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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <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></translator> <lang>it</lang> <cvs_file>stell_mecha_041_it_1597.xml</cvs_file> <cvs_version></cvs_version> <locator>041.xml</locator> </info> <text> <front> <section> <pb xlink:href="041/01/001.jpg" id="p.0001"></pb> <p id="N1001B" type="head"> <s id="N1001D">DE GLI <lb></lb>ELEMENTI <lb></lb>MECHANICI<lb></lb><figure id="id.041.01.001.1.jpg" xlink:href="041/01/001/1.jpg"></figure><lb></lb></s> <s>La ſtatera. </s> <s>Leua. <lb></lb></s> <s id="N1002B">Raggi nell'aſſe. </s> <s id="N1002D"> Rote vettiue <lb></lb></s> <s id="N10030">Taglia. </s> <s id="N10032">Rote motiue. <lb></lb></s> <s id="N10035">Cugno. </s> <s id="N10037"> Vite. </s></p><p id="N10039" type="head"> <s id="N1003B"><emph type="italics"></emph>DI C. ANTONIO STELLIOLA. <emph.end type="italics"></emph.end></s></p><figure id="id.041.01.001.2.jpg" xlink:href="041/01/001/2.jpg"></figure> <p id="N10044" type="head"> <s id="N10046">IN NAPOLI, Nella Stamparia à Porta Regale <lb></lb>M. D. XCVII. </s></p> </section> <pb xlink:href="041/01/002.jpg" pagenum="1"></pb> <section> <p id="N1004E" type="head"> <s id="N10050"><emph type="italics"></emph>PROPOSITIONE <lb></lb>di tutta l'opera. <emph.end type="italics"></emph.end></s></p><p id="N10058" type="main"> <s id="N1005A">Cerchiamo come poſſa la potenza <lb></lb>minore vincer di forza la maggiore: <lb></lb>e la potenza piu tarda, vincer di mo<lb></lb>uimento la piu veloce. </s> <s id="N10062">e queſto con <lb></lb>Leue, Taglie, Viti, Rote, e tutti inſtrumenti che <lb></lb>moltiplicar poſſono il momento, o della forza, <lb></lb>o della velocità. </s> <s id="N1006A">Qual ſoggetto communemente <lb></lb>gli antichi chiamarono Mechaniche. </s> <s id="N1006E">Il che tut<lb></lb>to ſi tratterà ſecondo le ſuppoſitioni fatte de mo<lb></lb>menti, o per linee parallelle, o per linee con<lb></lb>correnti ad vn ponto, o per circonferenze d'in<lb></lb>torno vn centro iſteſſo: e ſecondo il ſolito vſo de <lb></lb>mathematici deducendo le dimoſtrationi, e cau<lb></lb>ſe 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"></figure> </section> <pb xlink:href="041/01/003.jpg" pagenum="2"></pb> <section> <p id="N10083" type="head"> <s id="N10085"><emph type="italics"></emph>DEFINITIONI. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N1008D" type="main"> <s id="N1008F">Centro di peſo diciamo il ponto, per cui il corpo co<lb></lb>munque ſoſpeſo, non muta poſitione. </s></p><p id="N10093" type="head"> <s id="N10095">II. </s></p><p id="N10097" type="main"> <s id="N10099">Corpo egualmente diſteſo diciamo, che comunque <lb></lb>tagliato con pianezze parallele, fa figure ſuperficiali <lb></lb>eguali e ſimili. </s></p><p id="N1009F" type="head"> <s id="N100A1">III. </s></p><p id="N100A3" type="main"> <s id="N100A5">Applicarſi diciamo vn corpo ad vna linea, quando <lb></lb>detto corpo vgualmente diſteſo occupi la lunghezza di <lb></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></lb>ſo della grauezza da impedimento libera ſi moue. </s></p><p id="N100B5" type="head"> <s id="N100B7">V. </s></p><p id="N100B9" type="main"> <s id="N100BB">Libra ò ſtatera diciamo la linea a cui ſi applicano, ò <lb></lb>appendono le grauezze: e che ſia ſuſpeſa da vn ſol <lb></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 ſoſtenuta da due ponti, o ſo<lb></lb>ſtenuta da vn ponto e moſſa da vna poſſanza. </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 ſtatera e leua, il <lb></lb>ponto, nel quale s'incontra la linea del momento, con <lb></lb>la linea della ſtatera. </s></p><p id="N100D7" type="head"> <s id="N100D9">VIII. </s></p><p id="N100DB" type="main"> <s id="N100DD">E ponto di appenſione: il ponto, onde perde la gra<pb xlink:href="041/01/004.jpg" pagenum="3"></pb>uezza ſtaccata dalla ſtatera, o leua, nelquale iſteſſo pon<lb></lb>to s'intende hauer il ſuo 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 peſi la ſuperficie in cui le linee de <lb></lb>momenti tutte vanno perpendicolarmente. </s></p><p id="N100EF" type="head"> <s id="N100F1"><emph type="italics"></emph>Appendice. <emph.end type="italics"></emph.end></s></p><p id="N100F7" type="main"> <s id="N100F9">Dalche è manifeſto, che l'Horizonte de' momenti pa<lb></lb>ralleli, ſia ſuperficie piana: e delli concorrenti ſia ſuper<lb></lb>cie sferica. </s></p><p id="N100FF" type="head"> <s id="N10101"><emph type="italics"></emph>POSITIONI. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N10109" type="main"> <s id="N1010B">Pigliamo nelli corpi egualmente diſteſi il centro del <lb></lb>peſo eſſer nella ſuperficie, che diuide egualmente la <lb></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 appeſe o nell'iſteſſo ponto, o in <lb></lb>ponti della libra egualmente diſtanti dalla ſuſpenſione <lb></lb>della ſtatera, 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 iſteſſa natura ſia proportionale <lb></lb>il peſo 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 appeſa non ſi fermi, ſin che il <expan abbr="cẽtro">cen<lb></lb>tro</expan> del peſo non ſia nella perpendicolare del ponto del <lb></lb>ſoſtenimento. </s></p> </section> </front> <pb xlink:href="041/01/005.jpg" pagenum="4"></pb> <body> <chap id="N1013B"> <p id="N1013C" type="head"> <s id="N1013E"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N10146" type="main"> <s id="N10148">Se ſi togliono due quantità da due altre, che ſiano <lb></lb>eguali, e tra di loro, & alla compoſta delle due tolte: di<lb></lb>co che le reſtanti alle tolte ſcambieuolmente ſono egua<lb></lb>li. </s></p><figure id="id.041.01.005.1.jpg" xlink:href="041/01/005/1.jpg"></figure> <p id="N10153" type="head"> <s id="N10155"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N1015B" type="main"> <s id="N1015D"><emph type="italics"></emph>Siano le due quantità, A & B, & alla compoſta di ambe ſiano egua<lb></lb>li, la C D, & la E F; e dalla C D, togliaſi eguale ad A, che ſia, <lb></lb>C G, e dalla E F togliaſi eguale a B, che ſia E H. </s> <s id="N10165">dico che la reſtan<lb></lb>te H F, è vguale ad A; e la G D, eguale a B. </s> <s id="N10169">Si moſtra perciò <lb></lb>che eſſendo C D, eguale ad A e B inſieme: tolti dall'vna e l'altra ſum<lb></lb>ma le A, e C G eguali: le reſtanti, B, e G D di conſeguenza ſo<lb></lb>no eguali. </s> <s id="N10171">Similmente perche la E F ſi pone vguale alle A, & B <lb></lb>gionte inſieme; tolte la E H, & B vguali: le reſtanti, H F, e A ſo<lb></lb>no di conſeguenza eguali. </s> <s id="N10177">è adunque la H F eguale a C G: e la G D <lb></lb>eguale ad E H. </s> <s id="N1017B">il che hauea da moſtrarſi. <emph.end type="italics"></emph.end></s></p><p id="N1017F" type="head"> <s id="N10181"><emph type="italics"></emph>Appendice. <emph.end type="italics"></emph.end></s></p><p id="N10187" type="main"> <s id="N10189">Dalche è manifeſto, che le iſteſſe reſtanti ſcambieuol<lb></lb>mente ſono proportionali alle tolte. </s></p> <pb xlink:href="041/01/006.jpg" pagenum="5"></pb> <p id="N10190" type="main"> <s id="N10192">Percioche eſſendo le C G H F eguali. </s> <s id="N10194">e le G D E H <lb></lb>anco eguali: ma le eguali ſono proportionali: ſono <expan abbr="dũque">dunque</expan><lb></lb>come C G ad E H, coſi H F ad G D: ilche hauea da mo<lb></lb>ſtrarſi. </s></p> </chap> <chap id="N1019F"> <p id="N101A0" type="head"> <s id="N101A2"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>II. </s></p><p id="N101AA" type="main"> <s id="N101AC">Se alla linea della ſtatera ſi applicano continuatamen<lb></lb>te due corpi: li centri delli corpi applicati, ſono diſtanti <lb></lb>dal centro di tutto il compoſto, di diſtanze proportio<lb></lb>nali alli peſi, pigliati reciprocamente. </s></p><figure id="id.041.01.006.1.jpg" xlink:href="041/01/006/1.jpg"></figure> <p id="N101B7" type="head"> <s id="N101B9"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N101BF" type="main"> <s id="N101C1"><emph type="italics"></emph>Sia la linea della ſtatera, A B, l'vn delli corpi applicati ſia <lb></lb>B C, l'altro ſia C A, e l'applicatione del corpo B C occupi la parte di li<lb></lb>nea B D, e del corpo C A, la parte A D, e diuidaſi B D, in par<lb></lb>ti vguali al ponto E: & A D in parti eguali il ponto F: è manifesto <lb></lb>che del corpo applicato à B D, il ponto del momento ſia E, e del cor<lb></lb>po applicato a D A, il ponto del momento ſia F, dico che diuiſa B A <lb></lb>tutta per metà nel ponto G, che è ponto di <expan abbr="momẽto">momento</expan><lb></lb> della grauezza tut<lb></lb>ta compoſta di ambedue: c'habbia la diſtanza F G a G E la ragione <lb></lb>che'l peſo di B C al peſo di C A. </s> <s id="N101DA">Si moſtra percioche la ragione del <lb></lb>peſo di B C, al peſo di C A, e l'iſteſſa che delli corpi: e delli corpi<emph.end type="italics"></emph.end><emph.end type="italics"></emph.end><pb xlink:href="041/01/007.jpg" pagenum="6"></pb><emph type="italics"></emph>vgualmente diſteſi, e l'iſteſſa che delle linee: qual è della linea B D <lb></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></lb>& perche ſe due quantità compongono quantità, e le metà del<lb></lb>le componenti, compongono la metà della tutta: ma le metà delle li<lb></lb>nee componenti ſono A F, e B E, la metà della tutta, e coſi la B G <lb></lb>come la A G. </s> <s id="N101F2">perciò togliendo due quantità A F B E dalle due, <lb></lb>A G, G E eguali tra di loro, & alla compoſta di A F, B E. </s><lb></lb> <s id="N101F7">le reſtanti ſcambieuolmente ſono proportionali, e perciò F G, a G E <lb></lb>ſarà nell'iſteſſa ragione di B E, ad A F, cioè della doppia, B D, a D <lb></lb>A: qual è l'iſteſſa del corpo, B C, a C A: e della grauezza di B C, a <lb></lb>C A. </s> <s id="N101FF">la diſtanza dunque F G, alla diſtanza E G, ha la ragione che'l <lb></lb>peſo di B C, al peſo di C A, il che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N10205"> <p id="N10206" type="head"> <s id="N10208"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>III. </s></p><p id="N10210" type="main"> <s id="N10212">Se ad vn vna ſtatera ſiano appeſe due grauezze, e l' <lb></lb>interuallo delli ponti della ſoſpenſione ſi diuida nella <lb></lb>ragione delle grauezze: ſoſpeſa la ſtatera dal ponto del<lb></lb>la diuiſione, ſta in equilibrio. </s></p><figure id="id.041.01.007.1.jpg" xlink:href="041/01/007/1.jpg"></figure> <p id="N1021D" type="head"> <s id="N1021F"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10225" type="main"> <s id="N10227"><emph type="italics"></emph>Sia la ſtatera A B: le grauezze in eſſa ſoſpeſe: C, & D: la C, dal <lb></lb>ponto, A, & la D, dal ponto, ‘B, & in quellaragione che ha la gra<lb></lb>uezza D, alla grauezza C, ſi diuida A B nel ponto E dico che ſoſpeſa <emph.end type="italics"></emph.end><pb xlink:href="041/01/008.jpg" pagenum="7"></pb><emph type="italics"></emph>la ſtatera nel ponto E, ſta in equilibrio. </s> <s id="N10237">Si moſtra alla linea, B E, ta<lb></lb>gliſi eguale la linea A F, dunque giunta communemente, F E, ſarà B <lb></lb>F, vguale ad A E, e perciò haurà B F, ad F A, l'iſteſſa ragione, che <lb></lb>D, a C. </s> <s id="N1023F">facciaſi alla B F, vguale, B G, & alla A F, vguale A H, <lb></lb>dunque ſe alla linea, G F, s'intenda applicato un corpo eguale di peſo <lb></lb>alla grauezza D, e tal corpo ſi allunghi nella isteſſa groſſezza fin ad H, <lb></lb>ſarà il corpo applicato ad F H, uguale di peſo a C: percio che hauendo <lb></lb>G F, ad F H, l'iſteſſa ragione che D, a C, e li corpi applicati l'iſteſ<lb></lb>ſa delle linee: ſono perciò come la grauezza D alla C, coſi il cor<lb></lb>po applicato ad F G, al corpo applicato ad H F: dunque mutando, ſo<lb></lb>no anco proportionali: ma il corpo applicato a G F, è di peſo uguale al <lb></lb>D, dunque l'applicato ad F H è vguale di peſo a C: & è delli due ap<lb></lb>plicati, il commune punto di momento in E. </s> <s id="N10253">Dunque delli D C inſie<lb></lb>me pigliati il commun momento è nel ponto iſteſſo: & percio la ſtatera <lb></lb>ſoſtenuta in E, ſta in equilibrio, ilche ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p><p id="N1025B" type="head"> <s id="N1025D"><emph type="italics"></emph>Appendice. <emph.end type="italics"></emph.end> I. </s></p><p id="N10264" type="main"> <s id="N10266">Dal che è manifeſto che'l centro commune di due <lb></lb>peſi è il ponto che diuide l'interuallo de'centri loro, re<lb></lb>ciprocamente. </s></p><p id="N1026C" type="head"> <s id="N1026E"><emph type="italics"></emph>Appendice. <emph.end type="italics"></emph.end> II. </s></p><p id="N10275" type="main"> <s id="N10277">E ſe due grauezze diuiſamente ſi appendono: che di<lb></lb>uiſo l'interuallo nella ragione delle grauezze recipro<lb></lb>camente: dette grauezze, fanno l'iſteſſo effetto nel <expan abbr="momẽto">mo<lb></lb>mento</expan>, che ſe in detto ponto <expan abbr="giuntamẽte">giuntamente</expan> fuſſero appeſe. </s></p> </chap> <chap id="N10287"> <p id="N10288" type="head"> <s id="N1028A"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>IIII. </s></p><p id="N10292" type="main"> <s id="N10294">Se due grauezze appeſe in due ponti facciano equi<lb></lb>pondio: e di nuouo appeſe in due altri ponti facciano <pb xlink:href="041/01/009.jpg" pagenum="8"></pb>equipondio; l'interualli delle ſoſpenſioni mutate, ſono <lb></lb>proportionali con li peſi reciprocamente. </s></p><figure id="id.041.01.009.1.jpg" xlink:href="041/01/009/1.jpg"></figure> <p id="N102A1" type="head"> <s id="N102A3"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N102A9" type="main"> <s id="N102AB"><emph type="italics"></emph>Sia la ſtatera A B: il ponto della ſoſpenſione C, li ponti onde ſono <lb></lb>appeſe le grauezze che fanno equipondio A & B le grauezze appeſe <lb></lb>D & E. </s> <s id="N102B3">Quali di nuouo appeſe nelli ponti F & G faccino equipondio: <lb></lb>dico che la F A interuallo delle due ſoſpenſioni di D, a B G, inter<lb></lb>uallo delle <expan abbr="ſuſpẽſioni">ſuſpenſioni</expan> di E; ha quella ragione che la grauezza c alla gra<lb></lb>uezza D. </s> <s id="N102BF">Si moſtra perche D et E grauezze nella <expan abbr="ſuſpẽſion">ſuſpenſion</expan> prima han<lb></lb>no equipondio: dunque la ragione della grauezza D ad E, è l'iſteſſa che <lb></lb>di B C a C A: e nella ſeconda ſuſpenſione la ragione di D ad E e l'iſteſ<lb></lb>ſa che di G C a C F. </s> <s id="N102CB">e perciò come B C à C A, coſi G C à C F, e per che <lb></lb>da due ſi togliono due altre nell'iſteſſa ragione, le reſtanti anco ſono nel<lb></lb>l'iſteſſa ragione. </s> <s id="N102D1">è dunque B G ad F A, come D ad E, ilche hauea da <lb></lb>moſtrarſi. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N102D7"> <p id="N102D8" type="head"> <s id="N102DA"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>V. </s></p><p id="N102E2" type="main"> <s id="N102E4">Se due grauezze facciano equipondio, e gionte ò tol<lb></lb>te due altre grauezze facciano anco equipondio: le gion<lb></lb>te ancora ò le tolte ſono nell'iſteſſa raggione. </s></p> <pb xlink:href="041/01/010.jpg" pagenum="9"></pb> <figure id="id.041.01.010.1.jpg" xlink:href="041/01/010/1.jpg"></figure> <p id="N102F0" type="head"> <s id="N102F2"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N102F8" type="main"> <s id="N102FA"><emph type="italics"></emph>Sia la ſtatera A B: il ponto della ſuſpenſione C: le grauezze appe<lb></lb>ſe D et E: che facciano equipondio: e di nouo aggiuntoui due altre F e G <lb></lb>facciano anco equipondio. </s> <s id="N10302">dico che la grauezza F a G, ha la ragione <lb></lb>che D ad E: qual'è l'iſteſſa che di B C a C A. </s> <s id="N10306">ſi moſtra perche D & E, <lb></lb>fanno equipondio. </s> <s id="N1030A">& F e G fanno equipondio: perciò ſarà, come B C <lb></lb>à C A coſi D F ad E G e nell'iſteſſa era D ad E dunque le reſtanti F e G <lb></lb>ſono anco nell'iſteſſa ragione: e non altrimente che nella ſuppoſition del<lb></lb>la compoſta, ſi moſtra nella ſuppoſition delli reſidui. </s> <s id="N10312">Haſſi dunque l'in<lb></lb>tento. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N10318"> <p id="N10319" type="head"> <s id="N1031B"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>VI. </s></p><p id="N10323" type="main"> <s id="N10325">Date quante ſi voglia grauezze appeſe in vn'iſteſſa <lb></lb>ſtatera, ritrouare il ponto del momento commune. </s></p><p id="N10329" type="head"> <s id="N1032B"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10331" type="main"> <s id="N10333"><emph type="italics"></emph>Sia la ſtatera A B dalli cui ponti A e B ſiano ſoſpeſe le grauezze C D <lb></lb>e ſiano in altri ponti ſoſpeſi altri peſi, come E nel ponto F: ſi cerca il pon<lb></lb>to del momento commune. </s> <s id="N1033B">diuidaſi la B A nella ragione di C a D reci<lb></lb>procamente ſe dunque il detto punto uiene in F eſſendo F il ponto del <emph.end type="italics"></emph.end><pb xlink:href="041/01/011.jpg" pagenum="10"></pb><figure id="id.041.01.011.1.jpg" xlink:href="041/01/011/1.jpg"></figure><lb></lb><emph type="italics"></emph>momento delle C D pigliate inſieme, ſarà ponto di momento commu<lb></lb>ne delle grauezze C E D, tutte. </s> <s id="N1034E">Et harraſſi l'intento. <emph.end type="italics"></emph.end></s></p><p id="N10352" type="main"> <s id="N10354"><emph type="italics"></emph>Ma ſe'l dato ponto caſchi altroue come in H, perche le grauezze <lb></lb>D, & C appeſe in A e B fanno l'iſteſſo effetto che ſe giuntamente fuſſe<lb></lb>ro appeſe in H: perciò ſe quella ragione che hà il compoſto di C D <lb></lb>ad E habbia reciprocamente F G a G H, ſarà G ponto di momento <lb></lb>commune di tutti. </s> <s id="N10360">con l'iſteſſo ordine ſi ritrouerà il centro di quante <lb></lb>altre ſi uogliano, il che ſi hauea da trouare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N10366"> <p id="N10367" type="head"> <s id="N10369"><emph type="italics"></emph>PROPOSITIONE<emph.end type="italics"></emph.end><lb></lb> VII. </s></p><p id="N10371" type="main"> <s id="N10373">Delle grauezze che fanno <expan abbr="equipõdio">equipondio</expan>, compoſte le ra<lb></lb>gioni delle grauezze e delle diſtanze, li eſtremi termini <lb></lb>ſono eguali. </s></p><p id="N1037D" type="head"> <s id="N1037F"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10385" type="main"> <s id="N10387"><emph type="italics"></emph>Sia la ſtatera A B il ponto del ſostenimento C le due grauezze che <lb></lb>fanno <expan abbr="equipõdio">equipondio</expan> D & E: de quali la D ſia ſoſpeſa dal ponto A la E dal <expan abbr="põto">pon<lb></lb>to</expan> B: dico che compoſtala ragione della grauezza D ad E: e della <expan abbr="diſtãza">diſtanza</expan><lb></lb> A C a C B: cioè fatto che la <expan abbr="quãtità">quantità</expan> F a G ſia come la grauezza D ad E e <lb></lb>la quantità G ad H come la diſtanza A C alla C B, che F & H <emph.end type="italics"></emph.end><pb xlink:href="041/01/012.jpg" pagenum="11"></pb><figure id="id.041.01.012.1.jpg" xlink:href="041/01/012/1.jpg"></figure><lb></lb><emph type="italics"></emph> eſtremi termini ſiano uguali. </s> <s id="N103AF">ſi moſtra: perche A C a C B ſi è poſta co<lb></lb>me G ad H: dunque riuoltando H à G, è come B C à C A. </s> <s>e per l'e<lb></lb>quipondio, come la diſtanza B C a C A coſi la grauezza D ad E, & <lb></lb>come D ad E coſi ſi è pigliato F a G: dnnque F a G e come B C a C A, <lb></lb>e nell'iſteſſa ragione era H a G. </s> <s id="N103B9">hanno dunque li due termini F et H l'i<lb></lb>ſteſſa ragione al termine G. </s> <s>e perciò li F & H ſono eguali tra di loro: <lb></lb>il che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N103C1"> <p id="N103C2" type="head"> <s id="N103C4"><emph type="italics"></emph>PROPOSITIONE<emph.end type="italics"></emph.end><lb></lb> VIII. </s></p><p id="N103CC" type="main"> <s id="N103CE">Li momenti delle grauezze uguali, appeſe in diſtan<lb></lb>ze ineguali, hanno fra di loro la proportione che le di<lb></lb>ſtanze. </s></p><p id="N103D4" type="head"> <s id="N103D6"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N103DC" type="main"> <s id="N103DE"><emph type="italics"></emph>Sia la ſtatera A B, il ponto del ſoſtenimento C, le grauezze uguali D <lb></lb>& E. </s> <s id="N103E4">de quali il D ſia appeſo in A, & l'c in F. </s> <s id="N103E6">dico che il momento <lb></lb>di D al momento di E, hà quella ragione che l'interuallo di A C all'<lb></lb>interuallo di F C. </s> <s id="N103EC">ſi moſtra, pigliato dall' altra parte del ſeſtem<lb></lb>mento C, qual ſi uoglia ponto B: intendaſi in eſſa appeſe due grauezze, <emph.end type="italics"></emph.end><pb xlink:href="041/01/013.jpg" pagenum="12"></pb><figure id="id.041.01.013.1.jpg" xlink:href="041/01/013/1.jpg"></figure><lb></lb><emph type="italics"></emph>vna che faccia <expan abbr="equipōdio">equipondio</expan> a D & ſia G: et vn'altra che faccia <expan abbr="equipōdio">equipondio</expan><lb></lb> ad E. </s> <s id="N10406">& ſia H. </s> <s id="N10408">perche dunque G a D ha quella ragione che A C a C B <lb></lb>& D ouero E ad H, hala ragione di B C a C F. </s> <s id="N1040C">dunque di pari il pri<lb></lb>mo termine A C all'ultimo F C, ha quella ragione, che il primo ter<lb></lb>mine G, al terzo H. </s> <s id="N10412">ſe dunque G ad H hal'iſteſſa ragione che la diſtan<lb></lb>za A C alla diſtanza F C: & il momento di G è uguale al momento <lb></lb>di D appeſo in A, & il momento di H vguale al momento di E appeſo <lb></lb>in F. </s> <s id="N1041A">dunque il momento di D al momento di E ha quella ragione che <lb></lb>la diſtanza A C alla diſtanza F C. </s> <s id="N1041E">il che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N10422"> <p id="N10423" type="head"> <s id="N10425"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>IX. </s></p><p id="N1042D" type="main"> <s id="N1042F">Li momenti delle grauezze ſoſpeſe in qual ſi uoglia <lb></lb>ponti della ſtatera, han tra di loro la ragion compoſta, <lb></lb>della ragion delle grauezze, e delle diſtanze. </s></p><p id="N10435" type="head"> <s id="N10437"><emph type="italics"></emph>Dimostratione. <emph.end type="italics"></emph.end></s></p><p id="N1043D" type="main"> <s id="N1043F"><emph type="italics"></emph>Sia la ſtatera A B il <expan abbr="pōto">ponto</expan> del <expan abbr="ſoſtenimẽto">ſoſtenimento</expan> C le grauezze appeſe D dal <lb></lb> <expan abbr="pōto">ponto</expan> A, & E dal ponto F. </s> <s id="N10450">dico che la ragione del <expan abbr="momẽto">momento</expan> D al <expan abbr="momẽto">momen<lb></lb>to</expan> E, e compoſta di due ragioni cioè della ragione della grauezza D <lb></lb>alla grauezza E, e della diſtanza di A C alla F C. </s> <s id="N1045E">ſi moſtra appenda <lb></lb>ſi da B la grauezza G che faccia equipondio. </s> <s id="N10462">a D, & il peſo H che fac<emph.end type="italics"></emph.end><pb xlink:href="041/01/014.jpg" pagenum="13"></pb><figure id="id.041.01.014.1.jpg" xlink:href="041/01/014/1.jpg"></figure><lb></lb><emph type="italics"></emph>cia <expan abbr="equipōdio">equipondio</expan> all' E: dico prima che la grauezza G alla grauezza H G ha <lb></lb>la ragion compoſta, di D ad E, e di A C ad F C. </s> <s id="N10477">per il che da moſtrare: in<lb></lb>tendaſi nell' A ſoſpeſa la grauezza I uguale alla grauezza E, è manifeſto <lb></lb>che'l momento I al momento E, hà quella ragione che l'interuallo <lb></lb>A C all'interuallo F C come nel paſſato habbiamo moſtrato: & il mo<lb></lb>mento di D al momento d'I hà la ragione che la grauezza D alla gra<lb></lb>uezza I: perche ſono da un'iſteſſo ponto ſoſpeſi. </s> <s id="N10483">eſſendo dunque tre ter<lb></lb>mini in continua habitudine il momento D, il momento I, & il <expan abbr="momẽto">momen<lb></lb>to</expan> E: la ragione del primo termine al terzo è compoſta della ragione <lb></lb>di primo a ſecondo e della ragione di ſecondo a terzo: ma di primo <lb></lb>a ſecondo è di grauezza a grauezza: di ſecondo a terzo è d'interuallo <lb></lb>ad'interuallo. </s> <s id="N10493">dunque, la ragione delli <expan abbr="momẽti">momenti</expan> di D ad E, che è l'iſteſſa <lb></lb>che della portione G alla portione H: è compoſta della ragione delle <lb></lb>grauezze e della ragione delle diſtanze. </s> <s id="N1049D">Il che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N104A1"> <p id="N104A2" type="head"> <s id="N104A4"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>X. </s></p><p id="N104AC" type="main"> <s id="N104AE">Data qual ſi uoglia grauezza, e li ponti della ſoſpen<lb></lb>ſion della ſtatera, e della grauezza: e dato il peſo del <lb></lb>marco, ritrouare il luogo, oue detto marco faccia e<lb></lb>quipondio con la grauezza data. </s></p> <pb xlink:href="041/01/015.jpg" pagenum="14"></pb> <figure id="id.041.01.015.1.jpg" xlink:href="041/01/015/1.jpg"></figure> <p id="N104BC" type="head"> <s id="N104BE"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N104C4" type="main"> <s id="N104C6"><emph type="italics"></emph>Sia la ſtatera A B il ponto della ſoſpenſione C, la grauezza data D, <lb></lb>qual poniamo che ſi ſoſpenda in A: il marco dato di peſo E: ſi cerca il <lb></lb>ponto oue detto marco appeſo faccia equipondio. </s> <s id="N104CE">per queſto: facciaſi <lb></lb>che quella ragione che hà il peſo E al peſo D, quella habbia la linea A <lb></lb>C a C F, dico che appeſo il marco in F fa equipondio, cioè che'l pon<lb></lb>to del momento commune delle grauezze D & e ſia il ponto della ſo<lb></lb>ſpenſione C: il che è manifeſto, percioche ſono li peſi reciprochi al<lb></lb>le diſtanze. </s> <s id="N104DA">Haſſi dunque l'intento. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N104DE"> <p id="N104DF" type="head"> <s id="N104E1"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>XI. </s></p><p id="N104E9" type="main"> <s id="N104EB">Data una ſtatera, a cui ſia ugualmente applicato un <lb></lb>corpo, e data una grauezza ſoſpeſa da un dato ponto, <lb></lb>e dato il peſo del marco, ritrouare il ponto onde detto <lb></lb>marco ſoſpeſo faccia equipondio con la grauezza. </s></p><p id="N104F3" type="head"> <s id="N104F5"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N104FB" type="main"> <s id="N104FD"><emph type="italics"></emph>Sia la ſtatera A B il ponto del ſoſtenimento C, il corpo applicato <lb></lb>A D E B, la grauezza ſoſpeſa H I; il ponto onde la grauezza è appe<lb></lb>ſa in A: il marco L, ſi cerca il ponto onde ſoſpeſo il marco, faccia e<lb></lb>quipondio con H I, Per queſto: facciaſi alla linea A C uguale la C F,<emph.end type="italics"></emph.end><pb xlink:href="041/01/016.jpg" pagenum="15"></pb><figure id="id.041.01.016.1.jpg" xlink:href="041/01/016/1.jpg"></figure><lb></lb><emph type="italics"></emph>dunque il corpo D F applicato ad A F ſtà in equilibrio nel ponto della <lb></lb>ſoſpenſione C. </s> <s>et diuiſa F B reſtante per metà nel ponto G: del reſtan<lb></lb>te corpo F E applicato alla linea F B, ſarà G, il ponto di momento. </s> <s id="N10518">ſe <lb></lb>dunque la ragione che hà C F ad F G, habbia la grauezza E F alla <lb></lb>parte del peſo I, ſtarà il corpo F E in <expan abbr="equipōdio">equipondio</expan> con I, e perciò ſe di nuo<lb></lb>uo la ragione che hà, il marco al reſtante H habbia la parte de ſtate<lb></lb>ra A C, a C M, ſoſpeſo il marco L da M, farà equipondio <lb></lb>con H: & il corpo F E facea equipondio con I: ſtarà dunque ogni coſa <lb></lb>in equilibrio. </s> <s id="N1052A">ſi è dunque ritrouato il ponto M, onde ſoſpeſo il marco <lb></lb>faccia equipondio con la grauezza data. </s> <s id="N1052E">Il che ſi hauea da ritrouare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N10532"> <p id="N10533" type="head"> <s id="N10535"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>XII. </s></p><p id="N1053D" type="main"> <s id="N1053F">Fatta alla linea della ſtatera application di corpo, e <lb></lb>ſoſpeſe in eſſa più grauezze che ſoſtentino un peſo, ri<lb></lb>trouare ciaſcuna grauezza quanto portion di peſo ſo<lb></lb>ſtenti. </s></p><p id="N10547" type="head"> <s id="N10549"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N1054F" type="main"> <s id="N10551"><emph type="italics"></emph>Sia la linea della ſtatera A B il ponto del ſoſtentimento C & alla <emph.end type="italics"></emph.end><pb xlink:href="041/01/017.jpg" pagenum="16"></pb><figure id="id.041.01.017.1.jpg" xlink:href="041/01/017/1.jpg"></figure><lb></lb><emph type="italics"></emph>linea A B ſia fatta application di corpo et in eſſa appeſe le grauezze D <lb></lb>E, F: D in A, E in G, F in H: e dette grauezze ſoſtentine il peſo I <lb></lb>K L M: il cui momento ſia nel ponto B: ſi cerca ciaſcuna di dette gra<lb></lb>uezze D, E, F, quanta portione di peſo ſoſtenti. </s> <s id="N10568">facciaſi per queſto alla <lb></lb>linea B C uguale la C N: e la reſtante N A ſi diuida in parti uguali nel <lb></lb> <expan abbr="pōto">ponto</expan> O, e quella ragione che hà B C a C O quell'habbia il corpo della ſta<lb></lb>tera applicato ad N A ad M: ſarà <expan abbr="dũque">dunque</expan> <expan abbr="equipõderãte">equiponderante</expan> <expan abbr="cõ">con</expan> M: c la par<lb></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></lb>po della ſtatera ſtà in <expan abbr="equipõdio">equipondio</expan> con la portione del peſo M: e le ragioni <lb></lb>delle grauezze D, E, F, e delle diſtanze A C, G C, H C, cioè la ragio<lb></lb>ne della grauezza Dad F con la ragione della diſtanza A C a G C, <lb></lb>compongon la ragion di P a <expan abbr="q.">que</expan> & la ragione della grauezza E ad F, <lb></lb>con la ragione della diſtanza G C ad H C, compongon la ragione <emph.end type="italics"></emph.end><pb xlink:href="041/01/018.jpg" pagenum="17"></pb><emph type="italics"></emph>di Q ad R, & in quella ragione che ſono le tre quantità, P Q R, <lb></lb>poſte in continua habitudine, nella iſteſſa ſi diſtribuiſca il peſo I K L: <lb></lb>è manifeſto per quel che ſi è visto, che, D fa equiponderanza con I, lo <lb></lb>E co'l K, e lo F con lo L: ilche ſi cercaua. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N105A9"> <p id="N105AA" type="head"> <s id="N105AC"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>XIII. </s></p><p id="N105B4" type="main"> <s id="N105B6">La ſtatera di grauezze appeſe, che facciano equipon<lb></lb>dio: quantunque dal ſito orizontale moſſa ſi ſtà. </s></p><figure id="id.041.01.018.1.jpg" xlink:href="041/01/018/1.jpg"></figure> <p id="N105BD" type="head"> <s id="N105BF"><emph type="italics"></emph>Dimostratione. <emph.end type="italics"></emph.end></s></p><p id="N105C5" type="main"> <s id="N105C7"><emph type="italics"></emph>Sia la ſtatera nel ſito orizontale A B, il ponto della ſoſpenſione C, <lb></lb>li peſi e ſue centri D & E, il centro commune di ambe le grauezze F; <lb></lb>e moſſa la statera del ſito orizontale, paſſi il ponto A in G, il B in H,<emph.end type="italics"></emph.end><pb xlink:href="041/01/019.jpg" pagenum="18"></pb><emph type="italics"></emph>ſi che habbia la ſtatera la poſitione di G C H: li peſi e ſui centri di, I e <lb></lb>K: dico, che la ſtatera G H ſtarà, e non ſi mouerà di ſito. </s> <s id="N105D9">ſi moſtra <lb></lb>percioche eſſendo la grauezza I appeſa, inalzata, il centro ſuo gi<lb></lb>rando verrà nella perpendicolare del ponto della ſoſpenſione: e perciò <lb></lb>I, verrà nella perpendicolare del ponto G e K del ponto H. </s> <s id="N105E1">ſono <lb></lb>dunque, G I H K parallele. </s> <s id="N105E5">e perche il centro commune de peſi, diui<lb></lb>de nell'iſteſſa ragione la I K, & la D E, eſſendo la ragione delli peſi <lb></lb>vn'iſteſſa, & la C F nell'vna, e nell'altra ſoſpenſione perpendicolare, <lb></lb>e parallela, coſi alle A D E B, come alle G I, K H. </s> <s id="N105ED">perciò diuidendo <lb></lb>C F perpendicolare ſimilmente la D E, & la I K: ſarà il ponto F luo<lb></lb>go del centro nell'vna, luogo anco di centro nell'altra. </s> <s id="N105F3">eſſendo dunque <lb></lb>il centro del peſo commune nella perpendicolare della ſoſpenſione, ſta<lb></lb>rà. </s> <s id="N105F9">Ilche ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N105FD"> <p id="N105FE" type="head"> <s id="N10600"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>XV. </s></p><p id="N10608" type="main"> <s id="N1060A">La ſtatera di grauezze attaccate, che facciano equi<lb></lb>pondio, ſe'l ponto della ſoſpenſione, non ſia nella linea <lb></lb>delli centri: moſſa dal ſito orizontale non ſtarà, ma ri<lb></lb>tornarà nell'iſteſſo. </s></p><figure id="id.041.01.019.1.jpg" xlink:href="041/01/019/1.jpg"></figure> <p id="N10615" type="head"> <s id="N10617"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N1061D" type="main"> <s id="N1061F"><emph type="italics"></emph>Sia la linea della ſtatera, che ſtia nel ſito horizontale A B, li pe<lb></lb>ſi attaccati, & li lor centri C e D, e diuidaſi G D ſecondo li peſi reci<emph.end type="italics"></emph.end><pb xlink:href="041/01/020.jpg" pagenum="19"></pb><emph type="italics"></emph>procamente nel ponto E: è manifeſto che'l ponto E ſia il centro com<lb></lb>mune di ambi li peſi, e che mentre la ſtatera ſta, che ſia detto centro <lb></lb>nella perpendicolare, che cala dal ponto F. </s> <s id="N10631">perche dunque li peſi ſono <lb></lb>alla statera affiſſi, e non mutano li centri poſitura con la linea A B, e <lb></lb>ſempre fanno con eſſa angoli retti le C A, D B, E F, perciò moſſa <lb></lb>la ſtatera dal ſito horizontale, non ſarà E centro <expan abbr="cōmune">commune</expan> nella perpen<lb></lb>dicolare della ſoſpenſione: ma girando vſcirà di detta perpendicolare, <lb></lb>e perciò la ſtatera non ſtarà, ſin che di nuouo il detto ponto non torne <lb></lb>nella perpendicolare. <emph.end type="italics"></emph.end></s></p><figure id="id.041.01.020.1.jpg" xlink:href="041/01/020/1.jpg"></figure> <pb xlink:href="041/01/021.jpg" pagenum="20"></pb> <p id="N1064B" type="head"> <s id="N1064D">VETTE, E <lb></lb>LEVA. </s></p><p id="N10651" type="head"> <s id="N10653"><emph type="italics"></emph>DEFINITION. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N1065B" type="main"> <s id="N1065D">Vette diciamo la linea, che ſoſtiene grauezza, <lb></lb>qual ſia nelli ſue ponti eſtremi ſoſtenuta. </s></p><p id="N10661" type="head"> <s id="N10663"><emph type="italics"></emph>DEFINITION. <emph.end type="italics"></emph.end><lb></lb>II. </s></p><p id="N1066B" type="main"> <s id="N1066D">Et altrimente, vette motiua e leua, la linea che ſo<lb></lb>ſtenga grauezza, ſtabilita in vn ponto che ſotto leua <lb></lb>diciamo, & in vn'altro ponto da poſſanza, o moſſa, o <lb></lb>ſoſtenuta. </s></p><p id="N10675" type="head"> <s id="N10677"><emph type="italics"></emph>POSITION. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N1067F" type="main"> <s id="N10681">Miſuriamo la poſſanza con vna grauezza equiualen<lb></lb>te, o appeſa nell'iſteſſo ponto della poſſanza, o nell'al<lb></lb>tro ponto egualmente dal ſottoleua diſcoſto. </s></p><p id="N10687" type="head"> <s id="N10689"><emph type="italics"></emph>POSITION. <emph.end type="italics"></emph.end><lb></lb>II. </s></p><p id="N10691" type="main"> <s id="N10693">Ciaſcuna poſſanza in quanto ſoſtiene, eſſere egua<lb></lb>le al peſo ſoſtenuto. </s></p><figure id="id.041.01.021.1.jpg" xlink:href="041/01/021/1.jpg"></figure> <pb xlink:href="041/01/022.jpg" pagenum="21"></pb> <p id="N1069D" type="head"> <s id="N1069F"><emph type="italics"></emph>PROPOSITION. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N106A7" type="main"> <s id="N106A9">S'il ſottoleua ſtia tra la grauezza, e la poſſanza che <lb></lb>ſoſtenga detta grauezza; ſarà tra la poſſanza, & il pe<lb></lb>ſo la ragione, che è tra le parti della leua, reciprocamen<lb></lb>te. </s></p><figure id="id.041.01.022.1.jpg" xlink:href="041/01/022/1.jpg"></figure> <p id="N106B4" type="head"> <s id="N106B6"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N106BC" type="main"> <s id="N106BE"><emph type="italics"></emph>Sia la linea A B, il ponto del ſottoleua C, la grauezza D ſoste<lb></lb>nuta nel ponto della leua A; la poſſanza che ſoſtenga detta grauez<lb></lb>za in B: dico che la poſſanza B al peſo D, ha quella ragione che ha <lb></lb>la parte di leua A C alla C B, qual è ragion reciproca. </s> <s id="N106C8">ſi moſtra: inten<lb></lb>daſi attaccato in B il peſo che faccia equipondio con D: è manifeſto che <lb></lb>detto peſo E ſia equiualente alla forza B, ma il peſo E al peſo D ha la <lb></lb>ragione che A C a C B, che è la ragione reciproca di grauezza, e di<lb></lb>ſtanze: dunque, la potenza ancora haue l'iſteſſa ragione. </s> <s id="N106D2">ilche ſi ha<lb></lb>uea da moſtrare. <emph.end type="italics"></emph.end></s></p><p id="N106D8" type="head"> <s id="N106DA"><emph type="italics"></emph>PROPOSITION. <emph.end type="italics"></emph.end><lb></lb>II. </s></p><p id="N106E2" type="main"> <s id="N106E4">Se due potenze ſoſtentino vna grauezza con vn vet<lb></lb>te, ciaſcuna ſoſtentarà la ſua portione, ſecondo l'inter<lb></lb>uallo del peſo dalle potenze, pigliato reciprocamente. </s></p> <pb xlink:href="041/01/023.jpg" pagenum="22"></pb> <figure id="id.041.01.023.1.jpg" xlink:href="041/01/023/1.jpg"></figure> <p id="N106F0" type="head"> <s id="N106F2"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N106F8" type="main"> <s id="N106FA"><emph type="italics"></emph>Sia il vette A B, dal cui ponto D, penda il peſo C: le potenze che <lb></lb>ſoſtengono dette grauezze ſiano A & B: dico che'l B, e lo A ſoſten<lb></lb>tano portioni proportionali all'interualli reciprocamente: cio è che <lb></lb>quella ragione c'ha l'interuallo, B D, a D A, quella hàbbia la por<lb></lb>tione ſoſtentata dall' A, alla portione ſoſtentata dal B, ſi dimoſtra: <lb></lb>tagliſi ad A D uguale B E, accoppiata dunque communemente la D <lb></lb>E, ſarà A E uguale a B D: aggiungaſi all' A e la A G, che le ſia egua<lb></lb>le, & ad E B la B F che ſimilmente le ſia eguale. </s> <s id="N1070C">ſarà di tutta la G F, <lb></lb>il ponto mezzano D, & della G E, il ponto mezzano A, & della E <lb></lb>F, il ponto mezzano B. </s> <s id="N10712">applicata dunque a tutta la G F, una grauez<lb></lb>za che ſia uguale a C, ſarà di detta grauezza il ponto di momento in D <lb></lb>& ſarà equiualente nella ſua operatione alla grauezza C, & di eſſa <lb></lb>la parte applicata a G E ha il ſuo momento in A, c la parte applica<lb></lb>ta ad E ha il ſuo momento in B. </s> <s id="N1071C">dunque della grauezza applicata <lb></lb>la potenza A, ne ſoſtentarà la portione applicata a G E: e la potenza <lb></lb>B, la portione applicata ad E F. </s> <s id="N10722">Ma G E ad E F, ha la ragione che <lb></lb>l'interuallo B D, a D A che è reciproca. </s> <s id="N10726">dunque le potenze ſoſtenta<lb></lb>no le portioni de'peſi proportionali, reciprocamente pigliate con l'inter<lb></lb>ualli. </s> <s id="N1072C">ilche ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/024.jpg" pagenum="23"></pb> </chap> <chap id="N10733"> <p id="N10734" type="head"> <s id="N10736"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>III. </s></p><p id="N1073E" type="main"> <s id="N10740">Se il ſottoleua ſia fuori della grauezza, e della poſ<lb></lb>ſanza, ſarà la ragion della poſſanza alla grauezza l'iſteſ<lb></lb>ſa, che dell'interualli da eſse al ſottoleua <expan abbr="reciprocamẽte">reciprocamen<lb></lb>te</expan> pigliati </s></p><figure id="id.041.01.024.1.jpg" xlink:href="041/01/024/1.jpg"></figure> <p id="N1074F" type="head"> <s id="N10751"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10757" type="main"> <s id="N10759"><emph type="italics"></emph>Sia la leua A B il ſotto leua A, la grauezza C, il ſuo momento in <lb></lb>D, la poſſanza che ſoſtiene in B: dico che la poſſanza alla grauezza <lb></lb>ha la ragione, che D A ad A B, che è la ragion delle diſtanze piglia<lb></lb>te dal ſottoleua reciprocamente: ſi moſtra: perche il peſo C, e ſoſten<lb></lb>tato dalla leua B A, e la leua è ſoſtentata in due ponti B & A. </s> <s id="N10765">dunque <lb></lb>il peſo è ſoſtentato dalle potenze in B & A compartitamente, cioe <lb></lb>la poſſanza B ſoſtenta tal portion di peſo, qual'è la diſtanza A D di A <lb></lb>B, & A, tal portione qual'è D B, di B A, e perche la poſſanza ſo<lb></lb>ſtenente è uguale al peſo che ſoſtiene, ſono ambe le poſſanze B & A <lb></lb>giuntamente pigliate uguali al peſo E; e la portione ſoſtentata da B: <lb></lb>al tutto harrà quella ragione che la portion della leua D A a tutta <lb></lb>la leua A B. </s> <s id="N10776">qual è l'iſteſſa che della diſtanza della grauezza, alla di<lb></lb>ſtanza della potenza. </s> <s id="N1077A">ſi ha dunque l'intento. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/025.jpg" pagenum="24"></pb> </chap> <chap id="N10781"> <p id="N10782" type="head"> <s id="N10784"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>IV. </s></p><p id="N1078C" type="main"> <s id="N1078E">Se vna grauezza ſia con vna leua ſoſtenuta da due <lb></lb>ponti; & accreſciuta la leua dall altra parte ſi appenda <lb></lb>grauezza equiponderante, & ſi traſmuti in ſtatera: ſo<lb></lb>itentarà il ſoſtenimento in tal commutatione peſo mag<lb></lb>giore, quale al peſo di prima ſoſtenuto, ha ragione com<lb></lb>poſta della ragione delle portioni di tutta la linea accre<lb></lb>ſciuta communicanti, alle portioni interuallate: fat<lb></lb>te le due diuiſioni al ponto del ſottoleua, & al ponto <lb></lb>del primo momento. </s></p><figure id="id.041.01.025.1.jpg" xlink:href="041/01/025/1.jpg"></figure> <p id="N107A3" type="head"> <s id="N107A5"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N107AB" type="main"> <s id="N107AD"><emph type="italics"></emph>Sia la leua A B, il ſottoleua in A: la grauezza ſoſtenuta in C, la <lb></lb>poſſanza che'l ſoſtiene in B. </s> <s>& allungata la B A in vn D, appenda<lb></lb>ſi in D, vna grauezza che ſoſtenti la grauezza C. </s> <s id="N107B5">dico che in queſta <lb></lb>commutatione il ſottoleua A ſoſtenti peſo maggiore, & che il peſo <lb></lb>ſoſtenuto in detta commutatione, al peſo ſoſtenuto di prima, ha la ra<lb></lb>gion compoſta delle D C, A D, parti communicanti, alle D A, a C <lb></lb>B, parti interuallate. </s> <s id="N107BF">ſi moſtra: perche la parte del peſo ſoſtenuto da <lb></lb>A, a tutto il peſo C, ha la ragione, che B C a B A: &. </s> <s id="N107C3">il peſo C, ad <lb></lb>ambi li peſi C & D, ha la ragione che D A a D C, ma la ragione del<emph.end type="italics"></emph.end><pb xlink:href="041/01/026.jpg" pagenum="25"></pb><emph type="italics"></emph>la portione ſoſtenuta da A, alla grauezza C, & di C, ad ambe CD, ſot<lb></lb>trattone il termine mezzano, compongono la ragione della portione ſoſte <lb></lb>nuta da A, ad ambe le C D, & la ragione di B C a BA, & di D A a D <lb></lb>C, fanno la ragione compoſta delle parti communicanti alle interuallate. </s><lb></lb> <s id="N107D6">Haſſi dunque l'intento: che'l peſo di prima ſoſtenuto, al peſo ſoſtenuto <lb></lb>dopo la commutatione, ha la ragion compoſta delle parti interuallate alle <lb></lb>communicanti. </s> <s id="N107DC">Ilche ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N107E0"> <p id="N107E1" type="head"> <s id="N107E3"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>V. </s></p><p id="N107EB" type="main"> <s id="N107ED">Date nell' eſtremità del vette due poſsanze c'habbia<lb></lb>no qualſiuoglia ragione tra di loro; e dato vn peſo, a det<lb></lb>te poſſanze giuntamente pigliate vguale, ritrouare il <lb></lb>ponto del vette, onde il dato peſo ſoſpeſo, ſia da det<lb></lb>te poſſanze ſoſtenuto. </s></p><figure id="id.041.01.026.1.jpg" xlink:href="041/01/026/1.jpg"></figure> <p id="N107FA" type="head"> <s id="N107FC"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10802" type="main"> <s id="N10804"><emph type="italics"></emph>Sia il vette AB: le poſſanze nelli ponti A e B, c'habbiano tradi lo<lb></lb>ro qualſiuoglia ragione: & il peſo ad ambe poſſanze giuntamente pi<lb></lb>gliate vguale ſia C: ſi cerca il ponto, onde detto peſo ſia da dette <expan abbr="poßãanze">poßan<lb></lb>ze</expan> ſoſtenuto. </s> <s id="N10812">per il che dico: che ſe in quella ragione, c' ha la poſſan<emph.end type="italics"></emph.end><pb xlink:href="041/01/027.jpg" pagenum="26"></pb><emph type="italics"></emph>za B, ſi diuida la vette AB in D, e ſia come la poſſanza A alla B: così, <lb></lb>la portione di vette B D a D A: dico che poſto il peſo C, in D: ſarà, <lb></lb>ſoſtenuto da dette poſſanze: percioche grauando il peſo nelli ponti B: & <lb></lb>A, che ſoſtentano compartitamente, ſecondo la ragion di BD a DA: <lb></lb>& hauendo la portion che graua in A, alla portion che graua in B, la <lb></lb>ragion che B D a D A: qual'è l'iſteſſa che della poſſanza A alla poſſanza <lb></lb>B: dunque la portione che graua in, A alla portione che graua in B, <lb></lb>e come la poſſanza A, alla B: e <expan abbr="permutãdo">permutando</expan> la portion che graua in A, a <lb></lb>la poſſanza A, ſarà come la portione che graua in B alla poſſanza B, <lb></lb>e componendo li antecedenti, tutto il peſo C, ad ambe le poſſanze giun<lb></lb>te, harrà l'iſteſſa ragione che vna advna: ma il peſo tutto C, è vgua<lb></lb>le ad ambe le poſſanze giuntamente pigliate: dunque diuiſamente le <lb></lb>portioni, ciaſcuna alla poſſanza oue graua, ſarà vguale: e percio ſa<lb></lb>rà del peſo ſoſtenuto, la portione che graua in A, vguale alla poſſanza <lb></lb>in A: e la portione che graua in B, vguale alla pòſſanza in B: e percio <lb></lb>le poſſanze ſoſtentaranno il detto peſo nel ponto D. </s> <s id="N1083E">Il che ſi hauea da <lb></lb>moſtrare,<emph.end type="italics"></emph.end></s></p><p id="N10844" type="head"> <s id="N10846"><emph type="italics"></emph>Appendice. <emph.end type="italics"></emph.end></s></p><p id="N1084C" type="main"> <s id="N1084E">Et è manifeſto che in ogni altro ponto del detto vet<lb></lb>te, il peſo non ſarà ſoſtenuto, ma aggrauerà più l'vna ò <lb></lb>l'altra poſſanza, verſo oue ſarà portato. </s></p> </chap> <chap id="N10854"> <p id="N10855" type="head"> <s id="N10857"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>VI. </s></p><p id="N1085F" type="main"> <s id="N10861">Se una leua ſia inalzata, o baſſata ſotto l'orizonte: <lb></lb>& da un ponto fuori di eſſa, ſi tireranno due perpendi<lb></lb>colari, l'vna ad eſſa leua, e l'altra all'orizonte: faran<lb></lb>no le due perpendicolari angolo tra di loro, vguale all' <lb></lb>angolo della leua con l'orizonte. </s></p> <pb xlink:href="041/01/028.jpg" pagenum="27"></pb> <figure id="id.041.01.028.1.jpg" xlink:href="041/01/028/1.jpg"></figure> <p id="N10871" type="head"> <s id="N10873"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10879" type="main"> <s id="N1087B"><emph type="italics"></emph>Sia la linea orizontale A B, la leua ſopra di eſſa inalzata o deſpreſ<lb></lb>ſa A C. </s> <s>il ponto fuori della leua E: da cui ſi tirino due perpendicolari <lb></lb>l'vna alla leua DE, l'altra all'orizonte D F, che ſeghi la leua in F, & <lb></lb>la linea orizontale in G. </s> <s id="N10885">dico che l'angolo fatto dalle due D E, D F <lb></lb>ſia vguale all'angolo fatto, dalle due A B, A C: ſi moſtra: percioche <lb></lb>le due A C, D G, ſi ſegano nel ponto F, ſaranno l'angoli A F G, et D <lb></lb>F E, d'incontro vguali: e gli angoli ad E & G ſono retti: dunque il tri<lb></lb>angolo D F E, è equiangolo al triangolo A F G, e l'angolo F D E, v<lb></lb>guale a l'angolo F A G. </s> <s id="N10891">Il che ſi hauea da mòſtrare. <emph.end type="italics"></emph.end></s></p><p id="N10895" type="head"> <s id="N10897"><emph type="italics"></emph>Appendice,<emph.end type="italics"></emph.end></s></p><p id="N1089D" type="main"> <s id="N1089F">Et è manifeſto che eſſendo detto ponto di ſopra la li<lb></lb>nea della leua inalzata, e di ſotto della leua baſſata; ſe<lb></lb>cherà detta linea in ponto più dalla poſſanza lontano. </s><lb></lb> <s id="N108A6">e per <expan abbr="cõtrario">contrario</expan> pigliandoſi detto ponto, o ſotto dell'alza<lb></lb>tà, o ſopra della baſſata, ſegherà in ponti più à detta pos<lb></lb>ſanza vicini. </s></p> <pb xlink:href="041/01/029.jpg" pagenum="28"></pb> </chap> <chap id="N108B3"> <p id="N108B4" type="head"> <s id="N108B6"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>VII. </s></p><p id="N108BE" type="main"> <s id="N108C0">Se'l centro del peſo attaccato ad eſſa leua ſia ſopra <lb></lb>della leua, inalzata la leua, la poſſanza ſoſtentarà minor <lb></lb>peſo. </s></p><figure id="id.041.01.029.1.jpg" xlink:href="041/01/029/1.jpg"></figure> <p id="N108C9" type="head"> <s id="N108CB"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N108D1" type="main"> <s id="N108D3"><emph type="italics"></emph>Sia la leua A B, a cui ſia attaccata vna grauezza, il cui centro ſia <lb></lb>C: & intendaſi detta leua in ſito dall'orizonte eleuato: dico che la po<lb></lb>tenza B, ſoſtenta del peſo della grauezza minor portione, che nel ſito <lb></lb>orizontale. </s> <s id="N108DD">ſi moſtra: tirinſi dal ponto C linee, l'vna perpendicolare <lb></lb>alla leua che ſia C D, & l'altra perpendicolare all'orizonte, che <lb></lb>ſia C E, che ſeghi la leua nel ponto E: è manifeſto che'l detto ponto <lb></lb>ſarà più diſcoſto dalla poſſanza, e più vicino al ponto del ſottoleua. </s> <s id="N108E5">ſe <lb></lb>dunque per lo ponto C, ſi tiri la linea G C F, parallela all'orizonte, & <lb></lb>per li ponti B & A, le linee B F, A G, perpendicolari <expan abbr="all'orizõte">all'orizonte</expan> è mani <lb></lb>feſto, che l'iſteſſo effetto fa la poſſanza in F che ſe fuſſe in B, e lo ſoſtegno <lb></lb>in A l'iſteſſo che ſe fuſſe in C: percioche ciaſcun momento opera ſecon<lb></lb>da la ſua perpendicolare: perche dunque poſta la poſſanza in F, e lo ſo<emph.end type="italics"></emph.end><pb xlink:href="041/01/030.jpg" pagenum="29"></pb><emph type="italics"></emph>ſtegno in G, la poſſanza F, ſoſtiene tal portione di tutto il peſo, qual <lb></lb>portione è G C, di G F: e qual'è G C, di tutta G F, tal'è A E di tutta <lb></lb>A B, perche le A G, C E, B F, ſono parallele: ſoſtenta dunque la poſ<lb></lb>ſanza B, del peſo tal portione, qual'è A E di tutta A B: ſe dunque <lb></lb>A E è minor portione di A B, che la A D, dell'iſteſſa A B: la poſſan<lb></lb>za con la leua inalzata il cui centro del peſo è ſopra, ſoſtenta minor <lb></lb>portione che nel ſito orizontale. </s> <s id="N10909">Il che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p><p id="N1090D" type="head"> <s id="N1090F"><emph type="italics"></emph>Appendice. </s> <s id="N10913">I. <emph.end type="italics"></emph.end></s></p><p id="N10917" type="main"> <s id="N10919">E per l'iſteſſo mezzo ſi moſtrerà che quanto più la le <lb></lb>ua s'inalza, tanto minor peſo ſoſtiene. </s></p><p id="N1091D" type="head"> <s id="N1091F"><emph type="italics"></emph>Appendice. </s> <s id="N10923">II. <emph.end type="italics"></emph.end></s></p><p id="N10927" type="main"> <s id="N10929">E che poſto il centro della grauezza ſotto la leua, <lb></lb>quanto più s'inalzi, magior portione di peſo ſoſtenga. </s></p><p id="N1092D" type="head"> <s id="N1092F"><emph type="italics"></emph>Appendice. </s> <s id="N10933">III. <emph.end type="italics"></emph.end></s></p><p id="N10937" type="main"> <s id="N10939">E che nelle leue baſſate ſotto l'orizonte, auuenga a <lb></lb>contrario. </s></p> </chap> <chap id="N1093D"> <p id="N1093E" type="head"> <s id="N10940"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>VIII. </s></p><p id="N10948" type="main"> <s id="N1094A">Dato nella leua il ponto di momento di una grauez<lb></lb>za, e data qualſivoglia ragione di poſſanza a grauez<lb></lb>za, ritrouar nella leua il ponto, oue la data poſſanza ſo<lb></lb>ſtenga la data grauezza. </s></p> <pb xlink:href="041/01/031.jpg" pagenum="30"></pb> <figure id="id.041.01.031.1.jpg" xlink:href="041/01/031/1.jpg"></figure> <p id="N10958" type="head"> <s id="N1095A"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10960" type="main"> <s id="N10962"><emph type="italics"></emph>Sia nella leua A B, il ponto del ſottoleua in A: il ponto di momen<lb></lb>to della data grauezza in C. </s> <s>et la ragion della poſſanza data alla grauez<lb></lb>za, come di E a D: ſi cerca nella leua il ponto, oue poſta la data poſ<lb></lb>ſanza ſoſtenga la data grauezza. </s> <s id="N1096C">per queſto: facciaſi come E a D, così <lb></lb>A C ad A F: & intendaſi la poſſanza in F. </s> <s id="N10970">dico che detta poſſanza in <lb></lb>F ſoſtiene la grauezza in C. </s> <s id="N10974">ſi moſtra: percioche eſſendo la ragion del<lb></lb>la poſſanza alla grauezza come E a D, e la ragion dell'interuallo del<lb></lb>la grauezza A C, all'interuallo della poſſanza A F, l'iſteſſa reciproca <lb></lb>mente: ſoſtentarà dunque la data poſſanza in F, la grauezza in C. </s> <s id="N1097C">Il <lb></lb>che ſi cercaua. <emph.end type="italics"></emph.end></s></p><p id="N10982" type="head"> <s id="N10984"><emph type="italics"></emph>Appendice. <emph.end type="italics"></emph.end></s></p><p id="N1098A" type="main"> <s id="N1098C">Et è manifeſto che in qual ſi uoglia altro ponto oltre <lb></lb>del termine del ſoſtenimento, la data poſſanza mouerà <lb></lb>la data grauezza: e tanto più facilmente quanto più ſi <lb></lb>ſcoſtarà. </s></p><figure id="id.041.01.031.2.jpg" xlink:href="041/01/031/2.jpg"></figure> <pb xlink:href="041/01/032.jpg" pagenum="31"></pb> <p id="N1099A" type="head"> <s id="N1099C">RAGGI NELL <lb></lb>ASSE. </s></p><p id="N109A0" type="head"> <s id="N109A2"><emph type="italics"></emph>SVPPOSITIONE. <emph.end type="italics"></emph.end></s></p><p id="N109A8" type="main"> <s id="N109AA">Svpponiamo, in vno iſteſſo aſſe, due rag<lb></lb>gi c'habbiano nelli ſuoi ſtremi li centri de peſi. </s></p><p id="N109AE" type="main"> <s id="N109B0">E detti raggi, o in vna pianezza, e che non facciano <lb></lb>angolo, o in due, e che facciano angolo. </s></p><p id="N109B4" type="head"> <s id="N109B6"><emph type="italics"></emph>POSITIONE. <emph.end type="italics"></emph.end></s></p><p id="N109BC" type="main"> <s id="N109BE">Pigliamo, il momento di ciaſcun peſo, ſecondo il pon<lb></lb>to, oue la perpendicolare del momento taglia la linea <lb></lb>orizontale, che paſſa per l'aſſe. </s></p><figure id="id.041.01.032.1.jpg" xlink:href="041/01/032/1.jpg"></figure> </chap> <chap id="N109C7"> <p id="N109C8" type="head"> <s id="N109CA"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N109D2" type="main"> <s id="N109D4">Delle grauezze poſte in raggi che non fanno tra di <lb></lb>loro angolo, in qualunque ſito poſte, li momenti tra di <lb></lb>loro hanno l'iſteſſa ragione. </s></p><p id="N109DA" type="head"> <s id="N109DC"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N109E2" type="main"> <s id="N109E4"><emph type="italics"></emph>Sial'aſſe A, a cui ſiano affiſſi li raggi A B, A C, quali s'intenda<emph.end type="italics"></emph.end><pb xlink:href="041/01/033.jpg" pagenum="32"></pb><figure id="id.041.01.033.1.jpg" xlink:href="041/01/033/1.jpg"></figure><lb></lb><emph type="italics"></emph>no eſſere nel ſito orizontale, & mouaſi dal detto ſito, sì che il B uen<lb></lb>ga in D, & il C venga in E: dico che li momenti delle grauezze in det<lb></lb>ti raggi quantunque moſſi di ſito, ſiano nell'iſteſſa ragione tra di loro. </s><lb></lb> <s id="N109FA">ſi moſtra: tiriſi per D la perpendicolare D F & per E la perpendicola <lb></lb>re E G; perche dunque F A ad A G, ha la ragione che D A ad A E, <lb></lb>perciò che ſono D F, E G, parallele: ma come D A ad A E, così B A ad <lb></lb>A C: perche ſono l'iſteſſi raggi, come dunque B A ad A C, così F A <lb></lb>ad A G: e perche la ragion delli momenti e compoſta della ragion delle <lb></lb>grauezze, e della ragion delle diſtanze dal centro: ma la ragione delle <lb></lb>grauezze è l'iſteſſa: e la ragione delle <expan abbr="diſtãze">diſtanze</expan> è l'iſteſſa: dunque la ragion <lb></lb>di ambe compoſte, è anco l'iſteſſa. </s> <s id="N10A0E">Il che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/034.jpg" pagenum="33"></pb> </chap> <chap id="N10A15"> <p id="N10A16" type="head"> <s id="N10A18"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>II. </s></p><p id="N10A20" type="main"> <s id="N10A22">Date qual ſi uoglia due grauezze, nelli raggi che fac<lb></lb>ciano angolo dato, ritrouar nelle loro circolationi, pon<lb></lb>ti oue facciano equipondio. </s></p><figure id="id.041.01.034.1.jpg" xlink:href="041/01/034/1.jpg"></figure> <p id="N10A2B" type="head"> <s id="N10A2D"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10A33" type="main"> <s id="N10A35"><emph type="italics"></emph>Sia l'aſſe A: li raggi che facciano <expan abbr="ãgolo">angolo</expan> dato A C, B A: & intendaſi <expan abbr="nelli">nel<lb></lb>li</expan> ponti B e C, eſſer li <expan abbr="cẽtri">centri</expan> delle grauezze: & le circonferenze che det<lb></lb>ti ponti girando attorno fanno, ſiano E B, C F: ſi cercano in dette cir<emph.end type="italics"></emph.end><pb xlink:href="041/01/035.jpg" pagenum="34"></pb><emph type="italics"></emph>conferenze li ponti, oue eſſendo dette grauezze, facciano equipondio. </s><lb></lb> <s id="N10A52">Diuidaſi la B C interuallo de centri, ſiche qual ragione ha la grauezza, <lb></lb>B, alla C, tal habbia la linea C D alla, D B: e tiriſi A D: e tirata <lb></lb>per A, la A E B perpendicolare all'Orizonte, facciaſi all'angolo D A <lb></lb>B, vguale lo E A G: & allo D A C, vguale E A H: dico che'l ponto <lb></lb>G, è oue portato il B, & H, oue portato il C, fanno equipondio. </s> <s id="N10A5C">E prima <lb></lb>che portato il B in G, venga il C in H, è manifeſto: percioche l'ango <lb></lb>B A C è vguale al G A H: e per l'iſteſſa ragione, è manifeſto che nell' <lb></lb>iſteſſo tempo il ponto D, ſia nella A E. </s> <s id="N10A64">ma il <expan abbr="põto">ponto</expan> D è il centro commu<lb></lb>ne di peſo di dette due grauezze. </s> <s id="N10A6C">E dunque il centro commune nel <lb></lb>la perpendicolare del ſoſtenimento: e perciò le grauezze ſtanno. </s> <s id="N10A70">Jl che <lb></lb>ſi cercaua. <emph.end type="italics"></emph.end></s></p><p id="N10A76" type="head"> <s id="N10A78"><emph type="italics"></emph>Appendice. </s> <s id="N10A7C">I. <emph.end type="italics"></emph.end></s></p><p id="N10A80" type="main"> <s id="N10A82">Et è manifeſto che nelli due ponti, oppoſti alli ritroua <lb></lb>ti, facciano equipondio: & non altroue: percioche in o<lb></lb>gni altra poſitura oltre di dette due, il centro commune <lb></lb>e fuori del perpendicolo. </s></p><p id="N10A8A" type="head"> <s id="N10A8C"><emph type="italics"></emph>Appendice. </s> <s id="N10A90">II. <emph.end type="italics"></emph.end></s></p><p id="N10A94" type="main"> <s id="N10A96">Et è manifeſto che nell'arco ſotto il ponto dell'equi<lb></lb>pondio la grauezza ha momento maggiore: e nell'arco <lb></lb>ſopra il ponto dell'equipondio ha momento minore. </s></p> </chap> <chap id="N10A9C"> <p id="N10A9D" type="head"> <s id="N10A9F"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N10AA7" type="main"> <s id="N10AA9">Dàte qual ſi uoglia due grauezze nelli dati raggi, che <lb></lb>fanno dato angolo: ritrouar nelle loro circolationi, pon<pb xlink:href="041/01/036.jpg" pagenum="35"></pb>ti oue il momento dell'uno, al <expan abbr="momẽto">momento</expan> dell'altro habbia <lb></lb>qual ſi voglia data ragione. </s></p><figure id="id.041.01.036.1.jpg" xlink:href="041/01/036/1.jpg"></figure> <p id="N10ABA" type="head"> <s id="N10ABC"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10AC2" type="main"> <s id="N10AC4"><emph type="italics"></emph>Siano le date grauezze A & B: li raggi AC, BC, fiſſi nell'aſſe C: <lb></lb>che facciano dato <expan abbr="ãgolo">angolo</expan>: e la circolation di A ſia, AD: di B ſia B E. </s> <s>e <lb></lb>la data qualſiuoglia ragione ſia di F a G: ſi cercano nella circolatione DA <lb></lb>e nella BE, ponti oue habbian li momenti di A e B, ragion di F a G. </s> <s id="N10AD2">In<lb></lb>tendaſi nella ragion di A a B, la quantità F ad H. </s> <s id="N10AD6">e nella reſtante ra<lb></lb>gione di H a G, ſi diuida A B in L. </s> <s id="N10ADA">& all'angolo L C A facciaſi vgua<emph.end type="italics"></emph.end><pb xlink:href="041/01/037.jpg" pagenum="36"></pb><emph type="italics"></emph>le il D C M, & allo L C B eguale il DCN: èmanifeſto, che portato A in <lb></lb>M: B verrà in N. </s> <s id="N10AE6">& il ponto L nella perpendicolare C D. </s> <s id="N10AE8">e ſe per il <lb></lb>ponto C ſi tiri la P C Q parallela all'Orizonte: e dalli ponti M & N ſi <lb></lb>tirino a queſta, perpendicolari le MQ NP: ſarà il momento della <lb></lb>grauezza in M, al momento della grauezza in N di ragion compo <lb></lb>ſta della grauezza A alla grauezza B, e della distanza Q C, alla CP, <lb></lb>che è l'iſteſſa che di A L ad L B.percioche queſta è l'iſteſſa che di M O ad <lb></lb>O N: cioè della compoſta delle ragioni di F ad H, e di H a G: ciò è di F a <lb></lb>G. </s> <s id="N10AF8"> <expan abbr="harrãno">harranno</expan> dunque li <expan abbr="momẽti">momenti</expan> di A & B, mentre ſiano poſti nelli ponti <lb></lb>M & N la ragion data di F a G. </s> <s id="N10B03">Il che ſi cercaua. <emph.end type="italics"></emph.end></s></p><p id="N10B07" type="head"> <s id="N10B09"><emph type="italics"></emph>Appendice. <emph.end type="italics"></emph.end></s></p><p id="N10B0F" type="main"> <s id="N10B11">Et è manifeſto che prodotte le linee del centro nelli <lb></lb>ponti oppoſti delle dette circonferenze, hauranno iui li <lb></lb>momenti delle date grauezze l'iſteſſa ragione: e non <lb></lb>altroue. </s></p><figure id="id.041.01.037.1.jpg" xlink:href="041/01/037/1.jpg"></figure> <pb xlink:href="041/01/038.jpg" pagenum="37"></pb> <p id="N10B1F" type="head"> <s id="N10B21">MOMENTI <lb></lb>CENTRALI</s></p><p id="N10B25" type="main"> <s id="N10B27">E qvanto delli momenti paralleli habbiamo <lb></lb>moſtrato, tutto ſi adatterà anco alli momenti con<lb></lb>correnti à centro: ſe in vece di linee dritte conſideria<lb></lb>mo le circolari d'intorno il centro oue li momenti con<lb></lb>corrono: & in dette circolari ſi faccia l'iſteſſa partitione: <lb></lb>e ſe in vece delli corpi terminati, da ſuperficie parallele, <lb></lb>s'intendano altri corpi terminati, parte da ſuperficie sfe<lb></lb>riche c'habbiano detto centro: parte da ſuperficie pia<lb></lb>ne che paſſino per eſſo. </s></p><figure id="id.041.01.038.1.jpg" xlink:href="041/01/038/1.jpg"></figure> <pb xlink:href="041/01/039.jpg" pagenum="38"></pb> <p id="N10B3F" type="head"> <s id="N10B41">ROTE VET<lb></lb>TIVE. </s></p><p id="N10B45" type="head"> <s id="N10B47"><emph type="italics"></emph>SVPPOSITIONE. <emph.end type="italics"></emph.end></s></p><p id="N10B4D" type="main"> <s id="N10B4F">Svpponiamo vna, o più rote congiogate, <lb></lb>muouerſi per piano, che ſia, o di poſitura orizontale, <lb></lb>o inchinata. </s></p><p id="N10B55" type="head"> <s id="N10B57"><emph type="italics"></emph>DEFINITION. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N10B5F" type="main"> <s id="N10B61">Cogiogation ſemplice, diciamo delle rote, che ſono <lb></lb>sù di vn'iſteſſo aſſe. </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 ſono in più aſſi. </s></p><p id="N10B6D" type="head"> <s id="N10B6F">III. </s></p><p id="N10B71" type="main"> <s id="N10B73">Portioni terminate dal ſoſtenimento diciamo nel cir<lb></lb>colo, le fatte dalla linea perpendicolare per lo ponto del <lb></lb>contatto, all'orizonte: e nel cilindro, dalla ſuperficie pia <lb></lb>na per la linea del contatto, perpendicolare ſimilmente <lb></lb>all'orizonte. </s></p> <pb xlink:href="041/01/040.jpg" pagenum="39"></pb> <p id="N10B80" type="head"> <s id="N10B82"><emph type="italics"></emph>POSITIONE. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N10B8A" type="main"> <s id="N10B8C">Poniamo ogni forza, o trattiua, o pulſiua, giunger mo<lb></lb>mento uerſo quella parte, oue tira, o ſpinge. </s></p><p id="N10B90" type="head"> <s id="N10B92">II. </s></p><p id="N10B94" type="main"> <s id="N10B96">E ſe'l centro del peſo ſia nell'iſteſſa linea dell'appendi <lb></lb>mento, o ſoſtenimento: che la grauezza non habbia mo<lb></lb>mento, ne uerſo l'vna, ne uerſo l'altra parte. </s></p><figure id="id.041.01.040.1.jpg" xlink:href="041/01/040/1.jpg"></figure> </chap> <chap id="N10B9F"> <p id="N10BA0" type="head"> <s id="N10BA2"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N10BAA" type="main"> <s id="N10BAC">Della rota vettiua, che ſi moue ſopra di vn piano ori<lb></lb>zontale, il centro del peſo ſempre è nella perpendicola<lb></lb>re del ſoſtenimento. </s></p><figure id="id.041.01.040.2.jpg" xlink:href="041/01/040/2.jpg"></figure> <pb xlink:href="041/01/041.jpg" pagenum="40"></pb> <p id="N10BB8" type="head"> <s id="N10BBA"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10BC0" type="main"> <s id="N10BC2"><emph type="italics"></emph>Sia la linea Orizontale A B: il circolo che rappreſenta la rota, CD: <lb></lb>il ponto, oue detta rota tocca il piano C: da cui ſi cacci ad angoli ret<lb></lb>ti la linea C D, è manifeſto che detta linea, è la perpendicolare del ſoſte<lb></lb>nimento: & perquelche nelli libri Giometrici ſi moſtra: che paſſa per il <lb></lb>centro del circolo, che è il centro della rota e grauezza: perilche diui<lb></lb>de il circolo il parti vguali, & equeponderanti: è dunque il centro <lb></lb>del peſo nella perpendicolare del ſoſtenimento. </s> <s id="N10BD2">Il che ſi hauea da <lb></lb>moſtrare,<emph.end type="italics"></emph.end></s></p><p id="N10BD8" type="head"> <s id="N10BDA"><emph type="italics"></emph>Appendice. </s> <s id="N10BDE">I. <emph.end type="italics"></emph.end></s></p><p id="N10BE2" type="main"> <s id="N10BE4">Et il ſimile ſi moſtra, nelle ſemplici rote congiogate, <lb></lb>ſopra l'aſſe de quali, poſi la grauezza. </s></p><p id="N10BE8" type="head"> <s id="N10BEA"><emph type="italics"></emph>Appendice, II. <emph.end type="italics"></emph.end></s></p><p id="N10BF0" type="main"> <s id="N10BF2">Et è manifeſto nelle rote, sù l'aſſe de quali poſi la <lb></lb>grauezza: che nel piano <expan abbr="orizõ">orizon</expan>tale, non habbian momen<lb></lb>to ne verſo l'vna, ne verſo l'altra parte. </s></p><p id="N10BFC" type="head"> <s id="N10BFE"><emph type="italics"></emph>Appendice. </s> <s id="N10C02">III. <emph.end type="italics"></emph.end></s></p><p id="N10C06" type="main"> <s id="N10C08">E che perciò qual ſi voglia poſſanza, le porterà così <lb></lb>nell'vna, come nell'altra parte, </s></p> <pb xlink:href="041/01/042.jpg" pagenum="41"></pb> </chap> <chap id="N10C0F"> <p id="N10C10" type="head"> <s id="N10C12"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>II. </s></p><p id="N10C1A" type="main"> <s id="N10C1C">Nella rota che ſi porta per piano inchinato, il centro <lb></lb>del peſo, è fuori della perpendicolare del ſoſtenimento. </s><lb></lb> <s id="N10C21">et il momento della rota appoggiata al piano, al momen<lb></lb>to della rota ſoſpeſa, la ha ragione, che l'ecceſſo delle <lb></lb>portioni del circolo, al circolo tutto. </s></p><figure id="id.041.01.042.1.jpg" xlink:href="041/01/042/1.jpg"></figure> <pb xlink:href="041/01/043.jpg" pagenum="42"></pb> <p id="N10C2D" type="head"> <s id="N10C2F"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10C35" type="main"> <s id="N10C37"><emph type="italics"></emph>Sia la linea che rappreſenta il piano orizontale A B: la linea del pia <lb></lb>no inchinato A C: il circolo della rota D E FG: il toccamento D: e dal <lb></lb>ponto D, tiriſi perpendicolare all'orizonte B D F: è manifeſto che detta <lb></lb>linea, ſia la perpendicolare del ſoſtenimento: dico che'l centro del peſo <lb></lb>è fuori di detta linea. </s> <s id="N10C43">Si moſtra: perche del triangolo D B A: l'angolo, <lb></lb>E, che fa la perpendicolare con l'orizonte, è retto: reſta l'angolo B D A, <lb></lb>a cuto: e perciò la portione D G F, e maggiore del ſemicircolo; & in eſ<lb></lb>ſa ſarà il centro del circolo, che è anco centro di peſo. </s> <s id="N10C4B">è dunque il cen<lb></lb>tro del peſo fuori della linea del <expan abbr="ſoſtenimẽto">ſoſtenimento</expan>. </s> <s id="N10C53">De ſcriuaſi alla D E, la por<lb></lb>tione di circolo D H F, ſimile a D E F; ſaranno dette portioni vgua<lb></lb>li, e faranno equipondio. </s> <s id="N10C59">reſta dunque la figura lunare ſenza equi<lb></lb>pondio: & il momento della rota appoggiata ſarà meno che della ro<lb></lb>ta ſoſpeſa, ſecondo la ragione della figura lunare a tutto il circolo: cio è <lb></lb>ſecondo la ragione dell'ecceſſo delle portioni, al circolo tutto. </s> <s id="N10C61">Il che <lb></lb>ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p><p id="N10C67" type="head"> <s id="N10C69"><emph type="italics"></emph>Appendice. </s> <s id="N10C6D">I. <emph.end type="italics"></emph.end></s></p><p id="N10C71" type="main"> <s id="N10C73">E l'iſteſſo che si è moſtrato nella rota c'ha grauezza; <lb></lb>si moſtra nelle rote al cui aſſe appoggi altro peſo. </s></p><p id="N10C77" type="main"> <s id="N10C79"><emph type="italics"></emph>Percio che ſe in vece del peſo appoggiato all'aſſe, intendiamo darſi <lb></lb>l'iſteſſo peſo alle rote: eſſendo peſi vguali con loro centri nell'iſteſſe li<lb></lb>nee, & la linea del ſoſtenimento l'iſteſſa, harranno li peſi l'iſteſſi <expan abbr="momẽti">momenti</expan> <emph.end type="italics"></emph.end></s></p><p id="N10C86" type="head"> <s id="N10C88"><emph type="italics"></emph>Appendice, II. <emph.end type="italics"></emph.end></s></p><p id="N10C8E" type="main"> <s id="N10C90">Et è manifeſto che detta rota correrà verſo la parte <lb></lb>del piano inferiore. </s></p><p id="N10C94" type="main"> <s id="N10C96"><emph type="italics"></emph>Percioche tirata dal centro I, la IG K perpendicolare del momento <lb></lb>tutto ſin che <expan abbr="s'incõtri">s'incontri</expan> col piano per oue camina: ſarà il ponto G della cir<emph.end type="italics"></emph.end><pb xlink:href="041/01/044.jpg" pagenum="43"></pb><emph type="italics"></emph>conferenza diſcoſto dal <expan abbr="põto">ponto</expan> K del piano per oue camina la rota: e <expan abbr="tãto">tanto</expan> <lb></lb> <expan abbr="maggiormẽte">maggiormente</expan> il <expan abbr="põto">ponto</expan> oue <expan abbr="s'incõtra">s'incontra</expan> la <expan abbr="perpẽdicolare">perpendicolare</expan> del <expan abbr="cẽtro">centro</expan> di peſo del<lb></lb>la figura lunare: la cui <expan abbr="diſtãza">diſtanza</expan> dalla linea del <expan abbr="ſoſtenimẽto">ſoſtenimento</expan>, è maggior che <lb></lb>la diſtanza del centro del circolo, ſecondo la ragion di tutto il circolo al <lb></lb>la figura lunare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N10CD4"> <p id="N10CD5" type="head"> <s id="N10CD7"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>III. </s></p><p id="N10CDF" type="main"> <s id="N10CE1">Se vn peſo ſia portato da due <expan abbr="cõgiogationi">congiogationi</expan> di rote, <lb></lb>ſarà il peſo ſoſtenuto dalli due aſſi compartitamente, ſe<lb></lb>condo la ragione delle diſtanze del momento da gli aſſi, <lb></lb>reciprocamente. </s></p><figure id="id.041.01.044.1.jpg" xlink:href="041/01/044/1.jpg"></figure> <p id="N10CF0" type="head"> <s id="N10CF2"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10CF8" type="main"> <s id="N10CFA"><emph type="italics"></emph>Siano le due congiogationi di rote rappreſentate con li due circoli, <lb></lb>de quali gli <expan abbr="cẽtri">centri</expan> ſono A e B ponti, che rappreſentano li due aſſi: e dal <lb></lb>ponto A al B, tiriſi la A B. </s> <s id="N10D06">& intendaſi il centro del peſo tutto appo<lb></lb>giato a detti due aſſi hauere il momento nel ponto C della detta linea. <emph.end type="italics"></emph.end></s> <pb xlink:href="041/01/045.jpg" pagenum="44"></pb> <s><emph type="italics"></emph><emph type="italics"></emph>Dico che'l detto peſo è ſoſtenuto da detti aſſi compartitamente, ſecondo <lb></lb>la ragione delle BC, AC: cioè che di tutto il peſo l'aſſe A. </s> <s id="N10D14">ne ſoſten<lb></lb>terà tal portione qual'è BC di B A, e B tale qual'è AC di AB, Si mo<lb></lb>ſtra intendaſi <expan abbr="prolõgata">prolongata</expan> la AB nell'vna e l'altra banda, farſi ad AC <lb></lb>vguale la BD: & alla BC, vguale la AE: ſaranno le EC, DC vguali: <lb></lb>e di nuouo fatto alla AC uguale la AE, ſaranno le DB, BF, e le AE <lb></lb>AF, vguali: e percio ſe alla linea DE, s'intenda fatta application di <lb></lb>corpo: il momento di tutto ſarà nel ponto C. </s> <s id="N10D26">di cui il detto aſſe A ne <lb></lb>ſoſtentarà la portione applicata ad EF: e l'aſſe B la portione applicata <lb></lb>a DF, la ragion de quali è l'iſteſſa: che di BC ad AC: ma del corpo ap<lb></lb>plicato il centro del peſo è l'iſteſſo, dall'iſteſſi ponti ſoſtenuto. </s> <s id="N10D2E">ſoſtengono <lb></lb>dunque gli aſſi il peſo compartitamente ſecondo la ragion di BC a C<lb></lb>A. </s> <s id="N10D34">Il che ſi hauea da mostrare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N10D38"> <p id="N10D39" type="head"> <s id="N10D3B"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>IIII. </s></p><p id="N10D43" type="main"> <s id="N10D45">Se'l peſo sia portato da due <expan abbr="congiõgationi">congiongationi</expan> di rote per <lb></lb>piano inchinato: <expan abbr="ſoſtẽntarà">ſoſtenntarà</expan> l'aſſe delle rote inferiori di <lb></lb>detto peſo, maggior portione che ſe fuſſe nel piano ori<lb></lb>zontale. </s></p><p id="N10D55" type="head"> <s id="N10D57"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10D5D" type="main"> <s id="N10D5F"><emph type="italics"></emph>Sia la linea del piano orizontale AB: del piano inchinato AC: li <lb></lb>centri de circoli delle rote D, & E: il centro della grauezza che sù <lb></lb>gli aſſi di dette rote appoggia F: Dico che di detta grauezza, dall'aſſe <lb></lb>D, ne ſarà ſostentata maggior portione: e dall'aſſe E, minore, che ſe <lb></lb>portata fuſſe per piano Orizontale. </s> <s id="N10D6B">Si moſtra: tiriſi da F perpendico<lb></lb>lare alla DE, che ſia FG: e perpendicolare all'orizonte che ſia FH: ſa <lb></lb>rà il ponto G, il ponto del momento nel ſito orizontale. </s> <s id="N10D71">& H, nell'in <lb></lb>chinato: e perche EH, è maggior portione di ED: che EG, e DH,<emph.end type="italics"></emph.end><pb xlink:href="041/01/046.jpg" pagenum="45"></pb><emph type="italics"></emph>minore che DG: ſoſtentarà la rota inferiore ſecondo la ragione di EH, <lb></lb>ad ED; e la ſuperiore <expan abbr="ſecõdo">ſecondo</expan> la ragione di DH ad ED: ſoſtenta <expan abbr="dũ">dun</expan>que <lb></lb>la rota inferiore, maggior portione di peſo: e la ſuperiore minor porti <lb></lb>one, che ſe nel ſito orizontale fuſſero. </s> <s id="N10D8B">Il che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p><figure id="id.041.01.046.1.jpg" xlink:href="041/01/046/1.jpg"></figure> </chap> <chap id="N10D92"> <p id="N10D93" type="head"> <s id="N10D95"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></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> ſotto il piano <expan abbr="orizõtale">orizon<pb xlink:href="041/01/047.jpg" pagenum="46"></pb>tale</expan>: e data qual si uoglia grauezza: ritrouare in vn rag<lb></lb>gio la diſtanza oltre di cui detta grauezza appeſa, ſol<lb></lb>leui detta rota. </s></p><figure id="id.041.01.047.1.jpg" xlink:href="041/01/047/1.jpg"></figure> <p id="N10DB8" type="head"> <s id="N10DBA"><emph type="italics"></emph>Dimostratione. <emph.end type="italics"></emph.end></s></p><p id="N10DC0" type="main"> <s id="N10DC2"><emph type="italics"></emph>Sia la linea del piano orizontale ABCD: la concauità in eſſa BE <lb></lb>C: la rota che affondi BCF: la grauezza data G. </s> <s id="N10DC8">ſi cerca in vn raggio <lb></lb>della rota, ponto oltre di cui ſoſpeſa la G, ſolleui detta rota. </s> <s id="N10DCC">Sia il cen<lb></lb>tro H: la linea del raggio prodotto HFI: qual ſia parallela all'ori<lb></lb>zonte: e dal <expan abbr="põto">ponto</expan> C, ſi tiri la CK perpendicolare che <expan abbr="affrõti">affronti</expan> la HF, in K: <lb></lb>e la ragion c'ha la grauezza G al peſo della rota, habbia HK a KI: è <lb></lb>manifeſto perche KC, è <expan abbr="perdendicolare">perpendicolare</expan> del ſoſtenimento, che dal ponto I <lb></lb>la grauezza G, fa equipondio alla rota. </s> <s id="N10DE0">e che da ogni ponto oltre, la ſol <lb></lb>leui, il che ſi cerca un. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/048.jpg" pagenum="47"></pb> <p id="N10DE9" type="head"> <s id="N10DEB">TAGLIA. </s><lb></lb> <s id="N10DEE"><emph type="italics"></emph>SVPPOSITIONE. <emph.end type="italics"></emph.end></s></p><p id="N10DF4" type="main"> <s id="N10DF6">Svpponiamo la taglia c'habbia in ſe una, o più <lb></lb>girelle, o sia in vno o più ordini. </s> <s id="N10DFA">Et delle taglie, ſta <lb></lb>bile diciamo, il cui collo sia legato ad vn termine: mo<lb></lb>bile il cui collo sia legato al peſo. </s> <s id="N10E00">Et <expan abbr="altrimẽte">altrimente</expan> mobile la <lb></lb>guidata da vna potenza, e che ad vn capo di eſſa ſia attac<lb></lb>cato il peſo. </s> <s id="N10E0A">In oltre ſupponiamo della corda auuolta il <lb></lb>capo andare, o alla taglia, o ad'vn termine fiſſo, o a poſ<lb></lb>ſanza, ò a peſo. </s></p><p id="N10E10" type="head"> <s id="N10E12"><emph type="italics"></emph>POSITIONE <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N10E1A" type="main"> <s id="N10E1C">Poniamo della girella a cui sia auuolta corda data <lb></lb>a pesi, & a poſſanze, mentre detta girella non volta il mo<lb></lb>mento de capi eſſere vguale. </s></p><p id="N10E22" type="head"> <s id="N10E24">II. </s></p><p id="N10E26" type="main"> <s id="N10E28">Ma ſe la girella volta, il momento di quella corda eſ<lb></lb>ſer maggiore, verſo 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 poſſanze e peſi vguali, <lb></lb>li momenti eſſere vguali. </s></p> <pb xlink:href="041/01/049.jpg" pagenum="48"></pb> </chap> <chap id="N10E39"> <p id="N10E3A" type="head"> <s id="N10E3C"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N10E44" type="main"> <s id="N10E46">Se delli due capi della girella, l'vna ſoſtenti peſo, l'al<lb></lb>tro ſia dato a poſſanza: la poſſanza del capo ſarà di mo<lb></lb>mento eguale al peſo. </s> <s id="N10E4C">e la poſſanza della taglia ſoſtenta <lb></lb>rà il doppio. </s></p><p id="N10E50" type="head"> <s id="N10E52"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10E58" type="main"> <s id="N10E5A"><emph type="italics"></emph>Sia la taglia AB: li capi della fune auuolta <emph.end type="italics"></emph.end><lb></lb><figure id="id.041.01.049.1.jpg" xlink:href="041/01/049/1.jpg"></figure><lb></lb><emph type="italics"></emph>A C, B D: dequali A C, ſoſtenti il peſo C: e B <lb></lb>D, ſia dato alla poſſanza in D: dico che la <lb></lb>poſſanza in D è di momento eguale al peſo: e <lb></lb>che la poſſanza in E, ſoſtenta il doppio. </s> <s id="N10E6F">Si <lb></lb>moſtra: e prima che'l momento di D, ſia v<lb></lb>guale al momento di C. </s> <s>è manifeſto: perche <lb></lb>ſe l'vn di loro fuſſe maggiore, la girella volte<lb></lb>rebbe verſo detto momento: Jl che è contro <lb></lb>il ſuppoſto. </s> <s id="N10E7B">Dico hora che la poſſanza della <lb></lb>taglia ſia doppia del peſo: percioche eſſendo <lb></lb>la poſſanza di D, equiualente al peſo C: ambi <lb></lb>C e D, ſono il doppio di eſſo C: ma la <expan abbr="poſsãza">poſsanza</expan> in <lb></lb>E, in quanto ſoſtiene, è vguale ad ambi: dun<lb></lb>que è doppia di vn di loro. </s> <s id="N10E8F">Haſſi dunque il <lb></lb>propoſto, che la poßanza D, ſia vguale al mo<lb></lb>mento di C: e che la E, ſoſtenti il doppio di eſſo. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/050.jpg" pagenum="49"></pb> </chap> <chap id="N10E9A"> <p id="N10E9B" type="head"> <s id="N10E9D"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>II. </s></p><p id="N10EA5" type="main"> <s id="N10EA7">Se li due capi di girella mobile, ſiano raccomanda<lb></lb>ti a due poſſanze: ſoſtentarà così l'vna, come l'altra poſ<lb></lb>ſanza, la metà del peſo. </s></p><p id="N10EAD" type="head"> <s id="N10EAF"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10EB5" type="main"> <s id="N10EB7"><emph type="italics"></emph>Sia la taglia A B, a cui ſia attaccato il pe<emph.end type="italics"></emph.end><lb></lb><figure id="id.041.01.050.1.jpg" xlink:href="041/01/050/1.jpg"></figure><lb></lb><emph type="italics"></emph>ſo C: li due capi della corda auuolta alla gi<lb></lb>rella A D, B E: le poſſanze in D, et E: dico <lb></lb>che così l'vna, come l'altra poſſanza ſoſten<lb></lb>ta la metà del peſo. </s> <s id="N10ECE">ſi moſtra: percioche ſtan<lb></lb>do la girella <expan abbr="sẽza">senza</expan> voltare, <expan abbr="ſecõdo">ſecondo</expan> il ſup. ſara <lb></lb>di conſeguenza il momento dell'vn capo v<lb></lb>guale al momento dell'altro: e perciò le poſ<lb></lb>ſanze anco eguali. </s> <s id="N10EE0">e perche ambe ſoſtenta<lb></lb>no il peſo C: e le poſſanze, in quanto ſoſten<lb></lb>gono, ſono eguali alli peſi. </s> <s id="N10EE6">ſono dunque am<lb></lb>be eguali al peſo C: e perciò diuiſamente l'v <lb></lb>na e l'altra ſarà la metà di detto peſo <lb></lb>al che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/051.jpg" pagenum="50"></pb> <p id="N10EF3" type="head"> <s id="N10EF5"><emph type="italics"></emph>Appendice,<emph.end type="italics"></emph.end></s></p><p id="N10EFB" type="main"> <s id="N10EFD">E perciò anco se l'vn capo sia raccomandato ad vn <lb></lb>termine fiſſo, l'altro a poſſanza: ſoſterrà la poſſanza la <lb></lb>metà del peſo. </s></p><p id="N10F03" type="main"> <s id="N10F05"><emph type="italics"></emph>Percioche mutato il termine in un'altra poſſanza: la poſſanza ſuppo<lb></lb>ſta ſoſterrà l'iſteſſa altra quantità di peſo che prima. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N10F0D"> <p id="N10F0E" type="head"> <s id="N10F10"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>III. </s></p><p id="N10F18" type="main"> <s id="N10F1A">Delle corde, che dalla taglia ſu<lb></lb>periore, & dalla poſſanza alla ta<lb></lb><figure id="id.041.01.051.1.jpg" xlink:href="041/01/051/1.jpg"></figure><lb></lb>glia inferiore peruengono: ciaſcu<lb></lb>na ſoſtiene egual parte dipeſo. </s></p><p id="N10F27" type="head"> <s id="N10F29"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N10F2F" type="main"> <s id="N10F31"><emph type="italics"></emph>Sia la taglia ſuperiore A B: l'inferiore <lb></lb>C D: la corda auuolta no tata con l'iſteße let<lb></lb>tere: e di lei l'vn termine vada a ſoſtenere <lb></lb>la taglia inferiore in E: l'altro ſia dato al<lb></lb>la poſſanza in F. </s> <s id="N10F3D">Dico che ciaſcuna corda <lb></lb>ſoſtiene egual parte di peſo. </s> <s id="N10F41">Si moſtra: <lb></lb>perche ſtando la girella A B, il momento <lb></lb>del capo B D è eguale al momento del ca<lb></lb>po A E: e del capo C F, al capo B D, per <lb></lb>la girella C D: ſono <expan abbr="dũque">dunque</expan> tutte di momen<lb></lb>to eguali: perciò ciaſcuna ſoſtentarà e<lb></lb>gual parte di peſo. </s> <s id="N10F53">e ſe il capo A E non fuſ<lb></lb>ſe ligato alla taglia, ma ad altro termine, ſa<lb></lb>rebbe l'iſteſſo, ma il numero delle corde di <lb></lb>vna meno. </s> <s id="N10F5B">Il che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/052.jpg" pagenum="51"></pb> </chap> <chap id="N10F62"> <p id="N10F63" type="head"> <s id="N10F65"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>IIII. </s></p><p id="N10F6D" type="main"> <s id="N10F6F">Se l'vn capo della fune auuolta <lb></lb><figure id="id.041.01.052.1.jpg" xlink:href="041/01/052/1.jpg"></figure><lb></lb>a girelle, ſia raccomandato alla ta<lb></lb>glia ſuperiore: il peſo ſoſtenuto <lb></lb>è diſtribuito in parti di numero <lb></lb>pare. </s></p><p id="N10F7E" type="head"> <s id="N10F80"><emph type="italics"></emph>Dimostratione. <emph.end type="italics"></emph.end></s></p><p id="N10F86" type="main"> <s id="N10F88"><emph type="italics"></emph>Sia la taglia inferiore e mobile AC B D: <lb></lb>la ſuperiore E F G H: la fune auuolta nota<lb></lb>ta <expan abbr="cõ">con</expan> l'isteſſe lettere: il termine del capo C 1, <lb></lb>attaccato alla taglia ſuperiore, <expan abbr="s'intẽda">s'intenda</expan> eſſe <lb></lb>re in I: l'altro termine raccomandato alla <lb></lb>poſſanza s'intenda eſſere o in K del capo B <lb></lb>K, che vien dalla taglia inferiore, o in L, del <lb></lb>capo G L, che vien dalla taglia ſuperiore. </s><lb></lb> <s id="N10FA4">Dico che, e nell'vno, e nell'altro modo, il pe<lb></lb>ſo è diſtribuito in parti di numero pare. </s><lb></lb> <s id="N10FA9">Si moſtra: percioche venendo alla girella <lb></lb>C D due corde, l'vna da taglia, l'altra da <lb></lb>girella E F: ſaranno detti capi di momen<lb></lb>ti eguali: perche ſi pone la girella non vol <lb></lb>tare. </s> <s id="N10FB3">ſimilmente perche alla girella A B <lb></lb>vengono due corde, l'vna dalla girella EF, <lb></lb>che è la corda E A, l'altra dalla poßanza <lb></lb>K, che è la corda KB: ſaranno dette corde <lb></lb>di <expan abbr="momẽti">momenti</expan> eguali. </s> <s id="N10FC1">ma la DF, è di momento <lb></lb>eguale alla A E, e alla B K: ſono dunque <lb></lb>tutte tra di loro di momento eguale: e <lb></lb>ſono di numero pare: percioche a cia<emph.end type="italics"></emph.end><pb xlink:href="041/01/053.jpg" pagenum="52"></pb><emph type="italics"></emph>ſcuna girella ne vengono due. </s> <s id="N10FD1">perche dunque il peſo è ſoſtenuto da <lb></lb>dette corde di <expan abbr="momẽto">momento</expan> eguale: perciò, mentre l'vn capo ſia attaccato <lb></lb>alla taglia ſuperiore, l'altro dato alla poſſanza, il <expan abbr="momẽto">momento</expan> del peſo è di<lb></lb>ſtribuito in parti di numero pare: ne altro auuiene, ſe la poßanza ſia in <lb></lb>L, nel capo, che viene dalla taglia ſuperiore: percioche il numero del<lb></lb>le corde, che alla taglia inferiore peruengono è l'iſteſſo. <emph.end type="italics"></emph.end></s></p><p id="N10FE7" type="head"> <s id="N10FE9"><emph type="italics"></emph>Appendice. </s> <s id="N10FED">I<emph.end type="italics"></emph.end></s></p><p id="N10FF1" type="main"> <s id="N10FF3">Et è manifeſto, che poſta vna girella meno nella ta<lb></lb>glia ſuperiore, ſi ſoſterrà dalla poſſanza l'iſteſſo che ſe <lb></lb>fuſſero le girelle ſuperiori di numero eguale alle in<lb></lb>feriori, è che per detta girella aggiunta, si muta ſo<lb></lb>lamente l'un momento nell'altro di ſpezie contraria. </s></p> </chap> <chap id="N10FFD"> <p id="N10FFE" type="head"> <s id="N11000"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>V. </s></p><p id="N11008" type="main"> <s id="N1100A">Se l'vn capo della fune auuolta a girelle, ſia racco<lb></lb>mandato alla taglia inferiore: il peſo ſoſtenuto è diſtri<lb></lb>buito in parti di numero ſpare. </s></p><p id="N11010" type="head"> <s id="N11012"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N11018" type="main"> <s id="N1101A"><emph type="italics"></emph>Sia la taglia ſuperiore e stabile AB, CD: l'inferiore e mobile E<lb></lb>F, G H: la fune auuolta notata con l'iſteſſe lettere: e di eſſa l'vn ter<lb></lb>mine I, che è del capo C I, ſia attaccato alla taglia inferiore: et il termi<lb></lb>ne K, del capo E K, raccomandato alla poſſanza in K: dico che'l peſo è<emph.end type="italics"></emph.end><pb xlink:href="041/01/054.jpg" pagenum="53"></pb><emph type="italics"></emph>diſtribuito in parti dinumero ſpare. </s> <s id="N1102C">Si <emph.end type="italics"></emph.end><lb></lb><figure id="id.041.01.054.1.jpg" xlink:href="041/01/054/1.jpg"></figure><lb></lb><emph type="italics"></emph>moſtra: percioche vengono due capi dalla <lb></lb>girella C D, alla taglia inferiore, e due <lb></lb>dalla A B, e ſimilmente da qual ſi voglia <lb></lb>altra girella: ſono dunque li capi, che dal <lb></lb>le girelle alla taglia vengono, di numero <lb></lb>pare. </s> <s id="N11043">et euui in oltre il capo della poſſan<lb></lb>za: ſono dunque tutti di numero ſpare. </s><lb></lb> <s id="N11048">e ſono, per quel che ſi è detto nelle prece <lb></lb>denti, tutte di momento eguale: dunque <lb></lb>il peſo è diſtribuito in parti di numero <lb></lb>ſpare. </s> <s id="N11050">Jl che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p><p id="N11054" type="head"> <s id="N11056"><emph type="italics"></emph>Appendice. </s> <s id="N1105A">I. <emph.end type="italics"></emph.end></s></p><p id="N1105E" type="main"> <s id="N11060">Et è manifeſto, che aggionta <lb></lb>alla taglia ſuperiore vna girel<lb></lb>la, si commuta ſolamente il mo<lb></lb>mento della poſſanza, in mo<lb></lb>mento di ſpezie contraria. </s></p><p id="N1106A" type="head"> <s id="N1106C"><emph type="italics"></emph>Appendice. </s> <s id="N11070">II. <emph.end type="italics"></emph.end></s></p><p id="N11074" type="main"> <s id="N11076">E raccogliamo, che ligato l'vn <lb></lb>capo alla taglia ſuperiore, puote <lb></lb>ſtar detta taglia con vna girella <lb></lb>meno: e ligata all'inferiore con <lb></lb>vna girella più. </s></p> <pb xlink:href="041/01/055.jpg" pagenum="54"></pb> </chap> <chap id="N11083"> <p id="N11084" type="head"> <s id="N11086"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>VI</s></p><p id="N1108E" type="main"> <s id="N11090">Se vn capo della taglia ſupe<lb></lb><figure id="id.041.01.055.1.jpg" xlink:href="041/01/055/1.jpg"></figure><lb></lb>riore sia raccomandato ad vn <lb></lb>termine fiſſo: ſarà il peſo diſtri<lb></lb>buito in parti di numero pare. </s></p><p id="N1109D" type="head"> <s id="N1109F"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N110A5" type="main"> <s id="N110A7"><emph type="italics"></emph>Sia la taglia ſuperiore A B C D, l'<lb></lb>inferiore E F G H: la fune auuolta a gi<lb></lb>relle notata con l'iſteſſe lettere: di cui <lb></lb>il capo D I dalla girella C D della taglia <lb></lb>ſuperiore ſia raccomandato ad I termi<lb></lb>ne fiſſo: & il capo F K, dalla girella E F, <lb></lb>della taglia inferiore, raccomandato al <lb></lb>la poſſanza in K. </s> <s id="N110B9">Dico che'l peſo è diſtri<lb></lb>buito in parti di numero pare. </s> <s id="N110BD">Si mo<lb></lb>ſtra: percio che venendo alla taglia infe<lb></lb>riore le corde ſolo delle girelle, & da cia<lb></lb>ſcuna girella due corde, quali tutte ſi è <lb></lb>moſtrato che ſoſtentino egual momento: <lb></lb>ſarà il peſo diſtribuito in corde di nume<lb></lb>ro pare, che egualmente ſoſtentano: e <lb></lb>perciò ſarà diſtribuito in dette parti. </s> <s id="N110CD">Il <lb></lb>che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p><p id="N110D3" type="head"> <s id="N110D5"><emph type="italics"></emph>Correlario. </s> <s id="N110D9">I. <emph.end type="italics"></emph.end></s></p><p id="N110DD" type="main"> <s id="N110DF">E manifeſto dunque che li<lb></lb>gato il capo di ſopra alla taglia <pb xlink:href="041/01/056.jpg" pagenum="55"></pb>inferiore, il peſo è diſtribuito in parti di numero ſpare, <lb></lb>et comunque altrimente, in parti di numero pare. </s></p><p id="N110E9" type="head"> <s id="N110EB"><emph type="italics"></emph>Correlario. </s> <s id="N110EF">II. <emph.end type="italics"></emph.end></s></p><p id="N110F3" type="main"> <s id="N110F5">Et attaccato il capo di girella inferiore alla taglia ſu<lb></lb>periore, o à qual si voglia termine fiſſo: che la taglia in<lb></lb>feriore habbia vna girella più. </s></p> </chap> <chap id="N110FB"> <p id="N110FC" type="head"> <s id="N110FE"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>VII. </s></p><figure id="id.041.01.056.1.jpg" xlink:href="041/01/056/1.jpg"></figure> <p id="N11109" type="main"> <s id="N1110B">Se'l peſo sia moſſo con ta<lb></lb>glie, quanto il peſo è moltepli<lb></lb>ce della poſſanza ſoſtenente, <lb></lb>tanto lo ſpatio, che detta <expan abbr="poſsãza">poſ<lb></lb>sanza</expan> camina, è molteplice del<lb></lb>lo ſpatio caminato dal peſo. </s></p><p id="N1111B" type="head"> <s id="N1111D"><emph type="italics"></emph>Dimostratione. <emph.end type="italics"></emph.end></s></p><p id="N11123" type="main"> <s id="N11125"><emph type="italics"></emph>Sia la girella della taglia ſuperiore <lb></lb>A B: della inferiore nella prima po<lb></lb>ſitione ſia C D: e la poſſanza che ſoſtie<lb></lb>ne il capo ſia in E: della ſeconda poſi <lb></lb>tione ſia in G H, e la poſſanza in I. </s> <s id="N11131">Di <lb></lb>co che lo ſpatio caminato dalla taglia <lb></lb>mobile e peſo, è tal parte dello ſpatio <lb></lb>E I, qual la poſſanza ſoſtenente in E <lb></lb>è parte del peſo. </s> <s id="N1113B">Si moſtra: perche <lb></lb>quante ſono la corde, che alla taglia <emph.end type="italics"></emph.end><pb xlink:href="041/01/057.jpg" pagenum="56"></pb><emph type="italics"></emph>inferiore peruengono, ſecondo tal numero la poſſanza che ſoſtiene è <lb></lb>parte del peſo: e perche nel mouimento della taglia ciaſcuna corda <lb></lb>ſi abbreuia egualmente, portata C D, in H G: le C G, D H parti del<lb></lb>la corda auuolta, quante ſi ſiano, pigliate inſieme, ſarano di lunghezza <lb></lb>tanto molteplici dello ſpatio caminato, quanto è il numero delle cor<lb></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></lb>tone di commune la F G H B A E, reſta le E I, eguale alla G C D H: <lb></lb>e percio E I, ſarà altre tanto molteplice dello ſpatio caminato, quan<lb></lb>to erano le corde C G, D H. </s> <s id="N11159">ciò è il peſo tutto del peſo da vna corda <lb></lb>ſostenuto. </s> <s id="N1115D">Il che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p> </chap> <chap id="N11161"> <p id="N11162" type="head"> <s id="N11164"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>VIII. </s></p><p id="N1116C" type="main"> <s id="N1116E"><emph type="italics"></emph>Problema. </s> <s id="N11172">I. <emph.end type="italics"></emph.end></s></p><p id="N11176" type="main"> <s id="N11178">Data qual si voglia grauezza, e poſſanza: ritroua <lb></lb>re il minor numero di girelle nella taglia, con quali <lb></lb>la data poſſanza moua il dato peſo. </s></p><p id="N1117E" type="head"> <s id="N11180"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N11186" type="main"> <s id="N11188"><emph type="italics"></emph>Sia la data grauezza A, la poſſanza B: a cui ſi pigli vn peſo e<lb></lb>quiualente C: e moltiplichiſi C, ſin che la prima volta ecceda la <lb></lb>grauezza A, il che ſia per il numero D. </s> <s id="N11190">ſe dunque D è pare piglin <lb></lb>ſi nella taglia inferiore altre tante girelle, quante vnità ſono nella <lb></lb>inetà del numero: è manifeſto che la poſſanza mouerà il peſo con le <lb></lb>date girelle: ma ſe D ſia ſpare, toltane vnità, piglinſi girelle quan<lb></lb>te vnità ſono nella metà del reſto, e ligheſi vn delli capi alla taglia: <lb></lb>è manifesto ſimilmente che mouerà la poſſanza la data grauezza. </s><lb></lb> <s id="N1119D">Il che ſi cercaua. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/058.jpg" pagenum="57"></pb> </chap> <chap id="N111A4"> <p id="N111A5" type="head"> <s id="N111A7"><emph type="italics"></emph>PROPOSITIONE. <emph.end type="italics"></emph.end><lb></lb>VIII. </s></p><p id="N111AF" type="main"> <s id="N111B1"><emph type="italics"></emph>Problema. </s> <s id="N111B5">II. <emph.end type="italics"></emph.end></s></p><p id="N111B9" type="main"> <s id="N111BB">Data qual ſi voglia velocità, e data la tardità della <lb></lb>poſsanza: applicar o vna taglia di più girelle, o più ta<lb></lb>glie di vna girella, ſi che la poſſanza moua il dato pe<lb></lb>ſo in velocità magior della data. </s></p><p id="N111C3" type="head"> <s id="N111C5"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N111CB" type="main"> <s id="N111CD"><emph type="italics"></emph>Pigliſi lo ſpatio che nel dato tem<emph.end type="italics"></emph.end><lb></lb><figure id="id.041.01.058.1.jpg" xlink:href="041/01/058/1.jpg"></figure><lb></lb><emph type="italics"></emph>po camini la poſſanza: e lo ſpatio che <lb></lb>vogliamo che la coſa camini, e ſi mol<lb></lb>tiplichi il minore fin che la prima vol<lb></lb>ta auanzi, e quanto queſto è molte<lb></lb>plice, tante corde ſiano nella taglia ſu<lb></lb>periore, pigliando la metà di girelle ſe <lb></lb>ſia pare, & ſe ſia ſpare, ligando vn ca<lb></lb>po ad eſſa taglia ſuperiore. </s> <s id="N111EA">Ligato dun<lb></lb>que il peſo ad vn capo, la poſſanza, che <lb></lb>tira la taglia, tirerà anco il peſo: e ca<lb></lb>minerà lo ſpatio moltiplice al moui<lb></lb>mento di eſſa poſſanza. </s> <s id="N111F4">Ma, ſe vo<lb></lb>gliamo far ciò con piú taglie di vna gi<lb></lb>rella, radoppiſi lo ſpatio, e di nuouo il <lb></lb>fatto dal radoppiamento ſi radoppij: e <lb></lb>ciò ſi torni a fare, si che l'ultimo radop<lb></lb>piamento auanzi lo ſpatio maggiore. </s><lb></lb> <s id="N11201">Se dunque, quante volte ſi è radoppia<lb></lb>to, tanto numero ditaglie ſi pigli, si mo<lb></lb>uer à il peſo ſecondo la ragion del radop<lb></lb>piamento dello ſpatio, e perciò ſi mo<lb></lb>uerà con maggior velocità della data. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/059.jpg" pagenum="58"></pb> <p id="N11210" type="head"> <s id="N11212">ROTE MO<lb></lb>TIVE. </s></p><figure id="id.041.01.059.1.jpg" xlink:href="041/01/059/1.jpg"></figure> <p id="N11219" type="head"> <s id="N1121B"><emph type="italics"></emph>SVPPOSITION. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N11223" type="main"> <s id="N11225">Svpponiamo il mouimento di rote in aſſi <lb></lb>che ſtanno co'l toccamento, communicarſi l'vna <lb></lb>all'altra il mouimento: e che'l momento della poſſan<lb></lb>za ſia per linea che faccia angolo retto co'l raggio di <lb></lb>eſſa rota: e de momenti altri eſſer concorrenti, altri <lb></lb>contrarij. </s></p><p id="N11231" type="head"> <s id="N11233"><emph type="italics"></emph>DEFINITION. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N1123B" type="main"> <s id="N1123D">Concorrenti momenti diciamo quelli, che portan<lb></lb>do verſo l'iſteſſa parte, ſi accreſcono. </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'impediſcono portando in <lb></lb>contrario. </s></p><p id="N1124B" type="head"> <s id="N1124D"><emph type="italics"></emph>POSITION. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N11255" type="main"> <s id="N11257">Poniamo, poſſanze eguali in circonferenze direte <lb></lb>eguali, hauer momenti eguali. </s></p> <pb xlink:href="041/01/060.jpg" pagenum="59"></pb> <p id="N1125E" type="head"> <s id="N11260"><emph type="italics"></emph>POSITION. <emph.end type="italics"></emph.end><lb></lb>II. </s></p><p id="N11268" type="main"> <s id="N1126A">Et in rote ineguali hauer momento ineguale, ſecon<lb></lb>do la ragion de ſemidiametri. </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 ſi annullano, l' <lb></lb>vno eſſere eguale all'altro. </s></p><p id="N11278" type="head"> <s id="N1127A"><emph type="italics"></emph>PROPOSITION. <emph.end type="italics"></emph.end><lb></lb>I. </s></p><p id="N11282" type="main"> <s id="N11284">Se quante ſi voglia rote, vna per aſſe, ſi tocchino: <lb></lb>e poſte le poſſanze l'vna nella circonferenza della pri<lb></lb>ma, e l'altra dell'vltima, ſi rattengano: ſaranno le poſ<lb></lb>ſanze eguali. </s></p><figure id="id.041.01.060.1.jpg" xlink:href="041/01/060/1.jpg"></figure> <pb xlink:href="041/01/061.jpg" pagenum="60"></pb> <p id="N11292" type="head"> <s id="N11294"><emph type="italics"></emph>Dimostratione. <emph.end type="italics"></emph.end></s></p><p id="N1129A" type="main"> <s id="N1129C"><emph type="italics"></emph>Siano quante ſi voglia rote ne gli aſſi A, B, C, che ſi tocchino: ciò <lb></lb>è che la A tocchi la B nel ponto D: e la B tocchila C nel ponto E: <lb></lb>& intendaſi nella circonferenza di A eſſer la potenza F: e nella cir<lb></lb>conferenza di C la potenza G: che l'una rattenga l'altra. </s> <s id="N112A6">Dico che <lb></lb>le potenze ſono eguali. </s> <s id="N112AA">Si moſtra: percio che la poßanza in F, è dell'<lb></lb>iſteſſo momento, che ſe fuſſe in D, dell'iſteſſa rota A: ma il ponto <lb></lb>D, è ponto commune a due rote: e la poſſanza in D della rota B, <lb></lb>è quanto fuſſe in E: ſarà dunque la poſſanza in F l'iſteſſo che ſi fuſſe <lb></lb>in E: perche <expan abbr="dũque">dunque</expan> la poſſanza in F ſi annulla con la poſſanza in G, ſo<lb></lb>no li loro momenti eguali. </s> <s id="N112BA">Ma le poſſanze che ſono in un'iſteſſa rota <lb></lb>di momenti eguali, ſono eguali: dunque la poſſanza in F è uguale alla <lb></lb>poſſanza in G. </s> <s id="N112C0">Jl che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p><p id="N112C4" type="head"> <s id="N112C6"><emph type="italics"></emph>PROPOSITION. <emph.end type="italics"></emph.end><lb></lb>II. </s></p><p id="N112CE" type="main"> <s id="N112D0">Delle due rote in vno aſſe la poſſanza, che fa egual <lb></lb>momento nella rota magiore è di valor minore: e nel <lb></lb>la minore è di valor maggiore, nella ragione de ſemi <lb></lb>diametri reciproca. </s></p><p id="N112D8" type="head"> <s id="N112DA"><emph type="italics"></emph>Dimostratione. <emph.end type="italics"></emph.end></s></p><p id="N112E0" type="main"> <s id="N112E2"><emph type="italics"></emph>Siano ſu l' aſſe A le rote A B, A C: & intendaſi la poſſanza B, <lb></lb>in circonferenza della rota maggiore, hauere egual momento alla <lb></lb>poſſanza C in circonferenza della rota minore. </s> <s id="N112EA">Dico che la poſſan<lb></lb>za B è minore della poſſanza C, ſecondo la ragione di C A ad A B. </s><lb></lb> <s id="N112EF">Si moſtra: intendaſi nell circonferenza di A C eſſer poſſanza eguale <lb></lb>a B, che ſia D: ſarà il momento di B al momento di D, nella ragion <emph.end type="italics"></emph.end><pb xlink:href="041/01/062.jpg" pagenum="61"></pb><figure id="id.041.01.062.1.jpg" xlink:href="041/01/062/1.jpg"></figure><lb></lb><emph type="italics"></emph>della linea dritta B A alla D A: ma il momento di B, è uguale al <lb></lb>momento di C: dunque il momento di C al momento di D, è come <lb></lb>B A ad A D. </s> <s id="N11304">Se dimque le poſſanze dell'iſteſſa rota ſono tra di loro <lb></lb>nella ragione delli momenti: ſarà di conſeguenza la poſſanza in D <lb></lb>alla poſſanza in C, come il ſemidiametro D A, al ſemidiame<lb></lb>tro A B, e del diametro tutto a tutto. </s> <s id="N1130C">Il che ſi hauea da mo<lb></lb>ſtrare. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/063.jpg" pagenum="62"></pb> <p id="N11315" type="head"> <s id="N11317"><emph type="italics"></emph>PROPOSITION. <emph.end type="italics"></emph.end><lb></lb>III. </s></p><p id="N1131F" type="main"> <s id="N11321">Se le rote, poſte a due in ciaſcun aſſe, ſi tocchino: <lb></lb>e le poſſanze, poſte l'vna nella prima, l'altra nell'vl<lb></lb>tima rota, ſi rattengano: ſarà la ragion dell'vna poſſan<lb></lb>za all'altra l'iſteſſa, che la ragion compoſta delli ſemi <lb></lb>diametri, che ſono ſu l'iſteſſo aſſe, pigliate reciproca<lb></lb>mente. </s></p><p id="N1132D" type="head"> <s id="N1132F"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N11335" type="main"> <s id="N11337"><emph type="italics"></emph>Siano ſu l'aſſe A, le due rote A B, A C: e ſu l'aſſe D, le rote D C <lb></lb>D F: & intendaſi la rota D C, eſſer toccata dalla A C nel ponto C: <lb></lb>c l'una poſſanza eſſere in B l'altra in E. </s> <s id="N1133F">Dico che la poſſanza in B,<emph.end type="italics"></emph.end><lb></lb><figure id="id.041.01.063.1.jpg" xlink:href="041/01/063/1.jpg"></figure><lb></lb><emph type="italics"></emph>alla poſſanza in F ha la ragion compoſta delle ragioni di F D a D C,<emph.end type="italics"></emph.end><pb xlink:href="041/01/064.jpg" pagenum="63"></pb><emph type="italics"></emph>e di C A ad A B, che ſono le ragioni de ſemidiametri reciprocamen<lb></lb>te pigliati. </s> <s id="N11356">Si moſtra: percioche eſſendo il momento in B uguale al <lb></lb>momento in C, perche ſono in vno iſteſſo aſſe: & il momento in C al <lb></lb>momento in F, per l'isteſſa ragione: & è la poſſanza in F, alla <lb></lb>poſſanza in C, come il diametro C D a D F: e la poſſanza in C, <lb></lb>alla poſſanza in B, co me B A ad A C. </s> <s id="N11360">Dunque la poſſanza in F alla <lb></lb>poſſanza in B, ha la ragion compoſta di C D a D F c di B A ad A C, <lb></lb>che è la ragion compoſta delle ragioni de diametri reciprocamente <lb></lb>pigliati. </s> <s id="N11368">Jl che ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p><p id="N1136C" type="head"> <s id="N1136E"><emph type="italics"></emph>DEFINITIONE. <emph.end type="italics"></emph.end></s></p><p id="N11374" type="main"> <s id="N11376">Momento della rota diciamo, il momento del pon<lb></lb>to poſto nella circonferenza di eſſa rota. </s></p><p id="N1137A" type="head"> <s id="N1137C"><emph type="italics"></emph>PROPOSITION. <emph.end type="italics"></emph.end><lb></lb>IIII. </s></p><p id="N11384" type="main"> <s id="N11386">Se in vna congiogation di rote ineguali, o in più, <lb></lb>che la minor dell'vna congiogatione tocchi la mag<lb></lb>gior dell'altra, ſi ponga la poſſanza in vna di dette <lb></lb>rote: ſarà il momento dell'vltima minor rota, maggior <lb></lb>del momento della prima maggior rota, ſecondo la ra<lb></lb>gion compoſta delli diametri. </s> <s id="N11392">e la velocità ſarà mino<lb></lb>re, ſecondo l'iſteſſa ragion de diametri. </s></p><p id="N11396" type="head"> <s id="N11398"><emph type="italics"></emph>Dimoſtratione. <emph.end type="italics"></emph.end></s></p><p id="N1139E" type="main"> <s id="N113A0"><emph type="italics"></emph>Siano le congiogationi di rote, de quali gli aſsi ſiano A e B: & in<lb></lb>tendaſi ſu l'aſſe A eſſer la rota maggiore A C, e la minore A D,<emph.end type="italics"></emph.end><pb xlink:href="041/01/065.jpg" pagenum="64"></pb><figure id="id.041.01.065.1.jpg" xlink:href="041/01/065/1.jpg"></figure><lb></lb><emph type="italics"></emph>e ſu l' aſſe B, eſſer la maggiore D B, e la minore B E: e ſia il contat<lb></lb>to della minore di vn'ordine, con la maggiore dell'altro, il ponto D: <lb></lb>e ſuppongaſi prima la poſſanza porſi nella circonferenza di A C. </s><lb></lb> <s id="N113B8">Dico che'l momento della rota A D, è maggiore del momento di <lb></lb>A C, secondo la ragione della linea C A ad A D. </s> <s id="N113BC">Si moſtra: per<lb></lb>cioche poſta in D una poſſanza di <expan abbr="momẽto">momento</expan> eguale alla poſſanza in C, <lb></lb>ſarà detta poſſanza in D, maggiore, che la poſſanza in C: ma il mo<lb></lb>mento della rota, oue è poſta la poſſanza, è uguale ad eſſa poſſan<lb></lb>za: ſarà dunque il <expan abbr="momẽto">momento</expan> della rota A D maggiore che della rota <lb></lb>A C ſecondo la ragion de diametri: queſto in una congiogatione <lb></lb>& in più: per che il momento della circonferenza di A D è l'iſteſſo <lb></lb>che della circonferenza di B D, per lo contatto, che fa communi<lb></lb>canza: ma il momento della circonferenza di B E, è di forza <lb></lb>maggiore che di B D <expan abbr="ſecõdo">ſecondo</expan> la ragione del diametro, B D a B E: <lb></lb>dunque fatta compoſitione de ragioni il momento della circonferen<lb></lb>za di B E, è maggiore del momento della circonferenza di C A ſe <lb></lb>condo la ragion compoſta di B D a B E, e di C A ad A D. </s> <s id="N113E2">Il che <lb></lb>ſi hauea da moſtrare. <emph.end type="italics"></emph.end></s></p><p id="N113E8" type="main"> <s id="N113EA"><emph type="italics"></emph>Dico che la uelocità è minore nella iſteſſa ragione: il che è mani<emph.end type="italics"></emph.end><pb xlink:href="041/01/066.jpg" pagenum="65"></pb><emph type="italics"></emph>feſto: percioche la velocità delle rote, che nell'iſteſſo tempo finiſcono <lb></lb>il circuito, è proportionale alle circonferenze di eſſe rote: e le circon<lb></lb>ferenze ſono di quantità proportionale alli diametri. </s> <s id="N113FA">Sono dunque le <lb></lb>velocità delle rote proportionali alli diametri. </s> <s id="N113FE">Jl che ſi hauea da <lb></lb>moſtrare. <emph.end type="italics"></emph.end></s></p><p id="N11404" type="head"> <s id="N11406"><emph type="italics"></emph>PROPOSITION. <emph.end type="italics"></emph.end><lb></lb>V. </s></p><p id="N1140E" type="main"> <s id="N11410">Date due poſſanze di momento contrario, l'vna mi<lb></lb>nore, e l'altra maggiore: e data la ragione dell'vna al<lb></lb>l'altra delle due rote congiogate: ritrouar il minor nu<lb></lb>mero de congiogationi, ſiche la data poſſanza minore <lb></lb>vinca la maggiore. </s></p><figure id="id.041.01.066.1.jpg" xlink:href="041/01/066/1.jpg"></figure> <pb xlink:href="041/01/067.jpg" pagenum="96"></pb> <p id="N11420" type="head"> <s id="N11422"><emph type="italics"></emph>Dimostratione. <emph.end type="italics"></emph.end></s></p><p id="N11428" type="main"> <s id="N1142A"><emph type="italics"></emph>Siano le date poſſanze di momento contrario A, B: De quali A <lb></lb>ſia la maggiore, c B la minore.: la ragion delle rote congiogate ſia di <lb></lb>C a D: ſi cerca il minor numero de congiogationi, ſiche la poſſanza <lb></lb>B minore vinca la A maggiore. </s> <s id="N11434">Piglinſi nella ragione di C a D con<lb></lb>tinuamente le C, D, E, F: ſiche la C ad F habbia maggior ragione <lb></lb>che l' A a B: & eguale di numero all'interualli de termini ſi piglino <lb></lb>le congiogationi di rote G, H, I: e ſiano ſu l'aſſe G, le rote G K, G <lb></lb>L, ſu l'aſſe H le rote H L, H M: e ſu l'aſſe I le rote M I, I N. </s> <s id="N1143E">E ma<lb></lb>nifeſto che'l momento della <expan abbr="poſsãza">poſsanza</expan> in K, al momento ſuo in N, ha la <lb></lb>ragion compoſta delle ragioni de ſemidiametri: e perciò poſta la poſ<lb></lb>ſanza maggiore A in N: e la minore B in K: ſara il momento della B <lb></lb>in K, maggiore che'l momento dell' A in N. </s> <s id="N1144C">Il che ſi hauea da <lb></lb>trouare. <emph.end type="italics"></emph.end></s></p><p id="N11452" type="head"> <s id="N11454"><emph type="italics"></emph>PROPOSITION. <emph.end type="italics"></emph.end><lb></lb>VI. </s></p><p id="N1145C" type="main"> <s id="N1145E">Data qualſivoglia tardità di poſſanza, & qualſivo<lb></lb>glia velocità: e data la ragion de diametri delle rote <expan abbr="cõgiogate">con<lb></lb>giogate</expan>: ritrouar vn minimo numero de congiogatio <lb></lb>ni, ſi che la data poſſanza moua la coſa con velocità <lb></lb>maggiore della data. </s></p><p id="N1146C" type="head"> <s id="N1146E"><emph type="italics"></emph>Dimostratione. <emph.end type="italics"></emph.end></s></p><p id="N11474" type="main"> <s id="N11476"><emph type="italics"></emph>Sia la poſſanza tarda A, la veloce B, lo ſpatio caminato da A in <lb></lb>vn dato tempo ſia C, lo C caminato da B nell'iſteſſo tempo ſia D: la <lb></lb>ragion de diametri congiogati ſia di E, ad F: biſogna ritrouare il <emph.end type="italics"></emph.end><pb xlink:href="041/01/068.jpg" pagenum="67"></pb><figure id="id.041.01.068.1.jpg" xlink:href="041/01/068/1.jpg"></figure><lb></lb><emph type="italics"></emph>minimo numero de <expan abbr="cõgiogationi">congiogationi</expan>, col quale la tarda A moua con ve<lb></lb>locità maggior che'l B. </s> <s id="N11491">Piglinſi le E, F, G, continuate nella ragion <lb></lb>de diametri, che la prima volta l'interuallo della prima all'vltima <lb></lb>dico di G ad E, ſia maggiore che di C a D: e quanti interualli ſono il <lb></lb>E, F, G: tante congiogationi di rote ſi piglino nella iſteſſa ragione: l'a<lb></lb>ſe de quali ſiano H, I: e nello aſſe H, la minor rota ſia H K, la maggio <lb></lb>re H L: e nell'aſſe I la minore I L, la maggiore L M. </s> <s>il contatto del<lb></lb>l'vna congiogatione all'altra il ponto L: è manifeſto che la veloci<lb></lb>tà del ponto M, alla velocità del ponto K, è compoſta della ragion del<lb></lb>li diametri M I, ad I L, & H L ad H K: che è l'iſteſſa, che di G ad E: <lb></lb>ma G ad E, è di maggior interuallo che di D a C. </s> <s id="N114A5"> <expan abbr="Dũque">Dunque</expan>, poſta la poſſan<lb></lb>za tarda in K, la coſa moſſa con la circonferenza M, ſi mouerà <expan abbr="cõ">con</expan> mag<lb></lb>gior velocità della data. </s> <s id="N114B2">Il che ſi hauea da trouare. <emph.end type="italics"></emph.end></s></p> <pb xlink:href="041/01/069.jpg" pagenum="68"></pb> <p id="N114B9" type="head"> <s id="N114BB"><emph type="italics"></emph>MOMENTI ACQVISTATI. <emph.end type="italics"></emph.end></s></p><p id="N114C1" type="main"> <s id="N114C3">Poniamo degli momenti, altri eſſer intrinſechi: al <lb></lb>tri acquiſtati, & altri miſti: & intrinſechi quelli, che <lb></lb>non da mouimento precedente dipendono: come ſono <lb></lb>gli mouimenti delle grauezze in giù, e del corpo leggiero <lb></lb>dentro l'humor più graue in sù. </s> <s id="N114CD">Acquiſtati quelli, che ſe<lb></lb>guono l'impreſsion fatta da precedente mouimento: come <lb></lb>il mouimento della coſa lanciata, che ſegue il <expan abbr="mouimẽto">mouimento</expan><lb></lb> del braccio, o della corda. </s> <s id="N114D8">Miſti, come il mouimento delle <lb></lb>grauezze dopo l'hauer dato principio a mouerſi: per il che <lb></lb>veggiamo li peſi di vicino laſciati, mouerſi con minor mo<lb></lb>mento, che laſciati di lontano: e molte coſe portate dalla <lb></lb>propria grauezza nell'aria penetrar ſotto l'accqua, con<lb></lb>tro di quel che porta l'intrinſeco momento: onde dopo <lb></lb>l'eſſere affondate da ſe ſteſsi ritornar á galla. </s> <s id="N114E6">Et il momen<lb></lb>to intrinſeco eſſer l'iſteſſo ſempre. </s> <s id="N114EA">l'acquiſtato, mancando <lb></lb>la cauſa di ponerſi, e con il tempo, e dall'impedimento che <lb></lb>le faccia reſiſtenza. <emph type="italics"></emph>CVGNO. <emph.end type="italics"></emph.end></s></p><p id="N114F5" type="main"> <s id="N114F7">Il cugno percoſſo, conſiderato in vn modo, rappreſenta <lb></lb>un piano inchinato, che ſi ſpinga ſotto il peſo. </s> <s id="N114FB">Et altrimen<lb></lb>te rappreſenta due leue, che nelle loro ſtremità, facciano <lb></lb>l'vna all'altra ſottoleua, & habbiano il peſo tra la poſſanza, <lb></lb>e'l ſottoleua. </s> <s id="N11503">Et altrimente rappreſenta leua nel cui ſtremo <lb></lb>ſia il peſo, & il ſottoleua tramezzo. <emph type="italics"></emph>VITE E CHIOCCIA. <emph.end type="italics"></emph.end></s></p><p id="N1150C" type="main"> <s id="N1150E">La vite, o chioccia rappreſenta vno o più piani auuolti <lb></lb>ad vn fuſello. </s> <s id="N11512">Sono e maſchia, e femina: de quali vna ſtan<lb></lb>do ferma, l'altra che gira ſoſtiene il peſo. </s> <s id="N11516">acquiſta dunque for<lb></lb>za, <expan abbr="ſecõdo">ſecondo</expan> la detta inchinazione, e ſecondo la lunghezza del <lb></lb>raggio che ſe le accompagna. </s> <s id="N11520">Vite perpetua diciamo vn <lb></lb>sympano con denti a vite, che girando tocchi rota dentata. </s><lb></lb> <s id="N11525">Per il che accreſce la forza, e per la proprieta della vite, <lb></lb>e della congiogatione delle rote. </s></p><p id="N11529" type="head"> <s id="N1152B"><emph type="italics"></emph>IL FINE. <emph.end type="italics"></emph.end></s></p> </chap> </body> <back></back> </text> </archimedes>