i in e&longs;&longs;a appe&longs;e due grauzze,<emph.end type="italics"/><pb pagenum="12"/><figure id="fig4"/><lb/><emph type="italics"/>vna che faccia <expan abbr="equipōdio">equipondio</expan> a D & &longs;ia G: et vn'altra che faccia <expan abbr="equipōdio">equipondio</expan> <lb/>ad E. <!-- KEEP S--></s> <s>& &longs;ia H. <!-- KEEP S--></s> <s>perche dunque G a D ha quella ragione che A C a C B <lb/>& D ouero E ad H, hala ragione di B C a C F. <!-- KEEP S--></s> <s>dunque di pari il pri­<lb/>mo termine A C all'ultimo F C, ha quella ragione, che il primo ter­<lb/>mine G, al terzo H. <!-- KEEP S--></s> <s>&longs;e dunque G ad H hal'i&longs;te&longs;&longs;a ragione che la di&longs;tan­ <lb/>za A C alla di&longs;tanza F C: & il momento di G è uguale al momento <lb/>di D appe&longs;o in A, & il momento di H vguale al momento di E appe&longs;o <lb/>in F. <!-- KEEP S--></s> <s>dunque il momento di D al momento di E ha quella ragione che <lb/>la di&longs;tanza A C alla di&longs;tanza F C. <!-- KEEP S--></s> <s>il che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>IX.</s></p><p type="main"> <s>Li momenti delle grauezze &longs;o&longs;pe&longs;e in qual &longs;i uoglia <lb/>ponti della &longs;tatera, han tra di loro la ragion compo&longs;ta, <lb/>della ragion delle grauezze, e delle di&longs;tanze. </s></p><p type="head"> <s><emph type="italics"/>Dimostratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la &longs;tatera A B il <expan abbr="pōto">ponto</expan> del <expan abbr="&longs;o&longs;tenim&etilde;to">&longs;o&longs;tenimento</expan> C le grauezze appe&longs;e D dal <lb/><expan abbr="pōto">ponto</expan> A, & E dal ponto F. <!-- KEEP S--></s> <s>dico che la ragione del <expan abbr="mom&etilde;to">momento</expan> D al <expan abbr="mom&etilde;">momen</expan> <lb/>to E, e compo&longs;ta di due ragioni cioè della ragione della grauezza D <lb/>alla grauezza E, e della di&longs;tanza di A C alla F C. <!-- KEEP S--></s> <s>&longs;i mo&longs;tra appenda <lb/>&longs;i da B la grauezza G che faccia equipondio. </s> <s>a D, & il pe&longs;o H che fac­<emph.end type="italics"/><pb pagenum="13"/><figure id="fig5"/><lb/><emph type="italics"/>cia <expan abbr="equipōdio">equipondio</expan> all' E: dico prima che la grauezza G alla grauezza H G ha <lb/>la ragion compo&longs;ta, di D ad E, e di A C ad F C. <!-- KEEP S--></s> <s>per il che da mo&longs;trare: in|­<lb/>tenda&longs;i nell' A &longs;o&longs;pe&longs;a la grauezza I uguale alla grauezza E, è manife&longs;to <lb/>che'l momento I al momento E, hà quella ragione che l'interuallo <lb/>A C all'interuallo F C come nel pa&longs;&longs;ato habbiamo mo&longs;trato: & il mo­<lb/>mento di D al momento d'I hà la ragione che la grauezza D alla gra­<lb/>uezza I: perche &longs;ono da un'i&longs;te&longs;&longs;o ponto &longs;o&longs;pe&longs;i. </s> <s>e&longs;&longs;endo dunque tre ter <lb/>mini in continua habitudine il momento D, il momento I, & il <expan abbr="mom&etilde;­">momen­</expan> <lb/>to E: la ragione del primo termine al terzo è compo&longs;ta della ragione <lb/>di primo a &longs;econdo e della ragione di &longs;econdo a terzo: ma di primo <lb/>a &longs;econdo è di grauezza a grauezza: di&longs;econdo a terzo è d'interuallo <lb/>ad'interuallo. </s> <s>dunque, la ragione delli <expan abbr="mom&etilde;ti">momenti</expan> di D ad E, che è l'i&longs;te&longs;&longs;a <lb/>che della portione G alla portione H: è compo&longs;ta della ragione delle <lb/>grauezze e della ragione delle di&longs;tanze. </s> <s>Il che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>X.</s></p><p type="main"> <s>Data qual &longs;i uoglia grauezza, e li ponti della &longs;o&longs;pen|­<lb/>&longs;ion della &longs;tatera, e della grauezza: e dato il pe&longs;o del <lb/>marco, ritrouare il luogo, oue detto marco faccia e­<lb/>quipondio con la grauezza data. </s></p><pb pagenum="14"/><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la &longs;tatera A B il ponto della &longs;o&longs;pen&longs;ione C, la grauezza data D, <lb/>qual poniamo che &longs;i &longs;o&longs;penda in A: il marco dato di pe&longs;o E: &longs;i cerca il <lb/>ponto oue detto marco appe&longs;o faccia equipondio. </s> <s>per que&longs;to: faccia&longs;i <lb/>che quella ragione che hà il pe&longs;o E al pe&longs;o D, quella habbia la linea A <lb/>C a C F, dico che appe&longs;o il marco in F fa equipondio, cioè che'l pon|­<lb/>to del momento commune delle grauezze D & e&longs;ia il ponto della &longs;o­<lb/>&longs;pen&longs;ione C: il che è manife&longs;to, percioche &longs;ono li pe&longs;i reciprochi al­<lb/>le di&longs;tanze. </s> <s>Ha&longs;&longs;i dunque l'intento.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>XI.</s></p><p type="main"> <s>Data una &longs;tatera, a cui &longs;ia ugualmente applicato un <lb/>corpo, e data una grauezza &longs;o&longs;pe&longs;a da un dato ponto, <lb/>e dato il pe&longs;o del marco, ritrouare il ponto onde detto <lb/>marco &longs;o&longs;pe&longs;o faccia equipondio con la grauezza. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la &longs;tatera A B il ponto del &longs;o&longs;tenimento C, il corpo applicato <lb/>A D E B, la grauezza &longs;o&longs;pe&longs;a H I; il ponto onde la grauezza è appe­<lb/>&longs;a in A: il marco L, &longs;i cerca il ponto onde &longs;o&longs;pe&longs;o il marco, faccia e­<lb/>quipondio con H I, Per que&longs;to: faccia&longs;i alla linea A C uguale la C F,<emph.end type="italics"/><pb pagenum="15"/><figure id="fig6"/><lb/><emph type="italics"/>dunque il corpo D F applicato ad A F &longs;tà in equilibrio nel ponto della <lb/>&longs;o&longs;pen&longs;ione C. et diui&longs;a F B re&longs;tante per metà nel ponto G: del re&longs;tan­<lb/>te corpo F E applicato alla linea F B, &longs;arà G, il ponto di momento. </s> <s>&longs;e <lb/>dunque la ragione che hà C F ad F G, habbia la grauezza E F alla <lb/>parte del pe&longs;o I, &longs;tarà il corpo F E in <expan abbr="equipōdio">equipondio</expan> con I, e perciò &longs;e di nuo|­<lb/>uo la ragione che hà, il marco al re&longs;tante H habbia la parte de &longs;tate <lb/>ra A C, a C M, &longs;o&longs;pe&longs;o il marco L da M, farà equipondio <lb/>con H: & il corpo F E facea equipondio con I: &longs;tarà dunque ogni co&longs;a <lb/>in equilibrio. </s> <s>&longs;i è dunque ritrouato il ponto M, onde &longs;o&longs;pe&longs;o il marco <lb/>faccia equipondio con la grauezza data. </s> <s>Il che &longs;i hauea da ritrouare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>XII.</s></p><p type="main"> <s>Fatta alla linea della &longs;tatera application di corpo, e <lb/>&longs;o&longs;pe&longs;e in e&longs;&longs;a più grauezze che &longs;o&longs;tentino un pe&longs;o, ri|­<lb/>trouare cia&longs;cuna grauezza quanto portion di pe&longs;o &longs;o­<lb/>&longs;tenti. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la linea della &longs;tatera A B il ponto del &longs;o&longs;tentimento C & alla<emph.end type="italics"/><pb pagenum="16"/><figure id="fig7"/><lb/><emph type="italics"/>linea A B &longs;ia fatta application di corpo et in e&longs;&longs;a appe&longs;e le grauezze D <lb/>E, F: D in A, E in G, F in H: e dette grauezze &longs;o&longs;tentine il pe&longs;o I <lb/>K L M: il cui momento &longs;ia nel ponto B: &longs;i cerca cia&longs;cuna di dette gra­<lb/>uezze D, E, F, quanta portione di pe&longs;o &longs;o&longs;tenti. </s> <s>faccia&longs;i per que&longs;to alla <lb/>linea B C uguale la C N: e la re&longs;tante N A &longs;i diuida in parti uguali nel <lb/><expan abbr="pōto">ponto</expan> O, e quella ragione che hà B C a C O quell'habbia il corpo della &longs;ta|­<lb/>tera applicato ad N A ad M: &longs;arà <expan abbr="dũque">dunque</expan> <expan abbr="equipõderãte">equiponderante</expan> <expan abbr="cõ">com</expan> M: c la par <lb/>te applicata ad N C è <expan abbr="equiponderãte">equiponderante</expan> alla applicata à B C: <expan abbr="dũque">dunque</expan> il cor­<lb/>po della &longs;tatera &longs;tà in <expan abbr="equipõdio">equipondio</expan> con la portione del pe&longs;o M: e le ragioni <lb/>delle grauezze D, E, F, e delle di&longs;tanze A C, G C, H C, cioè la ragio­<lb/>ne della grauezza Dad F con la ragione della di&longs;tanza A C a G C, <lb/>compongon la ragion di P a <expan abbr="q.">que</expan> & la ragione della grauezza E ad F, <lb/>con la ragione della di&longs;tanza G C ad H C, compongon la ragione<emph.end type="italics"/><pb pagenum="17"/><emph type="italics"/>di Q ad R, & in quella ragione che &longs;ono le tre quantità, P Q R, <lb/>po&longs;te in continua habitudine, nella i&longs;te&longs;&longs;a &longs;i di&longs;tribui&longs;ca il pe&longs;o I K L: <lb/>è manife&longs;to per quel che &longs;i è visto, che, D fa equiponderanza con I, lo <lb/>E co'l K, e lo F con lo L: ilche &longs;i cercaua.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>XIII.</s></p><p type="main"> <s>La &longs;tatera di grauezze appe&longs;e, che facciano equipon|­<lb/>dio: quantunque dal &longs;ito orizontale mo&longs;&longs;a &longs;i &longs;tà. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimostratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la &longs;tatera nel &longs;ito orizontale A B, il ponto della &longs;o&longs;pen&longs;ione C, <lb/>li pe&longs;ie &longs;ue centri D & E, il centro commune di ambe le grauezze F; <lb/>e mo&longs;&longs;a la statera del &longs;ito orizontale, pa&longs;&longs;i il ponto A in G, il B in H,<emph.end type="italics"/><pb pagenum="18"/><emph type="italics"/>&longs;i che habbia la &longs;tatera la po&longs;itione di G C H: li pe&longs;i e &longs;ui centri di, I e <lb/>K: dico, che la &longs;tatera G H &longs;tarà, e non &longs;i mouerà di &longs;ito. </s> <s>&longs;i mo&longs;tra <lb/>percioche e&longs;&longs;endo la grauezza I appe&longs;a, inalzata, il centro &longs;uo gi­<lb/>rando verrà nella perpendicolare del ponto della &longs;o&longs;pen&longs;ione: e perciò <lb/>I, verrà nella perpendicolare del ponto G e K del ponto H. <!-- KEEP S--></s> <s>&longs;ono <lb/>dunque, G I H K parallele. </s> <s>e perche il centro commune de pe&longs;i, diui­<lb/>de nell'i&longs;te&longs;&longs;a ragione la I K, & la D E, e&longs;&longs;endo la ragione delli pe&longs;i <lb/>vn'i&longs;te&longs;&longs;a, & la C F nell'vna, e nell'altra &longs;o&longs;pen&longs;ione perpendicolare, <lb/>e parallela, co&longs;i alle A D E B, come alle G I, K H. <!-- KEEP S--></s> <s>perciò diuidendo <lb/>C F perpendicolare &longs;imilmente la D E, & la I K: &longs;arà il ponto F luo­<lb/>go del centro nell'vna, luogo anco di centro nell'altra. </s> <s>e&longs;&longs;endo dunque <lb/>il centro del pe&longs;o commune nella perpendicolare della &longs;o&longs;pen&longs;ione, &longs;ta­<lb/>rà. </s> <s>Ilche &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>XV.</s></p><p type="main"> <s>La &longs;tatera di grauezze attaccate, che facciano equi­<lb/>pondio, &longs;e'l ponto della &longs;o&longs;pen&longs;ione, non &longs;ia nella linea <lb/>delli centri: mo&longs;&longs;a dal &longs;ito orizontale non &longs;tarà, ma ri­<lb/>tornarà nell'i&longs;te&longs;&longs;o. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la linea della &longs;tatera, che &longs;tia nel &longs;ito horizontale A B, li pe­<lb/>&longs;i attaccati, & li lor centri C e D, e diuida&longs;i G D &longs;econdo li pe&longs;ireci-<emph.end type="italics"/><pb pagenum="19"/><emph type="italics"/>pro camente nel ponto E: è manife&longs;to che'l ponto E &longs;ia il centro com­<lb/>mune di ambi li pe&longs;i, e che mentre la &longs;tatera &longs;ta, che &longs;ia detto centro <lb/>nella perpendicolare, che cala dal ponto F. <!-- KEEP S--></s> <s>perche dunque li pe&longs;i &longs;ono <lb/>alla statera affi&longs;&longs;i, e non mutano li centri po&longs;itura con la linea A B, e <lb/>&longs;empre fanno con e&longs;&longs;a angoli retti le C A, D B, E F, perciò mo&longs;&longs;a <lb/>la &longs;tatera dal &longs;ito horizontale, non &longs;arà E centro <expan abbr="cōmune">commune</expan> nella perpen­<lb/>dicolare della &longs;o&longs;pen&longs;ione: ma girando v&longs;cirà di detta perpendicolare, <lb/>e perciò la &longs;tatera non &longs;tarà, &longs;in che di nuouo il detto ponto non torne <lb/>nella perpendicolare.<emph.end type="italics"/></s></p><figure/><pb pagenum="20"/><p type="head"> <s>VETTE, E <lb/>LEVA.<!-- KEEP S--></s></p><p type="head"> <s><emph type="italics"/>DEFINITION.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Vette diciamo la linea, che &longs;o&longs;tiene grauezza, <lb/>qual &longs;ia nelli &longs;ue ponti e&longs;tremi &longs;o&longs;tenuta. </s></p><p type="head"> <s><emph type="italics"/>DEFINITION.<emph.end type="italics"/><lb/>II.<!-- KEEP S--></s></p><p type="main"> <s>Et altrimente, vette motiua e leua, la linea che &longs;o­<lb/>&longs;tenga grauezza, &longs;tabilita in vn ponto che &longs;otto leua <lb/>diciamo, & in vn'altro ponto da po&longs;&longs;anza, o mo&longs;&longs;a, o <lb/>&longs;o&longs;tenuta. </s></p><p type="head"> <s><emph type="italics"/>POSITION.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Mi&longs;uriamo la po&longs;&longs;anza con vna grauezza equiualen|­<lb/>te, o appe&longs;a nell'i&longs;te&longs;&longs;o ponto della po&longs;&longs;anza, o nell'al­<lb/>tro ponto egualmente dal &longs;ottoleua di&longs;co&longs;to. </s></p><p type="head"> <s><emph type="italics"/>POSITION.<emph.end type="italics"/><lb/>II.<!-- KEEP S--></s></p><p type="main"> <s>Cia&longs;cuna po&longs;&longs;anza in quanto &longs;o&longs;tiene, e&longs;&longs;ere egua­<lb/>le al pe&longs;o &longs;o&longs;tenuto. </s></p><figure/><pb pagenum="21"/><p type="head"> <s><emph type="italics"/>PROPOSITION.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>S'il &longs;ottoleua &longs;tia tra la grauezza, e la po&longs;&longs;anza che <lb/>&longs;o&longs;tenga detta grauezza; &longs;arà tra la po&longs;&longs;anza, & il pe­<lb/>&longs;o la ragione, che è tra le parti della leua, reciprocamen|­<lb/>te. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la linea A B, il ponto del &longs;ottoleua C, la grauezza D &longs;oste­<lb/>nuta nel ponto della leua A; la po&longs;&longs;anza che &longs;o&longs;tenga detta graue z­<lb/>za in B: dico che la po&longs;&longs;anza B al pe&longs;o D, ha quella ragione che ha <lb/>la parte di leua A C alla C B, qual è ragion reciproca. </s> <s>&longs;i mo&longs;tra: inten|­<lb/>da&longs;i attaccato in B il pe&longs;o che faccia equipondio con D: è manife&longs;to che <lb/>detto pe&longs;o E &longs;ia equiualente alla forza B, ma il pe&longs;o E al pe&longs;o Dhala <lb/>ragione che A C a C B, che è la ragione reciproca di grauezza, e di­<lb/>&longs;tanze: dunque, la potenza ancora haue l'i&longs;te&longs;&longs;a ragione. </s> <s>ilche&longs;i ha­<lb/>uea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITION.<emph.end type="italics"/><lb/>II.<!-- KEEP S--></s></p><p type="main"> <s>Se due potenze &longs;o&longs;tentino vna grauezza con vn vet­<lb/>te, cia&longs;cuna &longs;o&longs;tentarà la &longs;ua portione, &longs;econdo l'inter­<lb/>uallo del pe&longs;o dalle potenze, pigliato reciprocamente. </s></p><pb pagenum="22"/><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia il vette A B, dal cui ponto D, penda il pe&longs;o C: le potenze che <lb/>&longs;o&longs;tengono dette grauezze &longs;iano A & B: dico che'l B, e lo A &longs;o&longs;ten­<lb/>tano portioni proportionali all'interualli reciprocamente: cio è che <lb/>quella ragione c'ha l'interuallo, B D, a D A, quella hàbbia la por­<lb/>tione &longs;o&longs;tentata dall' A, alla portione &longs;o&longs;tentata dal B, &longs;i dimo&longs;tra: <lb/>tagli&longs;i ad A D uguale B E, accoppiata dunque communemente la D <lb/>E, &longs;arà A E uguale a B D: ag giunga&longs;i all' A e la A G, che le &longs;ia egua­<lb/>le, & ad E B la B F che &longs;imilmente le &longs;ia eguale. </s> <s>&longs;arà di tutta la G F, <lb/>il ponto mezzano D, & della G E, il ponto mezzano A, & della E <lb/>F, il ponto mezzano B. </s> <s>applicata dunque a tutta la G F, una grauez­<lb/>za che &longs;ia uguale a C, &longs;arà di detta grauezza il ponto di momento in D <lb/>& &longs;arà equiualente nella &longs;ua operatione alla grauezza C, & di e&longs;&longs;a <lb/>la parte applicata a G E ha il &longs;uo momento in A, c la parte applica­<lb/>ta ad E ha il &longs;uo momento in B. </s> <s>dunque della grauezza applicata <lb/>la potenza A, ne &longs;o&longs;tentarà la portione applicata a G E: e la potenza <lb/>B, la portione applicata ad E F. <!-- KEEP S--></s> <s>Ma G E ad E F, ha la ragione che <lb/>l'interuallo B D, a D A che è reciproca. </s> <s>dunque le potenze &longs;o&longs;tenta­<lb/>no le portioni de'pe&longs;i proportionali, reciprocamente pigliate con l'inter <lb/>ualli. </s> <s>ilche &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><pb pagenum="23"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>III.<!-- KEEP S--></s></p><p type="main"> <s>Se il &longs;ottoleua &longs;ia fuori della grauezza, e della po&longs;­<lb/>&longs;anza, &longs;arà la ragion della po&longs;&longs;anza alla grauezza l'i&longs;te&longs; <lb/>&longs;a, che dell'interualli da e&longs;se al &longs;ottoleua <expan abbr="reciprocam&etilde;-te">reciprocamen­<lb/>te</expan> pigliati </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la leua A B il &longs;otto lcua A, la grauezza C, il &longs;uo momento in <lb/>D, la po&longs;&longs;anza che &longs;o&longs;tiene in B: dico che la po&longs;&longs;anza alla grauezza <lb/>ha la ragione, che D A ad A B, che è la ragion delle di&longs;tanze piglia­<lb/>te dal &longs;ottoleua reciprocamente: &longs;i mo&longs;tra: perche il pe&longs;o C, e &longs;o&longs;ten­<lb/>tato dalla leua B A, e la leua è &longs;o&longs;tentata in due ponti B & A. <!-- KEEP S--></s> <s>dunque <lb/>il pe&longs;o è &longs;o&longs;tentato dalle potenze in B & A compartitamente, cio<gap/><lb/>la po&longs;&longs;anza B &longs;o&longs;tenta tal portion di pe&longs;o, qual'è la di&longs;tanza A D di A <lb/>B, & A, tal portione qual'è D B, di B A, e perche la po&longs;&longs;anza &longs;o­<lb/>&longs;tenente è uguale al pe&longs;o che &longs;o&longs;tiene, &longs;ono ambe le po&longs;&longs;anze B & A <lb/>giuntamente pigliate uguali al pe&longs;o E; e la portione &longs;o&longs;tentata da B: <lb/>al tutto harrà quella ragione che la portion della leua D A a tutta <lb/>la leua A B. </s> <s>qual è l'i&longs;te&longs;&longs;a che della di&longs;tanza della grauezza, alla di­<lb/>&longs;tanza della potenza. </s> <s>&longs;i ha dunque l'intento.<emph.end type="italics"/></s></p><pb pagenum="24"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>IV.</s></p><p type="main"> <s>Se vna grauezza &longs;ia con vna leua &longs;o&longs;tenuta da due <lb/>ponti; & accrefciuta la leua dall altra parte &longs;i appenda <lb/>grauezza equiponderante, & &longs;i tra &longs;muti in &longs;tatera: &longs;o­<lb/>itentarà il &longs;o&longs;tenimento in tal commutatione pe&longs;o mag <lb/>giore, quale al pe&longs;o di prima &longs;o&longs;tenuto, ha ragione com <lb/>po&longs;ta della ragione delle portioni di tutta la linea accre <lb/>&longs;ciuta communicanti, alle portioni interuallate: fat­<lb/>te le due diui&longs;ioni al ponto del &longs;ottoleua, & al ponto <lb/>del primo momento. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la leua A B, il &longs;ottoleua in A: lagrauezza &longs;o&longs;tenuta in C, la <lb/>po&longs;&longs;anza che'l &longs;o&longs;tiene in B. & allungata la B A in vn D, appenda­<lb/>&longs;i in D, vna grauezza che &longs;o&longs;tenti la grauezza C. <!-- KEEP S--></s> <s>dico che in que&longs;ta <lb/>commutatione il &longs;ottoleua A &longs;o&longs;tenti pe&longs;o maggiore, & che ilpe&longs;o <lb/>&longs;o&longs;tenuto in detta commutatione, alpe&longs;o &longs;o&longs;tenuto di prima, ha la ra­ <lb/>gion compo&longs;ta delle D C, A D, parti communicanti, alle D A, a C <lb/>B, parti interuallate. </s> <s>&longs;i mo&longs;tra: perche la parte del pe&longs;o &longs;o&longs;tenuto da <lb/>A, a tutto il pe&longs;o C, ha la ragione, che B C a B A: &. </s> <s>il pe&longs;o C, ad <lb/>ambi li pe&longs;i C & D, ha la ragione che D A a D C, ma la ragione del-<emph.end type="italics"/><pb pagenum="25"/><emph type="italics"/>la portione &longs;o&longs;tenuta da A, alla grauezza C, & di C, ad ambe CD, &longs;ot <lb/>trattone il termine mezzano, compongono la ragione della portione &longs;o&longs;te <lb/>nuta da A, ad ambe le C D, & la ragione di B C a BA, & di D A a D <lb/>C, fanno la ragione compo&longs;ta delle parti communicanti alle interuallate. </s><lb/> <s>Ha&longs;&longs;i dunque l'intento: che'l pe&longs;o di prima &longs;o&longs;tenuto, al pe&longs;o &longs;o&longs;tenuto <lb/>dopo la commutatione, ha la ragion compo&longs;ta delle parti interuallate alle <lb/>communicanti. </s> <s>Ilche &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>V.<!-- KEEP S--></s></p><p type="main"> <s>Date nell' e&longs;tremità del vette due po&longs;sanze c'habbia­<lb/>no qual&longs;iuoglia ragione tra di loro; e dato vn pe&longs;o, a det­<lb/>te po&longs;&longs;anze giuntamente pigliate vguale, ritrouare il <lb/>ponto del vette, onde il dato pe&longs;o &longs;o&longs;pe&longs;o, &longs;ia da det <lb/>te po&longs;&longs;anze &longs;o&longs;tenuto. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia il vette AB: le po&longs;&longs;anze nelli ponti A e B, c'habbiano tradi lo­<lb/>ro qual&longs;iuoglia ragione: & il pe&longs;o ad ambe po&longs;&longs;anze giuntamente pi­<lb/>gliate vguale &longs;ia C: &longs;i cerca il ponto, onde detto pe&longs;o &longs;ia da dette <expan abbr="po<32>ã">po<32>am</expan> <lb/>ze &longs;o&longs;tenuto. </s> <s>per il che dico: che &longs;e in quella ragione, c' ha la po&longs;&longs;an-<emph.end type="italics"/><pb pagenum="26"/><emph type="italics"/>za B, &longs;i diuida la vette AB in D, e &longs;ia come la po&longs;&longs;anza Aalla B: così, <lb/>la portione di vette B D a D A: dico che po&longs;to il pe&longs;o C, in D: &longs;arà, <lb/>&longs;o&longs;tenuto da dette po&longs;&longs;anze: percioche grauando il pe&longs;o nelli ponti B: & <lb/>A, cbe &longs;o&longs;tentano compartitamente, &longs;econdo la ragion di BD a DA: <lb/>& hauendo la portion che graua in A, alla portion che graua in B, la <lb/>ragion che B D a D A: qual'è l'i&longs;te&longs;&longs;a che della po&longs;&longs;anza A alla po&longs;&longs;anza <lb/>B: dunqne la portione che graua in, A alla portione che graua in B, <lb/>e come la po&longs;&longs;anza A, alla B: e <expan abbr="permutãdo">permutando</expan> la portion che graua in A, a <lb/>la po&longs;&longs;anza A, &longs;arà come la portione che graua in B alla po&longs;&longs;anza B, <lb/>e componendo li antecedenti, tutto il pe&longs;o C, ad ambe le po&longs;&longs;anze giun <lb/>te, harrà l'i&longs;te&longs;&longs;a ragione che vna advna: ma il pe&longs;o tutto C, è vgua­<lb/>le ad ambe le po&longs;&longs;anze giuntamente pigliate: dunque diui&longs;amente le <lb/>portioni, cia&longs;cuna alla po&longs;&longs;anza oue graua, &longs;arà vguale: e percio &longs;a­<lb/>rà del pe&longs;o &longs;o&longs;tenuto, la portione che graua in A, vguale alla po&longs;&longs;anza <lb/>in A: e la portione che graua in B, vguale alla pò&longs;&longs;anza iu B: e percio <lb/>le po&longs;&longs;anze &longs;o&longs;tentaranno il detto pe&longs;o nel ponto D. <!-- KEEP S--></s> <s>Il che &longs;i hauea da <lb/>mo&longs;trare,<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et è manife&longs;to che in ogni altro ponto del detto vet <lb/>te, il pe&longs;o non &longs;arà &longs;o&longs;tenuto, ma aggrauerà più l'vna ò <lb/>l'aitra po&longs;&longs;anza, ver&longs;o oue &longs;arà portato. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>VI.<!-- KEEP S--></s></p><p type="main"> <s>Se una leua &longs;ia inalzata, oba&longs;&longs;ata &longs;otto l'orizonte: <lb/>& da un ponto fuori di e&longs;&longs;a, &longs;i tireranno due perpendi­<lb/>colari, l'vna ad e&longs;&longs;a leua, e l'altra all'orizonte: faran <lb/>no le due perpendicolari angolo tra di loro, vguale all' <lb/>angolo della leua con l'orizonte. </s></p><pb pagenum="27"/><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la linea orizontale A B, la leua &longs;opra di e&longs;&longs;a inalzata o de&longs;pre&longs;­<lb/>&longs;a A C. il ponto fuori della leua E: da cui &longs;i tirino due perpendicolari <lb/>l'vna alla leua DE, l'altra all'orizonte D F, che &longs;eghi la leua in F, & <lb/>la linea orizontale in G. <!-- KEEP S--></s> <s>dico che l'angolo fatto dalle due D E, D F <lb/>&longs;ia vguale allangolo fatto, dalle due A B, A C: &longs;i mo&longs;tra: percioche <lb/>le due A C, D G, &longs;i &longs;egano nel ponto F, &longs;aranno l'angoli A F G, et D <lb/>F E, d'incontr vguali: e gli angoli ad E & G &longs;ono retti: dunque il tri­<lb/>angolo D F E, è equiangolo al triangolo A F G, e l'angolo F D E, v­<lb/>guale a l'angolo F A G. <!-- KEEP S--></s> <s>Il che &longs;i hauea da mò&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice,<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et è manife&longs;to che e&longs;&longs;endo detto ponto di &longs;opra la li <lb/>nea della leua inalzata, e di &longs;otto della leua ba&longs;&longs;ata; &longs;e­<lb/>cherà detta linea in ponto più dalla po&longs;&longs;anza lontano. </s><lb/> <s>e per <expan abbr="cõtrario">contrario</expan> pigliando&longs;i detto ponto, o &longs;otto dell'alza­<lb/>tà, o&longs;opra della ba&longs;&longs;ata, &longs;egherà in ponti più à detta pos­<lb/>&longs;anza vicini. </s></p><pb pagenum="28"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>VII.<!-- KEEP S--></s></p><p type="main"> <s>Se'l centro del pe&longs;o attaccato ad e&longs;&longs;a leua &longs;ia &longs;opra <lb/>della leua, inalzata la leua, la po&longs;&longs;anza &longs;o&longs;tentarà minor <lb/>pe&longs;o. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la leua A B, a cui &longs;ia attaccata vna grauezza, il cui centro &longs;ia <lb/>C: & intenda&longs;i detta leua in &longs;ito dall'orizonte eleuato: dico che la po­<lb/>tenza B, &longs;o&longs;tenta del pe&longs;o della grauezza minor portione, che nel &longs;ito <lb/>orizontale. </s> <s>&longs;i mo&longs;tra: tirin&longs;i dal ponto C linee, l'vna perpendicolare <lb/>alla leua che &longs;ia C D, & l'altra perpendicolare all'orizonte, che <lb/>&longs;ia C E, che &longs;eghi la leua nel ponto E: è manife&longs;to che'l detto ponto <lb/>&longs;arà più di&longs;co&longs;to dalla po&longs;&longs;anza, e più vicino al ponto del &longs;ottoleua. </s> <s>&longs;e <lb/>dunque per lo ponto C, &longs;i tiri la linea G C F, parallela all'orizonte, & <lb/>per li ponti B & A, lelinee B F, A G, perpendicolari <expan abbr="all'orizõte">all'orizonte</expan> è mani <lb/>fe&longs;to, che l'i&longs;te&longs;&longs;o effetto fa la po&longs;&longs;anza in F che &longs;e fu&longs;&longs;e in B, e lo &longs;o&longs;tegno <lb/>in A l'i&longs;teffo che &longs;e fu&longs;&longs;e in C: percioche cia&longs;cun momento opera &longs;econ <lb/>da la &longs;ua perpendicolare: perche dunque po&longs;ta la po&longs;&longs;anza in F, e lo &longs;o-<emph.end type="italics"/><pb pagenum="29"/><emph type="italics"/>&longs;tegno in G, la po&longs;&longs;anza F, &longs;o&longs;tiene tal portione di tutto ilpe&longs;o, qual <lb/>portione è G C, di G F: e qual'è G C, di tutta G F, tal'è A E di tutta <lb/>A B, perche le A G, C E, B F, &longs;ono parallele: &longs;o&longs;tenta dunque la po&longs;­<lb/>&longs;anza B, del pe&longs;o tal portione, qual'è A E ditutta A B: &longs;e dunque <lb/>A E è minor portione di A B, che la A D, dell'i&longs;te&longs;&longs;a A B: la po&longs;&longs;an <lb/>za con la leua inalzata il cui centro del pe&longs;o è &longs;opra, &longs;o&longs;tenta minor <lb/>portione che nel &longs;ito orizontale. </s> <s>Il che &longs;i hauea damo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>I.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>E per l'i&longs;te&longs;&longs;o mezzo &longs;i mo&longs;trerà che quanto più la le <lb/>ua s'inalza, tanto minor pe&longs;o &longs;o&longs;tiene. </s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>E che po&longs;to il centro della grauezza &longs;otto la leua, <lb/>quanto più s'inalzi, magior portione di pe&longs;o &longs;o&longs;tenga. </s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>III.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>E che nelle leue ba&longs;&longs;ate &longs;otto l'orizonte, auuenga a <lb/>contrario. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>VIII.<!-- KEEP S--></s></p><p type="main"> <s>Dato nella leua il ponto di momento di una grauez­<lb/>za, e data qual&longs;ivoglia ragione di po&longs;&longs;anza a grauez­<lb/>za, ritrouar nella leua il ponto, oue la data po&longs;&longs;anza &longs;o <lb/>&longs;tenga la data grauezza. </s></p><pb pagenum="30"/><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia nella leua A B, il ponto del &longs;ottoleua in A: il ponto di momen­<lb/>to della data grauezza in C. et la ragion della po&longs;&longs;anza data alla grauez <lb/>za, come di E a D: &longs;i cerca nella leua il ponto, oue po&longs;ta la data po&longs;­<lb/>&longs;anza &longs;o&longs;tenga la data grauezza. </s> <s>per que&longs;to: faccia&longs;i come E a D, così <lb/>A C ad A F: & intenda&longs;i la po&longs;&longs;anza in F. <!-- KEEP S--></s> <s>dico che detta po&longs;&longs;anza in <lb/>F &longs;o&longs;tiene la grauezza in C. <!-- KEEP S--></s> <s>&longs;i mo&longs;tra: percioche e&longs;&longs;endo la ragion del­<lb/>la po&longs;&longs;anza alla grauezza come E a D, e la ragion dell'interuallo del <lb/>la grauezza A C, all'interuallo della po&longs;&longs;anza A F, l'i&longs;te&longs;&longs;a reciproca <lb/>mente: &longs;o&longs;tentarà dunque la data po&longs;&longs;anza in F, la grauezza in C. <!-- KEEP S--></s> <s>Il <lb/>che &longs;i cercaua.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et è manife&longs;to che in qual &longs;i uoglia altro ponto oltre <lb/>del termine del &longs;o&longs;tenimento, la data po&longs;&longs;anza mouerà <lb/>la data grauezza: e tanto più facilmente quanto più &longs;i <lb/>&longs;co&longs;tarà. </s></p><figure/><pb pagenum="31"/><p type="head"> <s>RAGGINELL <lb/>ASSE.<!-- KEEP S--></s></p><p type="head"> <s><emph type="italics"/>SVPPOSITIONE.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Svpponiamo, in vno i&longs;te&longs;&longs;o a&longs;&longs;e, due rag <lb/>gic'habbiano nelli &longs;uoi &longs;tremi li centri de pe&longs;i. </s></p><p type="main"> <s>E detti raggi, o in vna pianezza, e che non facciano <lb/>angolo, o in due, e che facciano angolo. </s></p><p type="head"> <s><emph type="italics"/>POSITIONE.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Pigliamo, il momento di cia&longs;cun pe&longs;o, &longs;econdo il pon <lb/>to, oue la perpendicolare del momento taglia la linea <lb/>orizontale, che pa&longs;&longs;a per l'a&longs;&longs;e. </s></p><figure/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Delle grauezze po&longs;te in raggi che non fanno tra di <lb/>loro angolo, in qualunque &longs;ito po&longs;te, li momenti tra di <lb/>loro hanno l'i&longs;te&longs;&longs;a ragione. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sial'a&longs;&longs;e A, a cui &longs;iano affi&longs;&longs;i li raggi A B, A C, qualis'intenda-<emph.end type="italics"/><pb pagenum="32"/><figure id="fig8"/><lb/><emph type="italics"/>no e&longs;&longs;ere nel&longs;ito orizontale, & moua&longs;i dal detto &longs;ito, sì che il B uen <lb/>gain D, & il C venga in E: dico che li momenti delle grauezze in det <lb/>ti raggi quantunque mo&longs;&longs;i di &longs;ito, &longs;iano nell'i&longs;te&longs;&longs;a ragione tra di loro. </s><lb/> <s>&longs;i mo&longs;tra: tiri&longs;iper D la perpendicolare D F & per E la perpendicola <lb/>re E G; perche dunque F A ad A G, ha la ragione che D A ad A E, <lb/>perciò che &longs;ono D F, E G, parallele: ma come D A ad A E, così B A ad <lb/>A C: perche &longs;ono l'i&longs;te&longs;&longs;i raggi, come dunque B A ad A C, così F A <lb/>ad A G: e perche la ragion delli momentie compo&longs;ta della ragion delle <lb/>grauezze, e della ragion delle di&longs;tanze dal centro: ma la ragione delle <lb/>grauezze è l'i&longs;te&longs;&longs;a: e la ragione delle <expan abbr="di&longs;tãze">di&longs;tanze</expan> è l'i&longs;te&longs;&longs;a: dunque la ragion <lb/>di ambe compo&longs;te, è anco l'i&longs;te&longs;&longs;a. </s> <s>Il che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><pb pagenum="33"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>II.<!-- KEEP S--></s></p><p type="main"> <s>Date qual &longs;i uoglia due grauezze, nelli raggiche fac­<lb/>ciano angolo dato, ritrouar nelle loro circolationi, pon­<lb/>ti oue facciano equipondio. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia l'a&longs;&longs;e A: li raggi che facciano <expan abbr="ãgolo">angolo</expan> dato A C, B A: & <expan abbr="int&etilde;da&longs;inel-li">intenda&longs;inel­<lb/>li</expan> ponti B e C, e&longs;&longs;er li <expan abbr="c&etilde;tri">centri</expan> delle grauezze: & le circonferenze che det­<lb/>ti ponti girando attorno fanno, &longs;iano E B, C F: &longs;i cercano in dette cir-<emph.end type="italics"/><pb pagenum="34"/><emph type="italics"/>conferenze li ponti, oue e&longs;&longs;endo dette grauezze, facciano equipondio. </s><lb/> <s>Diuida&longs;i la B C interuallo de centri, &longs;iche qual ragione ha la grauczza, <lb/>B, alla C, tal habbia la linea C D alla, D B: e tiri&longs;i A D: e tirata <lb/>per A, la A E B perpendicolare all'Orizonte, faccia&longs;i all'angolo D A <lb/>B, vguale lo E A G: & allo D A C, vguale E A H: dico che'l ponto <lb/>G, è oue portato il B, & H, oue portato il C, fanno equipondio. </s> <s>E prima <lb/>che portato il B in G, venga il C in H, è manife&longs;to: percioche l'ango <lb/>B A C è vguale al G A H: e perl'i&longs;te&longs;&longs;a ragione, è manife&longs;to che nell' <lb/>i&longs;te&longs;&longs;o tempo il ponto D, &longs;ia nella A E. <!-- KEEP S--></s> <s>ma il <expan abbr="põto">ponto</expan> D è il centro commu­<lb/>ne di pe&longs;o di dette due grauezze. </s> <s>E dunque il centro commune nel <lb/>la perpendicolare del &longs;o&longs;tenimento: e perciò le grauezze &longs;tanno. </s> <s>Jl che <lb/>&longs;i cercaua.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>I.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et è manife&longs;to che nelli due ponti, oppo&longs;ti alli ritroua <lb/>ti, facciano equipondio: & non altroue: percioche in o­<lb/>gni altra po&longs;itura oltre di dette due, il centro commune <lb/>e fuori del perpendicolo. </s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et è manife&longs;to che nell'arco &longs;otto il ponto dell'equi <lb/>pondio la grauezza ha momento maggiore: e nell'arco <lb/>&longs;oprail ponto dell'equipondio ha momento minore. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Dàte qual &longs;i uoglia due grauezze nelli dati raggi, che <lb/>fanno dato angolo: ritrouar nelle loro circolationi, pon-<pb pagenum="35"/>ti oue ilmomento dell'uno, al <expan abbr="mom&etilde;to">momento</expan> dell'altro habbia <lb/>qual &longs;i voglia data ragione. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Siano le date grauezze A & B: liraggi AC, BC, fi&longs;&longs;i nell'a&longs;&longs;e C: <lb/>che facciano dato <expan abbr="ãgolo">angolo</expan>: e la circolation di A &longs;ia, AD: di B &longs;ia B E.e <lb/>la data qual&longs;iuoglia ragione &longs;ia diFaG: &longs;icercano nella circolatione DA <lb/>e nella BE, ponti oue habbian li momenti di A e B, ragion di F a G. <!-- KEEP S--></s> <s>In|­<lb/>tenda&longs;i nella ragion di A a B, la quantità F ad H. <!-- KEEP S--></s> <s>e nella re&longs;tante ra­<lb/>gione di H a G, &longs;i diuida A B in L. <!-- KEEP S--></s> <s>& all'angolo L C A faccia&longs;i vgua-<emph.end type="italics"/><pb pagenum="36"/><emph type="italics"/>le il D C M, & allo L C B eguale il DCN: èmanife&longs;to, che portato A in <lb/>M: B verrà in N. <!-- KEEP S--></s> <s>& il ponto L nella perpendicolare C D. <!-- KEEP S--></s> <s>e&longs;e per il <lb/>ponto C. &longs;i tiri la P C Q parallela all'Orizonte: e dalli ponti M & N &longs;i <lb/>tirino a que&longs;ta, perpendicolari le MQ NP: &longs;arà il momento della <lb/>grauezza in M, al momento della grauezza in N di ragion compo <lb/>&longs;ta della grauezza A alla grauezza B, e della distanza Q C, alla CP, <lb/>che è l'i&longs;te&longs;&longs;a che di A L ad L B.percioche que&longs;ta è l'i&longs;te&longs;&longs;a che di M O ad <lb/>O N: cioè della compo&longs;ta delle ragioni di F ad H, e di H a G: ciò è di F a <lb/>G. <!-- KEEP S--></s> <s><expan abbr="harrãno">harranno</expan> dunque li <expan abbr="mom&etilde;ti">momenti</expan> di A & B, mentre &longs;iano po&longs;ti nelli ponti <lb/>M & N la ragiondata di F a G. <!-- KEEP S--></s> <s>Il che &longs;i cercaua.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et è manife&longs;to che prodotte le linee del centro nelli <lb/>ponti oppo&longs;ti delle dette circonferenze, hauranno iui li <lb/>momenti delle date grauezze l'i&longs;te&longs;&longs;a ragione: enon <lb/>altroue. </s></p><figure/><pb pagenum="37"/><p type="head"> <s>MOMENTI <lb/>CENTRALI<!-- KEEP S--></s></p><p type="main"> <s>Eqvanto delli momenti paralleli habbiamo <lb/>mo&longs;trato, tutto &longs;i adatterà anco alli momenti con­<lb/>correnti à centro: &longs;e in vece di linee dritte con&longs;ideria­<lb/>mo le circolari d'intorno il centro oue li momenti con­<lb/>corrono: & in dettecircolari &longs;i faccia l'i&longs;te&longs;&longs;a partitione: <lb/>e &longs;e in vece delli corpi terminati, da &longs;uperficie parallele, <lb/>s'intendano altri corpi terminati, parte da &longs;uperficie sfe <lb/>riche c'habbiano detto centro: parte da &longs;uperficie pia <lb/>ne che pa&longs;&longs;ino per e&longs;&longs;o. </s></p><figure/><pb pagenum="38"/><p type="head"> <s>ROTE VET­<lb/>TIVE.<!-- KEEP S--></s></p><p type="head"> <s><emph type="italics"/>SVPPOSITIONE.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Svpponiamo vna, o più rote congiogate, <lb/>muouer&longs;i per piano, che &longs;ia, o di po&longs;itura orizontale, <lb/>o inchinata. </s></p><p type="head"> <s><emph type="italics"/>DEFINITION.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Cogiogation &longs;emplice, diciamo dellerote, che &longs;ono <lb/>sù di vn'i&longs;te&longs;&longs;o a&longs;&longs;e. </s></p><p type="head"> <s>I.<!-- KEEP S--></s></p><p type="main"> <s>Molteplice, delle rote che &longs;ono in più a&longs;&longs;i. </s></p><p type="head"> <s>III.<!-- KEEP S--></s></p><p type="main"> <s>Portioni terminate dal &longs;o&longs;tenimento diciamo nel cir <lb/>colo, le fatte dalla linea perpendicolare per lo ponto del <lb/>contatto, all'orizonte: e nel cilindro, dalla &longs;uperficie pia <lb/>na per la linea del contatto, perpendicolare &longs;imilmente <lb/>all'orizonte. </s></p><pb pagenum="39"/><p type="head"> <s><emph type="italics"/>POSITIONE.<emph.end type="italics"/><lb/>I.</s></p><p type="main"> <s>Poniamo ogni forza, o trattiua, o pul&longs;iua, giunger mo <lb/>mento uer&longs;o quella parte, oue tira, o &longs;pinge. </s></p><p type="head"> <s>II.<!-- KEEP S--></s></p><p type="main"> <s>E &longs;e'l centro del pe&longs;o &longs;ia nell'i&longs;te&longs;&longs;a linea dell'appendi <lb/>mento, o &longs;o&longs;tenimento: che la grauezza non habbia mo <lb/>mento, ne uer&longs;o l'vna, ne uer&longs;o l'altra parte. </s></p><figure/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Dellarota vettiua, che &longs;i moue &longs;opra di vn piano ori­<lb/>zontale, il centro del pe&longs;o &longs;empre è nella perpendicola­<lb/>re del &longs;o&longs;tenimento. </s></p><figure/><pb pagenum="40"/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la linea Orrzontale A B: il circolo che rappre &longs;enta la rota, CD: <lb/>il ponto, oue detta rota tocca il piano C: da cui &longs;i cacci ad angoli ret­<lb/>ti la linea C D, è manife&longs;to che detta linea, è la perpendicolare del &longs;o&longs;te <lb/>nimento: & perquelche nelli libri Giometrici &longs;i mo&longs;tra: che pa&longs;&longs;a per il <lb/>centro del circolo, che è il centro della rota e grauezza: perilche diui­<lb/>de il circolo il parti vguali, & equeponderanti: è dunque il centro <lb/>del pe&longs;o nella perpendicolare del &longs;o&longs;tenimento. </s> <s>Il che &longs;i hauea da <lb/>mo&longs;trare,<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>I.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et il &longs;imile &longs;i mo&longs;tra, nelle &longs;emplici rote congiogate, <lb/>&longs;opra l'a&longs;&longs;e de quali, po&longs;i la grauezza. </s></p><p type="head"> <s><emph type="italics"/>Appendice, II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et è manife&longs;to nelle rote, sù l'a&longs;&longs;e de quali po&longs;i la <lb/>grauezza: che nel piano <expan abbr="orizõ">orizom</expan> tale, non habbian momen­ <lb/>to ne ver&longs;o l'vna, ne ver&longs;o l'altra parte. </s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>III.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>E che perciò qual &longs;i voglia po&longs;&longs;anza, le porterà così <lb/>nell'vna, come nell'altra parte, </s></p><pb pagenum="41"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>II.<!-- KEEP S--></s></p><p type="main"> <s>Nella rota che &longs;i porta per piano inchinato, il centro <lb/>del pe&longs;o, è fuori della perpendicolare del &longs;o&longs;tenimento. </s><lb/> <s>et il momento della rota appoggiata al piano, al momen­ <lb/>to della rota &longs;o&longs;pe&longs;a, la ha ragione, che l'ecce&longs;&longs;o delle <lb/>portioni del circolo, al circolo tutto. </s></p><figure/><pb pagenum="42"/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la linea che rappre &longs;enta il piano orizontale A B: la linea del pia <lb/>no inchinato A C: il circolo della rota D E FG: il toccamento D: e dal <lb/>ponto D, tiri&longs;i perpendicolare all'orizonte B D F: è manife&longs;to che detta <lb/>linea, &longs;ia la perpendicolare del &longs;o&longs;tenimento: dico che'l centro del pe&longs;o <lb/>è fuori di detta linea. </s> <s>Si mo&longs;tra: perche del triangolo D B A: l'angolo, <lb/>E, che fa la perpendicolare con l'orizonte, è retto: re&longs;ta l'angolo B D A, <lb/>a cuto: e perciò la portione D G F, e maggiore del &longs;emicircolo; & in e&longs; <lb/>&longs;a &longs;arà il centro del circolo, che è anco centro di pe&longs;o. </s> <s>è dunque il cen­<lb/>tro del pe&longs;o fuori della linea del <expan abbr="&longs;o&longs;tenim&etilde;to">&longs;o&longs;tenimento</expan>. </s> <s>De &longs;criua&longs;i alla D E, la por <lb/>tione di cir colo D H F, &longs;imile a D E F; &longs;aranno dette portioni vgua­<lb/>li, e faranno equipondio. </s> <s>re&longs;ta dunque la figura lunare &longs;enza equi <lb/>pondio: & il momento della rota appoggiata &longs;arà meno che della ro <lb/>ta &longs;o&longs;pe&longs;a, &longs;econdo la ragione della figura lunare a tutto il circolo: cio è <lb/>&longs;econdo la ragione dell'ecce&longs;&longs;o delle portioni, al circolo tutto. </s> <s>Il che <lb/>&longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>I.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>E l'i&longs;te&longs;&longs;o che si è mo&longs;trato nella rota c'ha grauezza; <lb/>si mo&longs;tra nelle rote al cui a&longs;&longs;e appoggi altro pe&longs;o. </s></p><p type="main"> <s><emph type="italics"/>Percio che &longs;e in vece delpe&longs;o appoggiato all'a&longs;&longs;e, intendiamo dar&longs;i <lb/>l'i&longs;te&longs;&longs;o pe&longs;o alle rote: e&longs;&longs;endo pe&longs;i vguali con loro centri nell'i&longs;te&longs;&longs;e li­<lb/>nee, & la linea del &longs;o&longs;tenimento l'i&longs;te&longs;&longs;a, harranno li pe&longs;i l'i&longs;te&longs;&longs;i <expan abbr="mom&etilde;ti">momenti</expan><emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice, II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et è manife&longs;to che detta rota correrà ver&longs;o la parte <lb/>del piano inferiore. </s></p><p type="main"> <s><emph type="italics"/>Percioche tirata dal centro I, la IG K perpendicolare del momento <lb/>tutto &longs;in che <expan abbr="s'incõtri">s'incontri</expan> col piano per oue camina: &longs;arà il ponto G della cir<emph.end type="italics"/><pb pagenum="43"/><emph type="italics"/>conferenza di&longs;co&longs;to dal <expan abbr="põto">ponto</expan> K del piano per oue camina la rota: e <expan abbr="tãto">tanto</expan> <lb/><expan abbr="maggiorm&etilde;te">maggiormente</expan> il <expan abbr="põto">ponto</expan> oue <expan abbr="s'incõtra">s'incontra</expan> la <expan abbr="perp&etilde;dicolare">perpendicolare</expan> del <expan abbr="c&etilde;tro">centro</expan> di pe&longs;o del <lb/>la figura lunare: la cui <expan abbr="di&longs;tãza">di&longs;tanza</expan> dalla linea del <expan abbr="&longs;o&longs;tenim&etilde;to">&longs;o&longs;tenimento</expan>, è maggior che <lb/>la di&longs;tanza del centro del circolo, &longs;econdo la ragion di tutto il circolo al <lb/>la figura lunare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>III.<!-- KEEP S--></s></p><p type="main"> <s>Se vn pe&longs;o &longs;ia portato da due <expan abbr="cõgiogationi">congiogationi</expan> di rote, <lb/>&longs;arà il pe&longs;o &longs;o&longs;tenuto dalli due a&longs;&longs;i compartitamente, &longs;e­<lb/>condo la ragione delle di&longs;tanze del momento da gli a&longs;&longs;i, <lb/>reciprocamente. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Siano le due congiogationi di rote rappre&longs;entate con li due circoli, <lb/>de quali gli <expan abbr="c&etilde;tri">centri</expan> &longs;ono A e B ponti, che rappre&longs;entano li due a&longs;&longs;i: e dal <lb/>ponto A al B, tiri&longs;i la A B. </s> <s>& intenda&longs;i il centro del pe&longs;o tutto appo <lb/>giato a detti due a&longs;&longs;i hauere il momento nel ponto C della detta linea.<emph.end type="italics"/><pb pagenum="44"/><emph type="italics"/>Dico che'ldetto pe&longs;o è &longs;o&longs;tenuto da detti a&longs;&longs;i compartitamente, &longs;econdo <lb/>la ragione delle BC, AC: cioè che di tutto il pe&longs;o l'a&longs;&longs;e A. <!-- KEEP S--></s> <s>ne &longs;o&longs;ten­<lb/>terà tal portione qual'è BC di B A, e B tale qual'è AC di AB, Si mo <lb/>&longs;tra intenda&longs;i <expan abbr="prolõgata">prolongata</expan> la AB nell'vna e l'altra banda, far&longs;i ad AC <lb/>vguale la BD: & alla BC, vguale la AE: &longs;aranno le EC, DC vguali: <lb/>e di nuouo fatto alla AC uguale la AE, &longs;aranno le DB, BF, e le AE <lb/>AF, vguali: epercio &longs;e alla linea DE, s'intenda fatta application di <lb/>corpo: il momento di tutto &longs;arà nel ponto C. <!-- KEEP S--></s> <s>di cui il detto a&longs;&longs;e A ne <lb/>&longs;o&longs;tentarà la portione applicata ad EF: e l'a&longs;&longs;e B la portione applicata <lb/>a DF, la ragion de quali è l'i&longs;te&longs;&longs;a: che di BC ad AC: ma del corpo ap <lb/>plicato il centro del pe&longs;o è l'i&longs;te&longs;&longs;o, dall'i&longs;te&longs;&longs;i ponti &longs;o&longs;tenuto. </s> <s>&longs;o&longs;tengono <lb/>dunque gli a&longs;&longs;i il pe&longs;o compartitamente &longs;econdo la ragion di BC a C­<lb/>A. <!-- KEEP S--></s> <s>Il che &longs;i hauea da mostrare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>IIII.<!-- KEEP S--></s></p><p type="main"> <s>Se'l pe&longs;o sia portato da due <expan abbr="congiõgationi">congiongationi</expan> di rote per <lb/>piano inchinato: <expan abbr="&longs;o&longs;t&etilde;ntarà">&longs;o&longs;tenntarà</expan> l'a&longs;&longs;e dellerote inferiori di <lb/>detto pe&longs;o, ma ggior portione che &longs;e fu&longs;&longs;e nel piano ori <lb/>zontale. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la linea del piano orizontale AB: del piano inchinato AC: li <lb/>centri de circoli delle rote D, & E: il centro della grauezza che sù <lb/>gli a&longs;&longs;i di dette rote appoggia F: Dico che di detta grauezza, dall'a&longs;&longs;e <lb/>D, ne &longs;arà &longs;ostentata maggior portione: e dall'a&longs;&longs;e E, minore, che &longs;e <lb/>portata fu&longs;&longs;e per piano Orizontale. <!-- KEEP S--></s> <s>Si mo&longs;tra: tiri&longs;i da F perpendico <lb/>lare alla DE, che &longs;ia FG: e perpendicolare all'orizonte che &longs;ia FH: &longs;a <lb/>rà il ponto G, il ponto del momento nel &longs;ito orizontale. </s> <s>& H, nell'in <lb/>chinato: e perche EH, è maggior portione di ED: che EG, e DH,<emph.end type="italics"/><pb pagenum="45"/><emph type="italics"/>minore che DG: &longs;o&longs;tentarà la rota inferiore &longs;econdo la ragione di EH, <lb/>ad ED; e la &longs;uperiore &longs;ecodo la ragione di DH ad ED: &longs;o&longs;tenta <expan abbr="dũ">dum</expan> que <lb/>la rota inferiore, maggior portione di pe&longs;o: e la &longs;uperiore minor porti <lb/>one, che &longs;e nel &longs;ito orizontale fu&longs;&longs;ero. </s> <s>Il che &longs;i hauca da mo&longs;trare.<emph.end type="italics"/></s></p><figure/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>V.<!-- KEEP S--></s></p><p type="main"> <s>Data la rota che <expan abbr="affõdi">affondi</expan> in <expan abbr="cõcauita">concauita</expan> &longs;otto il piano <expan abbr="orizõ">orizom</expan> <pb pagenum="46"/>tale: e data qual si uoglia grauezza: ritrouare in vn rag<lb/>gio la di&longs;tanza oltre di cui detta grauezza appe&longs;a, &longs;ol­<lb/>leui detta rota. </s></p><figure/><p type="head"> <s><emph type="italics"/>Dimostratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la linea del piano orizontale ABCD: la concauità in e&longs;&longs;a BE <lb/>C: la rota che affondi BCF: la grauezza data G. <!-- KEEP S--></s> <s>&longs;i cerca in vn raggio <lb/>della rota, ponto oltre di cui &longs;o&longs;pe&longs;a la G, &longs;olleui detta rota. </s> <s>Sia il cen­<lb/>tro H: la linea del raggio prodotto HFI: qual &longs;ia parallela all'ori­<lb/>zonte: e dal <expan abbr="põto">ponto</expan> C, &longs;i tiri la CK perpendicolare che <expan abbr="affrõti">affronti</expan> la HF, in K: <lb/>e la ragion c'ha la grauezza G al pe&longs;o della rota, habbia HK a KI: è <lb/>manife&longs;to perche KC, è perdendicolare del &longs;o&longs;tenimento, che dal ponto I <lb/>la grauezza G, fa equipondio allarota. </s> <s>e che da ogni ponto oltre, la &longs;ol <lb/>leui, il che &longs;i cercaun.<emph.end type="italics"/></s></p><pb pagenum="47"/><p type="head"> <s>TAGLIA. </s><lb/> <s><emph type="italics"/>SVPPOSITIONE.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Svpponiamo la taglia c'habbia in &longs;e una, o più <lb/>girelle, o sia in vno o più ordini. </s> <s>Et delle taglie, &longs;ta <lb/>bile diciamo, il cui collo sia legato ad vn termine: mo­<lb/>bile il cui collo sia legato al pe&longs;o. </s> <s>Et <expan abbr="altrim&etilde;te">altrimente</expan> mobile la <lb/>guidata da vna potenza, e che ad vn capo di e&longs;&longs;a &longs;ia attac <lb/>cato il pe&longs;o. </s> <s>In oltre &longs;upponiamo della corda auuolta il <lb/>capo andare, o alla taglia, o ad'vn termine fi&longs;&longs;o, o a pof­<lb/>&longs;anza, ò a pe&longs;o. </s></p><p type="head"> <s><emph type="italics"/>POSITIONE<emph.end type="italics"/><lb/>I.</s></p><p type="main"> <s>Poniamo della girella a cui sia auuolta corda data <lb/>a pesi, & a po&longs;&longs;anze, mentre detta girella non volta il mo <lb/>mento de capi e&longs;&longs;ere vguale. </s></p><p type="head"> <s>II.<!-- KEEP S--></s></p><p type="main"> <s>Ma &longs;e la girella volta, il momento di quella corda e&longs;­<lb/>&longs;er maggiore, ver&longs;o di cui volta. </s></p><p type="head"> <s>III.<!-- KEEP S--></s></p><p type="main"> <s>E poniamo nelle girelle, di po&longs;&longs;anze e pe&longs;i vguali, <lb/>li momenti e&longs;&longs;ere vguali. </s></p><pb pagenum="48"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Se delli due capi della girella, l'vna &longs;o&longs;tenti pe&longs;o, l'al <lb/>tro &longs;ia dato a po&longs;&longs;anza: la po&longs;&longs;anza del capo &longs;arà di mo <lb/>mento eguale al pe&longs;o. </s> <s>e la po&longs;&longs;anza della taglia &longs;o&longs;tenta <lb/>rà il doppio. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la taglia AB: li capi della fune auuolta<emph.end type="italics"/><lb/><figure id="fig9"/><lb/><emph type="italics"/>A C, B D: dequali A C, &longs;o&longs;tenti il pe&longs;o C: e B <lb/>D, &longs;ia dato alla po&longs;&longs;anza in D: dico che la <lb/>po&longs;&longs;anza in D è di momento eguale al pefo: e <lb/>che la po&longs;&longs;anza in E, &longs;o&longs;tenta il doppio. </s> <s>Si <lb/>mo&longs;tra: e prima che'l momento di D, &longs;ia v­<lb/>guale al momento di C. è manife&longs;to: perche <lb/>&longs;e l'vn di loro fu&longs;&longs;e maggiore, la girella volte <lb/>rebbe ver&longs;o detto momento: Jl che è contro <lb/>il &longs;uppo&longs;to. </s> <s>Dico hora che la<gap/>&longs;&longs;<gap/>za della <lb/>taglia &longs;ia doppia del pe&longs;o: percioche e&longs;&longs;endo <lb/>la po&longs;&longs;anza di D, equiualente al pe&longs;o C: ambi <lb/>C e D, &longs;ono il aoppio di e&longs;&longs;o C: ma la <expan abbr="po&longs;sãza">po&longs;sanza</expan> in <lb/>E, in quanto &longs;o&longs;tiene, è vguale ad ambi: dun­<lb/>que è doppia di vn di loro. </s> <s>Ha&longs;&longs;i dunque il <lb/>propo&longs;to, che la po<32>anza D, &longs;ia vguale al mo <lb/>mento di C: e che la E, &longs;o&longs;tenti il doppio di e&longs;&longs;o.<emph.end type="italics"/></s></p><pb pagenum="49"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>II.<!-- KEEP S--></s></p><p type="main"> <s>Se li due capi di girella mobile, &longs;iano raccomanda­<lb/>ti a due po&longs;&longs;anze: &longs;o&longs;tentarà così l'vna, come l'altra po&longs; <lb/>&longs;anza, la metà del pe&longs;o. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la taglia A B, a cui &longs;ia attaccato il pe<emph.end type="italics"/><lb/><figure id="fig10"/><lb/><emph type="italics"/>&longs;o <gap/>: li due capi della corda auuolta alla gi­<lb/>rella A D, B E: le po&longs;&longs;anze in D, et E: dico <lb/>che così l'vna, come l'altra po&longs;&longs;anza &longs;o&longs;ten <lb/>ta la metà del pe&longs;o. </s> <s>&longs;i mo&longs;tra: percioche &longs;tan­ <lb/>do la girella <expan abbr="s&etilde;za">senza</expan> voltare, <expan abbr="&longs;ecõdo">&longs;econdo</expan> il &longs;up: &longs;ara <lb/>di con&longs;eguenza il momento dell'vn capo v­<lb/>guale al momento dell'altro: e perciò le po&longs; <lb/>&longs;anze anco eguali. </s> <s>e perche ambe &longs;o&longs;tenta­<lb/>no il pe&longs;o C: e le po&longs;&longs;anze, in quanto &longs;o&longs;ten­<lb/>gono, &longs;ono eguali alli pe&longs;i. </s> <s>&longs;ono dunque am <lb/>be eguali al pe&longs;o C: e perciò diui&longs;amente l'v <lb/>na e l'altra &longs;arà la metà di detto pe&longs;o­<lb/>al che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><pb pagenum="50"/><p type="head"> <s><emph type="italics"/>Appendice,<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>E perciò anco se l'vn capo sia raccomandato ad vn <lb/>termine fi&longs;&longs;o, l'altro a po&longs;&longs;anza: &longs;o&longs;terrà la po&longs;&longs;anza la <lb/>metà del pe&longs;o. </s></p><p type="main"> <s><emph type="italics"/>Percioche mutato il termine in un'altra po&longs;&longs;anza: la po&longs;&longs;anza &longs;uppo <lb/>&longs;ta &longs;o&longs;terrà l'i&longs;te&longs;&longs;a altra quantità di pe&longs;o che prima.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>III.<!-- KEEP S--></s></p><p type="main"> <s>Delle corde, che dalla taglia &longs;u <lb/>periore, & dalla po&longs;&longs;anza alla ta <lb/><figure id="fig11"/><lb/>glia inferiore peruengono: cia&longs;cu <lb/>na &longs;o&longs;tiene egual parte dipe&longs;o. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la taglia &longs;uperiore A B: l'inferiore <lb/>C D: la corda auuolta no tata con l'i&longs;te<32>e let <lb/>tere: e di lei l'vn termine vada a &longs;o&longs;tenere <lb/>la taglia inferiore in E: l'altro &longs;ia dato al­<lb/>la po&longs;&longs;anza in F. <!-- KEEP S--></s> <s>Dico che cia&longs;cuna corda <lb/>&longs;o&longs;tiene egual parte di pe&longs;o. </s> <s>Si mo&longs;tra: <lb/>perche &longs;tando la girella A B, il momento <lb/>del capo B D è eguale al momento del ca­<lb/>po A E: e del capo C F, al capo B D, per <lb/>la girella C D: &longs;ono <expan abbr="dũque">dunque</expan> tutte di momen­ <lb/>to eguali: perciò cia&longs;cuna &longs;o&longs;tentarà e­<lb/>gual parte di pe&longs;o. </s> <s>e &longs;e il capo A E non fu&longs; <lb/>&longs;e ligato alla taglia, ma ad altro termine, &longs;a <lb/>rebbe l'i&longs;te&longs;&longs;o, ma il numero delle corde di <lb/>vna meno. </s> <s>Il che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><pb pagenum="51"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>IIII.<!-- KEEP S--></s></p><p type="main"> <s>Se l'vn capo della fune auuolta <lb/><figure id="fig12"/><lb/>a girelle, &longs;ia raccomandato alla ta <lb/>glia &longs;uperiore: il pe&longs;o &longs;o&longs;tenuto <lb/>è di&longs;tribuito in parti di numero <lb/>pare. </s></p><p type="head"> <s><emph type="italics"/>Dimostratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la taglia inferiore e mobile AC B D: <lb/>la &longs;uperiore E F G H: la fune auuolta nota, <lb/>ta, <expan abbr="cõ">com</expan> l'iste&longs;&longs;e lettere: il termine del capo C 1, <lb/>attaccato alla taglia &longs;uperiore, <expan abbr="s'int&etilde;da">s'intenda</expan> e&longs;&longs;e <lb/>re in I: l'altro termine raccomandato alla <lb/>po&longs;&longs;anza s'intenda e&longs;&longs;ere o in K del capo B <lb/>K, che vien dalla taglia inferiore, o in L, del <lb/>capo G L, che vien dalla taglia &longs;uperiore. </s><lb/> <s>Dico che, e nell'vno, e nell'altro modo, il pe <lb/>&longs;o è di&longs;tribuito in parti di numero pare. </s><lb/> <s>Si mo&longs;tra: percioche venendo alla girella <lb/>C D due corde, l'vna da taglia, l'altra da­<lb/>girella E F: &longs;aranno detti capi di momen­ <lb/>ti eguali: perche &longs;i pone la girella non vol <lb/>tare. </s> <s>&longs;imilmente perche alla girella A B <lb/>vengono due corde, l'vna dalla girella EF, <lb/>che è la corda E A, l'altra dalla po<32>anza <lb/>K, che è la corda KB: &longs;aranno dette corde <lb/>di <expan abbr="mom&etilde;ti">momenti</expan> eguali. </s> <s>ma la DF, è di momento <lb/>eguale alla A E, e alla B K: &longs;ono dunque <lb/>tutte tra di loro di momento eguale: e <lb/>&longs;ono di numero pare: percioche a cia-<emph.end type="italics"/><pb pagenum="52"/><emph type="italics"/>&longs;cuna girella ne vengono due. </s> <s>perche dunque il pe&longs;o è &longs;o&longs;tenuto da <lb/>dette corde di <expan abbr="mom&etilde;to">momento</expan> eguale: perciò, mentre l'vn capo &longs;ia attac cato <lb/>alla taglia &longs;uperiore, l'altro dato alla po&longs;&longs;anza, il <expan abbr="mom&etilde;to">momento</expan> del pe&longs;o è di <lb/>&longs;tribuito in parti di numero pare: ne altro auuiene, &longs;e la po<32>anza &longs;ia in <lb/>L, nel capo, che viene dalla taglia &longs;uperiore: percioche il numero del <lb/>le corde, che alla taglia inferiore peruengono è l'i&longs;te&longs;&longs;o.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>I<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et è manife&longs;to, che po&longs;ta vna girella meno nella ta­<lb/>glia &longs;uperiore, &longs;i &longs;o&longs;terrà dalla po&longs;&longs;anza l'i&longs;te&longs;&longs;o che &longs;e <lb/>fu&longs;&longs;ero le girelle &longs;uperiori di numero eguale alle in­<lb/>feriori, è che per detta girella aggiunta, si muta &longs;o <lb/>lamente l'un momento nell'altro di &longs;pezie contraria. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>V.<!-- KEEP S--></s></p><p type="main"> <s>Se l'vn capo della fune auuolta a girelle, &longs;ia racco­<lb/>mandato alla taglia inferiore: il pe&longs;o &longs;o&longs;tenuto è di&longs;tri <lb/>buito in parti di numero &longs;pare. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la taglia &longs;uperiore e stabile AB, CD: l'inferiore e mobile E­<lb/>F, G H: la fune auuolta notata con l'i&longs;te&longs;&longs;e lettere: e di e&longs;&longs;a l'vn ter­<lb/>mine I, che è del capo C I, &longs;ia attaccato alla taglia inferiore: etil termi <lb/>ne K, del capo E K, rac comandato alla po&longs;&longs;anza in K: dico che'l pe&longs;o è<emph.end type="italics"/><pb pagenum="53"/><emph type="italics"/>di&longs;tribuito in parti dinumero &longs;pare. </s> <s>Si<emph.end type="italics"/><lb/><figure id="fig13"/><lb/><emph type="italics"/>mo&longs;tra: percioche vengono due capi dalla <lb/>girella C D, alla taglia inferiore, e due <lb/>dalla A B, e &longs;imilmente da qual &longs;i voglia <lb/>altra girella: &longs;ono dunque li capi, che dal <lb/>le girelle alla taglia vengono, di numero <lb/>pare. </s> <s>et euui in oltre il capo della po&longs;&longs;an <lb/>za: &longs;ono dunque tutti di numero &longs;pare. </s><lb/> <s>e &longs;ono, per quel che &longs;i è detto nelle prece <lb/>denti, tutte di momento eguale: dunque <lb/>il pe&longs;o è di&longs;tribuito in parti di numero <lb/>&longs;pare. </s> <s>Jl che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>I.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et è manife&longs;to, che aggionta <lb/>alla taglia &longs;uperiore vna girel­<lb/>la, si commuta &longs;olamente il mo­<lb/>mento della po&longs;&longs;anza, in mo­<lb/>mento di &longs;pezie contraria. </s></p><p type="head"> <s><emph type="italics"/>Appendice. <!-- KEEP S--></s> <s>II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>E raccogliamo, che ligato l'vn <lb/>capo alla taglia &longs;uperiore, puote <lb/>&longs;tar detta taglia con vna girella <lb/>meno: e ligata all'inferiore con <lb/>vna girella più. </s></p><pb pagenum="54"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>VI</s></p><p type="main"> <s>Se vn capo della taglia &longs;upe <lb/><figure id="fig14"/><lb/>riore sia raccomandato ad vn <lb/>termine fi&longs;&longs;o: &longs;arà il pe&longs;o di&longs;tri <lb/>buito in parti di numero pare. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la taglia &longs;uperiore A B C D, l'­<lb/>inferiore E F G H: la fune auuolta a gi <lb/>rellenotata con l'i&longs;te&longs;&longs;e lettere: di cui <lb/>il capo D I dalla girella C D della taglia <lb/>&longs;uperiore &longs;ia raccomandato ad I termi <lb/>ne fi&longs;&longs;o: & il capo F K, dalla girella E F, <lb/>della taglia inferiore, raccomandato al <lb/>la po&longs;&longs;anza in K. <!-- KEEP S--></s> <s>Dico che'l pe&longs;o è di&longs;tri <lb/>buito in parti di numero pare. </s> <s>Si mo­<lb/>&longs;tra: percio che venendo alla taglia infe <lb/>riore le corde &longs;olo delle girelle, & da cia <lb/>&longs;cuna girella due corde, quali tutte &longs;i è <lb/>mo&longs;trato che &longs;o&longs;tentino egual momento: <lb/>&longs;arà il pe&longs;o di&longs;tribuito in corde di nume <lb/>ro pare, che egualmente &longs;o&longs;tentano: e <lb/>perciò &longs;arà di&longs;tribuito in dette parti. </s> <s>Il <lb/>che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>Correlario. <!-- KEEP S--></s> <s>I.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>E manife&longs;to dunque che li <lb/>gatoil capo di &longs;opra alla taglia <pb pagenum="55"/>inferiore, il pe&longs;o è di&longs;tribuito in parti di numero &longs;pare, <lb/>et comunque altrimente, in parti di numero pare. </s></p><p type="head"> <s><emph type="italics"/>Correlario. <!-- KEEP S--></s> <s>II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Et attaccato il capo di girella inferiore alla taglia &longs;u <lb/>periore, o à qual si voglia termine fi&longs;&longs;o: che la taglia in <lb/>feriore habbia vna girella più. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>VII.<!-- KEEP S--></s></p><figure/><p type="main"> <s>Se'l pe&longs;o sia mo&longs;&longs;o con ta <lb/>glie, quanto il pe&longs;o è moltepli <lb/>ce della po&longs;&longs;anza &longs;o&longs;tenente, <lb/>tanto lo &longs;patio, che detta <expan abbr="po&longs;-sãza">po&longs;­<lb/>sanza</expan> camina, è molteplice del <lb/>lo &longs;patio caminato dal pe&longs;o. </s></p><p type="head"> <s><emph type="italics"/>Dimostratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la girella dellataglia &longs;uperiore <lb/>A B: della inferiore nella prima po­<lb/>&longs;itione &longs;ia C D: e lapo&longs;&longs;anza che &longs;o&longs;tie <lb/>ne il capo &longs;ia in E: della &longs;econda po&longs;i <lb/>tione &longs;ia in G H, e la po&longs;&longs;anza in I. <!-- KEEP S--></s> <s>Di <lb/>co che lo &longs;patio caminato dalla taglia <lb/>mobile e pe&longs;o, è talparte dello &longs;patio <lb/>E I, qual la po&longs;&longs;anza &longs;o&longs;tenente in E <lb/>è parte del pe&longs;o. </s> <s>Si mo&longs;tra: perche <lb/>quante &longs;ono la corde, che alla taglia<emph.end type="italics"/><pb pagenum="56"/><emph type="italics"/>inferiore peruengono, &longs;econdo tal numero la po&longs;&longs;anza che &longs;o&longs;tiene è <lb/>parte del pe&longs;o: e perche nel mouimento della taglia cia&longs;cuna corda <lb/>&longs;i abbreuia egualmente, portata C D, in H G: le C G, D H parti del <lb/>la corda auuolta, quante &longs;i &longs;iano, pigliate in&longs;ieme, &longs;arano di lunghezza <lb/>tanto molteplici dello &longs;patio caminato, quanto è il numero delle cor <lb/>de. </s> <s>ma la corda E A B D C F, è vguale alla I A B H G: dunque tol <lb/>tone di commune la F G H B A E, re&longs;ta le E I, eguale alla G C D H: <lb/>e percto E I, &longs;arà altre tanto molteplice dello &longs;patio caminato, quan <lb/>to erano le corde C G, D H. <!-- KEEP S--></s> <s>ciò è il pe&longs;o tutto del pe&longs;o da vna corda <lb/>&longs;ostenuto. </s> <s>Il che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>VIII.<!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Problema. <!-- KEEP S--></s> <s>I.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Data qual si voglia grauezza, e po&longs;&longs;anza: ritroua <lb/>re il minor numero di girelle nella taglia, con quali <lb/>la data po&longs;&longs;anza moua il dato pe&longs;o. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la data grauezza A, la po&longs;&longs;anza B: a cui &longs;i pigli vn pe&longs;o e­<lb/>quiualente C: e moltiplichi&longs;i C, &longs;in che la prima volta ecceda la <lb/>grauezza A, il che &longs;ia per il numero D. <!-- KEEP S--></s> <s>&longs;e dunque D è pare piglin <lb/>&longs;i nella taglia inferiore altre tante girelle, quante vnità &longs;ono nella <lb/>inetà del numero: è manife&longs;to che la po&longs;&longs;anza mouerà il pe&longs;o con le <lb/>date girelle: ma &longs;e D &longs;ia &longs;pare, toltane vnità, piglin&longs;i girelle quan <lb/>te vnità &longs;ono nella metà del re&longs;to, e lighe&longs;i vn delli capi alla taglia: <lb/>è manifesto &longs;i milmente che mouerà la po&longs;&longs;anza la data grauezza. </s><lb/> <s>Il che &longs;i cercaua.<emph.end type="italics"/></s></p><pb pagenum="57"/><p type="head"> <s><emph type="italics"/>PROPOSITIONE.<emph.end type="italics"/><lb/>VIII.<!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Problema. <!-- KEEP S--></s> <s>II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Data qual &longs;i voglia velocità, e data la tardità della <lb/>po&longs;sanza: applicar o vna taglia di più girelle, o più ta <lb/>glie di vna girella, &longs;i che la po&longs;&longs;anza moua il dato pe­<lb/>&longs;o in velocità magior della data. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Pigli&longs;i lo &longs;patio che nel dato tem-<emph.end type="italics"/><lb/><figure id="fig15"/><lb/><emph type="italics"/>po camini la po&longs;&longs;anza: e lo &longs;patio che <lb/>vogliamo che la co&longs;a camini, e &longs;i mol <lb/>tiplichi il minore fin che la prima vol <lb/>ta auanzi, e quanto que&longs;to è molte­<lb/>plice, tante corde &longs;iano nella taglia &longs;u <lb/>periore, pigliando la metà di girelle &longs;e <lb/>&longs;ia pare, & &longs;e &longs;ia &longs;pare, ligando vn ca <lb/>po ad e&longs;&longs;a taglia &longs;uperiore. </s> <s>Ligato dun <lb/>que il pe&longs;o ad vn capo, la po&longs;&longs;anza, che <lb/>tira la taglia, tirerà anco il pe&longs;o: e ca­<lb/>minerà lo &longs;patio moltiplice al moui­<lb/>mento di e&longs;&longs;a po&longs;&longs;anza. </s> <s>Ma, &longs;e vo <lb/>gliamo far ciò con piú taglie di vna gi <lb/>rella, radoppi&longs;i lo &longs;patio, e di nuouo il <lb/>fatto dal radoppiamento &longs;i radoppÿ: e <lb/>ciò &longs;i torni a fare, si che l'ultimo radop <lb/>piamento auanzi lo &longs;patio maggiore. </s><lb/> <s>Se dunque, quante volte &longs;i è radoppia­<lb/>to, tanto numero ditaglie &longs;i pigli, si mo<lb/>uer à ispe&longs;o &longs;econdo la ragion del radop <lb/>piamento dello &longs;patio, e perciò &longs;i mo­<lb/>uerà con maggior velocità della data.<emph.end type="italics"/></s></p><pb pagenum="58"/><p type="head"> <s>ROTE MO <lb/>TIVE.<!-- KEEP S--></s></p><figure/><p type="head"> <s><emph type="italics"/>SVTPOSITION.<emph.end type="italics"/><lb/>I.</s></p><p type="main"> <s>Svpponiamo il mouimento di rote in a&longs;&longs;i <lb/>che &longs;tanno co'l toccamento, communicar&longs;i l'vna <lb/>all'altrail mouimento: e che'l momento della po&longs;&longs;an­<lb/>za &longs;ia per linea chefaccia angolo retto co'l raggio di <lb/>e&longs;&longs;arota: e de momenti altri e&longs;&longs;er concorrenti, altri <lb/>contrarij. </s></p><p type="head"> <s><emph type="italics"/>DEFINITION.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Concorrenti momenti diciamo quelli, che portan­ <lb/>do ver&longs;o l'i&longs;te&longs;&longs;a parte, &longs;i accre&longs;cono. </s></p><p type="head"> <s>II.<!-- KEEP S--></s></p><p type="main"> <s>Contrarij quelli, che s'impedi&longs;cono portando in <lb/>contrario. </s></p><p type="head"> <s><emph type="italics"/>POSITION.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Poniamo, po&longs;&longs;anze eguali in circonferenze direte <lb/>eguali, hauer momenti eguali. </s></p><pb pagenum="59"/><p type="head"> <s><emph type="italics"/>POSITION.<emph.end type="italics"/><lb/>II.<!-- KEEP S--></s></p><p type="main"> <s>Et in rote ineguali hauer momento ineguale, &longs;econ <lb/>do la ragion de &longs;emidiametrj. </s></p><p type="head"> <s>III.<!-- KEEP S--></s></p><p type="main"> <s>E gli momenti contrarij, per quanto &longs;i annullano, l' <lb/>vno e&longs;&longs;ere eguale all'altro. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITION.<emph.end type="italics"/><lb/>I.<!-- KEEP S--></s></p><p type="main"> <s>Se quante &longs;i voglia rote, vna per a&longs;&longs;e, &longs;i tocchino: <lb/>e po&longs;te le po&longs;&longs;anze l'vna nella circonferenza della pri <lb/>ma, e l'altra dell'vltima, &longs;i rattengano: &longs;aranno le po&longs; <lb/>&longs;anze egualj. </s></p><figure/><pb pagenum="60"/><p type="head"> <s><emph type="italics"/>Dimostratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Siano quante &longs;i voglia rote ne gli a&longs;&longs;i A, B, C, che &longs;i tocchino: ciò <lb/>è che la A tocchi la B nelponto D: e la B tocchila C nel ponto E: <lb/>& intenda&longs;i nella circonferenza di A e&longs;&longs;er la potenza F: e nella cir <lb/>conferenza di C la potenza G: che l'una rattenga l'altra. </s> <s>Dico che <lb/>le potenze &longs;ono eguali. </s> <s>Si mo&longs;tra: percio che la po<32>anza in F, è dell'­<lb/>i&longs;te&longs;&longs;o momento, che &longs;e fu&longs;&longs;e in D, dell'i&longs;te&longs;&longs;a rota A: ma il ponto <lb/>D, è ponto commune a due rote: e la po&longs;&longs;anza in D della rota B, <lb/>è quanto fu&longs;&longs;e in E: &longs;arà dunque la po&longs;&longs;anza in F l'i&longs;te&longs;&longs;o che &longs;i fu&longs;&longs;e <lb/>in E: perche <expan abbr="dũque">dunque</expan> la po&longs;&longs;anza in F &longs;i annulla con la po&longs;&longs;anza in G, &longs;o <lb/>no li loro momenti eguali. </s> <s>Ma le po&longs;&longs;anze che &longs;ono in un'i&longs;te&longs;&longs;a rota <lb/>di momenti eguali, &longs;ono eguali: dunque la po&longs;&longs;anza in F è uguale alla <lb/>po&longs;&longs;anza in G. <!-- KEEP S--></s> <s>Jl che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITION.<emph.end type="italics"/><lb/>II.<!-- KEEP S--></s></p><p type="main"> <s>Delle due rote in vno a&longs;&longs;e la po&longs;&longs;anza, che fa egual <lb/>momento nella rota magiore è di valor minore: e nel <lb/>la minore è di valor maggiore, nella ragione de &longs;emi <lb/>diametri reciproca. </s></p><p type="head"> <s><emph type="italics"/>Dimostratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Siano &longs;u l' a&longs;&longs;e A le rote A B, A C: & intenda&longs;i la po&longs;&longs;anza B, <lb/>in circonferenza della rota maggiore, hauere egual momento alla <lb/>po&longs;&longs;anza C in circonferenza della rota minore. </s> <s>Dico che la po&longs;&longs;an­<lb/>za B è minore della po&longs;&longs;anza C, &longs;econdo la ragione di C A ad A B. </s><lb/> <s>Si mo&longs;tra: intenda&longs;i nell circonferenza di A C e&longs;&longs;er po&longs;&longs;anza eguale <lb/>a B, che &longs;ia D: &longs;arà il momento di B al momento di D, nella ragion<emph.end type="italics"/><pb pagenum="61"/><figure id="fig16"/><lb/><emph type="italics"/>della linea dritta B A alla D A: ma il momento di B, è uguale al <lb/>momento di C: dunque il momento di C al momento di D, è come <lb/>B A ad A D. <!-- KEEP S--></s> <s>Se dimque le po&longs;&longs;anze dell'i&longs;te&longs;&longs;a rota &longs;ono tra di loro <lb/>nella ragione delli momenti: &longs;arà di con&longs;eguenza la po&longs;&longs;anza in D <lb/>alla po&longs;&longs;anza in C, come il &longs;emidiametro D A, al &longs;emidiame­<lb/>tro A B, e del diametro tutto a tutto. </s> <s>Il che &longs;i hauea da mo­<lb/>&longs;trare.<emph.end type="italics"/></s></p><pb pagenum="62"/><p type="head"> <s><emph type="italics"/>PROPOSITION.<emph.end type="italics"/><lb/>III.</s></p><p type="main"> <s>Se le rote, po&longs;te a due in cia&longs;cun a&longs;&longs;e, &longs;i tocchino: <lb/>e le po&longs;&longs;anze, po&longs;te l'vna nella prima, l'altra nell'vl­<lb/>tima rota, &longs;i rattengano: &longs;arà la ragion dell'vna po&longs;&longs;an <lb/>za all'altra l'i&longs;te&longs;&longs;a, che la ragion compo&longs;ta delli &longs;emi <lb/>diametri, che &longs;ono &longs;u l'i&longs;te&longs;&longs;o a&longs;&longs;e, pigliate reciproca­<lb/>mente. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Siano &longs;u l'a&longs;&longs;e A, le due rote A B, A C: e &longs;u l'a&longs;&longs;e D, le rote D C <lb/>D F: & intenda&longs;i la rota D C, e&longs;&longs;er toccata dalla A C nel ponto C: <lb/>c l'una po&longs;&longs;anza e&longs;&longs;ere in B l'altra in E. <!-- KEEP S--></s> <s>Dicoche la po&longs;&longs;anza in B,<emph.end type="italics"/><lb/><figure id="fig17"/><lb/><emph type="italics"/>alla po&longs;&longs;anza in F ha la ragion compo&longs;ta delle ragioni di F D a D C,<emph.end type="italics"/><pb pagenum="63"/><emph type="italics"/>e di C A ad A B, che &longs;ono le ragioni de &longs;emidiametri reciprocamen <lb/>te pigliati. </s> <s>Si mo&longs;tra: percioche e&longs;&longs;endo il momento in B uguale al <lb/>momento in C, perche &longs;ono in vno i&longs;te&longs;&longs;o a&longs;&longs;e: & il momento in C al <lb/>momento in F, per l'iste&longs;&longs;a ragione: & è la po&longs;&longs;anza in F, alla <lb/>po&longs;&longs;anza in C, come il diametro C D a D F: e la po&longs;&longs;anza in C, <lb/>alla po&longs;&longs;anza in B, co me B A ad A C. <!-- KEEP S--></s> <s>Dunque la po&longs;&longs;anza in F alla <lb/>po&longs;&longs;anza in B, ha la ragion compo&longs;ta di C D a D F c di B A ad A C, <lb/>che è la ragion compo&longs;ta delle ragioni de diametri reciprocamente <lb/>pigliati. </s> <s>Jl che &longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>DEFINITIONE.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Momento della rota diciamo, il momento del pon <lb/>to po&longs;to nella circonferenza di e&longs;&longs;a rota. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITION.<emph.end type="italics"/><lb/>IIII.</s></p><p type="main"> <s>Se in vna congiogation di rote ineguali, o in più, <lb/>che la minor dell'vna congiogatione tocchi la mag­<lb/>gior dell'altra, &longs;i ponga la po&longs;&longs;anza in vna di dette <lb/>rote: &longs;arà il momento dell'vltima minor rota, maggior <lb/>del momento della prima maggior rota, &longs;econdo la ra­ <lb/>gion compo&longs;ta delli diametri. </s> <s>e la velocità &longs;arà mino<lb/>re, &longs;econdo l'i&longs;te&longs;&longs;a ragion de diametri. </s></p><p type="head"> <s><emph type="italics"/>Dimo&longs;tratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Siano le congiogationi di rote, de quali gli a&longs;si &longs;iano A e B: & in|­<lb/>tenda&longs;i &longs;u l'a&longs;&longs;e A e&longs;&longs;er la rota maggiore A C, e la minore A D,<emph.end type="italics"/><pb pagenum="64"/><figure id="fig18"/><lb/><emph type="italics"/>e &longs;u l' a&longs;&longs;e B, e&longs;&longs;er la maggiore D B, e la minore B E: e &longs;ia il contat <lb/>to della minore di vn'ordine, con la maggiore dell'altro, il ponto D: <lb/>e &longs;upponga&longs;i prima la po&longs;&longs;anza por&longs;i nella cir conferenza di A C. <!-- KEEP S--></s><lb/> <s>Dico che'l momento della rota A D, è maggiore del momento di <lb/>A C, secondo la ragione della linea C A ad A D. <!-- KEEP S--></s> <s>Si mo&longs;tra: per <lb/>cioche po&longs;ta in D una po&longs;&longs;anza di <expan abbr="mom&etilde;to">momento</expan> eguale alla po&longs;&longs;anza in C, <lb/>&longs;arà detta po&longs;&longs;anza in D, maggiore, che la po&longs;&longs;anza in C: ma ilmo <lb/>mento della rota, oue è po&longs;ta la po&longs;&longs;anza, è uguale ad e&longs;&longs;a po&longs;&longs;an­<lb/>za: &longs;arà dunque il <expan abbr="mom&etilde;to">momento</expan> della rota A D maggiore che della rota <lb/>A C &longs;econdo la ragion de diametri: que&longs;to in una congiogatione <lb/>& in più: per che il momento della circonferenza di A D è l'i&longs;te&longs;&longs;o <lb/>che della circonferenza di B D, per lo contatto, che fa communi­<lb/>canza: ma il momento della circonferenza di B E, è di forza <lb/>maggiore che di B D <expan abbr="&longs;ecõdo">&longs;econdo</expan> la ragione del diametro, B D a B E: <lb/>dunque fatta compo&longs;itione de ragioni il momento della circonferen <lb/>za di B E, è maggiore del momento della circonferenza di C A &longs;e <lb/>condo la ragion compo&longs;ta di B D a B E, e di C A ad A D. <!-- KEEP S--></s> <s>Il che <lb/>&longs;i hauea da mo&longs;trare.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Dico che la uelocità è minore nella i&longs;te&longs;&longs;a ragione: il che è mani-<emph.end type="italics"/><pb pagenum="65"/><emph type="italics"/>fe&longs;to: <expan abbr="percĩoche">percinoche</expan> la velocità delle rote, che nell'i&longs;te&longs;&longs;o tempo fini&longs;cono <lb/>il circuito, è proportionale alle circonferenze di e&longs;&longs;e rote: e le circon <lb/>ferenze &longs;ono di quantità proportionale alli diametri. </s> <s>Sono dunque le <lb/>velocità delle rote proportionali alli diametri. </s> <s>Jl che &longs;i hauea da <lb/>mo&longs;trare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITION.<emph.end type="italics"/><lb/>V.</s></p><p type="main"> <s>Date due po&longs;&longs;anze di momento contrario, l'vna mi<lb/>nore, el'altra maggiore: e data la ragione dell'vna al­<lb/>l'altra delle due rote congiogate: ritrouar il minor nu­<lb/>mero de congiogationi, &longs;iche la data po&longs;&longs;anza minore <lb/>vinca la maggiore. </s></p><figure/><pb pagenum="96"/><p type="head"> <s><emph type="italics"/>Dimostratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Siano le date po&longs;&longs;anze di momento contrario A, B: De quali A <lb/>&longs;ia la maggiore, c B la minore.: la ragion delle rote congiogate &longs;ia di <lb/>C a D: &longs;i cerca il minor numero de congiogationi, &longs;iche la po&longs;&longs;anza <lb/>B minore vinca la A maggiore. </s> <s>Piglin&longs;i nella ragione di C a D con <lb/>tinuamente le C, D, E, F: &longs;iche la C ad F habbia maggior ragione <lb/>che l' A a B: & eguale di numero all'interualli de termini &longs;i piglino <lb/>le congiogationi di rote G, H, I: e &longs;iano &longs;u l'a&longs;&longs;e G, le rote G K, G <lb/>L, &longs;u l'a&longs;&longs;e H le rote H L, H M: e &longs;u l'a&longs;&longs;e I le rote M I, I N. <!-- KEEP S--></s> <s>E ma <lb/>nife&longs;to che'l momento della <expan abbr="po&longs;sãza">po&longs;sanza</expan> in K, al momento &longs;uo in N, ha la <lb/>ragion compo&longs;ta delle ragioni de &longs;emidiametri: e perciò po&longs;ta la po&longs;­<lb/>&longs;anza maggiore A in N: e la minore B in K: &longs;ara il momento della B <lb/>in K, maggiore che'l momento dell' A in N. <!-- KEEP S--></s> <s>Il che &longs;i hauea da <lb/>trouare.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>PROPOSITION.<emph.end type="italics"/><lb/>VI.</s></p><p type="main"> <s>Data qual&longs;ivoglia tardità di po&longs;&longs;anza, & qual&longs;ivo <lb/>glia velocità: e data la ragion de diametri delle rote <expan abbr="cõ">com</expan> <lb/>giogate: ritrouar vn minimo numero de congiogatio <lb/>ni, &longs;i che la data po&longs;&longs;anza moua la co&longs;a con velocità <lb/>maggiore della data. </s></p><p type="head"> <s><emph type="italics"/>Dimostratione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sia la po&longs;&longs;anza tarda A, la veloce B, lo &longs;patio caminato da A in <lb/>vn dato tempo &longs;ia C, lo C caminato da B nell'i&longs;te&longs;&longs;o tempo &longs;ia D: la <lb/>ragion de diametri congiogati &longs;ia di E, ad F: bi&longs;ogna ritrouare il<emph.end type="italics"/><pb pagenum="67"/><figure id="fig19"/><lb/><emph type="italics"/>minimo numero de <expan abbr="cõgiogationi">congiogationi</expan>, col quale la tarda A moua con ve <lb/>locità maggior che'l B. </s> <s>Piglin&longs;i le E, F, G, continuate nella ragion <lb/>de diametri, che la prima volta l'interuallo della prima all'vltima <lb/>dico di G ad E, &longs;ia maggiore che di C a D: e quanti interualli &longs;ono il <lb/>E, F, G: tante congiogationi di rote &longs;i piglino nella i&longs;te&longs;&longs;a ragione: l'a <lb/>&longs;e de quali &longs;iano H, I: e nello a&longs;&longs;e H, la minor rota &longs;ia H K, la maggio <lb/>re H L: e nell'a&longs;&longs;e I laminore I L, la maggiore L M. il contatto del <lb/>l'vna congiogatione all'altra il ponto L: è manife&longs;to che la veloci­<lb/>tà del ponto M, alla velocità del ponto K, è compo&longs;ta della ragion del <lb/>li diametri M I, ad I L, & H L ad H K: che è l'i&longs;te&longs;&longs;a, che di G ad E: <lb/>ma G ad E, è di maggior interuallo che di D a C. <!-- KEEP S--></s> <s><expan abbr="Dũque">Dunque</expan>, po&longs;ta la po&longs;&longs;an <lb/>za tarda in K, la co&longs;a mo&longs;&longs;a con la rirconferenza M, &longs;i mouerà <expan abbr="cõ">com</expan> mag <lb/>gior velocità della data. </s> <s>Il che &longs;i hauea da trouare.<emph.end type="italics"/></s></p><pb pagenum="68"/><p type="head"> <s><emph type="italics"/>MOMENTI ACQVISTATI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Poniamo degli momenti, altri e&longs;&longs;er intrin&longs;echi: al <lb/>tri acqui&longs;tati, & altri mi&longs;ti: & intrin&longs;echi quelli, che <lb/>non da mouimento precedente dipendono: come &longs;ono <lb/>gli mouimenti delle grauezze in giù, e del corpo leggiero <lb/>dentro l'humor più graue in sù. </s> <s>Acqui&longs;tati quelli, che &longs;e­<lb/>guono l'impre&longs;sion fatta da precedente mouimento: come <lb/>il mouimento della co&longs;a lanciata, che &longs;egue il <expan abbr="mouim&etilde;to">mouimento</expan> <lb/>del braccio, o della corda. </s> <s>Mi&longs;ti, come il mouimento delle <lb/>grauezze dopo l'hauer dato principio a mouer&longs;i: per il che <lb/>veggiamo li pe&longs;i di vicino la&longs;ciati, mouer&longs;i con minor mo­<lb/>mento, che la&longs;ciati di lontano: e molte co&longs;e portate dalla <lb/>propria grauezza nell'aria penetrar &longs;otto l'accqua, con­<lb/>tro di quel che porta l'intrin&longs;eco momento: onde dopo <lb/>l'e&longs;&longs;ere affondate da &longs;e &longs;te&longs;si ritornar á galla. </s> <s>Et il momen­ <lb/>to intrin&longs;eco e&longs;&longs;er l'i&longs;te&longs;&longs;o &longs;empre. </s> <s>l'acqui&longs;tato, mancando <lb/>la cau&longs;a di poner&longs;i, e con il tempo, e dall'impedimento che <lb/>le faccia re&longs;i&longs;tenza. <emph type="italics"/>CVGNO.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Il cugno perco&longs;&longs;o, con&longs;iderato in vn modo, rappre&longs;enta <lb/>un piano inchinato, che &longs;i &longs;pinga &longs;otto il pe&longs;o. </s> <s>Et altrimen <lb/>te rappre&longs;enta due leue, che nelle loro &longs;tremità, facciano <lb/>l'vna all'altra &longs;ottoleua, & habbiano il pe&longs;o tra la po&longs;&longs;anza, <lb/>e'l &longs;ottoleua. </s> <s>Et altrimente rappre&longs;enta leua nel cui &longs;tremo <lb/>&longs;ia il pe&longs;o, & il &longs;ottoleua tramezzo. <emph type="italics"/>VITE E CHIOCCIA.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>La vite, o chioccia rappre&longs;enta vno o più piani auuolti <lb/>ad vn fu&longs;ello. </s> <s>Sono e ma&longs;chia, e femina: de quali vna &longs;tan­ <lb/>do ferma, l'altra che gira &longs;o&longs;tiene il pe&longs;o. </s> <s>acqui&longs;ta dunque for <lb/>za, <expan abbr="&longs;ecõdo">&longs;econdo</expan> la detta inchinazione, e &longs;econdo la lunghezza del <lb/>raggio che &longs;e le accompagna. </s> <s>Vite perpetua diciamo vn <lb/><gap/>ympano con denti a vite, che girando tocchi rota dentata. </s><lb/> <s>Per il che accre&longs;ce la forza, e per la proprieta della vite, <lb/>e della congiogatione delle rote. </s></p><p type="head"> <s><emph type="italics"/><gap/>L FINS.<emph.end type="italics"/><!-- KEEP S--></s></p> </chap> </body> <back/> 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