Mercurial > hg > mpdl-xml-content
view texts/archimedesOldCVSRepository/archimedes/xml/stell_mecha_041_it_1597.xml @ 12:f9a6b8344c3a
DESpecs 2.0 Autumn 2009
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 11:14:40 +0200 |
| parents | 22d6a63640c6 |
| children |
line wrap: on
line source
<?xml version="1.0" encoding="UTF-8"?> <!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_041_it_1597.xml</cvs_file> <cvs_version/> <locator>041.xml</locator> </info> <text> <front> <section> <pb xlink:href="041/01/001.jpg" id="p.0001"/> <p id="N1001B" type="head"> <s id="N1001D">DE GLI <lb/>ELEMENTI <lb/>MECHANICI<lb/><figure id="id.041.01.001.1.jpg" xlink:href="041/01/001/1.jpg"/><lb/></s> <s>La &longs;tatera. </s> <s>Leua. <lb/></s> <s id="N1002B">Raggi nell'a&longs;&longs;e. </s> <s id="N1002D"> Rote vettiue <lb/></s> <s id="N10030">Taglia. </s> <s id="N10032">Rote motiue. <lb/></s> <s id="N10035">Cugno. </s> <s id="N10037"> Vite. </s></p><p id="N10039" type="head"> <s id="N1003B"><emph type="italics"/>DI C. ANTONIO STELLIOLA. <emph.end type="italics"/></s></p><figure id="id.041.01.001.2.jpg" xlink:href="041/01/001/2.jpg"/> <p id="N10044" type="head"> <s id="N10046">IN NAPOLI, Nella Stamparia à Porta Regale <lb/>M. D. XCVII. </s></p> </section> <pb xlink:href="041/01/002.jpg" pagenum="1"/> <section> <p id="N1004E" type="head"> <s id="N10050"><emph type="italics"/>PROPOSITIONE <lb/>di tutta l'opera. <emph.end type="italics"/></s></p><p id="N10058" type="main"> <s id="N1005A">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 id="N10062">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 id="N1006A">Qual &longs;oggetto communemente <lb/>gli antichi chiamarono Mechaniche. </s> <s id="N1006E">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 id="id.041.01.002.1.jpg" xlink:href="041/01/002/1.jpg"/> </section> <pb xlink:href="041/01/003.jpg" pagenum="2"/> <section> <p id="N10083" type="head"> <s id="N10085"><emph type="italics"/>DEFINITIONI. <emph.end type="italics"/><lb/>I. </s></p><p id="N1008D" type="main"> <s id="N1008F">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 id="N10093" type="head"> <s id="N10095">II. </s></p><p id="N10097" type="main"> <s id="N10099">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 id="N1009F" type="head"> <s id="N100A1">III. </s></p><p id="N100A3" type="main"> <s id="N100A5">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 id="N100AB" type="head"> <s id="N100AD">IIII. </s></p><p id="N100AF" type="main"> <s id="N100B1">Linea di momento diciamo, per cui il centro di pe­<lb/>&longs;o della grauezza da impedimento libera &longs;i moue. </s></p><p id="N100B5" type="head"> <s id="N100B7">V. </s></p><p id="N100B9" type="main"> <s id="N100BB">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 id="N100C1" type="head"> <s id="N100C3">VI. </s></p><p id="N100C5" type="main"> <s id="N100C7">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 id="N100CB" type="head"> <s id="N100CD">VII. </s></p><p id="N100CF" type="main"> <s id="N100D1">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 id="N100D7" type="head"> <s id="N100D9">VIII. </s></p><p id="N100DB" type="main"> <s id="N100DD">E ponto di appen&longs;ione: il ponto, onde perde la gra­<pb xlink:href="041/01/004.jpg" 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 id="N100E5" type="head"> <s id="N100E7">IX. </s></p><p id="N100E9" type="main"> <s id="N100EB">Et Horizonte de pe&longs;i la &longs;uperficie in cui le linee de <lb/>momenti tutte vanno perpendicolarmente. </s></p><p id="N100EF" type="head"> <s id="N100F1"><emph type="italics"/>Appendice. <emph.end type="italics"/></s></p><p id="N100F7" type="main"> <s id="N100F9">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 id="N100FF" type="head"> <s id="N10101"><emph type="italics"/>POSITIONI. <emph.end type="italics"/><lb/>I. </s></p><p id="N10109" type="main"> <s id="N1010B">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. </s></p><p id="N10111" type="head"> <s id="N10113">II. </s></p><p id="N10115" type="main"> <s id="N10117">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 id="N1011D" type="head"> <s id="N1011F">III. </s></p><p id="N10121" type="main"> <s id="N10123">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 id="N10127" type="head"> <s id="N10129">IIII. </s></p><p id="N1012B" type="main"> <s id="N1012D">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> </section> </front> <pb xlink:href="041/01/005.jpg" pagenum="4"/> <body> <chap id="N1013B"> <p id="N1013C" type="head"> <s id="N1013E"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>I. </s></p><p id="N10146" type="main"> <s id="N10148">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 id="id.041.01.005.1.jpg" xlink:href="041/01/005/1.jpg"/> <p id="N10153" type="head"> <s id="N10155"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N1015B" type="main"> <s id="N1015D"><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. </s> <s id="N10165">dico che la re&longs;tan­<lb/>te H F, è vguale ad A; e la G D, eguale a B. </s> <s id="N10169">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/>ma le A, e C G eguali: le re&longs;tanti, B, e G D di con&longs;eguenza &longs;o­<lb/>no eguali. </s> <s id="N10171">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 id="N10177">è adunque la H F eguale a C G: e la G D <lb/>eguale ad E H. </s> <s id="N1017B">il che hauea da mo&longs;trar&longs;i. <emph.end type="italics"/></s></p><p id="N1017F" type="head"> <s id="N10181"><emph type="italics"/>Appendice. <emph.end type="italics"/></s></p><p id="N10187" type="main"> <s id="N10189">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 xlink:href="041/01/006.jpg" pagenum="5"/> <p id="N10190" type="main"> <s id="N10192">Percioche e&longs;&longs;endo le C G H F eguali. </s> <s id="N10194">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> </chap> <chap id="N1019F"> <p id="N101A0" type="head"> <s id="N101A2"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>II. </s></p><p id="N101AA" type="main"> <s id="N101AC">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 di tutto il compo&longs;to, di di&longs;tanze proportio­<lb/>nali alli pe&longs;i, pigliati reciprocamente. </s></p><figure id="id.041.01.006.1.jpg" xlink:href="041/01/006/1.jpg"/> <p id="N101B7" type="head"> <s id="N101B9"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N101BF" type="main"> <s id="N101C1"><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'l pe&longs;o di B C al pe&longs;o di C A. </s> <s id="N101DA">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"/><emph.end type="italics"/><pb xlink:href="041/01/007.jpg" 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. </s> <s id="N101E8">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. </s> <s id="N101F2">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. </s><lb/> <s id="N101F7">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. </s> <s id="N101FF">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> </chap> <chap id="N10205"> <p id="N10206" type="head"> <s id="N10208"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>III. </s></p><p id="N10210" type="main"> <s id="N10212">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;tatera dal ponto del­<lb/>la diui&longs;ione, &longs;ta in equilibrio. </s></p><figure id="id.041.01.007.1.jpg" xlink:href="041/01/007/1.jpg"/> <p id="N1021D" type="head"> <s id="N1021F"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10225" type="main"> <s id="N10227"><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 nel ponto E dico che &longs;o&longs;pe&longs;a <emph.end type="italics"/><pb xlink:href="041/01/008.jpg" pagenum="7"/><emph type="italics"/>la &longs;tatera nel ponto E, &longs;ta in equilibrio. </s> <s id="N10237">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. </s> <s id="N1023F">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. </s> <s id="N10253">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 id="N1025B" type="head"> <s id="N1025D"><emph type="italics"/>Appendice. <emph.end type="italics"/> I. </s></p><p id="N10264" type="main"> <s id="N10266">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 id="N1026C" type="head"> <s id="N1026E"><emph type="italics"/>Appendice. <emph.end type="italics"/> II. </s></p><p id="N10275" type="main"> <s id="N10277">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 <expan abbr="mo­m&etilde;to">mo­<lb/>mento</expan>, che &longs;e in detto ponto <expan abbr="giuntam&etilde;te">giuntamente</expan> fu&longs;&longs;ero appe&longs;e. </s></p> </chap> <chap id="N10287"> <p id="N10288" type="head"> <s id="N1028A"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>IIII. </s></p><p id="N10292" type="main"> <s id="N10294">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 xlink:href="041/01/009.jpg" 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 id="id.041.01.009.1.jpg" xlink:href="041/01/009/1.jpg"/> <p id="N102A1" type="head"> <s id="N102A3"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N102A9" type="main"> <s id="N102AB"><emph type="italics"/>Sia la &longs;tatera A B: il ponto della &longs;o&longs;pen&longs;ione C, li ponti onde &longs;ono <lb/>appe&longs;e le grauezze che fanno equipondio A & B le grauezze appe&longs;e <lb/>D & E. </s> <s id="N102B3">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. </s> <s id="N102BF">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. </s> <s id="N102CB">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 id="N102D1">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> </chap> <chap id="N102D7"> <p id="N102D8" type="head"> <s id="N102DA"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>V. </s></p><p id="N102E2" type="main"> <s id="N102E4">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 xlink:href="041/01/010.jpg" pagenum="9"/> <figure id="id.041.01.010.1.jpg" xlink:href="041/01/010/1.jpg"/> <p id="N102F0" type="head"> <s id="N102F2"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N102F8" type="main"> <s id="N102FA"><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 id="N10302">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. </s> <s id="N10306">&longs;i mo&longs;tra perche D & E, <lb/>fanno equipondio. </s> <s id="N1030A">& 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 id="N10312">Ha&longs;&longs;i dunque l'in­<lb/>tento. <emph.end type="italics"/></s></p> </chap> <chap id="N10318"> <p id="N10319" type="head"> <s id="N1031B"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>VI. </s></p><p id="N10323" type="main"> <s id="N10325">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 id="N10329" type="head"> <s id="N1032B"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10331" type="main"> <s id="N10333"><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 id="N1033B">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 xlink:href="041/01/011.jpg" pagenum="10"/><figure id="id.041.01.011.1.jpg" xlink:href="041/01/011/1.jpg"/><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 id="N1034E">Et harra&longs;&longs;i l'intento. <emph.end type="italics"/></s></p><p id="N10352" type="main"> <s id="N10354"><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 id="N10360">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> </chap> <chap id="N10366"> <p id="N10367" type="head"> <s id="N10369"><emph type="italics"/>PROPOSITIONE<emph.end type="italics"/><lb/> VII. </s></p><p id="N10371" type="main"> <s id="N10373">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 id="N1037D" type="head"> <s id="N1037F"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10385" type="main"> <s id="N10387"><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="põto">pon<lb/>to</expan> 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 xlink:href="041/01/012.jpg" pagenum="11"/><figure id="id.041.01.012.1.jpg" xlink:href="041/01/012/1.jpg"/><lb/><emph type="italics"/> e&longs;tremi termini &longs;iano uguali. </s> <s id="N103AF">&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. </s> <s>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. </s> <s id="N103B9">hanno dunque li due termini F et H l'i­<lb/>&longs;te&longs;&longs;a ragione al termine G. </s> <s>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> </chap> <chap id="N103C1"> <p id="N103C2" type="head"> <s id="N103C4"><emph type="italics"/>PROPOSITIONE<emph.end type="italics"/><lb/> VIII. </s></p><p id="N103CC" type="main"> <s id="N103CE">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 id="N103D4" type="head"> <s id="N103D6"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N103DC" type="main"> <s id="N103DE"><emph type="italics"/>Sia la &longs;tatera A B, il ponto del &longs;o&longs;tenimento C, le grauezze uguali D <lb/>& E. </s> <s id="N103E4">de quali il D &longs;ia appe&longs;o in A, & l'c in F. </s> <s id="N103E6">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. </s> <s id="N103EC">&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 grauezze, <emph.end type="italics"/><pb xlink:href="041/01/013.jpg" pagenum="12"/><figure id="id.041.01.013.1.jpg" xlink:href="041/01/013/1.jpg"/><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. </s> <s id="N10406">& &longs;ia H. </s> <s id="N10408">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. </s> <s id="N1040C">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. </s> <s id="N10412">&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. </s> <s id="N1041A">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. </s> <s id="N1041E">il che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p> </chap> <chap id="N10422"> <p id="N10423" type="head"> <s id="N10425"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>IX. </s></p><p id="N1042D" type="main"> <s id="N1042F">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 id="N10435" type="head"> <s id="N10437"><emph type="italics"/>Dimostratione. <emph.end type="italics"/></s></p><p id="N1043D" type="main"> <s id="N1043F"><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. </s> <s id="N10450">dico che la ragione del <expan abbr="mom&etilde;to">momento</expan> D al <expan abbr="mom&etilde;to">momen<lb/>to</expan> 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. </s> <s id="N1045E">&longs;i mo&longs;tra appenda <lb/>&longs;i da B la grauezza G che faccia equipondio. </s> <s id="N10462">a D, & il pe&longs;o H che fac­<emph.end type="italics"/><pb xlink:href="041/01/014.jpg" pagenum="13"/><figure id="id.041.01.014.1.jpg" xlink:href="041/01/014/1.jpg"/><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. </s> <s id="N10477">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 id="N10483">e&longs;&longs;endo dunque tre ter<lb/>mini in continua habitudine il momento D, il momento I, & il <expan abbr="mom&etilde;­to">momen­<lb/>to</expan> 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 id="N10493">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 id="N1049D">Il che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p> </chap> <chap id="N104A1"> <p id="N104A2" type="head"> <s id="N104A4"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>X. </s></p><p id="N104AC" type="main"> <s id="N104AE">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 xlink:href="041/01/015.jpg" pagenum="14"/> <figure id="id.041.01.015.1.jpg" xlink:href="041/01/015/1.jpg"/> <p id="N104BC" type="head"> <s id="N104BE"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N104C4" type="main"> <s id="N104C6"><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 id="N104CE">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 id="N104DA">Ha&longs;&longs;i dunque l'intento. <emph.end type="italics"/></s></p> </chap> <chap id="N104DE"> <p id="N104DF" type="head"> <s id="N104E1"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>XI. </s></p><p id="N104E9" type="main"> <s id="N104EB">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 id="N104F3" type="head"> <s id="N104F5"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N104FB" type="main"> <s id="N104FD"><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 xlink:href="041/01/016.jpg" pagenum="15"/><figure id="id.041.01.016.1.jpg" xlink:href="041/01/016/1.jpg"/><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. </s> <s>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 id="N10518">&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 id="N1052A">&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 id="N1052E">Il che &longs;i hauea da ritrouare. <emph.end type="italics"/></s></p> </chap> <chap id="N10532"> <p id="N10533" type="head"> <s id="N10535"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>XII. </s></p><p id="N1053D" type="main"> <s id="N1053F">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 id="N10547" type="head"> <s id="N10549"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N1054F" type="main"> <s id="N10551"><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 xlink:href="041/01/017.jpg" pagenum="16"/><figure id="id.041.01.017.1.jpg" xlink:href="041/01/017/1.jpg"/><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 id="N10568">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">con</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 xlink:href="041/01/018.jpg" 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> </chap> <chap id="N105A9"> <p id="N105AA" type="head"> <s id="N105AC"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>XIII. </s></p><p id="N105B4" type="main"> <s id="N105B6">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 id="id.041.01.018.1.jpg" xlink:href="041/01/018/1.jpg"/> <p id="N105BD" type="head"> <s id="N105BF"><emph type="italics"/>Dimostratione. <emph.end type="italics"/></s></p><p id="N105C5" type="main"> <s id="N105C7"><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;i e &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 xlink:href="041/01/019.jpg" 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 id="N105D9">&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. </s> <s id="N105E1">&longs;ono <lb/>dunque, G I H K parallele. </s> <s id="N105E5">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. </s> <s id="N105ED">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 id="N105F3">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 id="N105F9">Ilche &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p> </chap> <chap id="N105FD"> <p id="N105FE" type="head"> <s id="N10600"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>XV. </s></p><p id="N10608" type="main"> <s id="N1060A">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 id="id.041.01.019.1.jpg" xlink:href="041/01/019/1.jpg"/> <p id="N10615" type="head"> <s id="N10617"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N1061D" type="main"> <s id="N1061F"><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;i reci­<emph.end type="italics"/><pb xlink:href="041/01/020.jpg" pagenum="19"/><emph type="italics"/>procamente 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. </s> <s id="N10631">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 id="id.041.01.020.1.jpg" xlink:href="041/01/020/1.jpg"/> <pb xlink:href="041/01/021.jpg" pagenum="20"/> <p id="N1064B" type="head"> <s id="N1064D">VETTE, E <lb/>LEVA. </s></p><p id="N10651" type="head"> <s id="N10653"><emph type="italics"/>DEFINITION. <emph.end type="italics"/><lb/>I. </s></p><p id="N1065B" type="main"> <s id="N1065D">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 id="N10661" type="head"> <s id="N10663"><emph type="italics"/>DEFINITION. <emph.end type="italics"/><lb/>II. </s></p><p id="N1066B" type="main"> <s id="N1066D">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 id="N10675" type="head"> <s id="N10677"><emph type="italics"/>POSITION. <emph.end type="italics"/><lb/>I. </s></p><p id="N1067F" type="main"> <s id="N10681">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 id="N10687" type="head"> <s id="N10689"><emph type="italics"/>POSITION. <emph.end type="italics"/><lb/>II. </s></p><p id="N10691" type="main"> <s id="N10693">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 id="id.041.01.021.1.jpg" xlink:href="041/01/021/1.jpg"/> <pb xlink:href="041/01/022.jpg" pagenum="21"/> <p id="N1069D" type="head"> <s id="N1069F"><emph type="italics"/>PROPOSITION. <emph.end type="italics"/><lb/>I. </s></p><p id="N106A7" type="main"> <s id="N106A9">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 id="id.041.01.022.1.jpg" xlink:href="041/01/022/1.jpg"/> <p id="N106B4" type="head"> <s id="N106B6"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N106BC" type="main"> <s id="N106BE"><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 grauez­<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 id="N106C8">&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 D ha la <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 id="N106D2">ilche &longs;i ha­<lb/>uea da mo&longs;trare. <emph.end type="italics"/></s></p><p id="N106D8" type="head"> <s id="N106DA"><emph type="italics"/>PROPOSITION. <emph.end type="italics"/><lb/>II. </s></p><p id="N106E2" type="main"> <s id="N106E4">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 xlink:href="041/01/023.jpg" pagenum="22"/> <figure id="id.041.01.023.1.jpg" xlink:href="041/01/023/1.jpg"/> <p id="N106F0" type="head"> <s id="N106F2"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N106F8" type="main"> <s id="N106FA"><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: aggiunga&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 id="N1070C">&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 id="N10712">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 id="N1071C">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. </s> <s id="N10722">Ma G E ad E F, ha la ragione che <lb/>l'interuallo B D, a D A che è reciproca. </s> <s id="N10726">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 id="N1072C">ilche &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p> <pb xlink:href="041/01/024.jpg" pagenum="23"/> </chap> <chap id="N10733"> <p id="N10734" type="head"> <s id="N10736"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>III. </s></p><p id="N1073E" type="main"> <s id="N10740">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 id="id.041.01.024.1.jpg" xlink:href="041/01/024/1.jpg"/> <p id="N1074F" type="head"> <s id="N10751"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10757" type="main"> <s id="N10759"><emph type="italics"/>Sia la leua A B il &longs;otto leua 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. </s> <s id="N10765">dunque <lb/>il pe&longs;o è &longs;o&longs;tentato dalle potenze in B & A compartitamente, cioe <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 id="N10776">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 id="N1077A">&longs;i ha dunque l'intento. <emph.end type="italics"/></s></p> <pb xlink:href="041/01/025.jpg" pagenum="24"/> </chap> <chap id="N10781"> <p id="N10782" type="head"> <s id="N10784"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>IV. </s></p><p id="N1078C" type="main"> <s id="N1078E">Se vna grauezza &longs;ia con vna leua &longs;o&longs;tenuta da due <lb/>ponti; & accre&longs;ciuta 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 id="id.041.01.025.1.jpg" xlink:href="041/01/025/1.jpg"/> <p id="N107A3" type="head"> <s id="N107A5"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N107AB" type="main"> <s id="N107AD"><emph type="italics"/>Sia la leua A B, il &longs;ottoleua in A: la grauezza &longs;o&longs;tenuta in C, la <lb/>po&longs;&longs;anza che'l &longs;o&longs;tiene in B. </s> <s>& allungata la B A in vn D, appenda­<lb/>&longs;i in D, vna grauezza che &longs;o&longs;tenti la grauezza C. </s> <s id="N107B5">dico che in que&longs;ta <lb/>commutatione il &longs;ottoleua A &longs;o&longs;tenti pe&longs;o maggiore, & che il pe&longs;o <lb/>&longs;o&longs;tenuto in detta commutatione, al pe&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 id="N107BF">&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 id="N107C3">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 xlink:href="041/01/026.jpg" 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 id="N107D6">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 id="N107DC">Ilche &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p> </chap> <chap id="N107E0"> <p id="N107E1" type="head"> <s id="N107E3"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>V. </s></p><p id="N107EB" type="main"> <s id="N107ED">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 id="id.041.01.026.1.jpg" xlink:href="041/01/026/1.jpg"/> <p id="N107FA" type="head"> <s id="N107FC"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10802" type="main"> <s id="N10804"><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="poßãanze">poßan<lb/>ze</expan> &longs;o&longs;tenuto. </s> <s id="N10812">per il che dico: che &longs;e in quella ragione, c' ha la po&longs;&longs;an­<emph.end type="italics"/><pb xlink:href="041/01/027.jpg" pagenum="26"/><emph type="italics"/>za B, &longs;i diuida la vette AB in D, e &longs;ia come la po&longs;&longs;anza A alla 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, che &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: dunque 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 in B: e percio <lb/>le po&longs;&longs;anze &longs;o&longs;tentaranno il detto pe&longs;o nel ponto D. </s> <s id="N1083E">Il che &longs;i hauea da <lb/>mo&longs;trare,<emph.end type="italics"/></s></p><p id="N10844" type="head"> <s id="N10846"><emph type="italics"/>Appendice. <emph.end type="italics"/></s></p><p id="N1084C" type="main"> <s id="N1084E">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'altra po&longs;&longs;anza, ver&longs;o oue &longs;arà portato. </s></p> </chap> <chap id="N10854"> <p id="N10855" type="head"> <s id="N10857"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>VI. </s></p><p id="N1085F" type="main"> <s id="N10861">Se una leua &longs;ia inalzata, o ba&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 xlink:href="041/01/028.jpg" pagenum="27"/> <figure id="id.041.01.028.1.jpg" xlink:href="041/01/028/1.jpg"/> <p id="N10871" type="head"> <s id="N10873"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10879" type="main"> <s id="N1087B"><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. </s> <s>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. </s> <s id="N10885">dico che l'angolo fatto dalle due D E, D F <lb/>&longs;ia vguale all'angolo 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'incontro 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. </s> <s id="N10891">Il che &longs;i hauea da mò&longs;trare. <emph.end type="italics"/></s></p><p id="N10895" type="head"> <s id="N10897"><emph type="italics"/>Appendice,<emph.end type="italics"/></s></p><p id="N1089D" type="main"> <s id="N1089F">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 id="N108A6">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 xlink:href="041/01/029.jpg" pagenum="28"/> </chap> <chap id="N108B3"> <p id="N108B4" type="head"> <s id="N108B6"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>VII. </s></p><p id="N108BE" type="main"> <s id="N108C0">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 id="id.041.01.029.1.jpg" xlink:href="041/01/029/1.jpg"/> <p id="N108C9" type="head"> <s id="N108CB"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N108D1" type="main"> <s id="N108D3"><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 id="N108DD">&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 id="N108E5">&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, le linee 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;te&longs;&longs;o 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 xlink:href="041/01/030.jpg" pagenum="29"/><emph type="italics"/>&longs;tegno in G, la po&longs;&longs;anza F, &longs;o&longs;tiene tal portione di tutto il pe&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 di tutta 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 id="N10909">Il che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p><p id="N1090D" type="head"> <s id="N1090F"><emph type="italics"/>Appendice. </s> <s id="N10913">I. <emph.end type="italics"/></s></p><p id="N10917" type="main"> <s id="N10919">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 id="N1091D" type="head"> <s id="N1091F"><emph type="italics"/>Appendice. </s> <s id="N10923">II. <emph.end type="italics"/></s></p><p id="N10927" type="main"> <s id="N10929">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 id="N1092D" type="head"> <s id="N1092F"><emph type="italics"/>Appendice. </s> <s id="N10933">III. <emph.end type="italics"/></s></p><p id="N10937" type="main"> <s id="N10939">E che nelle leue ba&longs;&longs;ate &longs;otto l'orizonte, auuenga a <lb/>contrario. </s></p> </chap> <chap id="N1093D"> <p id="N1093E" type="head"> <s id="N10940"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>VIII. </s></p><p id="N10948" type="main"> <s id="N1094A">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 xlink:href="041/01/031.jpg" pagenum="30"/> <figure id="id.041.01.031.1.jpg" xlink:href="041/01/031/1.jpg"/> <p id="N10958" type="head"> <s id="N1095A"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10960" type="main"> <s id="N10962"><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. </s> <s>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 id="N1096C">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. </s> <s id="N10970">dico che detta po&longs;&longs;anza in <lb/>F &longs;o&longs;tiene la grauezza in C. </s> <s id="N10974">&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. </s> <s id="N1097C">Il <lb/>che &longs;i cercaua. <emph.end type="italics"/></s></p><p id="N10982" type="head"> <s id="N10984"><emph type="italics"/>Appendice. <emph.end type="italics"/></s></p><p id="N1098A" type="main"> <s id="N1098C">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 id="id.041.01.031.2.jpg" xlink:href="041/01/031/2.jpg"/> <pb xlink:href="041/01/032.jpg" pagenum="31"/> <p id="N1099A" type="head"> <s id="N1099C">RAGGI NELL <lb/>ASSE. </s></p><p id="N109A0" type="head"> <s id="N109A2"><emph type="italics"/>SVPPOSITIONE. <emph.end type="italics"/></s></p><p id="N109A8" type="main"> <s id="N109AA">Svpponiamo, in vno i&longs;te&longs;&longs;o a&longs;&longs;e, due rag<lb/>gi c'habbiano nelli &longs;uoi &longs;tremi li centri de pe&longs;i. </s></p><p id="N109AE" type="main"> <s id="N109B0">E detti raggi, o in vna pianezza, e che non facciano <lb/>angolo, o in due, e che facciano angolo. </s></p><p id="N109B4" type="head"> <s id="N109B6"><emph type="italics"/>POSITIONE. <emph.end type="italics"/></s></p><p id="N109BC" type="main"> <s id="N109BE">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 id="id.041.01.032.1.jpg" xlink:href="041/01/032/1.jpg"/> </chap> <chap id="N109C7"> <p id="N109C8" type="head"> <s id="N109CA"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>I. </s></p><p id="N109D2" type="main"> <s id="N109D4">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 id="N109DA" type="head"> <s id="N109DC"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N109E2" type="main"> <s id="N109E4"><emph type="italics"/>Sial'a&longs;&longs;e A, a cui &longs;iano affi&longs;&longs;i li raggi A B, A C, quali s'intenda­<emph.end type="italics"/><pb xlink:href="041/01/033.jpg" pagenum="32"/><figure id="id.041.01.033.1.jpg" xlink:href="041/01/033/1.jpg"/><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/>ga in 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 id="N109FA">&longs;i mo&longs;tra: tiri&longs;i per 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 momenti e 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 id="N10A0E">Il che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p> <pb xlink:href="041/01/034.jpg" pagenum="33"/> </chap> <chap id="N10A15"> <p id="N10A16" type="head"> <s id="N10A18"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>II. </s></p><p id="N10A20" type="main"> <s id="N10A22">Date qual &longs;i uoglia due grauezze, nelli raggi che fac­<lb/>ciano angolo dato, ritrouar nelle loro circolationi, pon­<lb/>ti oue facciano equipondio. </s></p><figure id="id.041.01.034.1.jpg" xlink:href="041/01/034/1.jpg"/> <p id="N10A2B" type="head"> <s id="N10A2D"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10A33" type="main"> <s id="N10A35"><emph type="italics"/>Sia l'a&longs;&longs;e A: li raggi che facciano <expan abbr="ãgolo">angolo</expan> dato A C, B A: & intenda&longs;i <expan abbr="nel­li">nel­<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 xlink:href="041/01/035.jpg" pagenum="34"/><emph type="italics"/>conferenze li ponti, oue e&longs;&longs;endo dette grauezze, facciano equipondio. </s><lb/> <s id="N10A52">Diuida&longs;i la B C interuallo de centri, &longs;iche qual ragione ha la grauezza, <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 id="N10A5C">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 per l'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. </s> <s id="N10A64">ma il <expan abbr="põto">ponto</expan> D è il centro commu­<lb/>ne di pe&longs;o di dette due grauezze. </s> <s id="N10A6C">E dunque il centro commune nel <lb/>la perpendicolare del &longs;o&longs;tenimento: e perciò le grauezze &longs;tanno. </s> <s id="N10A70">Jl che <lb/>&longs;i cercaua. <emph.end type="italics"/></s></p><p id="N10A76" type="head"> <s id="N10A78"><emph type="italics"/>Appendice. </s> <s id="N10A7C">I. <emph.end type="italics"/></s></p><p id="N10A80" type="main"> <s id="N10A82">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 id="N10A8A" type="head"> <s id="N10A8C"><emph type="italics"/>Appendice. </s> <s id="N10A90">II. <emph.end type="italics"/></s></p><p id="N10A94" type="main"> <s id="N10A96">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;opra il ponto dell'equipondio ha momento minore. </s></p> </chap> <chap id="N10A9C"> <p id="N10A9D" type="head"> <s id="N10A9F"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>I. </s></p><p id="N10AA7" type="main"> <s id="N10AA9">Dàte qual &longs;i uoglia due grauezze nelli dati raggi, che <lb/>fanno dato angolo: ritrouar nelle loro circolationi, pon­<pb xlink:href="041/01/036.jpg" pagenum="35"/>ti oue il momento dell'uno, al <expan abbr="mom&etilde;to">momento</expan> dell'altro habbia <lb/>qual &longs;i voglia data ragione. </s></p><figure id="id.041.01.036.1.jpg" xlink:href="041/01/036/1.jpg"/> <p id="N10ABA" type="head"> <s id="N10ABC"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10AC2" type="main"> <s id="N10AC4"><emph type="italics"/>Siano le date grauezze A & B: li raggi 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. </s> <s>e <lb/>la data qual&longs;iuoglia ragione &longs;ia di F a G: &longs;i cercano nella circolatione DA <lb/>e nella BE, ponti oue habbian li momenti di A e B, ragion di F a G. </s> <s id="N10AD2">In­<lb/>tenda&longs;i nella ragion di A a B, la quantità F ad H. </s> <s id="N10AD6">e nella re&longs;tante ra­<lb/>gione di H a G, &longs;i diuida A B in L. </s> <s id="N10ADA">& all'angolo L C A faccia&longs;i vgua­<emph.end type="italics"/><pb xlink:href="041/01/037.jpg" 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. </s> <s id="N10AE6">& il ponto L nella perpendicolare C D. </s> <s id="N10AE8">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. </s> <s id="N10AF8"> <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 ragion data di F a G. </s> <s id="N10B03">Il che &longs;i cercaua. <emph.end type="italics"/></s></p><p id="N10B07" type="head"> <s id="N10B09"><emph type="italics"/>Appendice. <emph.end type="italics"/></s></p><p id="N10B0F" type="main"> <s id="N10B11">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: e non <lb/>altroue. </s></p><figure id="id.041.01.037.1.jpg" xlink:href="041/01/037/1.jpg"/> <pb xlink:href="041/01/038.jpg" pagenum="37"/> <p id="N10B1F" type="head"> <s id="N10B21">MOMENTI <lb/>CENTRALI</s></p><p id="N10B25" type="main"> <s id="N10B27">E qvanto 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 dette circolari &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 id="id.041.01.038.1.jpg" xlink:href="041/01/038/1.jpg"/> <pb xlink:href="041/01/039.jpg" pagenum="38"/> <p id="N10B3F" type="head"> <s id="N10B41">ROTE VET­<lb/>TIVE. </s></p><p id="N10B45" type="head"> <s id="N10B47"><emph type="italics"/>SVPPOSITIONE. <emph.end type="italics"/></s></p><p id="N10B4D" type="main"> <s id="N10B4F">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 id="N10B55" type="head"> <s id="N10B57"><emph type="italics"/>DEFINITION. <emph.end type="italics"/><lb/>I. </s></p><p id="N10B5F" type="main"> <s id="N10B61">Cogiogation &longs;emplice, diciamo delle rote, che &longs;ono <lb/>sù di vn'i&longs;te&longs;&longs;o a&longs;&longs;e. </s></p><p id="N10B65" type="head"> <s id="N10B67">I. </s></p><p id="N10B69" type="main"> <s id="N10B6B">Molteplice, delle rote che &longs;ono in più a&longs;&longs;i. </s></p><p id="N10B6D" type="head"> <s id="N10B6F">III. </s></p><p id="N10B71" type="main"> <s id="N10B73">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 xlink:href="041/01/040.jpg" pagenum="39"/> <p id="N10B80" type="head"> <s id="N10B82"><emph type="italics"/>POSITIONE. <emph.end type="italics"/><lb/>I. </s></p><p id="N10B8A" type="main"> <s id="N10B8C">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 id="N10B90" type="head"> <s id="N10B92">II. </s></p><p id="N10B94" type="main"> <s id="N10B96">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 id="id.041.01.040.1.jpg" xlink:href="041/01/040/1.jpg"/> </chap> <chap id="N10B9F"> <p id="N10BA0" type="head"> <s id="N10BA2"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>I. </s></p><p id="N10BAA" type="main"> <s id="N10BAC">Della rota 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 id="id.041.01.040.2.jpg" xlink:href="041/01/040/2.jpg"/> <pb xlink:href="041/01/041.jpg" pagenum="40"/> <p id="N10BB8" type="head"> <s id="N10BBA"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10BC0" type="main"> <s id="N10BC2"><emph type="italics"/>Sia la linea Orizontale 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 id="N10BD2">Il che &longs;i hauea da <lb/>mo&longs;trare,<emph.end type="italics"/></s></p><p id="N10BD8" type="head"> <s id="N10BDA"><emph type="italics"/>Appendice. </s> <s id="N10BDE">I. <emph.end type="italics"/></s></p><p id="N10BE2" type="main"> <s id="N10BE4">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 id="N10BE8" type="head"> <s id="N10BEA"><emph type="italics"/>Appendice, II. <emph.end type="italics"/></s></p><p id="N10BF0" type="main"> <s id="N10BF2">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">orizon</expan>tale, non habbian momen­<lb/>to ne ver&longs;o l'vna, ne ver&longs;o l'altra parte. </s></p><p id="N10BFC" type="head"> <s id="N10BFE"><emph type="italics"/>Appendice. </s> <s id="N10C02">III. <emph.end type="italics"/></s></p><p id="N10C06" type="main"> <s id="N10C08">E che perciò qual &longs;i voglia po&longs;&longs;anza, le porterà così <lb/>nell'vna, come nell'altra parte, </s></p> <pb xlink:href="041/01/042.jpg" pagenum="41"/> </chap> <chap id="N10C0F"> <p id="N10C10" type="head"> <s id="N10C12"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>II. </s></p><p id="N10C1A" type="main"> <s id="N10C1C">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 id="N10C21">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 id="id.041.01.042.1.jpg" xlink:href="041/01/042/1.jpg"/> <pb xlink:href="041/01/043.jpg" pagenum="42"/> <p id="N10C2D" type="head"> <s id="N10C2F"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10C35" type="main"> <s id="N10C37"><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 id="N10C43">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 id="N10C4B">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 id="N10C53">De &longs;criua&longs;i alla D E, la por<lb/>tione di circolo D H F, &longs;imile a D E F; &longs;aranno dette portioni vgua­<lb/>li, e faranno equipondio. </s> <s id="N10C59">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 id="N10C61">Il che <lb/>&longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p><p id="N10C67" type="head"> <s id="N10C69"><emph type="italics"/>Appendice. </s> <s id="N10C6D">I. <emph.end type="italics"/></s></p><p id="N10C71" type="main"> <s id="N10C73">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 id="N10C77" type="main"> <s id="N10C79"><emph type="italics"/>Percio che &longs;e in vece del pe&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 id="N10C86" type="head"> <s id="N10C88"><emph type="italics"/>Appendice, II. <emph.end type="italics"/></s></p><p id="N10C8E" type="main"> <s id="N10C90">Et è manife&longs;to che detta rota correrà ver&longs;o la parte <lb/>del piano inferiore. </s></p><p id="N10C94" type="main"> <s id="N10C96"><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 xlink:href="041/01/044.jpg" 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> </chap> <chap id="N10CD4"> <p id="N10CD5" type="head"> <s id="N10CD7"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>III. </s></p><p id="N10CDF" type="main"> <s id="N10CE1">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 id="id.041.01.044.1.jpg" xlink:href="041/01/044/1.jpg"/> <p id="N10CF0" type="head"> <s id="N10CF2"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10CF8" type="main"> <s id="N10CFA"><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 id="N10D06">& 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"/></s> <pb xlink:href="041/01/045.jpg" pagenum="44"/> <s><emph type="italics"/><emph type="italics"/>Dico che'l detto 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. </s> <s id="N10D14">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: e percio &longs;e alla linea DE, s'intenda fatta application di <lb/>corpo: il momento di tutto &longs;arà nel ponto C. </s> <s id="N10D26">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 id="N10D2E">&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. </s> <s id="N10D34">Il che &longs;i hauea da mostrare. <emph.end type="italics"/></s></p> </chap> <chap id="N10D38"> <p id="N10D39" type="head"> <s id="N10D3B"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>IIII. </s></p><p id="N10D43" type="main"> <s id="N10D45">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 delle rote inferiori di <lb/>detto pe&longs;o, maggior portione che &longs;e fu&longs;&longs;e nel piano ori<lb/>zontale. </s></p><p id="N10D55" type="head"> <s id="N10D57"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10D5D" type="main"> <s id="N10D5F"><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. </s> <s id="N10D6B">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 id="N10D71">& H, nell'in <lb/>chinato: e perche EH, è maggior portione di ED: che EG, e DH,<emph.end type="italics"/><pb xlink:href="041/01/046.jpg" 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 <expan abbr="&longs;ecõdo">&longs;econdo</expan> la ragione di DH ad ED: &longs;o&longs;tenta <expan abbr="dũ">dun</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 id="N10D8B">Il che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p><figure id="id.041.01.046.1.jpg" xlink:href="041/01/046/1.jpg"/> </chap> <chap id="N10D92"> <p id="N10D93" type="head"> <s id="N10D95"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>V. </s></p><p id="N10D9D" type="main"> <s id="N10D9F">Data la rota che <expan abbr="affõdi">affondi</expan> in <expan abbr="cõcauita">concauita</expan> &longs;otto il piano <expan abbr="orizõtale">orizon<pb xlink:href="041/01/047.jpg" pagenum="46"/>tale</expan>: 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 id="id.041.01.047.1.jpg" xlink:href="041/01/047/1.jpg"/> <p id="N10DB8" type="head"> <s id="N10DBA"><emph type="italics"/>Dimostratione. <emph.end type="italics"/></s></p><p id="N10DC0" type="main"> <s id="N10DC2"><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. </s> <s id="N10DC8">&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 id="N10DCC">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 <expan abbr="perdendicolare">perpendicolare</expan> del &longs;o&longs;tenimento, che dal ponto I <lb/>la grauezza G, fa equipondio alla rota. </s> <s id="N10DE0">e che da ogni ponto oltre, la &longs;ol <lb/>leui, il che &longs;i cerca un. <emph.end type="italics"/></s></p> <pb xlink:href="041/01/048.jpg" pagenum="47"/> <p id="N10DE9" type="head"> <s id="N10DEB">TAGLIA. </s><lb/> <s id="N10DEE"><emph type="italics"/>SVPPOSITIONE. <emph.end type="italics"/></s></p><p id="N10DF4" type="main"> <s id="N10DF6">Svpponiamo la taglia c'habbia in &longs;e una, o più <lb/>girelle, o sia in vno o più ordini. </s> <s id="N10DFA">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 id="N10E00">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 id="N10E0A">In oltre &longs;upponiamo della corda auuolta il <lb/>capo andare, o alla taglia, o ad'vn termine fi&longs;&longs;o, o a po&longs;­<lb/>&longs;anza, ò a pe&longs;o. </s></p><p id="N10E10" type="head"> <s id="N10E12"><emph type="italics"/>POSITIONE <emph.end type="italics"/><lb/>I. </s></p><p id="N10E1A" type="main"> <s id="N10E1C">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 id="N10E22" type="head"> <s id="N10E24">II. </s></p><p id="N10E26" type="main"> <s id="N10E28">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 id="N10E2C" type="head"> <s id="N10E2E">III. </s></p><p id="N10E30" type="main"> <s id="N10E32">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 xlink:href="041/01/049.jpg" pagenum="48"/> </chap> <chap id="N10E39"> <p id="N10E3A" type="head"> <s id="N10E3C"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>I. </s></p><p id="N10E44" type="main"> <s id="N10E46">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 id="N10E4C">e la po&longs;&longs;anza della taglia &longs;o&longs;tenta <lb/>rà il doppio. </s></p><p id="N10E50" type="head"> <s id="N10E52"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10E58" type="main"> <s id="N10E5A"><emph type="italics"/>Sia la taglia AB: li capi della fune auuolta <emph.end type="italics"/><lb/><figure id="id.041.01.049.1.jpg" xlink:href="041/01/049/1.jpg"/><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 pe&longs;o: e <lb/>che la po&longs;&longs;anza in E, &longs;o&longs;tenta il doppio. </s> <s id="N10E6F">Si <lb/>mo&longs;tra: e prima che'l momento di D, &longs;ia v­<lb/>guale al momento di C. </s> <s>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 id="N10E7B">Dico hora che la po&longs;&longs;anza 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 doppio 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 id="N10E8F">Ha&longs;&longs;i dunque il <lb/>propo&longs;to, che la poßanza 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 xlink:href="041/01/050.jpg" pagenum="49"/> </chap> <chap id="N10E9A"> <p id="N10E9B" type="head"> <s id="N10E9D"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>II. </s></p><p id="N10EA5" type="main"> <s id="N10EA7">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 id="N10EAD" type="head"> <s id="N10EAF"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10EB5" type="main"> <s id="N10EB7"><emph type="italics"/>Sia la taglia A B, a cui &longs;ia attaccato il pe<emph.end type="italics"/><lb/><figure id="id.041.01.050.1.jpg" xlink:href="041/01/050/1.jpg"/><lb/><emph type="italics"/>&longs;o C: 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 id="N10ECE">&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 id="N10EE0">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 id="N10EE6">&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 xlink:href="041/01/051.jpg" pagenum="50"/> <p id="N10EF3" type="head"> <s id="N10EF5"><emph type="italics"/>Appendice,<emph.end type="italics"/></s></p><p id="N10EFB" type="main"> <s id="N10EFD">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 id="N10F03" type="main"> <s id="N10F05"><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> </chap> <chap id="N10F0D"> <p id="N10F0E" type="head"> <s id="N10F10"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>III. </s></p><p id="N10F18" type="main"> <s id="N10F1A">Delle corde, che dalla taglia &longs;u<lb/>periore, & dalla po&longs;&longs;anza alla ta<lb/><figure id="id.041.01.051.1.jpg" xlink:href="041/01/051/1.jpg"/><lb/>glia inferiore peruengono: cia&longs;cu<lb/>na &longs;o&longs;tiene egual parte dipe&longs;o. </s></p><p id="N10F27" type="head"> <s id="N10F29"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N10F2F" type="main"> <s id="N10F31"><emph type="italics"/>Sia la taglia &longs;uperiore A B: l'inferiore <lb/>C D: la corda auuolta no tata con l'i&longs;teße 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. </s> <s id="N10F3D">Dico che cia&longs;cuna corda <lb/>&longs;o&longs;tiene egual parte di pe&longs;o. </s> <s id="N10F41">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 id="N10F53">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 id="N10F5B">Il che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p> <pb xlink:href="041/01/052.jpg" pagenum="51"/> </chap> <chap id="N10F62"> <p id="N10F63" type="head"> <s id="N10F65"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>IIII. </s></p><p id="N10F6D" type="main"> <s id="N10F6F">Se l'vn capo della fune auuolta <lb/><figure id="id.041.01.052.1.jpg" xlink:href="041/01/052/1.jpg"/><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 id="N10F7E" type="head"> <s id="N10F80"><emph type="italics"/>Dimostratione. <emph.end type="italics"/></s></p><p id="N10F86" type="main"> <s id="N10F88"><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">con</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 id="N10FA4">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 id="N10FA9">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 id="N10FB3">&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 poßanza <lb/>K, che è la corda KB: &longs;aranno dette corde <lb/>di <expan abbr="mom&etilde;ti">momenti</expan> eguali. </s> <s id="N10FC1">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 xlink:href="041/01/053.jpg" pagenum="52"/><emph type="italics"/>&longs;cuna girella ne vengono due. </s> <s id="N10FD1">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 attaccato <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 poßanza &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 id="N10FE7" type="head"> <s id="N10FE9"><emph type="italics"/>Appendice. </s> <s id="N10FED">I<emph.end type="italics"/></s></p><p id="N10FF1" type="main"> <s id="N10FF3">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> </chap> <chap id="N10FFD"> <p id="N10FFE" type="head"> <s id="N11000"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>V. </s></p><p id="N11008" type="main"> <s id="N1100A">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 id="N11010" type="head"> <s id="N11012"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N11018" type="main"> <s id="N1101A"><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: et il termi<lb/>ne K, del capo E K, raccomandato alla po&longs;&longs;anza in K: dico che'l pe&longs;o è<emph.end type="italics"/><pb xlink:href="041/01/054.jpg" pagenum="53"/><emph type="italics"/>di&longs;tribuito in parti dinumero &longs;pare. </s> <s id="N1102C">Si <emph.end type="italics"/><lb/><figure id="id.041.01.054.1.jpg" xlink:href="041/01/054/1.jpg"/><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 id="N11043">et euui in oltre il capo della po&longs;&longs;an<lb/>za: &longs;ono dunque tutti di numero &longs;pare. </s><lb/> <s id="N11048">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 id="N11050">Jl che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p><p id="N11054" type="head"> <s id="N11056"><emph type="italics"/>Appendice. </s> <s id="N1105A">I. <emph.end type="italics"/></s></p><p id="N1105E" type="main"> <s id="N11060">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 id="N1106A" type="head"> <s id="N1106C"><emph type="italics"/>Appendice. </s> <s id="N11070">II. <emph.end type="italics"/></s></p><p id="N11074" type="main"> <s id="N11076">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 xlink:href="041/01/055.jpg" pagenum="54"/> </chap> <chap id="N11083"> <p id="N11084" type="head"> <s id="N11086"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>VI</s></p><p id="N1108E" type="main"> <s id="N11090">Se vn capo della taglia &longs;upe<lb/><figure id="id.041.01.055.1.jpg" xlink:href="041/01/055/1.jpg"/><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 id="N1109D" type="head"> <s id="N1109F"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N110A5" type="main"> <s id="N110A7"><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/>relle notata 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. </s> <s id="N110B9">Dico che'l pe&longs;o è di&longs;tri<lb/>buito in parti di numero pare. </s> <s id="N110BD">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 id="N110CD">Il <lb/>che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p><p id="N110D3" type="head"> <s id="N110D5"><emph type="italics"/>Correlario. </s> <s id="N110D9">I. <emph.end type="italics"/></s></p><p id="N110DD" type="main"> <s id="N110DF">E manife&longs;to dunque che li<lb/>gato il capo di &longs;opra alla taglia <pb xlink:href="041/01/056.jpg" 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 id="N110E9" type="head"> <s id="N110EB"><emph type="italics"/>Correlario. </s> <s id="N110EF">II. <emph.end type="italics"/></s></p><p id="N110F3" type="main"> <s id="N110F5">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> </chap> <chap id="N110FB"> <p id="N110FC" type="head"> <s id="N110FE"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>VII. </s></p><figure id="id.041.01.056.1.jpg" xlink:href="041/01/056/1.jpg"/> <p id="N11109" type="main"> <s id="N1110B">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 id="N1111B" type="head"> <s id="N1111D"><emph type="italics"/>Dimostratione. <emph.end type="italics"/></s></p><p id="N11123" type="main"> <s id="N11125"><emph type="italics"/>Sia la girella della taglia &longs;uperiore <lb/>A B: della inferiore nella prima po­<lb/>&longs;itione &longs;ia C D: e la po&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. </s> <s id="N11131">Di <lb/>co che lo &longs;patio caminato dalla taglia <lb/>mobile e pe&longs;o, è tal parte 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 id="N1113B">Si mo&longs;tra: perche <lb/>quante &longs;ono la corde, che alla taglia <emph.end type="italics"/><pb xlink:href="041/01/057.jpg" 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 id="N11151">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 percio E I, &longs;arà altre tanto molteplice dello &longs;patio caminato, quan<lb/>to erano le corde C G, D H. </s> <s id="N11159">ciò è il pe&longs;o tutto del pe&longs;o da vna corda <lb/>&longs;ostenuto. </s> <s id="N1115D">Il che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p> </chap> <chap id="N11161"> <p id="N11162" type="head"> <s id="N11164"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>VIII. </s></p><p id="N1116C" type="main"> <s id="N1116E"><emph type="italics"/>Problema. </s> <s id="N11172">I. <emph.end type="italics"/></s></p><p id="N11176" type="main"> <s id="N11178">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 id="N1117E" type="head"> <s id="N11180"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N11186" type="main"> <s id="N11188"><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. </s> <s id="N11190">&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;imilmente che mouerà la po&longs;&longs;anza la data grauezza. </s><lb/> <s id="N1119D">Il che &longs;i cercaua. <emph.end type="italics"/></s></p> <pb xlink:href="041/01/058.jpg" pagenum="57"/> </chap> <chap id="N111A4"> <p id="N111A5" type="head"> <s id="N111A7"><emph type="italics"/>PROPOSITIONE. <emph.end type="italics"/><lb/>VIII. </s></p><p id="N111AF" type="main"> <s id="N111B1"><emph type="italics"/>Problema. </s> <s id="N111B5">II. <emph.end type="italics"/></s></p><p id="N111B9" type="main"> <s id="N111BB">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 id="N111C3" type="head"> <s id="N111C5"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N111CB" type="main"> <s id="N111CD"><emph type="italics"/>Pigli&longs;i lo &longs;patio che nel dato tem­<emph.end type="italics"/><lb/><figure id="id.041.01.058.1.jpg" xlink:href="041/01/058/1.jpg"/><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 id="N111EA">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 id="N111F4">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 radoppij: e <lb/>ciò &longs;i torni a fare, si che l'ultimo radop<lb/>piamento auanzi lo &longs;patio maggiore. </s><lb/> <s id="N11201">Se dunque, quante volte &longs;i è radoppia­<lb/>to, tanto numero ditaglie &longs;i pigli, si mo<lb/>uer à il pe&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 xlink:href="041/01/059.jpg" pagenum="58"/> <p id="N11210" type="head"> <s id="N11212">ROTE MO<lb/>TIVE. </s></p><figure id="id.041.01.059.1.jpg" xlink:href="041/01/059/1.jpg"/> <p id="N11219" type="head"> <s id="N1121B"><emph type="italics"/>SVPPOSITION. <emph.end type="italics"/><lb/>I. </s></p><p id="N11223" type="main"> <s id="N11225">Svpponiamo il mouimento di rote in a&longs;&longs;i <lb/>che &longs;tanno co'l toccamento, communicar&longs;i l'vna <lb/>all'altra il mouimento: e che'l momento della po&longs;&longs;an­<lb/>za &longs;ia per linea che faccia angolo retto co'l raggio di <lb/>e&longs;&longs;a rota: e de momenti altri e&longs;&longs;er concorrenti, altri <lb/>contrarij. </s></p><p id="N11231" type="head"> <s id="N11233"><emph type="italics"/>DEFINITION. <emph.end type="italics"/><lb/>I. </s></p><p id="N1123B" type="main"> <s id="N1123D">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 id="N11241" type="head"> <s id="N11243">II. </s></p><p id="N11245" type="main"> <s id="N11247">Contrarij quelli, che s'impedi&longs;cono portando in <lb/>contrario. </s></p><p id="N1124B" type="head"> <s id="N1124D"><emph type="italics"/>POSITION. <emph.end type="italics"/><lb/>I. </s></p><p id="N11255" type="main"> <s id="N11257">Poniamo, po&longs;&longs;anze eguali in circonferenze direte <lb/>eguali, hauer momenti eguali. </s></p> <pb xlink:href="041/01/060.jpg" pagenum="59"/> <p id="N1125E" type="head"> <s id="N11260"><emph type="italics"/>POSITION. <emph.end type="italics"/><lb/>II. </s></p><p id="N11268" type="main"> <s id="N1126A">Et in rote ineguali hauer momento ineguale, &longs;econ<lb/>do la ragion de &longs;emidiametri. </s></p><p id="N1126E" type="head"> <s id="N11270">III. </s></p><p id="N11272" type="main"> <s id="N11274">E gli momenti contrarij, per quanto &longs;i annullano, l' <lb/>vno e&longs;&longs;ere eguale all'altro. </s></p><p id="N11278" type="head"> <s id="N1127A"><emph type="italics"/>PROPOSITION. <emph.end type="italics"/><lb/>I. </s></p><p id="N11282" type="main"> <s id="N11284">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 eguali. </s></p><figure id="id.041.01.060.1.jpg" xlink:href="041/01/060/1.jpg"/> <pb xlink:href="041/01/061.jpg" pagenum="60"/> <p id="N11292" type="head"> <s id="N11294"><emph type="italics"/>Dimostratione. <emph.end type="italics"/></s></p><p id="N1129A" type="main"> <s id="N1129C"><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 nel ponto 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 id="N112A6">Dico che <lb/>le potenze &longs;ono eguali. </s> <s id="N112AA">Si mo&longs;tra: percio che la poßanza 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 id="N112BA">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. </s> <s id="N112C0">Jl che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p><p id="N112C4" type="head"> <s id="N112C6"><emph type="italics"/>PROPOSITION. <emph.end type="italics"/><lb/>II. </s></p><p id="N112CE" type="main"> <s id="N112D0">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 id="N112D8" type="head"> <s id="N112DA"><emph type="italics"/>Dimostratione. <emph.end type="italics"/></s></p><p id="N112E0" type="main"> <s id="N112E2"><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 id="N112EA">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 id="N112EF">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 xlink:href="041/01/062.jpg" pagenum="61"/><figure id="id.041.01.062.1.jpg" xlink:href="041/01/062/1.jpg"/><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. </s> <s id="N11304">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 id="N1130C">Il che &longs;i hauea da mo­<lb/>&longs;trare. <emph.end type="italics"/></s></p> <pb xlink:href="041/01/063.jpg" pagenum="62"/> <p id="N11315" type="head"> <s id="N11317"><emph type="italics"/>PROPOSITION. <emph.end type="italics"/><lb/>III. </s></p><p id="N1131F" type="main"> <s id="N11321">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 id="N1132D" type="head"> <s id="N1132F"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N11335" type="main"> <s id="N11337"><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. </s> <s id="N1133F">Dico che la po&longs;&longs;anza in B,<emph.end type="italics"/><lb/><figure id="id.041.01.063.1.jpg" xlink:href="041/01/063/1.jpg"/><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 xlink:href="041/01/064.jpg" 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 id="N11356">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. </s> <s id="N11360">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 id="N11368">Jl che &longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p><p id="N1136C" type="head"> <s id="N1136E"><emph type="italics"/>DEFINITIONE. <emph.end type="italics"/></s></p><p id="N11374" type="main"> <s id="N11376">Momento della rota diciamo, il momento del pon<lb/>to po&longs;to nella circonferenza di e&longs;&longs;a rota. </s></p><p id="N1137A" type="head"> <s id="N1137C"><emph type="italics"/>PROPOSITION. <emph.end type="italics"/><lb/>IIII. </s></p><p id="N11384" type="main"> <s id="N11386">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 id="N11392">e la velocità &longs;arà mino<lb/>re, &longs;econdo l'i&longs;te&longs;&longs;a ragion de diametri. </s></p><p id="N11396" type="head"> <s id="N11398"><emph type="italics"/>Dimo&longs;tratione. <emph.end type="italics"/></s></p><p id="N1139E" type="main"> <s id="N113A0"><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 xlink:href="041/01/065.jpg" pagenum="64"/><figure id="id.041.01.065.1.jpg" xlink:href="041/01/065/1.jpg"/><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 circonferenza di A C. </s><lb/> <s id="N113B8">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. </s> <s id="N113BC">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 il mo<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. </s> <s id="N113E2">Il che <lb/>&longs;i hauea da mo&longs;trare. <emph.end type="italics"/></s></p><p id="N113E8" type="main"> <s id="N113EA"><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 xlink:href="041/01/066.jpg" pagenum="65"/><emph type="italics"/>fe&longs;to: percioche 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 id="N113FA">Sono dunque le <lb/>velocità delle rote proportionali alli diametri. </s> <s id="N113FE">Jl che &longs;i hauea da <lb/>mo&longs;trare. <emph.end type="italics"/></s></p><p id="N11404" type="head"> <s id="N11406"><emph type="italics"/>PROPOSITION. <emph.end type="italics"/><lb/>V. </s></p><p id="N1140E" type="main"> <s id="N11410">Date due po&longs;&longs;anze di momento contrario, l'vna mi<lb/>nore, e l'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 id="id.041.01.066.1.jpg" xlink:href="041/01/066/1.jpg"/> <pb xlink:href="041/01/067.jpg" pagenum="96"/> <p id="N11420" type="head"> <s id="N11422"><emph type="italics"/>Dimostratione. <emph.end type="italics"/></s></p><p id="N11428" type="main"> <s id="N1142A"><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 id="N11434">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. </s> <s id="N1143E">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. </s> <s id="N1144C">Il che &longs;i hauea da <lb/>trouare. <emph.end type="italics"/></s></p><p id="N11452" type="head"> <s id="N11454"><emph type="italics"/>PROPOSITION. <emph.end type="italics"/><lb/>VI. </s></p><p id="N1145C" type="main"> <s id="N1145E">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="cõgiogate">con<lb/>giogate</expan>: 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 id="N1146C" type="head"> <s id="N1146E"><emph type="italics"/>Dimostratione. <emph.end type="italics"/></s></p><p id="N11474" type="main"> <s id="N11476"><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 xlink:href="041/01/068.jpg" pagenum="67"/><figure id="id.041.01.068.1.jpg" xlink:href="041/01/068/1.jpg"/><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 id="N11491">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 la minore I L, la maggiore L M. </s> <s>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. </s> <s id="N114A5"> <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 circonferenza M, &longs;i mouerà <expan abbr="cõ">con</expan> mag<lb/>gior velocità della data. </s> <s id="N114B2">Il che &longs;i hauea da trouare. <emph.end type="italics"/></s></p> <pb xlink:href="041/01/069.jpg" pagenum="68"/> <p id="N114B9" type="head"> <s id="N114BB"><emph type="italics"/>MOMENTI ACQVISTATI. <emph.end type="italics"/></s></p><p id="N114C1" type="main"> <s id="N114C3">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 id="N114CD">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 id="N114D8">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 id="N114E6">Et il momen­<lb/>to intrin&longs;eco e&longs;&longs;er l'i&longs;te&longs;&longs;o &longs;empre. </s> <s id="N114EA">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"/></s></p><p id="N114F5" type="main"> <s id="N114F7">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 id="N114FB">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 id="N11503">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"/></s></p><p id="N1150C" type="main"> <s id="N1150E">La vite, o chioccia rappre&longs;enta vno o più piani auuolti <lb/>ad vn fu&longs;ello. </s> <s id="N11512">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 id="N11516">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 id="N11520">Vite perpetua diciamo vn <lb/>sympano con denti a vite, che girando tocchi rota dentata. </s><lb/> <s id="N11525">Per il che accre&longs;ce la forza, e per la proprieta della vite, <lb/>e della congiogatione delle rote. </s></p><p id="N11529" type="head"> <s id="N1152B"><emph type="italics"/>IL FINE. <emph.end type="italics"/></s></p> </chap> </body> <back/> </text> </archimedes>